Physics Formula Sheet

Physics FormularyBy ir. J.C.A. Wevers c (1995, 2009 J.C.A. Wevers Version: December 16, 2009 Dear reader, This document contains a 108 page L A T E Xfile which contains a lot equations in physics. It is written at advanced undergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics and often needs to look up equations. This, and a Dutch version of this file, can be obtained from the author, Johan Wevers ([email protected]). It can also be obtained on the WWW. See http://www.xs4all.nl/˜johanw/index.html, where also a Postscript version is available. If you find any errors or have any comments, please let me know. I am always open for suggestions and possible corrections to the physics formulary. The Physics Formulary is made with teT E X and L A T E X version 2.09. It can be possible that your L A T E X version has problems compiling the file. The most probable source of problems would be the use of large bezier curves and/or emT E X specials in pictures. If you prefer the notation in which vectors are typefaced in boldface, uncomment the redefinition of the `vec command in the T E X file and recompile the file. This work is licenced under the Creative Commons Attribution 3.0 Netherlands License. To view a copy of this licence, visit http://creativecommons.org/licenses/by/3.0/nl/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. Johan Wevers Contents Contents I Physical Constants 1 1 Mechanics 2 1.1 Point-kinetics in a fixed coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Point-dynamics in a fixed coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.1 Force, (angular)momentumand energy . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.2 Conservative force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.3 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.4 Orbital equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.5 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Point dynamics in a moving coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.1 Apparent forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.2 Tensor notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Dynamics of masspoint collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.1 The centre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.2 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Dynamics of rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6.1 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6.2 Principal axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6.3 Time dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Variational Calculus, Hamilton and Lagrange mechanics . . . . . . . . . . . . . . . . . . . . 6 1.7.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7.2 Hamilton mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7.3 Motion around an equilibrium, linearization . . . . . . . . . . . . . . . . . . . . . . . 7 1.7.4 Phase space, Liouville’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7.5 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Electricity & Magnetism 9 2.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Force and potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Energy of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5.1 Electromagnetic waves in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5.2 Electromagnetic waves in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.7 Electric currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.8 Depolarizing field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.9 Mixtures of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 II Physics Formulary by ir. J.C.A. Wevers 3 Relativity 13 3.1 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Red and blue shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.3 The stress-energy tensor and the field tensor . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Riemannian geometry, the Einstein tensor . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 The line element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.3 Planetary orbits and the perihelion shift . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.4 The trajectory of a photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.5 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.6 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Oscillations 18 4.1 Harmonic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Mechanic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Electric oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4 Waves in long conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5 Coupled conductors and transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.6 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Waves 20 5.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Solutions of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2.2 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2.3 Cylindrical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2.4 The general solution in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 The stationary phase method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.4 Green functions for the initial-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.5 Waveguides and resonating cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.6 Non-linear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6 Optics 24 6.1 The bending of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2 Paraxial geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2.1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2.2 Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2.3 Principal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2.4 Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.3 Matrix methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.4 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.5 Reflection and transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.7 Prisms and dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.8 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.9 Special optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.10 The Fabry-Perot interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7 Statistical physics 30 7.1 Degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.2 The energy distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.3 Pressure on a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.4 The equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.5 Collisions between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Physics Formulary by ir. J.C.A. Wevers III 7.6 Interaction between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8 Thermodynamics 33 8.1 Mathematical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.3 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.4 The laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.5 State functions and Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.6 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.7 Maximal work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 8.8 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 8.9 Thermodynamic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.10 Ideal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.11 Conditions for equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.12 Statistical basis for thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8.13 Application to other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9 Transport phenomena 39 9.1 Mathematical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9.3 Bernoulli’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.4 Characterising of flows by dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . 41 9.5 Tube flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 9.6 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 9.7 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.7.1 Flow boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.7.2 Temperature boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.8 Heat conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.9 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 9.10 Self organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 10 Quantum physics 45 10.1 Introduction to quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10.1.1 Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10.1.2 The Compton effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10.1.3 Electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10.2 Wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10.3 Operators in quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 10.4 The uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 10.5 The Schr¨ odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 10.6 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 10.7 The tunnel effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10.8 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10.9 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10.10 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 10.11 The Dirac formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 10.12 Atomic physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.12.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.12.2 Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.12.3 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.12.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 10.13 Interaction with electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 10.14 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 10.14.1 Time-independent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . 50 10.14.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 51 IV Physics Formulary by ir. J.C.A. Wevers 10.15 N-particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 10.15.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 10.15.2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 10.16 Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 11 Plasma physics 54 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 11.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 11.3 Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 11.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 11.3.2 The Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11.3.3 The induced dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11.3.4 The centre of mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11.3.5 Scattering of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11.4 Thermodynamic equilibrium and reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . 57 11.5 Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 11.5.1 Types of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 11.5.2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 11.6 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 11.7 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 11.8 Collision-radiative models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11.9 Waves in plasma’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 12 Solid state physics 62 12.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 12.2 Crystal binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 12.3 Crystal vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.3.1 A lattice with one type of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.3.2 A lattice with two types of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.3.3 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.3.4 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 12.4 Magnetic field in the solid state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.4.1 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.4.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.4.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.5 Free electron Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 12.5.1 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 12.5.2 Electric conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 12.5.3 The Hall-effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12.5.4 Thermal heat conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12.6 Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12.7 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12.8 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 12.8.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 12.8.2 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12.8.3 Flux quantisation in a superconducting ring . . . . . . . . . . . . . . . . . . . . . . . 69 12.8.4 Macroscopic quantum interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.8.5 The London equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.8.6 The BCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Physics Formulary by ir. J.C.A. Wevers V 13 Theory of groups 71 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 13.1.1 Definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 13.1.2 The Cayley table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 13.1.3 Conjugated elements, subgroups and classes . . . . . . . . . . . . . . . . . . . . . . . 71 13.1.4 Isomorfism and homomorfism; representations . . . . . . . . . . . . . . . . . . . . . 72 13.1.5 Reducible and irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . 72 13.2 The fundamental orthogonality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 13.2.1 Schur’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 13.2.2 The fundamental orthogonality theorem . . . . . . . . . . . . . . . . . . . . . . . . . 72 13.2.3 Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 13.3 The relation with quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.3.1 Representations, energy levels and degeneracy . . . . . . . . . . . . . . . . . . . . . 73 13.3.2 Breaking of degeneracy by a perturbation . . . . . . . . . . . . . . . . . . . . . . . . 73 13.3.3 The construction of a base function . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 13.3.4 The direct product of representations . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.3.5 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.3.6 Symmetric transformations of operators, irreducible tensor operators . . . . . . . . . . 74 13.3.7 The Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.4 Continuous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.4.1 The 3-dimensional translation group . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.4.2 The 3-dimensional rotation group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13.4.3 Properties of continuous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 13.5 The group SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 13.6 Applications to quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 13.6.1 Vectormodel for the addition of angular momentum . . . . . . . . . . . . . . . . . . . 77 13.6.2 Irreducible tensor operators, matrixelements and selection rules . . . . . . . . . . . . 78 13.7 Applications to particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 14 Nuclear physics 81 14.1 Nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 14.2 The shape of the nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.3 Radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.4 Scattering and nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 14.4.1 Kinetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 14.4.2 Quantum mechanical model for n-p scattering . . . . . . . . . . . . . . . . . . . . . . 83 14.4.3 Conservation of energy and momentum in nuclear reactions . . . . . . . . . . . . . . 84 14.5 Radiation dosimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 15 Quantum field theory & Particle physics 85 15.1 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 15.2 Classical and quantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 15.3 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 15.4 Real scalar field in the interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 15.5 Charged spin-0 particles, conservation of charge . . . . . . . . . . . . . . . . . . . . . . . . 87 15.6 Field functions for spin- 1 2 particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 15.7 Quantization of spin- 1 2 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 15.8 Quantization of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 15.9 Interacting fields and the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 15.10 Divergences and renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 15.11 Classification of elementary particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 15.12 P and CP-violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 15.13 The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 15.13.1 The electroweak theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 15.13.2 Spontaneous symmetry breaking: the Higgs mechanism . . . . . . . . . . . . . . . . 94 VI Physics Formulary by ir. J.C.A. Wevers 15.13.3 Quantumchromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 15.14 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 15.15 Unification and quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 16 Astrophysics 96 16.1 Determination of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 16.2 Brightness and magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 16.3 Radiation and stellar atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 16.4 Composition and evolution of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 16.5 Energy production in stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 The ∇-operator 99 The SI units 100 Physical Constants Name Symbol Value Unit Number π π 3.14159265358979323846 Number e e 2.71828182845904523536 Euler’s constant γ = lim n→∞ n ¸ k=1 1/k −ln(n) = 0.5772156649 Elementary charge e 1.60217733 10 −19 C Gravitational constant G, κ 6.67259 10 −11 m 3 kg −1 s −2 Fine-structure constant α = e 2 /2hcε 0 ≈ 1/137 Speed of light in vacuum c 2.99792458 10 8 m/s (def) Permittivity of the vacuum ε 0 8.854187 10 −12 F/m Permeability of the vacuum µ 0 4π 10 −7 H/m (4πε 0 ) −1 8.9876 10 9 Nm 2 C −2 Planck’s constant h 6.6260755 10 −34 Js Dirac’s constant ¯h = h/2π 1.0545727 10 −34 Js Bohr magneton µ B = e¯h/2m e 9.2741 10 −24 Am 2 Bohr radius a 0 0.52918 ˚ A Rydberg’s constant Ry 13.595 eV Electron Compton wavelength λ Ce = h/m e c 2.2463 10 −12 m Proton Compton wavelength λ Cp = h/m p c 1.3214 10 −15 m Reduced mass of the H-atom µ H 9.1045755 10 −31 kg Stefan-Boltzmann’s constant σ 5.67032 10 −8 Wm −2 K −4 Wien’s constant k W 2.8978 10 −3 mK Molar gasconstant R 8.31441 Jmol −1 K −1 Avogadro’s constant N A 6.0221367 10 23 mol −1 Boltzmann’s constant k = R/N A 1.380658 10 −23 J/K Electron mass m e 9.1093897 10 −31 kg Proton mass m p 1.6726231 10 −27 kg Neutron mass m n 1.674954 10 −27 kg Elementary mass unit m u = 1 12 m( 12 6 C) 1.6605656 10 −27 kg Nuclear magneton µ N 5.0508 10 −27 J/T Diameter of the Sun D 1392 10 6 m Mass of the Sun M 1.989 10 30 kg Rotational period of the Sun T 25.38 days Radius of Earth R A 6.378 10 6 m Mass of Earth M A 5.976 10 24 kg Rotational period of Earth T A 23.96 hours Earth orbital period Tropical year 365.24219879 days Astronomical unit AU 1.4959787066 10 11 m Light year lj 9.4605 10 15 m Parsec pc 3.0857 10 16 m Hubble constant H ≈ (75 ±25) kms −1 Mpc −1 Chapter 1 Mechanics 1.1 Point-kinetics in a fixed coordinate system 1.1.1 Definitions The position r, the velocity v and the acceleration a are defined by: r = (x, y, z), v = ( ˙ x, ˙ y, ˙ z), a = (¨ x, ¨ y, ¨ z). The following holds: s(t) = s 0 + [v(t)[dt ; r(t) = r 0 + v(t)dt ; v(t) = v 0 + a(t)dt When the acceleration is constant this gives: v(t) = v 0 +at and s(t) = s 0 +v 0 t + 1 2 at 2 . For the unit vectors in a direction ⊥ to the orbit e t and parallel to it e n holds: e t = v [v[ = dr ds ˙ e t = v ρ e n ; e n = ˙ e t [ ˙ e t [ For the curvature k and the radius of curvature ρ holds: k = de t ds = d 2 r ds 2 = dϕ ds ; ρ = 1 [k[ 1.1.2 Polar coordinates Polar coordinates are defined by: x = r cos(θ), y = r sin(θ). So, for the unit coordinate vectors holds: ˙ e r = ˙ θe θ , ˙ e θ = − ˙ θe r The velocity and the acceleration are derived from: r = re r , v = ˙ re r +r ˙ θe θ , a = (¨ r −r ˙ θ 2 )e r +(2 ˙ r ˙ θ +r ¨ θ)e θ . 1.2 Relative motion For the motion of a point D w.r.t. a point Q holds: r D = r Q + ω v Q ω 2 with QD = r D −r Q and ω = ˙ θ. Further holds: α = ¨ θ. means that the quantity is defined in a moving system of coordinates. In a moving system holds: v = v Q +v + ω r and a =a Q +a + α r +2ω v +ω (ω r ) with ω (ω r ) = −ω 2 r n 1.3 Point-dynamics in a fixed coordinate system 1.3.1 Force, (angular)momentum and energy Newton’s 2nd law connects the force on an object and the resulting acceleration of the object where the mo- mentum is given by p = mv: F(r, v, t) = d p dt = d(mv ) dt = m dv dt +v dm dt m=const = ma Chapter 1: Mechanics 3 Newton’s 3rd law is given by: F action = − F reaction . For the power P holds: P = ˙ W = F v. For the total energy W, the kinetic energy T and the potential energy U holds: W = T +U ; ˙ T = − ˙ U with T = 1 2 mv 2 . The kick S is given by: S = ∆ p = Fdt The work A, delivered by a force, is A = 2 1 F ds = 2 1 F cos(α)ds The torque τ is related to the angular momentum L: τ = ˙ L = r F; and L = r p = mv r, [ L[ = mr 2 ω. The following equation is valid: τ = − ∂U ∂θ Hence, the conditions for a mechanical equilibrium are: ¸ F i = 0 and ¸ τ i = 0. The force of friction is usually proportional to the force perpendicular to the surface, except when the motion starts, when a threshold has to be overcome: F fric = f F norm e t . 1.3.2 Conservative force fields A conservative force can be written as the gradient of a potential: F cons = − ∇U. From this follows that ∇ F = 0. For such a force field also holds: F ds = 0 ⇒ U = U 0 − r1 r0 F ds So the work delivered by a conservative force field depends not on the trajectory covered but only on the starting and ending points of the motion. 1.3.3 Gravitation The Newtonian law of gravitation is (in GRT one also uses κ instead of G): F g = −G m 1 m 2 r 2 e r The gravitational potential is then given by V = −Gm/r. From Gauss law it then follows: ∇ 2 V = 4πG. 1.3.4 Orbital equations If V = V (r) one can derive from the equations of Lagrange for φ the conservation of angular momentum: ∂L ∂φ = ∂V ∂φ = 0 ⇒ d dt (mr 2 φ) = 0 ⇒L z = mr 2 φ = constant For the radial position as a function of time can be found that: dr dt 2 = 2(W −V ) m − L 2 m 2 r 2 The angular equation is then: φ −φ 0 = r 0 ¸ mr 2 L 2(W −V ) m − L 2 m 2 r 2 ¸ −1 dr r −2 field = arccos 1 + 1 r − 1 r0 1 r0 +km/L 2 z If F = F(r): L =constant, if F is conservative: W =constant, if F ⊥ v then ∆T = 0 and U = 0. 4 Physics Formulary by ir. J.C.A. Wevers Kepler’s orbital equations In a force field F = kr −2 , the orbits are conic sections with the origin of the force in one of the foci (Kepler’s 1st law). The equation of the orbit is: r(θ) = 1 +ε cos(θ −θ 0 ) , or: x 2 +y 2 = ( −εx) 2 with = L 2 Gµ 2 M tot ; ε 2 = 1 + 2WL 2 G 2 µ 3 M 2 tot = 1 − a ; a = 1 −ε 2 = k 2W a is half the length of the long axis of the elliptical orbit in case the orbit is closed. Half the length of the short axis is b = √ a. ε is the excentricity of the orbit. Orbits with an equal ε are of equal shape. Now, 5 types of orbits are possible: 1. k < 0 and ε = 0: a circle. 2. k < 0 and 0 < ε < 1: an ellipse. 3. k < 0 and ε = 1: a parabole. 4. k < 0 and ε > 1: a hyperbole, curved towards the centre of force. 5. k > 0 and ε > 1: a hyperbole, curved away from the centre of force. Other combinations are not possible: the total energy in a repulsive force field is always positive so ε > 1. If the surface between the orbit covered between t 1 and t 2 and the focus C around which the planet moves is A(t 1 , t 2 ), Kepler’s 2nd law is A(t 1 , t 2 ) = L C 2m (t 2 −t 1 ) Kepler’s 3rd law is, with T the period and M tot the total mass of the system: T 2 a 3 = 4π 2 GM tot 1.3.5 The virial theorem The virial theorem for one particle is: 'mv r` = 0 ⇒'T` = − 1 2 F r = 1 2 r dU dr = 1 2 n'U` if U = − k r n The virial theorem for a collection of particles is: 'T` = − 1 2 ¸ particles F i r i + ¸ pairs F ij r ij ¸ These propositions can also be written as: 2E kin +E pot = 0. 1.4 Point dynamics in a moving coordinate system 1.4.1 Apparent forces The total force in a moving coordinate system can be found by subtracting the apparent forces from the forces working in the reference frame: F = F − F app . The different apparent forces are given by: 1. Transformation of the origin: F or = −ma a 2. Rotation: F α = −m α r 3. Coriolis force: F cor = −2mω v 4. Centrifugal force: F cf = mω 2 r n = − F cp ; F cp = − mv 2 r e r Chapter 1: Mechanics 5 1.4.2 Tensor notation Transformation of the Newtonian equations of motion to x α = x α (x) gives: dx α dt = ∂x α ∂¯ x β d¯ x β dt ; The chain rule gives: d dt dx α dt = d 2 x α dt 2 = d dt ∂x α ∂¯ x β d¯ x β dt = ∂x α ∂¯ x β d 2 ¯ x β dt 2 + d¯ x β dt d dt ∂x α ∂¯ x β so: d dt ∂x α ∂¯ x β = ∂ ∂¯ x γ ∂x α ∂¯ x β d¯ x γ dt = ∂ 2 x α ∂¯ x β ∂¯ x γ d¯ x γ dt This leads to: d 2 x α dt 2 = ∂x α ∂¯ x β d 2 ¯ x β dt 2 + ∂ 2 x α ∂¯ x β ∂¯ x γ d¯ x γ dt d¯ x β dt Hence the Newtonian equation of motion m d 2 x α dt 2 = F α will be transformed into: m d 2 x α dt 2 +Γ α βγ dx β dt dx γ dt = F α The apparent forces are taken from he origin to the effect side in the way Γ α βγ dx β dt dx γ dt . 1.5 Dynamics of masspoint collections 1.5.1 The centre of mass The velocity w.r.t. the centre of mass R is given by v − ˙ R. The coordinates of the centre of mass are given by: r m = ¸ m i r i ¸ m i In a 2-particle system, the coordinates of the centre of mass are given by: R = m 1 r 1 +m 2 r 2 m 1 +m 2 With r = r 1 −r 2 , the kinetic energy becomes: T = 1 2 M tot ˙ R 2 + 1 2 µ˙ r 2 , with the reduced mass µ given by: 1 µ = 1 m 1 + 1 m 2 The motion within and outside the centre of mass can be separated: ˙ L outside = τ outside ; ˙ L inside = τ inside p = mv m ; F ext = ma m ; F 12 = µu 1.5.2 Collisions With collisions, where B are the coordinates of the collision and C an arbitrary other position, holds: p = mv m is constant, and T = 1 2 mv 2 m is constant. The changes in the relative velocities can be derived from: S = ∆ p = µ(v aft −v before ). Further holds ∆ L C = CB S, p | S =constant and L w.r.t. B is constant. 6 Physics Formulary by ir. J.C.A. Wevers 1.6 Dynamics of rigid bodies 1.6.1 Moment of Inertia The angular momentum in a moving coordinate system is given by: L = I ω + L n where I is the moment of inertia with respect to a central axis, which is given by: I = ¸ i m i r i 2 ; T = W rot = 1 2 ωI ij e i e j = 1 2 Iω 2 or, in the continuous case: I = m V r 2 n dV = r 2 n dm Further holds: L i = I ij ω j ; I ii = I i ; I ij = I ji = − ¸ k m k x i x j Steiner’s theorem is: I w.r.t.D = I w.r.t.C +m(DM) 2 if axis C | axis D. Object I Object I Cavern cylinder I = mR 2 Massive cylinder I = 1 2 mR 2 Disc, axis in plane disc through m I = 1 4 mR 2 Halter I = 1 2 µR 2 Cavern sphere I = 2 3 mR 2 Massive sphere I = 2 5 mR 2 Bar, axis ⊥ through c.o.m. I = 1 12 ml 2 Bar, axis ⊥ through end I = 1 3 ml 2 Rectangle, axis ⊥ plane thr. c.o.m. I = 1 12 m(a 2 +b 2 ) Rectangle, axis | b thr. m I = ma 2 1.6.2 Principal axes Each rigid body has (at least) 3 principal axes which stand ⊥ to each other. For a principal axis holds: ∂I ∂ω x = ∂I ∂ω y = ∂I ∂ω z = 0 so L n = 0 The following holds: ˙ ω k = −a ijk ω i ω j with a ijk = I i −I j I k if I 1 ≤ I 2 ≤ I 3 . 1.6.3 Time dependence For torque of force τ holds: τ = I ¨ θ ; d L dt = τ −ω L The torque T is defined by: T = F d. 1.7 Variational Calculus, Hamilton and Lagrange mechanics 1.7.1 Variational Calculus Starting with: δ b a L(q, ˙ q, t)dt = 0 with δ(a) = δ(b) = 0 and δ du dx = d dx (δu) Chapter 1: Mechanics 7 the equations of Lagrange can be derived: d dt ∂L ∂ ˙ q i = ∂L ∂q i When there are additional conditions applying to the variational problem δJ(u) = 0 of the type K(u) =constant, the new problem becomes: δJ(u) −λδK(u) = 0. 1.7.2 Hamilton mechanics The Lagrangian is given by: L = ¸ T( ˙ q i ) − V (q i ). The Hamiltonian is given by: H = ¸ ˙ q i p i − L. In 2 dimensions holds: L = T −U = 1 2 m( ˙ r 2 +r 2 ˙ φ 2 ) −U(r, φ). If the used coordinates are canonical the Hamilton equations are the equations of motion for the system: dq i dt = ∂H ∂p i ; dp i dt = − ∂H ∂q i Coordinates are canonical if the following holds: ¦q i , q j ¦ = 0, ¦p i , p j ¦ = 0, ¦q i , p j ¦ = δ ij where ¦, ¦ is the Poisson bracket: ¦A, B¦ = ¸ i ¸ ∂A ∂q i ∂B ∂p i − ∂A ∂p i ∂B ∂q i The Hamiltonian of a Harmonic oscillator is given by H(x, p) = p 2 /2m + 1 2 mω 2 x 2 . With new coordinates (θ, I), obtained by the canonical transformation x = 2I/mωcos(θ) and p = − √ 2Imωsin(θ), with inverse θ = arctan(−p/mωx) and I = p 2 /2mω + 1 2 mωx 2 it follows: H(θ, I) = ωI. The Hamiltonian of a charged particle with charge q in an external electromagnetic field is given by: H = 1 2m p −q A 2 +qV This Hamiltonian can be derived from the Hamiltonian of a free particle H = p 2 /2m with the transformations p → p − q A and H → H − qV . This is elegant from a relativistic point of view: this is equivalent to the transformation of the momentum 4-vector p α → p α − qA α . A gauge transformation on the potentials A α corresponds with a canonical transformation, which make the Hamilton equations the equations of motion for the system. 1.7.3 Motion around an equilibrium, linearization For natural systems around equilibrium the following equations are valid: ∂V ∂q i 0 = 0 ; V (q) = V (0) +V ik q i q k with V ik = ∂ 2 V ∂q i ∂q k 0 With T = 1 2 (M ik ˙ q i ˙ q k ) one receives the set of equations M¨ q + V q = 0. If q i (t) = a i exp(iωt) is substituted, this set of equations has solutions if det(V − ω 2 M) = 0. This leads to the eigenfrequencies of the problem: ω 2 k = a T k V a k a T k Ma k . If the equilibrium is stable holds: ∀k that ω 2 k > 0. The general solution is a superposition if eigenvibrations. 1.7.4 Phase space, Liouville’s equation In phase space holds: ∇ = ¸ i ∂ ∂q i , ¸ i ∂ ∂p i so ∇ v = ¸ i ∂ ∂q i ∂H ∂p i − ∂ ∂p i ∂H ∂q i 8 Physics Formulary by ir. J.C.A. Wevers If the equation of continuity, ∂ t +∇ (v ) = 0 holds, this can be written as: ¦, H¦ + ∂ ∂t = 0 For an arbitrary quantity A holds: dA dt = ¦A, H¦ + ∂A ∂t Liouville’s theorem can than be written as: d dt = 0 ; or: pdq = constant 1.7.5 Generating functions Starting with the coordinate transformation: Q i = Q i (q i , p i , t) P i = P i (q i , p i , t) one can derive the following Hamilton equations with the new Hamiltonian K: dQ i dt = ∂K ∂P i ; dP i dt = − ∂K ∂Q i Now, a distinction between 4 cases can be made: 1. If p i ˙ q i −H = P i Q i −K(P i , Q i , t) − dF 1 (q i , Q i , t) dt , the coordinates follow from: p i = ∂F 1 ∂q i ; P i = − ∂F 1 ∂Q i ; K = H + ∂F 1 ∂t 2. If p i ˙ q i −H = − ˙ P i Q i −K(P i , Q i , t) + dF 2 (q i , P i , t) dt , the coordinates follow from: p i = ∂F 2 ∂q i ; Q i = ∂F 2 ∂P i ; K = H + ∂F 2 ∂t 3. If −˙ p i q i −H = P i ˙ Q i −K(P i , Q i , t) + dF 3 (p i , Q i , t) dt , the coordinates follow from: q i = − ∂F 3 ∂p i ; P i = − ∂F 3 ∂Q i ; K = H + ∂F 3 ∂t 4. If −˙ p i q i −H = −P i Q i −K(P i , Q i , t) + dF 4 (p i , P i , t) dt , the coordinates follow from: q i = − ∂F 4 ∂p i ; Q i = ∂F 4 ∂P i ; K = H + ∂F 4 ∂t The functions F 1 , F 2 , F 3 and F 4 are called generating functions. Chapter 2 Electricity & Magnetism 2.1 The Maxwell equations The classical electromagnetic field can be described by the Maxwell equations. Those can be written both as differential and integral equations: (( D n)d 2 A = Q free,included ∇ D = ρ free (( B n)d 2 A = 0 ∇ B = 0 E ds = − dΦ dt ∇ E = − ∂ B ∂t H ds = I free,included + dΨ dt ∇ H = J free + ∂ D ∂t For the fluxes holds: Ψ = ( D n)d 2 A, Φ = ( B n)d 2 A. The electric displacement D, polarization P and electric field strength E depend on each other according to: D = ε 0 E + P = ε 0 ε r E, P = ¸ p 0 /Vol, ε r = 1 +χ e , with χ e = np 2 0 3ε 0 kT The magnetic field strength H, the magnetization M and the magnetic flux density B depend on each other according to: B = µ 0 ( H + M) = µ 0 µ r H, M = ¸ m/Vol, µ r = 1 +χ m , with χ m = µ 0 nm 2 0 3kT 2.2 Force and potential The force and the electric field between 2 point charges are given by: F 12 = Q 1 Q 2 4πε 0 ε r r 2 e r ; E = F Q The Lorentzforce is the force which is felt by a charged particle that moves through a magnetic field. The origin of this force is a relativistic transformation of the Coulomb force: F L = Q(v B) = l( I B). The magnetic field in point P which results from an electric current is given by the law of Biot-Savart, also known as the law of Laplace. In here, d l | I and r points from d l to P: d B P = µ 0 I 4πr 2 d l e r If the current is time-dependent one has to take retardation into account: the substitution I(t) → I(t − r/c) has to be applied. The potentials are given by: V 12 = − 2 1 E ds and A = 1 2 B r. 10 Physics Formulary by ir. J.C.A. Wevers Here, the freedom remains to apply a gauge transformation. The fields can be derived from the potentials as follows: E = −∇V − ∂ A ∂t , B = ∇ A Further holds the relation: c 2 B = v E. 2.3 Gauge transformations The potentials of the electromagnetic fields transform as follows when a gauge transformation is applied: A = A −∇f V = V + ∂f ∂t so the fields E and B do not change. This results in a canonical transformation of the Hamiltonian. Further, the freedom remains to apply a limiting condition. Two common choices are: 1. Lorentz-gauge: ∇ A+ 1 c 2 ∂V ∂t = 0. This separates the differential equations for A and V : ✷V = − ρ ε 0 , ✷ A = −µ 0 J. 2. Coulomb gauge: ∇ A = 0. If ρ = 0 and J = 0 holds V = 0 and follows A from✷ A = 0. 2.4 Energy of the electromagnetic field The energy density of the electromagnetic field is: dW dVol = w = HdB + EdD The energy density can be expressed in the potentials and currents as follows: w mag = 1 2 J Ad 3 x , w el = 1 2 ρV d 3 x 2.5 Electromagnetic waves 2.5.1 Electromagnetic waves in vacuum The wave equation ✷Ψ(r, t) = −f(r, t) has the general solution, with c = (ε 0 µ 0 ) −1/2 : Ψ(r, t) = f(r, t −[r −r [/c) 4π[r −r [ d 3 r If this is written as: J(r, t) = J(r ) exp(−iωt) and A(r, t) = A(r ) exp(−iωt) with: A(r ) = µ 4π J(r ) exp(ik[r −r [) [r −r [ d 3 r , V (r ) = 1 4πε ρ(r ) exp(ik[r −r [) [r −r [ d 3 r A derivation via multipole expansion will show that for the radiated energy holds, if d, λ r: dP dΩ = k 2 32π 2 ε 0 c J ⊥ (r )e i k·r d 3 r 2 The energy density of the electromagnetic wave of a vibrating dipole at a large distance is: w = ε 0 E 2 = p 2 0 sin 2 (θ)ω 4 16π 2 ε 0 r 2 c 4 sin 2 (kr −ωt) , 'w` t = p 2 0 sin 2 (θ)ω 4 32π 2 ε 0 r 2 c 4 , P = ck 4 [ p [ 2 12πε 0 The radiated energy can be derived from the Poynting vector S: S = E H = cWe v . The irradiance is the time-averaged of the Poynting vector: I = '[ S [` t . The radiation pressure p s is given by p s = (1 + R)[ S [/c, where R is the coefficient of reflection. Chapter 2: Electricity & Magnetism 11 2.5.2 Electromagnetic waves in matter The wave equations in matter, with c mat = (εµ) −1/2 the lightspeed in matter, are: ∇ 2 −εµ ∂ 2 ∂t 2 − µ ρ ∂ ∂t E = 0 , ∇ 2 −εµ ∂ 2 ∂t 2 − µ ρ ∂ ∂t B = 0 give, after substitution of monochromatic plane waves: E = E exp(i( kr−ωt)) and B = Bexp(i( kr−ωt)) the dispersion relation: k 2 = εµω 2 + iµω ρ The first term arises from the displacement current, the second from the conductance current. If k is written in the form k := k +ik it follows that: k = ω 1 2 εµ 1 + 1 + 1 (ρεω) 2 and k = ω 1 2 εµ −1 + 1 + 1 (ρεω) 2 This results in a damped wave: E = E exp(−k n r ) exp(i(k n r −ωt)). If the material is a good conductor, the wave vanishes after approximately one wavelength, k = (1 +i) µω 2ρ . 2.6 Multipoles Because 1 [r −r [ = 1 r ∞ ¸ 0 r r l P l (cos θ) the potential can be written as: V = Q 4πε ¸ n k n r n For the lowest-order terms this results in: • Monopole: l = 0, k 0 = ρdV • Dipole: l = 1, k 1 = r cos(θ)ρdV • Quadrupole: l = 2, k 2 = 1 2 ¸ i (3z 2 i −r 2 i ) 1. The electric dipole: dipole moment: p = Qle, where e goes from ⊕ to , and F = ( p ∇) E ext , and W = − p E out . Electric field: E ≈ Q 4πεr 3 3 p r r 2 − p . The torque is: τ = p E out 2. The magnetic dipole: dipole moment: if r √ A: µ = I (Ae ⊥ ), F = (µ ∇) B out [µ[ = mv 2 ⊥ 2B , W = −µ B out Magnetic field: B = −µ 4πr 3 3µ r r 2 − µ . The moment is: τ = µ B out 2.7 Electric currents The continuity equation for charge is: ∂ρ ∂t +∇ J = 0. The electric current is given by: I = dQ dt = ( J n)d 2 A For most conductors holds: J = E/ρ, where ρ is the resistivity. 12 Physics Formulary by ir. J.C.A. Wevers If the flux enclosed by a conductor changes this results in an induced voltage V ind = −N dΦ dt . If the current flowing through a conductor changes, this results in a self-inductance which opposes the original change: V selfind = −L dI dt . If a conductor encloses a flux Φ holds: Φ = LI. The magnetic induction within a coil is approximated by: B = µNI √ l 2 +4R 2 where l is the length, R the radius and N the number of coils. The energy contained within a coil is given by W = 1 2 LI 2 and L = µN 2 A/l. The capacity is defined by: C = Q/V . For a capacitor holds: C = ε 0 ε r A/d where d is the distance between the plates and A the surface of one plate. The electric field strength between the plates is E = σ/ε 0 = Q/ε 0 A where σ is the surface charge. The accumulated energy is given by W = 1 2 CV 2 . The current through a capacity is given by I = −C dV dt . For most PTC resistors holds approximately: R = R 0 (1 + αT), where R 0 = ρl/A. For a NTC holds: R(T) = C exp(−B/T) where B and C depend only on the material. If a current flows through two different, connecting conductors x and y, the contact area will heat up or cool down, depending on the direction of the current: the Peltier effect. The generated or removed heat is given by: W = Π xy It. This effect can be amplified with semiconductors. The thermic voltage between 2 metals is given by: V = γ(T − T 0 ). For a Cu-Konstantane connection holds: γ ≈ 0.2 −0.7 mV/K. In an electrical net with only stationary currents, Kirchhoff ’s equations apply: for a knot holds: ¸ I n = 0, along a closed path holds: ¸ V n = ¸ I n R n = 0. 2.8 Depolarizing field If a dielectric material is placed in an electric or magnetic field, the field strength within and outside the material will change because the material will be polarized or magnetized. If the medium has an ellipsoidal shape and one of the principal axes is parallel with the external field E 0 or B 0 then the depolarizing is field homogeneous. E dep = E mat − E 0 = − ^ P ε 0 H dep = H mat − H 0 = −^ M ^ is a constant depending only on the shape of the object placed in the field, with 0 ≤ ^ ≤ 1. For a few limiting cases of an ellipsoid holds: a thin plane: ^ = 1, a long, thin bar: ^ = 0, a sphere: ^ = 1 3 . 2.9 Mixtures of materials The average electric displacement in a material which is inhomogenious on a mesoscopic scale is given by: 'D` = 'εE` = ε ∗ 'E` where ε ∗ = ε 1 1 − φ 2 (1 −x) Φ(ε ∗ /ε 2 ) −1 where x = ε 1 /ε 2 . For a sphere holds: Φ = 1 3 + 2 3 x. Further holds: ¸ i φ i ε i −1 ≤ ε ∗ ≤ ¸ i φ i ε i Chapter 3 Relativity 3.1 Special relativity 3.1.1 The Lorentz transformation The Lorentz transformation (x , t ) = (x (x, t), t (x, t)) leaves the wave equation invariant if c is invariant: ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 − 1 c 2 ∂ 2 ∂t 2 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 − 1 c 2 ∂ 2 ∂t 2 This transformation can also be found when ds 2 = ds 2 is demanded. The general form of the Lorentz transformation is given by: x = x + (γ −1)(x v )v [v[ 2 −γvt , t = γ t − x v c 2 where γ = 1 1 − v 2 c 2 The velocity difference v between two observers transforms according to: v = γ 1 − v 1 v 2 c 2 −1 v 2 +(γ −1) v 1 v 2 v 2 1 v 1 −γv 1 If the velocity is parallel to the x-axis, this becomes y = y, z = z and: x = γ(x −vt) , x = γ(x +vt ) t = γ t − xv c 2 , t = γ t + x v c 2 , v = v 2 −v 1 1 − v 1 v 2 c 2 If v = ve x holds: p x = γ p x − βW c , W = γ(W −vp x ) With β = v/c the electric field of a moving charge is given by: E = Q 4πε 0 r 2 (1 −β 2 )e r (1 −β 2 sin 2 (θ)) 3/2 The electromagnetic field transforms according to: E = γ( E +v B) , B = γ B − v E c 2 Length, mass and time transform according to: ∆t r = γ∆t 0 , m r = γm 0 , l r = l 0 /γ, with 0 the quantities in a co-moving reference frame and r the quantities in a frame moving with velocity v w.r.t. it. The proper time τ is defined as: dτ 2 = ds 2 /c 2 , so ∆τ = ∆t/γ. For energy and momentum holds: W = m r c 2 = γW 0 , 14 Physics Formulary by ir. J.C.A. Wevers W 2 = m 2 0 c 4 + p 2 c 2 . p = m r v = γm 0 v = Wv/c 2 , and pc = Wβ where β = v/c. The force is defined by F = d p/dt. 4-vectors have the property that their modulus is independent of the observer: their components can change after a coordinate transformation but not their modulus. The difference of two 4-vectors transforms also as a 4-vector. The 4-vector for the velocity is given by U α = dx α dτ . The relation with the “common” velocity u i := dx i /dt is: U α = (γu i , icγ). For particles with nonzero restmass holds: U α U α = −c 2 , for particles with zero restmass (so with v = c) holds: U α U α = 0. The 4-vector for energy and momentum is given by: p α = m 0 U α = (γp i , iW/c). So: p α p α = −m 2 0 c 2 = p 2 −W 2 /c 2 . 3.1.2 Red and blue shift There are three causes of red and blue shifts: 1. Motion: with e v e r = cos(ϕ) follows: f f = γ 1 − v cos(ϕ) c . This can give both red- and blueshift, also ⊥ to the direction of motion. 2. Gravitational redshift: ∆f f = κM rc 2 . 3. Redshift because the universe expands, resulting in e.g. the cosmic background radiation: λ 0 λ 1 = R 0 R 1 . 3.1.3 The stress-energy tensor and the field tensor The stress-energy tensor is given by: T µν = (c 2 +p)u µ u ν +pg µν + 1 c 2 F µα F α ν + 1 4 g µν F αβ F αβ The conservation laws can than be written as: ∇ ν T µν = 0. The electromagnetic field tensor is given by: F αβ = ∂A β ∂x α − ∂A α ∂x β with A µ := ( A, iV/c) and J µ := ( J, icρ). The Maxwell equations can than be written as: ∂ ν F µν = µ 0 J µ , ∂ λ F µν +∂ µ F νλ +∂ ν F λµ = 0 The equations of motion for a charged particle in an EM field become with the field tensor: dp α dτ = qF αβ u β 3.2 General relativity 3.2.1 Riemannian geometry, the Einstein tensor The basic principles of general relativity are: 1. The geodesic postulate: free falling particles move along geodesics of space-time with the proper time τ or arc length s as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds ds = 0. From δ ds = 0 the equations of motion can be derived: d 2 x α ds 2 +Γ α βγ dx β ds dx γ ds = 0 Chapter 3: Relativity 15 2. The principle of equivalence: inertial mass ≡ gravitational mass ⇒ gravitation is equivalent with a curved space-time were particles move along geodesics. 3. By a proper choice of the coordinate system it is possible to make the metric locally flat in each point x i : g αβ (x i ) = η αβ :=diag(−1, 1, 1, 1). The Riemann tensor is defined as: R µ ναβ T ν := ∇ α ∇ β T µ −∇ β ∇ α T µ , where the covariant derivative is given by ∇ j a i = ∂ j a i +Γ i jk a k and ∇ j a i = ∂ j a i −Γ k ij a k . Here, Γ i jk = g il 2 ∂g lj ∂x k + ∂g lk ∂x j − ∂g j k ∂x l , for Euclidean spaces this reduces to: Γ i jk = ∂ 2 ¯ x l ∂x j ∂x k ∂x i ∂¯ x l , are the Christoffel symbols. For a second-order tensor holds: [∇ α , ∇ β ]T µ ν = R µ σαβ T σ ν + R σ ναβ T µ σ , ∇ k a i j = ∂ k a i j −Γ l kj a i l +Γ i kl a l j , ∇ k a ij = ∂ k a ij −Γ l ki a lj −Γ l kj a jl and ∇ k a ij = ∂ k a ij +Γ i kl a lj +Γ j kl a il . The following holds: R α βµν = ∂ µ Γ α βν −∂ ν Γ α βµ + Γ α σµ Γ σ βν −Γ α σν Γ σ βµ . The Ricci tensor is a contraction of the Riemann tensor: R αβ := R µ αµβ , which is symmetric: R αβ = R βα . The Bianchi identities are: ∇ λ R αβµν +∇ ν R αβλµ +∇ µ R αβνλ = 0. The Einstein tensor is given by: G αβ := R αβ − 1 2 g αβ R, where R := R α α is the Ricci scalar, for which holds: ∇ β G αβ = 0. With the variational principle δ (L(g µν ) − Rc 2 /16πκ) [g[d 4 x = 0 for variations g µν →g µν +δg µν the Einstein field equations can be derived: G αβ = 8πκ c 2 T αβ , which can also be written as R αβ = 8πκ c 2 (T αβ − 1 2 g αβ T µ µ ) For empty space this is equivalent to R αβ = 0. The equation R αβµν = 0 has as only solution a flat space. The Einstein equations are 10 independent equations, which are of second order in g µν . From this, the Laplace equation from Newtonian gravitation can be derived by stating: g µν = η µν + h µν , where [h[ < 1. In the stationary case, this results in ∇ 2 h 00 = 8πκ/c 2 . The most general form of the field equations is: R αβ − 1 2 g αβ R +Λg αβ = 8πκ c 2 T αβ where Λ is the cosmological constant. This constant plays a role in inflatory models of the universe. 3.2.2 The line element The metric tensor in an Euclidean space is given by: g ij = ¸ k ∂¯ x k ∂x i ∂¯ x k ∂x j . In general holds: ds 2 = g µν dx µ dx ν . In special relativity this becomes ds 2 = −c 2 dt 2 + dx 2 + dy 2 + dz 2 . This metric, η µν :=diag(−1, 1, 1, 1), is called the Minkowski metric. The external Schwarzschild metric applies in vacuum outside a spherical mass distribution, and is given by: ds 2 = −1 + 2m r c 2 dt 2 + 1 − 2m r −1 dr 2 +r 2 dΩ 2 Here, m := Mκ/c 2 is the geometrical mass of an object with mass M, and dΩ 2 = dθ 2 + sin 2 θdϕ 2 . This metric is singular for r = 2m = 2κM/c 2 . If an object is smaller than its event horizon 2m, that implies that its escape velocity is > c, it is called a black hole. The Newtonian limit of this metric is given by: ds 2 = −(1 +2V )c 2 dt 2 +(1 −2V )(dx 2 +dy 2 +dz 2 ) where V = −κM/r is the Newtonian gravitation potential. In general relativity, the components of g µν are associated with the potentials and the derivatives of g µν with the field strength. The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metric near r = 2m. They are defined by: 16 Physics Formulary by ir. J.C.A. Wevers • r > 2m: u = r 2m −1 exp r 4m cosh t 4m v = r 2m −1 exp r 4m sinh t 4m • r < 2m: u = 1 − r 2m exp r 4m sinh t 4m v = 1 − r 2m exp r 4m cosh t 4m • r = 2m: here, the Kruskal coordinates are singular, which is necessary to eliminate the coordinate singularity there. The line element in these coordinates is given by: ds 2 = − 32m 3 r e −r/2m (dv 2 −du 2 ) +r 2 dΩ 2 The line r = 2m corresponds to u = v = 0, the limit x 0 →∞with u = v and x 0 →−∞with u = −v. The Kruskal coordinates are only singular on the hyperbole v 2 −u 2 = 1, this corresponds with r = 0. On the line dv = ±du holds dθ = dϕ = ds = 0. For the metric outside a rotating, charged spherical mass the Newman metric applies: ds 2 = 1 − 2mr −e 2 r 2 +a 2 cos 2 θ c 2 dt 2 − r 2 +a 2 cos 2 θ r 2 −2mr +a 2 −e 2 dr 2 −(r 2 +a 2 cos 2 θ)dθ 2 − r 2 +a 2 + (2mr −e 2 )a 2 sin 2 θ r 2 +a 2 cos 2 θ sin 2 θdϕ 2 + 2a(2mr −e 2 ) r 2 +a 2 cos 2 θ sin 2 θ(dϕ)(cdt) where m = κM/c 2 , a = L/Mc and e = κQ/ε 0 c 2 . A rotating charged black hole has an event horizon with R S = m+ √ m 2 −a 2 −e 2 . Near rotating black holes frame dragging occurs because g tϕ = 0. For the Kerr metric (e = 0, a = 0) then follows that within the surface R E = m + √ m 2 −a 2 cos 2 θ (de ergosphere) no particle can be at rest. 3.2.3 Planetary orbits and the perihelion shift To find a planetary orbit, the variational problemδ ds = 0 has to be solved. This is equivalent to the problem δ ds 2 = δ g ij dx i dx j = 0. Substituting the external Schwarzschild metric yields for a planetary orbit: du dϕ d 2 u dϕ 2 +u = du dϕ 3mu + m h 2 where u := 1/r and h = r 2 ˙ ϕ =constant. The term 3mu is not present in the classical solution. This term can in the classical case also be found from a potential V (r) = − κM r 1 + h 2 r 2 . The orbital equation gives r =constant as solution, or can, after dividing by du/dϕ, be solved with perturbation theory. In zeroth order, this results in an elliptical orbit: u 0 (ϕ) = A + Bcos(ϕ) with A = m/h 2 and B an arbitrary constant. In first order, this becomes: u 1 (ϕ) = A +Bcos(ϕ −εϕ) +ε A+ B 2 2A − B 2 6A cos(2ϕ) where ε = 3m 2 /h 2 is small. The perihelion of a planet is the point for which r is minimal, or u maximal. This is the case if cos(ϕ − εϕ) = 0 ⇒ ϕ ≈ 2πn(1 + ε). For the perihelion shift then follows: ∆ϕ = 2πε = 6πm 2 /h 2 per orbit. Chapter 3: Relativity 17 3.2.4 The trajectory of a photon For the trajectory of a photon (and for each particle with zero restmass) holds ds 2 = 0. Substituting the external Schwarzschild metric results in the following orbital equation: du dϕ d 2 u dϕ 2 +u −3mu = 0 3.2.5 Gravitational waves Starting with the approximation g µν = η µν + h µν for weak gravitational fields and the definition h µν = h µν − 1 2 η µν h α α it follows that ✷h µν = 0 if the gauge condition ∂h µν /∂x ν = 0 is satisfied. From this, it follows that the loss of energy of a mechanical system, if the occurring velocities are <c and for wavelengths the size of the system, is given by: dE dt = − G 5c 5 ¸ i,j d 3 Q ij dt 3 2 with Q ij = (x i x j − 1 3 δ ij r 2 )d 3 x the mass quadrupole moment. 3.2.6 Cosmology If for the universe as a whole is assumed: 1. There exists a global time coordinate which acts as x 0 of a Gaussian coordinate system, 2. The 3-dimensional spaces are isotrope for a certain value of x 0 , 3. Each point is equivalent to each other point for a fixed x 0 . then the Robertson-Walker metric can be derived for the line element: ds 2 = −c 2 dt 2 + R 2 (t) r 2 0 1 − kr 2 4r 2 0 (dr 2 +r 2 dΩ 2 ) For the scalefactor R(t) the following equations can be derived: 2 ¨ R R + ˙ R 2 +kc 2 R 2 = − 8πκp c 2 +Λ and ˙ R 2 +kc 2 R 2 = 8πκ 3 + Λ 3 where p is the pressure and the density of the universe. If Λ = 0 can be derived for the deceleration parameter q: q = − ¨ RR ˙ R 2 = 4πκ 3H 2 where H = ˙ R/R is Hubble’s constant. This is a measure of the velocity with which galaxies far away are moving away from each other, and has the value ≈ (75±25) kms −1 Mpc −1 . This gives 3 possible conditions for the universe (here, W is the total amount of energy in the universe): 1. Parabolical universe: k = 0, W = 0, q = 1 2 . The expansion velocity of the universe → 0 if t → ∞. The hereto related critical density is c = 3H 2 /8πκ. 2. Hyperbolical universe: k = −1, W < 0, q < 1 2 . The expansion velocity of the universe remains positive forever. 3. Elliptical universe: k = 1, W > 0, q > 1 2 . The expansion velocity of the universe becomes negative after some time: the universe starts collapsing. Chapter 4 Oscillations 4.1 Harmonic oscillations The general form of a harmonic oscillation is: Ψ(t) = ˆ Ψe i(ωt±ϕ) ≡ ˆ Ψcos(ωt ±ϕ), where ˆ Ψ is the amplitude. A superposition of several harmonic oscillations with the same frequency results in another harmonic oscillation: ¸ i ˆ Ψ i cos(α i ±ωt) = ˆ Φcos(β ±ωt) with: tan(β) = ¸ i ˆ Ψ i sin(α i ) ¸ i ˆ Ψ i cos(α i ) and ˆ Φ 2 = ¸ i ˆ Ψ 2 i +2 ¸ j>i ¸ i ˆ Ψ i ˆ Ψ j cos(α i −α j ) For harmonic oscillations holds: x(t)dt = x(t) iω and d n x(t) dt n = (iω) n x(t). 4.2 Mechanic oscillations For a construction with a spring with constant C parallel to a damping k which is connected to a mass M, to which a periodic force F(t) = ˆ F cos(ωt) is applied holds the equation of motion m¨ x = F(t) − k ˙ x − Cx. With complex amplitudes, this becomes −mω 2 x = F −Cx −ikωx. With ω 2 0 = C/m follows: x = F m(ω 2 0 −ω 2 ) +ikω , and for the velocity holds: ˙ x = F i √ Cmδ +k where δ = ω ω 0 − ω 0 ω . The quantity Z = F/ ˙ x is called the impedance of the system. The quality of the system is given by Q = √ Cm k . The frequency with minimal [Z[ is called velocity resonance frequency. This is equal to ω 0 . In the resonance curve [Z[/ √ Cmis plotted against ω/ω 0 . The width of this curve is characterized by the points where [Z(ω)[ = [Z(ω 0 )[ √ 2. In these points holds: R = X and δ = ±Q −1 , and the width is 2∆ω B = ω 0 /Q. The stiffness of an oscillating system is given by F/x. The amplitude resonance frequency ω A is the frequency where iωZ is minimal. This is the case for ω A = ω 0 1 − 1 2 Q 2 . The damping frequency ω D is a measure for the time in which an oscillating system comes to rest. It is given by ω D = ω 0 1 − 1 4Q 2 . A weak damped oscillation (k 2 < 4mC) dies out after T D = 2π/ω D . For a critical damped oscillation (k 2 = 4mC) holds ω D = 0. A strong damped oscillation (k 2 > 4mC) drops like (if k 2 4mC) x(t) ≈ x 0 exp(−t/τ). 4.3 Electric oscillations The impedance is given by: Z = R + iX. The phase angle is ϕ := arctan(X/R). The impedance of a resistor is R, of a capacitor 1/iωC and of a self inductor iωL. The quality of a coil is Q = ωL/R. The total impedance in case several elements are positioned is given by: Chapter 4: Oscillations 19 1. Series connection: V = IZ, Z tot = ¸ i Z i , L tot = ¸ i L i , 1 C tot = ¸ i 1 C i , Q = Z 0 R , Z = R(1 +iQδ) 2. parallel connection: V = IZ, 1 Z tot = ¸ i 1 Z i , 1 L tot = ¸ i 1 L i , C tot = ¸ i C i , Q = R Z 0 , Z = R 1 +iQδ Here, Z 0 = L C and ω 0 = 1 √ LC . The power given by a source is given by P(t) = V (t) I(t), so 'P` t = ˆ V eff ˆ I eff cos(∆φ) = 1 2 ˆ V ˆ I cos(φ v −φ i ) = 1 2 ˆ I 2 Re(Z) = 1 2 ˆ V 2 Re(1/Z), where cos(∆φ) is the work factor. 4.4 Waves in long conductors These cables are in use for signal transfer, e.g. coax cable. For them holds: Z 0 = dL dx dx dC . The transmission velocity is given by v = dx dL dx dC . 4.5 Coupled conductors and transformers For two coils enclosing each others flux holds: if Φ 12 is the part of the flux originating from I 2 through coil 2 which is enclosed by coil 1, than holds Φ 12 = M 12 I 2 , Φ 21 = M 21 I 1 . For the coefficients of mutual induction M ij holds: M 12 = M 21 := M = k L 1 L 2 = N 1 Φ 1 I 2 = N 2 Φ 2 I 1 ∼ N 1 N 2 where 0 ≤ k ≤ 1 is the coupling factor. For a transformer is k ≈ 1. At full load holds: V 1 V 2 = I 2 I 1 = − iωM iωL 2 +R load ≈ − L 1 L 2 = − N 1 N 2 4.6 Pendulums The oscillation time T = 1/f, and for different types of pendulums is given by: • Oscillating spring: T = 2π m/C if the spring force is given by F = C ∆l. • Physical pendulum: T = 2π I/τ with τ the moment of force and I the moment of inertia. • Torsion pendulum: T = 2π I/κ with κ = 2lm πr 4 ∆ϕ the constant of torsion and I the moment of inertia. • Mathematical pendulum: T = 2π l/g with g the acceleration of gravity and l the length of the pendu- lum. Chapter 5 Waves 5.1 The wave equation The general form of the wave equation is: ✷u = 0, or: ∇ 2 u − 1 v 2 ∂ 2 u ∂t 2 = ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 + ∂ 2 u ∂z 2 − 1 v 2 ∂ 2 u ∂t 2 = 0 where u is the disturbance and v the propagation velocity. In general holds: v = fλ. By definition holds: kλ = 2π and ω = 2πf. In principle, there are two types of waves: 1. Longitudinal waves: for these holds k | v | u. 2. Transversal waves: for these holds k | v ⊥ u. The phase velocity is given by v ph = ω/k. The group velocity is given by: v g = dω dk = v ph +k dv ph dk = v ph 1 − k n dn dk where n is the refractive index of the medium. If v ph does not depend on ω holds: v ph = v g . In a dispersive medium it is possible that v g > v ph or v g < v ph , and v g v f = c 2 . If one wants to transfer information with a wave, e.g. by modulation of an EM wave, the information travels with the velocity at with a change in the electromagnetic field propagates. This velocity is often almost equal to the group velocity. For some media, the propagation velocity follows from: • Pressure waves in a liquid or gas: v = κ/, where κ is the modulus of compression. • For pressure waves in a gas also holds: v = γp/ = γRT/M. • Pressure waves in a thin solid bar with diameter << λ: v = E/ • waves in a string: v = F span l/m • Surface waves on a liquid: v = gλ 2π + 2πγ λ tanh 2πh λ where h is the depth of the liquid and γ the surface tension. If h <λ holds: v ≈ √ gh. 5.2 Solutions of the wave equation 5.2.1 Plane waves In n dimensions a harmonic plane wave is defined by: u(x, t) = 2 n ˆ ucos(ωt) n ¸ i=1 sin(k i x i ) Chapter 5: Waves 21 The equation for a harmonic traveling plane wave is: u(x, t) = ˆ ucos( k x ±ωt +ϕ) If waves reflect at the end of a spring this will result in a change in phase. A fixed end gives a phase change of π/2 to the reflected wave, with boundary condition u(l) = 0. A lose end gives no change in the phase of the reflected wave, with boundary condition (∂u/∂x) l = 0. If an observer is moving w.r.t. the wave with a velocity v obs , he will observe a change in frequency: the Doppler effect. This is given by: f f 0 = v f −v obs v f . 5.2.2 Spherical waves When the situation is spherical symmetric, the homogeneous wave equation is given by: 1 v 2 ∂ 2 (ru) ∂t 2 − ∂ 2 (ru) ∂r 2 = 0 with general solution: u(r, t) = C 1 f(r −vt) r +C 2 g(r +vt) r 5.2.3 Cylindrical waves When the situation has a cylindrical symmetry, the homogeneous wave equation becomes: 1 v 2 ∂ 2 u ∂t 2 − 1 r ∂ ∂r r ∂u ∂r = 0 This is a Bessel equation, with solutions which can be written as Hankel functions. For sufficient large values of r these are approximated by: u(r, t) = ˆ u √ r cos(k(r ±vt)) 5.2.4 The general solution in one dimension Starting point is the equation: ∂ 2 u(x, t) ∂t 2 = N ¸ m=0 b m ∂ m ∂x m u(x, t) where b m ∈ IR. Substituting u(x, t) = Ae i(kx−ωt) gives two solutions ω j = ω j (k) as dispersion relations. The general solution is given by: u(x, t) = ∞ −∞ a(k)e i(kx−ω1(k)t) +b(k)e i(kx−ω2(k)t) dk Because in general the frequencies ω j are non-linear in k there is dispersion and the solution cannot be written any more as a sum of functions depending only on x ±vt: the wave front transforms. 