MAGNETIC MOMENT DEFINITIONS OF FIELD QUANTITIES AND MAGNETIZATION Discussion of the magnetic properties of materials begins by defining macroscopic field quantities.1 The MICHAEL E. MCHENRY AND DAVID E. LAUGHLIN two fundamental quantities are the magnetic induction, Carnegie Mellon University, Pittsburgh, PA, USA B~, and the magnetic field, H~, both of which are axial vector quantities. In many cases the induction and the field will be collinear (parallel) so that we can treat them INTRODUCTION as scalar quantities, B and H.2 In a vacuum, the magnetic induction, B~, is related to Magnetism has engaged explorers and scientists for over the magnetic field, H~: two millennia. The ancient Greeks and Turks noted the attraction between magnetite (lodestone) and iron ~ ¼ ~; ~ ¼ ~ ð Þ B m0H B H 1 (Cullity and Graham, 1978). Explorers used lodestone’s magnetization to construct compass needles used to where the permeability of the vacuum, m0,is point to the direction of the earth’s magnetic North 4p 10 7 H=m in SI (mksa) units. This quantity is taken Pole. This singularly important device aided in naviga- as 1 in cgs units. In cgs units, the induction and field tion and the exploration of the planet. Today, the have the same values. In SI (mksa) units we assign a miniaturization of magnetic sensors has propelled the permeability to the vacuum, so the two are proportional. magnetic recording industry as well as contributed to In a magnetic material the magnetic induction can far-reaching applications such as planetary exploration be enhanced or reduced by the material’s magnetization, (Diaz-Michelena, 2009). M~ (defined as net dipole moment per unit volume), Michael Faraday’s (1791–1867) discovery of electro- so that magnetic induction provides the principle for understand- ing the operation of electric generators, transformers, and B~ ¼ m ðH~ þ M~Þ; B~ ¼ H~ þ 4pM~ ð2Þ a variety of other magnetic devices. The quantum 0 mechanical description of the origin of atomic magnetic where the magnetization, M~, is expressed in linear dipole moments provides the basis for understanding a response theory as variety of disparate phenomena including the magnetic response of materials. ~ ¼ ~ ð Þ The intrinsic magnetic properties of materials refer to: M wmH 3

and the constant of proportionality is called the magnetic i. the origin, magnitude, and directions of atomic susceptibility, w . The magnetic susceptibility that magnetic dipole moments; m relates two vector quantities is a polar second-rank ii. the nature of the interaction between atomic tensor. For most discussions (whenever B and H are magnetic dipole moments on neighboring atoms collinear or when interested in the magnetization com- in the solid; ponent in the field direction) we treat the susceptibility iii. whether these result in collective magnetic phe- as a scalar. nomena; and We continue our discussion considering scalar induc- iv. the resulting temperature dependence of the tion, field, and magnetization. We can further express ¼ magnetization. B mrH as

¼ ð þ Þ ; ¼ð þ Þ ð Þ For materials exhibiting collective magnetic response, B m0 1 wm H B 1 4pwm H 4 otherintrinsicmagneticpropertiesincludethestrengthof the coupling of magnetic moments to one another and and we see that the relative permeability, mr, can be to crystallographic directions, magnetoelastic coupling expressed as coefficients, and the temperature(s) at which magnetic ¼ ð þ Þ; ¼ þ ð Þ phase transformations occur. The intrinsic magnetic mr m0 1 wm mr 1 4pwm 5 properties, of species at surfaces and interfaces, are known to be distinctly different from those of the bulk mr thus represents an enhancement factor of the flux in many cases. This article reviews the theory of intrinsic density in a magnetic material due to the magnetization < magnetic properties of dipole moment and magnetization that is an intrinsic material property. If we have wm 0, > as well as theory and examples of collective magnetic we speak of diamagnetic response, and for wm 0 (and response. The National Institute of Standards and Tech- no collective magnetism) we speak of paramagnetic nology (NIST) keeps an up-to-date compilation of units. ThesecanbefoundinR.B.GoldfarbandF.R.Fickett,U.S. 1 Selected formulas are introduced in SI (mksa) units followed DepartmentofCommerce,NationalBureauofStandards, by cgs units. Boulder, Colorado 80303, March 1985 NBS Special Pub- 2 For many discussions it is sufficient to treat field quantities as lication696forsale bythe SuperintendentofDocuments, scalars; when this is not the case, vector symbols will be explic- U.S.GovernmentPrintingOffice,Washington,DC20402. itly used.

Characterization of Materials, edited by Elton N. Kaufmann. Copyright Ó 2012 John Wiley & Sons, Inc. 2 MAGNETIC MOMENT AND MAGNETIZATION response.Asuperconductor is a material that acts as a iii. In quantum mechanics, every electron has a 1 dipole moment associated with its spinning perfect diamagnet so that w ¼1orw ¼ . m m 4p charge density (spin) and its orbit about the nucleus (orbit). Angular momentum (whether spin or orbital) is quantized in units of Planck’s MAGNETIC DIPOLE MOMENTS—DEFINITIONS h constant divided by 2p, that is, h ¼ ¼ 1:05 AND ATOMIC ORIGINS 2p 10 34 Js¼ 1:05 10 27 erg s. We define the fun- A magnetic dipole moment has its origin in circulating damental unit of magnetic dipole moment, the charge (Fig. 1). This concept is made more complicated Bohr magneton,as by the need to treat circulating charges of electrons within the framework of quantum mechanics. ¼ e ; ¼ ¼ e ð Þ mB h m mB h 8 Concepts relating circulating charge, angular 2m 2mc momentum, and dipole moments are: The Bohr magneton is calculated to have the following value: i. A dipole moment for a circulating charge is ¼ : 24 2 ð = Þ; defined formally as mB 9 27 10 Am J T ð m ¼ m ¼ 9:27 10 21 erg=G ð9Þ ~ ¼ ~ ¼ ~ ~ ð Þ B m IAu~r J~ r JdV 6 V iv. An atomic dipole moment is calculated by sum- where ~r is the position vector of the charged particle ming all of the electron dipole moments for a given about the origin for the rotation. ~r is the current atomic species. Quantum mechanical rules for density of the orbiting charge. I is the current due this summing called Hund’s rules are discussed to the circulating charge, A ¼ pr 2 is the area swept out below. ~ by the circulating charge, and V is the volume. u~r J~ is v. For a collection of identical atoms the magneti- a unit vector normal to the area, A. zation, M,is ii. We relate the magnetic dipole moment to the ¼ ð Þ M Namatom 10 angular momentum. Let P~ be a general angular momentum vector. In classical mechanics, the where Na is the number of dipole moments per unit angular momentum vector, expressed as volume and matom is the atomic dipole moment. P~ ¼ ~ ~ ¼ 2 r mv, has magnitude mvr mo0r , where vi. The potential energy of a dipole moment in the o0 is an angular frequency, and is directed nor- presence of a field is mal to the current loop (parallel to the dipole moment). The fundamental relationship between ~ Ep ¼ ~m B ¼ mB cosy ð11Þ magnetic dipole moment and the angular momentum vector is where y is the angle between the dipole moment and B~. This implies that magnetization (or other field e e m ¼ g P; m ¼ g P ð7Þ quantity) multiplied by another field has units of 2m 2mc energy per unit volume. It is important to begin to think of energy densities (energy per unit volume) for where g is called the Lande g-factor. For an orbiting magnetic systems. In quantum mechanical systems, electron the constant g ¼ 1. The dipole moment the component of the dipole moment vector projected associated with spin angular momentum has g ¼ 2. along the field direction is quantized and only partic- ular values of the angle y are allowed.

μ Closed Shell Diamagnetism: Langevin Theory of Diamagnetism The diamagnetic susceptibility of closed shells is dis- cussed in Box 1.

r Diamagnetism is the atomic magnetic response due to closed shell orbits of core electrons.

q v This is to be contrasted with the perfect diamagnetism of a superconductor. Magnetic flux is excluded from the Figure 1. Geometry of a charged particle orbiting at a distance r, interior of a superconductor and it is a consequence of with a linear velocity, v. The particle orbit sweeps out an area, A, 1 and gives rise to a dipole moment, ~m. B ¼ 0, which requires that w ¼1orw ¼ . m m 4p MAGNETIC MOMENT AND MAGNETIZATION 3

MAGNETIC SUSCEPTIBILITY OF A SIMPLE DIAMAGNET quantizes the projected orbital angular momentum. This has the further consequence that the orbital angular ~ Consider an atom with a closed electronic shell. For momentum vector, L , can precess about the field axis, filled shells, electrons orbit the nuclei, but the net z, only at a set of discrete angles, y: current associated with their motion is zero because ¼ ; ¼ þ ; of cancellation of their summed orbital angular Lz ml h ml l 0 l ~ ~ momentum, L (i.e., L ¼ 0). However, even for a closed L m y ¼ arccos z ¼ arccos l ð13Þ shell, in the presence of an applied field a net current L l is induced. By Lenz’s law this current results in a dipole moment that opposes the applied field. The Open Shell Atomic Dipole Moments—Hund’s Rules Larmor frequency, oL, is the characteristic frequency of this circulating induced current and has a value We now examine systems where the electrons responsi- ble for the dipoles exist in localized states assigned to a eH eH particular atom. In systems (typically ionic) where the o ¼ ; o ¼ ð12aÞ L m L mc atomic orbitals responsible for the magnetic dipole moments are localized on specific atoms in a solid, If we wish to construct an atomic dipole moment, we discrete magnetic states can be calculated using quan- must consider the moment due to Z electrons that tum mechanical rules called Hund’s rules (Hund, 1927). orbit the nucleus. Assuming that all Z electrons orbit This discussion is applicable in understanding the mag- the nucleus with the same angular frequency, oL,we netic dipoles in ionic systems such as oxides and salts of express the current, I, as follows: transition metals. A general angular momentum vector, P~, can have ~ ¼ dq ¼ ZeoL ð Þ contributions from orbital angular momentum, L , and I 12b ~ dt 2p spin angular momentum, S. Both moments are quan- tized in units of h . The fundamental atomic unit of dipole The induced moment is calculated as the current moment is the Bohr magneton. multiplied by the area and the orbital atomic mag- In addition to the orbital moment, there is an addi- netic dipole moment is then tional contribution to the magnetic moment of an elec- tron, due to spin. Spin is a purely quantum mechanical Zeo hr 2i m ¼ L ð12cÞ property though we can view it semiclassically consid- atom 2 3 ering an electron as a particle with a rotating surface current. The classical problem yields a spin moment h 2i where the minus sign reflects Lenz’s law and r is similar to that which is derived in the quantum mechan- 2 the average value of r for the orbit. The average value ical description (Cullity and Graham, 1978). of the square of the orbital radius is Determining atomic dipole moments requires sum- ming spin and orbital angular momenta over all elec- hr 2i¼hx 2iþhy2iþhz2ið12dÞ trons on an atom. The summed orbital and spin angular momenta is zero for closed shells. The closed shells then and for an isotropic environment: only contribute to the small diamagnetic moment of the previous section.3 In open shells we need to consider hr 2i rules for summing the spin and orbital angular momenta hx 2i¼hy2i¼hz2i¼ ð12eÞ 3 for all electrons in the open shell. Hund’s rules allow us to describe the ground-state multiplet including the ml This may be associated with the negative diamagnetic and ms eigenstates and allow us to calculate the com- susceptibility (for N atoms/volume): ponents of the orbital, L, spin, S, and total angular, J, momenta. The magnitudes of orbital and spin angular Nm NZe2 Nm NZe2 momenta are constructed by summing angular momen- w ¼ atom ¼ ; w ¼ atom ¼ m H 6m m H 6mc tum over a multielectron shell: ð12fÞ Xn Xn ¼ ð Þ ; ¼ ð Þ ð Þ L ml i h S 2 ms i h 14 which describes well the diamagnetism of core i¼1 i¼1 electrons and of closed shell systems. Typically molar The projection of the total angular momentum vector, w ¼ ~ diamagnetic susceptibilities are on the order of m J~ ¼ L~þ S, along the applied field direction is also 6 5 3= ¼ 12 11 3= 10 to 10 cm mol 10 to 10 m mol. subject to quantization conditions. Hund’s rules require that J (J ¼ L þ S)isjLSj for less than half-filled shells and jL þ Sj for greater than half-filled shells. To The magnitude, L ¼jL~j, of the orbital angular determine the occupation of eigenstates of S, L, and ~ momentum vector, L , and its projection, Lz, onto an axis 3 of quantization, z, by the application of a field, Bz,is For open shells this diamagnetic contribution is small enough quantized in units of h . For Lz the quantum number ml to ignore. 4 MAGNETIC MOMENT AND MAGNETIZATION

