Magnetic properties of materials1 Fundamental Relationships Coulomb's Law • a force exists between 2 magnetic poles: • where is the force is the permeability of free space, = , are the magnetic pole strength is the distance separating the poles is the unit radial vector • o o o o unlike gravity, poles come in 2 flavors: + (north-seeking) - (south-seeking) like poles repel (F is +, force is outward) unlike poles attract (F is -, force is inward) • • Magnetic Dipole A dipole consists of two poles of opposite polarity and equal strength. The strength of a dipole depends on strength of magnetization of poles and their separation, and is a vector quantity known as dipole moment, which is analogous to mass in gravity: M = ml where M is a vector directed from the negative pole to the positive pole The dipole moment is analogous to mass • Intensity of magnetization Magnetic dipole moment is an extensive quantity. In analogy with gravity, magnetic dipole moment per unit volume is an intensive quantity (like density). This is also called the intensity of magnetization, or I = M/volume = ml/volume = m/area where I and M are vector quantities. M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 2 Permeability Permeability, also called magnetic permeability, is a constant of proportionality that exists between magnetic induction and magnetic field intensity. This constant is equal to approximately 1.257 x 10-6 henry per meter (H/m) in free space (a vacuum). In other materials it can be much different, often substantially greater than the free-space value, which is symbolized µ o. The permeability factors of some substances change with rising or falling temperature, or with the intensity of the applied magnetic field. Relative permeability, sometimes denoted by the symbol µr, is the ratio of the permeability of a specific medium to the permeability of free space. If µ o represents the permeability of free space (that is, 1.257 x 10-6 H/m) and µ represents the permeability of the substance in question (also specified in henrys per meter), then the relative permeability, µr, is given by: µ r = µ / µ o = µ (7.958 x 105) Magnetic Induction, B as with gravity, we are interested in force Earth exerts on a unit pole (like acceleration, with g) o or, 'magnetic field intensity' o Analogous to gravitational acceleration (but not acceleration units!) o force per unit pole strength (force exerted on unit magnetic pole) o (In our analogy with gravity, m here is the Earth's "monopole" field, which is a fiction; Stacey incorrectly calls B "magnetic field, which is H) Magnetic Field Strength, H o if we only had to deal with a vacuum (or even air, since it has negligible magnetic susceptibility), we could always deal with H (magnetic field strength).. o however, in presence of "magnetizable" material, there is a magnetic polarization (or, simply, magnetization) of material which produces an additional field (J) which adds to H o combining the field strength, H, and the magnetic polarization (magnetization), J, is call the magnetic induction, B B = µ0H + J J = µ 0M where µ 0 = 4 x 10-7 H/m (Henry/meter) is the permeability of free space Units SI system o in SI, for force of 1 Newton and 1 unit pole strength: A/m (H), or Tesla (B) B magnetic induction tesla H J magnetic field magnetic polarization, magnetization T A/m T A/m amperes per meter tesla amperes per meter M magnetic dipole moment per unit volume M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 3 MAGNETIC MOMENTS OF ELECTRONS There are two kinds of electron motion, orbital and spin, and each has a magnetic moment associated with it. The orbital motion of an electron around the nucleus may be likened to a current in a loop of wire having no resistance; both are equivalent to a circulation of charge. The magnetic moment of an electron, due to this motion, may be calculated by an equation To evaluate m we must know the size and shape of the orbit and the electron velocity. In the original (1913) Bohr theory of the atom, the electron moved with velocity v in a circular orbit of radius r. In cgs units e is the charge on the electron in esu and c the velocity of light, so e/c is the charge in emu. In SI units, the charge of the electron is measured in coulombs. The current, or charge passing a given point per unit time, is then (e=c)(v=2pr) (cgs) or ev/2pr (SI). Therefore, or An additional postulate of the theory was that the angular momentum of the electron must be an integral multiple of h/2p, where h is Planck’s constant. Therefore, mvr=nh/2π where m is the mass of the electron. Combining these relations, we have for the magnetic moment of the electron in the first (n = 1) Bohr orbit. The spin of the electron was postulated in 1925 in order to explain certain