5.3 The stationary phase method Usually the Fourier integrals of the previous section cannot be calculated exactly. If ω j (k) ∈ IR the stationary phase method can be applied. Assuming that a(k) is only a slowly varying function of k, one can state that the parts of the k-axis where the phase of kx −ω(k)t changes rapidly will give no net contribution to the integral because the exponent oscillates rapidly there. The only areas contributing significantly to the integral are areas with a stationary phase, determined by d dk (kx −ω(k)t) = 0. Now the following approximation is possible: ∞ −∞ a(k)e i(kx−ω(k)t) dk ≈ N ¸ i=1 2π d 2 ω(ki) dk 2 i exp −i 1 4 π +i(k i x −ω(k i )t) 22 Physics Formulary by ir. J.C.A. Wevers 5.4 Green functions for the initial-value problem This method is preferable if the solutions deviate much fromthe stationary solutions, like point-like excitations. Starting with the wave equation in one dimension, with ∇ 2 = ∂ 2 /∂x 2 holds: if Q(x, x , t) is the solution with initial values Q(x, x , 0) = δ(x − x ) and ∂Q(x, x , 0) ∂t = 0, and P(x, x , t) the solution with initial values P(x, x , 0) = 0 and ∂P(x, x , 0) ∂t = δ(x − x ), then the solution of the wave equation with arbitrary initial conditions f(x) = u(x, 0) and g(x) = ∂u(x, 0) ∂t is given by: u(x, t) = ∞ −∞ f(x )Q(x, x , t)dx + ∞ −∞ g(x )P(x, x , t)dx P and Q are called the propagators. They are defined by: Q(x, x , t) = 1 2 [δ(x −x −vt) +δ(x −x +vt)] P(x, x , t) = 1 2v if [x −x [ < vt 0 if [x −x [ > vt Further holds the relation: Q(x, x , t) = ∂P(x, x , t) ∂t 5.5 Waveguides and resonating cavities The boundary conditions for a perfect conductor can be derived from the Maxwell equations. If n is a unit vector ⊥ the surface, pointed from 1 to 2, and K is a surface current density, than holds: n ( D 2 − D 1 ) = σ n ( E 2 − E 1 ) = 0 n ( B 2 − B 1 ) = 0 n ( H 2 − H 1 ) = K In a waveguide holds because of the cylindrical symmetry: E(x, t) = c(x, y)e i(kz−ωt) and B(x, t) = B(x, y)e i(kz−ωt) . From this one can now deduce that, if B z and c z are not ≡ 0: B x = i εµω 2 −k 2 k ∂B z ∂x −εµω ∂c z ∂y B y = i εµω 2 −k 2 k ∂B z ∂y +εµω ∂c z ∂x c x = i εµω 2 −k 2 k ∂c z ∂x +εµω ∂B z ∂y c y = i εµω 2 −k 2 k ∂c z ∂y −εµω ∂B z ∂x Now one can distinguish between three cases: 1. B z ≡ 0: the Transversal Magnetic modes (TM). Boundary condition: c z [ surf = 0. 2. E z ≡ 0: the Transversal Electric modes (TE). Boundary condition: ∂B z ∂n surf = 0. For the TE and TM modes this gives an eigenvalue problem for c z resp. B z with boundary conditions: ∂ 2 ∂x 2 + ∂ 2 ∂y 2 ψ = −γ 2 ψ with eigenvalues γ 2 := εµω 2 −k 2 This gives a discrete solution ψ with eigenvalue γ 2 : k = εµω 2 −γ 2 . For ω < ω , k is imaginary and the wave is damped. Therefore, ω is called the cut-off frequency. In rectangular conductors the following expression can be found for the cut-off frequency for modes TE m,n of TM m,n : λ = 2 (m/a) 2 +(n/b) 2 Chapter 5: Waves 23 3. E z and B z are zero everywhere: the Transversal electromagnetic mode (TEM). Than holds: k = ±ω √ εµ and v f = v g , just as if here were no waveguide. Further k ∈ IR, so there exists no cut-off frequency. In a rectangular, 3 dimensional resonating cavity with edges a, b and c the possible wave numbers are given by: k x = n 1 π a , k y = n 2 π b , k z = n 3 π c This results in the possible frequencies f = vk/2π in the cavity: f = v 2 n 2 x a 2 + n 2 y b 2 + n 2 z c 2 For a cubic cavity, with a = b = c, the possible number of oscillating modes N L for longitudinal waves is given by: N L = 4πa 3 f 3 3v 3 Because transversal waves have two possible polarizations holds for them: N T = 2N L . 5.6 Non-linear wave equations The Van der Pol equation is given by: d 2 x dt 2 −εω 0 (1 −βx 2 ) dx dt +ω 2 0 x = 0 βx 2 can be ignored for very small values of the amplitude. Substitution of x ∼ e iωt gives: ω = 1 2 ω 0 (iε ± 2 1 − 1 2 ε 2 ). The lowest-order instabilities grow as 1 2 εω 0 . While x is growing, the 2nd term becomes larger and diminishes the growth. Oscillations on a time scale ∼ ω −1 0 can exist. If x is expanded as x = x (0) + εx (1) + ε 2 x (2) + and this is substituted one obtains, besides periodic, secular terms ∼ εt. If it is assumed that there exist timescales τ n , 0 ≤ τ ≤ N with ∂τ n /∂t = ε n and if the secular terms are put 0 one obtains: d dt 1 2 dx dt 2 + 1 2 ω 2 0 x 2 ¸ = εω 0 (1 −βx 2 ) dx dt 2 This is an energy equation. Energy is conserved if the left-hand side is 0. If x 2 > 1/β, the right-hand side changes sign and an increase in energy changes into a decrease of energy. This mechanism limits the growth of oscillations. The Korteweg-De Vries equation is given by: ∂u ∂t + ∂u ∂x − au ∂u ∂x . .. . non−lin + b 2 ∂ 3 u ∂x 3 . .. . dispersive = 0 This equation is for example a model for ion-acoustic waves in a plasma. For this equation, soliton solutions of the following form exist: u(x −ct) = −d cosh 2 (e(x −ct)) with c = 1 + 1 3 ad and e 2 = ad/(12b 2 ). Chapter 6 Optics 6.1 The bending of light For the refraction at a surface holds: n i sin(θ i ) = n t sin(θ t ) where n is the refractive index of the material. Snell’s law is: n 2 n 1 = λ 1 λ 2 = v 1 v 2 If ∆n ≤ 1, the change in phase of the light is ∆ϕ = 0, if ∆n > 1 holds: ∆ϕ = π. The refraction of light in a material is caused by scattering from atoms. This is described by: n 2 = 1 + n e e 2 ε 0 m ¸ j f j ω 2 0,j −ω 2 −iδω where n e is the electron density and f j the oscillator strength, for which holds: ¸ j f j = 1. From this follows that v g = c/(1 +(n e e 2 /2ε 0 mω 2 )). From this the equation of Cauchy can be derived: n = a 0 +a 1 /λ 2 . More general, it is possible to expand n as: n = n ¸ k=0 a k λ 2k . For an electromagnetic wave in general holds: n = √ ε r µ r . The path, followed by a light ray in material can be found from Fermat’s principle: δ 2 1 dt = δ 2 1 n(s) c ds = 0 ⇒δ 2 1 n(s)ds = 0 6.2 Paraxial geometrical optics 6.2.1 Lenses The Gaussian lens formula can be deduced from Fermat’s principle with the approximations cos ϕ = 1 and sin ϕ = ϕ. For the refraction at a spherical surface with radius R holds: n 1 v − n 2 b = n 1 −n 2 R where [v[ is the distance of the object and [b[ the distance of the image. Applying this twice results in: 1 f = (n l −1) 1 R 2 − 1 R 1 where n l is the refractive index of the lens, f is the focal length and R 1 and R 2 are the curvature radii of both surfaces. For a double concave lens holds R 1 < 0, R 2 > 0, for a double convex lens holds R 1 > 0 and R 2 < 0. Further holds: 1 f = 1 v − 1 b Chapter 6: Optics 25 D := 1/f is called the dioptric power of a lens. For a lens with thickness d and diameter D holds to a good approximation: 1/f = 8(n −1)d/D 2 . For two lenses placed on a line with distance d holds: 1 f = 1 f 1 + 1 f 2 − d f 1 f 2 In these equations the following signs are being used for refraction at a spherical surface, as is seen by an incoming light ray: Quantity + − R Concave surface Convex surface f Converging lens Diverging lens v Real object Virtual object b Virtual image Real image 6.2.2 Mirrors For images of mirrors holds: 1 f = 1 v + 1 b = 2 R + h 2 2 1 R − 1 v 2 where h is the perpendicular distance from the point the light ray hits the mirror to the optical axis. Spherical aberration can be reduced by not using spherical mirrors. A parabolical mirror has no spherical aberration for light rays parallel with the optical axis and is therefore often used for telescopes. The used signs are: Quantity + − R Concave mirror Convex mirror f Concave mirror Convex mirror v Real object Virtual object b Real image Virtual image 6.2.3 Principal planes The nodal points N of a lens are defined by the figure on the right. If the lens is surrounded by the same medium on both sides, the nodal points are the same as the principal points H. The plane ⊥ the optical axis through the principal points is called the principal plane. If the lens is described by a matrix m ij than for the distances h 1 and h 2 to the boundary of the lens holds: h 1 = n m 11 −1 m 12 , h 2 = n m 22 −1 m 12 r r r N 1 N 2 O 6.2.4 Magnification The linear magnification is defined by: N = − b v The angular magnification is defined by: N α = − α syst α none where α sys is the size of the retinal image with the optical system and α none the size of the retinal image without the system. Further holds: N N α = 1. For a telescope holds: N = f objective /f ocular . The f-number is defined by f/D objective . 26 Physics Formulary by ir. J.C.A. Wevers 6.3 Matrix methods A light ray can be described by a vector (nα, y) with α the angle with the optical axis and y the distance to the optical axis. The change of a light ray interacting with an optical system can be obtained using a matrix multiplication: n 2 α 2 y 2 = M n 1 α 1 y 1 where Tr(M) = 1. M is a product of elementary matrices. These are: 1. Transfer along length l: M R = 1 0 l/n 1 2. Refraction at a surface with dioptric power D: M T = 1 −D 0 1 6.4 Aberrations Lenses usually do not give a perfect image. Some causes are: 1. Chromatic aberration is caused by the fact that n = n(λ). This can be partially corrected with a lens which is composed of more lenses with different functions n i (λ). Using N lenses makes it possible to obtain the same f for N wavelengths. 2. Spherical aberration is caused by second-order effects which are usually ignored; a spherical surface does not make a perfect lens. Incomming rays far from the optical axis will more bent. 3. Coma is caused by the fact that the principal planes of a lens are only flat near the principal axis. Further away of the optical axis they are curved. This curvature can be both positive or negative. 4. Astigmatism: from each point of an object not on the optical axis the image is an ellipse because the thickness of the lens is not the same everywhere. 5. Field curvature can be corrected by the human eye. 6. Distorsion gives abberations near the edges of the image. This can be corrected with a combination of positive and negative lenses. 6.5 Reflection and transmission If an electromagnetic wave hits a transparent medium part of the wave will reflect at the same angle as the incident angle, and a part will be refracted at an angle according to Snell’s law. It makes a difference whether the E field of the wave is ⊥ or | w.r.t. the surface. When the coefficients of reflection r and transmission t are defined as: r ≡ E 0r E 0i , r ⊥ ≡ E 0r E 0i ⊥ , t ≡ E 0t E 0i , t ⊥ ≡ E 0t E 0i ⊥ where E 0r is the reflected amplitude and E 0t the transmitted amplitude. Then the Fresnel equations are: r = tan(θ i −θ t ) tan(θ i +θ t ) , r ⊥ = sin(θ t −θ i ) sin(θ t +θ i ) t = 2 sin(θ t ) cos(θ i ) sin(θ t +θ i ) cos(θ t −θ i ) , t ⊥ = 2 sin(θ t ) cos(θ i ) sin(θ t +θ i ) The following holds: t ⊥ − r ⊥ = 1 and t + r = 1. If the coefficient of reflection R and transmission T are defined as (with θ i = θ r ): R ≡ I r I i and T ≡ I t cos(θ t ) I i cos(θ i ) Chapter 6: Optics 27 with I = '[ S[` it follows: R+T = 1. A special case is r = 0. This happens if the angle between the reflected and transmitted rays is 90 ◦ . From Snell’s law it then follows: tan(θ i ) = n. This angle is called Brewster’s angle. The situation with r ⊥ = 0 is not possible. 6.6 Polarization The polarization is defined as: P = I p I p +I u = I max −I min I max +I min where the intensity of the polarized light is given by I p and the intensity of the unpolarized light is given by I u . I max and I min are the maximum and minimum intensities when the light passes a polarizer. If polarized light passes through a polarizer Malus law applies: I(θ) = I(0) cos 2 (θ) where θ is the angle of the polarizer. The state of a light ray can be described by the Stokes-parameters: start with 4 filters which each transmits half the intensity. The first is independent of the polarization, the second and third are linear polarizers with the transmission axes horizontal and at +45 ◦ , while the fourth is a circular polarizer which is opaque for L-states. Then holds S 1 = 2I 1 , S 2 = 2I 2 −2I 1 , S 3 = 2I 3 −2I 1 and S 4 = 2I 4 −2I 1 . The state of a polarized light ray can also be described by the Jones vector: E = E 0x e iϕx E 0y e iϕy For the horizontal P-state holds: E = (1, 0), for the vertical P-state E = (0, 1), the R-state is given by E = 1 2 √ 2(1, −i) and the L-state by E = 1 2 √ 2(1, i). The change in state of a light beam after passage of optical equipment can be described as E 2 = M E 1 . For some types of optical equipment the Jones matrix M is given by: Horizontal linear polarizer: 1 0 0 0 Vertical linear polarizer: 0 0 0 1 Linear polarizer at +45 ◦ 1 2 1 1 1 1 Lineair polarizer at −45 ◦ 1 2 1 −1 −1 1 1 4 -λ plate, fast axis vertical e iπ/4 1 0 0 −i 1 4 -λ plate, fast axis horizontal e iπ/4 1 0 0 i Homogene circular polarizor right 1 2 1 i −i 1 Homogene circular polarizer left 1 2 1 −i i 1 6.7 Prisms and dispersion A light ray passing through a prism is refracted twice and aquires a deviation from its original direction δ = θ i + θ i + α w.r.t. the incident direction, where α is the apex angle, θ i is the angle between the incident angle and a line perpendicular to the surface and θ i is the angle between the ray leaving the prism and a line perpendicular to the surface. When θ i varies there is an angle for which δ becomes minimal. For the refractive index of the prism now holds: n = sin( 1 2 (δ min +α)) sin( 1 2 α) 28 Physics Formulary by ir. J.C.A. Wevers The dispersion of a prism is defined by: D = dδ dλ = dδ dn dn dλ where the first factor depends on the shape and the second on the composition of the prism. For the first factor follows: dδ dn = 2 sin( 1 2 α) cos( 1 2 (δ min +α)) For visible light usually holds dn/dλ < 0: shorter wavelengths are stronger bent than longer. The refractive index in this area can usually be approximated by Cauchy’s formula. 6.8 Diffraction Fraunhofer diffraction occurs far away from the source(s). The Fraunhofer diffraction of light passing through multiple slits is described by: I(θ) I 0 = sin(u) u 2 sin(Nv) sin(v) 2 where u = πb sin(θ)/λ, v = πd sin(θ)/λ. N is the number of slits, b the width of a slit and d the distance between the slits. The maxima in intensity are given by d sin(θ) = kλ. The diffraction through a spherical aperture with radius a is described by: I(θ) I 0 = J 1 (ka sin(θ)) ka sin(θ) 2 The diffraction pattern of a rectangular aperture at distance R with length a in the x-direction and b in the y-direction is described by: I(x, y) I 0 = sin(α ) α 2 sin(β ) β 2 where α = kax/2R and β = kby/2R. When X rays are diffracted at a crystal holds for the position of the maxima in intensity Bragg’s relation: 2d sin(θ) = nλ where d is the distance between the crystal layers. Close at the source the Fraunhofermodel is invalid because it ignores the angle-dependence of the reflected waves. This is described by the obliquity or inclination factor, which describes the directionality of the sec- ondary emissions: E(θ) = 1 2 E 0 (1 +cos(θ)) where θ is the angle w.r.t. the optical axis. Diffraction limits the resolution of a system. This is the minimum angle ∆θ min between two incident rays coming from points far away for which their refraction patterns can be detected separately. For a circular slit holds: ∆θ min = 1.22λ/D where D is the diameter of the slit. For a grating holds: ∆θ min = 2λ/(Na cos(θ m )) where a is the distance between two peaks and N the number of peaks. The minimum difference between two wavelengths that gives a separated diffraction pattern in a multiple slit geometry is given by ∆λ/λ = nN where N is the number of lines and n the order of the pattern. 6.9 Special optical effects • Birefringe and dichroism. D is not parallel with E if the polarizability P of a material is not equal in all directions. There are at least 3 directions, the principal axes, in which they are parallel. This results in 3 refractive indices n i which can be used to construct Fresnel’s ellipsoid. In case n 2 = n 3 = n 1 , which happens e.g. at trigonal, hexagonal and tetragonal crystals there is one optical axis in the direction of n 1 . Incident light rays can now be split up in two parts: the ordinary wave is linear polarized ⊥ the plane through the transmission direction and the optical axis. The extraordinary wave is linear polarized Chapter 6: Optics 29 in the plane through the transmission direction and the optical axis. Dichroism is caused by a different absorption of the ordinary and extraordinary wave in some materials. Double images occur when the incident ray makes an angle with the optical axis: the extraordinary wave will refract, the ordinary will not. • Retarders: waveplates and compensators. Incident light will have a phase shift of ∆ϕ = 2πd([n 0 − n e [)/λ 0 if an uniaxial crystal is cut in such a way that the optical axis is parallel with the front and back plane. Here, λ 0 is the wavelength in vacuum and n 0 and n e the refractive indices for the ordinary and extraordinary wave. For a quarter-wave plate holds: ∆ϕ = π/2. • The Kerr-effect: isotropic, transparent materials can become birefringent when placed in an electric field. In that case, the optical axis is parallel to E. The difference in refractive index in the two directions is given by: ∆n = λ 0 KE 2 , where K is the Kerr constant of the material. If the electrodes have an effective length and are separated by a distance d, the retardation is given by: ∆ϕ = 2πKV 2 /d 2 , where V is the applied voltage. • The Pockels or linear electro-optical effect can occur in 20 (froma total of 32) crystal symmetry classes, namely those without a centre of symmetry. These crystals are also piezoelectric: their polarization changes when a pressure is applied and vice versa: P = pd +ε 0 χ E. The retardation in a Pockels cell is ∆ϕ = 2πn 3 0 r 63 V/λ 0 where r 63 is the 6-3 element of the electro-optic tensor. • The Faraday effect: the polarization of light passing through material with length d and to which a magnetic field is applied in the propagation direction is rotated by an angle β = 1Bd where 1 is the Verdet constant. • ˘ Cerenkov radiation arises when a charged particle with v q > v f arrives. The radiation is emitted within a cone with an apex angle α with sin(α) = c/c medium = c/nv q . 6.10 The Fabry-Perot interferometer For a Fabry-Perot interferometer holds in general: T + R + A = 1 where T is the transmission factor, R the reflection factor and A the absorption factor. If F is given by F = 4R/(1 − R) 2 it follows for the intensity distribution: I t I i = ¸ 1 − A 1 −R 2 1 1 +F sin 2 (θ) The term [1 + F sin 2 (θ)] −1 := /(θ) is called the Airy function. ✛✲ Source Lens d Focussing lens Screen € € € €q The width of the peaks at half height is given by γ = 4/ √ F. The finesse T is defined as T = 1 2 π √ F. The maximum resolution is then given by ∆f min = c/2ndT. Chapter 7 Statistical physics 7.1 Degrees of freedom Amolecule consisting of n atoms has s = 3n degrees of freedom. There are 3 translational degrees of freedom, a linear molecule has s = 3n − 5 vibrational degrees of freedom and a non-linear molecule s = 3n − 6. A linear molecule has 2 rotational degrees of freedom and a non-linear molecule 3. Because vibrational degrees of freedom account for both kinetic and potential energy they count double. So, for linear molecules this results in a total of s = 6n −5. For non-linear molecules this gives s = 6n −6. The average energy of a molecule in thermodynamic equilibrium is 'E tot ` = 1 2 skT. Each degree of freedom of a molecule has in principle the same energy: the principle of equipartition. The rotational and vibrational energy of a molecule are: W rot = ¯h 2 2I l(l + 1) = Bl(l +1) , W vib = (v + 1 2 )¯ hω 0 The vibrational levels are excited if kT ≈ ¯hω, the rotational levels of a hetronuclear molecule are excited if kT ≈ 2B. For homonuclear molecules additional selection rules apply so the rotational levels are well coupled if kT ≈ 6B. 7.2 The energy distribution function The general form of the equilibrium velocity distribution function is P(v x , v y , v z )dv x dv y dv z = P(v x )dv x P(v y )dv y P(v z )dv z with P(v i )dv i = 1 α √ π exp − v 2 i α 2 dv i where α = 2kT/m is the most probable velocity of a particle. The average velocity is given by 'v` = 2α/ √ π, and v 2 = 3 2 α 2 . The distribution as a function of the absolute value of the velocity is given by: dN dv = 4N α 3 √ π v 2 exp − mv 2 2kT The general form of the energy distribution function then becomes: P(E)dE = c(s) kT E kT 1 2 s−1 exp − E kT dE where c(s) is a normalization constant, given by: 1. Even s: s = 2l: c(s) = 1 (l −1)! 2. Odd s: s = 2l +1: c(s) = 2 l √ π(2l − 1)!! Chapter 7: Statistical physics 31 7.3 Pressure on a wall The number of molecules that collides with a wall with surface A within a time τ is given by: d 3 N = ∞ 0 π 0 2π 0 nAvτ cos(θ)P(v, θ, ϕ)dvdθdϕ From this follows for the particle flux on the wall: Φ = 1 4 n'v`. For the pressure on the wall then follows: d 3 p = 2mv cos(θ)d 3 N Aτ , so p = 2 3 n'E` 7.4 The equation of state If intermolecular forces and the volume of the molecules can be neglected then for gases from p = 2 3 n'E` and 'E` = 3 2 kT can be derived: pV = n s RT = 1 3 Nm v 2 Here, n s is the number of moles particles and N is the total number of particles within volume V . If the own volume and the intermolecular forces cannot be neglected the Van der Waals equation can be derived: p + an 2 s V 2 (V −bn s ) = n s RT There is an isotherme with a horizontal point of inflection. In the Van der Waals equation this corresponds with the critical temperature, pressure and volume of the gas. This is the upper limit of the area of coexistence between liquid and vapor. From dp/dV = 0 and d 2 p/dV 2 = 0 follows: T cr = 8a 27bR , p cr = a 27b 2 , V cr = 3bn s For the critical point holds: p cr V m,cr /RT cr = 3 8 , which differs from the value of 1 which follows from the general gas law. Scaled on the critical quantities, with p ∗ := p/p cr , T ∗ = T/T cr and V ∗ m = V m /V m,cr with V m := V/n s holds: p ∗ + 3 (V ∗ m ) 2 V ∗ m − 1 3 = 8 3 T ∗ Gases behave the same for equal values of the reduced quantities: the law of the corresponding states. A virial expansion is used for even more accurate views: p(T, V m ) = RT 1 V m + B(T) V 2 m + C(T) V 3 m + The Boyle temperature T B is the temperature for which the 2nd virial coefficient is 0. In a Van der Waals gas, this happens at T B = a/Rb. The inversion temperature T i = 2T B . The equation of state for solids and liquids is given by: V V 0 = 1 +γ p ∆T −κ T ∆p = 1 + 1 V ∂V ∂T p ∆T + 1 V ∂V ∂p T ∆p 32 Physics Formulary by ir. J.C.A. Wevers 7.5 Collisions between molecules The collision probability of a particle in a gas that is translated over a distance dx is given by nσdx, where σ is the cross section. The mean free path is given by = v 1 nuσ with u = v 2 1 +v 2 2 the relative velocity between the particles. If m 1 < m 2 holds: u v 1 = 1 + m 1 m 2 , so = 1 nσ . If m 1 = m 2 holds: = 1 nσ √ 2 . This means that the average time between two collisions is given by τ = 1 nσv . If the molecules are approximated by hard spheres the cross section is: σ = 1 4 π(D 2 1 + D 2 2 ). The average distance between two molecules is 0.55n −1/3 . Collisions between molecules and small particles in a solution result in the Brownian motion. For the average motion of a particle with radius R can be derived: x 2 i = 1 3 r 2 = kTt/3πηR. A gas is called a Knudsen gas if the dimensions of the gas, something that can easily occur at low pressures. The equilibrium condition for a vessel which has a hole with surface A in it for which holds that A/π is: n 1 √ T 1 = n 2 √ T 2 . Together with the general gas law follows: p 1 / √ T 1 = p 2 / √ T 2 . If two plates move along each other at a distance d with velocity w x the viscosity η is given by: F x = η Aw x d . The velocity profile between the plates is in that case given by w(z) = zw x /d. It can be derived that η = 1 3 'v` where v is the thermal velocity. The heat conductance in a non-moving gas is described by: dQ dt = κA T 2 −T 1 d , which results in a temper- ature profile T(z) = T 1 +z(T 2 −T 1 )/d. It can be derived that κ = 1 3 C mV n 'v` /N A . Also holds: κ = C V η. A better expression for κ can be obtained with the Eucken correction: κ = (1 + 9R/4c mV )C V η with an error <5%. 7.6 Interaction between molecules For dipole interaction between molecules can be derived that U ∼ −1/r 6 . If the distance between two molecules approaches the molecular diameter D a repulsing force between the electron clouds appears. This force can be described by U rep ∼ exp(−γr) or V rep = +C s /r s with 12 ≤ s ≤ 20. This results in the Lennard-Jones potential for intermolecular forces: U LJ = 4 ¸ D r 12 − D r 6 ¸ with a minimum at r = r m . The following holds: D ≈ 0.89r m . For the Van der Waals coefficients a and b and the critical quantities holds: a = 5.275N 2 A D 3 , b = 1.3N A D 3 , kT kr = 1.2 and V m,kr = 3.9N A D 3 . A more simple model for intermolecular forces assumes a potential U(r) = ∞for r < D, U(r) = U LJ for D ≤ r ≤ 3D and U(r) = 0 for r ≥ 3D. This gives for the potential energy of one molecule: E pot = 3D D U(r)F(r)dr. with F(r) the spatial distribution function in spherical coordinates, which for a homogeneous distribution is given by: F(r)dr = 4nπr 2 dr. Some useful mathematical relations are: ∞ 0 x n e −x dx = n! , ∞ 0 x 2n e −x 2 dx = (2n)! √ π n!2 2n+1 , ∞ 0 x 2n+1 e −x 2 dx = 1 2 n! Chapter 8 Thermodynamics 8.1 Mathematical introduction If there exists a relation f(x, y, z) = 0 between 3 variables, one can write: x = x(y, z), y = y(x, z) and z = z(x, y). The total differential dz of z is than given by: dz = ∂z ∂x y dx + ∂z ∂y x dy By writing this also for dx and dy it can be obtained that ∂x ∂y z ∂y ∂z x ∂z ∂x y = −1 Because dz is a total differential holds dz = 0. A homogeneous function of degree m obeys: ε m F(x, y, z) = F(εx, εy, εz). For such a function Euler’s theorem applies: mF(x, y, z) = x ∂F ∂x +y ∂F ∂y +z ∂F ∂z 8.2 Definitions • The isochoric pressure coefficient: β V = 1 p ∂p ∂T V • The isothermal compressibility: κ T = − 1 V ∂V ∂p T • The isobaric volume coefficient: γ p = 1 V ∂V ∂T p • The adiabatic compressibility: κ S = − 1 V ∂V ∂p S For an ideal gas follows: γ p = 1/T, κ T = 1/p and β V = −1/V . 8.3 Thermal heat capacity • The specific heat at constant X is: C X = T ∂S ∂T X • The specific heat at constant pressure: C p = ∂H ∂T p • The specific heat at constant volume: C V = ∂U ∂T V 34 Physics Formulary by ir. J.C.A. Wevers For an ideal gas holds: C mp −C mV = R. Further, if the temperature is high enough to thermalize all internal rotational and vibrational degrees of freedom, holds: C V = 1 2 sR. Hence C p = 1 2 (s +2)R. For their ratio now follows γ = (2 + s)/s. For a lower T one needs only to consider the thermalized degrees of freedom. For a Van der Waals gas holds: C mV = 1 2 sR +ap/RT 2 . In general holds: C p −C V = T ∂p ∂T V ∂V ∂T p = −T ∂V ∂T 2 p ∂p ∂V T ≥ 0 Because (∂p/∂V ) T is always < 0, the following is always valid: C p ≥ C V . If the coefficient of expansion is 0, C p = C V , and also at T = 0K. 8.4 The laws of thermodynamics The zeroth law states that heat flows from higher to lower temperatures. The first law is the conservation of energy. For a closed system holds: Q = ∆U + W, where Q is the total added heat, W the work done and ∆U the difference in the internal energy. In differential form this becomes: dQ = dU +dW, where d means that the it is not a differential of a quantity of state. For a quasi-static process holds: dW = pdV . So for a reversible process holds: dQ = dU +pdV . For an open (flowing) system the first law is: Q = ∆H +W i +∆E kin +∆E pot . One can extract an amount of work W t from the system or add W t = −W i to the system. The second lawstates: for a closed systemthere exists an additive quantity S, called the entropy, the differential of which has the following property: dS ≥ dQ T If the only processes occurring are reversible holds: dS = dQ rev /T. So, the entropy difference after a reversible process is: S 2 −S 1 = 2 1 dQ rev T So, for a reversible cycle holds: dQ rev T = 0. For an irreversible cycle holds: dQ irr T < 0. The third law of thermodynamics is (Nernst): lim T→0 ∂S ∂X T = 0 From this it can be concluded that the thermal heat capacity → 0 if T → 0, so absolute zero temperature cannot be reached by cooling through a finite number of steps. 8.5 State functions and Maxwell relations The quantities of state and their differentials are: Internal energy: U dU = TdS −pdV Enthalpy: H = U +pV dH = TdS +V dp Free energy: F = U −TS dF = −SdT −pdV Gibbs free enthalpy: G = H −TS dG = −SdT +V dp Chapter 8: Thermodynamics 35 From this one can derive Maxwell’s relations: ∂T ∂V S = − ∂p ∂S V , ∂T ∂p S = ∂V ∂S p , ∂p ∂T V = ∂S ∂V T , ∂V ∂T p = − ∂S ∂p T From the total differential and the definitions of C V and C p it can be derived that: TdS = C V dT +T ∂p ∂T V dV and TdS = C p dT −T ∂V ∂T p dp For an ideal gas also holds: S m = C V ln T T 0 +Rln V V 0 +S m0 and S m = C p ln T T 0 −Rln p p 0 +S m0 Helmholtz’ equations are: ∂U ∂V T = T ∂p ∂T V −p , ∂H ∂p T = V −T ∂V ∂T p for an enlarged surface holds: dW rev = −γdA, with γ the surface tension. From this follows: γ = ∂U ∂A S = ∂F ∂A T 8.6 Processes The efficiency η of a process is given by: η = Work done Heat added The Cold factor ξ of a cooling down process is given by: ξ = Cold delivered Work added Reversible adiabatic processes For adiabatic processes holds: W = U 1 − U 2 . For reversible adiabatic processes holds Poisson’s equation: with γ = C p /C V one gets that pV γ =constant. Also holds: TV γ−1 =constant and T γ p 1−γ =constant. Adiabatics exhibit a greater steepness p-V diagram than isothermics because γ > 1. Isobaric processes Here holds: H 2 −H 1 = 2 1 C p dT. For a reversible isobaric process holds: H 2 −H 1 = Q rev . The throttle process This is also called the Joule-Kelvin effect and is an adiabatic expansion of a gas through a porous material or a small opening. Here H is a conserved quantity, and dS > 0. In general this is accompanied with a change in temperature. The quantity which is important here is the throttle coefficient: α H = ∂T ∂p H = 1 C p ¸ T ∂V ∂T p −V ¸ The inversion temperature is the temperature where an adiabatically expanding gas keeps the same tempera- ture. If T > T i the gas heats up, if T < T i the gas cools down. T i = 2T B , with for T B : [∂(pV )/∂p] T = 0. The throttle process is e.g. applied in refridgerators. The Carnotprocess The system undergoes a reversible cycle with 2 isothemics and 2 adiabatics: 1. Isothermic expansion at T 1 . The system absorbs a heat Q 1 from the reservoir. 2. Adiabatic expansion with a temperature drop to T 2 . 36 Physics Formulary by ir. J.C.A. Wevers 3. Isothermic compression at T 2 , removing Q 2 from the system. 4. Adiabatic compression to T 1 . The efficiency for Carnot’s process is: η = 1 − [Q 2 [ [Q 1 [ = 1 − T 2 T 1 := η C The Carnot efficiency η C is the maximal efficiency at which a heat machine can operate. If the process is applied in reverse order and the system performs a work −W the cold factor is given by: ξ = [Q 2 [ W = [Q 2 [ [Q 1 [ −[Q 2 [ = T 2 T 1 − T 2 The Stirling process Stirling’s cycle exists of 2 isothermics and 2 isochorics. The efficiency in the ideal case is the same as for Carnot’s cycle. 8.7 Maximal work Consider a system that changes from state 1 into state 2, with the temperature and pressure of the surroundings given by T 0 and p 0 . The maximum work which can be obtained from this change is, when all processes are reversible: 1. Closed system: W max = (U 1 −U 2 ) −T 0 (S 1 −S 2 ) +p 0 (V 1 − V 2 ). 2. Open system: W max = (H 1 −H 2 ) −T 0 (S 1 −S 2 ) −∆E kin −∆E pot . The minimal work needed to attain a certain state is: W min = −W max . 8.8 Phase transitions Phase transitions are isothermic and isobaric, so dG = 0. When the phases are indicated by α, β and γ holds: G α m = G β m and ∆S m = S α m −S β m = r βα T 0 where r βα is the transition heat of phase β to phase α and T 0 is the transition temperature. The following holds: r βα = r αβ and r βα = r γα −r γβ . Further S m = ∂G m ∂T p so G has a twist in the transition point. In a two phase system Clapeyron’s equation is valid: dp dT = S α m −S β m V α m −V β m = r βα (V α m −V β m )T For an ideal gas one finds for the vapor line at some distance from the critical point: p = p 0 e −r βα/RT There exist also phase transitions with r βα = 0. For those there will occur only a discontinuity in the second derivates of G m . These second-order transitions appear at organization phenomena. A phase-change of the 3rd order, so with e.g. [∂ 3 G m /∂T 3 ] p non continuous arises e.g. when ferromagnetic iron changes to the paramagnetic state. Chapter 8: Thermodynamics 37 8.9 Thermodynamic potential When the number of particles within a system changes this number becomes a third quantity of state. Because addition of matter usually takes place at constant p and T, Gis the relevant quantity. If a system exists of more components this becomes: dG = −SdT +V dp + ¸ i µ i dn i where µ = ∂G ∂n i p,T,nj is called the thermodynamic potential. This is a partial quantity. For V holds: V = c ¸ i=1 n i ∂V ∂n i nj,p,T := c ¸ i=1 n i V i where V i is the partial volume of component i. The following holds: V m = ¸ i x i V i 0 = ¸ i x i dV i where x i = n i /n is the molar fraction of component i. The molar volume of a mixture of two components can be a concave line in a V -x 2 diagram: the mixing contracts the volume. The thermodynamic potentials are not independent in a multiple-phase system. It can be derived that ¸ i n i dµ i = −SdT +V dp, this gives at constant p and T: ¸ i x i dµ i = 0 (Gibbs-Duhmen). Each component has as much µ’s as there are phases. The number of free parameters in a system with c components and p different phases is given by f = c +2 −p. 8.10 Ideal mixtures For a mixture of n components holds (the index 0 is the value for the pure component): U mixture = ¸ i n i U 0 i , H mixture = ¸ i n i H 0 i , S mixture = n ¸ i x i S 0 i +∆S mix where for ideal gases holds: ∆S mix = −nR ¸ i x i ln(x i ). For the thermodynamic potentials holds: µ i = µ 0 i +RT ln(x i ) < µ 0 i . A mixture of two liquids is rarely ideal: this is usually only the case for chemically related components or isotopes. In spite of this holds Raoult’s law for the vapour pressure holds for many binary mixtures: p i = x i p 0 i = y i p. Here is x i the fraction of the ith component in liquid phase and y i the fraction of the ith component in gas phase. A solution of one component in another gives rise to an increase in the boiling point ∆T k and a decrease of the freezing point ∆T s . For x 2 <1 holds: ∆T k = RT 2 k r βα x 2 , ∆T s = − RT 2 s r γβ x 2 with r βα the evaporation heat and r γβ < 0 the melting heat. For the osmotic pressure Π of a solution holds: ΠV 0 m1 = x 2 RT. 8.11 Conditions for equilibrium When a system evolves towards equilibrium the only changes that are possible are those for which holds: (dS) U,V ≥ 0 or (dU) S,V ≤ 0 or (dH) S,p ≤ 0 or (dF) T,V ≤ 0 or (dG) T,p ≤ 0. In equilibrium for each component holds: µ α i = µ β i = µ γ i . 38 Physics Formulary by ir. J.C.A. Wevers 8.12 Statistical basis for thermodynamics The number of possibilities P to distribute N particles on n possible energy levels, each with a g-fold degen- eracy is called the thermodynamic probability and is given by: P = N! ¸ i g ni i n i ! The most probable distribution, that with the maximum value for P, is the equilibrium state. When Stirling’s equation, ln(n!) ≈ nln(n) − n is used, one finds for a discrete system the Maxwell-Boltzmann distribution. The occupation numbers in equilibrium are then given by: n i = N Z g i exp − W i kT The state sum Z is a normalization constant, given by: Z = ¸ i g i exp(−W i /kT). For an ideal gas holds: Z = V (2πmkT) 3/2 h 3 The entropy can then be defined as: S = k ln(P) . For a system in thermodynamic equilibriumthis becomes: S = U T +kN ln Z N +kN ≈ U T +k ln Z N N! For an ideal gas, with U = 3 2 kT then holds: S = 5 2 kN +kN ln V (2πmkT) 3/2 Nh 3 8.13 Application to other systems Thermodynamics can be applied to other systems than gases and liquids. To do this the term dW = pdV has to be replaced with the correct work term, like dW rev = −Fdl for the stretching of a wire, dW rev = −γdA for the expansion of a soap bubble or dW rev = −BdM for a magnetic system. A rotating, non-charged black hole has a temparature of T = ¯ hc/8πkm. It has an entropy S = Akc 3 /4¯hκ with A the area of its event horizon. For a Schwarzschild black hole A is given by A = 16πm 2 . Hawkings area theorem states that dA/dt ≥ 0. Hence, the lifetime of a black hole ∼ m 3 . Chapter 9 Transport phenomena 9.1 Mathematical introduction An important relation is: if X is a quantity of a volume element which travels from position r to r + dr in a time dt, the total differential dX is then given by: dX = ∂X ∂x dx + ∂X ∂y dy + ∂X ∂z dz + ∂X ∂t dt ⇒ dX dt = ∂X ∂x v x + ∂X ∂y v y + ∂X ∂z v z + ∂X ∂t This results in general to: dX dt = ∂X ∂t +(v ∇)X . From this follows that also holds: d dt Xd 3 V = ∂ ∂t Xd 3 V + (X(v n)d 2 A where the volume V is surrounded by surface A. Some properties of the ∇ operator are: div(φv ) = φdivv +gradφ v rot(φv ) = φrotv +(gradφ) v rot gradφ = 0 div(u v ) = v (rotu ) −u (rotv ) rot rotv = grad divv −∇ 2 v div rotv = 0 div gradφ = ∇ 2 φ ∇ 2 v ≡ (∇ 2 v 1 , ∇ 2 v 2 , ∇ 2 v 3 ) Here, v is an arbitrary vector field and φ an arbitrary scalar field. Some important integral theorems are: Gauss: ((v n)d 2 A = (divv )d 3 V Stokes for a scalar field: (φ e t )ds = (n gradφ)d 2 A Stokes for a vector field: (v e t )ds = (rotv n)d 2 A This results in: ((rotv n)d 2 A = 0 Ostrogradsky: ((n v )d 2 A = (rotv )d 3 A ((φn)d 2 A = (gradφ)d 3 V Here, the orientable surface d 2 A is limited by the Jordan curve ds. 9.2 Conservation laws On a volume work two types of forces: 1. The force f 0 on each volume element. For gravity holds: f 0 = g. 2. Surface forces working only on the margins: t. For these holds: t = n T, where T is the stress tensor. 40 Physics Formulary by ir. J.C.A. Wevers T can be split in a part pI representing the normal tensions and a part T representing the shear stresses: T = T +pI, where I is the unit tensor. When viscous aspects can be ignored holds: divT= −gradp. When the flow velocity is v at position r holds on position r +dr: v(dr ) = v(r ) .... translation + dr (gradv ) . .. . rotation, deformation, dilatation The quantity L:=gradv can be split in a symmetric part D and an antisymmetric part W. L = D +W with D ij := 1 2 ∂v i ∂x j + ∂v j ∂x i , W ij := 1 2 ∂v i ∂x j − ∂v j ∂x i When the rotation or vorticity ω = rotv is introduced holds: W ij = 1 2 ε ijk ω k . ω represents the local rotation velocity: dr W = 1 2 ω dr. For a Newtonian liquid holds: T = 2ηD. Here, η is the dynamical viscosity. This is related to the shear stress τ by: τ ij = η ∂v i ∂x j For compressible media can be stated: T = (η divv )I + 2ηD. From equating the thermodynamical and mechanical pressure it follows: 3η +2η = 0. If the viscosity is constant holds: div(2D) = ∇ 2 v +grad divv. The conservation laws for mass, momentum and energy for continuous media can be written in both integral and differential form. They are: Integral notation: 1. Conservation of mass: ∂ ∂t d 3 V + ((v n)d 2 A = 0 2. Conservation of momentum: ∂ ∂t vd 3 V + (v(v n)d 2 A = f 0 d 3 V + (n Td 2 A 3. Conservation of energy: ∂ ∂t ( 1 2 v 2 +e)d 3 V + (( 1 2 v 2 +e)(v n)d 2 A = − ((q n)d 2 A + (v f 0 )d 3 V + ((v n T)d 2 A Differential notation: 1. Conservation of mass: ∂ ∂t +div (v ) = 0 2. Conservation of momentum: ∂v ∂t +(v ∇)v = f 0 +divT = f 0 −gradp +divT 3. Conservation of energy: T ds dt = de dt − p d dt = −divq +T : D Here, e is the internal energy per unit of mass E/m and s is the entropy per unit of mass S/m. q = −κ ∇T is the heat flow. Further holds: p = − ∂E ∂V = − ∂e ∂1/ , T = ∂E ∂S = ∂e ∂s so C V = ∂e ∂T V and C p = ∂h ∂T p with h = H/m the enthalpy per unit of mass. Chapter 9: Transport phenomena 41 From this one can derive the Navier-Stokes equations for an incompressible, viscous and heat-conducting medium: divv = 0 ∂v ∂t +(v ∇)v = g −gradp +η∇ 2 v C ∂T ∂t +C(v ∇)T = κ∇ 2 T +2ηD : D with C the thermal heat capacity. The force F on an object within a flow, when viscous effects are limited to the boundary layer, can be obtained using the momentum law. If a surface A surrounds the object outside the boundary layer holds: F = − ([pn +v(v n)]d 2 A 9.3 Bernoulli’s equations Starting with the momentum equation one can find for a non-viscous medium for stationary flows, with (v grad)v = 1 2 grad(v 2 ) +(rotv ) v and the potential equation g = −grad(gh) that: 1 2 v 2 +gh + dp = constant along a streamline For compressible flows holds: 1 2 v 2 + gh + p/ =constant along a line of flow. If also holds rotv = 0 and the entropy is equal on each streamline holds 1 2 v 2 + gh + dp/ =constant everywhere. For incompressible flows this becomes: 1 2 v 2 + gh + p/ =constant everywhere. For ideal gases with constant C p and C V holds, with γ = C p /C V : 1 2 v 2 + γ γ −1 p = 1 2 v 2 + c 2 γ −1 = constant With a velocity potential defined by v = gradφ holds for instationary flows: ∂φ ∂t + 1 2 v 2 +gh + dp = constant everywhere 9.4 Characterising of flows by dimensionless numbers The advantage of dimensionless numbers is that they make model experiments possible: one has to make the dimensionless numbers which are important for the specific experiment equal for both model and the real situation. One can also deduce functional equalities without solving the differential equations. Some dimensionless numbers are given by: Strouhal: Sr = ωL v Froude: Fr = v 2 gL Mach: Ma = v c Fourier: Fo = a ωL 2 P´ eclet: Pe = vL a Reynolds: Re = vL ν Prandtl: Pr = ν a Nusselt: Nu = Lα κ Eckert: Ec = v 2 c∆T Here, ν = η/ is the kinematic viscosity, c is the speed of sound and L is a characteristic length of the system. α follows fromthe equation for heat transport κ∂ y T = α∆T and a = κ/c is the thermal diffusion coefficient. These numbers can be interpreted as follows: • Re: (stationary inertial forces)/(viscous forces) 42 Physics Formulary by ir. J.C.A. Wevers • Sr: (non-stationary inertial forces)/(stationary inertial forces) • Fr: (stationary inertial forces)/(gravity) • Fo: (heat conductance)/(non-stationary change in enthalpy) • Pe: (convective heat transport)/(heat conductance) • Ec: (viscous dissipation)/(convective heat transport) • Ma: (velocity)/(speed of sound): objects moving faster than approximately Ma = 0,8 produce shock- waves which propagate with an angle θ with the velocity of the object. For this angle holds Ma= 1/ arctan(θ). • Pr and Nu are related to specific materials. Now, the dimensionless Navier-Stokes equation becomes, with x = x/L, v = v/V , grad = Lgrad, ∇ 2 = L 2 ∇ 2 and t = tω: Sr ∂v ∂t +(v ∇ )v = −grad p + g Fr + ∇ 2 v Re 9.5 Tube flows For tube flows holds: they are laminar if Re< 2300 with dimension of length the diameter of the tube, and turbulent if Re is larger. For an incompressible laminar flow through a straight, circular tube holds for the velocity profile: v(r) = − 1 4η dp dx (R 2 −r 2 ) For the volume flow holds: Φ V = R 0 v(r)2πrdr = − π 8η dp dx R 4 The entrance length L e is given by: 1. 500 < Re D < 2300: L e /2R = 0.056Re D 2. Re > 2300: L e /2R ≈ 50 For gas transport at low pressures (Knudsen-gas) holds: Φ V = 4R 3 α √ π 3 dp dx For flows at a small Re holds: ∇p = η∇ 2 v and divv = 0. For the total force on a sphere with radius R in a flow then holds: F = 6πηRv. For large Re holds for the force on a surface A: F = 1 2 C W Av 2 . 9.6 Potential theory The circulation Γ is defined as: Γ = (v e t )ds = (rotv ) nd 2 A = (ω n)d 2 A For non viscous media, if p = p() and all forces are conservative, Kelvin’s theorem can be derived: dΓ dt = 0 For rotationless flows a velocity potential v = gradφ can be introduced. In the incompressible case follows from conservation of mass ∇ 2 φ = 0. For a 2-dimensional flow a flow function ψ(x, y) can be defined: with Φ AB the amount of liquid flowing through a curve s between the points A and B: Φ AB = B A (v n)ds = B A (v x dy −v y dx) Chapter 9: Transport phenomena 43 and the definitions v x = ∂ψ/∂y, v y = −∂ψ/∂x holds: Φ AB = ψ(B) −ψ(A). In general holds: ∂ 2 ψ ∂x 2 + ∂ 2 ψ ∂y 2 = −ω z In polar coordinates holds: v r = 1 r ∂ψ ∂θ = ∂φ ∂r , v θ = − ∂ψ ∂r = 1 r ∂φ ∂θ For source flows with power Q in (x, y) = (0, 0) holds: φ = Q 2π ln(r) so that v r = Q/2πr, v θ = 0. For a dipole of strength Qin x = a and strength −Qin x = −a follows fromsuperposition: φ = −Qax/2πr 2 where Qa is the dipole strength. For a vortex holds: φ = Γθ/2π. If an object is surrounded by an uniform main flow with v = ve x and such a large Re that viscous effects are limited to the boundary layer holds: F x = 0 and F y = −Γv. The statement that F x = 0 is d’Alembert’s paradox and originates from the neglection of viscous effects. The lift F y is also created by η because Γ = 0 due to viscous effects. Henxe rotating bodies also create a force perpendicular to their direction of motion: the Magnus effect. 9.7 Boundary layers 9.7.1 Flow boundary layers If for the thickness of the boundary layer holds: δ <L holds: δ ≈ L/ √ Re. With v ∞ the velocity of the main flow it follows for the velocity v y ⊥ the surface: v y L ≈ δv ∞ . Blasius’ equation for the boundary layer is, with v y /v ∞ = f(y/δ): 2f +ff = 0 with boundary conditions f(0) = f (0) = 0, f (∞) = 1. From this follows: C W = 0.664 Re −1/2 x . The momentum theorem of Von Karman for the boundary layer is: d dx (ϑv 2 ) +δ ∗ v dv dx = τ 0 where the displacement thickness δ ∗ v and the momentum thickness ϑv 2 are given by: ϑv 2 = ∞ 0 (v −v x )v x dy , δ ∗ v = ∞ 0 (v −v x )dy and τ 0 = −η ∂v x ∂y y=0 The boundary layer is released from the surface if ∂v x ∂y y=0 = 0. This is equivalent with dp dx = 12ηv ∞ δ 2 . 9.7.2 Temperature boundary layers If the thickness of the temperature boundary layer δ T <L holds: 1. If Pr ≤ 1: δ/δ T ≈ √ Pr. 2. If Pr 1: δ/δ T ≈ 3 √ Pr. 9.8 Heat conductance For non-stationairy heat conductance in one dimension without flow holds: ∂T ∂t = κ c ∂ 2 T ∂x 2 +Φ where Φ is a source term. If Φ = 0 the solutions for harmonic oscillations at x = 0 are: T −T ∞ T max −T ∞ = exp − x D cos ωt − x D 44 Physics Formulary by ir. J.C.A. Wevers with D = 2κ/ωc. At x = πD the temperature variation is in anti-phase with the surface. The one- dimensional solution at Φ = 0 is T(x, t) = 1 2 √ πat exp − x 2 4at This is mathematical equivalent to the diffusion problem: ∂n ∂t = D∇ 2 n +P −A where P is the production of and A the discharge of particles. The flow density J = −D∇n. 9.9 Turbulence The time scale of turbulent velocity variations τ t is of the order of: τ t = τ √ Re/Ma 2 with τ the molecular time scale. For the velocity of the particles holds: v(t) = 'v` + v (t) with 'v (t)` = 0. The Navier-Stokes equation now becomes: ∂ 'v ` ∂t +('v ` ∇) 'v ` = − ∇'p` +ν∇ 2 'v ` + divS R where S Rij = − 'v i v j ` is the turbulent stress tensor. Boussinesq’s assumption is: τ ij = − v i v j . It is stated that, analogous to Newtonian media: S R = 2ν t 'D`. Near a boundary holds: ν t = 0, far away of a boundary holds: ν t ≈ νRe. 9.10 Self organization For a (semi) two-dimensional flow holds: dω dt = ∂ω ∂t +J(ω, ψ) = ν∇ 2 ω With J(ω, ψ) the Jacobian. So if ν = 0, ω is conserved. Further, the kinetic energy/mA and the enstrofy V are conserved: with v = ∇( kψ) E ∼ (∇ψ) 2 ∼ ∞ 0 c(k, t)dk = constant , V ∼ (∇ 2 ψ) 2 ∼ ∞ 0 k 2 c(k, t)dk = constant From this follows that in a two-dimensional flow the energy flux goes towards large values of k: larger struc- tures become larger at the expanse of smaller ones. In three-dimensional flows the situation is just the opposite. Chapter 10 Quantum physics 10.1 Introduction to quantum physics 10.1.1 Black body radiation Planck’s law for the energy distribution for the radiation of a black body is: w(f) = 8πhf 3 c 3 1 e hf/kT −1 , w(λ) = 8πhc λ 5 1 e hc/λkT −1 Stefan-Boltzmann’s law for the total power density can be derived from this: P = AσT 4 . Wien’s law for the maximum can also be derived from this: Tλ max = k W . 10.1.2 The Compton effect For the wavelength of scattered light, if light is considered to exist of particles, can be derived: λ = λ + h mc (1 −cos θ) = λ +λ C (1 −cos θ) 10.1.3 Electron diffraction Diffraction of electrons at a crystal can be explained by assuming that particles have a wave character with wavelength λ = h/p. This wavelength is called the Broglie-wavelength. 10.2 Wave functions The wave character of particles is described by a wavefunction ψ. This wavefunction can be described in normal or momentum space. Both definitions are each others Fourier transform: Φ(k, t) = 1 √ h Ψ(x, t)e −ikx dx and Ψ(x, t) = 1 √ h Φ(k, t)e ikx dk These waves define a particle with group velocity v g = p/m and energy E = ¯ hω. The wavefunction can be interpreted as a measure for the probability P to find a particle somewhere (Born): dP = [ψ[ 2 d 3 V . The expectation value 'f` of a quantity f of a system is given by: 'f(t)` = Ψ ∗ fΨd 3 V , 'f p (t)` = Φ ∗ fΦd 3 V p This is also written as 'f(t)` = 'Φ[f[Φ`. The normalizing condition for wavefunctions follows from this: 'Φ[Φ` = 'Ψ[Ψ` = 1. 10.3 Operators in quantum physics In quantummechanics, classical quantities are translated into operators. These operators are hermitian because their eigenvalues must be real: ψ ∗ 1 Aψ 2 d 3 V = ψ 2 (Aψ 1 ) ∗ d 3 V 46 Physics Formulary by ir. J.C.A. Wevers When u n is the eigenfunction of the eigenvalue equation AΨ = aΨfor eigenvalue a n , Ψ can be expanded into a basis of eigenfunctions: Ψ = ¸ n c n u n . If this basis is taken orthonormal, then follows for the coefficients: c n = 'u n [Ψ`. If the system is in a state described by Ψ, the chance to find eigenvalue a n when measuring A is given by [c n [ 2 in the discrete part of the spectrum and [c n [ 2 da in the continuous part of the spectrum between a and a + da. The matrix element A ij is given by: A ij = 'u i [A[u j `. Because (AB) ij = 'u i [AB[u j ` = 'u i [A ¸ n [u n ` 'u n [B[u j ` holds: ¸ n [u n `'u n [ = 1. The time-dependence of an operator is given by (Heisenberg): dA dt = ∂A ∂t + [A, H] i¯h with [A, B] ≡ AB − BA the commutator of A and B. For hermitian operators the commutator is always complex. If [A, B] = 0, the operators A and B have a common set of eigenfunctions. By applying this to p x and x follows (Ehrenfest): md 2 'x` t /dt 2 = −'dU(x)/dx`. The first order approximation 'F(x)` t ≈ F('x`), with F = −dU/dx represents the classical equation. Before the addition of quantummechanical operators which are a product of other operators, they should be made symmetrical: a classical product AB becomes 1 2 (AB +BA). 10.4 The uncertainty principle If the uncertainty ∆A in A is defined as: (∆A) 2 = ψ[A op −'A` [ 2 ψ = A 2 −'A` 2 it follows: ∆A ∆B ≥ 1 2 [ 'ψ[[A, B][ψ` [ From this follows: ∆E ∆t ≥ 1 2 ¯h, and because [x, p x ] = i¯h holds: ∆p x ∆x ≥ 1 2 ¯h, and ∆L x ∆L y ≥ 1 2 ¯hL z . 10.5 The Schr¨ odinger equation The momentum operator is given by: p op = −i¯h∇. The position operator is: x op = i¯h∇ p . The energy operator is given by: E op = i¯h∂/∂t. The Hamiltonian of a particle with mass m, potential energy U and total energy E is given by: H = p 2 /2m+U. From Hψ = Eψ then follows the Schr ¨ odinger equation: − ¯h 2 2m ∇ 2 ψ +Uψ = Eψ = i¯h ∂ψ ∂t The linear combination of the solutions of this equation give the general solution. In one dimension it is: ψ(x, t) = ¸ + dE c(E)u E (x) exp − iEt ¯h The current density J is given by: J = ¯h 2im (ψ ∗ ∇ψ −ψ∇ψ ∗ ) The following conservation law holds: ∂P(x, t) ∂t = −∇J(x, t) 10.6 Parity The parity operator in one dimension is given by {ψ(x) = ψ(−x). If the wavefunction is split in even and odd functions, it can be expanded into eigenfunctions of {: ψ(x) = 1 2 (ψ(x) +ψ(−x)) . .. . even: ψ + + 1 2 (ψ(x) −ψ(−x)) . .. . odd: ψ − [{, H] = 0. The functions ψ + = 1 2 (1 + {)ψ(x, t) and ψ − = 1 2 (1 − {)ψ(x, t) both satisfy the Schr¨ odinger equation. Hence, parity is a conserved quantity. Chapter 10: Quantum physics 47 10.7 The tunnel effect The wavefunction of a particle in an ∞ high potential step from x = 0 to x = a is given by ψ(x) = a −1/2 sin(kx). The energylevels are given by E n = n 2 h 2 /8a 2 m. If the wavefunction with energy W meets a potential well of W 0 > W the wavefunction will, unlike the classical case, be non-zero within the potential well. If 1, 2 and 3 are the areas in front, within and behind the potential well, holds: ψ 1 = Ae ikx +Be −ikx , ψ 2 = Ce ik x +De −ik x , ψ 3 = A e ikx with k 2 = 2m(W − W 0 )/¯h 2 and k 2 = 2mW. Using the boundary conditions requiring continuity: ψ = continuous and ∂ψ/∂x =continuous at x = 0 and x = a gives B, C and D and A expressed in A. The amplitude T of the transmitted wave is defined by T = [A [ 2 /[A[ 2 . If W > W 0 and 2a = nλ = 2πn/k holds: T = 1. 10.8 The harmonic oscillator For a harmonic oscillator holds: U = 1 2 bx 2 and ω 2 0 = b/m. The Hamiltonian H is then given by: H = p 2 2m + 1 2 mω 2 x 2 = 1 2 ¯hω +ωA † A with A = 1 2 mωx + ip √ 2mω and A † = 1 2 mωx − ip √ 2mω A = A † is non hermitian. [A, A † ] = ¯ h and [A, H] = ¯ hωA. A is a so called raising ladder operator, A † a lowering ladder operator. HAu E = (E −¯hω)Au E . There is an eigenfunction u 0 for which holds: Au 0 = 0. The energy in this ground state is 1 2 ¯hω: the zero point energy. For the normalized eigenfunctions follows: u n = 1 √ n! A † √ ¯h n u 0 with u 0 = 4 mω π¯h exp − mωx 2 2¯h with E n = ( 1 2 +n)¯ hω. 10.9 Angular momentum For the angular momentum operators L holds: [L z , L 2 ] = [L z , H] = [L 2 , H] = 0. However, cyclically holds: [L x , L y ] = i¯hL z . Not all components of L can be known at the same time with arbitrary accuracy. For L z holds: L z = −i¯h ∂ ∂ϕ = −i¯h x ∂ ∂y −y ∂ ∂x The ladder operators L ± are defined by: L ± = L x ±iL y . Now holds: L 2 = L + L − + L 2 z −¯hL z . Further, L ± = ¯ he ±iϕ ± ∂ ∂θ +i cot(θ) ∂ ∂ϕ From [L + , L z ] = −¯hL + follows: L z (L + Y lm ) = (m+1)¯ h(L + Y lm ). From [L − , L z ] = ¯ hL − follows: L z (L − Y lm ) = (m−1)¯ h(L − Y lm ). From [L 2 , L ± ] = 0 follows: L 2 (L ± Y lm ) = l(l +1)¯ h 2 (L ± Y lm ). Because L x and L y are hermitian (this implies L † ± = L ∓ ) and [L ± Y lm [ 2 > 0 follows: l(l +1) − m 2 −m ≥ 0 ⇒ −l ≤ m ≤ l. Further follows that l has to be integral or half-integral. Half-odd integral values give no unique solution ψ and are therefore dismissed. 48 Physics Formulary by ir. J.C.A. Wevers 10.10 Spin For the spin operators are defined by their commutation relations: [S x , S y ] = i¯hS z . Because the spin operators do not act in the physical space (x, y, z) the uniqueness of the wavefunction is not a criterium here: also half odd-integer values are allowed for the spin. Because [L, S] = 0 spin and angular momentum operators do not have a common set of eigenfunctions. The spin operators are given by S = 1 2 ¯h σ, with σ x = 0 1 1 0 , σ y = 0 −i i 0 , σ z = 1 0 0 −1 The eigenstates of S z are called spinors: χ = α + χ + + α − χ − , where χ + = (1, 0) represents the state with spin up (S z = 1 2 ¯h) and χ − = (0, 1) represents the state with spin down (S z = − 1 2 ¯h). Then the probability to find spin up after a measurement is given by [α + [ 2 and the chance to find spin down is given by [α − [ 2 . Of course holds [α + [ 2 +[α − [ 2 = 1. The electron will have an intrinsic magnetic dipole moment M due to its spin, given by M = −eg S S/2m, with g S = 2(1 +α/2π + ) the gyromagnetic ratio. In the presence of an external magnetic field this gives a potential energy U = − M B. The Schr¨ odinger equation then becomes (because ∂χ/∂x i ≡ 0): i¯h ∂χ(t) ∂t = eg S ¯h 4m σ Bχ(t) with σ = ( σ x , σ y , σ z ). If B = Be z there are two eigenvalues for this problem: χ ± for E = ±eg S ¯hB/4m = ±¯hω. So the general solution is given by χ = (ae −iωt , be iωt ). From this can be derived: 'S x ` = 1 2 ¯hcos(2ωt) and 'S y ` = 1 2 ¯hsin(2ωt). Thus the spin precesses about the z-axis with frequency 2ω. This causes the normal Zeeman splitting of spectral lines. The potential operator for two particles with spin ± 1 2 ¯h is given by: V (r) = V 1 (r) + 1 ¯h 2 ( S 1 S 2 )V 2 (r) = V 1 (r) + 1 2 V 2 (r)[S(S +1) − 3 2 ] This makes it possible for two states to exist: S = 1 (triplet) or S = 0 (Singlet). 10.11 The Dirac formalism If the operators for p and E are substituted in the relativistic equation E 2 = m 2 0 c 4 + p 2 c 2 , the Klein-Gordon equation is found: ∇ 2 − 1 c 2 ∂ 2 ∂t 2 − m 2 0 c 2 ¯h 2 ψ(x, t) = 0 The operator ✷ −m 2 0 c 2 /¯h 2 can be separated: ∇ 2 − 1 c 2 ∂ 2 ∂t 2 − m 2 0 c 2 ¯h 2 = γ λ ∂ ∂x λ − m 0 c ¯h γ µ ∂ ∂x µ + m 0 c ¯h where the Dirac matrices γ are given by: ¦γ λ , γ µ ¦ = γ λ γ µ +γ µ γ λ = 2δ λµ (In general relativity this becomes 2g λµ ). From this it can be derived that the γ are hermitian 4 4 matrices given by: γ k = 0 −iσ k iσ k 0 , γ 4 = I 0 0 −I With this, the Dirac equation becomes: γ λ ∂ ∂x λ + m 0 c ¯h ψ(x, t) = 0 where ψ(x) = (ψ 1 (x), ψ 2 (x), ψ 3 (x), ψ 4 (x)) is a spinor. Chapter 10: Quantum physics 49 10.12 Atomic physics 10.12.1 Solutions The solutions of the Schr¨ odinger equation in spherical coordinates if the potential energy is a function of r alone can be written as: ψ(r, θ, ϕ) = R nl (r)Y l,m l (θ, ϕ)χ ms , with Y lm = C lm √ 2π P m l (cos θ)e imϕ For an atom or ion with one electron holds: R lm (ρ) = C lm e −ρ/2 ρ l L 2l+1 n−l−1 (ρ) with ρ = 2rZ/na 0 with a 0 = ε 0 h 2 /πm e e 2 . The L j i are the associated Laguere functions and the P m l are the associated Legendre polynomials: P |m| l (x) = (1 −x 2 ) m/2 d |m| dx |m| (x 2 −1) l , L m n (x) = (−1) m n! (n −m)! e −x x −m d n−m dx n−m (e −x x n ) The parity of these solutions is (−1) l . The functions are 2 n−1 ¸ l=0 (2l +1) = 2n 2 -folded degenerated. 10.12.2 Eigenvalue equations The eigenvalue equations for an atom or ion with with one electron are: Equation Eigenvalue Range H op ψ = Eψ E n = µe 4 Z 2 /8ε 2 0 h 2 n 2 n ≥ 1 L zop Y lm = L z Y lm L z = m l ¯h −l ≤ m l ≤ l L 2 op Y lm = L 2 Y lm L 2 = l(l +1)¯ h 2 l < n S zop χ = S z χ S z = m s ¯h m s = ± 1 2 S 2 op χ = S 2 χ S 2 = s(s +1)¯ h 2 s = 1 2 10.12.3 Spin-orbit interaction The total momentum is given by J = L + M. The total magnetic dipole moment of an electron is then M = M L + M S = −(e/2m e )( L + g S S) where g S = 2.0023 is the gyromagnetic ratio of the electron. Further holds: J 2 = L 2 + S 2 + 2 L S = L 2 + S 2 + 2L z S z + L + S − + L − S + . J has quantum numbers j with possible values j = l ± 1 2 , with 2j +1 possible z-components (m J ∈ ¦−j, .., 0, .., j¦). If the interaction energy between S and L is small it can be stated that: E = E n +E SL = E n +a S L. It can then be derived that: a = [E n [Z 2 α 2 ¯h 2 nl(l +1)(l + 1 2 ) After a relativistic correction this becomes: E = E n + [E n [Z 2 α 2 n 3 4n − 1 j + 1 2 The fine structure in atomic spectra arises from this. With g S = 2 follows for the average magnetic moment: M av = −(e/2m e )g¯h J, where g is the Land´ e-factor: g = 1 + S J J 2 = 1 + j(j +1) +s(s +1) −l(l +1) 2j(j +1) For atoms with more than one electron the following limiting situations occur: 50 Physics Formulary by ir. J.C.A. Wevers 1. L − S coupling: for small atoms the electrostatic interaction is dominant and the state can be char- acterized by L, S, J, m J . J ∈ ¦[L − S[, ..., L + S − 1, L + S¦ and m J ∈ ¦−J, ..., J − 1, J¦. The spectroscopic notation for this interaction is: 2S+1 L J . 