Homework # 8Chapter 9 Kittel Phys 175A Dr. Ray Kwok SJSU Prob. 1 – Brillouin zones of rectangular lattice Daniel Wolpert 9.1 Brillouin zones of rectangular lattice. Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a, b=3a 2π/a 2π/3a Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a. b=3a .9.1 Brillouin zones of rectangular lattice. 9. Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a. b=3a 2π/a 2π/3a 2nd BZ First BZ .1 Brillouin zones of rectangular lattice. .rectangular lattice Gregory Kaminsky This is a Wigner-Seitz cell.Prob. 2 – Brillouin zone. b = 4 A0. a) Draw the first Brillouin zone.A two-dimensional metal has one atom of valency one in a simple rectangular primitive cell a = 2 A0 . Give it’s dimensions in cm-1. c) Draw this sphere to scale ona drawing of the first Brillouin zone. b) Calculate the radius of the free electron fermi sphere. . Calculation of the radius of the Fermi sphere 2 2 *π * kF 2 π * kF 2* 2 = N 2π L 4 *π 0 2 4*2*(A ) 2 =1 π kF = 2 * A0 = π 2 * 1012 cm −1 . Brilloin zone Radius of free electron fermi sphere = π 2 π *1012 cm −1 π 2 * 1012 cm −1 . Make another sketch to show the first few periods of the free electron band in the periodic zone scheme. Assume there is a small energy gap at the zone boundary. for both the first and second energy bands. . This is the first energy band . Second energy band . sketch carefully the electron and hole energy surfaces. In the almost free electron approximation. Hole Energy surface Electron Energy Surface BZ periodic scheme Second Zone periodic scheme . 4 – Brillouin Zones of Two-Dimensional Divalent Metal Victor Chikhani A two dimensional metal in the form of a square lattice has two conduction electrons per atom. For the electrons choose a zone scheme such that the Fermi surface is shown as closed.Prob. A magnetic field B = 10 −1T is normal to the place of the open orbit. . 5 – Open Orbits John Anzaldo An open orbit in a monovalent tetragonal metal connects opposite faces of the boundary of a Brillouin zone. (a) What is the order of magnitude of the period of thek motion in 8 Take v = 10 cm / s space? (b) Describe in real space the motion of an electron on this orbit in the presence of the magnetic field.Prob. The faces are separated by G = 2 ×108 cm −1. 315 ⋅10 −10 s evB 2 ⋅ π ⋅1. where I have decided to use SI units.62 ⋅10 Gh cm m s = = 1.5 From Eq. 25a we have .602 ⋅10 −19 C ⋅108 cm ⋅ 1m ⋅10 −1 kg s 100cm C⋅s Part b) The electron will travel along the Fermi surface as shown. . The velocity will change as the electron moves along the Fermi surface. v v v G d r q = − e dt = τ h = − ev B Letting we get = v .9. setting dk = G τ dt because v ⊥ B since B is normal to the Fermi surface. Solving for τ gives Gh = τ . Plugging in the givens we evB get v v dk dr v h = q ×B dt dt 2 2 ⋅108 100cm −34 kg ⋅ m ⋅ ⋅ 6. Mike Tuffley 5/12/09 . U(x) -a/2 a/2 x -U0 . . . Chapter 9 Problem 7 Adam Gray 1 (a) Calculate the period ∆( B ) expected for potassium on the free electron model. (b) What is the area in real space of the extremal orbit. for B = 10kG = 1T ? . 1 on pg. 139.75x108cm-1 . for potassium we find kf=0.Starting with equation 34: 1 2πe ∆( ) = B hcS Where S = πK 2 f Using Table 6. . Plugging in: 1 2πe ∆( ) = B hc(πK 2 f ) 1 2e ∆( ) = B hcK2 f Note: The equation 34 was for cgs units.803x10-10 erg1/2 cm1/2 . so all values used with this equation must be in this form.05459x10-27 erg s e=4. c=3x1010 cm/s h=1. Be ωc = mc Solve for r r = v f ωc P = mv = hk vf v f mc hk f c r = = = = ω c Be Be Be mc vf .55×10−9 G −1 B (b) To solve this part of the problem.This results in 1 ∆( ) = 5. go back to the equations we used for the cyclotron. Plugging in values from before and B=10kG r = 4.94x10-4 cm The orbit is circular.67 ×10 cm 2 −7 2 . so the area is πr = 7.