features of the optical spectra of hot gases, particularly gases subjected to a magnetic field (Zeeman effect), and it later found theoretical confirmation in wave mechanics. Spin is a universal property of electrons in all states of matter at all temperatures. The electron behaves as if it were in some sense spinning about its own axis, and associated with this spin are definite amounts of magnetic moment and angular momentum. It is found experimentally and theoretically that the magnetic moment due to electron spin is equal to On the basis of their magnetic properties different materials are classified as: • Diamagnetic substance • Paramagnetic substance • Ferromagnetic substance Diamagnetic Substance Michael Faraday discovered that a specimen of bismuth was repelled by a strong magnet. Diamagnetism occurs in all materials. These materials are those in which individual atoms do not possess any net magnetic moment. [Their orbital and spin magnetic moment add vectorially to become zero]. The atoms of such material however acquire an induced dipole moments when they are placed in an external magnetic field. Some important properties are: M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials • 4 When suspended in a uniform magnetic field they set their longest axis at right angles to the field as shown 2) In a non-uniform magnetic material, these substances move from stronger parts of the field to the weaker parts. For e.g.,. when diamagnetic liquid is put in a watch glass placed on the two pole pieces of an electromagnet and current is switched on the liquid accumulates on the sides. [Note on increasing the distance between the pole, the effect is reversed] 3) A diamagnetic liquid in a U shaped tube is depressed, when subjected to a magnetic field. 4) The lines of force do not prefer to pass through the specimen, since the ability of a material to permit the passage of magnetic lines of force through it is less. 5) The permeability of the substance, that is, mr < 1. 6) The substance loses its magnetization as soon as the magnetizing field is removed. 7) Such specimen cannot be easily magnetized and so their susceptibility is negative. Example: Bismuth, antimony, copper, gold, quartz, mercury, water, alcohol, air, hydrogen etc. Paramagnetic Substance Paramagnetic substances are attracted by a magnet very feebly. In a sample of a paramagnetic material, the atomic dipole moments initially are randomly oriented in space. When an external field is applied, the dipoles rotate into alignment with field as shown M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 5 • • The vector sum of the individual dipole moments is no longer zero. Some important properties are: The paramagnetic substance develops a weak magnetization in the direction of the field. When a paramagnetic rod is suspended freely in a uniform magnetic field, it aligns itself in the direction of magnetic field. • The lines of force prefer to pass through the material rather than air that is mr > 1 that is their permeability is greater than one. • • • As soon as the magnetizing field is removed the paramagnetics lose their magnetization. In a non-uniform magnetic, the specimen move from weaker parts of the field to the stronger parts (that is it accumulates in the middle). A paramagnetic liquid in U tube placed between two poles of a magnet is elevated. • The magnetization of paramagnetism decreases with increase in temperature. This is because the thermal motion of the atoms tends to disturb the alignment of the dipoles. Example: Aluminum, platinum, chromium, manganese, copper sulphate, oxygen etc., Ferromagnetic Substance Ferromagnetism, like paramagnetism, occurs in materials in which atoms have permanent magnetic dipole moments. The strong interaction between neighboring atomic dipole moments keeps them aligned even when the external magnetic field is removed. Some important properties are: These substances get strongly magnetized in the direction of field. The lines of force prefer to pass through the material rather than air that is mr>1 that is their permeability is greater than one. M Chaitanya Varma Dept. of Engineering Physics GITAM University • • Magnetic properties of materials 6 • • In a non-uniform magnetic, the specimen move from weaker parts of the field to the stronger parts (that is it accumulates in the middle). A paramagnetic liquid in U tube placed between two poles of a magnet is elevated. • For ferromagnetic materials mr is very large and so its susceptibility i.e., Xm is positive. Ferromagnetic substances retain their magnetism even after the magnetizing field is removed. • The effectiveness of coupling between the neighboring atoms that causes ferromagnetism decreases by increasing the temperature of the