2S +1 is the multiplicity of a multiplet. 2. j − j coupling: for larger atoms the electrostatic interaction is smaller than the L i s i interaction of an electron. The state is characterized by j i ...j n , J, m J where only the j i of the not completely filled subshells are to be taken into account. The energy difference for larger atoms when placed in a magnetic field is: ∆E = gµ B m J B where g is the Land´ e factor. For a transition between two singlet states the line splits in 3 parts, for ∆m J = −1, 0 + 1. This results in the normal Zeeman effect. At higher S the line splits up in more parts: the anomalous Zeeman effect. Interaction with the spin of the nucleus gives the hyperfine structure. 10.12.4 Selection rules For the dipole transition matrix elements follows: p 0 ∼ ['l 2 m 2 [ E r [l 1 m 1 `[. Conservation of angular mo- mentum demands that for the transition of an electron holds that ∆l = ±1. For an atom where L − S coupling is dominant further holds: ∆S = 0 (but not strict), ∆L = 0, ±1, ∆J = 0, ±1 except for J = 0 →J = 0 transitions, ∆m J = 0, ±1, but ∆m J = 0 is forbidden if ∆J = 0. For an atom where j −j coupling is dominant further holds: for the jumping electron holds, except ∆l = ±1, also: ∆j = 0, ±1, and for all other electrons: ∆j = 0. For the total atom holds: ∆J = 0, ±1 but no J = 0 →J = 0 transitions and ∆m J = 0, ±1, but ∆m J = 0 is forbidden if ∆J = 0. 10.13 Interaction with electromagnetic fields The Hamiltonian of an electron in an electromagnetic field is given by: H = 1 2µ ( p +e A) 2 −eV = − ¯h 2 2µ ∇ 2 + e 2µ B L + e 2 2µ A 2 −eV where µ is the reduced mass of the system. The term ∼ A 2 can usually be neglected, except for very strong fields or macroscopic motions. For B = Be z it is given by e 2 B 2 (x 2 +y 2 )/8µ. When a gauge transformation A = A − ∇f, V = V + ∂f/∂t is applied to the potentials the wavefunction is also transformed according to ψ = ψe iqef/¯ h with qe the charge of the particle. Because f = f(x, t), this is called a local gauge transformation, in contrast with a global gauge transformation which can always be applied. 10.14 Perturbation theory 10.14.1 Time-independent perturbation theory To solve the equation (H 0 +λH 1 )ψ n = E n ψ n one has to find the eigenfunctions of H = H 0 +λH 1 . Suppose that φ n is a complete set of eigenfunctions of the non-perturbed Hamiltonian H 0 : H 0 φ n = E 0 n φ n . Because φ n is a complete set holds: ψ n = N(λ) φ n + ¸ k=n c nk (λ)φ k When c nk and E n are being expanded into λ: c nk = λc (1) nk +λ 2 c (2) nk + E n = E 0 n +λE (1) n +λ 2 E (2) n + Chapter 10: Quantum physics 51 and this is put into the Schr¨ odinger equation the result is: E (1) n = 'φ n [H 1 [φ n ` and c (1) nm = 'φ m [H 1 [φ n ` E 0 n − E 0 m if m = n. The second-order correction of the energy is then given by: E (2) n = ¸ k=n [ 'φ k [H 1 [φ n ` [ 2 E 0 n −E 0 k . So to first order holds: ψ n = φ n + ¸ k=n 'φ k [λH 1 [φ n ` E 0 n −E 0 k φ k . In case the levels are degenerated the above does not hold. In that case an orthonormal set eigenfunctions φ ni is chosen for each level n, so that 'φ mi [φ nj ` = δ mn δ ij . Now ψ is expanded as: ψ n = N(λ) ¸ i α i φ ni +λ ¸ k=n c (1) nk ¸ i β i φ ki + E ni = E 0 ni + λE (1) ni is approximated by E 0 ni := E 0 n . Substitution in the Schr¨ odinger equation and taking dot product with φ ni gives: ¸ i α i 'φ nj [H 1 [φ ni ` = E (1) n α j . Normalization requires that ¸ i [α i [ 2 = 1. 10.14.2 Time-dependent perturbation theory From the Schr¨ odinger equation i¯h ∂ψ(t) ∂t = (H 0 +λV (t))ψ(t) and the expansion ψ(t) = ¸ n c n (t) exp −iE 0 n t ¯h φ n with c n (t) = δ nk +λc (1) n (t) + follows: c (1) n (t) = λ i¯h t 0 'φ n [V (t )[φ k ` exp i(E 0 n −E 0 k )t ¯h dt 10.15 N-particle systems 10.15.1 General Identical particles are indistinguishable. For the total wavefunction of a system of identical indistinguishable particles holds: 1. Particles with a half-odd integer spin (Fermions): ψ total must be antisymmetric w.r.t. interchange of the coordinates (spatial and spin) of each pair of particles. The Pauli principle results from this: two Fermions cannot exist in an identical state because then ψ total = 0. 2. Particles with an integer spin (Bosons): ψ total must be symmetric w.r.t. interchange of the coordinates (spatial and spin) of each pair of particles. For a system of two electrons there are 2 possibilities for the spatial wavefunction. When a and b are the quantum numbers of electron 1 and 2 holds: ψ S (1, 2) = ψ a (1)ψ b (2) +ψ a (2)ψ b (1) , ψ A (1, 2) = ψ a (1)ψ b (2) −ψ a (2)ψ b (1) Because the particles do not approach each other closely the repulsion energy at ψ A in this state is smaller. The following spin wavefunctions are possible: χ A = 1 2 √ 2[χ + (1)χ − (2) −χ + (2)χ − (1)] m s = 0 χ S = χ + (1)χ + (2) m s = +1 1 2 √ 2[χ + (1)χ − (2) +χ + (2)χ − (1)] m s = 0 χ − (1)χ − (2) m s = −1 Because the total wavefunction must be antisymmetric it follows: ψ total = ψ S χ A or ψ total = ψ A χ S . 52 Physics Formulary by ir. J.C.A. Wevers For N particles the symmetric spatial function is given by: ψ S (1, ..., N) = ¸ ψ(all permutations of 1..N) The antisymmetric wavefunction is given by the determinant ψ A (1, ..., N) = 1 √ N! [u Ei (j)[ 10.15.2 Molecules The wavefunctions of atoma and b are φ a and φ b . If the 2 atoms approach each other there are two possibilities: the total wavefunction approaches the bonding function with lower total energy ψ B = 1 2 √ 2(φ a + φ b ) or approaches the anti-bonding function with higher energy ψ AB = 1 2 √ 2(φ a − φ b ). If a molecular-orbital is symmetric w.r.t. the connecting axis, like a combination of two s-orbitals it is called a σ-orbital, otherwise a π-orbital, like the combination of two p-orbitals along two axes. The energy of a system is: E = 'ψ[H[ψ` 'ψ[ψ` . The energy calculated with this method is always higher than the real energy if ψ is only an approximation for the solutions of Hψ = Eψ. Also, if there are more functions to be chosen, the function which gives the lowest energy is the best approximation. Applying this to the function ψ = ¸ c i φ i one finds: (H ij − ES ij )c i = 0. This equation has only solutions if the secular determinant [H ij − ES ij [ = 0. Here, H ij = 'φ i [H[φ j ` and S ij = 'φ i [φ j `. α i := H ii is the Coulomb integral and β ij := H ij the exchange integral. S ii = 1 and S ij is the overlap integral. The first approximation in the molecular-orbital theory is to place both electrons of a chemical bond in the bonding orbital: ψ(1, 2) = ψ B (1)ψ B (2). This results in a large electron density between the nuclei and therefore a repulsion. A better approximation is: ψ(1, 2) = C 1 ψ B (1)ψ B (2)+C 2 ψ AB (1)ψ AB (2), with C 1 = 1 and C 2 ≈ 0.6. In some atoms, such as C, it is energetical more suitable to form orbitals which are a linear combination of the s, p and d states. There are three ways of hybridization in C: 1. SP-hybridization: ψ sp = 1 2 √ 2(ψ 2s ± ψ 2p z ). There are 2 hybrid orbitals which are placed on one line under 180 ◦ . Further the 2p x and 2p y orbitals remain. 2. SP 2 hybridization: ψ sp 2 = ψ 2s / √ 3 +c 1 ψ 2p z +c 2 ψ 2p y , where (c 1 , c 2 ) ∈ ¦( 2/3, 0), (−1/ √ 6, 1/ √ 2) , (−1/ √ 6, −1/ √ 2)¦. The 3 SP 2 orbitals lay in one plane, with symmetry axes which are at an angle of 120 ◦ . 3. SP 3 hybridization: ψ sp 3 = 1 2 (ψ 2s ±ψ 2p z ±ψ 2p y ±ψ 2p x ). The 4 SP 3 orbitals form a tetraheder with the symmetry axes at an angle of 109 ◦ 28 . 10.16 Quantum statistics If a system exists in a state in which one has not the disposal of the maximal amount of information about the system, it can be described by a density matrix ρ. If the probability that the system is in state ψ i is given by a i , one can write for the expectation value a of A: 'a` = ¸ i r i 'ψ i [A[ψ i `. If ψ is expanded into an orthonormal basis ¦φ k ¦ as: ψ (i) = ¸ k c (i) k φ k , holds: 'A` = ¸ k (Aρ) kk = Tr(Aρ) where ρ lk = c ∗ k c l . ρ is hermitian, with Tr(ρ) = 1. Further holds ρ = ¸ r i [ψ i `'ψ i [. The probability to find eigenvalue a n when measuring A is given by ρ nn if one uses a basis of eigenvectors of A for ¦φ k ¦. For the time-dependence holds (in the Schr¨ odinger image operators are not explicitly time-dependent): i¯h dρ dt = [H, ρ] Chapter 10: Quantum physics 53 For a macroscopic system in equilibrium holds [H, ρ] = 0. If all quantumstates with the same energy are equally probable: P i = P(E i ), one can obtain the distribution: P n (E) = ρ nn = e −En/kT Z with the state sum Z = ¸ n e −En/kT The thermodynamic quantities are related to these definitions as follows: F = −kT ln(Z), U = 'H` = ¸ n p n E n = − ∂ ∂kT ln(Z), S = −k ¸ n P n ln(P n ). For a mixed state of M orthonormal quantum states with probability 1/M follows: S = k ln(M). The distribution function for the internal states for a system in thermal equilibrium is the most probable func- tion. This function can be found by taking the maximum of the function which gives the number of states with Stirling’s equation: ln(n!) ≈ nln(n) − n, and the conditions ¸ k n k = N and ¸ k n k W k = W. For identical, indistinguishable particles which obey the Pauli exclusion principle the possible number of states is given by: P = ¸ k g k ! n k !(g k −n k )! This results in the Fermi-Dirac statistics. For indistinguishable particles which do not obey the exclusion principle the possible number of states is given by: P = N! ¸ k g n k k n k ! This results in the Bose-Einstein statistics. So the distribution functions which explain how particles are distributed over the different one-particle states k which are each g k -fold degenerate depend on the spin of the particles. They are given by: 1. Fermi-Dirac statistics: integer spin. n k ∈ ¦0, 1¦, n k = N Z g g k exp((E k −µ)/kT) +1 with ln(Z g ) = ¸ g k ln[1 +exp((E i −µ)/kT)]. 2. Bose-Einstein statistics: half odd-integer spin. n k ∈ IN, n k = N Z g g k exp((E k −µ)/kT) −1 with ln(Z g ) = − ¸ g k ln[1 −exp((E i −µ)/kT)]. Here, Z g is the large-canonical state sum and µ the chemical potential. It is found by demanding ¸ n k = N, and for it holds: lim T→0 µ = E F , the Fermi-energy. N is the total number of particles. The Maxwell-Boltzmann distribution can be derived from this in the limit E k −µ kT: n k = N Z exp − E k kT with Z = ¸ k g k exp − E k kT With the Fermi-energy, the Fermi-Dirac and Bose-Einstein statistics can be written as: 1. Fermi-Dirac statistics: n k = g k exp((E k −E F )/kT) + 1 . 2. Bose-Einstein statistics: n k = g k exp((E k −E F )/kT) −1 . Chapter 11 Plasma physics 11.1 Introduction The degree of ionization α of a plasma is defined by: α = n e n e +n 0 where n e is the electron density and n 0 the density of the neutrals. If a plasma contains also negative charged ions α is not well defined. The probability that a test particle collides with another is given by dP = nσdx where σ is the cross section. The collision frequency ν c = 1/τ c = nσv. The mean free path is given by λ v = 1/nσ. The rate coefficient K is defined by K = 'σv`. The number of collisions per unit of time and volume between particles of kind 1 and 2 is given by n 1 n 2 'σv` = Kn 1 n 2 . The potential of an electron is given by: V (r) = −e 4πε 0 r exp − r λ D with λ D = ε 0 kT e T i e 2 (n e T i +n i T e ) ≈ ε 0 kT e n e e 2 because charge is shielded in a plasma. Here, λ D is the Debye length. For distances < λ D the plasma cannot be assumed to be quasi-neutral. Deviations of charge neutrality by thermic motion are compensated by oscillations with frequency ω pe = n e e 2 m e ε 0 The distance of closest approximation when two equal charged particles collide for a deviation of π/2 is 2b 0 = e 2 /(4πε 0 1 2 mv 2 ). A “neat” plasma is defined as a plasma for which holds: b 0 < n −1/3 e < λ D < L p . Here L p := [n e /∇n e [ is the gradient length of the plasma. 11.2 Transport Relaxation times are defined as τ = 1/ν c . Starting with σ m = 4πb 2 0 ln(Λ C ) and with 1 2 mv 2 = kT it can be found that: τ m = 4πε 2 0 m 2 v 3 ne 4 ln(Λ C ) = 8 √ 2πε 2 0 √ m(kT) 3/2 ne 4 ln(Λ C ) For momentum transfer between electrons and ions holds for a Maxwellian velocity distribution: τ ee = 6π √ 3ε 2 0 √ m e (kT e ) 3/2 n e e 4 ln(Λ C ) ≈ τ ei , τ ii = 6π √ 3ε 2 0 √ m i (kT i ) 3/2 n i e 4 ln(Λ C ) The energy relaxation times for identical particles are equal to the momentum relaxation times. Because for e-i collisions the energy transfer is only ∼ 2m e /m i this is a slow process. Approximately holds: τ ee : τ ei : τ ie : τ E ie = 1 : 1 : m i /m e : m i /m e . The relaxation for e-o interaction is much more complicated. For T > 10 eV holds approximately: σ eo = 10 −17 v −2/5 e , for lower energies this can be a factor 10 lower. The resistivity η = E/J of a plasma is given by: η = n e e 2 m e ν ei = e 2 √ m e ln(Λ C ) 6π √ 3ε 2 0 (kT e ) 3/2 Chapter 11: Plasma physics 55 The diffusion coefficient D is defined by means of the flux Γ by Γ = nv diff = −D∇n. The equation of continuity is ∂ t n + ∇(nv diff ) = 0 ⇒ ∂ t n = D∇ 2 n. One finds that D = 1 3 λ v v. A rough estimate gives τ D = L p /D = L 2 p τ c /λ 2 v . For magnetized plasma’s λ v must be replaced with the cyclotron radius. In electrical fields also holds J = neµ E = e(n e µ e + n i µ i ) E with µ = e/mν c the mobility of the particles. The Einstein ratio is: D µ = kT e Because a plasma is electrically neutral electrons and ions are strongly coupled and they don’t diffuse inde- pendent. The coefficient of ambipolar diffusion D amb is defined by Γ = Γ i = Γ e = −D amb ∇n e,i . From this follows that D amb = kT e /e −kT i /e 1/µ e −1/µ i ≈ kT e µ i e In an external magnetic field B 0 particles will move in spiral orbits with cyclotron radius ρ = mv/eB 0 and with cyclotron frequency Ω = B 0 e/m. The helical orbit is perturbed by collisions. A plasma is called magnetized if λ v > ρ e,i . So the electrons are magnetized if ρ e λ ee = √ m e e 3 n e ln(Λ C ) 6π √ 3ε 2 0 (kT e ) 3/2 B 0 < 1 Magnetization of only the electrons is sufficient to confine the plasma reasonable because they are coupled to the ions by charge neutrality. In case of magnetic confinement holds: ∇p = J B. Combined with the two stationary Maxwell equations for the B-field these form the ideal magneto-hydrodynamic equations. For a uniformB-field holds: p = nkT = B 2 /2µ 0 . If both magnetic and electric fields are present electrons and ions will move in the same direction. If E = E r e r + E z e z and B = B z e z the E B drift results in a velocity u = ( E B)/B 2 and the velocity in the r, ϕ plane is ˙ r(r, ϕ, t) = u + ˙ ρ(t). 11.3 Elastic collisions 11.3.1 General The scattering angle of a particle in interaction with another particle, as shown in the figure at the right is: χ = π −2b ∞ ra dr r 2 1 − b 2 r 2 − W(r) E 0 Particles with an impact parameter between b and b + db, moving through a ring with dσ = 2πbdb leave the scattering area at a solid angle dΩ = 2π sin(χ)dχ. The differential cross section is then defined as: I(Ω) = dσ dΩ = b sin(χ) ∂b ∂χ ✻ ❄ ❅ ❅s ❘ χ M b b r a ϕ For a potential energy W(r) = kr −n follows: I(Ω, v) ∼ v −4/n . For low energies, O(1 eV), σ has a Ramsauer minimum. It arises from the interference of matter waves behind the object. I(Ω) for angles 0 < χ < λ/4 is larger than the classical value. 56 Physics Formulary by ir. J.C.A. Wevers 11.3.2 The Coulomb interaction For the Coulomb interaction holds: 2b 0 = q 1 q 2 /2πε 0 mv 2 0 , so W(r) = 2b 0 /r. This gives b = b 0 cot( 1 2 χ) and I(Ω = b sin(χ) ∂b ∂χ = b 2 0 4 sin 2 ( 1 2 χ) Because the influence of a particle vanishes at r = λ D holds: σ = π(λ 2 D − b 2 0 ). Because dp = d(mv) = mv 0 (1 −cos χ) a cross section related to momentum transfer σ m is given by: σ m = (1 −cos χ)I(Ω)dΩ = 4πb 2 0 ln 1 sin( 1 2 χ min ) = 4πb 2 0 ln λ D b 0 := 4πb 2 0 ln(Λ C ) ∼ ln(v 4 ) v 4 where ln(Λ C ) is the Coulomb-logarithm. For this quantity holds: Λ C = λ D /b 0 = 9n(λ D ). 11.3.3 The induced dipole interaction The induced dipole interaction, with p = α E, gives a potential V and an energy W in a dipole field given by: V (r) = p e r 4πε 0 r 2 , W(r) = − [e[p 8πε 0 r 2 = − αe 2 2(4πε 0 ) 2 r 4 with b a = 4 2e 2 α (4πε 0 ) 2 1 2 mv 2 0 holds: χ = π −2b ∞ ra dr r 2 1 − b 2 r 2 + b 4 a 4r 4 If b ≥ b a the charge would hit the atom. Repulsing nuclear forces prevent this to happen. If the scattering angle is a lot times 2π it is called capture. The cross section for capture σ orb = πb 2 a is called the Langevin limit, and is a lowest estimate for the total cross section. 11.3.4 The centre of mass system If collisions of two particles with masses m 1 and m 2 which scatter in the centre of mass system by an angle χ are compared with the scattering under an angle θ in the laboratory system holds: tan(θ) = m 2 sin(χ) m 1 +m 2 cos(χ) The energy loss ∆E of the incoming particle is given by: ∆E E = 1 2 m 2 v 2 2 1 2 m 1 v 2 1 = 2m 1 m 2 (m 1 +m 2 ) 2 (1 −cos(χ)) 11.3.5 Scattering of light Scattering of light by free electrons is called Thomson scattering. The scattering is free from collective effects if kλ D <1. The cross section σ = 6.65 10 −29 m 2 and ∆f f = 2v c sin( 1 2 χ) This gives for the scattered energy E scat ∼ nλ 4 0 /(λ 2 −λ 2 0 ) 2 with n the density. If λ λ 0 it is called Rayleigh scattering. Thomson sccattering is a limit of Compton scattering, which is given by λ − λ = λ C (1 − cos χ) with λ C = h/mc and cannot be used any more if relativistic effects become important. Chapter 11: Plasma physics 57 11.4 Thermodynamic equilibrium and reversibility Planck’s radiation law and the Maxwellian velocity distribution hold for a plasma in equilibrium: ρ(ν, T)dν = 8πhν 3 c 3 1 exp(hν/kT) −1 dν , N(E, T)dE = 2πn (πkT) 3/2 √ E exp − E kT dE “Detailed balancing” means that the number of reactions in one direction equals the number of reactions in the opposite direction because both processes have equal probability if one corrects for the used phase space. For the reaction ¸ forward X forward ← → ¸ back X back holds in a plasma in equilibrium microscopic reversibility: ¸ forward ˆ η forward = ¸ back ˆ η back If the velocity distribution is Maxwellian, this gives: ˆ η x = n x g x h 3 (2πm x kT) 3/2 e −E kin /kT where g is the statistical weight of the state and n/g := η. For electrons holds g = 2, for excited states usually holds g = 2j +1 = 2n 2 . With this one finds for the Boltzmann balance, X p +e − ← → X 1 +e − +(E 1p ): n B p n 1 = g p g 1 exp E p − E 1 kT e And for the Saha balance, X p +e − +(E pi ) ← → X + 1 +2e − : n S p g p = n + 1 g + 1 n e g e h 3 (2πm e kT e ) 3/2 exp E pi kT e Because the number of particles on the left-hand side and right-hand side of the equation is different, a factor g/V e remains. This factor causes the Saha-jump. From microscopic reversibility one can derive that for the rate coefficients K(p, q, T) := 'σv` pq holds: K(q, p, T) = g p g q K(p, q, T) exp ∆E pq kT 11.5 Inelastic collisions 11.5.1 Types of collisions The kinetic energy can be split in a part of and a part in the centre of mass system. The energy in the centre of mass system is available for reactions. This energy is given by E = m 1 m 2 (v 1 −v 2 ) 2 2(m 1 +m 2 ) Some types of inelastic collisions important for plasma physics are: 1. Excitation: A p +e − ← → A q +e − 2. Decay: A q ← → A p +hf 58 Physics Formulary by ir. J.C.A. Wevers 3. Ionisation and 3-particles recombination: A p +e − ← → A + +2e − 4. radiative recombination: A + +e − ← → A p +hf 5. Stimulated emission: A q + hf →A p +2hf 6. Associative ionisation: A ∗∗ +B ← → AB + +e − 7. Penning ionisation: b.v. Ne ∗ +Ar ← → Ar + +Ne + e − 8. Charge transfer: A + +B ← → A + B + 9. Resonant charge transfer: A + +A ← → A + A + 11.5.2 Cross sections Collisions between an electron and an atom can be approximated by a collision between an electron and one of the electrons of that atom. This results in dσ d(∆E) = πZ 2 e 4 (4πε 0 ) 2 E(∆E) 2 Then follows for the transition p →q: σ pq (E) = πZ 2 e 4 ∆E q,q+1 (4πε 0 ) 2 E(∆E) 2 pq For ionization from state p holds to a good approximation: σ p = 4πa 2 0 Ry 1 E p − 1 E ln 1.25βE E p For resonant charge transfer holds: σ ex = A[1 −Bln(E)] 2 1 +CE 3.3 11.6 Radiation In equilibrium holds for radiation processes: n p A pq . .. . emission + n p B pq ρ(ν, T) . .. . stimulated emission = n q B qp ρ(ν, T) . .. . absorption Here, A pq is the matrix element of the transition p →q, and is given by: A pq = 8π 2 e 2 ν 3 [r pq [ 2 3¯hε 0 c 3 with r pq = 'ψ p [r [ψ q ` For hydrogenic atoms holds: A p = 1.58 10 8 Z 4 p −4.5 , with A p = 1/τ p = ¸ q A pq . The intensity I of a line is given by I pq = hfA pq n p /4π. The Einstein coefficients B are given by: B pq = c 3 A pq 8πhν 3 and B pq B qp = g q g p A spectral line is broadened by several mechanisms: 1. Because the states have a finite life time. The natural life time of a state p is given by τ p = 1/ ¸ q A pq . From the uncertainty relation then follows: ∆(hν) τ p = 1 2 ¯h, this gives ∆ν = 1 4πτ p = ¸ q A pq 4π The natural line width is usually <than the broadening due to the following two mechanisms: Chapter 11: Plasma physics 59 2. The Doppler broadening is caused by the thermal motion of the particles: ∆λ λ = 2 c 2 ln(2)kT i m i This broadening results in a Gaussian line profile: k ν = k 0 exp(−[2 √ ln 2(ν −ν 0 )/∆ν D ] 2 ), with k the coefficient of absorption or emission. 3. The Stark broadening is caused by the electric field of the electrons: ∆λ 1/2 = ¸ n e C(n e , T e ) 2/3 with for the H-β line: C(n e , T e ) ≈ 3 10 14 ˚ A −3/2 cm −3 . The natural broadening and the Stark broadening result in a Lorentz profile of a spectral line: k ν = 1 2 k 0 ∆ν L /[( 1 2 ∆ν L ) 2 +(ν −ν 0 ) 2 ]. The total line shape is a convolution of the Gauss- and Lorentz profile and is called a Voigt profile. The number of transitions p →q is given by n p B pq ρ and by n p n hf 'σ a c` = n p (ρdν/hν)σ a c where dν is the line width. Then follows for the cross section of absorption processes: σ a = B pq hν/cdν. The background radiation in a plasma originates from two processes: 1. Free-Bound radiation, originating from radiative recombination. The emission is given by: ε fb = C 1 λ 2 z i n i n e √ kT e ¸ 1 −exp − hc λkT e ξ fb (λ, T e ) with C 1 = 1.63 10 −43 Wm 4 K 1/2 sr −1 and ξ the Biberman factor. 2. Free-free radiation, originating from the acceleration of particles in the EM-field of other particles: ε ff = C 1 λ 2 z i n i n e √ kT e exp − hc λkT e ξ ff (λ, T e ) 11.7 The Boltzmann transport equation It is assumed that there exists a distribution function F for the plasma so that F(r, v, t) = F r (r, t) F v (v, t) = F 1 (x, t)F 2 (y, t)F 3 (z, t)F 4 (v x , t)F 5 (v y , t)F 6 (v z , t) Then the BTE is: dF dt = ∂F ∂t +∇ r (Fv ) +∇ v (Fa) = ∂F ∂t coll−rad Assuming that v does not depend on r and a i does not depend on v i , holds ∇ r (Fv ) = v∇F and ∇ v (Fa) = a ∇ v F. This is also true in magnetic fields because ∂a i /∂x i = 0. The velocity is separated in a thermal velocity v t and a drift velocity w. The total density is given by n = Fdv and vFdv = n w. The balance equations can be derived by means of the moment method: 1. Mass balance: (BTE)dv ⇒ ∂n ∂t +∇ (n w) = ∂n ∂t cr 2. Momentum balance: (BTE)mvdv ⇒mn d w dt +∇T +∇p = mn'a ` + R 3. Energy balance: (BTE)mv 2 dv ⇒ 3 2 dp dt + 5 2 p∇ w +∇ q = Q 60 Physics Formulary by ir. J.C.A. Wevers Here, 'a ` = e/m( E + w B) is the average acceleration, q = 1 2 nm v 2 t v t the heat flow, Q = mv 2 t r ∂F ∂t cr dv the source term for energy production, R is a friction term and p = nkT the pressure. A thermodynamic derivation gives for the total pressure: p = nkT = ¸ i p i − e 2 (n e +z i n i ) 24πε 0 λ D For the electrical conductance in a plasma follows from the momentum balance, if w e w i : η J = E − J B +∇p e en e In a plasma where only elastic e-a collisions are important the equilibrium energy distribution function is the Druyvesteyn distribution: N(E)dE = Cn e E E 0 3/2 exp ¸ − 3m e m 0 E E 0 2 ¸ dE with E 0 = eEλ v = eE/nσ. 11.8 Collision-radiative models These models are first-moment equations for excited states. One assumes the Quasi-steady-state solution is valid, where ∀ p>1 [(∂n p /∂t = 0) ∧ (∇ (n p w p ) = 0)]. This results in: ∂n p>1 ∂t cr = 0 , ∂n 1 ∂t +∇ (n 1 w 1 ) = ∂n 1 ∂t cr , ∂n i ∂t +∇ (n i w i ) = ∂n i ∂t cr with solutions n p = r 0 p n S p +r 1 p n B p = b p n S p . Further holds for all collision-dominated levels that δb p := b p −1 = b 0 p −x eff with p eff = Ry/E pi and 5 ≤ x ≤ 6. For systems in ESP, where only collisional (de)excitation between levels p and p ± 1 is taken into account holds x = 6. Even in plasma’s far from equilibrium the excited levels will eventually reach ESP, so from a certain level up the level densities can be calculated. To find the population densities of the lower levels in the stationary case one has to start with a macroscopic equilibrium: Number of populating processes of level p = Number of depopulating processes of level p , When this is expanded it becomes: n e ¸ q<p n q K qp . .. . coll. excit. +n e ¸ q>p n q K qp . .. . coll. deexcit. + ¸ q>p n q A qp . .. . rad. deex. to + n 2 e n i K +p . .. . coll. recomb. + n e n i α rad . .. . rad. recomb = n e n p ¸ q<p K pq . .. . coll. deexcit. +n e n p ¸ q>p K pq . .. . coll. excit. + n p ¸ q<p A pq . .. . rad. deex. from +n e n p K p+ . .. . coll. ion. 11.9 Waves in plasma’s Interaction of electromagnetic waves in plasma’s results in scattering and absorption of energy. For electro- magnetic waves with complex wave number k = ω(n +iκ)/c in one dimension one finds: E x = E 0 e −κωx/c cos[ω(t −nx/c)]. The refractive index n is given by: n = c k ω = c v f = 1 − ω 2 p ω 2 Chapter 11: Plasma physics 61 For disturbances in the z-direction in a cold, homogeneous, magnetized plasma: B = B 0 e z + ˆ Be i(kz−ωt) and n = n 0 + ˆ ne i(kz−ωt) (external E fields are screened) follows, with the definitions α = ω p /ω and β = Ω/ω and ω 2 p = ω 2 pi +ω 2 pe : J = σ E , with σ = iε 0 ω ¸ s α 2 s ¸ ¸ ¸ ¸ 1 1 −β 2 s −iβ s 1 −β 2 s 0 iβ s 1 −β 2 s 1 1 −β 2 s 0 0 0 1 ¸ where the sum is taken over particle species s. The dielectric tensor c, with property: k ( c E) = 0 is given by c = I − σ/iε 0 ω. With the definitions S = 1 − ¸ s α 2 s 1 −β 2 s , D = ¸ s α 2 s β s 1 −β 2 s , P = 1 − ¸ s α 2 s follows: c = ¸ S −iD 0 iD S 0 0 0 P ¸ The eigenvalues of this hermitian matrix are R = S + D, L = S − D, λ 3 = P, with eigenvectors e r = 1 2 √ 2(1, i, 0), e l = 1 2 √ 2(1, −i, 0) and e 3 = (0, 0, 1). e r is connected with a right rotating field for which iE x /E y = 1 and e l is connected with a left rotating field for which iE x /E y = −1. When k makes an angle θ with B one finds: tan 2 (θ) = P(n 2 −R)(n 2 −L) S(n 2 −RL/S)(n 2 −P) where n is the refractive index. From this the following solutions can be obtained: A. θ = 0: transmission in the z-direction. 1. P = 0: E x = E y = 0. This describes a longitudinal linear polarized wave. 2. n 2 = L: a left, circular polarized wave. 3. n 2 = R: a right, circular polarized wave. B. θ = π/2: transmission ⊥the B-field. 1. n 2 = P: the ordinary mode: E x = E y = 0. This is a transversal linear polarized wave. 2. n 2 = RL/S: the extraordinary mode: iE x /E y = −D/S, an elliptical polarized wave. Resonance frequencies are frequencies for which n 2 → ∞, so v f = 0. For these holds: tan(θ) = −P/S. For R → ∞ this gives the electron cyclotron resonance frequency ω = Ω e , for L → ∞ the ion cyclotron resonance frequency ω = Ω i and for S = 0 holds for the extraordinary mode: α 2 1 − m i m e Ω 2 i ω 2 = 1 − m 2 i m 2 e Ω 2 i ω 2 1 − Ω 2 i ω 2 Cut-off frequencies are frequencies for which n 2 = 0, so v f →∞. For these holds: P = 0 or R = 0 or L = 0. In the case that β 2 1 one finds Alfv´ en waves propagating parallel to the field lines. With the Alfv´ en velocity v A = Ω e Ω i ω 2 pe +ω 2 pi c 2 follows: n = 1 +c/v A , and in case v A <c: ω = kv A . Chapter 12 Solid state physics 12.1 Crystal structure A lattice is defined by the 3 translation vectors a i , so that the atomic composition looks the same from each point r and r = r + T, where T is a translation vector given by: T = u 1 a 1 + u 2 a 2 +u 3 a 3 with u i ∈ IN. A lattice can be constructed from primitive cells. As a primitive cell one can take a parallellepiped, with volume V cell = [a 1 (a 2 a 3 )[ Because a lattice has a periodical structure the physical properties which are connected with the lattice have the same periodicity (neglecting boundary effects): n e (r + T ) = n e (r ) This periodicity is suitable to use Fourier analysis: n(r ) is expanded as: n(r ) = ¸ G n G exp(i G r ) with n G = 1 V cell cell n(r ) exp(−i G r )dV G is the reciprocal lattice vector. If G is written as G = v 1 b 1 + v 2 b 2 + v 3 b 3 with v i ∈ IN, it follows for the vectors b i , cyclically: b i = 2π a i+1 a i+2 a i (a i+1 a i+2 ) The set of G-vectors determines the R¨ ontgen diffractions: a maximum in the reflected radiation occurs if: ∆ k = G with ∆ k = k − k . So: 2 k G = G 2 . From this follows for parallel lattice planes (Bragg reflection) that for the maxima holds: 2d sin(θ) = nλ. The Brillouin zone is defined as a Wigner-Seitz cell in the reciprocal lattice. 12.2 Crystal binding A distinction can be made between 4 binding types: 1. Van der Waals bond 2. Ion bond 3. Covalent or homopolar bond 4. Metalic bond. For the ion binding of NaCl the energy per molecule is calculated by: E = cohesive energy(NaCl) – ionization energy(Na) + electron affinity(Cl) The interaction in a covalent bond depends on the relative spin orientations of the electrons constituing the bond. The potential energy for two parallel spins is higher than the potential energy for two antiparallel spins. Furthermore the potential energy for two parallel spins has sometimes no minimum. In that case binding is not possible. Chapter 12: Solid state physics 63 12.3 Crystal vibrations 12.3.1 A lattice with one type of atoms In this model for crystal vibrations only nearest-neighbour interactions are taken into account. The force on atom s with mass M can then be written as: F s = M d 2 u s dt 2 = C(u s+1 −u s ) +C(u s−1 −u s ) Assuming that all solutions have the same time-dependence exp(−iωt) this results in: −Mω 2 u s = C(u s+1 +u s−1 −2u s ) Further it is postulated that: u s±1 = u exp(isKa) exp(±iKa). This gives: u s = exp(iKsa). Substituting the later two equations in the fist results in a system of linear equations, which has only a solution if their determinant is 0. This gives: ω 2 = 4C M sin 2 ( 1 2 Ka) Only vibrations with a wavelength within the first Brillouin Zone have a physical significance. This requires that −π < Ka ≤ π. The group velocity of these vibrations is given by: v g = dω dK = Ca 2 M cos( 1 2 Ka) . and is 0 on the edge of a Brillouin Zone. Here, there is a standing wave. 12.3.2 A lattice with two types of atoms Now the solutions are: ω 2 = C 1 M 1 + 1 M 2 ±C 1 M 1 + 1 M 2 2 − 4 sin 2 (Ka) M 1 M 2 Connected with each value of K are two values of ω, as can be seen in the graph. The upper line describes the optical branch, the lower line the acoustical branch. In the optical branch, both types of ions oscillate in opposite phases, in the acoustical branch they oscillate in the same phase. This results in a much larger induced dipole moment for optical oscillations, and also a stronger emission and absorption of radiation. Furthermore each branch has 3 polarization directions, one longitudinal and two transversal. ✲ ✻ 0 K ω π/a 2C M2 2C M1 12.3.3 Phonons The quantum mechanical excitation of a crystal vibration with an energy ¯hω is called a phonon. Phonons can be viewed as quasi-particles: with collisions, they behave as particles with momentum ¯hK. Their total momentum is 0. When they collide, their momentum need not be conserved: for a normal process holds: K 1 + K 2 = K 3 , for an umklapp process holds: K 1 + K 2 = K 3 + G. Because phonons have no spin they behave like bosons. 64 Physics Formulary by ir. J.C.A. Wevers 12.3.4 Thermal heat capacity The total energy of the crystal vibrations can be calculated by multiplying each mode with its energy and sum over all branches K and polarizations P: U = ¸ K ¸ P ¯hω 'n k,p ` = ¸ λ D λ (ω) ¯hω exp(¯ hω/kT) −1 dω for a given polarization λ. The thermal heat capacity is then: C lattice = ∂U ∂T = k ¸ λ D(ω) (¯ hω/kT) 2 exp(¯ hω/kT) (exp(¯hω/kT) −1) 2 dω The dispersion relation in one dimension is given by: D(ω)dω = L π dK dω dω = L π dω v g In three dimensions one applies periodic boundary conditions to a cube with N 3 primitive cells and a volume L 3 : exp(i(K x x +K y y +K z z)) ≡ exp(i(K x (x +L) +K y (y +L) + K z (z +L))). Because exp(2πi) = 1 this is only possible if: K x , K y , K z = 0; ± 2π L ; ± 4π L ; ± 6π L ; ... ± 2Nπ L So there is only one allowed value of K per volume (2π/L) 3 in K-space, or: L 2π 3 = V 8π 3 allowed K-values per unit volume in K-space, for each polarization and each branch. The total number of states with a wave vector < K is: N = L 2π 3 4πK 3 3 for each polarization. The density of states for each polarization is, according to the Einstein model: D(ω) = dN dω = V K 2 2π 2 dK dω = V 8π 3 dA ω v g The Debye model for thermal heat capacities is a low-temperature approximation which is valid up to ≈ 50K. Here, only the acoustic phonons are taken into account (3 polarizations), and one assumes that v = ωK, independent of the polarization. From this follows: D(ω) = V ω 2 /2π 2 v 3 , where v is the speed of sound. This gives: U = 3 D(ω) 'n` ¯hωdω = ωD 0 V ω 2 2π 2 v 3 ¯hω exp(¯ hω/kT) −1 dω = 3V k 2 T 4 2π 2 v 3 ¯h 3 xD 0 x 3 dx e x −1 . Here, x D = ¯ hω D /kT = θ D /T. θ D is the Debye temperature and is defined by: θ D = ¯hv k 6π 2 N V 1/3 where N is the number of primitive cells. Because x D →∞for T →0 it follows from this: U = 9NkT T θ D 3 ∞ 0 x 3 dx e x −1 = 3π 4 NkT 4 5θ D ∼ T 4 and C V = 12π 4 NkT 3 5θ 3 D ∼ T 3 In the Einstein model for the thermal heat capacity one considers only phonons at one frequency, an approxi- mation for optical phonons. Chapter 12: Solid state physics 65 12.4 Magnetic field in the solid state The following graph shows the magnetization versus fieldstrength for different types of magnetism: diamagnetism ferro paramagnetism χ m = ∂M ∂H M M sat 0 H ✲ ✻ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤❤ 12.4.1 Dielectrics The quantum mechanical origin of diamagnetism is the Larmorprecession of the spin of the electron. Starting with a circular electron orbit in an atom with two electrons, there is a Coulomb force F c and a magnetic force on each electron. If the magnetic part of the force is not strong enough to significantly deform the orbit holds: ω 2 = F c (r) mr ± eB m ω = ω 2 0 ± eB m (ω 0 +δ) ⇒ω = ω 0 ± eB 2m 2 + ≈ ω 0 ± eB 2m = ω 0 ±ω L Here, ω L is the Larmor frequency. One electron is accelerated, the other decelerated. Hence there is a net circular current which results in a magnetic moment µ. The circular current is given by I = −Zeω L /2π, and 'µ` = IA = Iπ ρ 2 = 2 3 Iπ r 2 . If N is the number of atoms in the crystal it follows for the susceptibility, with M = µN: χ = µ 0 M B = − µ 0 NZe 2 6m r 2 12.4.2 Paramagnetism Starting with the splitting of energy levels in a weak magnetic field: ∆U m − µ B = m J gµ B B, and with a distribution f m ∼ exp(−∆U m /kT), one finds for the average magnetic moment 'µ` = ¸ f m µ/ ¸ f m . After linearization and because ¸ m J = 0, ¸ J = 2J + 1 and ¸ m 2 J = 2 3 J(J +1)(J + 1 2 ) it follows that: χ p = µ 0 M B = µ 0 N 'µ` B = µ 0 J(J +1)g 2 µ 2 B N 3kT This is the Curie law, χ p ∼ 1/T. 12.4.3 Ferromagnetism A ferromagnet behaves like a paramagnet above a critical temperature T c . To describe ferromagnetism a field B E parallel with M is postulated: B E = λµ 0 M. From there the treatment is analogous to the paramagnetic case: µ 0 M = χ p (B a +B E ) = χ p (B a +λµ 0 M) = µ 0 1 −λ C T M From this follows for a ferromagnet: χ F = µ 0 M B a = C T −T c which is Weiss-Curie’s law. If B E is estimated this way it results in values of about 1000 T. This is clearly unrealistic and suggests another mechanism. A quantum mechanical approach from Heisenberg postulates an interaction between two neigh- bouring atoms: U = −2J S i S j ≡ −µ B E . J is an overlap integral given by: J = 3kT c /2zS(S + 1), with z the number of neighbours. A distinction between 2 cases can now be made: 1. J > 0: S i and S j become parallel: the material is a ferromagnet. 66 Physics Formulary by ir. J.C.A. Wevers 2. J < 0: S i and S j become antiparallel: the material is an antiferromagnet. Heisenberg’s theory predicts quantized spin waves: magnons. Starting from a model with only nearest neigh- bouring atoms interacting one can write: U = −2J S p ( S p−1 + S p+1 ) ≈ µ p B p with B p = −2J gµ B ( S p−1 + S p+1 ) The equation of motion for the magnons becomes: d S dt = 2J ¯h S p ( S p−1 + S p+1 ) From here the treatment is analogous to phonons: postulate traveling waves of the type S p = uexp(i(pka − ωt)). This results in a system of linear equations with solution: ¯hω = 4JS(1 −cos(ka)) 12.5 Free electron Fermi gas 12.5.1 Thermal heat capacity The solution with period L of the one-dimensional Schr¨ odinger equation is: ψ n (x) = Asin(2πx/λn) with nλ n = 2L. From this follows E = ¯h 2 2m nπ L 2 In a linear lattice the only important quantum numbers are n and m s . The Fermi level is the uppermost filled level in the ground state, which has the Fermi-energy E F . If n F is the quantum number of the Fermi level, it can be expressed as: 2n F = N so E F = ¯ h 2 π 2 N 2 /8mL. In 3 dimensions holds: k F = 3π 2 N V 1/3 and E F = ¯h 2 2m 3π 2 N V 2/3 The number of states with energy ≤ E is then: N = V 3π 2 2mE ¯h 2 3/2 . and the density of states becomes: D(E) = dN dE = V 2π 2 2m ¯h 2 3/2 √ E = 3N 2E . The heat capacity of the electrons is approximately 0.01 times the classical expected value 3 2 Nk. This is caused by the Pauli exclusion principle and the Fermi-Dirac distribution: only electrons within an energy range ∼ kT of the Fermi level are excited thermally. There is a fraction ≈ T/T F excited thermally. The internal energy then becomes: U ≈ NkT T T F and C = ∂U ∂T ≈ Nk T T F A more accurate analysis gives: C electrons = 1 2 π 2 NkT/T F ∼ T. Together with the T 3 dependence of the thermal heat capacity of the phonons the total thermal heat capacity of metals is described by: C = γT +AT 3 . 12.5.2 Electric conductance The equation of motion for the charge carriers is: F = mdv/dt = ¯ hd k/dt. The variation of k is given by δ k = k(t) − k(0) = −e Et/¯h. If τ is the characteristic collision time of the electrons, δ k remains stable if t = τ. Then holds: 'v ` = µ E, with µ = eτ/m the mobility of the electrons. The current in a conductor is given by: J = nqv = σ E = E/ρ = neµ E. Because for the collision time holds: 1/τ = 1/τ L + 1/τ i , where τ L is the collision time with the lattice phonons and τ i the collision time with the impurities follows for the resistivity ρ = ρ L +ρ i , with lim T→0 ρ L = 0. Chapter 12: Solid state physics 67 12.5.3 The Hall-effect If a magnetic field is applied ⊥ to the direction of the current the charge carriers will be pushed aside by the Lorentz force. This results in a magnetic field ⊥ to the flow direction of the current. If J = Je x and B = Be z than E y /E x = µB. The Hall coefficient is defined by: R H = E y /J x B, and R H = −1/ne if J x = neµE x . The Hall voltage is given by: V H = Bvb = IB/neh where b is the width of the material and h de height. 12.5.4 Thermal heat conductivity With = v F τ the mean free path of the electrons follows from κ = 1 3 C 'v` : κ electrons = π 2 nk 2 Tτ/3m. From this follows for the Wiedemann-Franz ratio: κ/σ = 1 3 (πk/e) 2 T. 12.6 Energy bands In the tight-bond approximation it is assumed that ψ = e ikna φ(x − na). From this follows for the energy: 'E` = 'ψ[H[ψ` = E at − α − 2β cos(ka). So this gives a cosine superimposed on the atomic energy, which can often be approximated by a harmonic oscillator. If it is assumed that the electron is nearly free one can postulate: ψ = exp(i k r ). This is a traveling wave. This wave can be decomposed into two standing waves: ψ(+) = exp(iπx/a) + exp(−iπx/a) = 2 cos(πx/a) ψ(−) = exp(iπx/a) −exp(−iπx/a) = 2i sin(πx/a) The probability density [ψ(+)[ 2 is high near the atoms of the lattice and low in between. The probability density [ψ(−)[ 2 is low near the atoms of the lattice and high in between. Hence the energy of ψ(+) is also lower than the energy of ψ)(−). Suppose that U(x) = U cos(2πx/a), than the bandgap is given by: E gap = 1 0 U(x) [ψ(+)[ 2 −[ψ(−)[ 2 dx = U 12.7 Semiconductors The band structures and the transitions between them of direct and indirect semiconductors are shown in the figures below. Here it is assumed that the momentum of the absorbed photon can be neglected. For an indirect semiconductor a transition from the valence- to the conduction band is also possible if the energy of the absorbed photon is smaller than the band gap: then, also a phonon is absorbed. Direct transition ✻ ✻ ✞ ✝ ✞ ✝¤ ✆ ¤ ✆ E conduction band ω g ◦ • Indirect transition ✻ ✻ ✞ ✝ ✞ ✝¤ ✆ E ◦ • ✛ ω Ω This difference can also be observed in the absorption spectra: 68 Physics Formulary by ir. J.C.A. Wevers Direct semiconductor ✻ ✲ absorption E ¯hω g Indirect semiconductor ✻ ✲ absorption E E g +¯ hΩ ✡ ✡ ✡ . . . . . . . . . . . . . . . . . . . . . So indirect semiconductors, like Si and Ge, cannot emit any light and are therefore not usable to fabricate lasers. When light is absorbed holds: k h = − k e , E h ( k h ) = −E e ( k e ), v h = v e and m h = −m ∗ e if the conduction band and the valence band have the same structure. Instead of the normal electron mass one has to use the effective mass within a lattice. It is defined by: m ∗ = F a = dp/dt dv g /dt = ¯ h dK dv g = ¯ h 2 d 2 E dk 2 −1 with E = ¯ hω and v g = dω/dk and p = ¯ hk. With the distribution function f e (E) ≈ exp((µ − E)/kT) for the electrons and f h (E) = 1 − f e (E) for the holes the density of states is given by: D(E) = 1 2π 2 2m ∗ ¯h 2 3/2 E −E c with E c the energy at the edge of the conductance band. From this follows for the concentrations of the holes p and the electrons n: n = ∞ Ec D e (E)f e (E)dE = 2 m ∗ kT 2π¯h 2 3/2 exp µ −E c kT For the product np follows: np = 4 kT 2π¯h 2 3 m ∗ e m h exp − E g kT For an intrinsic (no impurities) semiconductor holds: n i = p i , for a n − type holds: n > p and in a p − type holds: n < p. An exciton is a bound electron-hole pair, rotating on each other as in positronium. The excitation energy of an exciton is smaller than the bandgap because the energy of an exciton is lower than the energy of a free electron and a free hole. This causes a peak in the absorption just under E g . 12.8 Superconductivity 12.8.1 Description Asuperconductor is characterized by a zero resistivity if certain quantities are smaller than some critical values: T < T c , I < I c and H < H c . The BCS-model predicts for the transition temperature T c : T c = 1.14Θ D exp −1 UD(E F ) while experiments find for H c approximately: H c (T) ≈ H c (T c ) 1 − T 2 T 2 c . Chapter 12: Solid state physics 69 Within a superconductor the magnetic field is 0: the Meissner effect. There are type I and type II superconductors. Because the Meissner effect implies that a superconductor is a perfect diamagnet holds in the superconducting state: H = µ 0 M. This holds for a type I superconductor, for a type II superconductor this only holds to a certain value H c1 , for higher values of H the superconductor is in a vortex state to a value H c2 , which can be 100 times H c1 . If H becomes larger than H c2 the superconductor becomes a normal conductor. This is shown in the figures below. Type I ✻ ✲ µ 0 M H H c Type II ✻ ✲ µ 0 M H H c1 H c2 The transition to a superconducting state is a second order thermodynamic state transition. This means that there is a twist in the T −S diagram and a discontinuity in the C X −T diagram. 12.8.2 The Josephson effect For the Josephson effect one considers two superconductors, separated by an insulator. The electron wave- function in one superconductor is ψ 1 , in the other ψ 2 . The Schr¨ odinger equations in both superconductors is set equal: i¯h ∂ψ 1 ∂t = ¯ hTψ 2 , i¯h ∂ψ 2 ∂t = ¯ hTψ 1 ¯hT is the effect of the coupling of the electrons, or the transfer interaction through the insulator. The electron wavefunctions are written as ψ 1 = √ n 1 exp(iθ 1 ) and ψ 2 = √ n 2 exp(iθ 2 ). Because a Cooper pair exist of two electrons holds: ψ ∼ √ n. From this follows, if n 1 ≈ n 2 : ∂θ 1 ∂t = ∂θ 2 ∂t and ∂n 2 ∂t = − ∂n 1 ∂t The Josephson effect results in a current density through the insulator depending on the phase difference as: J = J 0 sin(θ 2 − θ 1 ) = J 0 sin(δ), where J 0 ∼ T. With an AC-voltage across the junction the Schr¨ odinger equations become: i¯h ∂ψ 1 ∂t = ¯ hTψ 2 −eV ψ 1 and i¯h ∂ψ 2 ∂t = ¯ hTψ 1 +eV ψ 2 This gives: J = J 0 sin θ 2 −θ 1 − 2eV t ¯h . Hence there is an oscillation with ω = 2eV/¯h. 12.8.3 Flux quantisation in a superconducting ring For the current density in general holds: J = qψ ∗ vψ = nq m [¯ h ∇θ −q A] From the Meissner effect, B = 0 and J = 0, follows: ¯h ∇θ = q A ⇒ ∇θdl = θ 2 − θ 1 = 2πs with s ∈ IN. Because: Adl = (rot A, n)dσ = ( B, n)dσ = Ψ follows: Ψ = 2π¯hs/q. The size of a flux quantum follows by setting s = 1: Ψ = 2π¯h/e = 2.0678 10 −15 Tm 2 . 70 Physics Formulary by ir. J.C.A. Wevers 12.8.4 Macroscopic quantum interference From θ 2 −θ 1 = 2eΨ/¯h follows for two parallel junctions: δ b − δ a = 2eΨ ¯h , so J = J a +J b = 2J 0 sin δ 0 cos eΨ ¯h This gives maxima if eΨ/¯h = sπ. 12.8.5 The London equation A current density in a superconductor proportional to the vector potential A is postulated: J = − A µ 0 λ 2 L or rot J = − B µ 0 λ 2 L where λ L = ε 0 mc 2 /nq 2 . From this follows: ∇ 2 B = B/λ 2 L . The Meissner effect is the solution of this equation: B(x) = B 0 exp(−x/λ L ). Magnetic fields within a superconductor drop exponentially. 12.8.6 The BCS model The BCS model can explain superconductivity in metals. (So far there is no explanation for high-T c supercon- ductance). A new ground state where the electrons behave like independent fermions is postulated. Because of the in- teraction with the lattice these pseudo-particles exhibit a mutual attraction. This causes two electrons with opposite spin to combine to a Cooper pair. It can be proved that this ground state is perfect diamagnetic. The infinite conductivity is more difficult to explain because a ring with a persisting current is not a real equilibrium: a state with zero current has a lower energy. Flux quantization prevents transitions between these states. Flux quantization is related to the existence of a coherent many-particle wavefunction. A flux quantum is the equivalent of about 10 4 electrons. So if the flux has to change with one flux quantum there has to occur a transition of many electrons, which is very improbable, or the system must go through intermediary states where the flux is not quantized so they have a higher energy. This is also very improbable. Some useful mathematical relations are: ∞ 0 xdx e ax +1 = π 2 12a 2 , ∞ −∞ x 2 dx (e x +1) 2 = π 2 3 , ∞ 0 x 3 dx e x +1 = π 4 15 And, when ∞ ¸ n=0 (−1) n = 1 2 follows: ∞ 0 sin(px)dx = ∞ 0 cos(px)dx = 1 p . Chapter 13 Theory of groups 13.1 Introduction 13.1.1 Definition of a group ( is a group for the operation • if: 1. ∀ A,B∈G ⇒A• B ∈ (: ( is closed. 2. ∀ A,B,C∈G ⇒(A • B) • C = A• (B • C): ( obeys the associative law. 3. ∃ E∈G so that ∀ A∈G A• E = E • A = A: ( has a unit element. 4. ∀ A∈G ∃ A −1 ∈G so that A • A −1 = E: Each element in ( has an inverse. If also holds: 5. ∀ A,B∈G ⇒A • B = B • A the group is called Abelian or commutative. 13.1.2 The Cayley table Each element arises only once in each row and column of the Cayley- or multiplication table: because EA i = A −1 k (A k A i ) = A i each A i appears once. There are h positions in each row and column when there are h elements in the group so each element appears only once. 13.1.3 Conjugated elements, subgroups and classes B is conjugate to A if ∃ X∈G such that B = XAX −1 . Then A is also conjugate to B because B = (X −1 )A(X −1 ) −1 . If B and C are conjugate to A, B is also conjugate with C. A subgroup is a subset of ( which is also a group w.r.t. the same operation. A conjugacy class is the maximumcollection of conjugated elements. Each group can be split up in conjugacy classes. Some theorems: • All classes are completely disjoint. • E is a class itself: for each other element in this class would hold: A = XEX −1 = E. • E is the only class which is also a subgroup because all other classes have no unit element. • In an Abelian group each element is a separate class. The physical interpretation of classes: elements of a group are usually symmetry operations which map a symmetrical object into itself. Elements of one class are then the same kind of operations. The opposite need not to be true. 72 Physics Formulary by ir. J.C.A. Wevers 13.1.4 Isomorfism and homomorfism; representations Two groups are isomorphic if they have the same multiplication table. The mapping from group ( 1 to ( 2 , so that the multiplication table remains the same is a homomorphic mapping. It need not be isomorphic. A representation is a homomorphic mapping of a group to a group of square matrices with the usual matrix multiplication as the combining operation. This is symbolized by Γ. The following holds: Γ(E) = II , Γ(AB) = Γ(A)Γ(B) , Γ(A −1 ) = [Γ(A)] −1 For each group there are 3 possibilities for a representation: 1. A faithful representation: all matrices are different. 2. The representation A →det(Γ(A)). 3. The identical representation: A →1. An equivalent representation is obtained by performing an unitary base transformation: Γ (A) = S −1 Γ(A)S. 13.1.5 Reducible and irreducible representations If the same unitary transformation can bring all matrices of a representation Γ in the same block structure the representation is called reducible: Γ(A) = Γ (1) (A) 0 0 Γ (2) (A) This is written as: Γ = Γ (1) ⊕Γ (2) . If this is not possible the representation is called irreducible. The number of irreducible representations equals the number of conjugacy classes. 13.2 The fundamental orthogonality theorem 13.2.1 Schur’s lemma Lemma: Each matrix which commutes with all matrices of an irreducible representation is a constant II, where II is the unit matrix. The opposite is (of course) also true. Lemma: If there exists a matrix M so that for two irreducible representations of group (, γ (1) (A i ) and γ (2) (A i ), holds: Mγ (1) (A i ) = γ (2) (A i )M, than the representations are equivalent, or M = 0. 13.2.2 The fundamental orthogonality theorem For a set of unequivalent, irreducible, unitary representations holds that, if h is the number of elements in the group and i is the dimension of the i th ¯ representation: ¸ R∈G Γ (i)∗ µν (R)Γ (j) αβ (R) = h i δ ij δ µα δ νβ 13.2.3 Character The character of a representation is given by the trace of the matrix and is therefore invariant for base trans- formations: χ (j) (R) = Tr(Γ (j) (R)) Also holds, with N k the number of elements in a conjugacy class: ¸ k χ (i)∗ (C k )χ (j) (C k )N k = hδ ij Theorem: n ¸ i=1 2 i = h Chapter 13: Theory of groups 73 13.3 The relation with quantum mechanics 13.3.1 Representations, energy levels and degeneracy Consider a set of symmetry transformations x = Rx which leave the Hamiltonian H invariant. These trans- formations are a group. An isomorfic operation on the wavefunction is given by: P R ψ(x ) = ψ(R −1 x ). This is considered an active rotation. These operators commute with H: P R H = HP R , and leave the volume element unchanged: d(Rx) = dx. P R is the symmetry group of the physical system. It causes degeneracy: if ψ n is a solution of Hψ n = E n ψ n than also holds: H(P R ψ n ) = E n (P R ψ n ). A degeneracy which is not the result of a symmetry is called an accidental degeneracy. Assume an n -fold degeneracy at E n : then choose an orthonormal set ψ (n) ν , ν = 1, 2, . . . , n . The function P R ψ (n) ν is in the same subspace: P R ψ (n) ν = n ¸ κ=1 ψ (n) κ Γ (n) κν (R) where Γ (n) is an irreducible, unitary representation of the symmetry group ( of the system. Each n corre- sponds with another energy level. One can purely mathematical derive irreducible representations of a sym- metry group and label the energy levels with a quantum number this way. A fixed choice of Γ (n) (R) defines the base functions ψ (n) ν . This way one can also label each separate base function with a quantum number. Particle in a periodical potential: the symmetry operation is a cyclic group: note the operator describing one translation over one unit as A. Then: ( = ¦A, A 2 , A 3 , . . . , A h = E¦. The group is Abelian so all irreducible representations are one-dimensional. For 0 ≤ p ≤ h −1 follows: Γ (p) (A n ) = e 2πipn/h If one defines: k = − 2πp ah mod 2π a , so: P A ψ p (x) = ψ p (x − a) = e 2πip/h ψ p (x), this gives Bloch’s theorem: ψ k (x) = u k (x)e ikx , with u k (x ±a) = u k (x). 13.3.2 Breaking of degeneracy by a perturbation Suppose the unperturbed system has Hamiltonian H 0 and symmetry group ( 0 . The perturbed system has H = H 0 + 1, and symmetry group ( ⊂ ( 0 . If Γ (n) (R) is an irreducible representation of ( 0 , it is also a representation of ( but not all elements of Γ (n) in ( 0 are also in (. The representation then usually becomes reducible: Γ (n) = Γ (n1) ⊕ Γ (n2) ⊕ . . .. The degeneracy is then (possibly partially) removed: see the figure below. SpectrumH 0 SpectrumH n n3 = dim(Γ (n3) ) n2 = dim(Γ (n2) ) n1 = dim(Γ (n1) ) Theorem: The set of n degenerated eigenfunctions ψ (n) ν with energy E n is a basis for an n -dimensional irreducible representation Γ (n) of the symmetry group. 13.3.3 The construction of a base function Each function F in configuration space can be decomposed into symmetry types: F = n ¸ j=1 j ¸ κ=1 f (j) κ The following operator extracts the symmetry types: j h ¸ R∈G Γ (j)∗ κκ (R)P R F = f (j) κ 74 Physics Formulary by ir. J.C.A. Wevers This is expressed as: f (j) κ is the part of F that transforms according to the κ th ¯ row of Γ (j) . F can also be expressed in base functions ϕ: F = ¸ ajκ c ajκ ϕ (aj) κ . The functions f (j) κ are in general not transformed into each other by elements of the group. However, this does happen if c jaκ = c ja . Theorem: Two wavefunctions transforming according to non-equivalent unitary representations or according to different rows of an unitary irreducible representation are orthogonal: 'ϕ (i) κ [ψ (j) λ ` ∼ δ ij δ κλ , and 'ϕ (i) κ [ψ (i) κ ` is independent of κ. 13.3.4 The direct product of representations Consider a physical system existing of two subsystems. The subspace D (i) of the system transforms according to Γ (i) . Basefunctions are ϕ (i) κ (x i ), 1 ≤ κ ≤ i . Now form all 1 2 products ϕ (1) κ (x 1 )ϕ (2) λ (x 2 ). These define a space D (1) ⊗D (2) . These product functions transform as: P R (ϕ (1) κ (x 1 )ϕ (2) λ (x 2 )) = (P R ϕ (1) κ (x 1 ))(P R ϕ (2) λ (x 2 )) In general the space D (1) ⊗D (2) can be split up in a number of invariant subspaces: Γ (1) ⊗Γ (2) = ¸ i n i Γ (i) A useful tool for this reduction is that for the characters hold: χ (1) (R)χ (2) (R) = ¸ i n i χ (i) (R) 13.3.5 Clebsch-Gordan coefficients With the reduction of the direct-product matrix w.r.t. the basis ϕ (i) κ ϕ (j) λ one uses a new basis ϕ (aκ) µ . These base functions lie in subspaces D (ak) . The unitary base transformation is given by: ϕ (ak) µ = ¸ κλ ϕ (i) κ ϕ (j) λ (iκjλ[akµ) and the inverse transformation by: ϕ (i) κ ϕ (j) λ = ¸ akµ ϕ (aκ) µ (akµ[iκjλ) In essence the Clebsch-Gordan coefficients are dot products: (iκjλ[akµ) := 'ϕ (i) k ϕ (j) λ [ϕ (ak) µ ` 13.3.6 Symmetric transformations of operators, irreducible tensor operators Observables (operators) transform as follows under symmetry transformations: A = P R AP −1 R . If a set of operators A (j) κ with 0 ≤ κ ≤ j transform into each other under the transformations of ( holds: P R A (j) κ P −1 R = ¸ ν A (j) ν Γ (j) νκ (R) If Γ (j) is irreducible they are called irreducible tensor operators A (j) with components A (j) κ . An operator can also be decomposed into symmetry types: A = ¸ jk a (j) k , with: a (j) κ = j h ¸ R∈G Γ (j)∗ κκ (R) (P R AP −1 R ) Chapter 13: Theory of groups 75 Theorem: Matrix elements H ij of the operator H which is invariant under ∀ A∈G , are 0 between states which transformaccording to non-equivalent irreducible unitary representations or according to different rows of such a representation. Further 'ϕ (i) κ [H[ψ (i) κ ` is independent of κ. For H = 1 this becomes the previous theorem. This is applied in quantum mechanics in perturbation theory and variational calculus. Here one tries to diag- onalize H. Solutions can be found within each category of functions ϕ (i) κ with common i and κ: H is already diagonal in categories as a whole. Perturbation calculus can be applied independent within each category. With variational calculus the try func- tion can be chosen within a separate category because the exact eigenfunctions transform according to a row of an irreducible representation. 13.3.7 The Wigner-Eckart theorem Theorem: The matrix element 'ϕ (i) λ [A (j) κ [ψ (k) µ ` can only be = 0 if Γ (j) ⊗ Γ (k) = . . . ⊕ Γ (i) ⊕ . . .. If this is the case holds (if Γ (i) appears only once, otherwise one has to sum over a): 'ϕ (i) λ [A (j) κ [ψ (k) µ ` = (iλ[jκkµ)'ϕ (i) |A (j) |ψ (k) ` This theorem can be used to determine selection rules: the probability of a dipole transition is given by ( is the direction of polarization of the radiation): P D = 8π 2 e 2 f 3 [r 12 [ 2 3¯hε 0 c 3 with r 12 = 'l 2 m 2 [ r [l 1 m 1 ` Further it can be used to determine intensity ratios: if there is only one value of a the ratio of the matrix elements are the Clebsch-Gordan coefficients. For more a-values relations between the intensity ratios can be stated. However, the intensity ratios are also dependent on the occupation of the atomic energy levels. 13.4 Continuous groups Continuous groups have h = ∞. However, not all groups with h = ∞ are continuous, e.g. the translation group of an spatially infinite periodic potential is not continuous but does have h = ∞. 