1=2 J we use Hund’s rules that state that for a closed peff ¼ gðJ; L; SÞ½JðJ þ 1Þ ð17Þ electronic shell J ¼ L ¼ S ¼ 0. For an open shell multi- electron atom: As an example of the calculation of a Hund’s rule þ þ ground state, we consider the Ho3 multiplet. Ho3 has 10 i. We fill ml states (which are (2l þ 1)-fold degener- a4f open shell configuration. According to Hund’s ate) in such a way as to first maximize total spin. rules we occupy the 7 ml states with spin-up electrons ¼ ; ; ii. We fill ml states first in such a way as to first followed by ml 3 2 1 to account for all 10 outer maximize total spin. shell f electrons (note that the 2 outer shell s and 1 outer shell d electron of the atom are those that are lost in 3 þ 2 þ ¼ 7 3 ¼ We consider the ions of transition metal series, TM , ionization). For Ho , we see that S 2 2 2 and ¼j j¼ that is, ions that have given up 2s electrons to yield a 3dn L 3 2 1 6 and since the f shell is more than ¼ þ ¼ 3 þ outer shell configuration in Figure 2a. The ground-state half-filled, J L S 8. The term symbol for Ho 3 þ 5 J, L, and S quantum numbers for rare earth, RE , ions is therefore I8. The Lande g-factor can be calculated are shown in Figure 2b. to be Defining L, S, and J for a given element specifies the 3 1 2ð2 þ 1Þ6ð6 þ 1Þ ground-state multiplet. This multiplet is written more gðJ; L; SÞ¼ þ compactly in the spectroscopic term symbol as 2 2 8ð8 þ 1Þ þ 3 1 6 42 2S 1L ð15Þ ¼ þ ¼ 1:25 J 2 2 72 where L is the alphabetic symbol for orbital angular and the effective moment in units of m is momentum (L ¼ 0 ¼ S, L ¼ 1 ¼ P, L ¼ 2 ¼ D, B pffiffiffi L ¼ 3 ¼ F, etc.) and 2S þ 1 and J are the numerical = = þ ¼ ð ; ; Þ½ ð þ Þ1 2 ¼ : ½ 1 2 ¼ : values of the same. For example, Cr3 with L ¼ 3, peff g J L S J J 1 mB 1 25 72 mB 7 5 2 mB ¼ 3 ¼ 3 S 2, and J 2 would be assigned the term symbol 4 L3=2. We can further relate the permanent local atomic moment vector with the total angular momentum vector, Dipole Moments in Systems with Quenched Orbital Angular ~ J ,as Momentum In many systems of interest the orbital angular momen- ~ ¼ ~ ¼ ð ; ; Þ ~ ð Þ m ghJ g J L S mBJ 16a tum is said to be quenched. The quenched orbital angular momentum refers to the fact that the orbital angular ¼ where g is called the gyromagnetic factor and g moment vector is strongly tied to a crystalline easy ð ; ; Þ g J L S is called the Lande g-factor and is given by magnetization direction (EMD). For this reason to a good approximation we can take L ¼ 0 and J ¼ S. In this case 3 1 SðS þ 1ÞLðL þ 1Þ ¼ ¼ ½ ð þ Þ1=2 gðJ; L; SÞ¼ þ ð16bÞ g 2 and peff 2 S S 1 . This is true for many 2 2 JðJ þ 1Þ transition metal systems and also for simple oxides of the transition metals. The relationship between mag- Table 1 tabulates the ground-state multiplets for tran- netic dipole moment, m, and angular momentum vector sition metal and rare earth cation species that are prev- is given by m ¼ g e P, where P can refer to orbital, L~,or ~ 2m alent in many oxides and other interesting ionic systems. spin, S, angular momentum and g is the gyromagnetic The Lande g-factor accounts for precession of angular factor. In ferrites the d shells of transition metal cations momentum and quantum mechanical rules for projec- are of interest, and we have quenched orbital angular tion onto the field axis (Fig. 3) (Russell and Saunders, momentum (i.e., L~ ¼ 0) in the crystal. The spin angular 1925). For identical ions with angular momentum J we momentum for a single electron is quantized by the spin define an effective magnetic moment in units of m : ¼1 ¼h B quantum number, ms 2,tobemsh 2. For spin

5 10 S S L L 4 J 8 J

3 6

2 4

12 S, L, and J angular momentum 00S, L, and J angular momentum 0 2 4 6 8 10 0 2 4 6 8 10 12 14 Figure 2. Ground-state J, L, and S quantum # d electrons # f electrons numbers for the (a) transition metal, TM2 þ , (a) (b) and (b) rare earth, RE3 þ , ions. MAGNETIC MOMENT AND MAGNETIZATION 5

Table 1. Ground-State Multiplets of Common TM and RE Ions (Van Vleck, 1932)

neff Ion SL J = = Term g½JðJ þ 1Þ1 2 Observed g½SðS þ 1Þ1 2 d-Shell electrons 3 þ 4 þ 1 3 2 1Ti,V 2 2 2 D3=2 1.55 1.70 1.73 3 þ 3 2V132 F2 1.63 2.61 2.83 2 þ 3 þ 3 3 4 3V,Cr 2 3 2 F3=2 0.77 3.85 3.87 2 þ 3 þ 5 4Cr,Mn 220 D0 0 4.82 4.90 3 þ 3 þ 3 5 5 5Mn,Mn 2 0 2 S5=2 5.92 5.82 5.92 2 þ 5 6Fe224 D4 6.7 5.36 4.90 2 þ 3 9 4 7Co2 3 2 F9=2 6.63 4.90 3.87 2 þ 3 8Ni134 F4 5.59 3.12 2.83 2 þ 1 5 2 9Cu2 2 2 D5=2 3.55 1.83 1.73 þ 2 þ 1 10 Cu ,Zn 000 S0 000 f-Shell electrons 3 þ 1 5 2 1Ce2 3 2 F5=2 2.54 2.51 3 þ 3 2Pr154 H4 3.58 3.56 3 þ 3 9 4 3Nd2 6 2 I9=2 3.62 3.3 3 þ 5 4Pm264 I4 2.68 — 3 þ 5 5 6 5Sm2 5 2 H5=2 0.85 (1.6) 1.74 3 þ 7 6Eu330 F0 0 (3.4) 3.4 3 þ 3 þ 5 5 8 7Gd,Eu 2 0 2 S7=2 7.94 7.98 3 þ 7 8Tb336 F6 9.72 9.77 3 þ 5 15 6 9Dy2 5 2 H15=2 10.63 10.63 3 þ 5 10 Ho 268 I8 10.60 10.4 3 þ 3 15 4 11 Er 2 6 2 I15=2 9.59 9.5 3 þ 3 12 Tm 156 H6 7.57 7.61 3 þ 1 7 2 13 Yb 2 3 2 F7=2 4.53 4.5 3 þ 3 þ 1 14 Lu ,Yb 000 S0 0—

¼ ¼ only, the gyromagnetic factor is g 2, and the single Thus, m mB with electron dipole moment is ¼ : 24 2 ð = Þ : 21 = mB 9 27 10 Am J T or 9 27 10 erg G e h he e h he m ¼g ¼ ; g ¼ ð18aÞ ð18cÞ 2m 2 2m 2mc 2 2mc

e h e h For a multielectron atom, the total spin angular m ¼g or g ð18bÞ momentum is 2m 2 2mc 2 Xn ¼ ð Þ S ms i h z ¼ J i 1 with the sum over electrons in the outer shell. Hund’s rules determine the occupation of eigenstates of S. L The first states that for an open shell multielectron atom S we fill the (2l þ 1)-fold degenerate (for d electrons l ¼ 2, ð2l þ 1Þ¼5) orbital angular momentum states so as to maximize total spin. To do so, we must fill each of the five d states with a positive (spin-up) spin before returning to fill the negative (spin-down) spin. The total spin angu- μ μ L S lar momentum for 3d transition metal ions is summa- rized in Table 2. The dipole moments are all integral μ = μ + μ S L numbers of Bohr magnetons allowing for simplification of the analysis of the magnetization of ferrites that typ- μ J ically have quenched angular momentum.

Figure 3. Vector (analogous to planetary orbit) model for the Atomic Dipole Moments—Energy Band Theory addition of angular momentum with spin angular momentum precessing around the orbital moment that precesses about In systems with significant atomic overlap of the electron the total angular momentum vector. z is the field axis (axis of wave functions for orbitals responsible for magnetic quantization). dipole moments, the energy and angular momentum 6 MAGNETIC MOMENT AND MAGNETIZATION

Table 2. Transition Metal Ion Spin and Dipole Moments We integrate each density of states separately to yield a (L ¼ 0) different number of electrons per unit volume in spin-up and spin-down bands, respectively: ð Þ d Electrons Cations S m mB 3 þ 4 þ ð ð 1 eF eF 1Ti,V 2 1 N" N# 3 þ ¼ ¼ ð Þ ; ¼ ¼ ð Þ ð Þ 2V 12 n" g" e de n# g# e de 20 þ þ V V 3V2 ,Cr3 3 3 0 0 þ þ 2 4Cr2 ,Mn3 24 þ þ 5Mn2 ,Fe3 5 5 The magnetization, net dipole moment per unit volume, þ 2 6Fe2 24is then very simply 2 þ 3 7Co2 3 2 þ 8Ni12 M ¼ðn"n#Þm ð21Þ 2 þ 1 B 9Cu2 1 þ 2 þ 10 Cu ,Zn 00To calculate the thermodynamic properties of transition metals, one can calculate electronic structure and total energies using state-of-the-art local density functional states for these electrons are no longer discrete. Instead theory. The total energy of a crystal can be calculated energy levels4 form a continuum of states over a range of self-consistently and a potential function determined energies called an energy band. The distribution func- from the variation of the total energy with interatomic tion for these energy levels is called the density of states. spacing. From the potential curve the equilibrium lattice The density of states (per unit volume), gðeÞ, is formally spacing, the bulk modulus, cohesive energy, compress- ð Þ defined such that the quantity gðeÞde represents the ibility, etc., can be determined. g e is generated at the number of electronic states (per unit volume) in the equilibrium separation describing the ground-state range of energies from e to e þ de: electronic structure. Spin-polarized calculations can be performed to determine magnetic properties. With 1 dN increasing computational power and more sophisticated gðeÞde ¼ e de ð19Þ V de algorithms, it is possible to calculate these quantities accurately. Nevertheless, it is useful to have approxi- Note that this definition is not specific to the free mate analytic models for describing properties such as electron model. this or the Friedel model. Figure 4a shows the density of states for free electrons 1=2 with its characteristic e energy dependence. Many Pauli paramagnetism is a weak magnetism that is ð Þ other forms (shapes) for g e are possible given different associated with the conduction electrons in a solid. solutions to Schrodinger’s equation with different poten- tials. The quantity gðeÞde is viewed as an electronic state distribution function. Pauli paramagnetism does not involve permanent Implicit in the free electron theory is the ignoring of local dipole moments that gave rise to the Curie law. potential energy and therefore its influence on angular Instead it involves a magnetic moment that is caused by momentum. With more realistic potentials, we can cal- the application of a field. We now describe the electronic culate densities of states whose shape is influenced by density of states in a field. the orbital angular momentum. To first approximation The free electron density of states specifically ac- (and a relatively good approximation for transition counts for a spin degeneracy of two. If we instead defined metals), the orbital angular momentum can be consid- a spin-up and spin-down density of states with identical ered to be quenched and we concern ourselves only degenerate states as illustrated in Figure 4a, as in the with spin angular momentum. The formal definition Zeeman effect, the spin degeneracy is lifted in a field = is general while an e1 2 dependence results from the and for a free electron metal we assume the spin-up assumptions of the free electron model. states to be rigidly shifted by an amount, mBH,where In magnetic systems we are often interested in the H is the applied field and mB is the spin dipole moment. influence of an applied or internal (exchange) field on Similarly the spin-down states are rigidly shifted by an þ the distribution of energy states. Figure 4b shows the energy equal to mBH. Now the Fermi energy of elec- density of states for free electrons where the spin degen- trons in the spin-up and spin-down bands must eracy is broken by a Zeeman energy due to an applied or remain the same so we remember that internal (exchange) field. We divide the density of states, ð ð e e gðeÞde, by two, placing half the electrons in spin-up states N" F N# F " ¼ ¼ "ðeÞ e; # ¼ ¼ #ðeÞ e spin-down states n g d n g d and the other half in . Spin-up electrons V 0 V 0 have potential energy lowered by mBH, where H is an applied, Ha, or internal exchange, Hex, field. Spin-down and n" þ n# ¼ n is the electron density. The magnetiza- electrons have their potential energy increased by m H. ¼ð Þ B tion, M,isM n" n# mB. We determine the T-dependent magnetic susceptibil- 4 We use e to denote the energy per electron and not the total ity, wðTÞ, by performing a Taylor series expansion of the energy which would be integrated over all electrons. density of states in the presence of a perturbing field: MAGNETIC MOMENT AND MAGNETIZATION 7