substance. The temperature at which a ferromagnetic material becomes paramagnetic is called its curie temperature. For example the curie temperature of iron is 1418oF, which means above this temperature, iron is paramagnetic. Example: Iron, cobalt, nickel and number of alloys. • CLASSICAL THEORY OF PARAMAGNETISM It is well known that most of the materials, subject to magnetic fields, exhibits either diamagnetic or paramagnetic behavior. This reflects in a value of the magnetic permeability slightly different from the vacuum permeability µo. Conversely, few materials, like Fe, Ni, and Co behave differently and are referred to as ferromagnetic materials. Let us consider a medium whose elementary particles possess magnetic moment. Let us suppose that no external field is applied, and that the body is in thermodynamic equilibrium. Due to the random orientation of the elementary magnets, the magnetization vector M is zero everywhere in the medium. When an external field Ha is applied, equilibrium between the tendency of dipoles to align with the field and the thermal agitation establishes. This produces the magnetization of the body in the same direction and orientation as the external field. if no opposing force acts, complete alignment of the atomic moments would be produced and the specimen as a whole would acquire a very large moment in the direction of the field. But thermal agitation of the atoms opposes this tendency and tends to keep the atomic moments pointed at random. The result is only partial alignment in the field direction, and therefore a M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 7 small positive susceptibility. The effect of an increase in temperature is to increase the randomizing effect of thermal agitation and therefore to decrease the susceptibility. If we call µ the permanent magnetic moment of the generic dipole and between µ and Ha, the angle Let us consider a unit volume of material containing n atoms, each having a magnetic moment m. Let dn number of moments inclined at an angle between θ and θ+dθ to the field H. In the absence of a field the number of m vectors passing through unit area of the sphere surface is the same at any point on the sphere surface, and dn is proportional simply to the area dA by 2π sinθ dθ for a sphere of unit radius. But when a field is applied, the m vectors all shift toward the direction of the field. Each atomic moment then has a certain potential energy Ep in the field, given by (1) In a state of thermal equilibrium at temperature T, the probability of an atom having an energy Ep is proportional to the Boltzmann factor e-Ep/kT where k is the Boltzmann constant. The number of moments between θ and θ+dθ will now be proportional to dA, multiplied by the Boltzmann factor, or (2) where K is a proportionality factor, determined by the fact that For brevity we put a = µH/kT. We then have (3) The total magnetic moment in the direction of the field acquired by the unit volume under consideration, that is, the magnetization M, is given by multiplying the number of atoms dn by the contribution µcosθ of each atom and integrating over the total number: (4) Substituting the above equations into this equation we get (5) M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 8 But nµ is the maximum possible moment which the material can have. It corresponds to perfect alignment of all the atomic magnets parallel to the field, which is a state of complete saturation. Calling this quantity M0, we have (6) The expression on the right is called the Langevin function, usually abbreviated to L(a). Expressed as a series, it is (7) which is valid only for a ≤ 1. L(a) as a function of a is plotted in below figure. At large a, L(a) tends to 1; and for a less than about 0.5, it is practically a straight line with a slope of 1/3, as seen in Equation 7. The Langevin theory leads to two conclusions: 1. Saturation will occur if is large enough. This makes good physical sense, because large H or low T, or both, is necessary if the aligning tendency of the field is going to overcome the disordering effect of thermal agitation. 