13.4.1 The 3-dimensional translation group For the translation of wavefunctions over a distance a holds: P a ψ(x) = ψ(x − a). Taylor expansion near x gives: ψ(x −a) = ψ(x) −a dψ(x) dx + 1 2 a 2 d 2 ψ(x) dx 2 −+. . . Because the momentum operator in quantum mechanics is given by: p x = ¯h i ∂ ∂x , this can be written as: ψ(x −a) = e −iapx/¯ h ψ(x) 13.4.2 The 3-dimensional rotation group This group is called SO(3) because a faithful representation can be constructed from orthogonal 33 matrices with a determinant of +1. For an infinitesimal rotation around the x-axis holds: P δθx ψ(x, y, z) ≈ ψ(x, y +zδθ x , z −yδθ x ) = ψ(x, y, z) + zδθ x ∂ ∂y −yδθ x ∂ ∂z ψ(x, y, z) = 1 − iδθ x L x ¯h ψ(x, y, z) 76 Physics Formulary by ir. J.C.A. Wevers Because the angular momentum operator is given by: L x = ¯h i z ∂ ∂y −y ∂ ∂z . So in an arbitrary direction holds: Rotations: P α,n = exp(−iα(n J )/¯h) Translations: P a,n = exp(−ia(n p)/¯h) J x , J y and J z are called the generators of the 3-dim. rotation group, p x , p y and p z are called the generators of the 3-dim. translation group. The commutation rules for the generators can be derived from the properties of the group for multiplications: translations are interchangeable ↔p x p y −p y p x = 0. Rotations are not generally interchangeable: consider a rotation around axis n in the xz-plane over an angle α. Then holds: P α,n = P −θ,y P α,x P θ,y , so: e −iα(n· J )/¯ h = e iθJy/¯ h e −iαJx/¯ h e −iθJy/¯ h If α and θ are very small and are expanded to second order, and the corresponding terms are put equal with n J = J x cos θ +J z sin θ, it follows from the αθ term: J x J y −J y J x = i¯hJ z . 13.4.3 Properties of continuous groups The elements R(p 1 , ..., p n ) depend continuously on parameters p 1 , ..., p n . For the translation group this are e.g. an x , an y and an z . It is demanded that the multiplication and inverse of an element Rdepend continuously on the parameters of R. The statement that each element arises only once in each row and column of the Cayley table holds also for continuous groups. The notion conjugacy class for continuous groups is defined equally as for discrete groups. The notion representation is fitted by demanding continuity: each matrix element depends continuously on p i (R). Summation over all group elements is for continuous groups replaced by an integration. If f(R) is a function defined on (, e.g. Γ αβ (R), holds: G f(R)dR := p1 pn f(R(p 1 , ..., p n ))g(R(p 1 , ..., p n ))dp 1 dp n Here, g(R) is the density function. Because of the properties of the Cayley table is demanded: f(R)dR = f(SR)dR. This fixes g(R) except for a constant factor. Define new variables p by: SR(p i ) = R(p i ). If one writes: dV := dp 1 dp n holds: g(S) = g(E) dV dV Here, dV dV is the Jacobian: dV dV = det ∂p i ∂p j , and g(E) is constant. For the translation group holds: g(a) = constant = g( 0 ) because g(an)da = g( 0 )da and da = da. This leads to the fundamental orthogonality theorem: G Γ (i)∗ µν (R)Γ (j) αβ (R)dR = 1 i δ ij δ µα δ νβ G dR and for the characters hold: G χ (i)∗ (R)χ (j) (R)dR = δ ij G dR Compact groups are groups with a finite group volume: G dR < ∞. Chapter 13: Theory of groups 77 13.5 The group SO(3) One can take 2 parameters for the direction of the rotational axis and one for the angle of rotation ϕ. The parameter space is a collection points ϕn within a sphere with radius π. The diametrical points on this sphere are equivalent because R n,π = R n,−π . Another way to define parameters is by means of Eulers angles. If α, β and γ are the 3 Euler angles, defined as: 1. The spherical angles of axis 3 w.r.t. xyz are θ, ϕ := β, α. Now a rotation around axis 3 remains possible. 2. The spherical angles of the z-axis w.r.t. 123 are θ, ϕ := β, π −γ. then the rotation of a quantum mechanical system is described by: ψ →e −iαJz¯ h e −iβJy/¯ h e −iγJz/¯ h ψ. So P R = e −iε(n· J )/¯ h . All irreducible representations of SO(3) can be constructed from the behaviour of the spherical harmonics Y lm (θ, ϕ) with −l ≤ m ≤ l and for a fixed l: P R Y lm (θ, ϕ) = ¸ m Y lm (θ, ϕ)D (l) mm (R) D (l) is an irreducible representation of dimension 2l +1. The character of D (l) is given by: χ (l) (α) = l ¸ m=−l e imα = 1 +2 l ¸ k=0 cos(kα) = sin([l + 1 2 ]α) sin( 1 2 α) In the performed derivation α is the rotational angle around the z-axis. This expression is valid for all rotations over an angle α because the classes of SO(3) are rotations around the same angle around an axis with an arbitrary orientation. Via the fundamental orthogonality theorem for characters one obtains the following expression for the density function (which is normalized so that g(0) = 1): g(α) = sin 2 ( 1 2 α) ( 1 2 α) 2 With this result one can see that the given representations of SO(3) are the only ones: the character of another representation χ would have to be ⊥ to the already found ones, so χ (α) sin 2 ( 1 2 α) = 0∀α ⇒ χ (α) = 0∀α. This is contradictory because the dimension of the representation is given by χ (0). Because fermions have an half-odd integer spin the states ψ sms with s = 1 2 and m s = ± 1 2 constitute a 2-dim. space which is invariant under rotations. A problem arises for rotations over 2π: ψ1 2 ms →e −2πiSz/¯ h ψ1 2 ms = e −2πims ψ1 2 ms = −ψ1 2 ms However, in SO(3) holds: R z,2π = E. So here holds E → ±II. Because observable quantities can always be written as 'φ[ψ` or 'φ[A[ψ`, and are bilinear in the states, they do not change sign if the states do. If only one state changes sign the observable quantities do change. The existence of these half-odd integer representations is connected with the topological properties of SO(3): the group is two-fold coherent through the identification R 0 = R 2π = E. 13.6 Applications to quantum mechanics 13.6.1 Vectormodel for the addition of angular momentum If two subsystems have angular momentum quantum numbers j 1 and j 2 the only possible values for the total angular momentumare J = j 1 +j 2 , j 1 +j 2 −1, ..., [j 1 −j 2 [. This can be derived from group theory as follows: from χ (j1) (α)χ (j2) (α) = ¸ J n j χ (J) (α) follows: D (j1) ⊗D (j2) = D (j1+j2) ⊕D (j1+j2−1) ⊕... ⊕D (|j1−j2|) 78 Physics Formulary by ir. J.C.A. Wevers The states can be characterized by quantum numbers in two ways: with j 1 , m 1 , j 2 , m 2 and with j 1 , j 2 , J, M. The Clebsch-Gordan coefficients, for SO(3) called the Wigner coefficients, can be chosen real, so: ψ j1j2JM = ¸ m1m2 ψ j1m1j2m2 (j 1 m 1 j 2 m 2 [JM) ψ j1m1j2m2 = ¸ JM ψ j1j2JM (j 1 m 1 j 2 m 2 [JM) 13.6.2 Irreducible tensor operators, matrixelements and selection rules Some examples of the behaviour of operators under SO(3) 1. Suppose j = 0: this gives the identical representation with j = 1. This state is described by a scalar operator. Because P R A (0) 0 P −1 R = A (0) 0 this operator is invariant, e.g. the Hamiltonian of a free atom. Then holds: 'J M [H[JM` ∼ δ MM δ JJ . 2. A vector operator: A = (A x , A y , A z ). The cartesian components of a vector operator transform equally as the cartesian components of r by definition. So for rotations around the z-axis holds: D(R α,z ) = ¸ cos α −sinα 0 sin α cos α 0 0 0 1 ¸ The transformed operator has the same matrix elements w.r.t. P R ψ and P R φ: P R ψ[P R A x P −1 R [P R φ = 'ψ[A x [φ`, and χ(R α,z ) = 1 + 2 cos(α). According to the equation for characters this means one can choose base operators which transform like Y 1m (θ, ϕ). These turn out to be the spherical components: A (1) +1 = − 1 √ 2 (A x +iA y ), A (1) 0 = A z , A (1) −1 = 1 √ 2 (A x −iA y ) 3. A cartesian tensor of rank 2: T ij is a quantity which transforms under rotations like U i V j , where U and V are vectors. So T ij transforms like P R T ij P −1 R = ¸ kl T kl D ki (R)D lj (R), so like D (1) ⊗ D (1) = D (2) ⊕ D (1) ⊕ D (0) . The 9 components can be split in 3 invariant subspaces with dimension 1 (D (0) ), 3 (D (1) ) and 5 (D (2) ). The new base operators are: I. Tr(T ) = T xx +T yy +T zz . This transforms as the scalar U V , so as D (0) . II. The 3 antisymmetric components A z = 1 2 (T xy −T yx ), etc. These transform as the vector U V , so as D (1) . III. The 5 independent components of the traceless, symmetric tensor S: S ij = 1 2 (T ij +T ji ) − 1 3 δ ij Tr(T). These transform as D (2) . Selection rules for dipole transitions Dipole operators transform as D (1) : for an electric dipole transfer is the operator er, for a magnetic e( L + 2 S )/2m. From the Wigner-Eckart theorem follows: 'J M [A (1) κ [JM` = 0 except D (J ) is a part of D (1) ⊗ D (J) = D (J+1) ⊕ D (J) ⊕ D (|J−1|) . This means that J ∈ ¦J + 1, J, [J − 1[¦: J = J or J = J ± 1, except J = J = 0. Land´ e-equation for the anomalous Zeeman splitting According to Land´ e’s model the interaction between a magnetic moment with an external magnetic field is determined by the projection of M on J because L and S precede fast around J. This can also be understood Chapter 13: Theory of groups 79 from the Wigner-Eckart theorem: from this follows that the matrix elements from all vector operators show a certain proportionality. For an arbitrary operator A follows: 'αjm [ A[αjm` = 'αjm[ A J [αjm` j(j +1)¯ h 2 'αjm [ J [αjm` 13.7 Applications to particle physics The physics of a system does not change after performing a transformation ψ = e iδ ψ where δ is a constant. This is a global gauge transformation: the phase of the wavefunction changes everywhere by the same amount. There exists some freedomin the choice of the potentials A and φ at the same E and B: gauge transformations of the potentials do not change E and B (See chapter 2 and 10). The solution ψ of the Schr¨ odinger equation with the transformed potentials is: ψ = e −iqf(r,t) ψ. This is a local gauge transformation: the phase of the wavefunction changes different at each position. The physics of the system does not change if A and φ are also transformed. This is now stated as a guide principle: the “right of existence” of the electromagnetic field is to allow local gauge invariance. The gauge transformations of the EM-field form a group: U(1), unitary 1 1-matrices. The split-off of charge in the exponent is essential: it allows one gauge field for all charged particles, independent of their charge. This concept is generalized: particles have a “special charge” Q. The group elements now are P R = exp(−iQΘ). Other force fields than the electromagnetic field can also be understood this way. The weak interaction together with the electromagnetic interaction can be described by a force field that transforms according to U(1)⊗SU(2), and consists of the photon and three intermediary vector bosons. The colour force is described by SU(3), and has a gauge field that exists of 8 types of gluons. In general the group elements are given by P R = exp(−i T Θ), where Θ n are real constants and T n operators (generators), like Q. The commutation rules are given by [T i , T j ] = i ¸ k c ijk T k . The c ijk are the structure constants of the group. For SO(3) these constants are c ijk = ε ijk , here ε ijk is the complete antisymmetric tensor with ε 123 = +1. These constants can be found with the help of group product elements: because ( is closed holds: e i Θ· T e i Θ · T e −i Θ· T e −i Θ · T = e −i Θ · T . Taylor expansion and setting equal Θ n Θ m -terms results in the com- mutation rules. The group SU(2) has 3 free parameters: because it is unitary there are 4 real conditions over 4 complex parameters, and the determinant has to be +1, remaining 3 free parameters. Each unitary matrix U can be written as: U = e −iH . Here, H is a Hermitian matrix. Further it always holds that: det(U) = e −iTr(H) . For each matrix of SU(2) holds that Tr(H)=0. Each Hermitian, traceless 2 2 matrix can be written as a linear combination of the 3 Pauli-matrices σ i . So these matrices are a choice for the operators of SU(2). One can write: SU(2)=¦exp(− 1 2 iσ Θ)¦. In abstraction, one can consider an isomorphic group where only the commutation rules are considered to be known regarding the operators T i : [T 1 , T 2 ] = iT 3 , etc. In elementary particle physics the T i can be interpreted e.g. as the isospin operators. Elementary particles can be classified in isospin-multiplets, these are the irreducible representations of SU(2). The classification is: 1. The isospin-singlet ≡ the identical representation: e −i T· Θ = 1 ⇒T i = 0 2. The isospin-doublet ≡ the faithful representation of SU(2) on 2 2 matrices. 80 Physics Formulary by ir. J.C.A. Wevers The group SU(3) has 8 free parameters. (The group SU(N) has N 2 − 1 free parameters). The Hermitian, traceless operators are 3 SU(2)-subgroups in the e 1 e 2 , e 1 e 3 and the e 2 e 3 plane. This gives 9 matrices, which are not all 9 linear independent. By taking a linear combination one gets 8 matrices. In the Lagrange density for the colour force one has to substitute ∂ ∂x → D Dx := ∂ ∂x − 8 ¸ i=1 T i A i x The terms of 3rd and 4th power in A show that the colour field interacts with itself. Chapter 14 Nuclear physics 14.1 Nuclear forces The mass of a nucleus is given by: M nucl = Zm p +Nm n −E bind /c 2 The binding energy per nucleon is given in the figure at the right. The top is at 56 26 Fe, the most stable nucleus. With the constants a 1 = 15.760 MeV a 2 = 17.810 MeV a 3 = 0.711 MeV a 4 = 23.702 MeV a 5 = 34.000 MeV 0 1 2 3 4 5 6 7 8 9 (MeV) E ↑ 0 40 80 120 160 200 240 A → and A = Z +N, in the droplet or collective model of the nucleus the binding energy E bind is given by: E bind c 2 = a 1 A−a 2 A 2/3 −a 3 Z(Z −1) A 1/3 −a 4 (N −Z) 2 A +a 5 A −3/4 These terms arise from: 1. a 1 : Binding energy of the strong nuclear force, approximately ∼ A. 2. a 2 : Surface correction: the nucleons near the surface are less bound. 3. a 3 : Coulomb repulsion between the protons. 4. a 4 : Asymmetry term: a surplus of protons or neutrons has a lower binding energy. 5. a 5 : Pair off effect: nuclei with an even number of protons or neutrons are more stable because groups of two protons or neutrons have a lower energy. The following holds: Z even, N even: = +1, Z odd, N odd: = −1. Z even, N odd: = 0, Z odd, N even: = 0. The Yukawa potential can be derived if the nuclear force can to first approximation, be considered as an exchange of virtual pions: U(r) = − W 0 r 0 r exp − r r 0 With ∆E ∆t ≈ ¯h, E γ = m 0 c 2 and r 0 = c∆t follows: r 0 = ¯ h/m 0 c. In the shell model of the nucleus one assumes that a nucleon moves in an average field of other nucleons. Further, there is a contribution of the spin-orbit coupling ∼ L S: ∆V ls = 1 2 (2l + 1)¯ hω. So each level (n, l) is split in two, with j = l ± 1 2 , where the state with j = l + 1 2 has the lowest energy. This is just the opposite for electrons, which is an indication that the L − S interaction is not electromagnetical. The energy of a 3-dimensional harmonic oscillator is E = (N + 3 2 )¯ hω. N = n x + n y + n z = 2(n − 1) + l where n ≥ 1 is the main oscillator number. Because −l ≤ m ≤ l and m s = ± 1 2 ¯h there are 2(2l + 1) 82 Physics Formulary by ir. J.C.A. Wevers substates which exist independently for protons and neutrons. This gives rise to the so called magical numbers: nuclei where each state in the outermost level are filled are particulary stable. This is the case if N or Z ∈ ¦2, 8, 20, 28, 50, 82, 126¦. 14.2 The shape of the nucleus A nucleus is to first approximation spherical with a radius of R = R 0 A 1/3 . Here, R 0 ≈ 1.4 10 −15 m, constant for all nuclei. If the nuclear radius is measured including the charge distribution one obtains R 0 ≈ 1.2 10 −15 m. The shape of oscillating nuclei can be described by spherical harmonics: R = R 0 ¸ 1 + ¸ lm a lm Y m l (θ, ϕ) ¸ l = 0 gives rise to monopole vibrations, density vibrations, which can be applied to the theory of neutron stars. l = 1 gives dipole vibrations, l = 2 quadrupole, with a 2,0 = β cos γ and a 2,±2 = 1 2 √ 2β sin γ where β is the deformation factor and γ the shape parameter. The multipole moment is given by µ l = Zer l Y m l (θ, ϕ). The parity of the electric moment is Π E = (−1) l , of the magnetic moment Π M = (−1) l+1 . There are 2 contributions to the magnetic moment: M L = e 2m p L and M S = g S e 2m p S. where g S is the spin-gyromagnetic ratio. For protons holds g S = 5.5855 and for neutrons g S = −3.8263. The z-components of the magnetic moment are given by M L,z = µ N m l and M S,z = g S µ N m S . The resulting magnetic moment is related to the nuclear spin I according to M = g I (e/2m p ) I. The z-component is then M z = µ N g I m I . 14.3 Radioactive decay The number of nuclei decaying is proportional to the number of nuclei: ˙ N = −λN. This gives for the number of nuclei N: N(t) = N 0 exp(−λt). The half life time follows from τ 1 2 λ = ln(2). The average life time of a nucleus is τ = 1/λ. The probability that N nuclei decay within a time interval is given by a Poisson distribution: P(N)dt = N 0 λ N e −λ N! dt If a nucleus can decay into more final states then holds: λ = ¸ λ i . So the fraction decaying into state i is λ i / ¸ λ i . There are 5 types of natural radioactive decay: 1. α-decay: the nucleus emits a He 2+ nucleus. Because nucleons tend to order themselves in groups of 2p+2n this can be considered as a tunneling of a He 2+ nucleus through a potential barrier. The tunnel probability P is P = incoming amplitude outgoing amplitude = e −2G with G = 1 ¯h 2m [V (r) −E]dr G is called the Gamow factor. 2. β-decay. Here a proton changes into a neutron or vice versa: p + →n 0 +W + →n 0 +e + +ν e , and n 0 →p + +W − →p + +e − +ν e . 3. Electron capture: here, a proton in the nucleus captures an electron (usually from the K-shell). 4. Spontaneous fission: a nucleus breaks apart. 5. γ-decay: here the nucleus emits a high-energetic photon. The decay constant is given by λ = P(l) ¯hω ∼ E γ (¯ hc) 2 E γ R ¯hc 2l ∼ 10 −4l Chapter 14: Nuclear physics 83 where l is the quantum number for the angular momentum and P the radiated power. Usually the decay constant of electric multipole moments is larger than the one of magnetic multipole moments. The energy of the photon is E γ = E i − E f − T R , with T R = E 2 γ /2mc 2 the recoil energy, which can usually be neglected. The parity of the emitted radiation is Π l = Π i Π f . With I the quantum number of angular momentum of the nucleus, L = ¯ h I(I +1), holds the following selection rule: [ I i − I f [ ≤ ∆l ≤ [ I i + I f [. 14.4 Scattering and nuclear reactions 14.4.1 Kinetic model If a beam with intensity I hits a target with density n and length x (Rutherford scattering) the number of scatterings R per unit of time is equal to R = Inxσ. From this follows that the intensity of the beam decreases as −dI = Inσdx. This results in I = I 0 e −nσx = I 0 e −µx . Because dR = R(θ, ϕ)dΩ/4π = Inxdσ it follows: dσ dΩ = R(θ, ϕ) 4πnxI If N particles are scattered in a material with density n then holds: ∆N N = n dσ dΩ ∆Ω∆x For Coulomb collisions holds: dσ dΩ C = Z 1 Z 2 e 2 8πε 0 µv 2 0 1 sin 4 ( 1 2 θ) 14.4.2 Quantum mechanical model for n-p scattering The initial state is a beam of neutrons moving along the z-axis with wavefunction ψ init = e ikz and current density J init = v[ψ init [ 2 = v. At large distances from the scattering point they have approximately a spherical wavefunction ψ scat = f(θ)e ikr /r where f(θ) is the scattering amplitude. The total wavefunction is then given by ψ = ψ in +ψ scat = e ikz +f(θ) e ikr r The particle flux of the scattered particles is v[ψ scat [ 2 = v[f(θ)[ 2 dΩ. From this it follows that σ(θ) = [f(θ)[ 2 . The wavefunction of the incoming particles can be expressed as a sum of angular momentum wavefunctions: ψ init = e ikz = ¸ l ψ l The impact parameter is related to the angular momentum with L = bp = b¯hk, so bk ≈ l. At very low energy only particles with l = 0 are scattered, so ψ = ψ 0 + ¸ l>0 ψ l and ψ 0 = sin(kr) kr If the potential is approximately rectangular holds: ψ 0 = C sin(kr +δ 0 ) kr The cross section is then σ(θ) = sin 2 (δ 0 ) k 2 so σ = σ(θ)dΩ = 4π sin 2 (δ 0 ) k 2 At very low energies holds: sin 2 (δ 0 ) = ¯h 2 k 2 /2m W 0 +W with W 0 the depth of the potential well. At higher energies holds: σ = 4π k 2 ¸ l sin 2 (δ l ) 84 Physics Formulary by ir. J.C.A. Wevers 14.4.3 Conservation of energy and momentum in nuclear reactions If a particle P 1 collides with a particle P 2 which is in rest w.r.t. the laboratory system and other particles are created, so P 1 +P 2 → ¸ k>2 P k the total energy Q gained or required is given by Q = (m 1 +m 2 − ¸ k>2 m k )c 2 . The minimal required kinetic energy T of P 1 in the laboratory system to initialize the reaction is T = −Q m 1 +m 2 + ¸ m k 2m 2 If Q < 0 there is a threshold energy. 14.5 Radiation dosimetry Radiometric quantities determine the strength of the radiation source(s). Dosimetric quantities are related to the energy transfer from radiation to matter. Parameters describing a relation between those are called inter- action parameters. The intensity of a beam of particles in matter decreases according to I(s) = I 0 exp(−µs). The deceleration of a heavy particle is described by the Bethe-Bloch equation: dE ds ∼ q 2 v 2 The fluention is given by Φ = dN/dA. The flux is given by φ = dΦ/dt. The energy loss is defined by Ψ = dW/dA, and the energy flux density ψ = dΨ/dt. The absorption coefficient is given by µ = (dN/N)/dx. The mass absorption coefficient is given by µ/. The radiation dose X is the amount of charge produced by the radiation per unit of mass, with unit C/kg. An old unit is the R¨ ontgen: 1Ro= 2.58 10 −4 C/kg. With the energy-absorption coefficient µ E follows: X = dQ dm = eµ E W Ψ where W is the energy required to disjoin an elementary charge. The absorbed dose D is given by D = dE abs /dm, with unit Gy=J/kg. An old unit is the rad: 1 rad=0.01 Gy. The dose tempo is defined as ˙ D. It can be derived that D = µ E Ψ The Kerma K is the amount of kinetic energy of secundary produced particles which is produced per mass unit of the radiated object. The equivalent dose H is a weight average of the absorbed dose per type of radiation, where for each type radiation the effects on biological material is used for the weight factor. These weight factors are called the quality factors. Their unit is Sv. H = QD. If the absorption is not equally distributed also weight factors w per organ need to be used: H = ¸ w k H k . For some types of radiation holds: Radiation type Q R¨ ontgen, gamma radiation 1 β, electrons, mesons 1 Thermic neutrons 3 to 5 Fast neutrons 10 to 20 protons 10 α, fission products 20 Chapter 15 Quantum field theory & Particle physics 15.1 Creation and annihilation operators A state with more particles can be described by a collection occupation numbers [n 1 n 2 n 3 `. Hence the vacuum state is given by [000 `. This is a complete description because the particles are indistinguishable. The states are orthonormal: 'n 1 n 2 n 3 [n 1 n 2 n 3 ` = ∞ ¸ i=1 δ nin i The time-dependent state vector is given by Ψ(t) = ¸ n1n2··· c n1n2··· (t)[n 1 n 2 ` The coefficients c can be interpreted as follows: [c n1n2··· [ 2 is the probability to find n 1 particles with momen- tum k 1 , n 2 particles with momentum k 2 , etc., and 'Ψ(t)[Ψ(t)` = ¸ [c ni (t)[ 2 = 1. The expansion of the states in time is described by the Schr¨ odinger equation i d dt [Ψ(t)` = H[Ψ(t)` where H = H 0 + H int . H 0 is the Hamiltonian for free particles and keeps [c ni (t)[ 2 constant, H int is the interaction Hamiltonian and can increase or decrease a c 2 at the cost of others. All operators which can change occupation numbers can be expanded in the a and a † operators. a is the annihilation operator and a † the creation operator, and: a( k i )[n 1 n 2 n i ` = √ n i [n 1 n 2 n i −1 ` a † ( k i )[n 1 n 2 n i ` = √ n i +1 [n 1 n 2 n i +1 ` Because the states are normalized holds a[0` = 0 and a( k i )a † ( k i )[n i ` = n i [n i `. So aa † is an occupation number operator. The following commutation rules can be derived: [a( k i ), a( k j )] = 0 , [a † ( k i ), a † ( k j )] = 0 , [a( k i ), a † ( k j )] = δ ij Hence for free spin-0 particles holds: H 0 = ¸ i a † ( k i )a( k i )¯ hω ki 15.2 Classical and quantum fields Starting with a real field Φ α (x) (complex fields can be split in a real and an imaginary part), the Lagrange density L is a function of the position x = (x, ict) through the fields: L = L(Φ α (x), ∂ ν Φ α (x)). The La- grangian is given by L = L(x)d 3 x. Using the variational principle δI(Ω) = 0 and with the action-integral I(Ω) = L(Φ α , ∂ ν Φ α )d 4 x the field equation can be derived: ∂L ∂Φ α − ∂ ∂x ν ∂L ∂(∂ ν Φ α ) = 0 The conjugated field is, analogous to momentum in classical mechanics, defined as: Π α (x) = ∂L ∂ ˙ Φ α 86 Physics Formulary by ir. J.C.A. Wevers With this, the Hamilton density becomes H(x) = Π α ˙ Φ α −L(x). Quantization of a classical field is analogous to quantization in point mass mechanics: the field functions are considered as operators obeying certain commutation rules: [Φ α (x), Φ β (x )] = 0 , [Π α (x), Π β (x )] = 0 , [Φ α (x), Π β (x )] = iδ αβ (x −x ) 15.3 The interaction picture Some equivalent formulations of quantum mechanics are possible: 1. Schr¨ odinger picture: time-dependent states, time-independent operators. 2. Heisenberg picture: time-independent states, time-dependent operators. 3. Interaction picture: time-dependent states, time-dependent operators. The interaction picture can be obtained from the Schr¨ odinger picture by an unitary transformation: [Φ(t)` = e iH S 0 [Φ S (t)` and O(t) = e iH S 0 O S e −iH S 0 The index S denotes the Schr¨ odinger picture. From this follows: i d dt [Φ(t)` = H int (t)[Φ(t)` and i d dt O(t) = [O(t), H 0 ] 15.4 Real scalar field in the interaction picture It is easy to find that, with M := m 2 0 c 2 /¯h 2 , holds: ∂ ∂t Φ(x) = Π(x) and ∂ ∂t Π(x) = (∇ 2 −M 2 )Φ(x) From this follows that Φ obeys the Klein-Gordon equation (✷ − M 2 )Φ = 0. With the definition k 2 0 = k 2 +M 2 := ω 2 k and the notation k x −ik 0 t := kx the general solution of this equation is: Φ(x) = 1 √ V ¸ k 1 √ 2ω k a( k )e ikx +a † ( k )e −ikx , Π(x) = i √ V ¸ k 1 2 ω k −a( k )e ikx +a † ( k )e −ikx The field operators contain a volume V , which is used as normalization factor. Usually one can take the limit V →∞. In general it holds that the term with e −ikx , the positive frequency part, is the creation part, and the negative frequency part is the annihilation part. the coefficients have to be each others hermitian conjugate because Φ is hermitian. Because Φ has only one component this can be interpreted as a field describing a particle with spin zero. From this follows that the commutation rules are given by [Φ(x), Φ(x )] = i∆(x −x ) with ∆(y) = 1 (2π) 3 sin(ky) ω k d 3 k ∆(y) is an odd function which is invariant for proper Lorentz transformations (no mirroring). This is consistent with the previously found result [Φ(x, t, Φ(x , t)] = 0. In general holds that ∆(y) = 0 outside the light cone. So the equations obey the locality postulate. The Lagrange density is given by: L(Φ, ∂ ν Φ) = − 1 2 (∂ ν Φ∂ ν Φ +m 2 Φ 2 ). The energy operator is given by: H = H(x)d 3 x = ¸ k ¯hω k a † ( k )a( k ) Chapter 15: Quantum field theory & Particle physics 87 15.5 Charged spin-0 particles, conservation of charge The Lagrange density of charged spin-0 particles is given by: L = −(∂ ν Φ∂ ν Φ ∗ +M 2 ΦΦ ∗ ). Noether’s theorem connects a continuous symmetry of L and an additive conservation law. Suppose that L((Φ α ) , ∂ ν (Φ α ) ) = L(Φ α , ∂ ν Φ α ) and there exists a continuous transformation between Φ α and Φ α such as Φ α = Φ α +f α (Φ). Then holds ∂ ∂x ν ∂L ∂(∂ ν Φ α ) f α = 0 This is a continuity equation ⇒conservation law. Which quantity is conserved depends on the symmetry. The above Lagrange density is invariant for a change in phase Φ → Φe iθ : a global gauge transformation. The conserved quantity is the current density J µ (x) = −ie(Φ∂ µ Φ ∗ − Φ ∗ ∂ µ Φ). Because this quantity is 0 for real fields a complex field is needed to describe charged particles. When this field is quantized the field operators are given by Φ(x) = 1 √ V ¸ k 1 √ 2ω k a( k )e ikx +b † ( k )e −ikx , Φ † (x) = 1 √ V ¸ k 1 √ 2ω k a † ( k )e ikx +b( k )e −ikx Hence the energy operator is given by: H = ¸ k ¯hω k a † ( k )a( k ) +b † ( k )b( k ) and the charge operator is given by: Q(t) = −i J 4 (x)d 3 x ⇒Q = ¸ k e a † ( k )a( k ) −b † ( k )b( k ) From this follows that a † a := N + ( k ) is an occupation number operator for particles with a positive charge and b † b := N − ( k ) is an occupation number operator for particles with a negative charge. 15.6 Field functions for spin- 1 2 particles Spin is defined by the behaviour of the solutions ψ of the Dirac equation. A scalar field Φ has the property that, if it obeys the Klein-Gordon equation, the rotated field ˜ Φ(x) := Φ(Λ −1 x) also obeys it. Λ denotes 4-dimensional rotations: the proper Lorentz transformations. These can be written as: ˜ Φ(x) = Φ(x)e −in· L with L µν = −i¯h x µ ∂ ∂x ν −x ν ∂ ∂x µ For µ ≤ 3, ν ≤ 3 these are rotations, for ν = 4, µ = 4 these are Lorentz transformations. A rotated field ˜ ψ obeys the Dirac equation if the following condition holds: ˜ ψ(x) = D(Λ)ψ(Λ −1 x). This results in the condition D −1 γ λ D = Λ λµ γ µ . One finds: D = e in· S with S µν = −i 1 2 ¯hγ µ γ ν . Hence: ˜ ψ(x) = e −i(S+L) ψ(x) = e −iJ ψ(x) Then the solutions of the Dirac equation are given by: ψ(x) = u r ± ( p )e −i( p·x±Et) Here, r is an indication for the direction of the spin, and ± is the sign of the energy. With the notation v r ( p ) = u r − (− p ) and u r ( p ) = u r + ( p ) one can write for the dot products of these spinors: u r + ( p )u r + ( p ) = E M δ rr , u r − ( p )u r − ( p ) = E M δ rr , u r + ( p )u r − ( p ) = 0 88 Physics Formulary by ir. J.C.A. Wevers Because of the factor E/M this is not relativistic invariant. A Lorentz-invariant dot product is defined by ab := a † γ 4 b, where a := a † γ 4 is a row spinor. From this follows: u r ( p )u r ( p ) = δ rr , v r ( p )v r ( p ) = −δ rr , u r ( p )v r ( p ) = 0 Combinations of the type aa give a 4 4 matrix: 2 ¸ r=1 u r ( p )u r ( p ) = −iγ λ p λ +M 2M , 2 ¸ r=1 v r ( p )v r ( p ) = −iγ λ p λ −M 2M The Lagrange density which results in the Dirac equation and having the correct energy normalization is: L(x) = −ψ(x) γ µ ∂ ∂x µ +M ψ(x) and the current density is J µ (x) = −ieψγ µ ψ. 15.7 Quantization of spin- 1 2 fields The general solution for the fieldoperators is in this case: ψ(x) = M V ¸ p 1 √ E ¸ r c r ( p )u r ( p )e ipx +d † r ( p )v r ( p )e −ipx and ψ(x) = M V ¸ p 1 √ E ¸ r c † r ( p )u r ( p )e −ipx +d r ( p )v r ( p )e ipx Here, c † and c are the creation respectively annihilation operators for an electron and d † and d the creation respectively annihilation operators for a positron. The energy operator is given by H = ¸ p E p 2 ¸ r=1 c † r ( p )c r ( p ) −d r ( p )d † r ( p ) To prevent that the energy of positrons is negative the operators must obey anti commutation rules in stead of commutation rules: [c r ( p ), c † r ( p )] + = [d r ( p ), d † r ( p )] + = δ rr δ pp , all other anti commutators are 0. The field operators obey [ψ α (x), ψ β (x )] = 0 , [ψ α (x), ψ β (x )] = 0 , [ψ α (x), ψ β (x )] + = −iS αβ (x −x ) with S(x) = γ λ ∂ ∂x λ −M ∆(x) The anti commutation rules give besides the positive-definite energy also the Pauli exclusion principle and the Fermi-Dirac statistics: because c † r ( p )c † r ( p ) = −c † r ( p )c † r ( p ) holds: ¦c † r (p)¦ 2 = 0. It appears to be impossible to create two electrons with the same momentum and spin. This is the exclusion principle. Another way to see this is the fact that ¦N + r ( p )¦ 2 = N + r ( p ): the occupation operators have only eigenvalues 0 and 1. To avoid infinite vacuum contributions to the energy and charge the normal product is introduced. The expres- sion for the current density now becomes J µ = −ieN(ψγ µ ψ). This product is obtained by: • Expand all fields into creation and annihilation operators, • Keep all terms which have no annihilation operators, or in which they are on the right of the creation operators, • In all other terms interchange the factors so that the annihilation operators go to the right. By an inter- change of two fermion operators add a minus sign, by interchange of two boson operators not. Assume hereby that all commutators are zero. Chapter 15: Quantum field theory & Particle physics 89 15.8 Quantization of the electromagnetic field Starting with the Lagrange density L = − 1 2 ∂A ν ∂x µ ∂A ν ∂x µ it follows for the field operators A(x): A(x) = 1 √ V ¸ k 1 √ 2ω k 4 ¸ m=1 a m ( k ) m ( k )e ikx +a † ( k ) m ( k ) ∗ e −ikx The operators obey [a m ( k ), a † m ( k )] = δ mm δ kk . All other commutators are 0. m gives the polarization direction of the photon: m = 1, 2 gives transversal polarized, m = 3 longitudinal polarized and m = 4 timelike polarized photons. Further holds: [A µ (x), A ν (x )] = iδ µν D(x −x ) with D(y) = ∆(y)[ m=0 In spite of the fact that A 4 = iV is imaginary in the classical case, A 4 is still defined to be hermitian be- cause otherwise the sign of the energy becomes incorrect. By changing the definition of the inner product in configuration space the expectation values for A 1,2,3 (x) ∈ IR and for A 4 (x) become imaginary. If the potentials satisfy the Lorentz gauge condition ∂ µ A µ = 0 the E and B operators derived from these potentials will satisfy the Maxwell equations. However, this gives problems with the commutation rules. It is now demanded that only those states are permitted for which holds ∂A + µ ∂x µ [Φ` = 0 This results in: ∂A µ ∂x µ = 0. From this follows that (a 3 ( k ) − a 4 ( k ))[Φ` = 0. With a local gauge transformation one obtains N 3 ( k ) = 0 and N 4 ( k ) = 0. However, this only applies to free EM-fields: in intermediary states in interactions there can exist longitudinal and timelike photons. These photons are also responsible for the stationary Coulomb potential. 15.9 Interacting fields and the S-matrix The S(scattering)-matrix gives a relation between the initial and final states of an interaction: [Φ(∞)` = S[Φ(−∞)`. If the Schr¨ odinger equation is integrated: [Φ(t)` = [Φ(−∞)` −i t −∞ H int (t 1 )[Φ(t 1 )`dt 1 and perturbation theory is applied one finds that: S = ∞ ¸ n=0 (−i) n n! T ¦H int (x 1 ) H int (x n )¦ d 4 x 1 d 4 x n ≡ ∞ ¸ n=0 S (n) Here, the T-operator means a time-ordered product: the terms in such a product must be ordered in increasing time order from the right to the left so that the earliest terms act first. The S-matrix is then given by: S ij = 'Φ i [S[Φ j ` = 'Φ i [Φ(∞)`. The interaction Hamilton density for the interaction between the electromagnetic and the electron-positron field is: H int (x) = −J µ (x)A µ (x) = ieN(ψγ µ ψA µ ) When this is expanded as: H int = ieN (ψ + +ψ − )γ µ (ψ + +ψ − )(A + µ +A − µ ) 90 Physics Formulary by ir. J.C.A. Wevers eight terms appear. Each term corresponds with a possible process. The term ieψ + γ µ ψ + A − µ acting on [Φ` gives transitions where A − µ creates a photon, ψ + annihilates an electron and ψ + annihilates a positron. Only terms with the correct number of particles in the initial and final state contribute to a matrix element 'Φ i [S[Φ j `. Further the factors in H int can create and thereafter annihilate particles: the virtual particles. The expressions for S (n) contain time-ordered products of normal products. This can be written as a sum of normal products. The appearing operators describe the minimal changes necessary to change the initial state into the final state. The effects of the virtual particles are described by the (anti)commutator functions. Some time-ordened products are: T ¦Φ(x)Φ(y)¦ = N ¦Φ(x)Φ(y)¦ + 1 2 ∆ F (x −y) T ψ α (x)ψ β (y) ¸ = N ψ α (x)ψ β (y) ¸ − 1 2 S F αβ (x −y) T ¦A µ (x)A ν (y)¦ = N ¦A µ (x)A ν (y)¦ + 1 2 δ µν D F µν (x −y) Here, S F (x) = (γ µ ∂ µ −M)∆ F (x), D F (x) = ∆ F (x)[ m=0 and ∆ F (x) = 1 (2π) 3 e ikx ω k d 3 k if x 0 > 0 1 (2π) 3 e −ikx ω k d 3 k if x 0 < 0 The term 1 2 ∆ F (x − y) is called the contraction of Φ(x) and Φ(y), and is the expectation value of the time- ordered product in the vacuum state. Wick’s theorem gives an expression for the time-ordened product of an arbitrary number of field operators. The graphical representation of these processes are called Feynman diagrams. In the x-representation each diagram describes a number of processes. The contraction functions can also be written as: ∆ F (x) = lim →0 −2i (2π) 4 e ikx k 2 +m 2 −i d 4 k and S F (x) = lim →0 −2i (2π) 4 e ipx iγ µ p µ −M p 2 +M 2 −i d 4 p In the expressions for S (2) this gives rise to terms δ(p +k −p −k ). This means that energy and momentum is conserved. However, virtual particles do not obey the relation between energy and momentum. 15.10 Divergences and renormalization It turns out that higher orders contribute infinite terms because only the sum p + k of the four-momentum of the virtual particles is fixed. An integration over one of them becomes ∞. In the x-representation this can be understood because the product of two functions containing δ-like singularities is not well defined. This is solved by discounting all divergent diagrams in a renormalization of e and M. It is assumed that an electron, if there would not be an electromagnetical field, would have a mass M 0 and a charge e 0 unequal to the observed mass M and charge e. In the Hamilton and Lagrange density of the free electron-positron field appears M 0 . So this gives, with M = M 0 +∆M: L e−p (x) = −ψ(x)(γ µ ∂ µ +M 0 )ψ(x) = −ψ(x)(γ µ ∂ µ +M)ψ(x) +∆Mψ(x)ψ(x) and H int = ieN(ψγ µ ψA µ ) −i∆eN(ψγ µ ψA µ ). 15.11 Classification of elementary particles Elementary particles can be categorized as follows: 1. Hadrons: these exist of quarks and can be categorized in: I. Baryons: these exist of 3 quarks or 3 antiquarks. Chapter 15: Quantum field theory & Particle physics 91 II. Mesons: these exist of one quark and one antiquark. 2. Leptons: e ± , µ ± , τ ± , ν e , ν µ , ν τ , ν e , ν µ , ν τ . 3. Field quanta: γ, W ± , Z 0 , gluons, gravitons (?). An overview of particles and antiparticles is given in the following table: Particle spin (¯h) B L T T 3 S C B ∗ charge (e) m 0 (MeV) antipart. u 1/2 1/3 0 1/2 1/2 0 0 0 +2/3 5 u d 1/2 1/3 0 1/2 −1/2 0 0 0 −1/3 9 d s 1/2 1/3 0 0 0 −1 0 0 −1/3 175 s c 1/2 1/3 0 0 0 0 1 0 +2/3 1350 c b 1/2 1/3 0 0 0 0 0 −1 −1/3 4500 b t 1/2 1/3 0 0 0 0 0 0 +2/3 173000 t e − 1/2 0 1 0 0 0 0 0 −1 0.511 e + µ − 1/2 0 1 0 0 0 0 0 −1 105.658 µ + τ − 1/2 0 1 0 0 0 0 0 −1 1777.1 τ + ν e 1/2 0 1 0 0 0 0 0 0 0(?) ν e ν µ 1/2 0 1 0 0 0 0 0 0 0(?) ν µ ν τ 1/2 0 1 0 0 0 0 0 0 0(?) ν τ γ 1 0 0 0 0 0 0 0 0 0 γ gluon 1 0 0 0 0 0 0 0 0 0 gluon W + 1 0 0 0 0 0 0 0 +1 80220 W − Z 1 0 0 0 0 0 0 0 0 91187 Z graviton 2 0 0 0 0 0 0 0 0 0 graviton Here B is the baryon number and L the lepton number. It is found that there are three different lepton numbers, one for e, µ and τ, which are separately conserved. T is the isospin, with T 3 the projection of the isospin on the third axis, C the charmness, S the strangeness and B ∗ the bottomness. The anti particles have quantum numbers with the opposite sign except for the total isospin T. The composition of (anti)quarks of the hadrons is given in the following table, together with their mass in MeV in their ground state: π 0 1 2 √ 2(uu+dd) 134.9764 J/Ψ cc 3096.8 Σ + d d s 1197.436 π + ud 139.56995 Υ bb 9460.37 Ξ 0 u s s 1314.9 π − du 139.56995 p + u u d 938.27231 Ξ 0 u s s 1314.9 K 0 sd 497.672 p − u u d 938.27231 Ξ − d s s 1321.32 K 0 ds 497.672 n 0 u d d 939.56563 Ξ + d s s 1321.32 K + us 493.677 n 0 u d d 939.56563 Ω − s s s 1672.45 K − su 493.677 Λ u d s 1115.684 Ω + s s s 1672.45 D + cd 1869.4 Λ u d s 1115.684 Λ + c u d c 2285.1 D − dc 1869.4 Σ + u u s 1189.37 ∆ 2− u u u 1232.0 D 0 cu 1864.6 Σ − u u s 1189.37 ∆ 2+ u u u 1232.0 D 0 uc 1864.6 Σ 0 u d s 1192.55 ∆ + u u d 1232.0 F + cs 1969.0 Σ 0 u d s 1192.55 ∆ 0 u d d 1232.0 F − sc 1969.0 Σ − d d s 1197.436 ∆ − d d d 1232.0 Each quark can exist in two spin states. So mesons are bosons with spin 0 or 1 in their ground state, while baryons are fermions with spin 1 2 or 3 2 . There exist excited states with higher internal L. Neutrino’s have a helicity of − 1 2 while antineutrino’s have only + 1 2 as possible value. The quantum numbers are subject to conservation laws. These can be derived from symmetries in the La- grange density: continuous symmetries give rise to additive conservation laws, discrete symmetries result in multiplicative conservation laws. Geometrical conservation laws are invariant under Lorentz transformations and the CPT-operation. These are: 1. Mass/energy because the laws of nature are invariant for translations in time. 92 Physics Formulary by ir. J.C.A. Wevers 2. Momentum because the laws of nature are invariant for translations in space. 3. Angular momentum because the laws of nature are invariant for rotations. Dynamical conservation laws are invariant under the CPT-operation. These are: 1. Electrical charge because the Maxwell equations are invariant under gauge transformations. 2. Colour charge is conserved. 3. Isospin because QCD is invariant for rotations in T-space. 4. Baryon number and lepton number are conserved but not under a possible SU(5) symmetry of the laws of nature. 5. Quarks type is only conserved under the colour interaction. 6. Parity is conserved except for weak interactions. The elementary particles can be classified into three families: leptons quarks antileptons antiquarks 1st generation e − d e + d ν e u ν e u 2nd generation µ − s µ + s ν µ c ν µ c 3rd generation τ − b τ + b ν τ t ν τ t Quarks exist in three colours but because they are confined these colours cannot be seen directly. The color force does not decrease with distance. The potential energy will become high enough to create a quark- antiquark pair when it is tried to disjoin an (anti)quark from a hadron. This will result in two hadrons and not in free quarks. 15.12 P and CP-violation It is found that the weak interaction violates P-symmetry, and even CP-symmetry is not conserved. Some processes which violate P symmetry but conserve the combination CP are: 1. µ-decay: µ − →e − +ν µ +ν e . Left-handed electrons appear more than 1000as much as right-handed ones. 2. β-decay of spin-polarized 60 Co: 60 Co → 60 Ni +e − +ν e . More electrons with a spin parallel to the Co than with a spin antiparallel are created: (parallel−antiparallel)/(total)=20%. 3. There is no connection with the neutrino: the decay of the Λ particle through: Λ → p + + π − and Λ →n 0 +π 0 has also these properties. The CP-symmetry was found to be violated by the decay of neutral Kaons. These are the lowest possible states with a s-quark so they can decay only weakly. The following holds: C[K 0 ` = η[K 0 ` where η is a phase factor. Further holds P[K 0 ` = −[K 0 ` because K 0 and K 0 have an intrinsic parity of −1. From this follows that K 0 and K 0 are not eigenvalues of CP: CP[K 0 ` = [K 0 `. The linear combinations [K 0 1 ` := 1 2 √ 2([K 0 ` +[K 0 `) and [K 0 2 ` := 1 2 √ 2([K 0 ` − [K 0 `) are eigenstates of CP: CP[K 0 1 ` = +[K 0 1 ` and CP[K 0 2 ` = −[K 0 2 `. A base of K 0 1 and K 0 2 is practical while describing weak interactions. For colour interactions a base of K 0 and K 0 is practical because then the number u−number u is constant. The expansion postulate must be used for weak decays: [K 0 ` = 1 2 ('K 0 1 [K 0 ` +'K 0 2 [K 0 `) Chapter 15: Quantum field theory & Particle physics 93 The probability to find a final state with CP= −1 is 1 2 [ K 0 2 [K 0 [ 2 , the probability of CP=+1 decay is 1 2 [ K 0 1 [K 0 [ 2 . The relation between the mass eigenvalues of the quarks (unaccented) and the fields arising in the weak currents (accented) is (u , c , t ) = (u, c, t), and: ¸ d s b ¸ = ¸ 1 0 0 0 cos θ 2 sinθ 2 0 −sinθ 2 cos θ 2 ¸ ¸ 1 0 0 0 1 0 0 0 e iδ ¸ ¸ cos θ 1 sin θ 1 0 −sin θ 1 cos θ 1 0 0 0 1 ¸ ¸ 1 0 0 0 cos θ 3 sinθ 3 0 −sinθ 3 cos θ 3 ¸ ¸ d s b ¸ θ 1 ≡ θ C is the Cabibbo angle: sin(θ C ) ≈ 0.23 ±0.01. 15.13 The standard model When one wants to make the Lagrange density which describes a field invariant for local gauge transformations from a certain group, one has to perform the transformation ∂ ∂x µ → D Dx µ = ∂ ∂x µ −i g ¯h L k A k µ Here the L k are the generators of the gauge group (the “charges”) and the A k µ are the gauge fields. g is the matching coupling constant. The Lagrange density for a scalar field becomes: L = − 1 2 (D µ Φ ∗ D µ Φ +M 2 Φ ∗ Φ) − 1 4 F a µν F µν a and the field tensors are given by: F a µν = ∂ µ A a ν −∂ ν A a µ +gc a lm A l µ A m ν . 15.13.1 The electroweak theory The electroweak interaction arises from the necessity to keep the Lagrange density invariant for local gauge transformations of the group SU(2)⊗U(1). Right- and left-handed spin states are treated different because the weak interaction does not conserve parity. If a fifth Dirac matrix is defined by: γ 5 := γ 1 γ 2 γ 3 γ 4 = − ¸ ¸ ¸ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ¸ the left- and right- handed solutions of the Dirac equation for neutrino’s are given by: ψ L = 1 2 (1 +γ 5 )ψ and ψ R = 1 2 (1 −γ 5 )ψ It appears that neutrino’s are always left-handed while antineutrino’s are always right-handed. The hypercharge Y , for quarks given by Y = B +S +C +B ∗ +T , is defined by: Q = 1 2 Y +T 3 so [Y, T k ] = 0. The group U(1) Y ⊗SU(2) T is taken as symmetry group for the electroweak interaction because the generators of this group commute. The multiplets are classified as follows: e − R ν eL e − L u L d L u R d R T 0 1 2 1 2 0 0 T 3 0 1 2 − 1 2 1 2 − 1 2 0 0 Y −2 −1 1 3 4 3 − 2 3 94 Physics Formulary by ir. J.C.A. Wevers Now, 1 field B µ (x) is connected with gauge group U(1) and 3 gauge fields A µ (x) are connected with SU(2). The total Lagrange density (minus the fieldterms) for the electron-fermion field now becomes: L 0,EW = −(ψ νe,L , ψ eL )γ µ ∂ µ −i g ¯h A µ ( 1 2 σ) − 1 2 i g ¯h B µ (−1) ψ νe,L ψ eL − ψ eR γ µ ∂ µ − 1 2 i g ¯h (−2)B µ ψ eR Here, 1 2 σ are the generators of T and −1 and −2 the generators of Y . 15.13.2 Spontaneous symmetry breaking: the Higgs mechanism All leptons are massless in the equations above. Their mass is probably generated by spontaneous symmetry breaking. This means that the dynamic equations which describe the systemhave a symmetry which the ground state does not have. It is assumed that there exists an isospin-doublet of scalar fields Φ with electrical charges +1 and 0 and potential V (Φ) = −µ 2 Φ ∗ Φ + λ(Φ ∗ Φ) 2 . Their antiparticles have charges −1 and 0. The extra terms in L arising from these fields, L H = (D Lµ Φ) ∗ (D µ L Φ) − V (Φ), are globally U(1)⊗SU(2) symmetric. Hence the state with the lowest energy corresponds with the state Φ ∗ (x)Φ(x) = v = µ 2 /2λ =constant. The field can be written (were ω ± and z are Nambu-Goldstone bosons which can be transformed away, and m φ = µ √ 2) as: Φ = Φ + Φ 0 = iω + (v +φ −iz)/ √ 2 and '0[Φ[0` = 0 v/ √ 2 Because this expectation value = 0 the SU(2) symmetry is broken but the U(1) symmetry is not. When the gauge fields in the resulting Lagrange density are separated one obtains: W − µ = 1 2 √ 2(A 1 µ +iA 2 µ ) , W + µ = 1 2 √ 2(A 1 µ −iA 2 µ ) Z µ = gA 3 µ −g B µ g 2 +g 2 ≡ A 3 µ cos(θ W ) −B µ sin(θ W ) A µ = g A 3 µ +gB µ g 2 +g 2 ≡ A 3 µ sin(θ W ) +B µ cos(θ W ) where θ W is called the Weinberg angle. For this angle holds: sin 2 (θ W ) = 0.255 ± 0.010. Relations for the masses of the field quanta can be obtained from the remaining terms: M W = 1 2 vg and M Z = 1 2 v g 2 +g 2 , and for the elementary charge holds: e = gg g 2 +g 2 = g cos(θ W ) = g sin(θ W ) Experimentally it is found that M W = 80.022 ±0.26 GeV/c 2 and M Z = 91.187 ±0.007 GeV/c 2 . According to the weak theory this should be: M W = 83.0 ±0.24 GeV/c 2 and M Z = 93.8 ±2.0 GeV/c 2 . 15.13.3 Quantumchromodynamics Coloured particles interact because the Lagrange density is invariant for the transformations of the group SU(3) of the colour interaction. A distinction can be made between two types of particles: 1. “White” particles: they have no colour charge, the generator T = 0. 2. “Coloured” particles: the generators T are 8 3 3 matrices. There exist three colours and three anti- colours. The Lagrange density for coloured particles is given by L QCD = i ¸ k Ψ k γ µ D µ Ψ k + ¸ k,l Ψ k M kl Ψ l − 1 4 F a µν F µν a The gluons remain massless because this Lagrange density does not contain spinless particles. Because left- and right- handed quarks now belong to the same multiplet a mass term can be introduced. This term can be brought in the form M kl = m k δ kl . Chapter 15: Quantum field theory & Particle physics 95 15.14 Path integrals The development in time of a quantum mechanical system can, besides with Schr¨ odingers equation, also be described by a path integral (Feynman): ψ(x , t ) = F(x , t , x, t)ψ(x, t)dx in which F(x , t , x, t) is the amplitude of probability to find a system on time t in x if it was in x on time t. Then, F(x , t , x, t) = exp iS[x] ¯h d[x] where S[x] is an action-integral: S[x] = L(x, ˙ x, t)dt. The notation d[x] means that the integral has to be taken over all possible paths [x]: d[x] := lim n→∞ 1 N ¸ n ∞ −∞ dx(t n ) in which N is a normalization constant. To each path is assigned a probability amplitude exp(iS/¯h). The classical limit can be found by taking δS = 0: the average of the exponent vanishes, except where it is stationary. In quantum fieldtheory, the probability of the transition of a fieldoperator Φ(x, −∞) to Φ (x, ∞) is given by F(Φ (x, ∞), Φ(x, −∞)) = exp iS[Φ] ¯h d[Φ] with the action-integral S[Φ] = Ω L(Φ, ∂ ν Φ)d 4 x 15.15 Unification and quantum gravity The strength of the forces varies with energy and the reciprocal coupling constants approach each other with increasing energy. The SU(5) model predicts complete unification of the electromagnetical, weak and colour forces at 10 15 GeV. It also predicts 12 extra X bosons which couple leptons and quarks and are i.g. responsible for proton decay, with dominant channel p + → π 0 + e + , with an average lifetime of the proton of 10 31 year. This model has been experimentally falsified. Supersymmetric models assume a symmetry between bosons and fermions and predict partners for the cur- rently known particles with a spin which differs 1 2 . The supersymmetric SU(5) model predicts unification at 10 16 GeV and an average lifetime of the proton of 10 33 year. The dominant decay channels in this theory are p + →K + +ν µ and p + →K 0 +µ + . Quantum gravity plays only a role in particle interactions at the Planck dimensions, where λ C ≈ R S : m Pl = hc/G = 3 10 19 GeV, t Pl = h/m Pl c 2 = hG/c 5 = 10 −43 sec and r Pl = ct Pl ≈ 10 −35 m. Chapter 16 Astrophysics 16.1 Determination of distances The parallax is mostly used to determine distances in nearby space. The parallax is the angular difference between two measurements of the position of the object from different view-points. If the annual parallax is given by p, the distance R of the object is given by R = a/ sin(p), in which a is the radius of the Earth’s orbit. The clusterparallax is used to determine the distance of a group of stars by using their motion w.r.t. a fixed background. The tangential velocity v t and the radial velocity v r of the stars along the sky are given by v r = V cos(θ) , v t = V sin(θ) = ωR where θ is the angle between the star and the point of convergence and ˆ R the distance in pc. This results, with v t = v r tan(θ), in: R = v r tan(θ) ω ⇒ ˆ R = 1 p where p is the parallax in arc seconds. The parallax is then given by p = 4.74µ v r tan(θ) RR-Lyrae Type 2 Type 1 0,1 0,3 1 3 10 30 100 1 0 -1 -2 -3 -4 -5 P (days) → 'M` with µ de proper motion of the star in /yr. A method to determine the distance of objects which are somewhat further away, like galaxies and star clusters, uses the period-Brightness relation for Cepheids. This relation is shown in the above figure for different types of stars. 16.2 Brightness and magnitudes The brightness is the total radiated energy per unit of time. Earth receives s 0 = 1.374 kW/m 2 from the Sun. Hence, the brightness of the Sun is given by L = 4πr 2 s 0 = 3.82 10 26 W. It is also given by: L = 4πR 2 ∞ 0 πF ν dν where πF ν is the monochromatic radiation flux. At the position of an observer this is πf ν , with f ν = (R/r) 2 F ν if absorption is ignored. If A ν is the fraction of the flux which reaches Earth’s surface, the transmission factor is given by R ν and the surface of the detector is given by πa 2 , then the apparent brightness b is given by: b = πa 2 ∞ 0 f ν A ν R ν dν The magnitude m is defined by: b 1 b 2 = (100) 1 5 (m2−m1) = (2.512) m2−m1 Chapter 16: Astrophysics 97 because the human eye perceives lightintensities logaritmical. From this follows that m 2 − m 1 = 2.5 10 log(b 1 /b 2 ), or: m = −2.5 10 log(b) + C. The apparent brightness of a star if this star would be at a distance of 10 pc is called the absolute brightness B: B/b = (ˆ r/10) 2 . The absolute magnitude is then given by M = −2.5 10 log(B) +C, or: M = 5 +m−5 10 log(ˆ r). When an interstellar absorption of 10 −4 /pc is taken into account one finds: M = (m−4 10 −4 ˆ r) +5 −5 10 log(ˆ r) If a detector detects all radiation emitted by a source one would measure the absolute bolometric magnitude. If the bolometric correction BC is given by BC = 2.5 10 log Energy flux received Energy flux detected = 2.5 10 log f ν dν f ν A ν R ν dν holds: M b = M V −BC where M V is the visual magnitude. Further holds M b = −2.5 10 log L L +4.72 16.3 Radiation and stellar atmospheres The radiation energy passing through a surface dA is dE = I ν (θ, ϕ) cos(θ)dνdΩdAdt, where I µ is the monochromatical intensity [Wm −2 sr −1 Hz −1 ]. When there is no absorption the quantity I ν is independent of the distance to the source. Planck’s law holds for a black body: I ν (T) ≡ B ν (T) = c 4π w ν (T) = 2hν 3 c 2 1 exp(hν/kT) −1 The radiation transport through a layer can then be written as: dI ν ds = −I ν κ ν +j ν Here, j ν is the coefficient of emission and κ ν the coefficient of absorption. ds is the thickness of the layer. The optical thickness τ ν of the layer is given by τ ν = κ ν ds. The layer is optically thin if τ ν < 1, the layer is optically thick if τ ν 1. For a stellar atmosphere in LTE holds: j ν = κ ν B ν (T). Then also holds: I ν (s) = I ν (0)e −τν +B ν (T)(1 −e −τν ) 16.4 Composition and evolution of stars The structure of a star is described by the following equations: dM(r) dr = 4π(r)r 2 dp(r) dr = − GM(r)(r) r 2 L(r) dr = 4π(r)ε(r)r 2 dT(r) dr rad = − 3 4 L(r) 4πr 2 κ(r) 4σT 3 (r) , (Eddington), or dT(r) dr conv = T(r) p(r) γ −1 γ dp(r) dr , (convective energy transport) Further, for stars of the solar type, the composing plasma can be described as an ideal gas: p(r) = (r)kT(r) µm H 98 Physics Formulary by ir. J.C.A. Wevers where µ is the average molecular mass, usually well approximated by: µ = nm H = 1 2X + 3 4 Y + 1 2 Z where X is the mass fraction of H, Y the mass fraction of He and Z the mass fraction of the other elements. Further holds: κ(r) = f((r), T(r), composition) and ε(r) = g((r), T(r), composition) Convection will occur when the star meets the Schwartzschild criterium: dT dr conv < dT dr rad Otherwise the energy transfer takes place by radiation. For stars in quasi-hydrostatic equilibrium hold the approximations r = 1 2 R, M(r) = 1 2 M, dM/dr = M/R, κ ∼ and ε ∼ T µ (this last assumption is only valid for stars on the main sequence). For pp-chains holds µ ≈ 5 and for the CNO chains holds µ = 12 tot 18. It can be derived that L ∼ M 3 : the mass-brightness relation. Further holds: L ∼ R 4 ∼ T 8 eff . This results in the equation for the main sequence in the Hertzsprung-Russel diagram: 10 log(L) = 8 10 log(T eff ) +constant 16.5 Energy production in stars The net reaction from which most stars gain their energy is: 4 1 H → 4 He +2e + +2ν e +γ. This reaction produces 26.72 MeV. Two reaction chains are responsible for this reaction. The slowest, speed- limiting reaction is shown in boldface. The energy between brackets is the energy carried away by the neutrino. 1. The proton-proton chain can be divided into two subchains: 1 H+p + → 2 D+e + +ν e , and then 2 D + p → 3 He +γ. I. pp1: 3 He + 3 He →2p + + 4 He. There is 26.21 + (0.51) MeV released. II. pp2: 3 He +α → 7 Be +γ i. 7 Be +e − → 7 Li +ν, then 7 Li +p + →2 4 He +γ. 25.92 + (0.80) MeV. ii. 7 Be +p + → 8 B +γ, then 8 B +e + →2 4 He +ν. 19.5 + (7.2) MeV. Both 7 Be chains become more important with raising T. 2. The CNO cycle. The first chain releases 25.03 + (1.69) MeV, the second 24.74 + (1.98) MeV. The reactions are shown below. −→ ` → 15 N + p + →α + 12 C 15 N + p + → 16 O+γ ↓ ↓ 15 O+e + → 15 N +ν 12 C +p + → 13 N +γ 16 O+p + → 17 F +γ ↑ ↓ ↓ 14 N+p + → 15 O+γ 13 N → 13 C +e + +ν 17 F → 17 O+e + +ν ↓ ↓ ` ← 13 C +p + → 14 N +γ 17 O+p + →α + 14 N ←− The ∇ operator 99 The ∇-operator In cartesian coordinates (x, y, z) holds: ∇ = ∂ ∂x e x + ∂ ∂y e y + ∂ ∂z e z , gradf = ∇f = ∂f ∂x e x + ∂f ∂y e y + ∂f ∂z e z div a = ∇ a = ∂a x ∂x + ∂a y ∂y + ∂a z ∂z , ∇ 2 f = ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 + ∂ 2 f ∂z 2 rot a = ∇a = ∂a z ∂y − ∂a y ∂z e x + ∂a x ∂z − ∂a z ∂x e y + ∂a y ∂x − ∂a x ∂y e z In cylinder coordinates (r, ϕ, z) holds: ∇ = ∂ ∂r e r + 1 r ∂ ∂ϕ e ϕ + ∂ ∂z e z , gradf = ∂f ∂r e r + 1 r ∂f ∂ϕ e ϕ + ∂f ∂z e z div a = ∂a r ∂r + a r r + 1 r ∂a ϕ ∂ϕ + ∂a z ∂z , ∇ 2 f = ∂ 2 f ∂r 2 + 1 r ∂f ∂r + 1 r 2 ∂ 2 f ∂ϕ 2 + ∂ 2 f ∂z 2 rot a = 1 r ∂a z ∂ϕ − ∂a ϕ ∂z e r + ∂a r ∂z − ∂a z ∂r e ϕ + ∂a ϕ ∂r + a ϕ r − 1 r ∂a r ∂ϕ e z In spherical coordinates (r, θ, ϕ) holds: ∇ = ∂ ∂r e r + 1 r ∂ ∂θ e θ + 1 r sinθ ∂ ∂ϕ e ϕ gradf = ∂f ∂r e r + 1 r ∂f ∂θ e θ + 1 r sin θ ∂f ∂ϕ e ϕ div a = ∂a r ∂r + 2a r r + 1 r ∂a θ ∂θ + a θ r tanθ + 1 r sinθ ∂a ϕ ∂ϕ rot a = 1 r ∂a ϕ ∂θ + a θ r tanθ − 1 r sinθ ∂a θ ∂ϕ e r + 1 r sin θ ∂a r ∂ϕ − ∂a ϕ ∂r − a ϕ r e θ + ∂a θ ∂r + a θ r − 1 r ∂a r ∂θ e ϕ ∇ 2 f = ∂ 2 f ∂r 2 + 2 r ∂f ∂r + 1 r 2 ∂ 2 f ∂θ 2 + 1 r 2 tanθ ∂f ∂θ + 1 r 2 sin 2 θ ∂ 2 f ∂ϕ 2 General orthonormal curvelinear coordinates (u, v, w) can be obtained fromcartesian coordinates by the trans- formation x = x(u, v, w). The unit vectors are then given by: e u = 1 h 1 ∂x ∂u , e v = 1 h 2 ∂x ∂v , e w = 1 h 3 ∂x ∂w where the factors h i set the norm to 1. Then holds: gradf = 1 h 1 ∂f ∂u e u + 1 h 2 ∂f ∂v e v + 1 h 3 ∂f ∂w e w div a = 1 h 1 h 2 h 3 ∂ ∂u (h 2 h 3 a u ) + ∂ ∂v (h 3 h 1 a v ) + ∂ ∂w (h 1 h 2 a w ) rot a = 1 h 2 h 3 ∂(h 3 a w ) ∂v − ∂(h 2 a v ) ∂w e u + 1 h 3 h 1 ∂(h 1 a u ) ∂w − ∂(h 3 a w ) ∂u e v + 1 h 1 h 2 ∂(h 2 a v ) ∂u − ∂(h 1 a u ) ∂v e w ∇ 2 f = 1 h 1 h 2 h 3 ¸ ∂ ∂u h 2 h 3 h 1 ∂f ∂u + ∂ ∂v h 3 h 1 h 2 ∂f ∂v + ∂ ∂w h 1 h 2 h 3 ∂f ∂w 100 The SI units The SI units Basic units Quantity Unit Sym. Length metre m Mass kilogram kg Time second s Therm. temp. kelvin K Electr. current ampere A Luminous intens. candela cd Amount of subst. mol mol Extra units Plane angle radian rad solid angle sterradian sr Derived units with special names Quantity Unit Sym. Derivation Frequency hertz Hz s −1 Force newton N kg m s −2 Pressure pascal Pa N m −2 Energy joule J N m Power watt W J s −1 Charge coulomb C A s El. Potential volt V W A −1 El. Capacitance farad F C V −1 El. Resistance ohm Ω V A −1 El. Conductance siemens S A V −1 Mag. flux weber Wb V s Mag. flux density tesla T Wb m −2 Inductance henry H Wb A −1 Luminous flux lumen lm cd sr Illuminance lux lx lm m −2 Activity bequerel Bq s −1 Absorbed dose gray Gy J kg −1 Dose equivalent sievert Sv J kg −1 Prefixes yotta Y 10 24 giga G 10 9 deci d 10 −1 pico p 10 −12 zetta Z 10 21 mega M 10 6 centi c 10 −2 femto f 10 −15 exa E 10 18 kilo k 10 3 milli m 10 −3 atto a 10 −18 peta P 10 15 hecto h 10 2 micro µ 10 −6 zepto z 10 −21 tera T 10 12 deca da 10 nano n 10 −9 yocto y 10 −24 c 1995, 2009 J.C.A. Wevers Dear reader, Version: December 16, 2009 A This document contains a 108 page L TEX file which contains a lot equations in physics. It is written at advanced undergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics and often needs to look up equations. This, and a Dutch version of this file, ([email protected]). can be obtained from the author, Johan Wevers It can also be obtained on the WWW. See http://www.xs4all.nl/˜johanw/index.html, where also a Postscript version is available. If you find any errors or have any comments, please let me know. I am always open for suggestions and possible corrections to the physics formulary. A A The Physics Formulary is made with teTEX and LTEX version 2.09. It can be possible that your L TEX version has problems compiling the file. The most probable source of problems would be the use of large bezier curves and/or emTEX specials in pictures. If you prefer the notation in which vectors are typefaced in boldface, uncomment the redefinition of the \vec command in the TEX file and recompile the file. This work is licenced under the Creative Commons Attribution 3.0 Netherlands License. To view a copy of this licence, visit http://creativecommons.org/licenses/by/3.0/nl/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. Johan Wevers Contents Contents Physical Constants 1 Mechanics 1.1 Point-kinetics in a fixed coordinate system . . . . . . . . 