Spin down

ε 2μ B ( ) B g ) ε

( 0 g ε ε F ε F ε ( ) 0 2 4 6 8 10 g a b ε (eV) Spin up Figure 4. ( ) Free electron density of states and ( ) free electron density of states where the spin degeneracy is (a)(b) broken by a Zeeman energy due to an applied or internal (exchange) field.

@g bands can be approximated by a constant density of g"ðe; HÞ¼gðe þ m HÞ¼gðeÞþm H ð22Þ B B @e states over a bandwidth, W. This is equivalent to smooth- ing the more complicated density of states and para- @ ð ; Þ¼ ð Þ¼ ð Þ g ð Þ meterizing it in terms of the constant gðeÞ and the band- g# e H g e mBH g e mBH 23 @e width, W. This approximation will serve us quite well in our approximate description of the electronic structure and therefore of transition metals. Friedel’s model considers contribu- ð tions to the density of states of transitions metals due to eF @ ¼ð Þ ¼ mB ð ð Þþ gÞ the constant d electron density of states and the free M n" n# mB g e mBH @ de 2 0 e electron s states. The d density of states (Fig. 4a) is ð ð  eF @g eF @g centered at ed with a bandwidth, Wd, that is: ð ðeÞm Þ e¼m2 e ¼ m2 ðe Þ g BH @ d BH @ d BHg F 0 e 0 e 10 Wd Wd gðeÞde ¼ ; ed < e < ed þ ; and Wd 2 2 gðeÞde ¼ 0 otherwise ð26Þ ¼ 2 ð ÞðÞ w mBg eF 24 and we see that integrating gðeÞde over the entire range which is essentially invariant with temperature. So from e Wd to e þ Wd accounts for all 10 of the d elec- unlike local moment paramagnetism that obeys the d 2 d 2 trons. The s electron DOS begins at e ¼ 0 and ends at the Curie law, with a strong 1 dependence, free electron T Fermi level, e ¼ e , and obeys the functional dependence (Pauli) paramagnetism is nearly T independent. F g ðeÞ¼Ce1=2. A band theory of ferromagnetism can also be s The Fermi level is determined by superimposing the expressed in free electron theory. Building on the theory two densities of states and filling to count the total of Pauli paramagnetism, band theory considers number of electrons. Note that the atomic d and s elec- exchange interactions between spin-up and spin-down tron count usually is not conserved (but of course the electrons whereby electrons with parallel spins have a total must be). In the solid state we can use N and N to lower energy by V than antiparallel spins (i.e., V ¼ 0). d s ex designate the integrated number of d and s electrons, With B ¼ 0, the total energy is unstable with respect to respectively, such that exchange splitting when

ð ðð = Þþð = Þ eF Wd 2 NdWd 10 10 4eF ¼ ð Þ ¼ ; V > ð Þ Nd gd e de de ex 25 = = W 3N Wd 2 Wd 2 d ð This Stoner criterion determines when a system will have eF Ns ¼ gsðeÞde ð27Þ a lower energy with a spontaneous magnetization 0 (ferromagnetic) than without (paramagnetic). This free electron ferromagnetism is called itinerant ferromagne- A ferromagnetic Friedel model considers spin-up and tism. Topologically close-packed alloys can have elec- spin-down d bands shifted rigidly by an amount þ D tronic structures with sharp peaks in the density of with respect to the original nonmagnetic configuration states near the Fermi level. These peaks allow the mate- (Fig. 5b). Since the magnetism observed in transition rials to satisfy the Stoner criterion. This explains itiner- metals is predominantly determined by more localized d ant ferromagnetism observed in the Laves phase, ZrZn2. electrons, the Friedel model will give more illustrative The Friedel model for transition metal alloys also results than the free electron model. Consider the ide- describes the d electron density of states in transition alized density of states shown above in which a transi- metals (Harrison, 1989). The d states are generally more tion metal d band is modeled with a constant density localized and atomic-like, especially for the late transi- of states and the s band with a free electron density of tion metals with more filling of the d shells. The Friedel states. The atomic configuration for these atoms is given n2 2 model assumes that the density of states for d electron by d s and Nd ¼ n2. 8 MAGNETIC MOMENT AND MAGNETIZATION

g(ε ) g (ε )

d band 10/W 5/W s band ε ε F F 0 ε ε ε d – W/2 d F ε d + W/2

5/W g (ε ) Figure 5. (a) Transition metal d and s band densities of b (a)(b) states for the Friedel model and ( ) for a spin-polarized Friedel model (d band only).

For solids, the s electrons are counted by integrating the basis for explaining the Slater–Pauling curve (Fig. 6) the DOS and Nd ¼ nNs. A rough estimate approximates (Slater, 1937; Pauling, 1938). Figure 6b shows a that Ns 0:6 so the d count is usually higher. The more sophisticated band structure determination of average magnetic moment as a function of composition the Slater–Pauling curve for FeCo alloys. Figure 6b predicted by the Friedel model gives Slater–Pauling curve shows the band theory prediction of the average (spin- and alloy dipole moment data. Weak solutes, with a only) dipole moment in FeCo to be in good quantitative valence difference DZ 1, are explained by a rigid band agreement with the experimentally derived model.Avirtual bound state (VBS) model is employed Slater–Pauling curve. when the solute perturbing potential is strong, DZ 2. For transition metal impurities that are strongly per- In dilute alloys, solute atoms that are only weakly turbing, Friedel (1958) has proposed a VBS model to perturbing (i.e., having a valency difference DZ 1), a explain departure from the simple relationship for the rigid band model can be employed to explain alloying compositional dependence of dipole moment above. In effects on magnetic moment. Rigid band theory assumes this case the change in average magnetic moment and that d bands do not change much in alloys but just get suppression are predicted to be filled or emptied depending on composition (Fig. 5). In this model, the magnetic moment of the solvent matrix dm m ¼ m ðDZ þ 10ÞCm ; ¼ðDZ þ 10Þm ð29Þ remains independent of concentration. At the site of a matrix B dC B solute atom the locally mobile minority-spin electrons are responsible for ensuring that the solute nuclear Figure 7a shows the binary FeCo phase diagram. charge is exactly screened; thus, a moment reduction Figure 7b and c shows spin-resolved densities of states D of ZmB is to be expected at the solute site. The average for Co and Fe atoms, in an equiatomic FeCo alloy, as a magnetic moment per solvent atom is the concentration- function of energy (where the Fermi energy, eF, is taken weighted average of that of the matrix and solute: as the zero of energy). The number of spin-up and spin- down electrons in each band is calculated by integrating ¼ D ð Þ m mmatrix ZCmB 28 these densities of states as are the atom-resolved magnetic dipole moments. Knowledge of atomic where C is the solute concentration and DZ is the valency volumes allows for the direct calculation of the alloy difference between solute and solvent atoms. This is magnetization. )

n 3

μ S.P curve 2.3 Fe–V ) ( 2.5 Fe–Cr

m Fe–Ni Fe–Co 2.2 Ni–Co 2 Ni–Cu Ni–Zn 2.1 Ni–V Ni–Cr Ni–Min 1.5 Co–Cr 2.0 Co–Mn

Pure metals Moment 1 1.9

0.5 1.8

0 1.7 Magentization peratom ( Magentization peratom 24 25 26 27 28 29 0 20 40 60 80 100 (Cr) (Mn) (Fe) (Co) (Ni) (Cu) Co (at.%) Number of electrons Data taken from Bozorth, PR 79,887 (1950) (a)(b)

Figure 6. (a) Slater–Pauling curve for Fe alloys and (b) spin-only Slater–Pauling curve for an ordered Fe–Co alloy as determined from LKKR band structure calculations (MacLaren et al., 1999). MAGNETIC MOMENT AND MAGNETIZATION 9

80 2000 70 Liquid Up (Co) 60 Down (Co) bcc 50 tcc Al 40 1500 A2 30 Tc 20 10 bcc A1 0 A2 T c Density of states (arbitrary units) –0.4 –0.3 –0.2 –0.1 0 0.1 1000 Energy (Ha) (b) B2–A2 T bcc 70 tcc 60 Up (Co) Temperature (K) Temperature Down (Co) B2 50 500 40 30 20 10 0 0 0.5 1.0 0 Fe Co –0.4 –0.3 –0.2 –0.1 0 0.1 (a) Energy (Ha) (c)

Figure 7. Binary FeCo phase diagram (a), and spin- and atom-resolved densities of states for an FeCo equiatomic alloy: (b) spin up and (c) spin down.