2. At small a, the magnetization M varies linearly with H. As we shall see presently, a is small under “normal” conditions, and linear M, H curves are observed. The Langevin theory also leads to the Curie law. For small a, L(a)=1/3, and Equation 6 becomes (8) Therefore (9) The Langevin theory of paramagnetism, which leads to the Curie law, is based on the assumption that the individual carriers of magnetic moment (atoms or molecules) do not interact with one another, but are acted on only by the applied field and thermal M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 9 agitation. Many paramagnetics, however, do not obey this law; they obey instead the more general Curie–Weiss law, (10) Weiss theory of Ferromagnetism Weiss postulated that the elementary moments do interact with one another and suggested that this interaction could be expressed in terms of a fictitious internal field which he called the “molecular field” Hm and which acted in addition to the applied field H. The molecular field was thought to be in some way caused by the magnetization of the surrounding material. Weiss assumed that the intensity of the molecular field was directly proportional to the magnetization: (11) H m = λM where λ is called the molecular field constant. Therefore, the total field acting on the material is (12) Curie Weiss law may be written as nµ 2 H M= 3kT H in this expression must now be replaced by Ht: (13) nµ 2 ( H + H m ) M= 3kT (14) Solving for M and susceptibility χm we get (15) Therefore, θ (= Cλ) is a measure of the strength of the interaction because it is proportional to the molecular field constant γ. Curie–Weiss law, which many paramagnetic materials obey. We saw also that θ is directly related to the molecular field Hm, because θ (= Cλ) and H m = λM ,. If θ is positive, so is γ, which means that Hm and M are in the same direction or that the molecular field aids the applied field in magnetizing the substance. Above its Curie temperature Tc a ferromagnet becomes paramagnetic, and its susceptibility then follows the Curie–Weiss law, with a value of θ approximately equal to Tc. The value of u is therefore large and positive (over 1000K for iron), and so is the molecular field coefficient. This fact led Weiss to make the bold and brilliant assumption that a molecular field acts in a ferromagnetic substance below its Curie temperature as well as above, and that this field is so strong that it can magnetize the substance to saturation even in the absence of an applied field. The substance is then self-saturating, or “spontaneously magnetized.” M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 10 Formation of Domains In order to explain the fact that ferromagnetic materials with spontaneous magnetisation could exist in the demagnetised state Weiss proposed the concept of magnetic domains. The magnetisation within the domain is saturated and will always lie in the easy direction of magnetisation when there is no externally applied field. The direction of the domain alignment across a large volume of material is more or less random and hence the magnetisation of a specimen can be zero. Magnetic domains exist in order to reduce the energy of the system. A uniformly magnetised specimen as shown in figure 1(a) has a large magnetostatic energy associated with it. This is the result of the presence of magnetic free poles at the surface of the specimen generating a demagnetising field, Hd. From the convention adopted for the definition of the magnetic moment for a magnetic dipole the magnetisation within the specimen points from the south pole to the north pole, while the direction of the magnetic field points from north to south. Therefore, the demagnetising field is in opposition to the magnetisation of the specimen. The magnitude of Hd is dependent on the geometry and magnetisation of the specimen. In general if the sample has a high length to diameter ratio (and is magnetised in the long axis) then the demagnetising field and the magnetostatic energy will be low. The break up of the magnetisation into two domains as illustrated in figure1(b) reduces the magnetostatic energy by half. In fact if the magnet breaks down into N domains then the magnetostatic energy is reduced by a factor of 1/N, hence figure 1(c) has a quarter of the magnetostatic energy of figure 1(a). Figure 1(d) shows a closure domain structure where the magnetostatic energy is zero, however, this is only possible for materials that do not have a strong uniaxial anisotropy, and the neighbouring domains do not have to be at 180º to each other. (a) (b) (c) (d) Figure 1: Schematic illustration of the break up of magnetisation into domains (a) single domain, (b) two domains, (c) four domains and (d) closure domains. The introduction of a domain raises the overall energy of the system, therefore the division into domains only continues while the reduction in magnetostatic energy is greater than the energy required to form the domain wall. The energy associated a domain wall is proportional to its area. The schematic representation of the domain wall, shown in figure 6, illustrates that the dipole moments of the atoms within the wall are not pointing in the easy direction of magnetisation and hence are in a higher energy state. In addition, the atomic dipoles within the wall are not at 180º to each other and so the exchange energy is also raised within the wall. Therefore, the