1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . 1.1.2 Polar coordinates . . . . . . . . . . . . . . . . . 1.2 Relative motion . . . . . . . . . . . . . . . . . . . . . . 1.3 Point-dynamics in a fixed coordinate system . . . . . . . 1.3.1 Force, (angular)momentum and energy . . . . . 1.3.2 Conservative force fields . . . . . . . . . . . . . 1.3.3 Gravitation . . . . . . . . . . . . . . . . . . . . 1.3.4 Orbital equations . . . . . . . . . . . . . . . . . 1.3.5 The virial theorem . . . . . . . . . . . . . . . . 1.4 Point dynamics in a moving coordinate system . . . . . 1.4.1 Apparent forces . . . . . . . . . . . . . . . . . . 1.4.2 Tensor notation . . . . . . . . . . . . . . . . . . 1.5 Dynamics of masspoint collections . . . . . . . . . . . . 1.5.1 The centre of mass . . . . . . . . . . . . . . . . 1.5.2 Collisions . . . . . . . . . . . . . . . . . . . . . 1.6 Dynamics of rigid bodies . . . . . . . . . . . . . . . . . 1.6.1 Moment of Inertia . . . . . . . . . . . . . . . . 1.6.2 Principal axes . . . . . . . . . . . . . . . . . . . 1.6.3 Time dependence . . . . . . . . . . . . . . . . . 1.7 Variational Calculus, Hamilton and Lagrange mechanics 1.7.1 Variational Calculus . . . . . . . . . . . . . . . 1.7.2 Hamilton mechanics . . . . . . . . . . . . . . . 1.7.3 Motion around an equilibrium, linearization . . . 1.7.4 Phase space, Liouville’s equation . . . . . . . . 1.7.5 Generating functions . . . . . . . . . . . . . . . 2 Electricity & Magnetism 2.1 The Maxwell equations . . . . . . . . . . 2.2 Force and potential . . . . . . . . . . . . 2.3 Gauge transformations . . . . . . . . . . 2.4 Energy of the electromagnetic field . . . . 2.5 Electromagnetic waves . . . . . . . . . . 2.5.1 Electromagnetic waves in vacuum 2.5.2 Electromagnetic waves in matter . 2.6 Multipoles . . . . . . . . . . . . . . . . . 2.7 Electric currents . . . . . . . . . . . . . . 2.8 Depolarizing field . . . . . . . . . . . . . 2.9 Mixtures of materials . . . . . . . . . . . I 1 2 2 2 2 2 2 2 3 3 3 4 4 4 5 5 5 5 6 6 6 6 6 6 7 7 7 8 9 9 9 10 10 10 10 11 11 11 12 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . . . . . . . . . . .2. . . . . . . . . . .C. . . .3 The stationary phase method . . . . 6. . . . . . . . . . . . 3. . . . . . . . . . . 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Green functions for the initial-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . 5. . . .2. . 4. . . . . . . . . .6 Cosmology . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . .1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . .A. . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Paraxial geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Statistical physics 7. . . . . . . . . . .9 Special optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1 The bending of light . 6. 13 13 13 14 14 14 14 15 16 17 17 17 18 18 18 18 19 19 19 20 20 20 20 21 21 21 21 22 22 23 24 24 24 24 25 25 25 26 26 26 27 27 28 28 29 30 30 30 31 31 32 . . . . .2 Mechanic oscillations . . . . . . . . .3 The stress-energy tensor and the field tensor 3. . . . . . . . . . . . . . . . 3. . .1. . . . . .3 Principal planes . . . . . . .2. . . . . . . . . . . .II Physics Formulary by ir.4 Waves in long conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . 5. . . . . . . . . . . . . . . . . . . . . . . . . . Wevers 3 Relativity 3. . . . . . . . . . . . . . . . .2 The line element . . . . . . . . . . . . . . . . . . . . . . . .1 Plane waves . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . .2 Solutions of the wave equation . . . . . .5 Coupled conductors and transformers 4. . . . . . . . . . . . . . . . . . 6. . . . . . .1. . .3 Cylindrical waves . . . . . . . . . . . . . . . . . .3 Electric oscillations . . . . . . . . . . . . . .3 Matrix methods . . . . . . . . . .4 The equation of state . . . . . . . . . . . . . .5 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Pressure on a wall . . . . . . . . . . . . . . . .1.10 The Fabry-Perot interferometer . . . . . . . . . . . . . . . . 5. . .5 Reflection and transmission .2. . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Waves 5. . . . . . . 6. . . . . 4. . . . . . . . . . . . . . . . .2. . . . . . .4 Aberrations . . . . . . . . . . . . . . . . . . .2 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. . . . . . . . . . . . . .1 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Oscillations 4. . . . . . . . . . . . .2 General relativity . . . .1 The wave equation . . . . .1 Harmonic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 The general solution in one dimension 5. . . . . . . . . 5. . . 5. . . . . . . . . . . . . . . . . . . . . . . . 3. . . . .2 Red and blue shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 Diffraction . 6 Optics 6. . . . .5 Waveguides and resonating cavities . . . . . . . . . . . . . . .4 Magnification . . . . . . 3. . . . . . . . . . . . . . . . . . . . 4. . . 4. . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Mirrors . . . . . . . . . . . . . . . . . . . . . . . .1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . .6 Polarization . . . . . . . . . . .2 The energy distribution function 7. . . . . . . . . .2. . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . 7. . . . . . . 5. . . . .3 Planetary orbits and the perihelion shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Riemannian geometry. . . . . . . . . . . . . . the Einstein tensor . . . .5 Collisions between molecules . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Prisms and dispersion . . . . . . . . . . . 6. . . . . .1 Degrees of freedom .6 Non-linear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The trajectory of a photon . . . . . . . . . . 8. . . . . . . . . . . . . . 9. 9.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Turbulence . . . . 10. . . . . . . . . . . . . . . . . .1 Solutions . . . . . . . . . . . . . . . . . . . . . 10. . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. .1 Mathematical introduction . . . . 10 Quantum physics 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . . .10 Ideal mixtures . . . . . . . . . 8. . . . 9 Transport phenomena 9. . . . . . . . . . . . . . . 9. . . . . . . . . . . . 9. . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . .3 Operators in quantum physics . . . . . . . . . .14 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 Interaction with electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . 8. . . . . . . . . . .12. . . . . 8. . . . .12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 Self organization . . . . . . . . . .14. 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12. . .3 Electron diffraction . . .5 State functions and Maxwell relations 8.4 Characterising of flows by dimensionless numbers . . . . . . . . . . 10. . . . . . . . . 10. . . . . . . . . . . . . . . . . . . .3 Spin-orbit interaction . . . . . . . . 8. . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Wave functions .2 The Compton effect . . . . . . . . 10. . . . . . . . . . . . .12 Atomic physics . . . . . . . . . . . . . . o 10.1 Introduction to quantum physics . . . . . . .1 Time-independent perturbation theory 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Potential theory . 10. . . . . . . . . . . .2 Eigenvalue equations . . . . . . . . 8. . . . .7 Maximal work . . . . . . . . . . . . . . . . . . . . . .4 The uncertainty principle . . . . .14. . . . . . . . . . . . . . . . . . . . . .10 Spin . . . . Wevers III 7. . . . . 8. . . . . . . . . . . . . . . . . . . . 9. . . . . . . . 10. . .7 The tunnel effect . . . . . . . 10. . . . . . . . . . . . .3 Bernoulli’s equations . . . . . . . . . 32 33 33 33 33 34 34 35 36 36 37 37 37 38 38 39 39 39 41 41 42 42 43 43 43 43 44 44 45 45 45 45 45 45 45 46 46 46 47 47 47 48 48 49 49 49 49 50 50 50 50 51 8 Thermodynamics 8. . . . . . .A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. . . . . . . .4 Selection rules . . . . . . . . . . . . . . . . . . 8. . . . . . 10. . . . . . . .6 Interaction between molecules . . . . . . .3 Thermal heat capacity . . . . . .2 Time-dependent perturbation theory . .7 Boundary layers . . . . . . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. . . . . . . . . . . . . . . .2 Conservation laws . . . . .7. . . . . . . . . . .12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. .1 Black body radiation . . . . . . . . . . . . . . . . . . . . . .9 Angular momentum . . . . . . . . . . . . . . . . . . . . .6 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Thermodynamic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 Phase transitions . . . . . . . . . . . . . . . . . . . . .8 Heat conductance . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . .1 Mathematical introduction . . . . . . . . . . . . J. . . .1 Flow boundary layers . . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Definitions . . . . . . .8 The harmonic oscillator . . . . . . . . . . . . . . .5 The Schr¨ dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 The Dirac formalism . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C. . . 9. . . . . . . . . . . . . . . . . . . .11 Conditions for equilibrium . . . . . . . . . . .5 Tube flows . . . . . . . . . . . . . . . . . . . .4 The laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. . . . . . . 9. . . . . . . . . . . . . . . . . . . . . . . . .6 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . .Physics Formulary by ir. .13 Application to other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 Statistical basis for thermodynamics . . . . .2 Temperature boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. . . . . . . . . . . . . . . . . . . .5 The London equation . . . . . . . . . . . . . . . . . . .15 N-particle systems 10. . . . . . . . . . . . . . . . .8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C.4 Thermal heat capacity . . . . . . . . . . . 12. . . 12. .1 Description . . . . . . . . . . . . . . . . . .5 Scattering of light . . . . . . . . . . . . . . . . . . 12. . . . . . . . . . . . . 11. . . . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . .8. . . . 12. . . . . . . . . .3. . . . .2 The Coulomb interaction . . . . . . . . . . . . . . . .5. 11. . . . . . . 12 Solid state physics 12. . . 11. . .3. . . . . . . . .16 Quantum statistics . . .15. . . . . . . . .3 The Hall-effect . . . . . . . . . 11. . . . . . . . . . . . . . . . . . . . . . . . . . .5 Free electron Fermi gas . . . . . . . . . . . .2 Crystal binding . . . . . 12. . . . . . . .6 The BCS model . . . . .3. . . . . . . . . . . . . . . . . . .1 Thermal heat capacity . . . . 12. . . . . . . . . . . . . . . . . . . .2 The Josephson effect . . 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Crystal structure . . .5. . . 12. . . . . . . . . . . . .4 Magnetic field in the solid state . . . . . . . . . . . 10. . . . .4 Macroscopic quantum interference . . Wevers 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. . . . . . .1 General . . . . . .4. . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . .2 Transport . . . . . . . . . . 12. . . . . . . . . . . . . . . . 12. . . . . . . .4 Thermodynamic equilibrium and reversibility 11. . . . . . 12. . . 12. . . . . . . . . . . . . . 12. . . . . . . . . . . . . . . .1 Dielectrics . . . . . . . . . . . . . . . . . . . . 11. . . . . . . . 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8. . 12. . . . . . . . . . . . . . . . . . . .8 Superconductivity . . . . . . . 12. . . . . . . . . . . . . . . . . . . . . . .1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. . .2 Molecules 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Inelastic collisions . . . . . . J. . . . . . . . . . . . . . . . . . . . . . . . . .2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Energy bands . . .5. . . . . . . . . . . . . . . . 12. . . . . . . . . . . . . . . . . . . . . .4. . . . . . . . . . . . 12. . 11. . . . 51 51 52 52 54 54 54 55 55 56 56 56 56 57 57 57 58 58 59 60 60 62 62 62 63 63 63 63 64 65 65 65 65 66 66 66 67 67 67 67 68 68 69 69 70 70 70 11 Plasma physics 11. . . . . . . . . . . . . . . . . . . . . . . . . . .1 Types of collisions . . . 11. . . . . . . 11. . . . . . . . . . . . 11. . . . . . . . .5. . . . . . . . . . . . . . . . . . . . .3. . . . . . . . . . . . .3. 11. . . . . . . . . . . . . . . . . . .2 Paramagnetism .4 The centre of mass system . . . . . . . . . . . . . . . . . .3 Crystal vibrations . . . . .15. . . .3 Flux quantisation in a superconducting ring 12. . . . . . 11. . . .3. . . . . . . . . . . . .1 A lattice with one type of atoms . . . . . . . . .7 Semiconductors . . . . . . . . . . . . . . . . . . . 12. . . . . . . .4 Thermal heat conductivity . . . . . 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Waves in plasma’s . . . . . 12. . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Radiation . . . . . . . . . . . . . . . . . . . .3 The induced dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . 12. . . . . . . . . . . . . . . . .A. . . . . . .2 Electric conductance . . . . . . . . . .7 The Boltzmann transport equation . . .8. . . .2 A lattice with two types of atoms . . . . . . . . . . . . . . .8 Collision-radiative models . . . . . . . . . . . . . . . . . . .3 Elastic collisions . . . . . . . . . . . . . . . 12. .IV Physics Formulary by ir.5. . . . . . . . . . . . . . . . .8. . . . . . . . . . . . . . .3 Ferromagnetism . . .5. . .3 Phonons . . . . . . . .8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. . . . . . . . . . . .1. . . . . 13. . . .4 Isomorfism and homomorfism. . . .6. . . . . . . . . . . .1 Vectormodel for the addition of angular momentum . . . . . . . . . . . . . . . . . . .2 The Cayley table . . . . . . . . . . . .1 Representations. . . . . . . . . .C. . . . . . . 13. . . . . . . . . . . .3. . . . . . 13. . . . . . . . . 13. . . . . 15 Quantum field theory & Particle physics 15. . . . .6 Applications to quantum mechanics . . . . . . . . . . . . . . . .5 Radiation dosimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 The relation with quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 15. . . . 71 71 71 71 71 72 72 72 72 72 72 73 73 73 73 74 74 74 75 75 75 75 76 77 77 77 78 79 81 81 82 82 83 83 83 84 84 85 85 85 86 86 87 87 88 89 89 90 90 92 93 93 94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 13. . .2 particles . . . . . . . . . . .4 The direct product of representations . . . . . . . .3 Conjugated elements.3 The interaction picture . . . . . . . . . . . . .1. 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . . . . . . . . . .6 Symmetric transformations of operators. . . . . . . . . . . . . . . . .2 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. . energy levels and degeneracy . . . . . . . . . . . . . 15. . . . . . . . . . subgroups and classes . . . . . . . . . . . .4. . . . . .A. . . .4. . . .3. . . . . .5 Reducible and irreducible representations . . . matrixelements and selection rules . 14. . . . . . . . . . . . . . . . . .12 P and CP-violation . . . .4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. 13. . 13.2 Irreducible tensor operators. . . . . . . . . . . . . . . .7 Applications to particle physics . . . . . . . . . . 14. . . . . . .Physics Formulary by ir. . . . . .1 Nuclear forces . . 13. . .6. . . . . . . . . . . . . . . . . . . . . . . . . .2 Breaking of degeneracy by a perturbation . . . .1. . . . . . . . . . . . . . . 1 15.4. . . . . . . . . . . . . . . 15. . . . . . . . . .3 Conservation of energy and momentum in nuclear reactions 14. . . . . . . . representations .13. . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . .3 Character . . . . . . . . . . . . . 13. . . . . . 14. . . . 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Kinetic model . . . . . . .3 The construction of a base function . . . . . . . . . . .2. . . . . . . . . . . . . 13. . . . . .7 Quantization of spin. . . . . . . . . . . . . . . . . . . . . . . . 15. . 15.3 Radioactive decay . . . . . . . .11 Classification of elementary particles . . . . . . . . . . . . . . . . . .2 The fundamental orthogonality theorem .3.1 Definition of a group . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . .10 Divergences and renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . .2 Quantum mechanical model for n-p scattering . . . . . . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . . . . .1 Creation and annihilation operators . . . . . . . .5 The group SO(3) . . . . . . . . . . . . . . . . . . . . . . . 14 Nuclear physics 14. . .3. . J.4. . . . . . . . . . 15. . . 14. . . . . . . . . . . . . . . . . . . . . . . . .8 Quantization of the electromagnetic field .1 The 3-dimensional translation group . . . . . 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. . . . . 13. . . . . . . . . . . . . . 15. . . . . . . . . . . . . . . . .5 Clebsch-Gordan coefficients .4 Real scalar field in the interaction picture . . . .1 The electroweak theory . . . . . . . . . .1 Schur’s lemma .9 Interacting fields and the S-matrix . . . . 13. irreducible tensor operators . . . . . . . . . . . . . . . . . . . . .3. . . . . . . .2 The shape of the nucleus . . . . . . . .1 Introduction . . . . . . . . . . . . . . . . . . . . . . 15. . 14. . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conservation of charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Spontaneous symmetry breaking: the Higgs mechanism . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . . . 1 15. . . . . . .3 Properties of continuous groups . . . 13.5 Charged spin-0 particles. . . . . . .2 The fundamental orthogonality theorem . . . .13 The standard model . . . .4 Scattering and nuclear reactions . . . . . . . . . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . . . . 15. . . . . . . . . . . . . . . . . . . . . . . Wevers V 13 Theory of groups 13. . . . . . . . . . .3. 14. . . . . . . . . . . . . . . . . .1. . . . . .7 The Wigner-Eckart theorem . . . . . . .4 Continuous groups . . . . 15. . . . . . . . . . . . . .2 Classical and quantum fields . . . . . . . . . . . . . . . . . . . . . . 13. . . . . . . . . . . . . . . . . . .13. . . . . . . . . . . . . . . . . . . . . . . . . .6 Field functions for spin. . . . . . . . . . . . . . . .2 The 3-dimensional rotation group . . . . . . . . . . . . . . . . . . . 16. . . . . .A. . . . . . . . 16. . . . . . . . . . . . . . . . . . . . . . . . . 15. . . . . . . . .C. . J. . . 15. . . . .3 Quantumchromodynamics . . . . . . . . . . . . . . . .3 Radiation and stellar atmospheres 16. . . . . . . . . .1 Determination of distances . . .13. . . . . . . . . . 16 Astrophysics 16. . . . . . . . . . . The ∇-operator The SI units 94 95 95 96 96 96 97 97 98 99 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Composition and evolution of stars 16. . . . . . . . . . . . .14 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VI Physics Formulary by ir. . . .15 Unification and quantum gravity . . . . . . . . . . . . . . . .2 Brightness and magnitudes . . . . . . . . . .5 Energy production in stars . . . . . . . . . . . . . . . . . . . . . . . . Wevers 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2741 · 10−24 0.24219879 1.0221367 · 1023 1.378 · 106 5.674954 · 10−27 1.67032 · 10 −8 2.5772156649 k=1 Unit γ = lim n→∞ e G.60217733 · 10 −19 6.854187 · 10−12 4π · 10−7 8.6605656 · 10−27 5.0508 · 10−27 1392 · 106 1.14159265358979323846 2.67259 · 10 −11 ≈ 1/137 2.380658 · 10−23 9.71828182845904523536 1/k − ln(n) = 0.976 · 1024 23.31441 6.595 2.8978 · 10−3 8.38 6.6726231 · 10−27 1.99792458 · 10 8 8.0545727 · 10−34 9.6260755 · 10 −34 1.3214 · 10−15 9.4605 · 10 15 3. κ α = e 2 /2hcε0 c ε0 µ0 h h ¯ = h/2π µ B = e¯ /2me h a0 Ry λ Ce = h/me c λ Cp = h/mp c µH σ kW R NA k = R/N A me mp mn mu = µN 1.0857 · 1016 ≈ (75 ± 25) C m3 kg−1 s−2 m/s (def) F/m H/m Nm2 C−2 Js Js Am2 ˚ A eV m m kg Wm−2 K−4 mK J·mol −1 ·K−1 mol−1 J/K kg kg kg kg J/T m kg days m kg hours days m m m km·s −1 ·Mpc−1 1 12 12 m( 6 C) D M T RA MA TA Tropical year AU lj pc H .96 365.1093897 · 10−31 1.52918 13.2463 · 10−12 1.989 · 1030 25.4959787066 · 10 11 9.Physical Constants Name Number π Number e Euler’s constant Elementary charge Gravitational constant Fine-structure constant Speed of light in vacuum Permittivity of the vacuum Permeability of the vacuum (4πε0 )−1 Planck’s constant Dirac’s constant Bohr magneton Bohr radius Rydberg’s constant Electron Compton wavelength Proton Compton wavelength Reduced mass of the H-atom Stefan-Boltzmann’s constant Wien’s constant Molar gasconstant Avogadro’s constant Boltzmann’s constant Electron mass Proton mass Neutron mass Elementary mass unit Nuclear magneton Diameter of the Sun Mass of the Sun Rotational period of the Sun Radius of Earth Mass of Earth Rotational period of Earth Earth orbital period Astronomical unit Light year Parsec Hubble constant Symbol π e n Value 3.9876 · 109 6.1045755 · 10−31 5. z ).1. e˙θ = −θer ˙ ˙ ¨ The velocity and the acceleration are derived from: r = re r . ˙ ˙ ˙ x ¨ ¨ The following holds: s(t) = s0 + |v(t)|dt . a = (¨. v. So. ω2 ¨ Further holds: α = θ. means that the quantity is defined in a moving system of coordinates. z).3. y.1 Definitions The position r. (angular)momentum and energy Newton’s 2nd law connects the force on an object and the resulting acceleration of the object where the momentum is given by p = mv: F (r.r. the velocity v and the acceleration a are defined by: r = (x.1 Force. ρ= ds ds ds |k| 1.t. v(t) = v0 + a(t)dt When the acceleration is constant this gives: v(t) = v 0 + at and s(t) = s0 + v0 t + 1 at2 .1. In a moving system holds: v = vQ + v + ω × r and a = aQ + a + α × r + 2ω × v + ω × (ω × r ) with ω × (ω × r ) = −ω 2 r n 1.1 Point-kinetics in a fixed coordinate system 1.Chapter 1 Mechanics 1.2 Polar coordinates Polar coordinates are defined by: x = r cos(θ). 2 For the unit vectors in a direction ⊥ to the orbit e t and parallel to it e n holds: et = dr ˙ v v e˙t = et = en . v = (x. for the unit coordinate vectors holds: ˙ ˙ e˙r = θeθ .3 Point-dynamics in a fixed coordinate system 1. a point Q holds: r D = rQ + ω × vQ ˙ with QD = rD − rQ and ω = θ. en = |v| ds ρ |e˙t | For the curvature k and the radius of curvature ρ holds: k= d2 r det 1 dϕ = 2 = . y.2 Relative motion For the motion of a point D w. z). y . r(t) = r0 + v(t)dt . y = r sin(θ). t) = d(mv ) dv dm m=const dp = =m +v = ma dt dt dt dt . ˙ r ˙˙ 1. a = (¨ − rθ2 )er + (2r θ + rθ)eθ . v = rer + rθeθ . The following equation is valid: τ =− Hence. From Gauss law it then follows: ∇ 2 V = 4πG . delivered by a force.3. if F is conservative: W =constant. 1. T = −U with T = 1 mv 2 . . the conditions for a mechanical equilibrium are: ∂U ∂θ Fi = 0 and τi = 0. ˙ For the power P holds: P = W = F · v. the kinetic energy T and the potential energy ˙ ˙ U holds: W = T + U . For the total energy W .3. 1. |L| = mr2 ω. 1. and L = r × p = mv × r. except when the motion starts.4 Orbital equations If V = V (r) one can derive from the equations of Lagrange for φ the conservation of angular momentum: ∂L ∂V d = = 0 ⇒ (mr2 φ) = 0 ⇒ Lz = mr2 φ = constant ∂φ ∂φ dt For the radial position as a function of time can be found that: dr dt The angular equation is then: r 2 = L2 2(W − V ) − 2 2 m m r −1 φ − φ0 = 0 mr2 L L2 2(W − V ) − 2 2 m m r dr r −2 field = arccos 1 + 1 r 1 r0 − 1 r0 + km/L2 z If F = F (r): L =constant.Chapter 1: Mechanics 3 Newton’s 3rd law is given by: Faction = −Freaction . is A = 1 F · ds = 1 F cos(α)ds ˙ The torque τ is related to the angular momentum L: τ = L = r × F .3 Gravitation The Newtonian law of gravitation is (in GRT one also uses κ instead of G): Fg = −G m1 m2 er r2 The gravitational potential is then given by V = −Gm/r. The force of friction is usually proportional to the force perpendicular to the surface. if F ⊥ v then ∆T = 0 and U = 0. 2 The kick S is given by: S = ∆p = F dt 2 2 The work A.3. when a threshold has to be overcome: F fric = f · Fnorm · et . From this follows that ∇ × F = 0.2 Conservative force fields A conservative force can be written as the gradient of a potential: Fcons = −∇U . For such a force field also holds: r1 F · ds = 0 ⇒ U = U0 − r0 F · ds So the work delivered by a conservative force field depends not on the trajectory covered but only on the starting and ending points of the motion. t2 ).A. Orbits with an equal ε are of equal shape. 5. k < 0 and ε = 1: a parabole. J. k < 0 and ε = 0: a circle. Other combinations are not possible: the total energy in a repulsive force field is always positive so ε > 1. curved towards the centre of force.C. t2 ) = 2m Kepler’s 3rd law is. Kepler’s 2nd law is LC (t2 − t1 ) A(t1 . Wevers Kepler’s orbital equations In a force field F = kr −2 . a = = 2M 2 Gµ tot G µ Mtot a 1−ε 2W a is half the√ length of the long axis of the elliptical orbit in case the orbit is closed.5 The virial theorem The virial theorem for one particle is: mv · r = 0 ⇒ T = − 1 F · r = 2 The virial theorem for a collection of particles is: T = −1 2 particles 1 2 r dU dr = 1 n U if U = − 2 k rn Fi · ri + pairs Fij · rij These propositions can also be written as: 2E kin + Epot = 0. Centrifugal force: Fcf = mω 2 rn = −Fcp .3. or: x2 + y 2 = ( − εx)2 2W L2 k L2 .1 Apparent forces The total force in a moving coordinate system can be found by subtracting the apparent forces from the forces working in the reference frame: F = F − Fapp . 1. with T the period and M tot the total mass of the system: T2 4π 2 = 3 a GMtot 1.4. The different apparent forces are given by: 1.4 Physics Formulary by ir. Fcp = − mv 2 er r .4 Point dynamics in a moving coordinate system 1. The equation of the orbit is: r(θ) = with 1 + ε cos(θ − θ0 ) . Now. the orbits are conic sections with the origin of the force in one of the foci (Kepler’s 1st law). Half the length of the short axis is b = a . curved away from the centre of force. If the surface between the orbit covered between t 1 and t2 and the focus C around which the planet moves is A(t1 . k < 0 and 0 < ε < 1: an ellipse. 5 types of orbits are possible: = 1. Transformation of the origin: F or = −maa 2. k < 0 and ε > 1: a hyperbole. 4. Rotation: Fα = −mα × r 3. ε2 = 1 + 2 3 2 = 1 − . ε is the excentricity of the orbit. k > 0 and ε > 1: a hyperbole. 3. 2. Coriolis force: Fcor = −2mω × v 4. F12 = µu 1.r. the coordinates of the centre of mass are given by: R= m1 r1 + m2 r2 m1 + m2 ˙ With r = r1 − r2 .t.2 Collisions With collisions. with the reduced mass µ given by: 2 2 ˙ 1 1 1 = + µ m1 m2 The motion within and outside the centre of mass can be separated: ˙ ˙ Loutside = τoutside .t.1 The centre of mass ˙ The velocity w. dt dt d¯β x dt This leads to: Hence the Newtonian equation of motion m will be transformed into: m d2 xα dxβ dxγ + Γα βγ 2 dt dt dt The apparent forces are taken from he origin to the effect side in the way Γ α βγ 1. p S =constant and L w. the centre of mass R is given by v − R. Fext = mam .Chapter 1: Mechanics 5 1.5 Dynamics of masspoint collections 1.5.5. Further holds ∆ LC = CB × S. The changes in the relative velocities can be derived from: S = ∆p = 2 µ(vaft − vbefore ). The coordinates of the centre of mass are given by: rm = mi ri mi In a 2-particle system.4. where B are the coordinates of the collision and C an arbitrary other position. . Linside = τinside p = mvm . holds: p = mv m 2 is constant. the kinetic energy becomes: T = 1 Mtot R2 + 1 µr2 .r. and T = 1 mvm is constant. dt ∂ xβ dt ¯ The chain rule gives: d2 xα d dxα d = = 2 dt dt dt dt so: x ∂xα d¯β β dt ∂x ¯ = ¯ d¯β d x ∂xα d2 xβ + β dt2 ∂x ¯ dt dt ∂xα ∂ xβ ¯ ∂ 2 xα d¯γ x x ∂ ∂xα d¯γ d ∂xα = = β γ ∂ xβ dt dt ∂ x ¯ ∂x ¯ ¯ ∂ xβ ∂ xγ dt ¯ ¯ d2 xα ¯ x ∂xα d2 xβ ∂ 2 xα d¯γ = + β γ 2 β dt2 dt ∂x ¯ ∂ x ∂ x dt ¯ ¯ d2 xα = Fα dt2 = Fα dxβ dxγ .2 Tensor notation Transformation of the Newtonian equations of motion to x α = xα (x) gives: dxα ∂xα d¯β x = . B is constant. 1 Variational Calculus Starting with: b δ a L(q.6. axis ⊥ through end + b2 ) Rectangle.1 Moment of Inertia The angular momentum in a moving coordinate system is given by: L = Iω + Ln where I is the moment of inertia with respect to a central axis. The torque T is defined by: T = F × d. t)dt = 0 with δ(a) = δ(b) = 0 and δ ˙ du dx = d (δu) dx . which is given by: I= i mi ri 2 .t. m I I = 1 mR2 2 I = 1 µR2 2 I = 2 mR2 5 I = 1 ml2 3 I = ma 2 1.C.6. Object Cavern cylinder Disc.o.2 Principal axes Each rigid body has (at least) 3 principal axes which stand ⊥ to each other.6 Physics Formulary by ir.r. in the continuous case: I= Further holds: Li = I ij ωj . c.6. J. d L =τ −ω×L dt 1.m. For a principal axis holds: ∂I ∂I ∂I = = = 0 so Ln = 0 ∂ωx ∂ωy ∂ωz The following holds: ω k = −aijk ωi ωj with aijk = ˙ Ii − Ij if I1 ≤ I2 ≤ I3 . axis in plane disc through m Cavern sphere Bar.t.7.D = Iw.C + m(DM )2 if axis C axis D. Wevers 1.A. T = Wrot = 1 ωIij ei ej = 1 Iω 2 2 2 or.7 Variational Calculus. axis ⊥ through c.r. I I = mR 2 I = 1 mR2 4 I = 2 mR2 3 I= I= 1 2 12 ml 1 2 12 m(a Object Massive cylinder Halter Massive sphere Bar. Iii = Ii .3 Time dependence For torque of force τ holds: ¨ τ = Iθ . Iij = Iji = − k m V r n dV = 2 r n dm mk xi xj 2 Steiner’s theorem is: Iw.m. axis b thr. Ik 1. q. Hamilton and Lagrange mechanics 1. Rectangle. axis ⊥ plane thr.o.6 Dynamics of rigid bodies 1. ˙ 2 qi pi − L. {pi . ∂qi i ∂ ∂pi so ∇ · v = i ∂ ∂H ∂ ∂H − ∂qi ∂pi ∂pi ∂qi . If q i (t) = ai exp(iωt) is substituted. This leads to the eigenfrequencies of the problem: aT V ak 2 k 2 . qj } = 0. The general solution is a superposition if ωk = T ak M ak eigenvibrations.4 Phase space. With new coordinates 2 √ (θ. obtained by the canonical transformation x = 2I/mω cos(θ) and p = − 2Imω sin(θ). Liouville’s equation In phase space holds: ∇= i ∂ . } is the Poisson bracket: ∂A ∂B ∂A ∂B {A.7.Chapter 1: Mechanics 7 the equations of Lagrange can be derived: ∂L d ∂L = dt ∂ qi ˙ ∂qi When there are additional conditions applying to the variational problem δJ(u) = 0 of the type K(u) =constant. with inverse θ = arctan(−p/mωx) and I = p2 /2mω + 1 mωx2 it follows: H(θ. linearization For natural systems around equilibrium the following equations are valid: ∂V ∂qi = 0 . dt ∂pi ∂H dpi =− dt ∂qi Coordinates are canonical if the following holds: {q i . 1. V (q) = V (0) + Vik qi qk with Vik = 0 ∂2V ∂qi ∂qk 0 With T = 1 (Mik qi qk ) one receives the set of equations M q + V q = 0. I). 1. I) = ωI.7. {qi . B} = − ∂qi ∂pi ∂pi ∂qi i The Hamiltonian of a Harmonic oscillator is given by H(x.7. p) = p 2 /2m + 1 mω 2 x2 . 1. 2 The Hamiltonian of a charged particle with charge q in an external electromagnetic field is given by: H= 1 p − qA 2m 2 + qV This Hamiltonian can be derived from the Hamiltonian of a free particle H = p 2 /2m with the transformations p → p − q A and H → H − qV . ˙ ˙ ¨ 2 this set of equations has solutions if det(V − ω 2 M ) = 0.3 Motion around an equilibrium. This is elegant from a relativistic point of view: this is equivalent to the transformation of the momentum 4-vector p α → pα − qAα . In 2 ˙ If the used coordinates are canonical the Hamilton equations are the equations of motion for the system: ∂H dqi = . which make the Hamilton equations the equations of motion for the system. the new problem becomes: δJ(u) − λδK(u) = 0. pj } = δij where {. φ). The Hamiltonian is given by: H = ˙ ˙ dimensions holds: L = T − U = 1 m(r2 + r2 φ2 ) − U (r.2 Hamilton mechanics The Lagrangian is given by: L = T (qi ) − V (qi ). A gauge transformation on the potentials A α corresponds with a canonical transformation. pj } = 0. If the equilibrium is stable holds: ∀k that ω k > 0. Qi . this can be written as: { . the coordinates follow from: dt ∂K dPi =− dt ∂Qi ∂F1 ∂F1 ∂F1 . K =H+ ∂pi ∂Qi ∂t dF4 (pi . t) . the coordinates follow from: dt ˙ 3. . F3 and F4 are called generating functions. If −pi qi − H = Pi Qi − K(Pi . t) Pi = Pi (qi .8 Physics Formulary by ir. J. Pi = − . t) ˙ . Qi = . or: dt pdq = constant ∂ =0 ∂t 1. If pi qi − H = Pi Qi − K(Pi . K=H+ ∂qi ∂Pi ∂t dF3 (pi .5 Generating functions Starting with the coordinate transformation: Qi = Qi (qi . pi . Qi . ∂ t + ∇ · ( v ) = 0 holds. pi . t) + dt pi = ∂F2 ∂F2 ∂F2 . Qi = . t) + ˙ qi = − ∂F4 ∂F4 ∂F4 .7. t) − ˙ pi = dF1 (qi . Qi . H} + For an arbitrary quantity A holds: dA ∂A = {A. K =H+ ∂qi ∂Qi ∂t dF2 (qi . t) . Wevers If the equation of continuity. Pi . the coordinates follow from: dt 4.C. Qi . a distinction between 4 cases can be made: 1. If −pi qi − H = −Pi Qi − K(Pi . K=H+ ∂pi ∂Pi ∂t The functions F 1 . Pi . Qi . t) . Qi . the coordinates follow from: ˙ 2. H} + dt ∂t Liouville’s theorem can than be written as: d = 0 . t) one can derive the following Hamilton equations with the new Hamiltonian K: dQi ∂K = . If pi qi − H = −Pi Qi − K(Pi . dt ∂Pi Now.A. t) + ˙ qi = − ∂F3 ∂F3 ∂F3 . F2 . Pi = − . µr = 1 + χm .included (B · n )d2 A = 0 E · ds = − dΦ dt dΨ dt ∇ · D = ρfree ∇·B = 0 ∇×E =− ∂B ∂t ∂D ∂t H · ds = Ifree.Chapter 2 Electricity & Magnetism 2. also known as the law of Laplace. with χe = np2 0 3ε0 kT The magnetic field strength H. d l I and r points from d l to P : dBP = µ0 I dl × er 4πr2 If the current is time-dependent one has to take retardation into account: the substitution I(t) → I(t − r/c) has to be applied. εr = 1 + χe . the magnetization M and the magnetic flux density B depend on each other according to: B = µ0 (H + M ) = µ0 µr H. In here. M = m/Vol. 2 The potentials are given by: V 12 = − 1 E · ds and A = 1 B × r. P = p0 /Vol. The origin of this force is a relativistic transformation of the Coulomb force: FL = Q(v × B ) = l(I × B ). polarization P and electric field strength E depend on each other according to: D = ε0 E + P = ε0 εr E. with χm = µ0 nm2 0 3kT 2.included + For the fluxes holds: Ψ = (D · n )d2 A.2 Force and potential The force and the electric field between 2 point charges are given by: F12 = Q1 Q2 F er . The magnetic field in point P which results from an electric current is given by the law of Biot-Savart. Φ = ∇ × H = Jfree + (B · n )d2 A. 2 . Those can be written both as differential and integral equations: (D · n )d2 A = Qfree. E = 4πε0 εr r2 Q The Lorentzforce is the force which is felt by a charged particle that moves through a magnetic field. The electric displacement D.1 The Maxwell equations The classical electromagnetic field can be described by the Maxwell equations. A. If ρ = 0 and J = 0 holds V = 0 and follows A from ✷A = 0. Further. The fields can be derived from the potentials as follows: ∂A E = −∇V − . 16π 2 ε0 r2 c4 w t = p2 sin2 (θ)ω 4 ck 4 |p |2 0 . The radiation pressure p s is given by ps = (1 + R)|S |/c. t) = −f (r. 2. λ 2 J⊥ (r )eik·r d3 r The energy density of the electromagnetic wave of a vibrating dipole at a large distance is: w = ε0 E 2 = p2 sin2 (θ)ω 4 0 sin2 (kr − ωt) . . the freedom remains to apply a limiting condition.10 Physics Formulary by ir.1 Electromagnetic waves in vacuum The wave equation ✷Ψ(r. Lorentz-gauge: ∇ · A + 2 c ∂t ε0 ✷A = −µ0 J. where R is the coefficient of reflection. with c = (ε 0 µ0 )−1/2 : Ψ(r. Coulomb gauge: ∇ · A = 0. J. 1. Wevers Here. This results in a canonical transformation of the Hamiltonian. V (r ) = |r − r | 4πε k2 dP = dΩ 32π 2 ε0 c ρ(r ) A derivation via multipole expansion will show that for the radiated energy holds. B = ∇×A ∂t Further holds the relation: c 2 B = v × E. The irradiance is the time-averaged of the Poynting vector: I = | S | t .4 Energy of the electromagnetic field The energy density of the electromagnetic field is: dW = w = HdB + EdD dVol The energy density can be expressed in the potentials and currents as follows: wmag = 1 2 J · A d3 x . if d. This separates the differential equations for A and V : ✷V = − . wel = 1 2 ρV d3 x 2. t) has the general solution.5.3 Gauge transformations The potentials of the electromagnetic fields transform as follows when a gauge transformation is applied:   A = A − ∇f ∂f  V =V + ∂t so the fields E and B do not change. t − |r − r |/c) 3 d r 4π|r − r | exp(ik|r − r |) 3 d r |r − r | r: If this is written as: J(r. P = 2 ε r 2 c4 32π 0 12πε0 The radiated energy can be derived from the Poynting vector S: S = E × H = cW ev . t) = A(r ) exp(−iωt) with: A(r ) = µ 4π J(r ) exp(ik|r − r |) 3 1 d r . the freedom remains to apply a gauge transformation.5 Electromagnetic waves 2. t) = f (r.C. Two common choices are: ρ 1 ∂V = 0. t) = J(r ) exp(−iωt) and A(r. 2. 2. k1 = ρdV r cos(θ)ρdV 1 2 i 2 2 (3zi − ri ) • Quadrupole: l = 2. where ρ is the resistivity. The moment is: τ = µ × Bout 4πr3 r2 2. W = −µ × Bout |µ| = 2B −µ 3µ · r Magnetic field: B = − µ . are: ∇2 − εµ ∂2 µ ∂ − ∂t2 ρ ∂t E =0. 3p · r Q Electric field: E ≈ − p . If the material is a good conductor. . and W = −p · Eout . where e goes from ⊕ to .6 Multipoles Because 1 1 = |r − r | r ∞ 0 r r l Pl (cos θ) the potential can be written as: V = Q 4πε n kn rn For the lowest-order terms this results in: • Monopole: l = 0. If k is written in the form k := k + ik it follows that: 1 2 εµ k =ω 1+ 1+ 1 and k = ω (ρεω)2 1 2 εµ −1 + 1+ 1 (ρεω)2 This results in a damped wave: E = E exp(−k n · r ) exp(i(k n · r − ωt)). k = (1 + i) 2ρ 2.7 Electric currents The continuity equation for charge is: ∂ρ + ∇ · J = 0.Chapter 2: Electricity & Magnetism 11 2. The electric dipole: dipole moment: p = Qle. and F = (p · ∇)Eext .2 Electromagnetic waves in matter The wave equations in matter.5. The magnetic dipole: dipole moment: if r A: µ = I × (Ae⊥ ). with c mat = (εµ)−1/2 the lightspeed in matter. k 2 = 1. The torque is: τ = p × Eout 4πεr3 r2 √ 2. F = (µ · ∇)Bout 2 mv⊥ . the second from the conductance current. The electric current is given by: ∂t I= dQ = dt (J · n )d2 A For most conductors holds: J = E/ρ. the wave vanishes after approximately one wavelength. µω . ∇2 − εµ ∂2 µ ∂ − ∂t2 ρ ∂t B=0 give. after substitution of monochromatic plane waves: E = E exp(i(k ·r −ωt)) and B = B exp(i(k ·r −ωt)) the dispersion relation: iµω k 2 = εµω 2 + ρ The first term arises from the displacement current. k 0 = • Dipole: l = 1. this results in a self-inductance which opposes the original change: dI Vselfind = −L . the contact area will heat up or cool down. If a current flows through two different. If the medium has an ellipsoidal shape and one of the principal axes is parallel with the external field E0 or B0 then the depolarizing is field homogeneous. with 0 ≤ N ≤ 1. a long. If the current If the flux enclosed by a conductor changes this results in an induced voltage V ind = −N dt flowing through a conductor changes. The generated or removed heat is given by: W = Πxy It. The thermic voltage between 2 metals is given by: V = γ(T − T 0 ). For a NTC holds: R(T ) = C exp(−B/T ) where B and C depend only on the material. connecting conductors x and y. depending on the direction of the current: the Peltier effect. This effect can be amplified with semiconductors. J. where R0 = ρl/A. the field strength within and outside the material will change because the material will be polarized or magnetized. The energy contained within a coil is given by W = 1 LI 2 and L = µN 2 A/l.9 Mixtures of materials The average electric displacement in a material which is inhomogenious on a mesoscopic scale is given by: −1 φ2 (1 − x) where x = ε1 /ε2 . capacity is given by I = −C dt For most PTC resistors holds approximately: R = R 0 (1 + αT ).A. For a few limiting cases of an ellipsoid holds: a thin plane: N = 1.8 Depolarizing field If a dielectric material is placed in an electric or magnetic field. In = 0.7 mV/K. The current through a 2 dV . dt µN I where l is the length. For a capacitor holds: C = ε 0 εr A/d where d is the distance between the plates and A the surface of one plate. Kirchhoff’s equations apply: for a knot holds: along a closed path holds: Vn = In Rn = 0. a sphere: N = 1 . If a conductor encloses a flux Φ holds: Φ = LI. 2. thin bar: N = 0.2 − 0. 3 2. Edep = Emat − E0 = − NP ε0 Hdep = Hmat − H0 = −N M N is a constant depending only on the shape of the object placed in the field.12 Physics Formulary by ir. In an electrical net with only stationary currents. Wevers dΦ . 2 The capacity is defined by: C = Q/V . Further holds: φi εi −1 ≤ ε∗ ≤ i φi εi i . R the radius The magnetic induction within a coil is approximated by: B = √ 2 + 4R2 l and N the number of coils.C. The electric field strength between the plates is E = σ/ε 0 = Q/ε0 A where σ is the surface charge. For a Cu-Konstantane connection holds: γ ≈ 0. For a sphere holds: Φ = D = εE = ε∗ E where ε∗ = ε1 1 − Φ(ε∗ /ε2 ) 1 2 3 + 3 x. The accumulated energy is given by W = 1 CV 2 . The proper time τ is defined as: dτ 2 = ds2 /c2 . x = γ(x + vt ) xv xv t =γ t− 2 . For energy and momentum holds: W = m r c2 = γW0 . z = z and: x = γ(x − vt) . W = γ(W − vpx ) v2 − v1 v1 v2 1− 2 c .t. with 0 the quantities in a co-moving reference frame and r the quantities in a frame moving with velocity v w.1 The Lorentz transformation The Lorentz transformation (x . mass and time transform according to: ∆t r = γ∆t0 . t=γ t + 2 c c If v = vex holds: px = γ px − βW c . lr = l0 /γ. t). it. v = With β = v/c the electric field of a moving charge is given by: E= Q (1 − β 2 )er 4πε0 r2 (1 − β 2 sin2 (θ))3/2 The electromagnetic field transforms according to: E = γ(E + v × B ) .Chapter 3 Relativity 3. t = γ t − 2 |v|2 c 1 2 1 − v2 c The velocity difference v between two observers transforms according to: v = γ 1− v1 · v2 c2 −1 v2 + (γ − 1) v1 · v2 2 v1 − γv1 v1 If the velocity is parallel to the x-axis. mr = γm0 . so ∆τ = ∆t/γ. t ) = (x (x. B = γ B− v×E c2 Length.1. .r. t (x. t)) leaves the wave equation invariant if c is invariant: ∂2 ∂2 1 ∂2 ∂2 ∂2 ∂2 1 ∂2 ∂2 + 2+ 2− 2 2 = + + − 2 2 ∂x2 ∂y ∂z c ∂t ∂x 2 ∂y 2 ∂z 2 c ∂t This transformation can also be found when ds 2 = ds 2 is demanded. The general form of the Lorentz transformation is given by: x =x+ where γ= (γ − 1)(x · v )v x·v − γvt . this becomes y = y.1 Special relativity 3. 1.2. and pc = W β where β = v/c. also ⊥ to the direction of motion.1. The force is defined by 0 F = dp/dt. Redshift because the universe expands. The 4-vector for energy and momentum is given by: pα = m0 U α = (γpi . iW/c).g. the Einstein tensor The basic principles of general relativity are: 1. κM ∆f = 2. resulting in e. 0 3. icγ). f rc 2. for particles with zero restmass (so with v = c) holds: U α Uα = 0. 4-vectors have the property that their modulus is independent of the observer: their components can change after a coordinate transformation but not their modulus. So: pα pα = −m2 c2 = p2 − W 2 /c2 .3 The stress-energy tensor and the field tensor The stress-energy tensor is given by: Tµν = ( c2 + p)uµ uν + pgµν + 1 α Fµα Fν + 1 gµν F αβ Fαβ 4 c2 The conservation laws can than be written as: ∇ ν T µν = 0. The Maxwell equations can than be written as: ∂ν F µν = µ0 J µ . the cosmic background radiation: R0 λ0 = .C. For particles with nonzero restmass holds: U α Uα = −c2 . iV /c) and Jµ := (J.14 Physics Formulary by ir. The geodesic postulate: free falling particles move along geodesics of space-time with the proper time τ or arc length s as parameter. ∂λ Fµν + ∂µ Fνλ + ∂ν Fλµ = 0 The equations of motion for a charged particle in an EM field become with the field tensor: dpα = qFαβ uβ dτ 3.A. Wevers W 2 = m2 c4 + p2 c2 . The difference of two 4-vectors transforms also as dxα . Motion: with e v · er = cos(ϕ) follows: v cos(ϕ) f =γ 1− .and blueshift. The electromagnetic field tensor is given by: Fαβ = ∂Aβ ∂Aα − α ∂x ∂xβ with Aµ := (A. For particles with zero rest mass (photons). The relation with the “common” velocity a 4-vector. the use of a free parameter is required because for them holds ds = 0. Gravitational redshift: 3. f c This can give both red. λ1 R1 3. The 4-vector for the velocity is given by U α = dτ i i α i u := dx /dt is: U = (γu . icρ). From δ ds = 0 the equations of motion can be derived: d2 xα dxβ dxγ =0 + Γα βγ 2 ds ds ds . p = mr v = γm0 v = W v/c2 .2 Red and blue shift There are three causes of red and blue shifts: 1. J.1 Riemannian geometry.2 General relativity 3. α The Einstein tensor is given by: G αβ := Rαβ − 1 g αβ R. it is called a black hole. which can also be written as R αβ = 8πκ µ (Tαβ − 1 gαβ Tµ ) 2 c2 For empty space this is equivalent to R αβ = 0. 1. the Laplace equation from Newtonian gravitation can be derived by stating: g µν = ηµν + hµν . the components of g µν are associated with the potentials and the derivatives of g µν with the field strength. µ The Riemann tensor is defined as: R ναβ T ν := ∇α ∇β T µ − ∇β ∇α T µ . m := M κ/c2 is the geometrical mass of an object with mass M . j ∂xk ∂ xl ∂x ¯ µ µ σ σ µ are the Christoffel symbols. for Euclidean spaces this reduces to: Γ i = jk ¯ ∂ 2 xl ∂xi . σµ βν σν βµ βν βµ µ The Ricci tensor is a contraction of the Riemann tensor: R αβ := Rαµβ . ij jk Γi = jk g il 2 ∂g k ∂glj ∂glk + − jl k j ∂x ∂x ∂x . where |h| 1. and dΩ 2 = dθ2 + sin2 θdϕ2 . 1). The most general form of the field equations is: R αβ − 1 gαβ R + Λgαβ = 2 3. 1. 8πκ Tαβ c2 where Λ is the cosmological constant. 1). 1. ηµν :=diag(−1. This metric. In general relativity. The principle of equivalence: inertial mass ≡ gravitational mass ⇒ gravitation is equivalent with a curved space-time were particles move along geodesics. Here. The external Schwarzschild metric applies in vacuum outside a spherical mass distribution. If an object is smaller than its event horizon 2m. In the stationary case. ∇k ai = j j il i l i i l l l ij ij i lj ∂k aj − Γkj al + Γkl aj . ∂xi ∂xj In general holds: ds = gµν dx dx . This metric is singular for r = 2m = 2κM/c 2 . With the variational principle δ (L(gµν ) − Rc2 /16πκ) |g|d4 x = 0 for variations gµν → gµν + δgµν the Einstein field equations can be derived: Gαβ = 8πκ Tαβ c2 . and is given by: ds2 = −1 + 2m r c2 dt2 + 1 − 2m r −1 dr2 + r2 dΩ2 Here. In special relativity this becomes ds 2 = −c2 dt2 + dx2 + dy 2 + dz 2 . where R := Rα is the Ricci scalar. is called the Minkowski metric. 1. this results in ∇2 h00 = 8πκ /c2 . ∇k aij = ∂k aij − Γki alj − Γkj ajl and ∇k a = ∂k a + Γkl a + Γkl a .2. The equation R αβµν = 0 has as only solution a flat space. which are of second order in g µν . By a proper choice of the coordinate system it is possible to make the metric locally flat in each point xi : gαβ (xi ) = ηαβ :=diag(−1.2 The line element The metric tensor in an Euclidean space is given by: g ij = k 2 µ ν ∂ xk ∂ xk ¯ ¯ . The following α holds: Rβµν = ∂µ Γα − ∂ν Γα + Γα Γσ − Γα Γσ . They are defined by: . which is symmetric: Rαβ = Rβα . that implies that its escape velocity is > c. For a second-order tensor holds: [∇ α . The Bianchi identities are: ∇ λ Rαβµν + ∇ν Rαβλµ + ∇µ Rαβνλ = 0. The Einstein equations are 10 independent equations. 3. From this. ∇β ]Tν = Rσαβ Tν + Rναβ Tσ .Chapter 3: Relativity 15 2. The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metric near r = 2m. for which 2 holds: ∇β Gαβ = 0. where the covariant derivative is given by ∇j ai = ∂j ai + Γi ak and ∇j ai = ∂j ai − Γk ak . The Newtonian limit of this metric is given by: ds2 = −(1 + 2V )c2 dt2 + (1 − 2V )(dx2 + dy 2 + dz 2 ) where V = −κM/r is the Newtonian gravitation potential. This constant plays a role in inflatory models of the universe. √ A rotating charged black hole has an event horizon with R S = m + m2 − a2 − e2 . this becomes: u1 (ϕ) = A + B cos(ϕ − εϕ) + ε A + B2 B2 − cos(2ϕ) 2A 6A where ε = 3m2 /h2 is small. a = 0) then √ follows that within the surface R E = m + m2 − a2 cos2 θ (de ergosphere) no particle can be at rest. which is necessary to eliminate the coordinate singularity there. the Kruskal coordinates are singular. In zeroth order. This is the case if cos(ϕ − εϕ) = 0 ⇒ ϕ ≈ 2πn(1 + ε). Substituting the external Schwarzschild metric yields for a planetary orbit: du dϕ d2 u +u dϕ2 = m du 3mu + 2 dϕ h where u := 1/r and h = r 2 ϕ =constant. This is equivalent to the problem δ ds2 = δ gij dxi dxj = 0.2.A.3 Planetary orbits and the perihelion shift To find a planetary orbit. J.16 Physics Formulary by ir. r r The orbital equation gives r =constant as solution. 3. . On the line dv = ±du holds dθ = dϕ = ds = 0. this corresponds with r = 0. The term 3mu is not present in the classical solution. be solved with perturbation theory. For the metric outside a rotating. a = L/M c and e = κQ/ε0 c2 . For the Kerr metric (e = 0. This term can ˙ κM h2 in the classical case also be found from a potential V (r) = − 1+ 2 . In first order. after dividing by du/dϕ. Near rotating black holes frame dragging occurs because g tϕ = 0. Wevers • r > 2m:    u        v  = r r − 1 exp cosh 2m 4m r r − 1 exp sinh 2m 4m 1− 1− r r exp sinh 2m 4m r r exp cosh 2m 4m t 4m t 4m t 4m t 4m = • r < 2m:    u        v  = = • r = 2m: here. For the perihelion shift then follows: ∆ϕ = 2πε = 6πm2 /h2 per orbit. The Kruskal coordinates are only singular on the hyperbole v 2 − u2 = 1.C. the limit x 0 → ∞ with u = v and x0 → −∞ with u = −v. or u maximal. or can. the variational problem δ ds = 0 has to be solved. charged spherical mass the Newman metric applies: ds2 = 1− 2mr − e2 r2 + a2 cos2 θ c2 dt2 − r2 + a2 cos2 θ r2 − 2mr + a2 − e2 sin2 θdϕ2 + dr2 − (r2 + a2 cos2 θ)dθ2 − sin2 θ(dϕ)(cdt) r 2 + a2 + (2mr − e2 )a2 sin2 θ r2 + a2 cos2 θ 2a(2mr − e2 ) r2 + a2 cos2 θ where m = κM/c2 . The line element in these coordinates is given by: ds2 = − 32m3 −r/2m 2 e (dv − du2 ) + r2 dΩ2 r The line r = 2m corresponds to u = v = 0. this results in an elliptical orbit: u 0 (ϕ) = A + B cos(ϕ) with A = m/h2 and B an arbitrary constant. The perihelion of a planet is the point for which r is minimal. The expansion velocity of the universe remains 3. W < 0. q < positive forever. 3. 3 3.6 Cosmology If for the universe as a whole is assumed: 1.j (xi xj − 1 δij r2 )d3 x the mass quadrupole moment. 2. Hyperbolical universe: k = −1. . W > 0. The expansion velocity of the universe → 0 if t → ∞. q > 1 . W = 0.2. 2. There exists a global time coordinate which acts as x 0 of a Gaussian coordinate system. The 3-dimensional spaces are isotrope for a certain value of x 0 . if the occurring velocities are c and for wavelengths the size of the system.5 Gravitational waves Starting with the approximation g µν = ηµν + hµν for weak gravitational fields and the definition h µν = hµν − 1 ηµν hα it follows that ✷hµν = 0 if the gauge condition ∂h µν /∂xν = 0 is satisfied. This gives 3 possible conditions for the universe (here. 1 2.2. q = 1 . The expansion velocity of the universe becomes negative 2 after some time: the universe starts collapsing. If Λ = 0 can be derived for the deceleration q=− ¨ RR 4πκ = ˙2 3H 2 R ˙ where H = R/R is Hubble’s constant.2.Chapter 3: Relativity 17 3. Substituting the external Schwarzschild metric results in the following orbital equation: du dϕ d2 u + u − 3mu dϕ2 =0 3. 2 The hereto related critical density is c = 3H 2 /8πκ. then the Robertson-Walker metric can be derived for the line element: ds2 = −c2 dt2 + 2 r0 R2 (t) kr2 1− 2 4r0 (dr2 + r2 dΩ2 ) For the scalefactor R(t) the following equations can be derived: ˙ ˙ ¨ R2 + kc2 2R R2 + kc2 Λ 8πκp 8πκ + + = − 2 + Λ and = 2 2 R R c R 3 3 where p is the pressure and parameter q: the density of the universe. Elliptical universe: k = 1. and has the value ≈ (75 ± 25) km·s −1 ·Mpc−1 . This is a measure of the velocity with which galaxies far away are moving away from each other. Parabolical universe: k = 0.4 The trajectory of a photon For the trajectory of a photon (and for each particle with zero restmass) holds ds 2 = 0. W is the total amount of energy in the universe): 1. From this. it α 2 follows that the loss of energy of a mechanical system. is given by: G dE =− 5 dt 5c with Qij = d3 Qij dt3 2 i. Each point is equivalent to each other point for a fixed x 0 . The phase angle is ϕ := arctan(X/R). of a capacitor 1/iωC and of a self inductor iωL. The stiffness of an oscillating system is given by F/x. In these points holds: R = X and δ = ±Q −1 . It is given 1 . The amplitude resonance frequency ω A is the frequency where iωZ is minimal.1 Harmonic oscillations ˆ ˆ The general form of a harmonic oscillation is: Ψ(t) = Ψei(ωt±ϕ) ≡ Ψ cos(ωt ± ϕ). A weak damped oscillation (k 2 < 4mC) dies out after T D = 2π/ωD . The width of this curve is characterized by the points where |Z(ω)| = √ |Z(ω0 )| 2. The quality of a coil is Q = ωL/R. The total impedance in case several elements are positioned is given by: . ˆ where Ψ is the amplitude. k The frequency with minimal |Z| is called velocity resonance frequency. This is equal to ω 0 . to ˆ which a periodic force F (t) = F cos(ωt) is applied holds the equation of motion m¨ = F (t) − k x − Cx. 4. x ˙ 2 With complex amplitudes.Chapter 4 Oscillations 4. A superposition of several harmonic oscillations with the same frequency results in another harmonic oscillation: ˆ ˆ Ψi cos(αi ± ωt) = Φ cos(β ± ωt) i with: tan(β) = i ˆ Ψi sin(αi ) ˆ Ψi cos(αi ) i ˆ and Φ2 = i ˆ Ψ2 + 2 i j>i i ˆ ˆ Ψi Ψj cos(αi − αj ) For harmonic oscillations holds: x(t)dt = dn x(t) x(t) and = (iω)n x(t). In the resonance √ curve |Z|/ Cm is plotted against ω/ω0 . The quantity Z = F/x is called the impedance of the system. and the width is 2∆ω B = ω0 /Q. The impedance of a resistor is R. 2 The damping frequency ω D is a measure for the time in which an oscillating system comes to rest. this becomes −mω 2 x = F − Cx − ikωx. The quality of the system ω0 √ ω Cm is given by Q = . A strong damped oscillation (k 2 > 4mC) drops like (if k2 4mC) x(t) ≈ x0 exp(−t/τ ).3 Electric oscillations The impedance is given by: Z = R + iX.2 Mechanic oscillations For a construction with a spring with constant C parallel to a damping k which is connected to a mass M . With ω0 = C/m follows: x= where δ = 2 m(ω0 F F . For a critical by ωD = ω0 1 − 4Q2 damped oscillation (k 2 = 4mC) holds ωD = 0. This is the case for ω A = ω0 1 − 1 Q2 . and for the velocity holds: x = √ ˙ 2 ) + ikω −ω i Cmδ + k ω ω0 ˙ − . iω dtn 4. Chapter 4: Oscillations 19 1. Series connection: V = IZ, Ztot = i Zi , Ltot = i Li , 1 = Ctot i 1 Z0 , Z = R(1 + iQδ) , Q= Ci R 2. parallel connection: V = IZ, 1 = Ztot Here, Z0 = 1 1 , = Zi Ltot 1 , Ctot = Li Ci , Q = i i i R R , Z= Z0 1 + iQδ L 1 . and ω0 = √ C LC ˆ ˆ The power given by a source is given by P (t) = V (t) · I(t), so P t = Veff Ieff cos(∆φ) 1ˆˆ 1 ˆ2 1 ˆ2 = 2 V I cos(φv − φi ) = 2 I Re(Z) = 2 V Re(1/Z), where cos(∆φ) is the work factor. 4.4 Waves in long conductors These cables are in use for signal transfer, e.g. coax cable. For them holds: Z 0 = The transmission velocity is given by v = dx dx . dL dC dL dx . dx dC 4.5 Coupled conductors and transformers For two coils enclosing each others flux holds: if Φ 12 is the part of the flux originating from I 2 through coil 2 which is enclosed by coil 1, than holds Φ 12 = M12 I2 , Φ21 = M21 I1 . For the coefficients of mutual induction Mij holds: N 1 Φ1 N 2 Φ2 M12 = M21 := M = k L1 L2 = = ∼ N1 N2 I2 I1 where 0 ≤ k ≤ 1 is the coupling factor. For a transformer is k ≈ 1. At full load holds: V1 I2 iωM = =− ≈− V2 I1 iωL2 + Rload L1 N1 =− L2 N2 4.6 Pendulums The oscillation time T = 1/f , and for different types of pendulums is given by: • Oscillating spring: T = 2π • Physical pendulum: T = 2π • Torsion pendulum: T = 2π m/C if the spring force is given by F = C · ∆l. I/τ with τ the moment of force and I the moment of inertia. I/κ with κ = 2lm the constant of torsion and I the moment of inertia. πr4 ∆ϕ • Mathematical pendulum: T = 2π lum. l/g with g the acceleration of gravity and l the length of the pendu- Chapter 5 Waves 5.1 The wave equation The general form of the wave equation is: ✷u = 0, or: ∇2 u − 1 ∂2u ∂2u ∂2u ∂2u 1 ∂2u = + 2 + 2 − 2 2 =0 2 ∂t2 2 v ∂x ∂y ∂z v ∂t where u is the disturbance and v the propagation velocity. In general holds: v = f λ. By definition holds: kλ = 2π and ω = 2πf . In principle, there are two types of waves: 1. Longitudinal waves: for these holds k 2. Transversal waves: for these holds k v u. v ⊥ u. The phase velocity is given by v ph = ω/k. The group velocity is given by: vg = dω dvph k dn = vph + k = vph 1 − dk dk n dk where n is the refractive index of the medium. If v ph does not depend on ω holds: v ph = vg . In a dispersive medium it is possible that vg > vph or vg < vph , and vg · vf = c2 . If one wants to transfer information with a wave, e.g. by modulation of an EM wave, the information travels with the velocity at with a change in the electromagnetic field propagates. This velocity is often almost equal to the group velocity. For some media, the propagation velocity follows from: • Pressure waves in a liquid or gas: v = κ/ , where κ is the modulus of compression. γp/ = γRT /M. E/ • For pressure waves in a gas also holds: v = • Pressure waves in a thin solid bar with diameter << λ: v = • waves in a string: v = Fspan l/m • Surface waves on a liquid: v = gλ 2πγ 2πh + tanh 2π λ λ where h is the depth of the liquid and γ the surface tension. If h λ holds: v ≈ √ gh. 5.2 Solutions of the wave equation 5.2.1 Plane waves In n dimensions a harmonic plane wave is defined by: n u(x, t) = 2n u cos(ωt) ˆ i=1 sin(ki xi ) Chapter 5: Waves 21 The equation for a harmonic traveling plane wave is: u(x, t) = u cos( k · x ± ωt + ϕ) ˆ If waves reflect at the end of a spring this will result in a change in phase. A fixed end gives a phase change of π/2 to the reflected wave, with boundary condition u(l) = 0. A lose end gives no change in the phase of the reflected wave, with boundary condition (∂u/∂x) l = 0. If an observer is moving w.r.t. the wave with a velocity v obs , he will observe a change in frequency: the f vf − vobs Doppler effect. This is given by: = . f0 vf 5.2.2 Spherical waves When the situation is spherical symmetric, the homogeneous wave equation is given by: 1 ∂ 2 (ru) ∂ 2 (ru) − =0 v 2 ∂t2 ∂r2 with general solution: u(r, t) = C1 f (r − vt) g(r + vt) + C2 r r 5.2.3 Cylindrical waves When the situation has a cylindrical symmetry, the homogeneous wave equation becomes: 1 ∂2u 1 ∂ − v 2 ∂t2 r ∂r r ∂u ∂r =0 This is a Bessel equation, with solutions which can be written as Hankel functions. For sufficient large values of r these are approximated by: u ˆ u(r, t) = √ cos(k(r ± vt)) r 5.2.4 The general solution in one dimension Starting point is the equation: ∂m ∂ 2 u(x, t) = bm m 2 ∂t ∂x m=0 N u(x, t) R. where bm ∈ I Substituting u(x, t) = Aei(kx−ωt) gives two solutions ω j = ωj (k) as dispersion relations. The general solution is given by: ∞ u(x, t) = −∞ a(k)ei(kx−ω1 (k)t) + b(k)ei(kx−ω2 (k)t) dk Because in general the frequencies ω j are non-linear in k there is dispersion and the solution cannot be written any more as a sum of functions depending only on x ± vt: the wave front transforms. 