There has been considerable growth in the field of where V is the volume of the material.5 Magnetization interfacial, surface, and multilayer magnetism is an extrinsic material property that depends on the (MacLaren et al.,1990; McHenry et al.,1990). Funda- constituent atoms in a system, their respective dipole mental interest in these materials stems from moments, and how the dipole moments add together. predictions of enhanced local moments and phenomena Because dipole moments are vectors, even if they are associated with two-dimensional (2D) magnetism. collinear, they can add or subtract depending on Technological interest in these lies in their potential whether they are parallel or antiparallel. This can give importance in thin film recording, spin electronics, etc. rise to many interesting types of collective magnetism. For a magnetic monolayer as a result of reduced mag- netic coordination the magnetic d states become more paramagnet localized and atomic-like, often causing the moment to A is a material where permanent local atomic dipole moments are aligned randomly. grow since the exchange interaction greatly exceeds the bandwidth. This behavior is mimicked in buried (sandwiched) single layers or in multilayers of a mag- In the absence of an applied field, Ha, the magnetiza- netic species and a noninteracting host. Transition tion of a paramagnet is precisely zero since the sum of metal/noble metal systems are examples of such sys- randomly oriented vectors is zero. This emphasizes the tems (McHenry and MacLaren, 1991). importance of the word “net” in the definition of magne- tization. A permanent nonzero magnetization does not Magnetization and Dipolar Interactions necessarily follow from having permanent dipole We now turn to the definition of the magnetization, M. moments. It is only through a coupling mechanism that This is the single most important concept in the chapter! acts to align the dipoles in the absence of a field that a macroscopic magnetization is possible. Two dipole moments interact through dipolar inter- Magnetization, M, is the net dipole moment per unit actions that are described by an interaction force anal- volume. ogous to the Coulomb interaction between charges. If we consider two coplanar (xy plane) magnetic dipole moments (Fig. 8a), ~m and ~m , separated by a distance It is expressed as 1 2

S m M ¼ atoms atom ð30Þ 5 Magnetization can also be reported as specific magnetization, V which is net dipole moment per unit weight. 10 MAGNETIC MOMENT AND MAGNETIZATION

y

+++++ N

S N S θ1 θ2 x - - - - - Figure 8. (a) Geometry of two coplanar dipole r12 moments used to define dipolar interactions, (b) net surface dipole moments at the poles of a permanent (a) (b) (c) magnet with a single domain, and (c) flux return path for magnetic dipoles in a horseshoe magnet.

r12 where the first dipole moment is inclined at an angle interact with the magnetization vector. Fe is an example ~ y1 with respect to the position vector r 12 and the second of a soft magnetic material with a large magnetization. To at an angle y2, then the field components of field com- use Fe as a permanent magnet it is often shaped into a ~ ~ ~ ponents m1 due to m2 acting at m1 can be written as horseshoe magnet (Fig. 8c) where the return path for flux lines is spatially far from the material’s magnetization. ¼ m2 2 cosy2 ; ¼ 2 cosy2 ð Þ H1x 3 H1x m2 3 31a 4pm0 r12 r12 Magnetization in Superconductors The superconducting state is a state of a material in m y y ¼ 2 sin 2 ; ¼ sin 2 ð Þ which it has no resistance to the flow of an electric H1y 3 H1y m2 3 31b 4pm0 r12 r12 current. The discovery of superconductivity by Onnes (1911) followed his successful liquification of and a similar expression follows for the field components He in 1908 (Onnes and Clay, 1908). The interplay ~ ~ ~ m2 due to m1 acting at m2. In general, the interaction between transport and magnetic properties in super- potential energy between the two dipoles is given by conductors was further elucidated in the Meissner effect (Meissner and Ochsenfeld, 1933) (Fig. 9), where the ¼ 1 ðð~ ~ Þ 3 ð~ ~Þð~ ~ÞÞ ð Þð Þ phenomena of flux expulsion and perfect diamagnetism Up 3 m1 m2 2 m1 r m2 r SI 32a 4pm0r12 r12 of superconductors were demonstrated. London (1950) gave a quantum mechanical motivation for an electro- ¼ 1 ðð~ ~ Þ 3 ð~ ~Þð~ ~ÞÞ ð ÞðÞ dynamic model of the superconducting wave function. Up 3 m1 m2 2 m1 r m2 r cgs 32b r12 r12 This work coincided with the Ginzburg–Landau (GL) theory (Ginzburg and Landau, 1950). which for collinear dipoles pointing in a direction The phenomenological GL theory combined an ~ perpendicular to r 12 reduces to expansion of the free energy in terms of powers of the superconducting electron density (the modulus of the m m m m superconducting electron wave function) with tempera- E ¼ 1 2 ; E ¼ 1 2 ð32cÞ p 3 p 3 ture-dependent coefficients and incorporation of elec- 4pm0r12 r12 trodynamic terms in the energy. The original GL theory which is the familiar Coulomb’s law.6 described the intermediate state in type I superconduc- Like net charge in a conductor, free dipoles collect at tors arising from geometric (demagnetization) effects. surfaces to form the North and South Poles of a magnet (Fig. 8b). Magnetic flux lines travel from the N to the S pole of a permanent magnet. This causes a self-field, the demagnetization field, Hd, outside of the magnet. Hd can act to demagnetize a material for which the magnetiza- tion is not strongly tied to an easy magnetization direc- tion (EMD). This allows us to distinguish between a soft magnet and a hard magnet. Most hard magnets have a large magnetocrystalline anisotropy that acts to strongly fix the magnetization vector along particular crystal axes. As a result, they are ZFC FC FC difficult to demagnetize. A soft magnet can be used as permanent magnet, if its shape is engineered to limit the Superconductor Perfect conductor size or control the path of the demagnetization field to not (a) (b)

Figure 9. Flux densities for a superconductor (a), zero-field 6 Some authors (Cullity and Graham, 2009, e.g.) use p to denote cooled (ZFC) or field cooled (FC) to T < Tc, and for a perfect dipole moment. conductor (b), which is FC. MAGNETIC MOMENT AND MAGNETIZATION 11

The GL theory also allowed for Abrikosov’s (1957) the free energies for the same material in the super- description of the mixed state and type II superconduc- conducting (type I) and normal metal states, respec- tivity in hard superconductors. tively, and where the Meissner effect (B ¼ 0) has been A fundamental experimental manifestation of explicitly used in expressing gs. These require that in superconductivity is the Meissner effect (exclusion of zero field: flux from a superconducting material). This identifies the superconducting state as a true thermodynamic H 2ðTÞ Dg ¼ g g ¼ c ð35Þ state and distinguishes it from perfect conductivity. n s 8p This is illustrated in Figure 9 that compares the mag- netic flux distribution near a perfect conductor and a These free energy densities imply a latent heat for the superconductor, for conditions where the materials are superconducting transition: cooled in a field (FC) and for cooling in zero field with  subsequent application of a field (ZFC). TH @H L ¼ c c ð36Þ After cooling (ZFC) the superconductor and perfect 4p @T Tc conductor have the same response. Both the perfect conductor and superconductor exclude magnetic flux which is less than 0. Further, the superconducting lines in the ZFC case because of diamagnetic screening transition is second order with a specific heat currents that oppose flux changes in accordance with difference: Lenz’s law. The first case (FC) distinguishes a super- conductor from a perfect conductor. The flux profile in a "# @2 @ 2 perfect conductor does not change on cooling below a T Hc Hc Dc ¼ cscn ¼ Hc ð37Þ hypothetical temperature where perfect conductivity 4p @T 2 @T occurs. However, a superconductor expels magnetic flux lines on field cooling, distinguishing it from perfect implying a specific heat jump at the transition temper- conductivity. In a clean superconductor this Meissner ature Tc: effect implies  T @H 2 B ¼ 0 ¼ H þ 4pM ð33Þ Dc ¼ c ð38Þ 4p @T T¼Tc ¼ M ¼ 1 and that the magnetic susceptibility, w H 4p for a superconductor. Figure 10a shows the typical temperature dependence Further evidence of the superconducting state as a of the thermodynamic critical field for a type I distinct thermodynamic state is given by observation of superconductor. the return to a normal resistive state for fields exceeding For a type II superconductor in a zero or constant a thermodynamic critical field, Hc(T), for type I super- field, Ha, the superconducting phase transition is sec- conductors. Early descriptions of the T dependence ond order. A type II superconductor has two critical ð Þ (Fig. 10) of Hc (e.g., Tuyn’s law (Tuyn, 1929), fields, the lower critical field, Hc1 T , and upper critical 2 T H ðTÞ¼H ð1t Þ, where t ¼ ) suggested the supercon- field, Hc2ðTÞ. On heating, a type II superconductor, in a c 0 Tc ducting transition to be second order. Thermodynamic field, Ha < Hc1 (0K), a transition is observed both from considerations of the Gibb’s free energy density give the Meissner to the mixed state and from the mixed to the normal states at temperatures T1 (Ha ¼ Hc1) and T2 2ð Þ 2 (H ¼ H ). Since dM ¼ ‘ at H ¼ H , it can be shown that Hc T H a c2 dH a c1 gs ¼ jðTÞ; gn ¼ jðTÞþ ð34Þ 8p 8p the entropy changes continuously at approaching T1 from below. The entropy has an infinite temperature derivative in the mixed state, that is, approaching T1 from above. The second-order transition at Hc1ðT1Þ H manifests itself in a l-type specific heat anomaly. As the Type I Type II temperature is further increased to T (i.e., H ¼ H ðT Þ) H H 2 a c2 2 c2 the entropy in the mixed state increases with increasing Vortex state T and a specific heat jump is observed at T2 consistent Hc Hc2(T) Hc(T) with a second-order phase transition. Only a single transition is observed at Hc2 if the constant applied field Meissner H Hc1(T) exceeds H (0 K). state c1 c1 Meissner state The equilibrium magnetization (M vs. H) curve, shown in Figure 11a, for a type I superconductor reflects the T T T T c c Meissner effect slope, 1 in the superconducting state, (a) (b) 4p and the diamagnetic moment disappearing above Hc. Figure 10. Critical field temperature dependence for (a) type I The Meissner effect provides a description of the mag- netic response of a type I superconductor, for H < H , superconductor showing the thermodynamic critical field Hc(T) c and (b) type II superconductors showing the lower critical field only for long cylindrical geometries with H parallel to Hc1ðTÞ and upper critical field Hc2ðTÞ. the long axis. For cylinders in a transverse field or for 12 MAGNETIC MOMENT AND MAGNETIZATION

–4πM –4πM study the distribution of pinning energies in HTSCs (Maley et al.,1990). Large anisotropies in layered superconductors lead to new physical models of the H–T phase diagram in HTSCs (Nelson, 1988). This includes the notions of vortex liquid and vortex glass phases (Fisher, 1989). Examples of proposed H–T phase diagrams for HTSC materials are illustrated in Figure 12. In anisotropic materials, flux Η Η Η Η Η c c1 c2 pinning and Jc values are different for fields aligned (a) (b) parallel and perpendicular to a crystal’s c-axis. The pinning of vortex pancakes is different than for Abriko- Figure 11. Equilibrium M(H) curves for (a) type I and (b) type II sov vortices. The concept of intrinsic pinning has been superconductors. used to describe low-energy positions for fluxons in regions between Cu–O planes (for field parallel to the (001) plane in anisotropic materials). Models based on noncylindrical geometries demagnetization effects need weakly coupled pancake vortices have been proposed for to be considered. The internal field and magnetization fields parallel to the c-axis for layered oxides. The vortex can be expressed as melting line of Figure 12a and b is intimately related to the state of disorder as provided by flux pinning sites. 1 H ¼ H DM; M ¼ H ð39Þ Figure 12c shows schematics of intrinsic pinning and i a 4pð1DÞ Figure 12d shows pinning of pancake vortices in an anisotropic superconductor. 1 1 where D is the demagnetization factor (3 for a sphere, 2 for a cylinder or infinite slab in a transverse field, etc.). For noncylindrical geometries, demagnetization COUPLING OF MAGNETIC DIPOLE MOMENTS: effects imply that the internal field can be concentrated MEAN FIELD THEORY to values that exceed Hc before the applied field Ha exceeds Hc. Since the field energy density is not greater Dipolar interactions are important in defining demag- than the critical thermodynamic value, the entire super- netization effects. However, they are much too weak to conductor is unstable with respect to breaking down explain the existence of a spontaneous magnetization in into a lamellar intermediate state with alternating super- conducting and normal regions (de Gennes, 1966). The HH intermediate state becomes possible for fields exceeding H H H T c2 H T c2 *( ) ð1DÞHc, for example, a superconducting sphere exists *( ) < 2 H (T) in the Meissner state for H 3 Hc, in the intermediate Vortex c2 2 Vortex state for Hc < H < Hc, and in the normal state for H T glass 3 lattice c2( ) H > Hc. Observations of the equilibrium magnetization, M(H), Vortex Vortex liquid liquid and the H–T phase diagrams of type II superconductors H H c1 are different. As shown in Figure 11b, the Meissner state c1 Meissner phase (perfect diamagnetism) persists only to a lower critical Meissner phase T c field, Hc1. However, superconductivity persists until a H T H T c1( ) c1( ) much larger upper critical field, Hc2, is achieved. The (a)(b) persistence of superconductivity above Hc1 is explained in terms of a mixed state in which the superconductor H coexists with quantized units of magnetic flux called a H vortices or fluxons. The H–T phase diagram therefore b exhibits a single Meissner phase for H < Hc1ðTÞ, a mixed state, H < H < H in which the superconducting and CuO2 c1 c2 planes normal states coexist, and the single normal state phase for H > Hc2ðTÞ. For applications it is important to pin magnetic flux lines. This flux pinning determines the critical current Defect pinning of density, Jc. Jc represents the current density that frees magnetic flux lines from pinning sites (McHenry and H II ab H II c Sutton, 1994). Studies of high-temperature supercon- (c)(d) ductors have elucidated the crucial role played by Figure 12. Vortex lattices with a melting transition for an ani- crystalline anisotropy in determining properties. sotropic material with weak random pinning (a) and a material Thermally activated dissipation in flux creep has been with strong random pinning (b). Schematics of intrinsic pinning identified as an important limitation for Jc. The time- (c) and pinning of pancake vortices (d) in an anisotropic dependent decay of the magnetization has been used to superconductor. MAGNETIC MOMENT AND MAGNETIZATION 13 a material at any appreciable temperature. This is the arrangement of the dipole moments is precisely because thermal energy at relatively low temperature ordered. will destroy the alignment of dipoles. To explain a spon- taneous magnetization it is necessary to describe the ferrimagnet origin of an internal magnetic field or other strong mag- A is a material having two (or more) sub- netic interaction that acts to align atomic dipoles in the lattices, for which the magnetic dipole moments of unequal absence of a field. magnitude on adjacent nearest neighbor atomic sites (or planes) are also arranged in an anti- parallel fashion. A ferromagnet is a material for which an internal field or equivalent exchange interaction acts to align atomic dipole moments parallel to one another in the Ferrimagnets have nonzero magnetization in the absence of an applied field (H ¼ 0). absence of an applied field because their adjacent dipole moments do not cancel.7 All of the collective magnets described thus far are collinear magnets, mean- Ferromagnetism is a collective phenomenon since ing that their dipole moments are either parallel or individual atomic dipole moments interact to promote antiparallel. It is possible to have ordered magnets for parallel alignment with one another. The interaction which the dipole moments are not randomly arranged, giving rise to the collective phenomenon of ferromagne- but are not parallel or antiparallel. The helimagnet of tism has been explained by two models: Figure 13d is a noncollinear ordered magnet. Other examples of noncollinear ordered magnetic states i. Mean field theory: considers the existence of a include the triangular spin arrangements in some fer- nonlocal internal magnetic field, called the Weiss rites (Yafet and Kittel, 1952). field,whichactstoalignmagneticdipolemoments even in the absence of an applied field, Ha. Classical and Quantum Theories of Paramagnetism ii. Heisenberg exchange theory: considers a local The phenomenon of paramagnetism results from the (nearest neighbor) interaction between atomic existence of permanent magnetic dipole moments on moments (spins) mediated by direct or indirect atoms. We have shown that in the absence of a field, a overlap of the atomic orbitals responsible for the permanent atomic dipole moment results from incom- dipole moments. This acts to align adjacent plete cancellation of the electron’s angular momentum moments in the absence of an applied field, Ha. vector.