domain M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 11 wall energy is an intrinsic property of a material depending on the degree of magnetocrystalline anisotropy and the strength of the exchange interaction between neighbouring atoms. The thickness of the wall will also vary in relation to these parameters, as strong magnetocrystalline anisotropy will favour a narrow wall, whereas a strong exchange interaction will favour a wider wall. Figure 2 : Schematic representation of a 180º domain wall. A minimum energy can therefore be achieved with a specific number of domains within a specimen. This number of domains will depend on the size and shape of the sample (which will affect the magnetostatic energy) and the intrinsic magnetic properties of the material (which will affect the magnetostatic energy and the domain wall energy). The Hysteresis Loop and Magnetic Properties A great deal of information can be learned about the magnetic properties of a material by studying its hysteresis loop. A hysteresis loop shows the relationship between the induced magnetic flux density (B) and the magnetizing force (H). It is often referred to as the B-H loop. An example hysteresis loop is shown below. The loop is generated by measuring the magnetic flux of a ferromagnetic material while the magnetizing force is changed. A ferromagnetic material that has never been previously magnetized or has been thoroughly demagnetized will follow the dashed line as H is increased. As the line demonstrates, the greater the amount of current applied (H+), the stronger the magnetic field in the component (B+). At point "a" almost all of the magnetic domains are aligned and an additional increase in the M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 12 magnetizing force will produce very little increase in magnetic flux. The material has reached the point of magnetic saturation. When H is reduced to zero, the curve will move from point "a" to point "b." At this point, it can be seen that some magnetic flux remains in the material even though the magnetizing force is zero. This is referred to as the point of retentivity on the graph and indicates the remanence or level of residual magnetism in the material. (Some of the magnetic domains remain aligned but some have lost their alignment.) As the magnetizing force is reversed, the curve moves to point "c", where the flux has been reduced to zero. This is called the point of coercivity on the curve. (The reversed magnetizing force has flipped enough of the domains so that the net flux within the material is zero.) The force required to remove the residual magnetism from the material is called the coercive force or coercivity of the material. As the magnetizing force is increased in the negative direction, the material will again become magnetically saturated but in the opposite direction (point "d"). Reducing H to zero brings the curve to point "e." It will have a level of residual magnetism equal to that achieved in the other direction. Increasing H back in the positive direction will return B to zero. Notice that the curve did not return to the origin of the graph because some force is required to remove the residual magnetism. The curve will take a different path from point "f" back to the saturation point where it with complete the loop. Hysteresis and domain mechanism The domain picture is a good one for ferromagnetic solids, when the domain size is much greater than the domain wall width. Domain wall motion and domain rotation are the two basic magnetization processes in any multidomain solid. For small external fields, the domain walls, being pinned at some defects, just bulge out in the proper directions to increase favorably oriented domains and decrease the others. The magnetization (or the magnetic flux B) increases about linearly with H At larger external fields, the domain walls overcome the pinning and move in the right direction where they will become pinned by other defects. Turning the field of will not drive the walls back; the movement is irreversible. M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 13 After just one domain is left over (or one big one and some little ones), increasing the field even more will turn the atomic dipoles in field direction. Finally, saturation is reached. All magnetic dipoles are fully oriented in field direction, no further increase is possible. ‘Soft’ and ‘hard’ magnets The wide variety of magnetic materials can be rather sharply divided into two groups, the magnetically soft (easy to magnetize and demagnetize) and the magnetically hard (hard to magnetize and demagnetize). The distinguishing characteristic of the first group is high permeability and low coercivity. Magnetically hard materials, on the other