5.3 The stationary phase method Usually the Fourier integrals of the previous section cannot be calculated exactly. If ω j (k) ∈ I the stationary R phase method can be applied. Assuming that a(k) is only a slowly varying function of k, one can state that the parts of the k-axis where the phase of kx − ω(k)t changes rapidly will give no net contribution to the integral because the exponent oscillates rapidly there. The only areas contributing significantly to the integral are areas d (kx − ω(k)t) = 0. Now the following approximation is possible: with a stationary phase, determined by dk ∞ N a(k)ei(kx−ω(k)t) dk ≈ −∞ i=1 2π d2 ω(ki ) 2 dki 1 exp −i 4 π + i(ki x − ω(ki )t) Bz ≡ 0: the Transversal Magnetic modes (TM). like point-like excitations. t) = ∂P (x. Ez ≡ 0: the Transversal Electric modes (TE). x .n : λ = 2 (m/a)2 + (n/b)2 . Boundary condition: E z |surf = 0. surf By = ∂Bz i ∂Ez k + εµω εµω 2 − k 2 ∂y ∂x ∂Bz i ∂Ez − εµω k Ey = εµω 2 − k 2 ∂y ∂x For the TE and TM modes this gives an eigenvalue problem for E z resp. Bz with boundary conditions: ∂2 ∂2 + 2 ∂x2 ∂y ψ = −γ 2 ψ with eigenvalues γ 2 := εµω 2 − k 2 This gives a discrete solution ψ with eigenvalue γ 2 : k = εµω 2 − γ 2 . y)ei(kz−ωt) and B(x. t) = P (x. x . ω is called the cut-off frequency.A. y)ei(kz−ωt) . x .5 Waveguides and resonating cavities The boundary conditions for a perfect conductor can be derived from the Maxwell equations. For ω < ω . Boundary condition: ∂Bz ∂n = 0. then the solution of the wave equation with arbitrary initial P (x. t) = B(x. 0) and g(x) = is given by: ∂t ∞ ∞ u(x. Starting with the wave equation in one dimension.n of TMm. 0) = 0. 0) conditions f (x) = u(x. t)dx P and Q are called the propagators. if B z and Ez are not ≡ 0: ∂Bz i ∂Ez k − εµω εµω 2 − k 2 ∂x ∂y ∂Bz i ∂Ez + εµω k Ex = εµω 2 − k 2 ∂x ∂y Bx = Now one can distinguish between three cases: 1. 0) = δ(x − x ). k is imaginary and the wave is damped.22 Physics Formulary by ir. and P (x. If n is a unit vector ⊥ the surface. and K is a surface current density. x . x . t) = −∞ f (x )Q(x. with ∇ 2 = ∂ 2 /∂x2 holds: if Q(x. x . In rectangular conductors the following expression can be found for the cut-off frequency for modes TE m. t) = E(x. t) = 1 2 [δ(x − x − vt) + δ(x − x + vt)] if |x − x | < vt if |x − x | > vt 1 2v 0 Further holds the relation: Q(x. x . 0) = 0 and ∂t ∂u(x. 0) = δ(x − x ) and ∂t ∂P (x. From this one can now deduce that.4 Green functions for the initial-value problem This method is preferable if the solutions deviate much from the stationary solutions. pointed from 1 to 2. x . x . x . t) ∂t 5. than holds: n · (D2 − D1 ) = σ n · (B2 − B1 ) = 0 n × (E2 − E1 ) = 0 n × (H2 − H1 ) = K In a waveguide holds because of the cylindrical symmetry: E(x. They are defined by: Q(x. Wevers 5. t) the solution with initial values initial values Q(x. t) is the solution with ∂Q(x. Therefore.C. x . 2. x . t)dx + −∞ g(x )P (x. J. just as if here were no waveguide. kz = This results in the possible frequencies f = vk/2π in the cavity: by: kx = a b c f= v 2 n2 n2 n2 y x z + 2 + 2 a2 b c For a cubic cavity. ky = . Than holds: k = √ ±ω εµ and vf = vg . While x is growing. In a rectangular. The Korteweg-De Vries equation is given by: ∂u ∂3u ∂u ∂u + − au + b2 3 = 0 ∂t ∂x ∂x ∂x non−lin dispersive This equation is for example a model for ion-acoustic waves in a plasma. the right-hand side changes sign and an increase in energy changes into a decrease of energy. Oscillations on a time scale ∼ ω 0 can exist. 0 ≤ τ ≤ N with ∂τn /∂t = εn and if the secular terms are put 0 one obtains: d dt 1 2 dx dt 2 2 + 1 ω0 x2 2 = εω0 (1 − βx2 ) dx dt 2 This is an energy equation. Ez and Bz are zero everywhere: the Transversal electromagnetic mode (TEM). with a = b = c. If x 2 > 1/β. b and c the possible wave numbers are given n1 π n2 π n3 π . frequency. This mechanism limits the growth of oscillations. Substitution of x ∼ e iωt gives: ω = 2 1− 1 2 2 ε ). the 2nd term becomes larger −1 and diminishes the growth. 5. the possible number of oscillating modes N L for longitudinal waves is given by: 4πa3 f 3 NL = 3v 3 Because transversal waves have two possible polarizations holds for them: N T = 2NL . 3 . Energy is conserved if the left-hand side is 0. For this equation. 1 2 ω0 (iε ± The lowest-order instabilities grow as 1 2 εω0 . secular terms ∼ εt. If it is assumed that there exist timescales τn . If x is expanded as x = x (0) + (1) 2 (2) εx + ε x + · · · and this is substituted one obtains.6 Non-linear wave equations The Van der Pol equation is given by: dx d2 x 2 + ω0 x = 0 − εω0 (1 − βx2 ) dt2 dt βx2 can be ignored for very small values of the amplitude. besides periodic. 3 dimensional resonating cavity with edges a. soliton solutions of the following form exist: −d u(x − ct) = cosh2 (e(x − ct)) with c = 1 + 1 ad and e2 = ad/(12b2 ).Chapter 5: Waves 23 3. Further k ∈ I so there exists no cut-off R. The refraction of light in a material is caused by scattering from atoms.j − ω 2 − iδω fj = 1. For the refraction at a spherical surface with radius R holds: n2 n1 − n2 n1 − = v b R where |v| is the distance of the object and |b| the distance of the image.1 Lenses The Gaussian lens formula can be deduced from Fermat’s principle with the approximations cos ϕ = 1 and sin ϕ = ϕ. for which holds: that vg = c/(1 + (ne e2 /2ε0 mω 2 )). if ∆n > 1 holds: ∆ϕ = π. For a double concave lens holds R 1 < 0. From this the equation of Cauchy can be derived: n = a 0 + a1 /λ2 . Snell’s law is: n2 λ1 v1 = = n1 λ2 v2 If ∆n ≤ 1. From this follows j j where ne is the electron density and f j the oscillator strength. The path. Further holds: 1 1 1 = − f v b .2. f is the focal length and R 1 and R2 are the curvature radii of both surfaces. λ2k k=0 For an electromagnetic wave in general holds: n = √ εr µr . This is described by: n2 = 1 + ne e 2 ε0 m fj 2 ω0. the change in phase of the light is ∆ϕ = 0.1 The bending of light For the refraction at a surface holds: n i sin(θi ) = nt sin(θt ) where n is the refractive index of the material. Applying this twice results in: 1 = (nl − 1) f 1 1 − R2 R1 where nl is the refractive index of the lens. it is possible to expand n as: n = .2 Paraxial geometrical optics 6. More n ak general. followed by a light ray in material can be found from Fermat’s principle: 2 2 2 δ 1 dt = δ 1 n(s) ds = 0 ⇒ δ c 1 n(s)ds = 0 6.Chapter 6 Optics 6. for a double convex lens holds R 1 > 0 and R2 < 0. R2 > 0. 2. The plane ⊥ the optical axis through the principal points is called the principal plane. A parabolical mirror has no spherical aberration for light rays parallel with the optical axis and is therefore often used for telescopes. For a telescope holds: N = f objective /focular. The used signs are: Quantity R f v b + Concave mirror Concave mirror Real object Real image − Convex mirror Convex mirror Virtual object Virtual image 6. . If the lens is surrounded by the same medium on both sides. The f-number is defined by f /Dobjective. For a lens with thickness d and diameter D holds to a good approximation: 1/f = 8(n − 1)d/D 2 . Spherical aberration can be reduced by not using spherical mirrors. h2 = n m12 m12 N1 r r r O N2 6. as is seen by an incoming light ray: Quantity R f v b + Concave surface Converging lens Real object Virtual image − Convex surface Diverging lens Virtual object Real image 6.3 Principal planes The nodal points N of a lens are defined by the figure on the right. If the lens is described by a matrix m ij than for the distances h1 and h2 to the boundary of the lens holds: h1 = n m11 − 1 m22 − 1 .2 Mirrors For images of mirrors holds: 1 1 2 h2 1 = + = + f v b R 2 1 1 − R v 2 where h is the perpendicular distance from the point the light ray hits the mirror to the optical axis. For two lenses placed on a line with distance d holds: 1 1 1 d = + − f f1 f2 f1 f2 In these equations the following signs are being used for refraction at a spherical surface.2.Chapter 6: Optics 25 D := 1/f is called the dioptric power of a lens. Further holds: N · N α = 1.4 Magnification The linear magnification is defined by: N = − b v αsyst αnone The angular magnification is defined by: N α = − where αsys is the size of the retinal image with the optical system and α none the size of the retinal image without the system. the nodal points are the same as the principal points H.2. 4. Field curvature can be corrected by the human eye. Then the Fresnel equations are: r = tan(θi − θt ) sin(θt − θi ) .t.3 Matrix methods A light ray can be described by a vector (nα. Coma is caused by the fact that the principal planes of a lens are only flat near the principal axis. Transfer along length l: M R = 1 0 l/n 1 1 0 −D 1 2. Further away of the optical axis they are curved. The change of a light ray interacting with an optical system can be obtained using a matrix multiplication: n1 α1 n2 α2 =M y2 y1 where Tr(M ) = 1. 3.A. 6. Chromatic aberration is caused by the fact that n = n(λ). Refraction at a surface with dioptric power D: M T = 6. r⊥ ≡ . J. t⊥ ≡ r ≡ E0i E0i ⊥ E0i E0i ⊥ where E0r is the reflected amplitude and E 0t the transmitted amplitude. t⊥ = sin(θt + θi ) cos(θt − θi ) sin(θt + θi ) The following holds: t ⊥ − r⊥ = 1 and t + r = 1. It makes a difference whether the E field of the wave is ⊥ or w. 5. When the coefficients of reflection r and transmission t are defined as: E0r E0r E0t E0t .r. Some causes are: 1.26 Physics Formulary by ir. the surface. Spherical aberration is caused by second-order effects which are usually ignored. This curvature can be both positive or negative. 6. M is a product of elementary matrices. and a part will be refracted at an angle according to Snell’s law. a spherical surface does not make a perfect lens. These are: 1.5 Reflection and transmission If an electromagnetic wave hits a transparent medium part of the wave will reflect at the same angle as the incident angle. y) with α the angle with the optical axis and y the distance to the optical axis. If the coefficient of reflection R and transmission T are defined as (with θi = θr ): Ir It cos(θt ) R≡ and T ≡ Ii Ii cos(θi ) . Astigmatism: from each point of an object not on the optical axis the image is an ellipse because the thickness of the lens is not the same everywhere. This can be corrected with a combination of positive and negative lenses. t ≡ . Wevers 6. Using N lenses makes it possible to obtain the same f for N wavelengths.C. Incomming rays far from the optical axis will more bent. This can be partially corrected with a lens which is composed of more lenses with different functions n i (λ). r⊥ = tan(θi + θt ) sin(θt + θi ) t = 2 sin(θt ) cos(θi ) 2 sin(θt ) cos(θi ) .4 Aberrations Lenses usually do not give a perfect image. 2. Distorsion gives abberations near the edges of the image. fast axis horizontal e iπ/4 e iπ/4 1 2 1 2 Homogene circular polarizor right Homogene circular polarizer left 6. The state of a light ray can be described by the Stokes-parameters: start with 4 filters which each transmits half the intensity. −i) and the L-state by E = 1 2(1. S2 = 2I2 − 2I1 . A special case is r = 0. θ i is the angle between the incident angle and a line perpendicular to the surface and θ i is the angle between the ray leaving the prism and a line perpendicular to the surface. From Snell’s law it then follows: tan(θ i ) = n.6 Polarization The polarization is defined as: P = Ip Imax − Imin = Ip + Iu Imax + Imin where the intensity of the polarized light is given by I p and the intensity of the unpolarized light is given by Iu . The situation with r ⊥ = 0 is not possible. while the fourth is a circular polarizer which is opaque for L-states. for the vertical P -state E = (0. where α is the apex angle. For the refractive index of the prism now holds: sin( 1 (δmin + α)) 2 n= sin( 1 α) 2 . When θ i varies there is an angle for which δ becomes minimal. Then holds S 1 = 2I1 .r. S3 = 2I3 − 2I1 and S4 = 2I4 − 2I1 . The change in state of a light beam after passage of 2 2 optical equipment can be described as E2 = M · E1 . Imax and Imin are the maximum and minimum intensities when the light passes a polarizer. If polarized light passes through a polarizer Malus law applies: I(θ) = I(0) cos 2 (θ) where θ is the angle of the polarizer. The first is independent of the polarization. This angle is called Brewster’s angle.t. i). The state of a polarized light ray can also be described by the Jones vector: E= E0x eiϕx E0y eiϕy For the √ horizontal P -state holds: E = (1.7 Prisms and dispersion A light ray passing through a prism is refracted twice and aquires a deviation from its original direction δ = θi + θi + α w. This happens if the angle between the reflected and transmitted rays is 90 ◦ . the second and third are linear polarizers with the transmission axes horizontal and at +45 ◦ . 0). the incident direction. 1). fast axis vertical plate. 6. the R-state is given by √ E = 1 2(1. For some types of optical equipment the Jones matrix M is given by: Horizontal linear polarizer: Vertical linear polarizer: Linear polarizer at +45 ◦ Lineair polarizer at −45 ◦ 1 4 -λ 1 4 -λ 1 2 1 2 1 0 0 0 0 0 0 1 1 1 1 1 1 −1 −1 1 1 0 0 −i 1 0 0 i 1 i −i 1 1 i −i 1 plate.Chapter 6: Optics 27 with I = |S| it follows: R + T = 1. A. For the first factor follows: 2 sin( 1 α) dδ 2 = dn cos( 1 (δmin + α)) 2 D= For visible light usually holds dn/dλ < 0: shorter wavelengths are stronger bent than longer.r. The diffraction through a spherical aperture with radius a is described by: I(θ) = I0 J1 (ka sin(θ)) ka sin(θ) 2 The diffraction pattern of a rectangular aperture at distance R with length a in the x-direction and b in the y-direction is described by: 2 2 sin(α ) sin(β ) I(x.t.g. which describes the directionality of the secondary emissions: E(θ) = 1 E0 (1 + cos(θ)) where θ is the angle w. N is the number of slits. The minimum difference between two wavelengths that gives a separated diffraction pattern in a multiple slit geometry is given by ∆λ/λ = nN where N is the number of lines and n the order of the pattern. which happens e. in which they are parallel. The refractive index in this area can usually be approximated by Cauchy’s formula. Close at the source the Fraunhofermodel is invalid because it ignores the angle-dependence of the reflected waves.9 Special optical effects • Birefringe and dichroism.28 Physics Formulary by ir. D is not parallel with E if the polarizability P of a material is not equal in all directions. The Fraunhofer diffraction of light passing through multiple slits is described by: 2 2 sin(u) sin(N v) I(θ) = · I0 u sin(v) where u = πb sin(θ)/λ.C. the optical axis. This is the minimum angle ∆θ min between two incident rays coming from points far away for which their refraction patterns can be detected separately.22λ/D where D is the diameter of the slit. The maxima in intensity are given by d sin(θ) = kλ. 2 Diffraction limits the resolution of a system. hexagonal and tetragonal crystals there is one optical axis in the direction of n1 . Wevers The dispersion of a prism is defined by: dδ dδ dn = dλ dn dλ where the first factor depends on the shape and the second on the composition of the prism. 6. In case n 2 = n3 = n1 . This is described by the obliquity or inclination factor. y) = I0 α β where α = kax/2R and β = kby/2R.8 Diffraction Fraunhofer diffraction occurs far away from the source(s). b the width of a slit and d the distance between the slits. When X rays are diffracted at a crystal holds for the position of the maxima in intensity Bragg’s relation: 2d sin(θ) = nλ where d is the distance between the crystal layers. For a circular slit holds: ∆θmin = 1. This results in 3 refractive indices n i which can be used to construct Fresnel’s ellipsoid. Incident light rays can now be split up in two parts: the ordinary wave is linear polarized ⊥ the plane through the transmission direction and the optical axis. v = πd sin(θ)/λ. For a grating holds: ∆θ min = 2λ/(N a cos(θm )) where a is the distance between two peaks and N the number of peaks. There are at least 3 directions. at trigonal. J. The extraordinary wave is linear polarized . the principal axes. 6. 0 • The Faraday effect: the polarization of light passing through material with length d and to which a magnetic field is applied in the propagation direction is rotated by an angle β = VBd where V is the Verdet constant. In that case. The 2 maximum resolution is then given by ∆f min = c/2ndF . where V is the applied voltage. Incident light will have a phase shift of ∆ϕ = 2πd(|n 0 − ne |)/λ0 if an uniaxial crystal is cut in such a way that the optical axis is parallel with the front and back plane. The retardation in a Pockels cell is ∆ϕ = 2πn3 r63 V /λ0 where r63 is the 6-3 element of the electro-optic tensor. transparent materials can become birefringent when placed in an electric field. Double images occur when the incident ray makes an angle with the optical axis: the extraordinary wave will refract. the ordinary will not. the retardation is given by: ∆ϕ = 2πK V 2 /d2 . 6. The term [1 + F sin2 (θ)]−1 := A(θ) is called the Airy function. .10 The Fabry-Perot interferometer For a Fabry-Perot interferometer holds in general: T + R + A = 1 where T is the transmission factor. Here. the optical axis is parallel to E. • The Kerr-effect: isotropic. If F is given by F = 4R/(1 − R)2 it follows for the intensity distribution: A It = 1− Ii 1−R 2 1 1 + F sin2 (θ) €€ € q € Source Lens ✛✲ d Screen Focussing lens √ √ The width of the peaks at half height is given by γ = 4/ F . If the electrodes have an effective length and are separated by a distance d. The radiation is emitted within a cone with an apex angle α with sin(α) = c/c medium = c/nvq . ˘ • Cerenkov radiation arises when a charged particle with v q > vf arrives. λ0 is the wavelength in vacuum and n 0 and ne the refractive indices for the ordinary and extraordinary wave. where K is the Kerr constant of the material. The finesse F is defined as F = 1 π F . The difference in refractive index in the two directions is given by: ∆n = λ0 KE 2 . namely those without a centre of symmetry. • The Pockels or linear electro-optical effect can occur in 20 (from a total of 32) crystal symmetry classes. For a quarter-wave plate holds: ∆ϕ = π/2.Chapter 6: Optics 29 in the plane through the transmission direction and the optical axis. R the reflection factor and A the absorption factor. Dichroism is caused by a different absorption of the ordinary and extraordinary wave in some materials. These crystals are also piezoelectric: their polarization changes when a pressure is applied and vice versa: P = pd + ε0 χE. • Retarders: waveplates and compensators. Odd s: s = 2l + 1: c(s) = √ .2 The energy distribution function The general form of the equilibrium velocity distribution function is P (vx . Wvib = (v + 1 )¯ ω0 2 h 2I The vibrational levels are excited if kT ≈ ¯ ω. The average energy of a molecule in thermodynamic equilibrium is E tot = 1 skT . vz )dvx dvy dvz = P (vx )dvx · P (vy )dvy · P (vz )dvz with P (vi )dvi = v2 1 √ exp − i2 α α π dvi where α = 2kT /m is the most probable velocity of a particle. given by: 1. So. A linear molecule has 2 rotational degrees of freedom and a non-linear molecule 3. 7. Because vibrational degrees of freedom account for both kinetic and potential energy they count double. For homonuclear molecules additional selection rules apply so the rotational levels are well coupled if kT ≈ 6B. Each degree of freedom of a 2 molecule has in principle the same energy: the principle of equipartition. vy . The average velocity is given by v = √ 2α/ π. and v 2 = 3 α2 . The rotational and vibrational energy of a molecule are: Wrot = h ¯2 l(l + 1) = Bl(l + 1) . There are 3 translational degrees of freedom. The distribution as a function of the absolute value of the velocity is given by: 2 4N mv 2 dN = 3 √ v 2 exp − dv α π 2kT The general form of the energy distribution function then becomes: c(s) P (E)dE = kT where c(s) is a normalization constant.Chapter 7 Statistical physics 7. Even s: s = 2l: c(s) = 1 (l − 1)! 2l π(2l − 1)!! E kT 1 2 s−1 exp − E kT dE 2. a linear molecule has s = 3n − 5 vibrational degrees of freedom and a non-linear molecule s = 3n − 6. for linear molecules this results in a total of s = 6n − 5. the rotational levels of a hetronuclear molecule are excited if h kT ≈ 2B.1 Degrees of freedom A molecule consisting of n atoms has s = 3n degrees of freedom. For non-linear molecules this gives s = 6n − 6. The equation of state for solids and liquids is given by: V 1 = 1 + γp ∆T − κT ∆p = 1 + V0 V ∂V ∂T ∆T + p 1 V ∂V ∂p ∆p T .4 The equation of state If intermolecular forces and the volume of the molecules can be neglected then for gases from p = and E = 3 kT can be derived: 2 1 pV = ns RT = N m v 2 3 2 3n E Here. θ.cr /RTcr = general gas law.3 Pressure on a wall The number of molecules that collides with a wall with surface A within a time τ is given by: ∞ π 2π d N= 0 0 0 3 nAvτ cos(θ)P (v. In a Van der Waals gas. A virial expansion is used for even more accurate views: p(T. ns is the number of moles particles and N is the total number of particles within volume V . This is the upper limit of the area of coexistence between liquid and vapor. Vcr = 3bns 27bR 27b2 3 8. ϕ)dvdθdϕ From this follows for the particle flux on the wall: Φ = d3 p = 1 4n v .cr with Vm := V /ns holds: p∗ + 3 ∗ (Vm )2 ∗ Vm − 1 3 = 8T∗ 3 Gases behave the same for equal values of the reduced quantities: the law of the corresponding states. this happens at T B = a/Rb. From dp/dV = 0 and d 2 p/dV 2 = 0 follows: Tcr = 8a a . with p ∗ := p/pcr . Vm ) = RT 1 B(T ) C(T ) + + + ··· 2 3 Vm Vm Vm The Boyle temperature T B is the temperature for which the 2nd virial coefficient is 0. pcr = . pressure and volume of the gas. In the Van der Waals equation this corresponds with the critical temperature. For the critical point holds: p cr Vm. T ∗ = T /Tcr and Vm = Vm /Vm.Chapter 7: Statistical physics 31 7. For the pressure on the wall then follows: 2 2mv cos(θ)d3 N . so p = n E Aτ 3 7. which differs from the value of 1 which follows from the ∗ Scaled on the critical quantities. The inversion temperature T i = 2TB . If the own volume and the intermolecular forces cannot be neglected the Van der Waals equation can be derived: p+ an2 s V2 (V − bns ) = ns RT There is an isotherme with a horizontal point of inflection. Some useful mathematical relations are: ∞ ∞ x e 0 n −x dx = n! .55n −1/3. The equilibrium condition for a vessel which has a hole with surface A in it for which holds that √ √ √ √ A/π is: n1 T1 = n2 T2 . D with F (r) the spatial distribution function in spherical coordinates. kTkr = 1. The following holds: D ≈ 0. For the average motion of a particle with radius R can be derived: x2 = 1 r2 = kT t/3πηR. which for a homogeneous distribution is given by: F (r)dr = 4nπr 2 dr. J.32 Physics Formulary by ir. For the Van der Waals coefficients a and b 2 and the critical quantities holds: a = 5. If two plates move along each other at a distance d with velocity w x the viscosity η is given by: Fx = η Awx . It can be derived that κ = 1 CmV n v /NA . If the distance between two molecules approaches the molecular diameter D a repulsing force between the electron clouds appears. U (r) = U LJ for D ≤ r ≤ 3D and U (r) = 0 for r ≥ 3D. This means . If m1 m2 holds: = 1+ . 0 x e 2n −x2 √ (2n)! π dx = .9NA D3 . Collisions between molecules and small particles in a solution result in the Brownian motion. It can be derived that η = 1 v where v is the thermal velocity. If m1 = m2 holds: = the particles. i 3 A gas is called a Knudsen gas if the dimensions of the gas. 3 T2 − T1 dQ = κA . b = 1.2 and Vm. This results in the Lennard-Jones potential for intermolecular forces: ULJ = 4 D r 12 − D r 6 with a minimum at r = rm . something that can easily occur at low pressures. where σ is v1 2 2 with u = v1 + v2 the relative velocity between the cross section. Also holds: κ = CV η. n!22n+1 ∞ x2n+1 e−x dx = 1 n! 2 0 2 .C. A more simple model for intermolecular forces assumes a potential U (r) = ∞ for r < D. Wevers 7.89r m . so = v1 m2 nσ nσ 2 1 .275N AD3 . 3 A better expression for κ can be obtained with the Eucken correction: κ = (1 + 9R/4c mV )CV · η with an error <5%. This force can be described by U rep ∼ exp(−γr) or Vrep = +Cs /rs with 12 ≤ s ≤ 20. The mean free path is given by = nuσ u m1 1 1 √ .6 Interaction between molecules For dipole interaction between molecules can be derived that U ∼ −1/r 6 .5 Collisions between molecules The collision probability of a particle in a gas that is translated over a distance dx is given by nσdx. The average distance between two molecules is 0. d The velocity profile between the plates is in that case given by w(z) = zw x /d. Together with the general gas law follows: p 1 / T1 = p2 / T2 .3NA D3 . If the molecules are approximated by hard that the average time between two collisions is given by τ = nσv 1 2 2 spheres the cross section is: σ = 4 π(D1 + D2 ). This gives for the potential energy of one molecule: E pot = 3D U (r)F (r)dr. which results in a temperdt d ature profile T (z) = T 1 + z(T2 − T1 )/d. The heat conductance in a non-moving gas is described by: 7.kr = 3.A. z). εz).Chapter 8 Thermodynamics 8.2 Definitions • The isochoric pressure coefficient: β V = • The isothermal compressibility: κ T = − • The isobaric volume coefficient: γ p = 1 V 1 V 1 p 1 V ∂p ∂T ∂V ∂p ∂V ∂T ∂V ∂p V T p • The adiabatic compressibility: κ S = − S For an ideal gas follows: γ p = 1/T . For such a function Euler’s theorem applies: ∂F ∂F ∂F +y +z mF (x. z) and z = z(x. y. z) = x ∂x ∂y ∂z 8. z) = F (εx. The total differential dz of z is than given by: dz = ∂z ∂x dx + y ∂z ∂y dy x By writing this also for dx and dy it can be obtained that ∂x ∂y Because dz is a total differential holds · z ∂y ∂z · x ∂z ∂x = −1 y dz = 0. z) = 0 between 3 variables.3 Thermal heat capacity • The specific heat at constant X is: C X = T • The specific heat at constant pressure: C p = • The specific heat at constant volume: C V = ∂S ∂T ∂H ∂T ∂U ∂T X p V . one can write: x = x(y. y. y. κT = 1/p and βV = −1/V . εy. y). A homogeneous function of degree m obeys: ε m F (x. y = y(x.1 Mathematical introduction If there exists a relation f (x. 8. For a Van der Waals gas holds: CmV = 1 sR + ap/RT 2. the entropy difference after a reversible process is: 2 S2 − S1 = 1 d Qrev T So. The first law is the conservation of energy.34 Physics Formulary by ir. for a reversible cycle holds: For an irreversible cycle holds: d Qrev = 0. One can extract an amount of work Wt from the system or add W t = −Wi to the system. where Q is the total added heat. called the entropy.5 State functions and Maxwell relations The quantities of state and their differentials are: Internal energy: Enthalpy: Free energy: Gibbs free enthalpy: U H = U + pV F = U − TS G = H − TS dU = T dS − pdV dH = T dS + V dp dF = −SdT − pdV dG = −SdT + V dp . So. holds: C V = 1 sR. T d Qirr < 0.A. 2 In general holds: Cp − CV = T ∂p ∂T · V ∂V ∂T = −T p ∂V ∂T 2 p ∂p ∂V ≥0 T Because (∂p/∂V )T is always < 0. For a lower T one needs only to consider the thermalized degrees of freedom. If the coefficient of expansion is 0. For their ratio now 2 2 follows γ = (2 + s)/s. where d means that the it is not a differential of a quantity of state. Cp = CV . Hence Cp = 1 (s + 2)R. and also at T = 0K. Further. For a closed system holds: Q = ∆U + W . if the temperature is high enough to thermalize all internal rotational and vibrational degrees of freedom. 8. The second law states: for a closed system there exists an additive quantity S. T The third law of thermodynamics is (Nernst): lim ∂S ∂X =0 T T →0 From this it can be concluded that the thermal heat capacity → 0 if T → 0. the following is always valid: C p ≥ CV . the differential of which has the following property: dQ dS ≥ T If the only processes occurring are reversible holds: dS = d Q rev /T .C. For a quasi-static process holds: d W = pdV .4 The laws of thermodynamics The zeroth law states that heat flows from higher to lower temperatures. J. For an open (flowing) system the first law is: Q = ∆H + W i + ∆Ekin + ∆Epot . In differential form this becomes: d Q = dU + d W . W the work done and ∆U the difference in the internal energy. So for a reversible process holds: d Q = dU + pdV . 8. Wevers For an ideal gas holds: C mp − CmV = R. so absolute zero temperature cannot be reached by cooling through a finite number of steps. In general this is accompanied with a change in temperature. For a reversible isobaric process holds: H 2 − H1 = Qrev . Also holds: T V γ−1 =constant and T γ p1−γ =constant. V ∂H ∂p =V −T T ∂V ∂T p for an enlarged surface holds: d W rev = −γdA. If T > Ti the gas heats up. The Carnotprocess The system undergoes a reversible cycle with 2 isothemics and 2 adiabatics: 1. and dS > 0. applied in refridgerators. Isothermic expansion at T 1 .Chapter 8: Thermodynamics 35 From this one can derive Maxwell’s relations: ∂T ∂V =− S ∂p ∂S .6 Processes The efficiency η of a process is given by: η = Work done Heat added Cold delivered Work added The Cold factor ξ of a cooling down process is given by: ξ = Reversible adiabatic processes For adiabatic processes holds: W = U 1 − U2 . V ∂T ∂p = S ∂V ∂S . Here H is a conserved quantity. with for TB : [∂(pV )/∂p]T = 0. 1 T Cp ∂V ∂T −V p The inversion temperature is the temperature where an adiabatically expanding gas keeps the same temperature. p ∂p ∂T = V ∂S ∂V . Adiabatics exhibit a greater steepness p-V diagram than isothermics because γ > 1. From this follows: γ= ∂U ∂A = S ∂F ∂A T 8. Isobaric processes Here holds: H2 − H1 = The throttle process This is also called the Joule-Kelvin effect and is an adiabatic expansion of a gas through a porous material or a small opening.g. 2. Adiabatic expansion with a temperature drop to T 2 . . The system absorbs a heat Q 1 from the reservoir. The quantity which is important here is the throttle coefficient: αH = ∂T ∂p = H 2 1 Cp dT . T i = 2TB . if T < T i the gas cools down. For reversible adiabatic processes holds Poisson’s equation: with γ = Cp /CV one gets that pV γ =constant. The throttle process is e. T ∂V ∂T =− p ∂S ∂p T From the total differential and the definitions of C V and Cp it can be derived that: T dS = CV dT + T For an ideal gas also holds: Sm = CV ln T T0 + R ln V V0 + Sm0 and Sm = Cp ln T T0 − R ln p p0 + Sm0 ∂p ∂T dV and T dS = Cp dT − T V ∂V ∂T dp p Helmholtz’ equations are: ∂U ∂V =T T ∂p ∂T −p . with γ the surface tension. A. The minimal work needed to attain a certain state is: W min = −Wmax . with the temperature and pressure of the surroundings given by T 0 and p0 . Wevers 3. For those there will occur only a discontinuity in the second derivates of Gm .g. 4.7 Maximal work Consider a system that changes from state 1 into state 2. Adiabatic compression to T 1 .8 Phase transitions Phase transitions are isothermic and isobaric. |Q2 | T2 |Q2 | = = W |Q1 | − |Q2 | T1 − T2 8. Closed system: Wmax = (U1 − U2 ) − T0 (S1 − S2 ) + p0 (V1 − V2 ).C.g. The maximum work which can be obtained from this change is. The efficiency in the ideal case is the same as for Carnot’s cycle. [∂ 3 Gm /∂T 3 ]p non continuous arises e. J. so with e. β and γ holds: Gα = Gβ and m m rβα α β ∆Sm = Sm − Sm = T0 where rβα is the transition heat of phase β to phase α and T 0 is the transition temperature. A phase-change of the 3rd order. so dG = 0. Open system: Wmax = (H1 − H2 ) − T0 (S1 − S2 ) − ∆Ekin − ∆Epot . . 2. 8. If the process is applied in reverse order and the system performs a work −W the cold factor is given by: ξ= The Stirling process Stirling’s cycle exists of 2 isothermics and 2 isochorics.36 Physics Formulary by ir. Further Sm = ∂Gm ∂T p so G has a twist in the transition point. In a two phase system Clapeyron’s equation is valid: β S α − Sm dp rβα = m = α−Vβ α − V β )T dT Vm (Vm m m For an ideal gas one finds for the vapor line at some distance from the critical point: p = p0 e−rβα/RT There exist also phase transitions with r βα = 0. The following holds: rβα = rαβ and rβα = rγα − rγβ . When the phases are indicated by α. These second-order transitions appear at organization phenomena. Isothermic compression at T 2 . removing Q 2 from the system. when ferromagnetic iron changes to the paramagnetic state. The efficiency for Carnot’s process is: η =1− T2 |Q2 | =1− := ηC |Q1 | T1 The Carnot efficiency η C is the maximal efficiency at which a heat machine can operate. when all processes are reversible: 1. Smixture = n i i 0 xi Si + ∆Smix ni Ui0 . 8. The number of free parameters in a system with c components and p different phases is given by f = c + 2 − p. The thermodynamic potentials are not independent in a multiple-phase system. A solution of one component in another gives rise to an increase in the boiling point ∆T k and a decrease of the freezing point ∆T s . In equilibrium for each component holds: µ α = µβ = µγ . ∆Ts = − s x2 rβα rγβ with rβα the evaporation heat and r γβ < 0 the melting heat. i i i . 8. A mixture of two liquids is rarely ideal: i i this is usually only the case for chemically related components or isotopes.T. For x2 1 holds: ∆Tk = 2 RTk RT 2 x2 .nj c V = i=1 ni ∂V ∂ni c := nj .p.Chapter 8: Thermodynamics 37 8. In spite of this holds Raoult’s law for the vapour pressure holds for many binary mixtures: p i = xi p0 = yi p. This is a partial quantity.p ≤ 0. Hmixture = where for ideal gases holds: ∆S mix = −nR i xi ln(xi ). For the osmotic pressure Π of a solution holds: 0 ΠVm1 = x2 RT . It can be derived that ni dµi = −SdT + V dp.T i=1 ni Vi where Vi is the partial volume of component i. If a system exists of more components this becomes: µi dni dG = −SdT + V dp + i where µ = ∂G ∂ni is called the thermodynamic potential.9 Thermodynamic potential When the number of particles within a system changes this number becomes a third quantity of state. G is the relevant quantity. i i Each component has as much µ’s as there are phases.11 Conditions for equilibrium When a system evolves towards equilibrium the only changes that are possible are those for which holds: (dS)U. The molar volume of a mixture of two components can be a concave line in a V -x 2 diagram: the mixing contracts the volume.p ≤ 0 or (dF )T. this gives at constant p and T : xi dµi = 0 (Gibbs-Duhmen). The following holds: Vm 0 = i xi Vi xi dVi i = where xi = ni /n is the molar fraction of component i.V ≤ 0 or (dG)T.V ≥ 0 or (dU )S. Because addition of matter usually takes place at constant p and T . For V holds: p.V ≤ 0 or (dH)S.10 Ideal mixtures For a mixture of n components holds (the index Umixture = i 0 is the value for the pure component): ni Hi0 . Here is xi the fraction of the ith i component in liquid phase and y i the fraction of the ith component in gas phase. For the thermodynamic potentials holds: µ i = µ0 + RT ln(xi ) < µ0 . For a system in thermodynamic equilibrium this becomes: S= U + kN ln T Z N + kN ≈ U + k ln T ZN N! For an ideal gas. When Stirling’s equation. d W rev = −γdA for the expansion of a soap bubble or d W rev = −BdM for a magnetic system. A rotating.A. . Hawkings area theorem states that dA/dt ≥ 0. one finds for a discrete system the Maxwell-Boltzmann distribution. non-charged black hole has a temparature of T = hc/8πkm. is the equilibrium state.13 Application to other systems Thermodynamics can be applied to other systems than gases and liquids.38 Physics Formulary by ir. To do this the term d W = pdV has to be replaced with the correct work term. For a Schwarzschild black hole A is given by A = 16πm 2 . that with the maximum value for P . For an ideal gas holds: i The state sum Z is a normalization constant. Hence.C. J. like d W rev = −F dl for the stretching of a wire. Wevers 8. It has an entropy S = Akc 3 /4¯ κ ¯ h with A the area of its event horizon. given by: Z = Z= V (2πmkT )3/2 h3 The entropy can then be defined as: S = k ln(P ) . ln(n!) ≈ n ln(n) − n is used. The occupation numbers in equilibrium are then given by: ni = N Wi gi exp − Z kT gi exp(−Wi /kT ). the lifetime of a black hole ∼ m 3 .12 Statistical basis for thermodynamics The number of possibilities P to distribute N particles on n possible energy levels. each with a g-fold degeneracy is called the thermodynamic probability and is given by: P = N! i n gi i ni ! The most probable distribution. with U = 3 kT then holds: S = 5 kN + kN ln 2 2 V (2πmkT )3/2 N h3 8. v is an arbitrary vector field and φ an arbitrary scalar field.Chapter 9 Transport phenomena 9. ∇2 v2 . the orientable surface d2 A is limited by the Jordan curve 9. . Surface forces working only on the margins: t.1 Mathematical introduction An important relation is: if X is a quantity of a volume element which travels from position r to r + dr in a time dt. Some important integral theorems are: Gauss: Stokes for a scalar field: Stokes for a vector field: This results in: Ostrogradsky: (v · n )d2 A = (φ · et )ds = (v · et )ds = (divv )d3 V (n × gradφ)d2 A (rotv · n )d2 A (rotv · n )d2 A = 0 (n × v )d2 A = (φn )d2 A = (rotv )d3 A (gradφ)d3 V ds. Here. The force f0 on each volume element. Some properties of the ∇ operator are: div(φv ) = φdivv + gradφ · v div(u × v ) = v · (rotu ) − u · (rotv ) div gradφ = ∇2 φ rot(φv ) = φrotv + (gradφ) × v rot rotv = grad divv − ∇2 v ∇2 v ≡ (∇2 v1 .2 Conservation laws On a volume work two types of forces: 1. 2. the total differential dX is then given by: dX = ∂X ∂X ∂X dX ∂X ∂X ∂X ∂X ∂X dx + dy + dz + dt ⇒ = vx + vy + vz + ∂x ∂y ∂z ∂t dt ∂x ∂y ∂z ∂t ∂X dX = + (v · ∇)X . For these holds: t = n T. where T is the stress tensor. ∇2 v3 ) rot gradφ = 0 div rotv = 0 Here. For gravity holds: f0 = g. dt ∂t d dt Xd3 V = ∂ ∂t Xd3 V + X(v · n )d2 A This results in general to: From this follows that also holds: where the volume V is surrounded by surface A. From equating the thermodynamical and mechanical pressure it follows: 3η + 2η = 0. ω represents the local rotation 2 velocity: dr · W = 1 ω × dr. η is the dynamical viscosity. The conservation laws for mass. Wevers T can be split in a part pI representing the normal tensions and a part T representing the shear stresses: T = T + pI.C. e is the internal energy per unit of mass E/m and s is the entropy per unit of mass S/m. Further holds: ∂e ∂E ∂e ∂E =− . Conservation of energy: ∂ ∂t − vd3 V + ( 1 v 2 + e) d3 V + 2 ( 1 v 2 + e) (v · n )d2 A = 2 (v · n T)d2 A (q · n )d2 A + (v · f0 )d3 V + Differential notation: 1. momentum and energy for continuous media can be written in both integral and differential form. This is related to the shear stress τ by: ∂vi τij = η ∂xj For compressible media can be stated: T = (η divv )I + 2ηD. q = −κ ∇T is the heat flow. When the flow velocity is v at position r holds on position r + dr: v(dr ) = v(r ) translation + dr · (gradv ) rotation. deformation. 2 For a Newtonian liquid holds: T = 2ηD. T = = p=− ∂V ∂1/ ∂S ∂s so ∂e ∂h CV = and Cp = ∂T V ∂T p with h = H/m the enthalpy per unit of mass. . J. where I is the unit tensor. Here. dilatation The quantity L:=gradv can be split in a symmetric part D and an antisymmetric part W.40 Physics Formulary by ir. Conservation of mass: ∂ ∂t ∂ ∂t d3 V + (v · n )d2 A = 0 v(v · n )d2 A = f0 d3 V + n · T d2 A 2. Conservation of mass: ∂ + div · ( v ) = 0 ∂t ∂v + ( v · ∇)v = f0 + divT = f0 − gradp + divT ∂t 2. Conservation of energy: T de p d ds = − = −divq + T : D dt dt dt Here. Conservation of momentum: 3. Conservation of momentum: 3. Wij := 1 2 ∂vj ∂vi − ∂xj ∂xi When the rotation or vorticity ω = rotv is introduced holds: W ij = 1 εijk ωk . L = D + W with Dij := 1 2 ∂vj ∂vi + ∂xj ∂xi .A. If the viscosity is constant holds: div(2D) = ∇ 2 v + grad divv. When viscous aspects can be ignored holds: divT= −gradp. They are: Integral notation: 1. Chapter 9: Transport phenomena 41 From this one can derive the Navier-Stokes equations for an incompressible. For incompressible 2 flows this becomes: 1 v 2 + gh + p/ =constant everywhere. when viscous effects are limited to the boundary layer.3 Bernoulli’s equations Starting with the momentum equation one can find for a non-viscous medium for stationary flows. c is the speed of sound and L is a characteristic length of the system. The force F on an object within a flow. If also holds rotv = 0 and 2 the entropy is equal on each streamline holds 1 v 2 + gh + dp/ =constant everywhere. Some dimensionless numbers are given by: Strouhal: Sr = Fourier: Prandtl: ωL v a Fo = ωL2 ν Pr = a v2 gL vL P´ clet: Pe = e a Lα Nusselt: Nu = κ Froude: Fr = Mach: v c vL Reynolds: Re = ν v2 Eckert: Ec = c∆T Ma = Here. viscous and heat-conducting medium: divv ∂v + (v · ∇)v ∂t C ∂T + C(v · ∇)T ∂t = = = 0 g − gradp + η∇2 v κ∇2 T + 2ηD : D with C the thermal heat capacity. α follows from the equation for heat transport κ∂ y T = α∆T and a = κ/ c is the thermal diffusion coefficient. These numbers can be interpreted as follows: • Re: (stationary inertial forces)/(viscous forces) . ν = η/ is the kinematic viscosity. One can also deduce functional equalities without solving the differential equations. can be obtained using the momentum law.4 Characterising of flows by dimensionless numbers The advantage of dimensionless numbers is that they make model experiments possible: one has to make the dimensionless numbers which are important for the specific experiment equal for both model and the real situation. with (v · grad)v = 1 grad(v 2 ) + (rotv ) × v 2 and the potential equation g = −grad(gh) that: 1 2 2v + gh + dp = constant along a streamline For compressible flows holds: 1 v 2 + gh + p/ =constant along a line of flow. 2 with γ = Cp /CV : γ p c2 1 2 = 1 v2 + = constant 2v + γ − 1 2 γ −1 With a velocity potential defined by v = gradφ holds for instationary flows: ∂φ 1 2 + 2 v + gh + ∂t dp = constant everywhere 9. For ideal gases with constant C p and CV holds. If a surface A surrounds the object outside the boundary layer holds: F =− [pn + v(v · n )]d2 A 9. For an incompressible laminar flow through a straight. For this angle holds Ma= 1/ arctan(θ).42 Physics Formulary by ir. with x = x/L. Wevers • Sr: (non-stationary inertial forces)/(stationary inertial forces) • Fr: (stationary inertial forces)/(gravity) • Fo: (heat conductance)/(non-stationary change in enthalpy) • Pe: (convective heat transport)/(heat conductance) • Ec: (viscous dissipation)/(convective heat transport) • Ma: (velocity)/(speed of sound): objects moving faster than approximately Ma = 0. For large Re holds for the force on a surface A: F = 1 CW A v 2 . Re > 2300: Le /2R ≈ 50 9.A.5 Tube flows For tube flows holds: they are laminar if Re< 2300 with dimension of length the diameter of the tube. v = v/V .6 Potential theory The circulation Γ is defined as: Γ = (v · et )ds = (rotv ) · nd2 A = (ω · n )d2 A For non viscous media. For a 2-dimensional flow a flow function ψ(x. grad = Lgrad. and turbulent if Re is larger. y) can be defined: with ΦAB the amount of liquid flowing through a curve s between the points A and B: B B ΦAB = A (v · n )ds = A (vx dy − vy dx) .056ReD √ 4R3 α π dp 3 dx 2 For flows at a small Re holds: ∇p = η∇ v and divv = 0. the dimensionless Navier-Stokes equation becomes. if p = p( ) and all forces are conservative. 500 < ReD < 2300: Le /2R = 0. 2 For gas transport at low pressures (Knudsen-gas) holds: Φ V = 2. J. For the total force on a sphere with radius R in a flow then holds: F = 6πηRv. Now. • Pr and Nu are related to specific materials. circular tube holds for the velocity profile: 1 dp 2 (R − r2 ) v(r) = − 4η dx R For the volume flow holds: Φ V = 0 v(r)2πrdr = − π dp 4 R 8η dx The entrance length L e is given by: 1.C. Kelvin’s theorem can be derived: dΓ =0 dt For rotationless flows a velocity potential v = gradφ can be introduced. ∇ 2 = L2 ∇2 and t = tω: g ∂v ∇ 2v + (v · ∇ )v = −grad p + Sr + ∂t Fr Re 9. In the incompressible case follows from conservation of mass ∇ 2 φ = 0.8 produce shockwaves which propagate with an angle θ with the velocity of the object. Blasius’ equation for the boundary layer is. Henxe rotating bodies also create a force perpendicular to their direction of motion: the Magnus effect. vθ = 0. x The momentum theorem of Von Karman for the boundary layer is: τ0 d dv (ϑv 2 ) + δ ∗ v = dx dx where the displacement thickness δ ∗ v and the momentum thickness ϑv 2 are given by: ∞ ∞ ∗ ϑv = 0 2 (v − vx )vx dy .1 Flow boundary layers √ If for the thickness of the boundary layer holds: δ L holds: δ ≈ L/ Re. The statement that Fx = 0 is d’Alembert’s paradox and originates from the neglection of viscous effects. 2. From this follows: CW = 0. In general holds: ∂2ψ ∂2ψ + = −ωz 2 ∂x ∂y 2 In polar coordinates holds: ∂φ 1 ∂φ 1 ∂ψ ∂ψ = .8 Heat conductance For non-stationairy heat conductance in one dimension without flow holds: κ ∂2T ∂T = +Φ ∂t c ∂x2 where Φ is a source term. y) = (0.7.2 Temperature boundary layers If the thickness of the temperature boundary layer δ T √ L holds: 1. For a vortex holds: φ = Γθ/2π. This is equivalent with y=0 12ηv∞ dp = . If Pr 1: δ/δT ≈ 3 Pr. vθ = − = r ∂θ ∂r ∂r r ∂θ Q ln(r) so that vr = Q/2πr. with vy /v∞ = f (y/δ): 2f + f f = 0 with boundary conditions f (0) = f (0) = 0. For source flows with power Q in (x.664 Re−1/2 . With v∞ the velocity of the main flow it follows for the velocity v y ⊥ the surface: vy L ≈ δv∞ . If Pr ≤ 1: δ/δ T ≈ √ Pr. dx δ2 9. 9.Chapter 9: Transport phenomena 43 and the definitions v x = ∂ψ/∂y.7. vy = −∂ψ/∂x holds: ΦAB = ψ(B) − ψ(A). f (∞) = 1. If Φ = 0 the solutions for harmonic oscillations at x = 0 are: x T − T∞ x cos ωt − = exp − Tmax − T∞ D D .7 Boundary layers 9. 9. The lift F y is also created by η because Γ = 0 due to viscous effects. δ v = 0 (v − vx )dy and τ0 = −η ∂vx ∂y ∂vx ∂y y=0 The boundary layer is released from the surface if = 0. 0) holds: φ = 2π vr = For a dipole of strength Q in x = a and strength −Q in x = −a follows from superposition: φ = −Qax/2πr 2 where Qa is the dipole strength. If an object is surrounded by an uniform main flow with v = ve x and such a large Re that viscous effects are limited to the boundary layer holds: F x = 0 and Fy = − Γv. far away of a boundary holds: ν t ≈ νRe.10 Self organization ∂ω dω = + J(ω. analogous to Newtonian media: S R = 2 νt D . . t) = √ exp − 4at 2 πat This is mathematical equivalent to the diffusion problem: ∂n = D∇2 n + P − A ∂t where P is the production of and A the discharge of particles. ψ) = ν∇2 ω dt ∂t With J(ω. 9. In three-dimensional flows the situation is just the opposite. Near a boundary holds: ν t = 0. 9. The Navier-Stokes equation now becomes: ∂ v ∇ p divSR + ( v · ∇) v = − + ν∇2 v + ∂t where SR ij = − vi vj is the turbulent stress tensor. ψ) the Jacobian. V ∼ (∇ ψ) ∼ 2 2 0 k 2 E(k.44 Physics Formulary by ir. t)dk = constant From this follows that in a two-dimensional flow the energy flux goes towards large values of k: larger structures become larger at the expanse of smaller ones. So if ν = 0.9 Turbulence √ The time scale of turbulent velocity variations τ t is of the order of: τ t = τ Re/Ma2 with τ the molecular time scale. The onedimensional solution at Φ = 0 is 1 x2 T (x. For the velocity of the particles holds: v(t) = v + v (t) with v (t) = 0. Further. The flow density J = −D∇n.A.C. At x = πD the temperature variation is in anti-phase with the surface. ω is conserved. the kinetic energy/mA and the enstrofy V are conserved: with v = ∇ × ( kψ) For a (semi) two-dimensional flow holds: ∞ ∞ E ∼ (∇ψ) ∼ 2 0 E(k. J. Wevers with D = 2κ/ω c. It is stated that. t)dk = constant . Boussinesq’s assumption is: τ ij = − vi vj . t)eikx dk These waves define a particle with group velocity v g = p/m and energy E = hω. t)e−ikx dx and Ψ(x. fp (t) = Φ∗ f Φd3 Vp This is also written as f (t) = Φ|f |Φ . if light is considered to exist of particles.3 Electron diffraction Diffraction of electrons at a crystal can be explained by assuming that particles have a wave character with wavelength λ = h/p. Wien’s law for the maximum can also be derived from this: T λ max = kW . 10. t) = √ h Φ(k.3 Operators in quantum physics In quantum mechanics. This wavefunction can be described in normal or momentum space. Both definitions are each others Fourier transform: 1 Φ(k. t) = √ h 1 Ψ(x.1.1 Introduction to quantum physics 10.2 Wave functions The wave character of particles is described by a wavefunction ψ. These operators are hermitian because their eigenvalues must be real: ∗ ψ1 Aψ2 d3 V = ψ2 (Aψ1 )∗ d3 V . ¯ The wavefunction can be interpreted as a measure for the probability P to find a particle somewhere (Born): dP = |ψ|2 d3 V . 10. 10.1. classical quantities are translated into operators.1 Black body radiation Planck’s law for the energy distribution for the radiation of a black body is: w(f ) = 1 1 8πhf 3 8πhc .1. can be derived: λ =λ+ h (1 − cos θ) = λ + λC (1 − cos θ) mc 10.2 The Compton effect For the wavelength of scattered light. The expectation value f of a quantity f of a system is given by: f (t) = Ψ∗ f Ψd3 V . This wavelength is called the Broglie-wavelength.Chapter 10 Quantum physics 10. w(λ) = 3 5 ehc/λkT − 1 hf /kT − 1 c λ e Stefan-Boltzmann’s law for the total power density can be derived from this: P = AσT 4 . The normalizing condition for wavefunctions follows from this: Φ|Φ = Ψ|Ψ = 1. Ψ can be expanded into a basis of eigenfunctions: Ψ = cn un . t) both satisfy the Schr¨ dinger o 2 2 equation. . with F = −dU/dx represents the classical equation. t) The following conservation law holds: = −∇J(x. they should be made symmetrical: a classical product AB becomes 1 (AB + BA). 2 10. B] = 0. The energy h h operator is given by: E op = i¯ ∂/∂t. ui |A |un un |B|uj holds: n n n The time-dependence of an operator is given by (Heisenberg): dA ∂A [A.5 The Schr¨ dinger equation o The momentum operator is given by: p op = −i¯ ∇. If [A. The position operator is: x op = i¯ ∇p . H] = 0. p x ] = i¯ holds: ∆px · ∆x ≥ 1 ¯ . The first order approximation F (x) t ≈ F ( x ).4 The uncertainty principle If the uncertainty ∆A in A is defined as: (∆A) 2 = ψ|Aop − A |2 ψ = A2 − A ∆A · ∆B ≥ 1 | ψ|[A. By applying this to p x and x follows (Ehrenfest): md 2 x t /dt2 = − dU (x)/dx . H] = + dt ∂t i¯ h with [A. If this basis is taken orthonormal. In one dimension it is: ψ(x. parity is a conserved quantity. The functions ψ + = 1 (1 + P)ψ(x. the chance to find eigenvalue a n when measuring A is given by |c n |2 in the discrete part of the spectrum and |c n |2 da in the continuous part of the spectrum between a and a + da. Before the addition of quantummechanical operators which are a product of other operators. The Hamiltonian of a particle with mass m. t) ∂t 10. t) and ψ − = 1 (1 − P)ψ(x. The matrix element A ij is given by: Aij = ui |A|uj .A. Hence. For hermitian operators the commutator is always complex. h 2h 2h 2h 2 it follows: 10. B] ≡ AB − BA the commutator of A and B. t) = The current density J is given by: J = + dE c(E)uE (x) exp − iEt h ¯ h ¯ (ψ ∗ ∇ψ − ψ∇ψ ∗ ) 2im ∂P (x. If the system is in a state described by Ψ. potential energy U and total h energy E is given by: H = p 2 /2m + U . it can be expanded into eigenfunctions of P: ψ(x) = 1 (ψ(x) + ψ(−x)) + 1 (ψ(x) − ψ(−x)) 2 2 even: ψ+ odd: ψ− [P. the operators A and B have a common set of eigenfunctions. If the wavefunction is split in even and odd functions. and ∆Lx · ∆Ly ≥ 1 ¯ Lz . B]|ψ | 2 From this follows: ∆E · ∆t ≥ 1 ¯ .6 Parity The parity operator in one dimension is given by Pψ(x) = ψ(−x). and because [x. Wevers When un is the eigenfunction of the eigenvalue equation AΨ = aΨ for eigenvalue a n .C. Because (AB)ij = ui |AB|uj = |un un | = 1. From Hψ = Eψ then follows the Schr odinger equation: ¨ − h ¯2 2 ∂ψ ∇ ψ + U ψ = Eψ = i¯ h 2m ∂t The linear combination of the solutions of this equation give the general solution.46 Physics Formulary by ir. J. then follows for the coefficients: cn = un |Ψ . Lz ] = −¯ L+ follows: Lz (L+ Ylm ) = (m + 1)¯ (L+ Ylm ). ψ3 = A eikx with k 2 = 2m(W − W0 )/¯ 2 and k 2 = 2mW .8 The harmonic oscillator 2 For a harmonic oscillator holds: U = 1 bx2 and ω0 = b/m. Further. L± ] = 0 follows: L2 (L± Ylm ) = l(l + 1)¯ 2 (L± Ylm ). Now holds: L2 = L+ L− + L2 − ¯ Lz . If 1. L2 ] = [Lz . . C and D and A expressed in A. The energylevels are given by E n = n2 h2 /8a2 m. For the normalized eigenfunctions follows: 2h 1 un = √ n! h with En = ( 1 + n)¯ ω.9 Angular momentum For the angular momentum operators L holds: [L z . Further follows that l has to be integral or half-integral. be non-zero within the potential well. There is an eigenfunction u 0 for which holds: Au 0 = 0. Not all components of L can be known at the same time with arbitrary accuracy.Chapter 10: Quantum physics 47 10. Using the boundary conditions requiring continuity: ψ = h continuous and ∂ψ/∂x =continuous at x = 0 and x = a gives B. HAu E = (E − ¯ ω)AuE . ¯ h From [L2 . The Hamiltonian H is then given by: 2 H= with A= 1 2 mωx p2 + 1 mω 2 x2 = 1 ¯ ω + ωA† A 2h 2m 2 ip and A† = +√ 2mω 1 2 mωx ip −√ 2mω A = A† is non hermitian. unlike the classical case. Ly ] = i¯ Lz . For L z h holds: ∂ ∂ ∂ Lz = −i¯ = −i¯ x h −y h ∂ϕ ∂y ∂x The ladder operators L ± are defined by: L ± = Lx ± iLy . 2 A† √ h ¯ n u0 with u0 = 4 mωx2 mω exp − π¯ h 2¯ h 10. h The energy in this ground state is 1 ¯ ω: the zero point energy. Half-odd integral values give no unique solution ψ and are therefore dismissed. H] = hωA. h Because Lx and Ly are hermitian (this implies L † = L∓ ) and |L± Ylm |2 > 0 follows: l(l + 1) − m2 − m ≥ ± 0 ⇒ −l ≤ m ≤ l. If W > W0 and 2a = nλ = 2πn/k holds: T = 1. The amplitude T of the transmitted wave is defined by T = |A |2 /|A|2 . holds: ψ1 = Aeikx + Be−ikx . 10. Lz ] = hL− follows: Lz (L− Ylm ) = (m − 1)¯ (L− Ylm ). [A. A † ] = h and [A. If the wavefunction with energy W meets a potential well of W 0 > W the wavefunction will. h z ¯ L± = he±iϕ ± ∂ ∂ + i cot(θ) ∂θ ∂ϕ h h From [L+ . However. H] = [L2 . A † a ¯ ¯ lowering ladder operator. 2 and 3 are the areas in front. From [L− . within and behind the potential well. ψ2 = Ceik x + De−ik x . A is a so called raising ladder operator.7 The tunnel effect The wavefunction of a particle in an ∞ high potential step from x = 0 to x = a is given by ψ(x) = a−1/2 sin(kx). H] = 0. cyclically holds: [Lx . C. with 2h σx = 0 1 1 0 . The Schr¨ dinger equation then becomes (because ∂χ/∂x i ≡ 0): o i¯ h egS ¯ ∂χ(t) h = σ · Bχ(t) ∂t 4m with σ = (σ x . The spin operators are given by S = 1 ¯ σ. ψ3 (x). the Dirac equation becomes: γλ m0 c ∂ + ∂xλ h ¯ ψ(x. the Klein-Gordon 0 equation is found: 1 ∂2 m 2 c2 ∇2 − 2 2 − 02 ψ(x. . ψ2 (x). y.48 Physics Formulary by ir.A. The electron will have an intrinsic magnetic dipole moment M due to its spin. Because the spin operators h do not act in the physical space (x.11 The Dirac formalism If the operators for p and E are substituted in the relativistic equation E 2 = m2 c4 + p2 c2 . σ z ). σz = 1 0 0 −1 The eigenstates of S z are called spinors: χ = α+ χ+ + α− χ− . Of course holds |α+ |2 + |α− |2 = 1. ψ4 (x)) is a spinor. t) = 0 c ∂t h ¯ The operator ✷ − m 2 c2 /¯ 2 can be separated: h 0 ∇2 − 1 ∂2 m 2 c2 − 02 = 2 ∂t2 c h ¯ γλ m0 c ∂ − ∂xλ h ¯ γµ m0 c ∂ + ∂xµ h ¯ where the Dirac matrices γ are given by: {γ λ . Because [L. Wevers 10. beiωt ). From this can be derived: S x = 1 ¯ cos(2ωt) h h 2 and Sy = 1 ¯ sin(2ωt). Thus the spin precesses about the z-axis with frequency 2ω. From this it can be derived that the γ are hermitian 4 × 4 matrices given by: γk = With this. z) the uniqueness of the wavefunction is not a criterium here: also half odd-integer values are allowed for the spin. So the general solution is given by χ = (ae −iωt . S] = 0 spin and angular momentum operators do not have a common set of eigenfunctions. where χ+ = (1. σ y .10 Spin For the spin operators are defined by their commutation relations: [S x . σy = 0 −i i 0 . given by M = −egS S/2m. This causes the normal 2h Zeeman splitting of spectral lines. Sy ] = i¯ Sz . 1) represents the state with spin down (S z = − 1 ¯ ). γ4 = I 0 0 −I where ψ(x) = (ψ1 (x). Then the probability 2h 2h to find spin up after a measurement is given by |α + |2 and the chance to find spin down is given by |α − |2 . In the presence of an external magnetic field this gives a potential energy U = − M · B. If B = Bez there are two eigenvalues for this problem: χ ± for E = ±egS ¯ B/4m = h ±¯ ω. 0) represents the state with spin up (Sz = 1 ¯ ) and χ− = (0. The potential operator for two particles with spin ± 1 ¯ is given by: 2h V (r) = V1 (r) + 1 (S1 · S2 )V2 (r) = V1 (r) + 1 V2 (r)[S(S + 1) − 3 ] 2 2 h ¯2 This makes it possible for two states to exist: S = 1 (triplet) or S = 0 (Singlet). t) = 0 0 iσk −iσk 0 . with gS = 2(1 + α/2π + · · ·) the gyromagnetic ratio. 10. J. γµ } = γλ γµ + γµ γλ = 2δλµ (In general relativity this becomes 2gλµ ). . with Clm Ylm = √ Plm (cos θ)eimϕ 2π For an atom or ion with one electron holds: R lm (ρ) = Clm e−ρ/2 ρl L2l+1 (ρ) n−l−1 with ρ = 2rZ/na0 with a0 = ε0 h2 /πme e2 .12 Atomic physics 10. 