Both of these theories help to explain the T dependence of the magnetization. In a paramagnetic material in the absence of a field, The Heisenberg theory lends itself to convenient the local atomic moments are uncoupled. representations of other collective magnetic phenomena such as antiferromagnetism, ferrimagnetism, helimag- For a collection of atoms, in the absence of a field, netism, etc., illustrated in Figure 13. these atomic moments will be aligned in random direc- h~ i¼ 8 ¼ tions so that matom 0 and therefore M 0. We now An antiferromagnet is a material for which dipoles of wish to consider the influence of an applied magnetic equal magnitude on adjacent nearest neighbor field on these moments. Consider the induction vector, ~ atomic sites (or planes) are arranged in an antipar- B, in our paramagnetic material arising from an applied ~ allel fashion in the absence of an applied field. field vector, H . Each individual atomic dipole moment has a potential energy9:

Antiferromagnets also have zero magnetization in ~ Up ¼~m B ¼mB cosy ð40aÞ the absence of an applied field because of the vector cancellation of adjacent moments. They exhibit temper- The distinction between the classical and quantum the- ature-dependent collective magnetism, though, because ories of paramagnetism lies in the fact that a continuum of values of y and therefore continuous projections of M~ on the field axis are allowed in the classical theory. In the quantum theory only discrete y and projected moments, m, are allowed consistent with the quantization of angu- (a)(b) lar momentum. Notice that in either case the potential

7 Ferrimagnets are named after a class of magnetic oxides called (c)(d) ferrites. 8 We will use the symbol hai to denote the spatial average of the Figure 13. Atomic dipole moment configurations in a variety of quantity a. magnetic ground states: (a) ferromagnet, (b) antiferromagnet, 9 This is the Zeeman energy and represents the internal poten- (c) ferrimagnet, and (d) noncollinear spins in a helimagnet. tial energy for electrons in a field. 14 MAGNETIC MOMENT AND MAGNETIZATION energy is minimized when local atomic moments and and integrating ~ induction, B, are parallel. ð We are now interested in determining the tempera- 2p ¼ ð Þ ture-dependent magnetic susceptibility for a collection N dn 40f 0 of local atomic moments in a paramagnetic material. In the presence of an applied field, at 0K, all gives the number of dipoles, N. We now calculate an atomic moments in a paramagnetic material will align average projected moment (along the axis of B~)as themselves with the induction vector, B~,soasto Ð lower their potential energy, U . At finite temperature, 2p p m cosy dn however, thermal fluctuations will cause misalignment, h~ i¼ 0 Ð matom 2p that is, thermal activation over the potential 0 dn hi energy barrier leading to a T dependence of the Ð ð Þ 2p matom m0H cosy susceptibility, wðTÞ. To determine wðTÞ for a classical 0 C exp k T m cosy 2p siny dy ¼ Ð hiB ð40gÞ paramagnet, we express the total energy of a 2p ð Þ C exp matom m0H cosy 2p siny dy collection of equal atomic magnetic dipole moments in 0 kBT SI units as and using the substitution x ¼ matomm0H, we have X kBT tot ¼ ð Þ ð Þ Ð Up matom m0H cosyi 40b 2p ; h~m i exp½x cosycosypsiny dy atoms i atom ¼ 0 Ð ð40hÞ m 2p ½ atom 0 exp x cosy p siny dy where yi is the angle between the ith atomic dipole moment and B~. We consider a field along the z-axis and and evaluation of the integrals reveals ¼ ~m the set of vectors ni m, which are unit vectors in the direction of the ith atomic moment, to determine wðTÞ;we h~m i 1 atom ¼ cothðxÞ ¼ LðxÞð40iÞ wish to discover the temperature distribution function of matom x the angle yi . The probability of any particular potential energy state being occupied is given by Boltzmann sta- where L(x) is called the Langevin function (Lange- tistics, in SI units as vin, 1907). The Langevin function has two interesting attributes as illustrated in Figure 15: ð Þ ¼ Up ¼ matom m0H cosy ð Þ ð Þ p C exp C exp 40c ð Þ¼ ; dL x ¼ 1 kBT kBT lim L x 1 lim x ! ‘ x ! 0 dx 3 where p ¼ pðUpÞ¼pðyÞ. As shown in Figure 14, the To calculate the magnetization we remember that M is number of dipoles for a given y at T ¼ 0K and H ¼ 0 defined as the total dipole moment per unit volume (we i ~ ~ is given by are interested in the component of M parallel to B); thus: ¼ h~ i¼ ð ÞðÞ dn ¼ 2p siny dy ð40dÞ M Nm matom NmmatomL x 41a since all angles are equally probable. At finite T and H: where Nm is the number of magnetic dipoles per unit x LðxÞ¼ volume. In the large limit 1, and we infer that the m ðm HÞcosy saturation magnetization, Ms, is given by dn ¼ C exp atom 0 2p siny dy ð40eÞ k T B ¼ ð Þ Ms Nmmatom 41b

and the temperature (and field) dependence of the mag- H netization can be expressed as dφ R dφ M ¼ LðxÞð41cÞ R dθ Ms dA ð Þx In the low a limit (low field, high temperature), L x 3, and θ θ + dθ  m m H M ¼ M 0 atom ð41dÞ s k T T R 3 B

and

2 Figure 14. Distribution of moment vector angles with respect to M Nmm ðm Þ C w ¼ ¼ 0 atom ¼ ð41eÞ the field axis. H 3kBT T MAGNETIC MOMENT AND MAGNETIZATION 15

1.0 0.6 L(x) 0.8 0.5 x/3 L(x) 0.4

s 0.6 s M M / / 0.3 M 0.4 M 0.2 0.2 0.1

0.0 0.0 0 5 10 15 20 25 30 35 40 0.0 0.5 1.0 1.5 2.0 2.5 x x Figure 15. (a) Langevin function and (b) its (a) (b) low-temperature limiting form. which is the Curie law of paramagnetism. Notice that if To further describe the response of the quantum we know Nm (the concentration of magnetic atoms) from paramagnet we again consider an atom left with a per- ¼ an independent experimental measurement, then an manent magnetic dipole moment of magnitude matom m, experimental determination of wðTÞ allows us to solve due to its unfilled shells. We can now consider its for C (as shown in Fig. 16) as the slope of w versus 1 and magnetic behavior in an applied field. The magnetic T ~ therefore matom. matom is associated with the local effective induction, B, will align the atomic dipole moments. moment as given by Hund’s rules. In materials with a The potential energy of a dipole oriented at an angle y phase transition the dipoles may order below a critical with respect to the magnetic induction is temperature, Tc. For these the paramagnetic response is observed for T > T and the Curie law is replaced by a ¼~ ~ ¼ ð~ ~ÞðÞ c Up m B gmB J B 42a Curie–Weiss law: where g is the gyromagnetic factor. Now our quantum ð Þ2 ð Þ2 ¼ M ¼ Nmm0 matom ¼ C ; ¼ Nm matom ð Þ mechanical description of angular momentum tells us w C cgs j j H 3kBðTTcÞ TTc 3kB that Jz , the projection of the total angular momentum ð41fÞ on the field axis, must be quantized, that is: ¼ð Þ ð Þ This ordering temperature can be associated with an Up gmB mJ B 42b internal Weiss molecular field, H , through the follow- int ¼ ; ; ... ¼j~j¼j~þ ~j ing expression: where mJ J J 1 J and J J L S that may take on integral or half-integral values. In the case of spin only, m ¼ m ¼1. The quantization of J Tc ¼ lC; Hint ¼ lM ð41gÞ J s 2 z requires that only certain angles y are possible for the ~ ~ where l is called the molecular field constant. orientation of m with respect to B. The ground state corresponds to ~mjjB~. However, with increasing thermal 5000 energy, it is possible to misalign~m so as to occupy excited angular momentum states. If we consider the simple system with spin only, the Zeeman splitting between the 4000 eigenstates is mBB, the lower lying state corresponding to ¼ ¼1 mJ ms 2 with the spin moment parallel to the field. ¼ ¼ 1 The higher energy state corresponds to mJ ms 2 and 3000 an antiparallel spin moment. For this simple two-level

χ system we can use Boltzmann statistics to describe the

1/ population of these two states. With N isolated atoms per unit volume in a field, we define 2000 ð Þ mB m0H N1 ¼ N" ¼ A exp ; kBT 1000 ð Þ mB m0H bcc Fe N2 ¼ N# ¼ A exp ð43aÞ kBT 0 800 820 840 860 880 900 920 Recognizing that N ¼ N1 þ N2 and the net magnetization T (ºC) (dipole moment/volume) is M ¼ðN1N2Þm, we have