hand, are made into permanent magnets; here a high coercivity is a primary requirement because a permanent magnet, once magnetized, must be able to resist the demagnetizing action of stray fields, including its own. A good soft magnetic material exhibits minimal hysteresis with low coercivity, high magnetization and the largest possible permeability. Hard magnetic materials are characterized by a field dependence of the magnetization showing a broad hysteresis loop and a concomitant high coercivity. The remanence Br determines the flux density that remains after removal of the magnetizing field and hence is a measure of the strength of the magnet, whereas the coercivity Hc is a measure of the resistance of the magnet against demagnetizing fields. Soft materials may be used for static or AC applications. The main static and lowfrequency AC applications are flux guidance and concentration in magnetic circuits, including cores for transformers and inductors operating at mains frequency (50 or 60 Hz). M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 14 Permanent magnets deliver magnetic flux into a region of space known as the air gap, with no expenditure of energy. Hard ferrite and rare-earth magnets are ideally suited to generate flux densities comparable in magnitude to their spontaneous magnetization Ms. Applications are classified by the nature of the flux distribution, which may be static or time-dependent, as well as spatially uniform or nonuniform. Applications are also discussed in terms of the physical effect exploited (force, torque, induced emf, Zeeman splitting, magnetoresistance). The most important uses of permanent magnets are in electric motors, generators and actuators. Their power ranges from microwatts for wristwatch motors to hundreds of kilowatts for industrial drives. Annual production for some consumer applications runs to tens or even hundreds of millions of motors. Ferrite These ferrites have the spinel structure and are sometimes called ferrospinels, because their crystal structure is closely related to that of the mineral spinel, MgO.Al2O3. The structure is complex, in that there are eight formula units per unit cell. The large M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 15 oxygen ions (radius about 0.13 nm) are packed quite close together in a face-centered cubic arrangement, and the much smaller metal ions (radii from about 0.07 to 0.08 nm) occupy the spaces between them. These spaces are of two kinds. One is called a tetrahedral or A site, because it is located at the center of a tetrahedron whose corners are occupied by oxygen ions. The other is called an octahedral or B site, because the oxygen ions around it occupy the corners of an octahedron. The crystallographic environments of the A and B sites are therefore distinctly different. Crystal structure of ferrite Not all of the available sites are actually occupied by metal ions. Only one-eighth of the A sites and one-half of the B sites are occupied, as shown in Table. In the mineral spinel, the Mg2þ ions are in A sites and the Al3þ ions are in B sites. Some ferrites MO.Fe2O3 have exactly this structure, with M2þ in A sites and Fe3þ in B sites. This is called the normal spinel structure. Both zinc and cadmium ferrite have this structure and they are both nonmagnetic, i.e., paramagnetic. Many other ferrites, however, have the inverse spinel structure, in which the divalent ions are on B sites, and the trivalent ions are equally divided between A and B sites. The divalent and trivalent ions normally occupy the B sites in a random fashion, i.e., they are M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 16 disordered. Iron, cobalt, and nickel ferrites have the inverse structure, and they are all ferrimagnetic. SATURATION MAGNETIZATION We can calculate the saturation magnetization of a ferrite at 0K, knowing (a) the moment on each ion, (b) the distribution of the ions between A and B sites, and (c) the fact that the exchange interaction between A and B sites is negative. Actually, the AB, AA, and BB interactions all tend to be negative, but they cannot all be negative simultaneously. The AB interaction is usually the strongest, so that all the A moments are parallel to one another and antiparallel to the B moments. Ferrite These ferrites have the spinel structure and are sometimes called ferrospinels, because their crystal structure is closely related to that of the mineral spinel, MgO.Al2O3. The structure is complex, in that there are eight formula units per unit cell. The large oxygen ions (radius about 0.13 nm) are packed quite close together in a face-centered cubic arrangement, and the much smaller metal ions (radii from about 0.07 to 0.08 nm) occupy the spaces between them. These spaces are