0. with 2j + 1 possible z-components (m J ∈ {−j.12. It can then be derived that: |En |Z 2 α2 a= 2 h ¯ nl(l + 1)(l + 1 ) 2 After a relativistic correction this becomes: E = En + |En |Z 2 α2 n 3 1 − 4n j + 1 2 The fine structure in atomic spectra arises from this. .Chapter 10: Quantum physics 49 10. ϕ)χms . The Lj are the associated Laguere functions and the P lm are the i associated Legendre polynomials: Pl |m| (x) = (1 − x2 )m/2 d|m| (x2 − 1)l dx|m| . ϕ) = R nl (r)Yl.1 Solutions The solutions of the Schr¨ dinger equation in spherical coordinates if the potential energy is a function of r o alone can be written as: ψ(r. J has quantum numbers j with possible values j = l ± 1 . If the interaction 2 energy between S and L is small it can be stated that: E = E n + ESL = En + aS · L. Further holds: J 2 = L2 + S 2 + 2L · S = L2 + S 2 + 2Lz Sz + L+ S− + L− S+ . θ.. The functions are 2 (2l + 1) = 2n2 -folded degenerated.ml (θ. 10. Lm (x) = n n−1 l=0 (−1)m n! −x −m dn−m −x n e x (e x ) (n − m)! dxn−m The parity of these solutions is (−1) l .. The total magnetic dipole moment of an electron is then M = ML + MS = −(e/2me )(L + gS S) where gS = 2.12. where g is the Land´ -factor: h e g =1+ S·J j(j + 1) + s(s + 1) − l(l + 1) =1+ J2 2j(j + 1) For atoms with more than one electron the following limiting situations occur: .3 Spin-orbit interaction The total momentum is given by J = L + M . . j}).12..2 Eigenvalue equations The eigenvalue equations for an atom or ion with with one electron are: Equation Hop ψ = Eψ Lzop Ylm = Lz Ylm L2 Ylm op = L Ylm 2 Eigenvalue En = µe4 Z 2 /8ε2 h2 n2 0 L z = ml ¯ h L = l(l + 1)¯ h Sz = ms ¯ h S 2 = s(s + 1)¯ 2 h 2 2 Range n≥1 −l ≤ ml ≤ l l<n ms = ± 1 2 s= 1 2 Szop χ = Sz χ 2 Sop χ = S 2 χ 10.0023 is the gyromagnetic ratio of the electron. With g S = 2 follows for the average magnetic moment: Mav = −(e/2me )g¯ J. J ∈ {|L − S|. 2.4 Selection rules For the dipole transition matrix elements follows: p 0 ∼ | l2 m2 |E · r |l1 m1 |. Wevers 1. For B = Bez it is given by e2 B 2 (x2 + y 2 )/8µ.. except for very strong fields or macroscopic motions. j − j coupling: for larger atoms the electrostatic interaction is smaller than the L i · si interaction of an electron. For an atom where L − S coupling is dominant further holds: ∆S = 0 (but not strict). 0 + 1. ±1 except for J = 0 → J = 0 transitions. ±1.14. except ∆l = ±1. t)...13 Interaction with electromagnetic fields The Hamiltonian of an electron in an electromagnetic field is given by: H= e2 2 1 h ¯2 e (p + eA)2 − eV = − ∇2 + A − eV B·L+ 2µ 2µ 2µ 2µ where µ is the reduced mass of the system. ∆J = 0. J − 1.jn . S.12.1 Time-independent perturbation theory To solve the equation (H 0 + λH1 )ψn = En ψn one has to find the eigenfunctions of H = H 0 + λH1 . but ∆mJ = 0 is forbidden if ∆J = 0. ∆m J = 0. The spectroscopic notation for this interaction is: 2S+1 LJ . J}. 10. Because f = f (x. The state is characterized by j i . this is called a local gauge transformation. ∆L = 0. J. 10.C. L + S − 1. . Conservation of angular momentum demands that for the transition of an electron holds that ∆l = ±1. The term ∼ A 2 can usually be neglected. For an atom where j − j coupling is dominant further holds: for the jumping electron holds. also: ∆j = 0.. and for all other electrons: ∆j = 0. ±1. mJ where only the j i of the not completely filled subshells are to be taken into account. Suppose 0 that φn is a complete set of eigenfunctions of the non-perturbed Hamiltonian H 0 : H0 φn = En φn . V = V + ∂f /∂t is applied to the potentials the wavefunction h is also transformed according to ψ = ψeiqef /¯ with qe the charge of the particle.. J... . L + S} and mJ ∈ {−J. Because φn is a complete set holds:     cnk (λ)φk ψn = N (λ) φn +   k=n When cnk and En are being expanded into λ: c nk = λcnk + λ2 cnk + · · · (1) (2) 0 En = En + λEn + λ2 En + · · · (1) (2) . mJ . 10. ±1 but no J = 0 → J = 0 transitions and ∆mJ = 0.A. When a gauge transformation A = A − ∇f . The energy difference for larger atoms when placed in a magnetic field is: ∆E = gµ B mJ B where g is the Land´ factor. At higher S the line splits up in more parts: the anomalous Zeeman effect.14 Perturbation theory 10. ±1. Interaction with the spin of the nucleus gives the hyperfine structure.50 Physics Formulary by ir.. for ∆m J = −1. J. but ∆mJ = 0 is forbidden if ∆J = 0. 2S + 1 is the multiplicity of a multiplet. For the total atom holds: ∆J = 0. This e results in the normal Zeeman effect. in contrast with a global gauge transformation which can always be applied. For a transition between two singlet states the line splits in 3 parts. ±1. L − S coupling: for small atoms the electrostatic interaction is dominant and the state can be characterized by L. Now ψ is expanded as:     (1) ψn = N (λ) αi φni + λ cnk βi φki + · · ·   i k=n i 0 0 0 Eni = Eni + λEni is approximated by E ni := En . Substitution in the Schr¨ dinger equation and taking dot o (1) product with φni gives: αi φnj |H1 |φni = En αj . Normalization requires that |αi |2 = 1.1 General Identical particles are indistinguishable.t. ψA (1. 2. In that case an orthonormal set eigenfunctions φ ni is chosen for each level n. so that φ mi |φnj = δmn δij .2 Time-dependent perturbation theory From the Schr¨ dinger equation i¯ o h and the expansion ψ(t) = n t ∂ψ(t) = (H0 + λV (t))ψ(t) ∂t 0 −iEn t h ¯ cn (t) exp φn with cn (t) = δnk + λcn (t) + · · · (1) follows: c(1) (t) n λ = i¯ h 0 φn |V (t )|φk exp 0 0 i(En − Ek )t h ¯ dt 10. . interchange of the coordinates (spatial and spin) of each pair of particles. interchange of the coordinates (spatial and spin) of each pair of particles.r. For the total wavefunction of a system of identical indistinguishable particles holds: 1.15. When a and b are the quantum numbers of electron 1 and 2 holds: ψS (1. i i (1) 10. The second-order correction of the energy is then given by: nm 0 0 En − Em | φk |H1 |φn |2 φk |λH1 |φn (2) En = . 2) = ψa (1)ψb (2) − ψa (2)ψb (1) Because the particles do not approach each other closely the repulsion energy at ψ A in this state is smaller.14. The Pauli principle results from this: two Fermions cannot exist in an identical state because then ψ total = 0. Particles with a half-odd integer spin (Fermions): ψ total must be antisymmetric w.15 N-particle systems 10. So to first order holds: ψ n = φn + φk . Particles with an integer spin (Bosons): ψ total must be symmetric w.r. The following spin wavefunctions are possible: √ χA = 1 2[χ+ (1)χ− (2) − χ+ (2)χ− (1)] ms = 0 2  ms = +1  χ+ (1)χ+ (2) √ 1 χS = 2[χ+ (1)χ− (2) + χ+ (2)χ− (1)] ms = 0  2 χ− (1)χ− (2) ms = −1 Because the total wavefunction must be antisymmetric it follows: ψ total = ψS χA or ψtotal = ψA χS .Chapter 10: Quantum physics 51 and this is put into the Schr¨ dinger equation the result is: E n = φn |H1 |φn and o φm |H1 |φn c(1) = if m = n. 2) = ψa (1)ψb (2) + ψa (2)ψb (1) .t. For a system of two electrons there are 2 possibilities for the spatial wavefunction. 0 0 − E0 0 En En − Ek k k=n k=n (1) In case the levels are degenerated the above does not hold. t. The 3 SP2 orbitals lay in one plane. A better approximation is: ψ(1. 3. Applying this to the function ψ = ci φi one finds: (Hij − ESij )ci = 0. In some atoms. J. such as C... with Tr(ρ) = 1.. Further holds ρ = k eigenvalue an when measuring A is given by ρ nn if one uses a basis of eigenvectors of A for {φ k }. (−1/ 6. p and d states. SP2 hybridization: ψ sp2 = ψ2s / 3 + c1 ψ2pz + c2 ψ2py . N ) = ψ(all permutations of 1.6.16 Quantum statistics If a system exists in a state in which one has not the disposal of the maximal amount of information about the system.15. If the probability that the system is in state ψ i is given by ai . Wevers For N particles the symmetric spatial function is given by: ψS (1. .r. N ) = √ |uEi (j)| N! 10. where (c1 . Also. −1/ 2)}.52 Physics Formulary by ir. SP-hybridization: ψ sp = 1 2(ψ2s ± ψ2pz ). one can write for the expectation value a of A: a = ri ψi |A|ψi .A. SP3 hybridization: ψ sp3 = 1 (ψ2s ± ψ2pz ± ψ2py ± ψ2px ).. like a combination of two s-orbitals it is called a σ-orbital. c2 ) ∈ {( 2/3. This results in a large electron density between the nuclei and therefore a repulsion. Hij = φi |H|φj and Sij = φi |φj . The first approximation in the molecular-orbital theory is to place both electrons of a chemical bond in the bonding orbital: ψ(1.. 0)..N ) 1 The antisymmetric wavefunction is given by the determinant ψ A (1. The probability to find where ρlk = c∗ cl .2 Molecules The wavefunctions of atom a and b are φ a and φb . 10. with C1 = 1 and C2 ≈ 0. S ii = 1 and Sij is the overlap integral. . the connecting axis. with symmetry axes which are at an angle of 120◦ .C. Here. ψ|ψ The energy calculated with this method is always higher than the real energy if ψ is only an approximation for the solutions of Hψ = Eψ. The energy of a system is: E = ψ|H|ψ . √ √ √ 2. the function which gives the lowest energy is the best approximation. For the time-dependence holds (in the Schr¨ dinger image operators are not explicitly time-dependent): o i¯ h dρ = [H. 2) = ψ B (1)ψB (2). like the combination of two p-orbitals along two axes. This equation has only solutions if the secular determinant |H ij − ESij | = 0. ρ] dt .. There are 2 hybrid orbitals which are placed on one line 2 under 180◦ . if there are more functions to be chosen. ρ is hermitian. 2) = C 1 ψB (1)ψB (2) + C2 ψAB (1)ψAB (2). it is energetical more suitable to form orbitals which are a linear combination of the s. (−1/ 6. otherwise a π-orbital. The 4 SP3 orbitals form a tetraheder with the 2 symmetry axes at an angle of 109 ◦28 . There are three ways of hybridization in C: √ 1. Further the 2p x and 2py orbitals remain. If the 2 atoms approach each other there are two possibilities: √ the total wavefunction approaches the bonding function with lower total energy ψ B = 1 2(φa + φb ) or 2 √ approaches the anti-bonding function with higher energy ψ AB = 1 2(φa − φb ). αi := Hii is the Coulomb integral and β ij := Hij the exchange integral. If a molecular-orbital is 2 symmetric w. i If ψ is expanded into an orthonormal basis {φ k } as: ψ (i) = k ck φk . it can be described by a density matrix ρ. holds: (i) A = k (Aρ)kk = Tr(Aρ) ri |ψi ψi |. 1/ 2) √ √ . nk = with ln(Zg ) = gk ln[1 + exp((Ei − µ)/kT )]. Stirling’s equation: ln(n!) ≈ n ln(n) − n. Fermi-Dirac statistics: nk = gk . Bose-Einstein statistics: nk = . It is found by demanding and for it holds: lim µ = EF . U = H = ∂ ln(Z). Zg is the large-canonical state sum and µ the chemical potential. For identical. S = −k Pn ln(Pn ). N is the total number of particles. Bose-Einstein statistics: half odd-integer spin. If all quantumstates with the same energy are equally probable: P i = P (Ei ). ρ] = 0. The Maxwell-Boltzmann distribution can be derived from this in the limit E k − µ nk = Ek N exp − Z kT T →0 kT : gk exp − k with Z = Ek kT With the Fermi-energy. gk N Zg exp((Ek − µ)/kT ) + 1 gk N Zg exp((Ek − µ)/kT ) − 1 2. This function can be found by taking the maximum of the function which gives the number of states with nk = N and nk Wk = W . The distribution function for the internal states for a system in thermal equilibrium is the most probable function. and the conditions k k indistinguishable particles which obey the Pauli exclusion principle the possible number of states is given by: P = k gk ! nk !(gk − nk )! This results in the Fermi-Dirac statistics. They are given by: 1. exp((Ek − EF )/kT ) + 1 gk . exp((Ek − EF )/kT ) − 1 2. So the distribution functions which explain how particles are distributed over the different one-particle states k which are each g k -fold degenerate depend on the spin of the particles. one can obtain the distribution: Pn (E) = ρnn = e−En /kT with the state sum Z = Z e−En /kT n The thermodynamic quantities are related to these definitions as follows: F = −kT ln(Z). nk = N . nk = N with ln(Zg ) = − gk ln[1 − exp((Ei − µ)/kT )]. Here. n k ∈ I . the Fermi-energy. 1}. For indistinguishable particles which do not obey the exclusion principle the possible number of states is given by: P = N! k n gk k nk ! This results in the Bose-Einstein statistics. Fermi-Dirac statistics: integer spin. For a mixed state of M orthonormal quantum states with pn En = − ∂kT n n probability 1/M follows: S = k ln(M ). n k ∈ {0.Chapter 10: Quantum physics 53 For a macroscopic system in equilibrium holds [H. the Fermi-Dirac and Bose-Einstein statistics can be written as: 1. Here. The mean free path is given by λ v = 1/nσ. Because for e-i collisions the energy transfer is only ∼ 2m e /mi this is a slow process. for lower energies this can be a factor 10 lower. τii = τee = ne e4 ln(ΛC ) ni e4 ln(ΛC ) The energy relaxation times for identical particles are equal to the momentum relaxation times. If a plasma contains also negative charged ions α is not well defined. Starting with σm = 4πb2 ln(ΛC ) and with 1 mv 2 = kT it can be 0 2 found that: √ 2 2 3 2√ 3/2 8 2πε0 m(kT ) 4πε m v τm = 4 0 = ne ln(ΛC ) ne4 ln(ΛC ) For momentum transfer between electrons and ions holds for a Maxwellian velocity distribution: √ √ √ √ 6π 3ε2 me (kTe )3/2 6π 3ε2 mi (kTi )3/2 0 0 ≈ τei . 2 Here Lp := |ne /∇ne | is the gradient length of the plasma. 10−17 ve The resistivity η = E/J of a plasma is given by: √ e2 me ln(ΛC ) ne e 2 √ 2 η= = me νei 6π 3ε0 (kTe )3/2 . Approximately holds: τ ee : τei : E τie : τie = 1 : 1 : mi /me : mi /me . A “neat” plasma is defined as a plasma for which holds: b 0 < ne λD Lp . For distances < λ D the plasma cannot be assumed to be quasi-neutral. Deviations of charge neutrality by thermic motion are compensated by oscillations with frequency ωpe = ne e 2 me ε 0 The distance of closest approximation when two equal charged particles collide for a deviation of π/2 is −1/3 2b0 = e2 /(4πε0 1 mv 2 ).1 Introduction ne ne + n0 where ne is the electron density and n 0 the density of the neutrals. The relaxation for e-o interaction is much more complicated. The collision frequency ν c = 1/τc = nσv.2 Transport Relaxation times are defined as τ = 1/ν c . 11. λ D is the Debye length. For T > 10 eV holds approximately: σ eo = −2/5 . The degree of ionization α of a plasma is defined by: α = The probability that a test particle collides with another is given by dP = nσdx where σ is the cross section. The rate coefficient K is defined by K = σv . The number of collisions per unit of time and volume between particles of kind 1 and 2 is given by n 1 n2 σv = Kn1 n2 . The potential of an electron is given by: V (r) = −e r exp − 4πε0 r λD with λD = ε0 kTe Ti ≈ e T i + ni T e ) ε0 kTe ne e 2 e2 (n because charge is shielded in a plasma.Chapter 11 Plasma physics 11. 1 General The scattering angle of a particle in interaction with another particle. One finds that D = 1 λv v.Chapter 11: Plasma physics 55 The diffusion coefficient D is defined by means of the flux Γ by Γ = nvdiff = −D∇n. The equation of continuity is ∂ t n + ∇(nvdiff ) = 0 ⇒ ∂t n = D∇2 n. t) = u + ρ(t). v) ∼ v −4/n . . If both magnetic and electric fields are present electrons and ions will move in the same direction. If E = Er er + Ez ez and B = Bz ez the E × B drift results in a velocity u = ( E × B )/B 2 and the velocity in the ˙ r. For magnetized plasma’s λ v must be replaced with the cyclotron radius. ˙ 11. It arises from the interference of matter waves behind the object. The helical orbit is perturbed by collisions. From this follows that kTe /e − kTi /e kTe µi Damb = ≈ 1/µe − 1/µi e In an external magnetic field B 0 particles will move in spiral orbits with cyclotron radius ρ = mv/eB 0 and with cyclotron frequency Ω = B 0 e/m. ϕ plane is r(r. I(Ω) for angles 0 < χ < λ/4 is larger than the classical value. The coefficient of ambipolar diffusion D amb is defined by Γ = Γi = Γe = −Damb ∇ne. Combined with the two stationary Maxwell equations for the B-field these form the ideal magneto-hydrodynamic equations. moving through a ring with dσ = 2πbdb leave the scattering area at a solid angle dΩ = 2π sin(χ)dχ. In electrical p v fields also holds J = neµE = e(ne µe + ni µi )E with µ = e/mνc the mobility of the particles. ϕ.3. A plasma is called magnetized if λv > ρe.i .3 Elastic collisions 11. In case of magnetic confinement holds: ∇p = J × B. A rough estimate gives 3 τD = Lp /D = L2 τc /λ2 . O(1 eV). σ has a Ramsauer minimum. For a uniform B-field holds: p = nkT = B 2 /2µ0 . The differential cross section is then defined as: I(Ω) = b ∂b dσ = dΩ sin(χ) ∂χ For a potential energy W (r) = kr −n follows: I(Ω. For low energies. So the electrons are magnetized if √ me e3 ne ln(ΛC ) ρe √ 2 = <1 λee 6π 3ε0 (kTe )3/2 B0 Magnetization of only the electrons is sufficient to confine the plasma reasonable because they are coupled to the ions by charge neutrality. The Einstein ratio is: kT D = µ e Because a plasma is electrically neutral electrons and ions are strongly coupled and they don’t diffuse independent. as shown in the figure at the right is: ∞ b s ❅ ❘ ❅ ra b ✻ ❄ ϕ M χ χ = π − 2b ra dr r2 1− b2 W (r) − r2 E0 Particles with an impact parameter between b and b + db.i . Because dp = d(mv) = 0 D mv0 (1 − cos χ) a cross section related to momentum transfer σ m is given by: σm = (1 − cos χ)I(Ω)dΩ = 4πb2 ln 0 1 sin( 1 χmin ) 2 = 4πb2 ln 0 λD b0 := 4πb2 ln(ΛC ) ∼ 0 ln(v 4 ) v4 where ln(ΛC ) is the Coulomb-logarithm. The scattering is free from collective effects if kλD 1. with p = α E. If λ λ 0 it is called Rayleigh 0 0 scattering. This gives b = b0 cot( 1 χ) and 2 I(Ω = b ∂b b2 0 = sin(χ) ∂χ 4 sin2 ( 1 χ) 2 Because the influence of a particle vanishes at r = λ D holds: σ = π(λ2 − b2 ). For this quantity holds: Λ C = λD /b0 = 9n(λD ).3 The induced dipole interaction The induced dipole interaction.2 The Coulomb interaction 2 For the Coulomb interaction holds: 2b 0 = q1 q2 /2πε0 mv0 .4 The centre of mass system If collisions of two particles with masses m 1 and m2 which scatter in the centre of mass system by an angle χ are compared with the scattering under an angle θ in the laboratory system holds: tan(θ) = m2 sin(χ) m1 + m2 cos(χ) The energy loss ∆E of the incoming particle is given by: ∆E = E 1 2 2 m2 v2 1 2 2 m1 v1 = 2m1 m2 (1 − cos(χ)) (m1 + m2 )2 11. W (r) = − =− 4πε0 r2 8πε0 r2 2(4πε0 )2 r4 ∞ with ba = 4 2e2 α holds: χ = π − 2b 2 (4πε0 )2 1 mv0 2 dr r2 1− b2 b4 + a4 2 r 4r ra If b ≥ ba the charge would hit the atom. Wevers 11. If the scattering angle is a lot times 2π it is called capture. and is a lowest estimate for the total cross section.3.56 Physics Formulary by ir. gives a potential V and an energy W in a dipole field given by: V (r) = |e|p αe2 p · er .5 Scattering of light Scattering of light by free electrons is called Thomson scattering. .C.65 · 10 −29 m2 and ∆f 2v = sin( 1 χ) 2 f c This gives for the scattered energy E scat ∼ nλ4 /(λ2 − λ2 )2 with n the density. The cross section for capture σ orb = πb2 is called the Langevin a limit. Thomson sccattering is a limit of Compton scattering. which is given by λ − λ = λC (1 − cos χ) with λC = h/mc and cannot be used any more if relativistic effects become important. The cross section σ = 6. J. so W (r) = 2b0 /r.3. 11. Repulsing nuclear forces prevent this to happen. 11.3.A.3. T )dE = E exp − 3 3/2 c exp(hν/kT ) − 1 kT (πkT ) dE “Detailed balancing” means that the number of reactions in one direction equals the number of reactions in the opposite direction because both processes have equal probability if one corrects for the used phase space. a factor g/Ve remains. T ) := σv K(q. p.4 Thermodynamic equilibrium and reversibility Planck’s radiation law and the Maxwellian velocity distribution hold for a plasma in equilibrium: ρ(ν.5 Inelastic collisions 11. This factor causes the Saha-jump. Excitation: Ap + e− → Aq + e− ← 2. This energy is given by E= m1 m2 (v1 − v2 )2 2(m1 + m2 ) Some types of inelastic collisions important for plasma physics are: 1.1 Types of collisions The kinetic energy can be split in a part of and a part in the centre of mass system. T ) = ∆Epq gp K(p. T )dν = 2πn √ 1 E 8πhν 3 dν . From microscopic reversibility one can derive that for the rate coefficients K(p. T ) exp gq kT pq holds: 11. this gives: ηx = ˆ h3 nx e−Ekin /kT gx (2πmx kT )3/2 where g is the statistical weight of the state and n/g := η. Decay: Aq → Ap + hf ← . The energy in the centre of mass system is available for reactions. For electrons holds g = 2. For the reaction Xforward → Xback ← forward back holds in a plasma in equilibrium microscopic reversibility: ηforward = ˆ forward back ηback ˆ If the velocity distribution is Maxwellian. X p + e− → X1 + e− + (E1p ): ← nB gp p = exp n1 g1 Ep − E1 kTe And for the Saha balance. for excited states usually holds g = 2j + 1 = 2n2 . q. N (E.Chapter 11: Plasma physics 57 11.5. q. X p + e− + (Epi ) → X+ + 2e− : ← 1 nS n+ ne h3 p 1 = + exp gp g1 ge (2πme kTe )3/2 Epi kTe Because the number of particles on the left-hand side and right-hand side of the equation is different. With this one finds for the Boltzmann balance. Because the states have a finite life time. T ) emission stimulated emission absorption Here. The intensity I of a line is q For hydrogenic atoms holds: A p = 1. Resonant charge transfer: A + + A → A + A+ ← 11. This results in πZ 2 e4 dσ = d(∆E) (4πε0 )2 E(∆E)2 Then follows for the transition p → q: σ pq (E) = πZ 2 e4 ∆Eq.25βE Ep For ionization from state p holds to a good approximation: σ p = 4πa2 Ry 0 For resonant charge transfer holds: σ ex = A[1 − B ln(E)]2 1 + CE 3. this gives 2h ∆ν = The natural line width is usually 1 = 4πτp Apq q 4π than the broadening due to the following two mechanisms: . Ionisation and 3-particles recombination: A p + e− → A+ + 2e− ← 4. The natural life time of a state p is given by τ p = 1/ q Apq .5 .6 Radiation In equilibrium holds for radiation processes: np Apq + np Bpq ρ(ν. Stimulated emission: Aq + hf → Ap + 2hf 6.3 11.A. and is given by: Apq = 8π 2 e2 ν 3 |rpq |2 with rpq = ψp |r |ψq 3¯ ε0 c3 h Apq . J. The Einstein coefficients B are given by: Bpq = c3 Apq Bpq gq and = 8πhν 3 Bqp gp A spectral line is broadened by several mechanisms: 1. Ne ∗ + Ar → Ar+ + Ne + e− ← 8.q+1 (4πε0 )2 E(∆E)2 pq 1 1 − Ep E ln 1.C. Charge transfer: A + + B → A + B+ ← 9. Wevers 3.2 Cross sections Collisions between an electron and an atom can be approximated by a collision between an electron and one of the electrons of that atom. with Ap = 1/τp = given by Ipq = hf Apq np /4π. From the uncertainty relation then follows: ∆(hν) · τ p = 1 ¯ .5. Associative ionisation: A∗∗ + B → AB+ + e− ← 7. Apq is the matrix element of the transition p → q.58 · 108 Z 4 p−4. T ) = nq Bqp ρ(ν.58 Physics Formulary by ir.v. radiative recombination: A + + e− → Ap + hf ← 5. Penning ionisation: b. originating from radiative recombination.Chapter 11: Plasma physics 59 2. The velocity is separated in a thermal velocity vt and a drift velocity w. The background radiation in a plasma originates from two processes: 1. Te ) 11. Then follows for the cross section of absorption processes: σ a = Bpq hν/cdν. t)F4 (vx .7 The Boltzmann transport equation It is assumed that there exists a distribution function F for the plasma so that F (r. originating from the acceleration of particles in the EM-field of other particles: εf f = C1 zi ni ne hc √ exp − λ2 kTe λkTe ξf f (λ. Momentum balance: 3. The total line shape is a convolution of the Gauss. Te ) ≈ 3 · 1014 A−3/2 cm−3 . 2. t) Then the BTE is: ∂F dF = + ∇r · (F v ) + ∇v · (F a ) = dt ∂t ∂F ∂t coll−rad Assuming that v does not depend on r and a i does not depend on v i . Mass balance: (BTE)dv ⇒ ∂n + ∇ · (nw) = ∂t ∂n ∂t cr 2. The Doppler broadening is caused by the thermal motion of the particles: ∆λ 2 = λ c 2 ln(2)kTi mi This broadening results in a Gaussian line profile: √ kν = k0 exp(−[2 ln 2(ν − ν0 )/∆νD ]2 ). The number of transitions p → q is given by n p Bpq ρ and by np nhf σa c = np (ρdν/hν)σa c where dν is the line width. Te ) 2/3 ˚ with for the H-β line: C(n e . v. t) = F1 (x. Te ) with C1 = 1. t)F6 (vz . The total density is given by n = F dv and vF dv = nw. This is also true in magnetic fields because ∂a i /∂xi = 0. t) · Fv (v. The emission is given by: εf b = C1 zi ni ne hc √ 1 − exp − 2 λ λkTe kTe ξf b (λ. The natural broadening and the Stark broadening result in a Lorentz profile of a spectral line: 1 kν = 1 k0 ∆νL /[( 2 ∆νL )2 + (ν − ν0 )2 ].63 · 10−43 Wm4 K1/2 sr−1 and ξ the Biberman factor. holds ∇r ·(F v ) = v·∇F and ∇v ·(F a ) = a · ∇v F . t)F3 (z. The balance equations can be derived by means of the moment method: 1. with k the coefficient of absorption or emission. Free-Bound radiation. t)F5 (vy . t) = Fr (r. t)F2 (y.and Lorentz profile 2 and is called a Voigt profile. Free-free radiation. 3. The Stark broadening is caused by the electric field of the electrons: ∆λ1/2 = ne C(ne . Energy balance: (BTE)mvdv ⇒ mn dw + ∇T + ∇p = mn a + R dt (BTE)mv 2 dv ⇒ 3 dp 5 + p∇ · w + ∇ · q = Q 2 dt 2 . coll. recomb rad. Even in plasma’s far from equilibrium the excited levels will eventually reach ESP. coll. nq Kqp + ne q>p nq Kqp + q>p nq Aqp + n2 ni K+p + ne ni αrad = e coll. For electromagnetic waves with complex wave number k = ω(n + iκ)/c in one dimension one finds: Ex = E0 e−κωx/c cos[ω(t − nx/c)]. deex. excit. 0 1 rp nS +rp nB p p To find the population densities of the lower levels in the stationary case one has to start with a macroscopic equilibrium: Number of populating processes of level p = Number of depopulating processes of level p . 2 2 mvt ∂F Q = dv the source term for energy production. One assumes the Quasi-steady-state solution is valid. q = 1 nm vt2 vt the heat flow. cr ∂n1 + ∇ · (n1 w1 ) = ∂t b p nS .9 Waves in plasma’s Interaction of electromagnetic waves in plasma’s results in scattering and absorption of energy. to coll. excit. rad. a = e/m(E + w × B ) is the average acceleration. so from a certain level up the level densities can be calculated. where ∀p>1 [(∂np /∂t = 0) ∧ (∇ · (np wp ) = 0)]. deexcit. When this is expanded it becomes: ne q<p coll.A. E E0 3/2 exp − 3me m0 E E0 2 dE 11. deex. where only collisional (de)excitation between levels p and p ± 1 is taken into account holds x = 6.C. The refractive index n is given by: n=c c k = = ω vf 1− 2 ωp ω2 . R is a friction term and p = nkT the r ∂t cr pressure. ne np q<p Kpq + ne np q>p Kpq + np q<p Apq + ne np Kp+ coll. A thermodynamic derivation gives for the total pressure: p = nkT = i pi − e2 (ne + zi ni ) 24πε0 λD wi : For the electrical conductance in a plasma follows from the momentum balance.8 Collision-radiative models These models are first-moment equations for excited states. J. from 11. ion. cr ∂ni + ∇ · (ni wi ) = ∂t ∂ni ∂t cr with solutions np = = Further holds for all collision-dominated levels that δb p := bp −1 = −x b0 peff with peff = Ry/Epi and 5 ≤ x ≤ 6. For systems in ESP. recomb. rad. p ∂n1 ∂t . Wevers Here. deexcit.60 Physics Formulary by ir. if w e ηJ = E − J × B + ∇pe ene In a plasma where only elastic e-a collisions are important the equilibrium energy distribution function is the Druyvesteyn distribution: N (E)dE = Cne with E0 = eEλv = eE/nσ. This results in: ∂np>1 ∂t =0 . The dielectric tensor E. With the Alfv´ n velocity e e vA = follows: n = 1 + c/vA . For these holds: P = 0 or R = 0 or L = 0. with σ = iε0 ω s   α2  s  1 2 1 − βs iβs 2 1 − βs 0 −iβs 2 1 − βs 1 2 1 − βs 0  0   0   1 where the sum is taken over particle species s. n2 = L: a left. so vf = 0. 1). homogeneous. so vf → ∞. B. magnetized plasma: B = B0 ez + Bei(kz−ωt) and i(kz−ωt) n = n0 + ne ˆ (external E fields are screened) follows. L = S − D. n2 = P : the ordinary mode: E x = Ey = 0. n2 = RL/S: the extraordinary mode: iE x /Ey = −D/S. P = 0: Ex = Ey = 0. 2. and in case vA 2 ωpe Ωe Ωi 2 2 c + ωpi c: ω = kvA . i. er is connected with a right rotating field for which iEx /Ey = 1 and el is connected with a left rotating field for which iE x /Ey = −1. n2 = R: a right. This is a transversal linear polarized wave. λ 3 = P . 1.Chapter 11: Plasma physics 61 ˆ For disturbances in the z-direction in a cold. For these holds: tan(θ) = −P/S. P =1− 2 1 − βs −iD S 0  0 0  P α2 s s follows: S E =  iD 0 The eigenvalues of this hermitian matrix are R = S + D. an elliptical polarized wave. 0. θ = 0: transmission in the z-direction. D= 2 1 − βs  s α2 βs s . 0) and e3 = (0. From this the following solutions can be obtained: A. for L → ∞ the ion cyclotron resonance frequency ω = Ω i and for S = 0 holds for the extraordinary mode: α2 1 − m i Ω2 i me ω 2 = 1− m 2 Ω2 i i m2 ω 2 e 1− Ω2 i ω2 Cut-off frequencies are frequencies for which n 2 = 0. 1. . Resonance frequencies are frequencies for which n 2 → ∞. circular polarized wave. In the case that β 2 1 one finds Alfv´ n waves propagating parallel to the field lines. with the definitions α = ω p /ω and β = Ω/ω 2 2 2 and ωp = ωpi + ωpe:  J = σ E . 2. el = 2 2(1. When k makes an angle θ with B one finds: P (n2 − R)(n2 − L) tan2 (θ) = S(n2 − RL/S)(n2 − P ) where n is the refractive index. With the definitions S = 1 − s α2 s . circular polarized wave. with property: k · (E · E) = 0 is given by E = I − σ/iε0 ω. 3. This describes a longitudinal linear polarized wave. θ = π/2: transmission ⊥ the B-field. 0). −i. with eigenvectors e r = √ √ 1 1 2 2(1. For R → ∞ this gives the electron cyclotron resonance frequency ω = Ω e . 2 Crystal binding A distinction can be made between 4 binding types: 1. As a primitive cell one can take a parallellepiped. A N lattice can be constructed from primitive cells. From this follows for parallel lattice planes (Bragg reflection) that for the maxima holds: 2d sin(θ) = nλ. If G is written as G = v1 b1 + v2 b2 + v3 b3 with vi ∈ I . The potential energy for two parallel spins is higher than the potential energy for two antiparallel spins. Metalic bond. Ion bond 3. Van der Waals bond 2. with volume Vcell = |a1 · (a2 × a3 )| Because a lattice has a periodical structure the physical properties which are connected with the lattice have the same periodicity (neglecting boundary effects): ne (r + T ) = ne (r ) This periodicity is suitable to use Fourier analysis: n(r ) is expanded as: n(r ) = G nG exp(iG · r ) with nG = 1 Vcell cell n(r ) exp(−iG · r )dV G is the reciprocal lattice vector. 12. . In that case binding is not possible. Covalent or homopolar bond 4. Furthermore the potential energy for two parallel spins has sometimes no minimum.1 Crystal structure A lattice is defined by the 3 translation vectors a i . where T is a translation vector given by: T = u1 a1 + u2 a2 + u3 a3 with ui ∈ I . it follows for the N vectors bi . so that the atomic composition looks the same from each point r and r = r + T .Chapter 12 Solid state physics 12. The Brillouin zone is defined as a Wigner-Seitz cell in the reciprocal lattice. cyclically: ai+1 × ai+2 bi = 2π ai · (ai+1 × ai+2 ) The set of G-vectors determines the R¨ ntgen diffractions: a maximum in the reflected radiation occurs if: o ∆k = G with ∆k = k − k . For the ion binding of NaCl the energy per molecule is calculated by: E = cohesive energy(NaCl) – ionization energy(Na) + electron affinity(Cl) The interaction in a covalent bond depends on the relative spin orientations of the electrons constituing the bond. So: 2k · G = G2 . Here. one longitudinal and two transversal. The force on atom s with mass M can then be written as: Fs = M d2 us = C(us+1 − us ) + C(us−1 − us ) dt2 Assuming that all solutions have the same time-dependence exp(−iωt) this results in: −M ω 2 us = C(us+1 + us−1 − 2us ) Further it is postulated that: u s±1 = u exp(isKa) exp(±iKa). for an umklapp process holds: K 1 + K2 = K3 + G. and also a stronger emission and absorption of radiation. which has only a solution if their determinant is 0. . Phonons h can be viewed as quasi-particles: with collisions.3 Crystal vibrations 12.3. Substituting the later two equations in the fist results in a system of linear equations. The upper line describes the optical branch. This requires that −π < Ka ≤ π.3 Phonons The quantum mechanical excitation of a crystal vibration with an energy ¯ ω is called a phonon. This results in a much larger induced dipole moment for optical oscillations. Because phonons have no spin they behave like bosons.Chapter 12: Solid state physics 63 12.3.2 A lattice with two types of atoms Now the solutions are: ω2 = C 1 1 + M1 M2 ±C 1 1 + M1 M2 2 ω ✻ − 4 sin2 (Ka) M1 M2 2C M2 2C M1 Connected with each value of K are two values of ω. both types of ions oscillate in opposite phases. in the acoustical branch they oscillate in the same phase.3. Their total h momentum is 0. The group velocity of these vibrations is given by: vg = dω = dK Ca2 cos( 1 Ka) . In the optical branch. This gives: us = exp(iKsa). Furthermore each branch has 3 polarization directions. the lower line the acoustical branch. This gives: ω2 = 4C sin2 ( 1 Ka) 2 M Only vibrations with a wavelength within the first Brillouin Zone have a physical significance. 2 M and is 0 on the edge of a Brillouin Zone. When they collide. there is a standing wave. 0 ✲ K π/a 12. as can be seen in the graph. their momentum need not be conserved: for a normal process holds: K1 + K2 = K3 .1 A lattice with one type of atoms In this model for crystal vibrations only nearest-neighbour interactions are taken into account. 12. they behave as particles with momentum ¯ K. or: L 2π 3 = V 8π 3 allowed K-values per unit volume in K-space.A. Because exp(2πi) = 1 this is only possible if: Kx . xD = hωD /kT = θD /T . ± . an approximation for optical phonons.. Here. and one assumes that v = ωK. Because x D → ∞ for T → 0 it follows from this: U = 9N kT T θD 3 ∞ 0 3π 4 N kT 4 x3 dx 12π 4 N kT 3 = ∼ T 4 and CV = ∼ T3 3 x−1 e 5θD 5θD In the Einstein model for the thermal heat capacity one considers only phonons at one frequency. . ± . This gives: ωD U =3 D(ω) n ¯ ωdω = h 0 3V k 2 T 4 V ω2 h ¯ω dω = 2π 2 v 3 exp(¯ ω/kT ) − 1 h 2π 2 v 3 ¯ 3 h xD 0 x3 dx . ± 4π 6π 2N π 2π .3.C. ex − 1 Here. according to the Einstein model: D(ω) = dN = dω V K2 2π 2 dK V = dω 8π 3 dAω vg The Debye model for thermal heat capacities is a low-temperature approximation which is valid up to ≈ 50K. Ky .64 Physics Formulary by ir. only the acoustic phonons are taken into account (3 polarizations). The total number of states with a wave vector < K is: 3 L 4πK 3 N= 2π 3 for each polarization. J. independent of the polarization.p = λ Dλ (ω) h ¯ω dω exp(¯ ω/kT ) − 1 h for a given polarization λ. ± L L L L So there is only one allowed value of K per volume (2π/L) 3 in K-space. for each polarization and each branch.4 Thermal heat capacity The total energy of the crystal vibrations can be calculated by multiplying each mode with its energy and sum over all branches K and polarizations P : U= K P h ¯ ω nk. Wevers 12. From this follows: D(ω) = V ω 2 /2π 2 v 3 . The density of states for each polarization is. The thermal heat capacity is then: Clattice = ∂U =k ∂T D(ω) λ h (¯ ω/kT )2 exp(¯ ω/kT ) h dω (exp(¯ ω/kT ) − 1)2 h The dispersion relation in one dimension is given by: D(ω)dω = L dK L dω dω = π dω π vg In three dimensions one applies periodic boundary conditions to a cube with N 3 primitive cells and a volume L3 : exp(i(Kx x + Ky y + Kz z)) ≡ exp(i(Kx (x + L) + Ky (y + L) + Kz (z + L))). θD is the Debye temperature and is defined by: ¯ θD = h ¯v k 6π 2 N V 1/3 where N is the number of primitive cells. where v is the speed of sound. .. Kz = 0. 4. and with a fm µ/ fm . Ba T − Tc If BE is estimated this way it results in values of about 1000 T. J is an overlap integral given by: J = 3kT c /2zS(S + 1). . After distribution fm ∼ exp(−∆Um /kT ). one finds for the average magnetic moment µ = linearization and because mJ = 0. µ0 N µ µ0 J(J + 1)g 2 µ2 N µ0 M B = = B B 3kT 12. Hence there is a net circular current which results in a magnetic moment µ. J > 0: Si and Sj become parallel: the material is a ferromagnet. A quantum mechanical approach from Heisenberg postulates an interaction between two neighbouring atoms: U = −2J Si · Sj ≡ −µ · BE . J = 2J + 1 and m2 = 2 J(J + 1)(J + 1 ) it follows that: J 3 2 χp = This is the Curie law. If N is the number of atoms in the crystal it follows for the susceptibility.4 Magnetic field in the solid state The following graph shows the magnetization versus fieldstrength for different types of magnetism: M ✻ Msat χm ∂M = ∂H ferro paramagnetism 0 ❤❤❤❤ ❤❤❤ diamagnetism ❤❤❤❤ ❤ ✲ H 12. with z the number of neighbours. To describe ferromagnetism a field BE parallel with M is postulated: BE = λµ0 M . the other decelerated.2 Paramagnetism Starting with the splitting of energy levels in a weak magnetic field: ∆U m − µ · B = mJ gµB B. One electron is accelerated. 3 with M = µN : µ0 N Ze2 2 µ0 M =− r χ= B 6m 12.4. A distinction between 2 cases can now be made: 1.Chapter 12: Solid state physics 65 12. From there the treatment is analogous to the paramagnetic case: C µ0 M = χp (Ba + BE ) = χp (Ba + λµ0 M ) = µ0 1 − λ M T From this follows for a ferromagnet: χ F = µ0 M C = which is Weiss-Curie’s law. ωL is the Larmor frequency. Starting with a circular electron orbit in an atom with two electrons. If the magnetic part of the force is not strong enough to significantly deform the orbit holds: ω2 = Fc (r) eB eB 2 ± ω = ω0 ± (ω0 + δ) ⇒ ω = mr m m ω0 ± eB 2m 2 + · · · ≈ ω0 ± eB = ω0 ± ωL 2m Here.3 Ferromagnetism A ferromagnet behaves like a paramagnet above a critical temperature T c .1 Dielectrics The quantum mechanical origin of diamagnetism is the Larmorprecession of the spin of the electron. The circular current is given by I = −Zeω L /2π. there is a Coulomb force F c and a magnetic force on each electron. and µ = IA = Iπ ρ2 = 2 Iπ r2 . χp ∼ 1/T . This is clearly unrealistic and suggests another mechanism.4. 12.1 Thermal heat capacity The solution with period L of the one-dimensional Schr¨ dinger equation is: ψ n (x) = A sin(2πx/λn) with o nλn = 2L. The internal energy then becomes: T ∂U T ≈ Nk and C = U ≈ N kT TF ∂T TF A more accurate analysis gives: C electrons = 1 π 2 N kT /TF ∼ T . J < 0: Si and Sj become antiparallel: the material is an antiferromagnet. Because for the collision time holds: 1/τ = 1/τL + 1/τi .5 Free electron Fermi gas 12. √ 3N . The Fermi level is the uppermost filled level in the ground state. Then holds: v = µ E. Wevers 2. This is caused 2 by the Pauli exclusion principle and the Fermi-Dirac distribution: only electrons within an energy range ∼ kT of the Fermi level are excited thermally. From this follows h ¯ 2 nπ 2 E= 2m L In a linear lattice the only important quantum numbers are n and m s .A.66 Physics Formulary by ir. δ k remains stable if h t = τ . with lim ρL = 0. If τ is the characteristic collision time of the electrons. 2E V dN = dE 2π 2 E= The heat capacity of the electrons is approximately 0. with µ = eτ /m the mobility of the electrons. Together with the T 3 dependence of the 2 thermal heat capacity of the phonons the total thermal heat capacity of metals is described by: C = γT + AT 3 . which has the Fermi-energy E F . Heisenberg’s theory predicts quantized spin waves: magnons. In 3 dimensions holds: ¯ kF = 3π 2 N V 1/3 and EF = V 3π 2 h ¯2 2m 3π 2 N V 3/2 2/3 The number of states with energy ≤ E is then: N = and the density of states becomes: D(E) = 2mE h ¯2 2m h ¯2 3/2 . T →0 .01 times the classical expected value 3 N k. The variation of k is given by ¯ δ k = k(t) − k(0) = −eEt/¯ . where τL is the collision time with the lattice phonons and τ i the collision time with the impurities follows for the resistivity ρ = ρ L + ρi . There is a fraction ≈ T /T F excited thermally.5. If nF is the quantum number of the Fermi level.2 Electric conductance The equation of motion for the charge carriers is: F = mdv/dt = hdk/dt. Starting from a model with only nearest neighbouring atoms interacting one can write: U = −2J Sp · (Sp−1 + Sp+1 ) ≈ µp · Bp with Bp = −2J (Sp−1 + Sp+1 ) gµB The equation of motion for the magnons becomes: 2J dS = Sp × (Sp−1 + Sp+1 ) dt h ¯ From here the treatment is analogous to phonons: postulate traveling waves of the type Sp = u exp(i(pka − ωt)). This results in a system of linear equations with solution: hω = 4JS(1 − cos(ka)) ¯ 12. it can be expressed as: 2n F = N so EF = h2 π 2 N 2 /8mL.C. J. The current in a conductor is given by: J = nqv = σ E = E/ρ = neµE.5. 4 Thermal heat conductivity With = vF τ the mean free path of the electrons follows from κ = 1 C v 3 From this follows for the Wiedemann-Franz ratio: κ/σ = 1 (πk/e)2 T .6 Energy bands In the tight-bond approximation it is assumed that ψ = e ikna φ(x − na).3 The Hall-effect If a magnetic field is applied ⊥ to the direction of the current the charge carriers will be pushed aside by the Lorentz force. The Hall coefficient is defined by: R H = Ey /Jx B. This is a traveling wave. The Hall voltage is given by: V H = Bvb = IB/neh where b is the width of the material and h de height. Suppose that U (x) = U cos(2πx/a). Hence the energy of ψ(+) is also lower than the energy of ψ)(−). So this gives a cosine superimposed on the atomic energy. than the bandgap is given by: 1 Egap = 0 U (x) |ψ(+)|2 − |ψ(−)|2 dx = U 12. which can often be approximated by a harmonic oscillator. 12. also a phonon is absorbed. Here it is assumed that the momentum of the absorbed photon can be neglected. From this follows for the energy: E = ψ|H|ψ = Eat − α − 2β cos(ka). 3 : κelectrons = π 2 nk 2 T τ /3m. 12. This wave can be decomposed into two standing waves: ψ(+) = ψ(−) = exp(iπx/a) + exp(−iπx/a) = 2 cos(πx/a) exp(iπx/a) − exp(−iπx/a) = 2i sin(πx/a) The probability density |ψ(+)| 2 is high near the atoms of the lattice and low in between. For an indirect semiconductor a transition from the valence. This results in a magnetic field ⊥ to the flow direction of the current. If J = Jex and B = Bez than Ey /Ex = µB.7 Semiconductors The band structures and the transitions between them of direct and indirect semiconductors are shown in the figures below. and RH = −1/ne if Jx = neµEx .5.5. The probability density |ψ(−)|2 is low near the atoms of the lattice and high in between. If it is assumed that the electron is nearly free one can postulate: ψ = exp(ik · r ).Chapter 12: Solid state physics 67 12. conduction E band ✻ • ¤ ✻ ✞✆ ω ✝¤ g ✞✆ ✝ ◦ E ✻ ✛ • Ω ✞ ✻ ω ✝¤ ✞✆ ✝ ◦ Direct transition This difference can also be observed in the absorption spectra: Indirect transition .to the conduction band is also possible if the energy of the absorbed photon is smaller than the band gap: then. for a n − type holds: n > p and in a p − type holds: n < p. ✡ . When light is absorbed holds: kh = −ke . cannot emit any light and are therefore not usable to fabricate lasers. The excitation energy of an exciton is smaller than the bandgap because the energy of an exciton is lower than the energy of a free electron and a free hole. For the product np follows: np = 4 An exciton is a bound electron-hole pair.. rotating on each other as in positronium.8.14ΘD exp while experiments find for H c approximately: Hc (T ) ≈ Hc (Tc ) 1 − T2 2 Tc . The BCS-model predicts for the transition temperature T c : Tc = 1. ✡ ... vh = ve and mh = −m∗ if the e conduction band and the valence band have the same structure.. Instead of the normal electron mass one has to use the effective mass within a lattice. Eh (kh ) = −Ee (ke ). J. From this follows for the concentrations of the holes p and the electrons n: ∞ n= Ec De (E)fe (E)dE = 2 3 m∗ kT 2π¯ 2 h 3/2 exp µ − Ec kT kT Eg m∗ mh exp − e kT 2π¯ 2 h For an intrinsic (no impurities) semiconductor holds: n i = pi . .A. Eg + hΩ ¯ h ¯ ωg ✲E ✲ E Direct semiconductor Indirect semiconductor So indirect semiconductors. . I < Ic and H < Hc . It is defined by: m∗ = dp/dt dK F = =h ¯ = h2 ¯ a dvg /dt dvg d2 E dk 2 −1 with E = hω and vg = dω/dk and p = hk. .C. . . This causes a peak in the absorption just under E g ... Wevers absorption ✻ absorption ✻ ..8 Superconductivity 12. ¯ ¯ With the distribution function f e (E) ≈ exp((µ − E)/kT ) for the electrons and f h (E) = 1 − fe (E) for the holes the density of states is given by: D(E) = 1 2π 2 2m∗ h ¯2 3/2 E − Ec with Ec the energy at the edge of the conductance band.68 Physics Formulary by ir.1 Description A superconductor is characterized by a zero resistivity if certain quantities are smaller than some critical values: T < Tc . . −1 U D(EF ) . . . like Si and Ge. 12. ✡ . . This holds for a type I superconductor. n )dσ = Ψ follows: Ψ = 2π¯ s/q. h . for a type II superconductor this only holds to a certain value H c1 . This is shown in the figures below. The electron √ √ wavefunctions are written as ψ 1 = n1 exp(iθ1 ) and ψ2 = n2 exp(iθ2 ). for higher values of H the superconductor is in a vortex state to a value H c2 .0678 · 10 −15 Tm2 . From this follows. h ¯ Hence there is an oscillation with ω = 2eV /¯ . This means that there is a twist in the T − S diagram and a discontinuity in the C X − T diagram. follows: ¯ ∇θ = q A ⇒ ∇θdl = θ2 − θ1 = 2πs with s ∈ I .8. The size of a flux quantum h follows by setting s = 1: Ψ = 2π¯ /e = 2. If H becomes larger than H c2 the superconductor becomes a normal conductor. where J0 ∼ T . or the transfer interaction through the insulator. Because a Cooper pair exist of two √ electrons holds: ψ ∼ n. h N Because: Adl = (rotA.8.3 Flux quantisation in a superconducting ring For the current density in general holds: J = qψ ∗ vψ = nq [¯ ∇θ − q A ] h m From the Meissner effect. With an AC-voltage across the junction the Schr¨ dinger o equations become: ∂ψ1 ∂ψ2 i¯ h = hT ψ2 − eV ψ1 and i¯ ¯ = hT ψ1 + eV ψ2 ¯ h ∂t ∂t This gives: J = J0 sin θ2 − θ1 − 2eV t .2 The Josephson effect For the Josephson effect one considers two superconductors.Chapter 12: Solid state physics 69 Within a superconductor the magnetic field is 0: the Meissner effect. B = 0 and J = 0. The Schr¨ dinger equations in both superconductors is o set equal: ∂ψ2 ∂ψ1 = hT ψ2 . 12. µ0 M ✻                 Hc Type I µ0 M ✻ ·  · ·  · · ·   · · · · ·   · · · ·   Hc1 Type II ✲H ✲ H Hc2 The transition to a superconducting state is a second order thermodynamic state transition. n )dσ = (B. There are type I and type II superconductors. Because the Meissner effect implies that a superconductor is a perfect diamagnet holds in the superconducting state: H = µ0 M . The electron wavefunction in one superconductor is ψ 1 . separated by an insulator. in the other ψ 2 . h 12. if n 1 ≈ n2 : ∂θ1 ∂θ2 ∂n2 ∂n1 = and =− ∂t ∂t ∂t ∂t The Josephson effect results in a current density through the insulator depending on the phase difference as: J = J0 sin(θ2 − θ1 ) = J0 sin(δ). which can be 100 times H c1 . i¯ ¯ = hT ψ1 ¯ h i¯ h ∂t ∂t h ¯ T is the effect of the coupling of the electrons. Flux quantization is related to the existence of a coherent many-particle wavefunction. h 12. p . Magnetic fields within a superconductor drop exponentially.A.70 Physics Formulary by ir. From this follows: ∇2 B = B/λ2 . Because of the interaction with the lattice these pseudo-particles exhibit a mutual attraction.C.5 The London equation A current density in a superconductor proportional to the vector potential A is postulated: J= where λL = −A −B or rotJ = 2 µ0 λL µ0 λ2 L ε0 mc2 /nq 2 . Some useful mathematical relations are: ∞ ∞ ∞ π2 xdx = . This causes two electrons with opposite spin to combine to a Cooper pair. ax + 1 e 12a2 ∞ 0 ∞ −∞ x2 dx π2 . So if the flux has to change with one flux quantum there has to occur a transition of many electrons. L The Meissner effect is the solution of this equation: B(x) = B0 exp(−x/λL ). It can be proved that this ground state is perfect diamagnetic.6 The BCS model The BCS model can explain superconductivity in metals. = x + 1)2 (e 3 ∞ π4 x3 dx = x+1 e 15 0 And. 12.4 Macroscopic quantum interference From θ2 − θ1 = 2eΨ/¯ follows for two parallel junctions: δ b − δa = h J = Ja + Jb = 2J0 sin δ0 cos eΨ h ¯ 2eΨ . when (−1) = n=0 n 1 2 follows: 0 sin(px)dx = 0 cos(px)dx = 1 . or the system must go through intermediary states where the flux is not quantized so they have a higher energy. (So far there is no explanation for high-T c superconductance). Wevers 12. so h ¯ This gives maxima if eΨ/¯ = sπ. The infinite conductivity is more difficult to explain because a ring with a persisting current is not a real equilibrium: a state with zero current has a lower energy.8. A new ground state where the electrons behave like independent fermions is postulated.8. This is also very improbable. Flux quantization prevents transitions between these states. which is very improbable. J.8. A flux quantum is the equivalent of about 10 4 electrons. A conjugacy class is the maximum collection of conjugated elements.1. ∀A.1 Introduction 13. A subgroup is a subset of G which is also a group w.B∈G ⇒ A • B ∈ G: G is closed.1. There are h positions in each row and column when there are h k elements in the group so each element appears only once. ∀A. . 13.B.or multiplication table: because EA i = A−1 (Ak Ai ) = Ai each Ai appears once. subgroups and classes B is conjugate to A if ∃ X∈G such that B = XAX −1 . ∃E∈G so that ∀A∈G A • E = E • A = A: G has a unit element.r. The physical interpretation of classes: elements of a group are usually symmetry operations which map a symmetrical object into itself.Chapter 13 Theory of groups 13. ∀A∈G ∃A−1 ∈G so that A • A−1 = E: Each element in G has an inverse. Elements of one class are then the same kind of operations. • E is a class itself: for each other element in this class would hold: A = XEX −1 = E.t. If B and C are conjugate to A. 2. ∀A. • In an Abelian group each element is a separate class. 13. Each group can be split up in conjugacy classes. The opposite need not to be true.1. If also holds: 5.2 The Cayley table Each element arises only once in each row and column of the Cayley. 3.3 Conjugated elements. Some theorems: • All classes are completely disjoint. the same operation. 4.1 Definition of a group G is a group for the operation • if: 1.C∈G ⇒ (A • B) • C = A • (B • C): G obeys the associative law. • E is the only class which is also a subgroup because all other classes have no unit element. B is also conjugate with C.B∈G ⇒ A • B = B • A the group is called Abelian or commutative. Then A is also conjugate to B because B = (X −1 )A(X −1 )−1 . γ (1) (Ai ) and γ (2) (Ai ). 3.4 Isomorfism and homomorfism.1 Schur’s lemma Lemma: Each matrix which commutes with all matrices of an irreducible representation is a constant ×I I. J. so that the multiplication table remains the same is a homomorphic mapping.1. The opposite is (of course) also true. The identical representation: A → 1. or M = 0. The number of irreducible representations equals the number of conjugacy classes. The mapping from group G 1 to G2 .3 Character The character of a representation is given by the trace of the matrix and is therefore invariant for base transformations: χ(j) (R) = Tr(Γ(j) (R)) Also holds. holds: M γ (1) (Ai ) = γ (2) (Ai )M .2 The fundamental orthogonality theorem For a set of unequivalent. if h is the number of elements in the group and i is the dimension of the i th representation: ¯ (i)∗ Γµν (R)Γαβ (R) = R∈G (j) h i δij δµα δνβ 13. A representation is a homomorphic mapping of a group to a group of square matrices with the usual matrix multiplication as the combining operation.5 Reducible and irreducible representations If the same unitary transformation can bring all matrices of a representation Γ in the same block structure the representation is called reducible: Γ(A) = Γ(1) (A) 0 0 Γ(2) (A) This is written as: Γ = Γ(1) ⊕ Γ(2) .2.2. unitary representations holds that. where I is the unit matrix. This is symbolized by Γ.2. irreducible. If this is not possible the representation is called irreducible. The following holds: Γ(E) = I . A faithful representation: all matrices are different. I Lemma: If there exists a matrix M so that for two irreducible representations of group G.72 Physics Formulary by ir.2 The fundamental orthogonality theorem 13.A. 2. 13.C. with Nk the number of elements in a conjugacy class: k n 2 i i=1 χ(i)∗ (Ck )χ(j) (Ck )Nk = hδij Theorem: =h . Wevers 13. Γ(AB) = Γ(A)Γ(B) . 13. Γ(A−1 ) = [Γ(A)]−1 I For each group there are 3 possibilities for a representation: 1. than the representations are equivalent. It need not be isomorphic. The representation A → det(Γ(A)). representations Two groups are isomorphic if they have the same multiplication table. 13.1. An equivalent representation is obtained by performing an unitary base transformation: Γ (A) = S −1 Γ(A)S. 3. unitary representation of the symmetry group G of the system. This is considered an active rotation. If Γ(n) (R) is an irreducible representation of G 0 .3 The construction of a base function n j Each function F in configuration space can be decomposed into symmetry types: F = j=1 κ=1 (j) fκ The following operator extracts the symmetry types: j h Γ(j)∗ (R)PR κκ R∈G (j) F = fκ . .. with energy En is a basis for an n -dimensional 13. For 0 ≤ p ≤ h − 1 follows: Γ(p) (An ) = e2πipn/h If one defines: k = − 2πp 2π mod . The representation then usually becomes reducible: Γ(n) = Γ(n1 ) ⊕ Γ(n2 ) ⊕ . Assume an PR ψν (n) n -fold degeneracy at E n : then choose an orthonormal set ψ ν . .2 Breaking of degeneracy by a perturbation Suppose the unperturbed system has Hamiltonian H 0 and symmetry group G 0 . It causes degeneracy: if ψ n is a solution of Hψn = En ψn than also holds: H(PR ψn ) = En (PR ψn ). Each n corresponds with another energy level. ν = 1. it is also a representation of G but not all elements of Γ (n) in G0 are also in G. .3. Ah = E}. The perturbed system has H = H0 + V.1 Representations. An isomorfic operation on the wavefunction is given by: P R ψ(x ) = ψ(R−1 x ).Chapter 13: Theory of groups 73 13. n1 n n2 n3 = dim(Γ(n1 ) ) = dim(Γ(n2 ) ) = dim(Γ(n3 ) ) Spectrum H0 Spectrum H (n) Theorem: The set of n degenerated eigenfunctions ψ ν irreducible representation Γ (n) of the symmetry group. A 2 . . Particle in a periodical potential: the symmetry operation is a cyclic group: note the operator describing one translation over one unit as A. . . One can purely mathematical derive irreducible representations of a symmetry group and label the energy levels with a quantum number this way. Then: G = {A. A degeneracy which is not the result of a symmetry is called an accidental degeneracy. so: PA ψp (x) = ψp (x − a) = e2πip/h ψp (x). . with uk (x ± a) = uk (x). The group is Abelian so all irreducible representations are one-dimensional. this gives Bloch’s ah a theorem: ψk (x) = uk (x)eikx . energy levels and degeneracy Consider a set of symmetry transformations x = Rx which leave the Hamiltonian H invariant. and leave the volume element unchanged: d(Rx ) = dx. A fixed choice of Γ (n) (R) defines (n) the base functions ψ ν .3 The relation with quantum mechanics 13. These transformations are a group. . These operators commute with H: P R H = HPR . The degeneracy is then (possibly partially) removed: see the figure below. n (n) n. . The function (n) is in the same subspace: PR ψν = κ=1 (n) (n) ψκ Γ(n) (R) κν where Γ is an irreducible. 13. A3 .3. 2. PR is the symmetry group of the physical system. . This way one can also label each separate base function with a quantum number. and symmetry group G ⊂ G 0 . and ϕκ |ψκ is independent of κ. this does happen if c jaκ = cja . These base ϕ(i) ϕλ (iκjλ|akµ) κ ϕ(aκ) (akµ|iκjλ) µ akµ (i) (j) (ak) (j) and the inverse transformation by: ϕ (i) ϕλ = κ (j) In essence the Clebsch-Gordan coefficients are dot products: (iκjλ|akµ) := ϕ k ϕλ |ϕµ 13. If a set of (j) operators Aκ with 0 ≤ κ ≤ j transform into each other under the transformations of G holds: −1 PR A(j) PR = κ ν A(j) Γ(j) (R) ν νκ (j) If Γ(j) is irreducible they are called irreducible tensor operators A (j) with components A κ .4 The direct product of representations Consider a physical system existing of two subsystems.5 Clebsch-Gordan coefficients With the reduction of the direct-product matrix w. An operator can also be decomposed into symmetry types: A = jk ak . J.t.3.3. However.C. the basis ϕ κ ϕλ one uses a new basis ϕµ functions lie in subspaces D (ak) . irreducible tensor operators −1 Observables (operators) transform as follows under symmetry transformations: A = PR APR . The subspace D (i) of the system transforms according (i) (1) (2) to Γ(i) . Theorem: Two wavefunctions transforming according to non-equivalent unitary representations or according to different rows of an unitary irreducible representation are orthogonal: (i) (j) (i) (i) ϕκ |ψλ ∼ δij δκλ . These define a space D (1) ⊗ D(2) . 13. Wevers This is expressed as: fκ is the part of F that transforms according to the κ th row of Γ(j) . with: (j) a(j) = κ j h −1 Γ(j)∗ (R) (PR APR ) κκ R∈G . ¯ (j) F can also be expressed in base functions ϕ: F = ajκ cajκ ϕκ (aj) . These product functions transform as: PR (ϕ(1) (x1 )ϕλ (x2 )) = (PR ϕ(1) (x1 ))(PR ϕλ (x2 )) κ κ In general the space D (1) ⊗ D(2) can be split up in a number of invariant subspaces: Γ(1) ⊗ Γ(2) = i (2) (2) ni Γ(i) A useful tool for this reduction is that for the characters hold: χ(1) (R)χ(2) (R) = i ni χ(i) (R) 13.A. Basefunctions are ϕκ (xi ). The functions f κ (j) are in general not transformed into each other by elements of the group.74 Physics Formulary by ir.r. Now form all 1 × 2 products ϕκ (x1 )ϕλ (x2 ).3. The unitary base transformation is given by: (ak) ϕµ = κλ (i) (j) (aκ) .6 Symmetric transformations of operators. 1 ≤ κ ≤ i . 4 Continuous groups Continuous groups have h = ∞.4.. otherwise one has to sum over a): (k) ϕλ |A(j) |ψµ = (iλ|jκkµ) ϕ(i) A(j) ψ (k) κ (i) (i) (j) (k) This theorem can be used to determine selection rules: the probability of a dipole transition is given by ( is the direction of polarization of the radiation): PD = 8π 2 e2 f 3 |r12 |2 with r12 = l2 m2 | · r |l1 m1 3¯ ε0 c3 h Further it can be used to determine intensity ratios: if there is only one value of a the ratio of the matrix elements are the Clebsch-Gordan coefficients. this can be written as: Because the momentum operator in quantum mechanics is given by: p x = i ∂x h ψ(x − a) = e−iapx /¯ ψ(x) 13. the intensity ratios are also dependent on the occupation of the atomic energy levels. z) . For more a-values relations between the intensity ratios can be stated.1 The 3-dimensional translation group For the translation of wavefunctions over a distance a holds: P a ψ(x) = ψ(x − a). not all groups with h = ∞ are continuous. Solutions can be found within each category of functions ϕ κ with common i and κ: H is already diagonal in categories as a whole. z) + zδθx − yδθx ∂y ∂z iδθx Lx ψ(x. y. .4. are 0 between states which transform according to non-equivalent irreducible unitary representations or according to different rows of such (i) (i) a representation. y. 13. Taylor expansion near x gives: dψ(x) 1 2 d2 ψ(x) ψ(x − a) = ψ(x) − a + a − +. the translation group of an spatially infinite periodic potential is not continuous but does have h = ∞. y. . z − yδθx ) ∂ ∂ ψ(x. Further ϕ κ |H|ψκ is independent of κ.Chapter 13: Theory of groups 75 Theorem: Matrix elements Hij of the operator H which is invariant under ∀ A∈G . However. However. 