Figure 16. Curie–Weiss law fit to paramagnetic magnetization ¼ Y ¼ ¼ N ð ð ÞÞ data for bcc Fe above its ordering temperature Tc 794K M ð þ ð ÞÞ A expx exp x m (measurements of Sucksmith and Pearce). A expx exp x 16 MAGNETIC MOMENT AND MAGNETIZATION x ðxÞ ¼ exp exp ¼ ð Þ The effective local moment can be contrasted with the Nm þ ð Þ Nm tanhx 43b expx exp x maximum value that occurs when all of the dipoles ð Þ are aligned with the magnetic field. This has the where x ¼ mB m0H . Notice that for small values of x, kBT following value: tanhðxÞx and we can approximate M and determine w: ¼ ð Þ mH gJmB 45e m ðm HÞ Nm2 ðm HÞ M Nm m2 M ¼ Nm B 0 ¼ B 0 ; w ¼ ¼ 0 B ð43cÞ kBT kBT H kBT Experimentally derived magnetic susceptibility versus T 1 data can be plotted as w versus T to determine C (from the 1 The expression relating w to T is called the paramagnetic slope) and therefore peff (if the concentration of para- Curie law. magnetic ions is known). Figure 17 shows the behavior of For the case where we have both spin and orbital H M versus T for a paramagnetic material. At low temper- angular momenta, we are interested in the J quantum ð Þx ð Þ ature M x 3. M H is well described by a Brillouin number and the 2J þ 1 possible values of mJ , each giving H function with a characteristic T scaling bringing curves a different projected value (Jz)ofJ along the field (z) axis. from different temperatures into coincidence. In this case we no longer have a two-level system but instead a ð2J þ 1Þ-level system. The 2J þ 1 projections Mean Field Theory—Ferromagnetism are equally spaced in energy. Again considering a Boltzmann distribution to describe the thermal occupa- Ferromagnetic response is distinct from paramagnetic ~ ~ ¼ J response, in that local atomic moments are coupled in tion of the excited states, we find that m B mBB and Jz the absence of an applied field. A ferromagnetic material P hi m ðm HÞ possesses a nonzero magnetization over a macroscopic J m exp J B 0 X ð Þ J Jz B Jz kBT J m m H volume, called a domain, containing many atomic m ¼ hi; N ¼ B 0 P ð Þ exp ¼ J mB m0H J k T sites, even for H 0. Ferromagnetism is a collective exp J z B J Jz kBT phenomenon since individual atomic moments interact ð44aÞ so as to promote alignment with one another. The inter- action between individual atomic moments gives rise to so that finally the collective phenomenon of ferromagnetism that hi can be explained in terms of mean field theory or Heisen- P ð Þ J J mB m0H berg exchange theory. We consider the mean field theory J J J mB exp J k T ¼ ¼ z hiz B ¼ ð Þ here and the Heisenberg exchange theory (Heisen- M Nm N P ð Þ NgJmBBJ x exp J mB m0H J Jz kBT berg, 1928) below. The two theories do lead to different pictures of certain aspects of the collective phenomena ð44bÞ and the ferromagnetic phase transformation. The Hei-

gJmBB senberg theory also lends itself to quite convenient where x ¼ and BJ ðxÞ is called the Brillouin function kBT representations of other collective magnetic phenomena and is expressed as such as antiferromagnetism, ferrimagnetism, helimag- hi netism, etc. 2J þ 1 ð2J þ 1Þx 1 x BJ ðxÞ¼ coth coth ð44cÞ The mean field theory of ferromagnetism was intro- 2J 2J 2J 2J duced by Weiss (1907). Weiss postulated the existence of an internal magnetic field (the Weiss field), H~ , which ¼ 1 ð Þ¼ ð Þ int For J 2, BJ x tanh x as before. The small x expan- ð Þ sion for BJ x is 20 4K xðJ þ 1Þ 5K B ðxÞ¼ ; x 1 ð45aÞ 15 J 3 6K 10 7K For small x we then see that 10K 5 100K 2ð Þ 2 ð þ Þ 2 ð Þ Ng m0H mBJ J 1 Npeff m0H M ¼ ¼ ð45bÞ (emu/g) 0

3kBT 3kBT M –5 where –10 ¼ ½ ð þ Þ1=2 ð Þ peff g J J 1 mB 45c –15 is called the effective local moment. This expression is a –20 Curie law with –1.5 –10 –0.5 0.0 0.5 1.0 1.5 H/T (T/K) 2 C Npeff 3 þ w ¼ ; C ¼ ð45dÞ Figure 17. Paramagnetic response of Gd ions in Gd2C3 nano- T 3kBT H crystals with T scaling (Diggs et al., 1994). MAGNETIC MOMENT AND MAGNETIZATION 17 acts to align the atomic moments even in the absence of which is dimensionless (and Mð0Þ¼MsLðbÞ) and define: an external applied field, Ha. The main assumption of mean field theory is that the internal field is directly ð Þ2 ¼ Nmm0 matom l ð Þ proportional to the magnetization of the sample: Tc 46f 3kB ~ ¼ ~ ð Þ H int lM 46a Notice that Tc has units of temperature. Notice also that b 3 Mð0Þ where the constant of proportionality, l,iscalledthe ¼ ð46gÞ Weiss molecular field constant (we can think of it as a Tc T Ms crystal field constant). We write the field quantities as vectors because it is possible to have situations in which so that themagneticdipolesarenoncollinearbutstillordered(i.e., a helimagnet where the ordering is wave-like, or a trian- Mð0Þ bT ¼ ¼ LðbÞð46hÞ gular spin structure). In the cases discussed below, fer- Ms 3Tc romagnetism, antiferromagnetism, and ferrimagnetism, ð Þ the dipoles are collinear, either parallel or antiparallel. The reduced magnetization equations (i.e., M 0 ¼ bT ð Þ Ms 3Tc We now wish to consider the effects on ferromagnetic and M 0 ¼ LðbÞ) can be solved graphically, for any choice Ms response of application of an applied field, Ha, and the ofT by considering the intersection of the two functions randomizing effects of temperature. We can treat this b T and L(b). As is shown in Figure 18, for T T 3 Tc c problem identically to that of a paramagnet but now the only solutions for which the two equations are considering the superposition of the applied and internal simultaneously satisfied are when M ¼ 0, that is, no magnetic fields. By analogy we conclude that spontaneous magnetization and paramagnetic response. For T < Tc we obtain solutions with a nonzero, h~ i matom ¼ ð 0Þ 1 ¼ ð 0Þ; M ¼ ð 0Þð Þ spontaneous, magnetization, the defining feature of a coth x 0 L x L x 46b ¼ ¼ matom x Ms ferromagnet. For T 0toT Tc we can determine the spontaneous magnetization graphically as the intersec- tion of our two functions b T and L(b). This allows us where 3 Tc to determine the zero-field magnetization, Mð0Þ,asa 0 m0matom x ¼ ½Ha þ lM ð46cÞ fraction of the spontaneous magnetization as a function k T B of temperature. As shown in the phase diagram of Mð0;TÞ Figure 18c, M decreases monotonically from 1, at for a collection of classical dipole moments. Similarly, ¼ s ¼ h~ i 0K, to 0 at T Tc, where Tc is called the ferromagnetic M Nm matom and Curie temperature.AtT ¼ Tc, we have a phase transfor-  M M m m mation from ferromagnetic to paramagnetic response ¼ ¼ L 0 atom ½H þ lM ð46dÞ that can be shown to be second order in the absence N m M k T a m atom s B of a field. In summary, mean field theory for ferromag- nets predicts: where this simple expression represents a formidable transcendental equation to solve. Under appropriate i. For T < T , collective magnetic response gives conditions, this leads to solutions for which there is a c rise to a spontaneous magnetization even in the nonzero magnetization (spontaneous magnetization) absence of a field. This spontaneous magnetiza- even in the absence of an applied field. We can show this tion is the defining feature of a ferromagnet. graphically considering MðH ¼ 0Þ defining the variables: ii. For T > Tc, the misaligning effects of temperature m m serve to completely randomize the direction of the b ¼ 0 atom ½lM ð46eÞ kBT atomic moments in the absence of a field. The loss

1.0 1.0 M m(t) Phase 0.9 T > T 0.9 a diagram 0.8 c T = Tc T < Tc 0.8 ′ Ma L(x ) 0.7 ′ 0.7 s L(x ) s 0.6 0.6 M Paramagnet M M c / 0.5 / 0.5 M 0.4 M 0.4 = 0.3 = 0.3 m m Ferromagnet 0.2 0.2 0.1 0.1 0.0 0.0 0 1 2 3 4 5 0 2 4 6 8 10 1 t = T/T x′ x′ c (a)(b)(c)  b T Figure 18. (a) Intersection between the curves and L(b) for T < Tc gives a nonzero, stable 3 Tc ferromagnetic state and (b) the locus of M(T) determined by intersections at temperatures T < T ;(c) reduced magnetization, m, versus reduced temperature, t ¼ T , as derived from (b). c Tc 18 MAGNETIC MOMENT AND MAGNETIZATION

Table 3. Structures, Room-Temperature, and 0 K Saturation Magnetizations and Curie Temperatures for Elemental Ferromagnets (O’Handley, 1987)

ð Þ 3 ð Þ 3 ð Þ Element Structure Ms 290K (emu/cm ) Ms 0K (emu/cm ) nB mB Tc (K) Fe bcc 1707 1740 2.22 1043 Co hcp, fcc 1440 1446 1.72 1388 Ni fcc 485 510 1.72 627 Gd hcp — 2060 7.63 292 Dy hcp — 2920 10.2 88

of the spontaneous magnetization defines the terms of more than one molecular field constant, return to paramagnetic response. each multiplied by the magnetization due to a different iii. In the absence of a field, the ferromagnetic to sublattice. An antiferromagnet has dipole moments paramagnetic phase transition is second order on adjacent atomic sites arranged in an antiparallel (first order in a field). fashion below an ordering temperature, TN, called the Neel temperature. Cr is an example of an antiferromag- Table 3 summarizes structures, room-temperature, and net. The susceptibility of an antiferromagnet does not 0K saturation magnetizations and Curie temperatures diverge at the ordering temperature but instead has a for elemental ferromagnets (O’Handley, 2000). weak cusp. The mean field theory for antiferromagnets As a last ramification of the mean field theory we considers two sublattices, an A sublattice for which consider the behavior of magnetic dipoles in the para- the spin moment is down. We can express, in mean field theory, the internal fields on the A and B sites, magnetic state T > Tc with the application of a small field, H. We wish to examine the expression for the paramag- respectively: ð Þ¼MðH;TÞ netic susceptibility, w T H . Here again we can ~int ¼ ~ ; ~int ¼ ~ ð Þ assume that we are in the small argument regime for H A lBAM B H B lABM A 48a describing the Langevin function so that ~ ~ 0 where by symmetry lBA ¼ lBA, and M A and M B are the MðH; TÞ 0 a m m l ¼ Lðx Þ¼ ¼ 0 atom ðH þ lMÞð47aÞ magnetizations of the A and B sublattices. The mean H k T 3 3 B field theory thus considers a field at the B atoms due to the magnetization of the A atoms and vice versa. Using and hi ð Þ2 the paramagnetic susceptibilities wA and wB, which are Nmm0 matom H 3kB CH the same and both equal w , we can express the high- M ¼ ¼ ð47bÞ p N m ðm lÞ2 temperature magnetization for each sublattice as T m 0 atom T Tc 3kB ~ ¼ ð ~ þ ~intÞ¼CA ð ~ ~ ÞðÞ Thus, the susceptibility, wðTÞ, is described by the so- M A wp H a H H a lBAM B 48b A T called Curie–Weiss law: int C M C M~ ¼ w ðH~ þ H~ Þ¼ B ðH~ l M~ Þð48cÞ w ¼ ¼ ð47cÞ B p a B T a AB A H TTc Now for an antiferromagnet the moments on the A and B where, as before, C is the Curie constant and Tc is the sublattices are equal and opposite so that CA ¼ CB ¼ C Curie temperature. Notice that and on rearranging we get "# 2 Nmm ðm Þ ~ þ ~ ¼ ~ ð Þ C ¼ 0 atom ð47dÞ TM A ClABM B CH a 48d 3kB

~ ~ ~ It is possible to determine the atomic moment and molec- ClABM A þ TM B ¼ CH a ð48eÞ ð Þ 1 ular field constant from w T data. PlottingÀÁw versus T ¼ 1 ! allows for determination of a slope C , from which In the limit as Ha 0 these two equations have Tc ¼ matom can be determined, and an intercept l. The nonzero solutions (spontaneous magnetizations) for C ~ ~ ferromagnetic to paramagnetic phase transition is not as M A and M B if the determinant of the coefficients sharp at T ¼ Tc in a field, exhibiting a Curie tail that vanishes, that is: reflects the ordering influence of field in the high-tem- perature paramagnetic phase. T lC ¼ 0 ð48fÞ lCT Mean Field Theory for Antiferromagnetism and Ferrimagnetism and we can solve for the ordering temperature: Mean field theories for antiferromagnetic and ferrimag- ¼ ð Þ netic ordering require expanding the internal field in TN lC 48g MAGNETIC MOMENT AND MAGNETIZATION 19

χ 1/χ

T T Figure 19. (a) T dependence of the magnetic –TN susceptibility for an antiferromagnetic TN material and (b) inverse susceptibility as a (a)(b) function of T.