of two kinds. One is called a tetrahedral or A site, because it is located at the center of a tetrahedron whose corners are occupied by oxygen ions. The other is called an octahedral or B site, because the oxygen ions around it occupy the corners of an octahedron. The crystallographic environments of the A and B sites are therefore distinctly different. M Chaitanya Varma Dept. of Engineering Physics GITAM University Magnetic properties of materials 17 Crystal structure of ferrite Not all of the available sites are actually occupied by metal ions. Only one-eighth of the A sites and one-half of the B sites are occupied. In the normal spinel-structure ferrites, the divalent M2+ ions are all on A sites and the Fe3+ ions occupy octahedral B sites. Examples of such ferrites include ZnO·Fe2O3 and CdO·Fe2O3. The dominant interaction determining the magnetic ordering in ferrites is an antiferromagnetic interaction betweenAandB site cations; however, since Zn2+ and Cd2+ do not have magnetic moments to mediate magnetic interactions, the net Fe3+–Fe3+ interaction in these materials is very weak, and they are paramagnetic. In the inverse spinels, the Fe3+ ions are divided equally between A and B sites, with the divalent ions (previously on the A sites) displaced to the remaining B sites; examples include Fe-, Co-, and Ni ferrite, all of which are ferrimagnetic. Again the dominant interaction determining the magnetic ordering is the antiferromagnetic A–B interaction. As a result, the spin moments of all the Fe3+ ions on the octahedral sites are aligned parallel to one another, but directed oppositely to the spin moments of the Fe3+ ions occupying the tetrahedral positions. Therefore the magnetic moments of all Fe3+ ions cancel and make no net contribution to the magnetization of the solid. However, all the divalent ions have their moments aligned parallel to one another, and it is this total moment which is responsible for the net magnetization. Thus the saturation magnetization of a ferrimagnetic solid can be calculated from the product of the net spin magnetic moment of each divalent cation and the concentration of divalent cations. The cubic ferrites are magnetically soft, and so are easily magnetized and demagnetized. Combined with their high permeability and saturation magnetization, and low electrical conductivity, this makes them particularly appropriate as cores for induction coils operating at high frequencies. Their high permeability concentrates flux density inside the coil and enhances the inductance, and their high electrical resistivity reduces the formation of undesirable eddy currents. The garnets The garnets have the chemical formula 3M2O3·5Fe2O3, where M is yttrium or one of the smaller rare earths towards the right-hand side of the lanthanide series (Gd to Lu). All cations in garnets are trivalent, in contrast to the ferrites, which contain some divalent and some trivalent cations. Since all of the cations have the same valence, the likelihood of electrons hopping through the material, say from 2+ ions (leaving them 3+) to 3+ ions (making them 2+) is very low, and so the resistivity of garnets is extremely high. Therefore they are used in very high frequency (microwave) applications, where even the ferrites would be too conductive. Ferromagnetic garnets are assigned to cubic structure, every cell contains molecules, and the ion distribution structure can be represented by writing the garnet formula as , {}, [], () are represented for M Chaitanya Varma Dept. of Engineering Physics GITAM University 24c (dodecahedral), 16a (octahedral) and 24d (tetrahedral) respectively. Yttrium iron Magnetic properties of materials 18 garnet (YIG) is a kind of synthetic garnet, with chemical composition Y3Fe2(FeO4)3, or Y3Fe5O12. It is a ferrimagnetic material with Curie temperature 550 K. In YIG, the five iron(III) ions occupy two octahedral and three tetrahedral sites, with the yttrium(III) ions coordinated by eight oxygen ions in an irregular cube. The yttrium does not have a magnetic moment (since it does not have any f electrons), so the net moment is due entirely to the unequal distribution of Fe3+ ions in up- and down-spin sites. When rare-earth ions with unfilled 4f n shell are substituted into the c sublattice to form solid solutions of RE3Fe5O12 and Y3Fe5O12, the magnetic properties of the iron garnets take on a remarkably different character. The net magnetization will be given by The antiferromagnetic superexchange interaction results in three up-spin electrons for every two down-spin electrons, and a net magnetic moment of 5µB per formula unit. Since the formula unit is very large, this leads to a small magnetization per unit volume. M Chaitanya Varma Dept. of Engineering Physics GITAM University