13..2 The 3-dimensional rotation group This group is called SO(3) because a faithful representation can be constructed from orthogonal 3 × 3 matrices with a determinant of +1. y + zδθx ..7 The Wigner-Eckart theorem Theorem: The matrix element ϕ λ |Aκ |ψµ can only be = 0 if Γ (j) ⊗ Γ(k) = .3. e. ⊕ Γ(i) ⊕ . . If this is the case holds (if Γ (i) appears only once. z) ≈ = = ψ(x. With variational calculus the try function can be chosen within a separate category because the exact eigenfunctions transform according to a row of an irreducible representation. y. 13. This is applied in quantum mechanics in perturbation theory and variational calculus. . z) 1− h ¯ ψ(x. Perturbation calculus can be applied independent within each category. For an infinitesimal rotation around the x-axis holds: Pδθx ψ(x. dx 2 dx2 h ¯ ∂ .g. For H = 1 this becomes the previous theorem. Here one tries to diag(i) onalize H. . This leads to the fundamental orthogonality theorem: (i)∗ Γµν (R)Γαβ (R)dR = G (j) 1 i δij δµα δνβ G dR and for the characters hold: χ(i)∗ (R)χ(j) (R)dR = δij G G G dR Compact groups are groups with a finite group volume: dR < ∞. translation group.. Because of the properties of the Cayley table is demanded: f (R)dR = f (SR)dR. pn ))dp1 · · · dpn Here. The notion conjugacy class for continuous groups is defined equally as for discrete groups. anx .. J. pn ))g(R(p1 . This fixes g(R) except for a constant factor. g(R) is the density function. and the corresponding terms are put equal with n · J = Jx cos θ + Jz sin θ. rotation group. h 13. Summation over all group elements is for continuous groups replaced by an integration... The notion representation is fitted by demanding continuity: each matrix element depends continuously on pi (R).y Pα.76 Physics Formulary by ir.. . . p x .A. Rotations are not generally interchangeable: consider a rotation around axis n in the xz-plane over an angle α.. .. For the translation group holds: g(a) = constant = g( 0 ) because g(an )da = g(0 )da and da = da.3 Properties of continuous groups The elements R(p1 . For the translation group this are e. If f (R) is a function defined on G.y . Define new variables p by: SR(pi ) = R(pi ). Wevers Because the angular momentum operator is given by: L x = So in an arbitrary direction holds: Rotations: Translations: h ¯ i z ∂ ∂ −y . py and pz are called the generators of the 3-dim. ∂y ∂z P α.. It is demanded that the multiplication and inverse of an element R depend continuously on the parameters of R. it follows from the αθ term: J x Jy − Jy Jx = i¯ Jz .n = exp(−ia(n · p )/¯ ) h Jx . . Then holds: Pα. .g. and g(E) is constant. If one writes: dV := dp1 · · · dpn holds: g(S) = g(E) dV dV is the Jacobian: = det dV dV ∂pi ∂pj dV dV Here. e. Γ αβ (R). The statement that each element arises only once in each row and column of the Cayley table holds also for continuous groups.n = exp(−iα(n · J )/¯ ) h Pa.. pn ) depend continuously on parameters p 1 . The commutation rules for the generators can be derived from the properties of the group for multiplications: translations are interchangeable ↔ p x py − py px = 0. holds: f (R)dR := G p1 ··· pn f (R(p1 .n = P−θ. any and anz .g..C. Jy and Jz are called the generators of the 3-dim. .4. so: h h h h e−iα(n·J )/¯ = eiθJy /¯ e−iαJx /¯ e−iθJy /¯ If α and θ are very small and are expanded to second order. pn ..x Pθ. then the rotation of a quantum mechanical system is described by: h ¯ h h ψ → e−iαJz h e−iβJy /¯ e−iγJz /¯ ψ. 2 space which is invariant under rotations. 123 are θ. A problem arises for rotations over 2π: h ψ 1 ms → e−2πiSz /¯ ψ 1 ms = e−2πims ψ 1 ms = −ψ 1 ms 2 2 2 2 However. The spherical angles of the z-axis w.Chapter 13: Theory of groups 77 13. and are bilinear in the states.−π . The existence of these half-odd integer representations is connected with the topological properties of SO(3): the group is two-fold coherent through the identification R 0 = R2π = E. This can be derived from group theory as follows: from χ(j1 ) (α)χ(j2 ) (α) = nj χ(J) (α) follows: J D(j1 ) ⊗ D(j2 ) = D(j1 +j2 ) ⊕ D(j1 +j2 −1) ⊕ . Another way to define parameters is by means of Eulers angles. 2. So PR = e−iε(n·J )/¯ . ϕ := β. ϕ) = m Ylm (θ.1 Vectormodel for the addition of angular momentum If two subsystems have angular momentum quantum numbers j 1 and j2 the only possible values for the total angular momentum are J = j 1 + j2 .6...t. So here holds E → ±I Because observable quantities can always be I.r.t. α. ϕ)Dmm (R) (l) D (l) is an irreducible representation of dimension 2l + 1.. All irreducible representations of SO(3) can be constructed from the behaviour of the spherical harmonics Ylm (θ. β and γ are the 3 Euler angles. they do not change sign if the states do. The character of D (l) is given by: l l χ(l) (α) = m=−l eimα = 1 + 2 k=0 cos(kα) = sin([l + 1 ]α) 2 sin( 1 α) 2 In the performed derivation α is the rotational angle around the z-axis. 2 This is contradictory because the dimension of the representation is given by χ (0). written as φ|ψ or φ|A|ψ . ⊕ D(|j1 −j2 |) . defined as: 1.π = Rn. 13. xyz are θ.. The parameter space is a collection points ϕn within a sphere with radius π. Now a rotation around axis 3 remains possible. If only one state changes sign the observable quantities do change. . Via the fundamental orthogonality theorem for characters one obtains the following expression for the density function (which is normalized so that g(0) = 1): g(α) = sin2 ( 1 α) 2 ( 1 α)2 2 With this result one can see that the given representations of SO(3) are the only ones: the character of another representation χ would have to be ⊥ to the already found ones. π − γ. in SO(3) holds: R z.2π = E.6 Applications to quantum mechanics 13. This expression is valid for all rotations over an angle α because the classes of SO(3) are rotations around the same angle around an axis with an arbitrary orientation. j1 + j2 − 1.. |j1 − j2 |. The spherical angles of axis 3 w. so χ (α) sin2 ( 1 α) = 0∀α ⇒ χ (α) = 0∀α. The diametrical points on this sphere are equivalent because R n.r. If α. 1 Because fermions have an half-odd integer spin the states ψ sms with s = 1 and ms = ± 2 constitute a 2-dim. ϕ) with −l ≤ m ≤ l and for a fixed l: PR Ylm (θ.5 The group SO(3) One can take 2 parameters for the direction of the rotational axis and one for the angle of rotation ϕ. ϕ := β. II.z ) = 1 + 2 cos(α). This transforms as the scalar U · V . where U and −1 V are vectors. so like D (1) ⊗ D(1) = D(2) ⊕ D(1) ⊕ D(0) . The 5 independent components of the traceless. except J = J = 0. P R ψ and PR φ: −1 PR ψ|PR Ax PR |PR φ = ψ|Ax |φ . ϕ).z ) =  sin α cos α 0  0 0 1 The transformed operator has the same matrix elements w. The 3 antisymmetric components A z = 1 (Txy − Tyx ). m2 and with j1 . j2 . the Hamiltonian of a free atom. According to the equation for characters this means one can choose base operators which transform like Y 1m (θ. e. for SO(3) called the Wigner coefficients. A−1 = √ (Ax − iAy ) 2 2 3. So for rotations around the z-axis holds:   cos α − sin α 0 D(Rα. Suppose j = 0: this gives the identical representation with j = 1. so: ψj1 j2 JM ψj1 m1 j2 m2 = m1 m2 ψj1 m1 j2 m2 (j1 m1 j2 m2 |JM ) = JM ψj1 j2 JM (j1 m1 j2 m2 |JM ) 13. This means that J ∈ {J + 1. The 9 components can be split in 3 invariant subspaces with dimension 1 (D (0) ). This can also be understood (1) kl . This state is described by a (0) −1 (0) scalar operator.6.t. Wevers The states can be characterized by quantum numbers in two ways: with j 1 . symmetric tensor S: Sij = 1 (Tij + Tji ) − 1 δij Tr(T ). III. From the Wigner-Eckart theorem follows: J M |Aκ |JM = 0 except D (J ) is a part of D (1) ⊗ D(J) = D(J+1) ⊕ D(J) ⊕ D(|J−1|) . These transform as the vector U × V . m1 . A0 = Az . Because P R A0 PR = A0 this operator is invariant. Az ). j2 .g. The new base operators are: I.2 Irreducible tensor operators. The Clebsch-Gordan coefficients. for a magnetic e( L + 2S )/2m. matrixelements and selection rules Some examples of the behaviour of operators under SO(3) 1. Tr(T ) = Txx + Tyy + Tzz .C. These transform as D (2) . etc. Ay .r. J. so as D (0) . Then holds: J M |H|JM ∼ δMM δJJ . A vector operator: A = (Ax . 3 (D (1) ) and 5 (D (2) ). M . J. These turn out to be the spherical components: 1 1 (1) (1) (1) A+1 = − √ (Ax + iAy ). Land´ -equation for the anomalous Zeeman splitting e According to Land´ ’s model the interaction between a magnetic moment with an external magnetic field is e determined by the projection of M on J because L and S precede fast around J. |J − 1|}: J = J or J = J ± 1. 2 so as D(1) .78 Physics Formulary by ir. 2. and χ(Rα. 2 3 Selection rules for dipole transitions Dipole operators transform as D (1) : for an electric dipole transfer is the operator er.A. So T ij transforms like PR Tij PR = Tkl Dki (R)Dlj (R). can be chosen real. A cartesian tensor of rank 2: T ij is a quantity which transforms under rotations like U i Vj . The cartesian components of a vector operator transform equally as the cartesian components of r by definition. J. T2 ] = iT3 . The physics of the system does not change if A and φ are also transformed. unitary 1 × 1-matrices. For each matrix of SU(2) holds that Tr(H)=0. Each Hermitian. Taylor expansion and setting equal Θ n Θ m -terms results in the commutation rules. one can consider an isomorphic group where only the commutation rules are considered to be known regarding the operators T i : [T1 . Tj ] = i cijk Tk . The gauge transformations of the EM-field form a group: U(1). This concept is generalized: particles have a “special charge” Q. here εijk is the complete antisymmetric tensor with ε123 = +1. For an arbitrary operator A follows: αjm |A|αjm = αjm|A · J |αjm j(j + 1)¯ 2 h αjm |J |αjm 13. In elementary particle physics the T i can be interpreted e.7 Applications to particle physics The physics of a system does not change after performing a transformation ψ = eiδ ψ where δ is a constant. This is a local gauge transformation: the phase of the wavefunction changes different at each position. The weak interaction together with the electromagnetic interaction can be described by a force field that transforms according to U(1)⊗SU(2).t) ψ. etc. The group SU(2) has 3 free parameters: because it is unitary there are 4 real conditions over 4 complex parameters. In general the group elements are given by P R = exp(−iT · Θ). Other force fields than the electromagnetic field can also be understood this way. This is a global gauge transformation: the phase of the wavefunction changes everywhere by the same amount. independent of their charge. Further it always holds that: det(U ) = e−iTr(H) . The group elements now are PR = exp(−iQΘ). These constants can be found with the help of group product elements: because G is closed holds: eiΘ·T eiΘ ·T e−iΘ·T e−iΘ ·T = e−iΘ ·T . traceless 2 × 2 matrix can be written as a linear combination of the 3 Pauli-matrices σ i . The solution ψ of the Schr¨ dinger equation with the transformed potentials is: ψ = e−iqf (r. Each unitary matrix U can be written as: U = e −iH . Elementary particles can be classified in isospin-multiplets. H is a Hermitian matrix. The classification is: 1. remaining 3 free parameters. and has a gauge field that exists of 8 types of gluons. There exists some freedom in the choice of the potentials A and φ at the same E and B: gauge transformations o of the potentials do not change E and B (See chapter 2 and 10). The isospin-doublet ≡ the faithful representation of SU(2) on 2 × 2 matrices. where Θn are real constants and T n operators (generators). Here. The isospin-singlet ≡ the identical representation: e −iT ·Θ = 1 ⇒ Ti = 0 2. The split-off of charge in the exponent is essential: it allows one gauge field for all charged particles. So these matrices are a choice for the operators of SU(2). One can write: SU(2)={exp(− 1 iσ · Θ)}. . as the isospin operators.Chapter 13: Theory of groups 79 from the Wigner-Eckart theorem: from this follows that the matrix elements from all vector operators show a certain proportionality. these are the irreducible representations of SU(2). 2 In abstraction. The colour force is described by SU(3). For SO(3) these constants are c ijk = εijk . like Q. and the determinant has to be +1.g. and consists of the photon and three intermediary vector bosons. This is now stated as a guide principle: the “right of existence” of the electromagnetic field is to allow local gauge invariance. The cijk are the structure k constants of the group. The commutation rules are given by [T i . In the Lagrange density for the colour force one has to substitute D ∂ ∂ → := − Ti Ai x ∂x Dx ∂x i=1 8 The terms of 3rd and 4th power in A show that the colour field interacts with itself. which are not all 9 linear independent.80 Physics Formulary by ir. J.C.A. The Hermitian. (The group SU(N ) has N 2 − 1 free parameters). . By taking a linear combination one gets 8 matrices. traceless operators are 3 SU(2)-subgroups in the e 1 e2 . e1 e3 and the e2 e3 plane. This gives 9 matrices. Wevers The group SU(3) has 8 free parameters. with j = l ± 1 . a4 : Asymmetry term: a surplus of protons or neutrons has a lower binding energy.1 Nuclear forces The mass of a nucleus is given by: Mnucl = Zmp + N mn − Ebind /c2 The binding energy per nucleon is given in the figure at the right. 3. So each level h 2 (n. Z odd.810 0. Eγ = m0 c2 and r0 = c∆t follows: r0 = h/m0 c. ¯ ¯ In the shell model of the nucleus one assumes that a nucleon moves in an average field of other nucleons.702 34. Z even. N odd: = −1. be considered as an exchange of virtual pions: r W0 r0 exp − U (r) = − r r0 With ∆E · ∆t ≈ h. a1 : Binding energy of the strong nuclear force. With the constants a1 a2 a3 a4 a5 = = = = = 15.000 MeV MeV MeV MeV MeV 9 8 ↑ 7 E 6 (MeV) 5 4 3 2 1 0 0 40 80 120 160 A→ 200 240 and A = Z + N . The energy of a 3-dimensional harmonic oscillator is E = (N + 3 )¯ ω. l) is split in two. 4. there is a contribution of the spin-orbit coupling ∼ L · S: ∆Vls = 1 (2l + 1)¯ ω. where the state with j = l + 1 has the lowest energy. which is an indication that the L − S interaction is not electromagnetical. 26 the most stable nucleus. Further. The top is at 56 Fe. Because −l ≤ m ≤ l and m s = ± 1 ¯ there are 2(2l + 1) 2h . N odd: = 0. approximately ∼ A. in the droplet or collective model of the nucleus the binding energy E bind is given by: Z(Z − 1) (N − Z)2 Ebind + a5 A−3/4 = a1 A − a2 A2/3 − a3 − a4 2 1/3 c A A These terms arise from: 1. This is just 2 2 the opposite for electrons. a2 : Surface correction: the nucleons near the surface are less bound. The following holds: Z even. The Yukawa potential can be derived if the nuclear force can to first approximation. N = nx + ny + nz = 2(n − 1) + l 2 h where n ≥ 1 is the main oscillator number. Z odd.Chapter 14 Nuclear physics 14. 5.711 23. N even: = 0.760 17. a5 : Pair off effect: nuclei with an even number of protons or neutrons are more stable because groups of two protons or neutrons have a lower energy. a3 : Coulomb repulsion between the protons. N even: = +1. 2. 5. The resulting magnetic moment is related to the nuclear spin I according to M = gI (e/2mp)I. The probability that N nuclei decay within a time interval is given by a Poisson distribution: λN e−λ dt P (N )dt = N0 N! If a nucleus can decay into more final states then holds: λ = λi .2 · 10−15 m.82 Physics Formulary by ir. Here. The parity of the electric moment is Π E = (−1)l . For protons holds g S = 5. The tunnel probability P is P = 1 incoming amplitude = e−2G with G = outgoing amplitude h ¯ 2m [V (r) − E]dr G is called the Gamow factor. α-decay: the nucleus emits a He 2+ nucleus. J. The z-components of the magnetic moment are given by M L. β-decay. So the fraction decaying into state i is λi / λi . e e L and MS = gS S. This is the case if N or Z ∈ {2. If the nuclear radius is measured including the charge distribution one obtains R 0 ≈ 1. 28. 14.0 = β cos γ and a2. Here a proton changes into a neutron or vice versa: p+ → n0 + W+ → n0 + e+ + νe . γ-decay: here the nucleus emits a high-energetic photon. 14. 20. which can be applied to the theory of neutron stars. Electron capture: here. The shape of oscillating nuclei can be described by spherical harmonics: R = R0 1 + lm alm Ylm (θ. This gives rise to the so called magical numbers: nuclei where each state in the outermost level are filled are particulary stable. 50.±2 = 1 2β sin γ where β is the 2 deformation factor and γ the shape parameter. Spontaneous fission: a nucleus breaks apart. 126}. The multipole moment is given by µ l = Zerl Ylm (θ. constant for all nuclei.5855 and for neutrons g S = −3.z = gS µN mS . Wevers substates which exist independently for protons and neutrons. R0 ≈ 1. with a 2. 3.z = µN ml and MS. density vibrations. a proton in the nucleus captures an electron (usually from the K-shell). The half life time follows from τ 1 λ = ln(2). and n0 → p+ + W− → p+ + e− + ν e . ϕ) l = 0 gives rise to monopole vibrations. l = 2 quadrupole. This gives for the number of nuclei N : N (t) = N0 exp(−λt). 8. The z-component is then Mz = µN gI mI .8263. The decay constant is given by λ= P (l) Eγ ∼ h ¯ω (¯ c)2 h Eγ R h ¯c 2l ∼ 10−4l . 82.C. ϕ).3 Radioactive decay ˙ The number of nuclei decaying is proportional to the number of nuclei: N = −λN .A. 2. of the magnetic moment Π M = (−1)l+1 . There are 2 contributions to the magnetic moment: ML = 2mp 2mp where gS is the spin-gyromagnetic ratio. The average life time 2 of a nucleus is τ = 1/λ.2 The shape of the nucleus A nucleus is to first approximation spherical with a radius of R = R 0 A1/3 . Because nucleons tend to order themselves in groups of 2p+2n this can be considered as a tunneling of a He 2+ nucleus through a potential barrier. There are 5 types of natural radioactive decay: 1. √ l = 1 gives dipole vibrations. 4.4·10−15 m. 2 Quantum mechanical model for n-p scattering The initial state is a beam of neutrons moving along the z-axis with wavefunction ψ init = eikz and current density Jinit = v|ψinit |2 = v.4. so ψ = ψ0 + l>0 ψl and ψ0 = sin(kr) kr If the potential is approximately rectangular holds: ψ 0 = C The cross section is then σ(θ) = sin2 (δ0 ) so σ = k2 h ¯ 2 k 2 /2m W0 + W sin(kr + δ0 ) kr 4π sin2 (δ0 ) k2 σ(θ)dΩ = At very low energies holds: sin 2 (δ0 ) = with W0 the depth of the potential well.4.Chapter 14: Nuclear physics 83 where l is the quantum number for the angular momentum and P the radiated power. The wavefunction of the incoming particles can be expressed as a sum of angular momentum wavefunctions: ψinit = eikz = l ψl The impact parameter is related to the angular momentum with L = bp = b¯ k.1 Kinetic model If a beam with intensity I hits a target with density n and length x (Rutherford scattering) the number of scatterings R per unit of time is equal to R = Inxσ. The parity of the emitted radiation is Π l = Πi · Πf . Usually the decay constant of electric multipole moments is larger than the one of magnetic multipole moments. L = h I(I + 1). At very low energy h only particles with l = 0 are scattered. At higher energies holds: σ = 4π k2 sin2 (δl ) l . which can usually be neglected. With I the quantum number of angular momentum of the nucleus. At large distances from the scattering point they have approximately a spherical wavefunction ψ scat = f (θ)eikr /r where f (θ) is the scattering amplitude. From this follows that the intensity of the beam decreases as −dI = Inσdx. ϕ) dσ = dΩ 4πnxI dσ ∆N = n ∆Ω∆x N dΩ If N particles are scattered in a material with density n then holds: For Coulomb collisions holds: dσ dΩ = C 1 Z1 Z2 e2 2 4 1 8πε0 µv0 sin ( 2 θ) 14. From this it follows that σ(θ) = |f (θ)| 2 . holds the following selection rule: ¯ |Ii − If | ≤ ∆l ≤ |Ii + If |. so bk ≈ l. with TR = Eγ /2mc2 the recoil energy.4 Scattering and nuclear reactions 14. Because dR = R(θ. This results in I = I0 e−nσx = I0 e−µx . The total wavefunction is then given by eikr ψ = ψin + ψscat = eikz + f (θ) r The particle flux of the scattered particles is v|ψ scat |2 = v|f (θ)|2 dΩ. ϕ)dΩ/4π = Inxdσ it follows: R(θ. 2 The energy of the photon is E γ = Ei − Ef − TR . 14. C. If the absorption is not equally distributed also weight factors w per organ need to be used: H = wk Hk . Wevers 14. The radiation dose X is the amount of charge produced by the radiation per unit of mass. where for each type radiation the effects on biological material is used for the weight factor. and the energy flux density ψ = dΨ/dt. It can be derived that D= µE Ψ The Kerma K is the amount of kinetic energy of secundary produced particles which is produced per mass unit of the radiated object.r. The intensity of a beam of particles in matter decreases according to I(s) = I 0 exp(−µs). The deceleration of a heavy particle is described by the Bethe-Bloch equation: dE q2 ∼ 2 ds v The fluention is given by Φ = dN/dA.4. For some types of radiation holds: Radiation type R¨ ntgen.t. These weight factors are called the quality factors. mesons Thermic neutrons Fast neutrons protons α. The absorption coefficient is given by µ = (dN/N )/dx.58 · 10 −4 C/kg. The absorbed dose D is given by D = dE abs /dm.84 Physics Formulary by ir. electrons. gamma radiation o β. The minimal required kinetic energy T of P 1 in the laboratory system to initialize the reaction is T = −Q If Q < 0 there is a threshold energy. Their unit is Sv. with unit Gy=J/kg.5 Radiation dosimetry Radiometric quantities determine the strength of the radiation source(s). so Pk P1 + P2 → k>2 the total energy Q gained or required is given by Q = (m 1 + m2 − k>2 mk )c2 . the laboratory system and other particles are created. An old unit is the rad: 1 rad=0. fission products Q 1 1 3 to 5 10 to 20 10 20 . The flux is given by φ = dΦ/dt. J. The equivalent dose H is a weight average of the absorbed dose per type of radiation. The energy loss is defined by Ψ = dW/dA.3 Conservation of energy and momentum in nuclear reactions If a particle P1 collides with a particle P2 which is in rest w. Parameters describing a relation between those are called interaction parameters. m1 + m2 + 2m2 mk 14. The mass absorption coefficient is given by µ/ . An old unit is the R¨ ntgen: 1Ro= 2.A.01 Gy. H = QD. With the energy-absorption coefficient µ E follows: o X= dQ eµE = Ψ dm W where W is the energy required to disjoin an elementary charge. ˙ The dose tempo is defined as D. with unit C/kg. Dosimetric quantities are related to the energy transfer from radiation to matter. Hence the vacuum state is given by |000 · · · . Using the variational principle δI(Ω) = 0 and with the action-integral I(Ω) = L(Φα . ∂ν Φα )d4 x the field equation can be derived: ∂ ∂L ∂L =0 − α ∂Φ ∂xν ∂(∂ν Φα ) The conjugated field is. [a(ki ). All operators which can change occupation numbers can be expanded in the a and a † operators. This is a complete description because the particles are indistinguishable. a(kj )] = 0 . Hint is the interaction Hamiltonian and can increase or decrease a c 2 at the cost of others. the Lagrange density L is a function of the position x = (x. The Lagrangian is given by L = L(x)d3 x. The expansion of the states tum k1 . a† (kj )] = δij Hence for free spin-0 particles holds: H 0 = i a† (ki )a(ki )¯ ωki h 15. etc. n2 particles with momentum k2 .Chapter 15 Quantum field theory & Particle physics 15. and Ψ(t)|Ψ(t) = in time is described by the Schr¨ dinger equation o i d |Ψ(t) = H|Ψ(t) dt where H = H0 + Hint .. a† (kj )] = 0 . a is the annihilation operator and a † the creation operator. defined as: Πα (x) = ∂L ˙ ∂ Φα . The following commutation rules can be derived: [a(ki ). ict) through the fields: L = L(Φ α (x). So aa† is an occupation number operator. ∂ν Φα (x)).2 Classical and quantum fields Starting with a real field Φ α (x) (complex fields can be split in a real and an imaginary part). [a† (ki ). analogous to momentum in classical mechanics. H0 is the Hamiltonian for free particles and keeps |c ni (t)|2 constant.1 Creation and annihilation operators A state with more particles can be described by a collection occupation numbers |n 1 n2 n3 · · · . The states are orthonormal: ∞ n1 n2 n3 · · · |n1 n2 n3 · · · = i=1 δni ni The time-dependent state vector is given by Ψ(t) = n1 n2 ··· cn1 n2 ··· (t)|n1 n2 · · · The coefficients c can be interpreted as follows: |c n1 n2 ··· |2 is the probability to find n 1 particles with momen|cni (t)|2 = 1. and: √ ni |n1 n2 · · · ni − 1 · · · a(ki )|n1 n2 · · · ni · · · = √ † ni + 1 |n1 n2 · · · ni + 1 · · · a (ki )|n1 n2 · · · ni · · · = Because the states are normalized holds a|0 = 0 and a( ki )a† (ki )|ni = ni |ni . 3. t)] = 0. In general it holds that the term with e −ikx . ∂ ν Φ) = − 1 (∂ν Φ∂ν Φ + m2 Φ2 ). With the definition k0 = 2 2 2 k + M := ωk and the notation k · x − ik0 t := kx the general solution of this equation is: 1 Φ(x) = √ V k 1 √ a(k )eikx + a† (k )e−ikx 2ωk i . time-dependent operators. with M := m 2 c2 /¯ 2 . Quantization of a classical field is analogous to quantization in point mass mechanics: the field functions are considered as operators obeying certain commutation rules: [Φα (x). From this follows that the commutation rules are given by [Φ(x). J. and the negative frequency part is the annihilation part. o 2. Usually one can take the limit V → ∞. So the equations obey the locality postulate. Πβ (x )] = 0 .3 The interaction picture Some equivalent formulations of quantum mechanics are possible: 1. holds: h 0 ∂ ∂ Φ(x) = Π(x) and Π(x) = (∇2 − M 2 )Φ(x) ∂t ∂t 2 From this follows that Φ obeys the Klein-Gordon equation (✷ − M 2 )Φ = 0. time-dependent operators. Φβ (x )] = 0 . The Lagrange density is given by: L(Φ. The interaction picture can be obtained from the Schr¨ dinger picture by an unitary transformation: o |Φ(t) = eiH0 |ΦS (t) and O(t) = eiH0 OS e−iH0 The index S denotes the Schr¨ dinger picture. which is used as normalization factor.4 Real scalar field in the interaction picture It is easy to find that. [Πα (x).C. the Hamilton density becomes H(x) = Π α Φα − L(x). is the creation part. From this follows: o i d d |Φ(t) = Hint (t)|Φ(t) and i O(t) = [O(t). Φ(x . This is consistent with the previously found result [Φ(x. Φ(x )] = i∆(x − x ) with ∆(y) = 1 (2π)3 sin(ky) 3 d k ωk ∆(y) is an odd function which is invariant for proper Lorentz transformations (no mirroring). The energy operator is given by: 2 H= H(x)d3 x = k h ¯ ωk a† (k )a(k ) . H0 ] dt dt S S S 15. Π(x) = √ V 1 2 ωk k −a(k )eikx + a† (k )e−ikx The field operators contain a volume V . time-independent operators. In general holds that ∆(y) = 0 outside the light cone. Interaction picture: time-dependent states.A. Schr¨ dinger picture: time-dependent states. Wevers ˙ With this. t.86 Physics Formulary by ir. the coefficients have to be each others hermitian conjugate because Φ is hermitian. Heisenberg picture: time-independent states. [Φα (x). Because Φ has only one component this can be interpreted as a field describing a particle with spin zero. Πβ (x )] = iδαβ (x − x ) 15. the positive frequency part. Noether’s theorem connects a continuous symmetry of L and an additive conservation law.1 particles 2 Spin is defined by the behaviour of the solutions ψ of the Dirac equation. Φ† (x) = √ V 1 √ a† (k )eikx + b(k )e−ikx 2ωk k k Hence the energy operator is given by: H= k h ¯ ωk a† (k )a(k ) + b† (k )b(k ) and the charge operator is given by: Q(t) = −i J4 (x)d3 x ⇒ Q = k e a† (k )a(k ) − b† (k )b(k ) From this follows that a † a := N+ (k ) is an occupation number operator for particles with a positive charge and b† b := N− (k ) is an occupation number operator for particles with a negative charge.5 Charged spin-0 particles. and ± is the sign of the energy. if it obeys the Klein-Gordon equation. ν ≤ 3 these are rotations. These can be written as: ∂ ∂ ˜ Φ(x) = Φ(x)e−in·L with Lµν = −i¯ xµ h − xν ∂xν ∂xµ For µ ≤ 3. ˜ ˜ A rotated field ψ obeys the Dirac equation if the following condition holds: ψ(x) = D(Λ)ψ(Λ−1 x). r is an indication for the direction of the spin. This 1 −1 in·S results in the condition D γλ D = Λλµ γµ . 15. conservation of charge The Lagrange density of charged spin-0 particles is given by: L = −(∂ ν Φ∂ν Φ∗ + M 2 ΦΦ∗ ). ∂ν (Φα ) ) = L (Φα . Suppose that L ((Φα ) . µ = 4 these are Lorentz transformations. With the notation v r (p ) = ur (−p ) and ur (p ) = ur (p ) one can write for the dot products of these spinors: − + ur (p )ur (p ) = + + E E δrr . the rotated field Φ(x) := Φ(Λ−1 x) also obeys it. ur (p )ur (p ) = 0 − − − + M M .6 Field functions for spin. The above Lagrange density is invariant for a change in phase Φ → Φe iθ : a global gauge transformation. Because this quantity is 0 for real fields a complex field is needed to describe charged particles. ∂ν Φα ) and there exists a continuous transformation between Φ α and Φα such as Φα = Φα + f α (Φ). Λ denotes 4-dimensional rotations: the proper Lorentz transformations. Hence: h ˜ ψ(x) = e−i(S+L) ψ(x) = e−iJ ψ(x) Then the solutions of the Dirac equation are given by: ψ(x) = ur (p )e−i(p·x±Et) ± Here. for ν = 4. One finds: D = e with Sµν = −i 2 ¯ γµ γν . The conserved quantity is the current density J µ (x) = −ie(Φ∂µ Φ∗ − Φ∗ ∂µ Φ).Chapter 15: Quantum field theory & Particle physics 87 15. A scalar field Φ has the property ˜ that. Then holds ∂L ∂ fα = 0 ∂xν ∂(∂ν Φα ) This is a continuity equation ⇒ conservation law. ur (p )ur (p ) = δrr . Which quantity is conserved depends on the symmetry. When this field is quantized the field operators are given by 1 Φ(x) = √ V 1 √ a(k )eikx + b† (k )e−ikx 2ωk 1 . A Lorentz-invariant dot product is defined by ab := a† γ4 b. by interchange of two boson operators not. v r (p )v r (p ) = −δrr . ψβ (x )] = 0 . The expression for the current density now becomes J µ = −ieN (ψγµ ψ).C. ∂ +M ∂xµ ψ(x) 15. ψβ (x )]+ = −iSαβ (x − x ) ∂ − M ∆(x) ∂xλ The anti commutation rules give besides the positive-definite energy also the Pauli exclusion principle and the Fermi-Dirac statistics: because c† (p )c† (p ) = −c† (p )c† (p ) holds: {c† (p)}2 = 0. ψβ (x )] = 0 . c and c are the creation respectively annihilation operators for an electron and d † and d the creation respectively annihilation operators for a positron. This is the exclusion principle. • In all other terms interchange the factors so that the annihilation operators go to the right.7 Quantization of spin. [ψα (x). all other anti commutators are 0. r r The field operators obey [ψα (x). • Keep all terms which have no annihilation operators. The energy operator is given by 2 H= p Ep r=1 c† (p )cr (p ) − dr (p )d† (p ) r r To prevent that the energy of positrons is negative the operators must obey anti commutation rules in stead of commutation rules: [cr (p ).1 fields 2 The general solution for the fieldoperators is in this case: ψ(x) = and ψ(x) = † M V M V p 1 √ E 1 √ E cr (p )ur (p )eipx + d† (p )v r (p )e−ipx r r c† (p )ur (p )e−ipx + dr (p )v r (p )eipx r r p Here.88 Physics Formulary by ir. c† (p )]+ = [dr (p ). with S(x) = γλ To avoid infinite vacuum contributions to the energy and charge the normal product is introduced. where a := a† γ4 is a row spinor. This product is obtained by: • Expand all fields into creation and annihilation operators. ur (p )v r (p ) = 0 Combinations of the type aa give a 4 × 4 matrix: 2 u r=1 r (p )ur (p ) −iγλ pλ + M . It appears to be impossible r r r r r to create two electrons with the same momentum and spin. = 2M 2 v r (p )v r (p ) = r=1 −iγλ pλ − M 2M The Lagrange density which results in the Dirac equation and having the correct energy normalization is: L(x) = −ψ(x) γµ and the current density is J µ (x) = −ieψγµ ψ. By an interchange of two fermion operators add a minus sign. From this follows: ur (p )ur (p ) = δrr . Wevers Because of the factor E/M this is not relativistic invariant. d† (p )]+ = δrr δpp . [ψα (x). or in which they are on the right of the creation operators. Assume hereby that all commutators are zero. J.A. Another way to see + + this is the fact that {Nr (p )}2 = Nr (p ): the occupation operators have only eigenvalues 0 and 1. . 3 (x) ∈ I and for A4 (x) become imaginary. The interaction Hamilton density for the interaction between the electromagnetic and the electron-positron field is: Hint (x) = −Jµ (x)Aµ (x) = ieN (ψγµ ψAµ ) When this is expanded as: H int = ieN (ψ + + ψ − )γµ (ψ + + ψ − )(A+ + A− ) µ µ . However. Aν (x )] = iδµν D(x − x ) with D(y) = ∆(y)|m=0 In spite of the fact that A 4 = iV is imaginary in the classical case. By changing the definition of the inner product in R configuration space the expectation values for A 1. this gives problems with the commutation rules. a† (k )] = δmm δkk . the T -operator means a time-ordered product: the terms in such a product must be ordered in increasing time order from the right to the left so that the earliest terms act first. A 4 is still defined to be hermitian because otherwise the sign of the energy becomes incorrect.2. All other commutators are 0. If the potentials satisfy the Lorentz gauge condition ∂ µ Aµ = 0 the E and B operators derived from these potentials will satisfy the Maxwell equations. From this follows that (a 3 (k ) − a4 (k ))|Φ = 0. It is now demanded that only those states are permitted for which holds ∂A+ µ |Φ = 0 ∂xµ This results in: ∂Aµ ∂xµ = 0. These photons are also responsible for the stationary Coulomb potential. However. this only applies to free EM-fields: in intermediary states in interactions there can exist longitudinal and timelike photons. Further holds: [Aµ (x).8 Quantization of the electromagnetic field Starting with the Lagrange density L = − 1 2 it follows for the field operators A(x): 1 A(x) = √ V 1 √ 2ωk 4 ∂Aν ∂Aν ∂xµ ∂xµ am (k ) m=1 m (k )eikx + a† (k ) m (k )∗ e−ikx k The operators obey [a m (k ).9 Interacting fields and the S-matrix The S(scattering)-matrix gives a relation between the initial and final states of an interaction: |Φ(∞) = S|Φ(−∞) . m gives the polarization m direction of the photon: m = 1. The S-matrix is then given by: S ij = Φi |S|Φj = Φi |Φ(∞) . If the Schr¨ dinger equation is integrated: o t |Φ(t) = |Φ(−∞) − i −∞ Hint (t1 )|Φ(t1 ) dt1 and perturbation theory is applied one finds that: S= (−i)n n! n=0 ∞ ∞ ··· T {Hint (x1 ) · · · Hint (xn )} d4 x1 · · · d4 xn ≡ n=0 S (n) Here. 15. 2 gives transversal polarized. With a local gauge transformation one obtains N 3 (k ) = 0 and N4 (k ) = 0.Chapter 15: Quantum field theory & Particle physics 89 15. m = 3 longitudinal polarized and m = 4 timelike polarized photons. 15. This can be written as a sum of normal products. The term ieψ + γµ ψ + A− acting on |Φ µ gives transitions where A − creates a photon. virtual particles do not obey the relation between energy and momentum. would have a mass M 0 and a charge e 0 unequal to the observed mass M and charge e. The contraction functions can also be written as: ∆F (x) = lim −2i →0 (2π)4 −2i eikx d4 k and S F (x) = lim →0 (2π)4 k 2 + m2 − i eipx iγµ pµ − M 4 d p p2 + M 2 − i In the expressions for S (2) this gives rise to terms δ(p + k − p − k ). However. This is solved by discounting all divergent diagrams in a renormalization of e and M . In the x-representation each diagram describes a number of processes.C. 15. Hadrons: these exist of quarks and can be categorized in: I. Some time-ordened products are: T {Φ(x)Φ(y)} T ψα (x)ψβ (y) T {Aµ (x)Aν (y)} = N {Φ(x)Φ(y)} + 1 ∆F (x − y) 2 F = N ψα (x)ψβ (y) − 1 Sαβ (x − y) 2 F = N {Aµ (x)Aν (y)} + 1 δµν Dµν (x − y) 2 Here. The appearing operators describe the minimal changes necessary to change the initial state into the final state. In the x-representation this can be understood because the product of two functions containing δ-like singularities is not well defined. The expressions for S (n) contain time-ordered products of normal products. J. In the Hamilton and Lagrange density of the free electron-positron field appears M 0 .11 Classification of elementary particles Elementary particles can be categorized as follows: 1. with M = M0 + ∆M : Le−p (x) = −ψ(x)(γµ ∂µ + M0 )ψ(x) = −ψ(x)(γµ ∂µ + M )ψ(x) + ∆M ψ(x)ψ(x) and Hint = ieN (ψγµ ψAµ ) − i∆eN (ψγµ ψAµ ). ψ + annihilates an electron and ψ + annihilates a positron. S F (x) = (γµ ∂µ − M )∆F (x). Wick’s theorem gives an expression for the time-ordened product of an arbitrary number of field operators. . Each term corresponds with a possible process. Further the factors in H int can create and thereafter annihilate particles: the virtual particles. It is assumed that an electron. if there would not be an electromagnetical field. The effects of the virtual particles are described by the (anti)commutator functions. and is the expectation value of the time2 ordered product in the vacuum state. Only µ terms with the correct number of particles in the initial and final state contribute to a matrix element Φ i |S|Φj .A. So this gives. Wevers eight terms appear.10 Divergences and renormalization It turns out that higher orders contribute infinite terms because only the sum p + k of the four-momentum of the virtual particles is fixed. D F (x) = ∆F (x)|m=0 and  1 eikx 3   d k   (2π)3 ωk  F ∆ (x) =   1 e−ikx 3    d k 3 (2π) ωk if x0 > 0 if x0 < 0 The term 1 ∆F (x − y) is called the contraction of Φ(x) and Φ(y).90 Physics Formulary by ir. This means that energy and momentum is conserved. An integration over one of them becomes ∞. The graphical representation of these processes are called Feynman diagrams. Baryons: these exist of 3 quarks or 3 antiquarks. τ ± .9764 139.56995 139. Mass/energy because the laws of nature are invariant for translations in time. µ and τ .45 2285.8 9460.684 1189.511 105. An overview of particles and antiparticles is given in the following table: Particle u d s c b t e− µ− τ− νe νµ ντ γ gluon W+ Z graviton spin (¯ ) B h 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 0 1/2 0 1/2 0 1/2 0 1/2 0 1/2 0 1 0 1 0 1 0 1 0 2 0 L 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 T 1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T3 1/2 −1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 B∗ 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 charge (e) +2/3 −1/3 −1/3 +2/3 −1/3 +2/3 −1 −1 −1 0 0 0 0 0 +1 0 0 m 0 (MeV) 5 9 175 1350 4500 173000 0. W± .672 493. These can be derived from symmetries in the Lagrange density: continuous symmetries give rise to additive conservation laws.6 1864.45 1672. which are separately conserved.1 1232.56995 497. Neutrino’s have a 2 2 1 1 helicity of − 2 while antineutrino’s have only + 2 as possible value.37 1189. one for e.436 Σ+ Ξ0 0 Ξ − Ξ Ξ+ Ω− Ω+ Λ+ c ∆ 2− ∆ 2+ ∆+ ∆0 ∆− dds uss uss dss dss sss sss udc uuu uuu uud udd ddd 1197.0 1232.4 1864.37 938. 3.0 1969.0 1232. Geometrical conservation laws are invariant under Lorentz transformations and the CPT-operation. Z0 . The quantum numbers are subject to conservation laws.32 1321.684 1115. 2. µ± . S the strangeness and B ∗ the bottomness.9 1314.1 0(?) 0(?) 0(?) 0 0 80220 91187 0 antipart. while baryons are fermions with spin 1 or 3 .27231 939. . ντ .0 1232.55 1197. discrete symmetries result in multiplicative conservation laws. So mesons are bosons with spin 0 or 1 in their ground state. Mesons: these exist of one quark and one antiquark.55 1192.56563 1115.0 1232. These are: 1. The anti particles have quantum numbers with the opposite sign except for the total isospin T. gravitons (?).6 1969. ν τ . Leptons: e± . with T 3 the projection of the isospin on the third axis.Chapter 15: Quantum field theory & Particle physics 91 II.436 1314.658 1777.9 1321.37 1192.677 1869.27231 938. u d s c b t e+ µ+ τ+ νe νµ ντ γ gluon W− Z graviton Here B is the baryon number and L the lepton number.672 497.0 J/Ψ Υ p+ p− n0 n0 Λ Λ Σ+ Σ− Σ0 Σ0 Σ− cc bb uud uud udd udd uds uds uus uus uds uds dds 3096. It is found that there are three different lepton numbers. together with their mass in MeV in their ground state: π0 π+ π− K0 K0 K+ K− D+ D− D0 D0 F+ F− 1 2 √ 2(uu+dd) ud du sd ds us su cd dc cu uc cs sc 134. ν µ . νµ .4 1869. Field quanta: γ. There exist excited states with higher internal L.677 493. C the charmness.0 Each quark can exist in two spin states. The composition of (anti)quarks of the hadrons is given in the following table.32 1672. νe . gluons. ν e . T is the isospin.56563 939. Isospin because QCD is invariant for rotations in T-space. Quarks type is only conserved under the colour interaction. Electrical charge because the Maxwell equations are invariant under gauge transformations. 3. Left-handed electrons appear more than 1000× as much as right-handed ones. 3. Baryon number and lepton number are conserved but not under a possible SU(5) symmetry of the laws of nature.C. These are the lowest possible states with a s-quark so they can decay only weakly. The following holds: C|K 0 = η|K0 where η is a phase factor. From this follows that K 0 and K0 are not eigenvalues of CP: CP|K 0 = |K0 .92 Physics Formulary by ir. 15. 2. Further holds P|K 0 = −|K0 because K0 and K0 have an intrinsic parity of −1. and even CP-symmetry is not conserved. Parity is conserved except for weak interactions. Some processes which violate P symmetry but conserve the combination CP are: 1. µ-decay: µ− → e− + νµ + ν e . A base of K0 and K0 is practical while 1 1 2 2 1 2 describing weak interactions. The potential energy will become high enough to create a quarkantiquark pair when it is tried to disjoin an (anti)quark from a hadron.A. Colour charge is conserved. There is no connection with the neutrino: the decay of the Λ particle through: Λ → p + + π − and Λ → n0 + π 0 has also these properties. For colour interactions a base of K 0 and K0 is practical because then the number u−number u is constant. 4. 2. β-decay of spin-polarized 60 Co: 60 Co →60 Ni + e− + ν e . The expansion postulate must be used for weak decays: |K0 = 1 ( K0 |K0 + K0 |K0 ) 1 2 2 . Dynamical conservation laws are invariant under the CPT-operation. The color force does not decrease with distance.12 P and CP-violation It is found that the weak interaction violates P-symmetry. This will result in two hadrons and not in free quarks. More electrons with a spin parallel to the Co than with a spin antiparallel are created: (parallel−antiparallel)/(total)=20%. Momentum because the laws of nature are invariant for translations in space. 3. The linear combinations √ √ |K0 := 1 2(|K0 + |K0 ) and |K0 := 1 2(|K0 − |K0 ) 1 2 2 2 are eigenstates of CP: CP|K0 = +|K0 and CP|K0 = −|K0 . 5. These are: 1. The CP-symmetry was found to be violated by the decay of neutral Kaons. 6. The elementary particles can be classified into three families: leptons 1st generation 2nd generation 3rd generation e νe − quarks d u s c b t antileptons e νe + antiquarks d u s c b t µ− νµ τ− ντ µ+ νµ τ+ ντ Quarks exist in three colours but because they are confined these colours cannot be seen directly. Wevers 2. J. Angular momentum because the laws of nature are invariant for rotations. g is the µ matching coupling constant. the probability of CP=+1 decay is 2 The relation between the mass eigenvalues of the quarks (unaccented) and the fields arising in the weak currents (accented) is (u . The multiplets are classified as follows: e− R T T3 Y 0 0 −2 1 2 νeL e− L 1 2 uL dL 1 2 1 2 uR 0 dR 0 0 −2 3 − −1 1 2 − 1 3 1 2 0 4 3 .and left-handed spin states are treated different because the weak interaction does not conserve parity. c. t ) = (u. for quarks given by Y = B + S + C + B ∗ + T . t).1 The electroweak theory The electroweak interaction arises from the necessity to keep the Lagrange density invariant for local gauge transformations of the group SU(2)⊗U(1). one has to perform the transformation ∂ D ∂ g → = − i Lk Ak µ ∂xµ Dxµ ∂xµ h ¯ Here the Lk are the generators of the gauge group (the “charges”) and the A k are the gauge fields.01. 2 | K1 |K 1 2| K0 |K0 |2 .handed solutions of the Dirac equation for neutrino’s are given by: ψL = 1 (1 + γ5 )ψ and ψR = 1 (1 − γ5 )ψ 2 2 It appears that neutrino’s are always left-handed while antineutrino’s are always right-handed. The Lagrange density for a scalar field becomes: a µν 1 L = − 2 (Dµ Φ∗ Dµ Φ + M 2 Φ∗ Φ) − 1 Fµν Fa 4 a and the field tensors are given by: F µν = ∂µ Aa − ∂ν Aa + gca Al Am . is defined by: Q = 1 Y + T3 2 so [Y. If a fifth Dirac matrix is defined by:   0 0 1 0  0 0 0 1   γ5 := γ1 γ2 γ3 γ4 = −   1 0 0 0  0 1 0 0 the left.13 The standard model When one wants to make the Lagrange density which describes a field invariant for local gauge transformations from a certain group.Chapter 15: Quantum field theory & Particle physics 93 The probability to find a final state with CP= −1 is 1 0 0 2 | .23 ± 0. Tk ] = 0. Right. 15. c . ν µ lm µ ν 15. The hypercharge Y .and right. The group U(1) Y ⊗SU(2)T is taken as symmetry group for the electroweak interaction because the generators of this group commute. and:       sin θ1 0 1 0 0 1 0 0 cos θ1 d  s  =  0 cos θ2 sin θ2   0 1 0   − sin θ1 cos θ1 0  b 0 − sin θ2 cos θ2 0 0 1 0 0 eiδ    d 1 0 0  0 cos θ3 sin θ3   s  0 − sin θ3 cos θ3 b θ1 ≡ θC is the Cabibbo angle: sin(θ C ) ≈ 0.13. Their antiparticles have charges −1 and 0.94 Physics Formulary by ir. and √ mφ = µ 2) as: Φ= Φ+ Φ0 = iω + √ (v + φ − iz)/ 2 and 0|Φ|0 = 0 √ v/ 2 Because this expectation value = 0 the SU(2) symmetry is broken but the U(1) symmetry is not.0 ± 0. 2 15.187 ± 0. The field can be written (were ω ± and z are Nambu-Goldstone bosons which can be transformed away.handed quarks now belong to the same multiplet a mass term can be introduced. 1 σ are the generators of T and −1 and −2 the generators of Y . It is assumed that there exists an isospin-doublet of scalar fields Φ with electrical charges +1 and 0 and potential V (Φ) = −µ 2 Φ∗ Φ + λ(Φ∗ Φ)2 .255 ± 0. According to the weak theory this should be: M W = 83. The total Lagrange density (minus the fieldterms) for the electron-fermion field now becomes: L0. There exist three colours and three anticolours. This means that the dynamic equations which describe the system have a symmetry which the ground state does not have. Hence the state with the lowest energy corresponds with the state Φ ∗ (x)Φ(x) = v = µ2 /2λ =constant.022 ± 0.26 GeV/c2 and MZ = 91. ψeL )γ µ ∂µ − i Aµ · ( 1 σ) − 1 i Bµ · (−1) 2 2 ¯ h ¯ h g ψeR γ µ ∂µ − 1 i (−2)Bµ ψeR 2 ¯ h ψνe.L . A distinction can be made between two types of particles: 1. .24 GeV/c2 and MZ = 93.007 GeV/c2 .A. When the gauge fields in the resulting Lagrange density are separated one obtains: √ √ − + Wµ = 1 2(A1 + iA2 ) .8 ± 2.13. 2. 1 field Bµ (x) is connected with gauge group U(1) and 3 gauge fields Aµ (x) are connected with SU(2). This term can be brought in the form M kl = mk δkl .l a µν Ψk Mkl Ψl − 1 Fµν Fa 4 The gluons remain massless because this Lagrange density does not contain spinless particles. 2 2 gg and for the elementary charge holds: e = = g cos(θW ) = g sin(θW ) g2 + g 2 Experimentally it is found that M W = 80. Wevers Now. “Coloured” particles: the generators T are 8 3 × 3 matrices. “White” particles: they have no colour charge.EW = g g −(ψνe. L H = (DLµ Φ)∗ (DL Φ) − V (Φ).L ψeL − Here. Wµ = 1 2(A1 − iA2 ) µ µ µ µ 2 2 Zµ Aµ = = gA3 − g Bµ µ g2 + g 2 g A3 + gBµ µ g2 + g 2 ≡ A3 cos(θW ) − Bµ sin(θW ) µ ≡ A3 sin(θW ) + Bµ cos(θW ) µ where θW is called the Weinberg angle.3 Quantumchromodynamics Coloured particles interact because the Lagrange density is invariant for the transformations of the group SU(3) of the colour interaction. For this angle holds: sin 2 (θW ) = 0. J. 15. are globally U(1)⊗SU(2) symmetric. Because leftand right.2 Spontaneous symmetry breaking: the Higgs mechanism All leptons are massless in the equations above.0 GeV/c2 . Their mass is probably generated by spontaneous symmetry breaking.C. The Lagrange density for coloured particles is given by LQCD = i k Ψ k γ µ Dµ Ψ k + k. the generator T = 0.010.13. Relations for the masses of the field quanta can be obtained from the remaining terms: M W = 1 vg and MZ = 1 v g 2 + g 2 . The extra µ terms in L arising from these fields. Chapter 15: Quantum field theory & Particle physics 95 15.14 Path integrals The development in time of a quantum mechanical system can, besides with Schr¨ dingers equation, also be o described by a path integral (Feynman): ψ(x , t ) = F (x , t , x, t)ψ(x, t)dx in which F (x , t , x, t) is the amplitude of probability to find a system on time t in x if it was in x on time t. Then, iS[x] F (x , t , x, t) = exp d[x] h ¯ where S[x] is an action-integral: S[x] = taken over all possible paths [x]: L(x, x, t)dt. The notation d[x] means that the integral has to be ˙   n ∞ 1 d[x] := lim n→∞ N  −∞   dx(tn )  in which N is a normalization constant. To each path is assigned a probability amplitude exp(iS/¯ ). The h classical limit can be found by taking δS = 0: the average of the exponent vanishes, except where it is stationary. In quantum fieldtheory, the probability of the transition of a fieldoperator Φ(x, −∞) to Φ (x, ∞) is given by iS[Φ] F (Φ (x, ∞), Φ(x, −∞)) = exp d[Φ] h ¯ with the action-integral S[Φ] = Ω L(Φ, ∂ν Φ)d4 x 15.15 Unification and quantum gravity The strength of the forces varies with energy and the reciprocal coupling constants approach each other with increasing energy. The SU(5) model predicts complete unification of the electromagnetical, weak and colour forces at 1015 GeV. It also predicts 12 extra X bosons which couple leptons and quarks and are i.g. responsible for proton decay, with dominant channel p + → π 0 + e+ , with an average lifetime of the proton of 10 31 year. This model has been experimentally falsified. Supersymmetric models assume a symmetry between bosons and fermions and predict partners for the currently known particles with a spin which differs 1 . The supersymmetric SU(5) model predicts unification at 2 1016 GeV and an average lifetime of the proton of 10 33 year. The dominant decay channels in this theory are p+ → K+ + ν µ and p+ → K0 + µ+ . Quantum gravity plays only a role in particle interactions at the Planck dimensions, where λ C ≈ RS : mPl = hc/G = 3 · 1019 GeV, tPl = h/mPlc2 = hG/c5 = 10−43 sec and rPl = ctPl ≈ 10−35 m. Chapter 16 Astrophysics 16.1 Determination of distances The parallax is mostly used to determine distances in nearby space. The parallax is the angular difference between two measurements of the position of the object from different view-points. If the annual parallax is given by p, the distance R of the object is given by R = a/ sin(p), in which a is the radius of the Earth’s orbit. The clusterparallax is used to determine the distance of a group of stars by using their motion w.r.t. a fixed background. The tangential velocity v t and the radial velocity v r of the stars along the sky are given by vr = V cos(θ) , vt = V sin(θ) = ωR ˆ where θ is the angle between the star and the point of convergence and R the distance in pc. This results, with v t = vr tan(θ), in: R= 1 vr tan(θ) ˆ ⇒ R= ω p -5 -4 M -3 -2 -1 0 1 Type 1 Type 2 RR-Lyrae 0,1 0,3 1 where p is the parallax in arc seconds. The parallax is then given by p= 4.74µ vr tan(θ) P (days) → 3 10 30 100 with µ de proper motion of the star in /yr. A method to determine the distance of objects which are somewhat further away, like galaxies and star clusters, uses the period-Brightness relation for Cepheids. This relation is shown in the above figure for different types of stars. 16.2 Brightness and magnitudes The brightness is the total radiated energy per unit of time. Earth receives s 0 = 1.374 kW/m2 from the Sun. Hence, the brightness of the Sun is given by L = 4πr2 s0 = 3.82 · 1026 W. It is also given by: ∞ L = 4πR 2 0 πFν dν where πFν is the monochromatic radiation flux. At the position of an observer this is πf ν , with fν = (R/r)2 Fν if absorption is ignored. If A ν is the fraction of the flux which reaches Earth’s surface, the transmission factor is given by Rν and the surface of the detector is given by πa 2 , then the apparent brightness b is given by: ∞ b = πa 2 0 fν Aν Rν dν The magnitude m is defined by: b1 1 = (100) 5 (m2 −m1 ) = (2.512)m2 −m1 b2 Chapter 16: Astrophysics 97 because the human eye perceives lightintensities logaritmical. From this follows that m 2 − m1 = 2.5 ·10 log(b1 /b2 ), or: m = −2.5 ·10 log(b) + C. The apparent brightness of a star if this star would be at a distance of 10 pc is called the absolute brightness B: B/b = (ˆ/10) 2 . The absolute magnitude is then given by r M = −2.5 ·10 log(B) + C, or: M = 5 + m − 5 ·10 log(ˆ). When an interstellar absorption of 10 −4 /pc is taken r into account one finds: ˆ r M = (m − 4 · 10−4 r ) + 5 − 5 ·10 log(ˆ) If a detector detects all radiation emitted by a source one would measure the absolute bolometric magnitude. If the bolometric correction BC is given by BC = 2.5 ·10 log Energy flux received Energy flux detected = 2.5 ·10 log fν dν fν Aν Rν dν holds: Mb = MV − BC where MV is the visual magnitude. Further holds Mb = −2.5 ·10 log L L + 4.72 16.3 Radiation and stellar atmospheres The radiation energy passing through a surface dA is dE = I ν (θ, ϕ) cos(θ)dνdΩdAdt, where Iµ is the monochromatical intensity [Wm −2 sr−1 Hz−1 ]. When there is no absorption the quantity I ν is independent of the distance to the source. Planck’s law holds for a black body: Iν (T ) ≡ Bν (T ) = 1 c 2hν 3 wν (T ) = 2 4π c exp(hν/kT ) − 1 The radiation transport through a layer can then be written as: dIν = −Iν κν + jν ds Here, jν is the coefficient of emission and κ ν the coefficient of absorption. ds is the thickness of the layer. 1, the layer The optical thickness τ ν of the layer is given by τ ν = κν ds. The layer is optically thin if τ ν is optically thick if τν 1. For a stellar atmosphere in LTE holds: j ν = κν Bν (T ). Then also holds: Iν (s) = Iν (0)e−τν + Bν (T )(1 − e−τν ) 16.4 Composition and evolution of stars The structure of a star is described by the following equations: dM (r) dr dp(r) dr L(r) dr dT (r) dr rad dT (r) dr conv = = = = = 4π (r)r2 − GM (r) (r) r2 4π (r)ε(r)r2 − 3 L(r) κ(r) , (Eddington), or 4 4πr2 4σT 3 (r) T (r) γ − 1 dp(r) , (convective energy transport) p(r) γ dr Further, for stars of the solar type, the composing plasma can be described as an ideal gas: p(r) = (r)kT (r) µmH I. 25. The slowest. pp2: 3 He + α → 7 Be + γ i. pp1: 3 He +3 He → 2p+ + 4 He.74 + (1.92 + (0. M (r) = 1 M . Two reaction chains are responsible for this reaction.98 Physics Formulary by ir. 7 Be + p+ → 8 B + γ.03 + (1.51) MeV released. speedlimiting reaction is shown in boldface. 19. → O + e+ → ↑ 14 N + p+ → 15 15 N+ν O+γ ← 15 −→ N + p+ → α +12 C ↓ 12 C + p+ → 13 N + γ ↓ 13 N → 13 C + e+ + ν ↓ 13 C + p+ → 14 N + γ ←− 15 15 N + p+ → 16 O + γ ↓ 16 O + p+ → 17 F + γ ↓ 17 F → 17 O + e+ + ν ↓ 17 O + p+ → α + 14 N .21 + (0. T (r). κ ∼ and ε ∼ T µ (this last assumption is only 2 2 valid for stars on the main sequence). then 7 Li + p+ → 24 He + γ.A. For stars in quasi-hydrostatic equilibrium hold the approximations r = 1 R. Wevers where µ is the average molecular mass. This reaction produces 26. ii. composition) and ε(r) = g( (r). The CNO cycle. 7 Be + e− → 7 Li + ν. 8 It can be derived that L ∼ M 3 : the mass-brightness relation. There is 26. 2.69) MeV. The reactions are shown below.72 MeV.5 Energy production in stars The net reaction from which most stars gain their energy is: 4 1 H → 4 He + 2e+ + 2νe + γ. dM/dr = M/R. Further holds: κ(r) = f ( (r). II. Y the mass fraction of He and Z the mass fraction of the other elements.80) MeV.2) MeV.5 + (7. The first chain releases 25. usually well approximated by: µ= nmH = 1 2X + 3 4Y + 1Z 2 where X is the mass fraction of H. The proton-proton chain can be divided into two subchains: 1 H + p+ → 2 D + e+ + νe . Both 7 Be chains become more important with raising T . The energy between brackets is the energy carried away by the neutrino. composition) Convection will occur when the star meets the Schwartzschild criterium: dT dr < conv dT dr rad Otherwise the energy transfer takes place by radiation. the second 24. T (r). Further holds: L ∼ R 4 ∼ Teff .98) MeV. For pp-chains holds µ ≈ 5 and for the CNO chains holds µ = 12 tot 18. This results in the equation for the main sequence in the Hertzsprung-Russel diagram: 10 log(L) = 8 ·10 log(Teff ) + constant 16.C. then 8 B + e+ → 24 He + ν. and then 2 D + p → 3 He + γ. J. 1. Then holds: gradf div a rot a = = = 1 ∂f 1 ∂f 1 ∂f eu + ev + ew h1 ∂u h2 ∂v h3 ∂w ∂ 1 ∂ ∂ (h2 h3 au ) + (h3 h1 av ) + (h1 h2 aw ) h1 h2 h3 ∂u ∂v ∂w 1 ∂(h3 aw ) ∂(h2 av ) ∂(h1 au ) ∂(h3 aw ) 1 − − eu + h2 h3 ∂v ∂w h3 h1 ∂w ∂u ∂(h2 av ) ∂(h1 au ) 1 − ew h1 h2 ∂u ∂v ∂ h2 h3 ∂f h1 h2 ∂f 1 ∂ h3 h1 ∂f ∂ + + h1 h2 h3 ∂u h1 ∂u ∂v h2 ∂v ∂w h3 ∂w ev + ∇2 f = . y. gradf = ∇f = ex + ey + ez ∂x ∂y ∂z ∂x ∂y ∂z ∂ay ∂az ∂ax ∂2f ∂2f ∂2f + + . v. ∇2 f = + 2 + 2 2 ∂x ∂y ∂z ∂x ∂y ∂z ex + ∂ax ∂az − ∂z ∂x ey + ∂ay ∂ax − ∂x ∂y ez div a = ∇ · a = rot a = ∇ × a = ∂az ∂ay − ∂y ∂z In cylinder coordinates (r. θ. z) holds: ∇= div a = rot a = ∂ 1 ∂ ∂ ∂f 1 ∂f ∂f er + eϕ + ez . ϕ. v. gradf = er + eϕ + ez ∂r r ∂ϕ ∂z ∂r r ∂ϕ ∂z ar 1 ∂aϕ ∂az 1 ∂2f ∂ar ∂2f 1 ∂f ∂2f + + + . The unit vectors are then given by: eu = 1 ∂x 1 ∂x 1 ∂x .The ∇ operator 99 The ∇-operator In cartesian coordinates (x. ϕ) holds: ∇ = gradf div a rot a = = = ∂ ∂ 1 ∂ 1 er + eθ + eϕ ∂r r ∂θ r sin θ ∂ϕ ∂f 1 ∂f 1 ∂f er + eθ + eϕ ∂r r ∂θ r sin θ ∂ϕ 2ar 1 ∂aθ aθ 1 ∂aϕ ∂ar + + + + ∂r r r ∂θ r tan θ r sin θ ∂ϕ 1 ∂aϕ 1 ∂ar aθ 1 ∂aθ ∂aϕ aϕ + − − − er + r ∂θ r tan θ r sin θ ∂ϕ r sin θ ∂ϕ ∂r r ∂aθ aθ 1 ∂ar + − eϕ ∂r r r ∂θ ∂2f 1 ∂2f ∂f 1 ∂2f 2 ∂f 1 + 2 2 + 2 + 2 2 + ∂r2 r ∂r r ∂θ r tan θ ∂θ r sin θ ∂ϕ2 eθ + ∇2 f = General orthonormal curvelinear coordinates (u. w) can be obtained from cartesian coordinates by the transformation x = x(u. ∇2 f = + 2 + + 2 2 2 ∂r r r ∂ϕ ∂z ∂r r ∂r r ∂ϕ ∂z 1 ∂az ∂aϕ − r ∂ϕ ∂z er + ∂ar ∂az − ∂z ∂r eϕ + ∂aϕ aϕ 1 ∂ar + − ∂r r r ∂ϕ ez In spherical coordinates (r. ev = . w). z) holds: ∇= ∂ ∂ ∂ ∂f ∂f ∂f ex + ey + ez . ew = h1 ∂u h2 ∂v h3 ∂w where the factors h i set the norm to 1. 100 The SI units The SI units Basic units Quantity Length Mass Time Therm. flux density Inductance Luminous flux Illuminance Activity Absorbed dose Dose equivalent Unit hertz newton pascal joule watt coulomb volt farad ohm siemens weber tesla henry lumen lux bequerel gray sievert Sym. current Luminous intens. Resistance El. flux Mag. temp. Unit metre kilogram second kelvin ampere candela mol Sym. Capacitance El. Conductance Mag. Potential El. m kg s K A cd mol Derived units with special names Quantity Frequency Force Pressure Energy Power Charge El. Electr. Hz N Pa J W C V F Ω S Wb T H lm lx Bq Gy Sv Derivation s −1 kg · m · s −2 N · m−2 N·m J · s−1 A·s W · A−1 C · V −1 V · A−1 A · V −1 V·s Wb · m −2 Wb · A −1 cd · sr lm · m −2 s −1 J · kg −1 J · kg −1 Extra units Plane angle solid angle radian sterradian rad sr Prefixes yotta zetta exa peta tera Y Z E P T 1024 1021 1018 1015 1012 giga mega kilo hecto deca G M k h da 109 106 103 102 10 deci centi milli micro nano d c m µ n 10−1 10−2 10−3 10−6 10 −9 pico femto atto zepto yocto p f a z y 10−12 10−15 10−18 10−21 10 −24 . Amount of subst.