For T > TN the susceptibility is given by Since the signs of all of the Weiss interactions are nega- tive, if the A and B sites couple antiferromagnetically, 2CT2lC2 2C 2C then the A–A and B–B pairs align parallel. w ¼ ¼ ¼ ð48hÞ 2 þ þ The mean field theory of magnetic ordering in spinels T 2ðlCÞ T lC T TN considers the magnetic exchange interactions between Figure 19 shows that this susceptibility has a cusp cations. The mean field theory of ferrimagnetism can be at the Neel temperature. Table 4 summarizes Neel extended to account for A–A, A–B, and B–B interactions temperatures for some simple antiferromagnets in ferrites. The Neel theory of two-sublattice ferrimag- (O’Handley, 2000). netism is used to consider A–A, A–B, and B–B interac- tions instead of the simple A–B interactions discussed Ferrites earlier. In this case the mean field theory is expressed as

The mean field theory for antiferromagnets is easily int int H~ ¼l M~ l M~ ; H~ ¼l M~ l M~ ð50Þ generalized to simple AB ferrimagnetic alloys. Here the A AA A BA B B AB A BB B magnitude of the moment on the A and B sublattices With three mean field parameters, solutions to the need not be the same and therefore C 6¼ C in the limit A B mean field equations give rise to a variety of different as H ! 0; the determinant of the coefficients is a temperature dependences for the magnetization, M(T).

Louis Neel solved the mean field equations to describe T lCB ¼ 0 ð49aÞ six different T dependences for the magnetization. Pos- lC T A sible ground-state configurations for the dipole moments can be made even further complicated if one and the ferrimagnetic order temperature is determined allows for noncollinear dipole moments (not parallel or to be antiparallel). This is considered in the Yafet–Kittel theory = that describes triangular spin configurations in ferrites. T ¼ðlC lC Þ1 2 ð49bÞ N A B Table 5 summarizes room-temperature saturation mag- netizations and Neel temperatures for selected spinel The magnetic susceptibility for T > T becomes N ferrites (O’Handley, 1987). ðC þ C ÞT2lC C w ¼ A B A B ð49cÞ 2 2 T TN EXCHANGE THEORY

1 with curvature in w versus T characteristic of a Exchange is an atomic quantum mechanical phenome- ferrimagnet. non that describes the origins of the internal fields. The mean field theory for spinel ferrimagnets can be Heisenberg exchange theory considers a local (nearest evenricher.Inthespinelstructure,magneticcationsmay occupy octahedral A or tetrahedral B sites and are close Table 5. Room-Temperature Saturation Magnetizations and enough that A–A, A–B, and B–B mean fields are possible. Neel Temperatures for Selected Spinel Ferrites (O’Handley, 2000) Table 4. Neel Temperatures for Some Simple

Antiferromagnets Msð290KÞ 3 Element (emu/cm ) nB (mB) TN (K) Material TN ðKÞ (Mn O)Fe2O3 410 5.0 573 NiO 600 (FeO)Fe2O3 480 4.1 858 Cr 311 (CoO)Fe2O3 — 3.2 — Mn 95 (NiO)Fe2O3 270 2.4 858 FeO 198 (CuO)Fe2O3 135 1.3 728 20 MAGNETIC MOMENT AND MAGNETIZATION neighbor) interaction between atomic moments (spins) which for Jex > 0 favors parallel spins. For a linear that acts to align adjacent moments in the absence of an chain of N spins (where N is large) or exploiting periodic, applied field. Various types of exchange interactions Born–Von Karmon BC, the total internal energy is exist in materials. These can be divided into direct exchange and mediated (indirect) exchange. Direct XN ¼ 2 ð Þ exchange results from the direct overlap of the orbitals Up 2JexS cosyi;i þ 1 51c responsible for atomic dipole moments. Indirect i¼1 exchange mechanisms include those mediated by over- which for J > 0 is minimized for a configuration in lap between the magnetic orbitals and nonmagnetic ex which all the spins are aligned in a parallel ferromagnetic orbitals on the same or other species. The superex- change mechanism involves overlap between magnetic configuration. Combining mean field theory and the Heisenberg orbitals and the p orbitals on intervening oxygen or other model, the Curie temperature can be estimated. A anions. RKKY interactions are those mediated through the conduction electrons. Exchange determines the statistical mechanical description of exchange has been developed within the context of the Ising model. strength of the coupling between dipoles and therefore One of the results of this model allows us to associate the temperature dependence of the magnetization. Weiss molecular field The term random exchange refers to the weakening of the exchange interaction with the of mean field theory. This results in the following exchange interactions by disorder and its consequent relationship: effects on the temperature dependence of the magnetization. ZJ l ¼ ex ð52Þ m m2 Heisenberg Exchange Theory 4N 0 B

Heisenberg exchange theory (Heisenberg, 1928) consid- Thus, larger exchange interactions result in higher Curie ers a local (nearest neighbor) interaction between atomic temperatures. This allows us to use the results of mean moments (spins) that acts to align adjacent moments field theory to express the T dependence of the magne- even in the absence of a field. The Heisenberg model tization. For spin-only angular momentum this is considers ferromagnetism and the defining spontane-  ous magnetization to result from nearest neighbor ð Þ¼ m0mB þ ZJexM ð Þ exchange interactions that act to align spins in a parallel M T M0 tanh H 53 kBT 4Nm m configuration. The Heisenberg model can be further 0 B generalized to account for atomic moments of different and similar expressions for the Langevin and Brillouin magnitude, that is, in alloys, and for exchange interac- functions for classical and total angular momentum tions that act to align nearest neighbor moments in an models. antiparallel fashion or in a noncollinear relationship. Exchange interactions result from the spatially Let us consider first the Heisenberg ferromagnet. Here dependent energy functional for electrons with parallel we assume that the atomic moments on nearest neigh- or antiparallel spins (or canted spins in more compli- bor sites are coupled by a nearest neighbor exchange cated models). For the hydrogen molecule, for example, interaction giving rise to a potential energy: as shown in Figure 20a the configuration with electron spins aligned parallel is stable at larger interatomic ¼ ~ ~ ð Þ Up JexS i S i þ 1 51a separations and that aligned antiparallel at smaller interatomic separations. The choice of spin configura- between identical spins at sites i and i þ 1 in a 1D lattice. tion depends on the relationship between the crossover For identical spins: radius and the equilibrium separation of the atoms. The famous Bethe–Slater curve (Bethe and Sommer- 2 Up ¼2JexS cosyi;i þ 1 ð51bÞ feld, 1984; Slater, 1930) (Fig. 20b) predicts the sign of the

2.0 Parallel + 1.6 Co α-Fe 1.2 Ni Gd 0 (arbitrary) U 0.8 γ-Fe (a) Energies for parallel and Exchange integral Mn Figure 20. 0.4 Antiparallel antiparallel spins in the hydrogen molecule as a function of interatomic separation and 0.0 − 0 1 2 3 4 5 6 (b) the Bethe–Slater curve predicting the 1.5 1.75 2.0 2.25 r (arbitrary) sign of the exchange interaction in 3d Interatomic distance (D/d) transition metal solids (Bethe and (a) (b) Sommerfeld, 1984; Slater, 1930). MAGNETIC MOMENT AND MAGNETIZATION 21 exchange interaction in 3d transition metal solids. The AB1 AB2 interplay between electron–electron Coulomb interac- 135º 155º tions and the constraints of the Pauli exclusion principle determine the sign of the exchange interaction. In tran- 0.25 0.25 0.44 sition metal solids a measure of the overlap between 0.22 H Large nearest neighbor d orbitals is given by the ratio of the AB atomic to the 3d ionic (or nearest neighbor) radius. The Bethe–Slater curve provides an empirical descrip- BB1 BB 2 125º A site B site tion of the exchange integral as a function of composition 90º (or more accurately, with interatomic spacing to 3d 0.29 0.25 orbital ratio). The ratio of the 3d ionic radius to the near 0.25 0.25 neighbor distance describes the amount of orbital over- HBB Moderate lap (or exchange). At this point it is useful to repeat the Heisenberg AA model predictions for other magnetic ground states. For 80º < 0.22 example, notice that if Jex 0, an antiparallel configu- HAASmall ration for adjacent spins in a 1D chain is predicted, 0.44 consistent with an antiferromagnetic ground state as shown in Figure 13b. We can deduce a ferrimagnetic ground state as illustrated in Figure 13c and observed, Figure 22. Geometry of the superexchange interactions in the for example, for ferrites when we have Jex < 0 and two spinel structure. magnetic sublattices, for example, a and b, for which 6¼ ma mb. In 3D systems it is possible to relax the restric- tions of nearest neighbor exchange and it is also possible illustrates the geometry of the superexchange to have noncollinear exchange interactions as shown interactions in the spinel structure. The strongest in Figure 13d for a helimagnet. In certain ferrite systems superexchange interactions occur for bond angles (e.g., Mn2O3) it has been observed that Mn atoms in approaching 180 . the octahedral and tetrahedral sites in fcc interstices of the oxygen anion sublattice couple in a Yafet–Kittel RKKY Exchange triangular spin configuration (Yafet and Kittel, 1952). Another form of indirect exchange has recently been shown to be important in, for example, rare earth metal Superexchange systems and in magnetic/nonmagnetic multilayers. In the context of the ferrites and other oxides we may This exchange is the oscillatory RKKY exchange distinguish between direct and indirect exchange. We (Fig. 23a) (Ruderman and Kittel, 1954; Kasuya, 1984; have described ferrimagnetic and Yafet–Kittel triangular Yosida, 1957) that is mediated through the conduction spin configurations between neighboring magnetic cat- electron gas often associated with nonmagnetic atoms ion sites. This is an example of an indirect exchange but sometimes associated, for example, with sp conduc- mechanism since it must be transmitted through inter- tion electrons of the magnetic atoms in rare earths. This vening nearest neighbor oxygen sites. In fact, the indirect exchange is transmitted by polarization of the exchange interaction is transmitted through overlap free electron gas. The free electron sea can be charac- between magnetic d orbitals on the cation sites and the terized by a wavevector called the Fermi wavevector that p orbitals of oxygen (Neel, 1932). This particular p orbital is defined as transmitted indirect exchange interaction is given the 1=3 name superexchange and illustrated in Figure 21. 3p2N The size of superexchange interactions depends on kF ¼ ð54Þ V the distance and angles of the d–p–d bonds in the crystal. These are determined by the crystallography. Figure 22 N where V is the number of free electrons per unit volume. Examples of magnetic systems coupling through RKKY interactions include the rare earths. In these the Hund’s rule ground state is determined by 4f elec- – trons that are very localized, that is, close to the nucleus and shielded by conduction electrons. Therefore, there is – – – no direct exchange since 4f states do not overlap from site to site. Instead coupling occurs through free – electrons. 4f dipole moments in rare earth elements polarize the free electrons around them; these in turn communicate the information to 4f dipole moments d p d on adjacent sites. Coupling through the conduction Figure 21. Superexchange interaction of magnetic cation d electrons is influenced by Friedel oscillations in the orbitals mediated by an O p orbital. conduction electron spin density. 22 MAGNETIC MOMENT AND MAGNETIZATION

Jex Ferromagnetic monolayer

+ Conducting layers + 51015– 20 Figure 23. (a) Spatial dependence of – k r Ferromagnetic monolayer F the RKKY exchange interaction and (b) magnetic bilayer system with intervening conducting layer that (a)(b) exhibits an RKKY exchange interaction.

In describing RKKY interactions we consider a polar- dence of the magnetization in amorphous alloys has ization plane wave emanating from a magnetic ion at a been proposed by Kobe (1969) and Handrich (1969). position r ¼ 0. The first waves influenced by a polarizing In this Handrich–Kobe theory was proposed an expres- ¼ ð Þ¼ MðTÞ field are those with wavevector k kF and therefore the sion for the reduced magnetization m T Mð0KÞ that sign of the exchange interaction oscillates spatially like consisted of a modified Brillouin function where a single ~ ~ cosðk F r Þ as well as decaying exponentially as a function exchange parameter was replaced by an exchange of r. To determine the sign of the indirect exchange parameter reflecting the distribution of nearest neighbor interaction between the magnetic ion at r ¼ 0 and at positions in the amorphous phase. This is expressed ~ ~ r ¼ r we calculate cosðk F r Þ. The RKKY exchange is as follows: weaker in magnitude and oscillates in sign spatially and can be described as 1 mðTÞ¼ ðB ½ð1 þ dÞx þ B ½Þð1dÞx ð56aÞ 2 s s ~ ~ JRKKY ¼ J0ðrÞcosðk F r Þð55Þ where where the prefactor J0ðrÞ decays exponentially with r. sffiffiffiffiffiffiffiffiffiffiffiffiffiffi RKKY interactions are important in a variety of thin 3S m T hDJ 2 i x ¼ ; t ¼ ; d ¼ ex ð56bÞ film magnetic multilayer devices such as spin valves. þ 2 S 1 T Tc hJexi These have interactions between 2D ferromagnetic layers that are coupled through a conducting layer The first expression is the argument of a conventional (Fig. 23b). By varying the thickness of the conducting (spin-only) Brillouin function. The exchange fluctuation layer, the sign and magnitude of the exchange interac- parameter, d, is defined by the second expression. This tion can be varied. parameterizes the root mean square (rms) fluctuation in the exchange interaction. The root mean square Random Exchange exchange fluctuation has been suggested to have a T ¼ ð 2Þ Because of large deviations in interatomic spacings in dependence of the form (Bhatnagar, 1984) d d0 1 t . amorphous alloys as compared with bulk crystalline, Figure 24a compares mean field results for M(T) in the they have distributed exchange interactions that alter classical, spin-only, and total angular momentum repre- the mean field description of the temperature depen- sentations. Figure 24b shows mean field results for M(T) dence of the magnetization, M(T) (Chien, 1978; taking into account the disorder-induced fluctuations Kaneyoshi, 1984). A mean field theory for the T depen- in the exchange parameter in hypothetical amorphous

1.0 δ = 0 (crystal) J = S = 1/2 1.0

0.8 0.8 (0) (0)

M 0.6 M 0.6 )/ )/ J = 7/2 (Gd3+) T T

( ( High δ M 0.4 M 0.4 = = m J = ∞ (classical) m Figure 24. Comparison of magnetization as a 0.2 0.2 δ = 0, 0.4, 0.6, 0.8 function of reduced temperature t (a) in a spin-only ¼ 1 J = 1/2 ferromagnet (using a J 2 Brillouin function), for 0.0 0.0 J ¼ 1 and for the classical limit J ¼ ‘, and (b)in 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 2 1.0 an amorphous magnet with J ¼ 1 and different t = T/T t = T/T 2 c c values of the exchange fluctuation parameter d (a) (b) (figure courtesy of Hiro Iwanabe). MAGNETIC MOMENT AND MAGNETIZATION 23 alloys. This construction then predicts a quite remark- able change in the mean field theory thermomagnetic response for amorphous magnets. +++++ N The T dependence of the magnetization for Fe88Zr7B4Cu amorphous alloy has been measured by S Gallagher et al. (1999). A two-parameter exchange - - - - - fluctuation mean field theory is shown to give signifi- cantly better fits than the single-parameter Handrich–Kobe model to m(T) for these amorphous alloys. The deviation in atomic nearest neighbor (NN) Figure 26. Reduction in the demagnetization field as a result of the introduction of magnetic domains into a ferromagnetic distances in this amorphous alloy was estimated from sample. X-ray scattering data. The Bethe–Slater curve was used to estimate fluctuations in the exchange interac- tion. An explanation for the relative invariance of M(T)in material a macroscopic volume contains many domains. disordered Co-based alloys as compared with Fe-based Each domain has a spontaneous magnetization of mag- alloys was proposed and good qualitative agreement of nitude, Ms. In the absence of an applied field the mag- the model and experimental data has been demon- netization vectors are randomly aligned from domain to strated. The modification made to the Handrich–Kobe domain (just like in a paramagnet atomic dipoles were (Kobe, 1969; Handrich, 1969) equation allowed for two d random). Taking a vector sum of the magnetization over parameters, d þ and d (Fig. 25). The new equation is as many domains yields zero sample magnetization follows: because of vector cancellation. We can qualitatively understand this by recognizing 1 that magnetic flux line leaves the north pole of a magnet mðTÞ¼ ðB ½þð1 þ d þ Þx B ½Þðð1dÞx 57Þ 2 s s and enters the south pole. This gives rise to a field outside the magnet, the demagnetization field, Hd, which where d þ and d are not necessarily the same and would like to misalign the dipole moments in the fer- therefore can act as a first-order fit to an asymmetric romagnet. It requires internal energy to maintain the distribution function. The Gallagher (Gallagher alignment of the dipoles. et al.,1999) model gives a better quantitative fit to the A configuration for which the demagnetization field is temperature dependence of the magnetization. reduced will lower the total energy of the system. For two domains (Fig. 26) we significantly reduce the return path that is necessary to be taken by fringing fields. By apply- MICROSCOPIC MAGNETIZATION AND DOMAINS ing successively more domains we can further reduce the magnetostatic self-energy to nearly zero. In the case A magnetic domain is macroscopic volumes over which where we have two long domains and two closure atomic magnetic moments are aligned. For a ferromag- domains the magnetization makes a nearly circuitous ¼ net, when Ha 0, the existence of a spontaneous mag- path reducing the demagnetization field to nearly zero. netization requires the existence of domains. It is per- There is no free lunch, though. Each boundary between haps surprising that ferromagnetic materials can exist domains requires that we pay an energy associated with in a “virgin state” for which the magnetization is zero in a domain wall. The configuration of domains and walls the absence of an applied field. This is understood by ultimately depends on the balancing of these two ferromagnetic domain theory. In a typical magnetic energies.

1.0 CONCLUSIONS

0.8 The basic notions of magnetic dipole moments and

(0) magnetization in solids have been summarized. In par-

M 0.6

)/ ticular, the origin of magnetic dipoles from an atomic, a T ( band structure, and a shielding current (superconduc- m

= 0.4 tors) picture has been described. The coupling of m magnetic dipoles, the field and temperature dependence 0.2 of the magnetization, and simple magnetic phase transi- tions have been illustrated. This serves to define 0.0 certain basic magnetic phenomena that will be elabo- 0.4 0.6 0.0 0.2 0.8 1.0 rated on, illustrated, and extended in subsequent t = T/Tc units. Magnetic phenomena are rich and varied; the Figure 25. Reduced magnetization, m, as a function of reduced fundamental principles of the origin, coupling, and temperature, t, in an amorphous Fe88Zr7B4 alloy fit with vector summation of magnetic dipole moments are at two asymmetric exchange fluctuation parameters, d þ and d the heart of a comprehensive understanding of magnetic (Gallagher et al., 1999). phenomena. 24 MAGNETIC MOMENT AND MAGNETIZATION

BIBLIOGRAPHY MacLaren, J. M., Schulthess, T. C., Butler, W. H., Sutton, R. A., and McHenry, M. E. 1999. Calculated Exchange Interactions and Curie Temperature of Equiatomic B2 FeCo. J. Appl. LITERATURE CITED Phys. 85:4833–35. Maley, M. P., Willis, J. O., Lessure, H., and McHenry, M. E. Abrikosov, A. A. 1957. On the magnetic properties of super- 1990. Dependence of flux creep activation energy upon conductors of the second group. Sov. Phys. JETP-USSR current density in grain-aligned YBa2Cu3O7-x. Phys. Rev. 5 (6):1174–1183. B 42:2369. Bhatnagar, K., Prasad, B. B. and Jagannathan, R. 1984. Phys. McHenry, M. E. and MacLaren, J. M. 1991. Iron and chromium Rev. B, 29. 4896. monolayer magnetism in noble-metal hosts: Systematics Bethe, H. A. and Sommerfeld, A. 1933. Handbuch der Physik, of local moment variation with structure. Phys. Rev. B Vol. 24. Springer, Berlin. 43:10611. Chien, C. L, 1978. Mossbauer study of a binary amorphous McHenry, M. E., MacLaren, J. M., Eberhart, M. E., and Crampin, ferromagnet: Fe80 B20, Phys. Rev. B. 18:1003–15. S. 1990. Electronic and magnetic properties of Fe/Au multi- Cullity, B. D. and Graham, C. D. 2009. Introduction to layers and interfaces. J. Magn. Magn. Met. 88:134. Magnetic Materials, 2nd ed. IEEE Press/John Wiley and McHenry, M. E. and Sutton, R. A. 1994. Flux pinning and Sons, Hoboken, NJ. dissipation in high temperature oxide superconductors. de Gennes, P. G. 1966. Superconductivity of Metals and Alloys. Prog. Met. Sci. 38:159. Benjamin, New York. Meissner, W. and Ochsenfeld, R. 1933. Ein neuer Effekt bei Diaz-Michelena, M. 2009. Small magnetic sensors for space Eintritt der Supraleitfahigkeit. Naturwissenschaften applications. Sensors 9:2271–2288. 21:787–788. Diggs, B., Zhou, A., Silva, C., Kirkpattrick, S., McHenry, M. E., Neel, L. 1932. Influences des Fluctuations du Champ Molecu- Petasis, D., Majetich, S. A., Brunett, B., Artmen, J. O., and laire sur les Proprietes Magnetiques des Corps. Ann. Phys. Staley, S. W. 1994. Magnetic properties of carbon-coated 18:5–105. rare earth carbide nanocrystals produced by a carbon arc Nelson, D. R. 1988. Phys. Rev. Lett. 60:1973. method. J. Appl. Phys. 75:5879. Onnes, H. K. 1911. The superconductivity of mercury. Commun. Fisher, M. P. A. 1989. Vortex-glass superconductivity: A possible Phys. Lab. Univ. Leiden 122, 124. new phase in bulk high-Tc oxides. Phys. Rev. Lett. 62:1415. Onnes, H. K. and Clay, J. 1908. On the change of the resistance Friedel, J. 1958. Theory of magnetism in transition metals. of the metals at very low temperatures and the influence on it Nuevo Cim. Suppl. 7:287. by small amounts of admixtures. Commun. Phys. Lab. Univ. Gallagher, K. A., Willard, M. A., Laughlin, D. E., and Leiden 99c:17–26. McHenry, M. E. 1999. Distributed magnetic exchange inter- O’Handley, R. C. 1987. Physics of ferromagnetic amorphous actions and mean field theory description of temperature alloys. J. Appl. Phys. 62 (10):R15–R49. dependent magnetization in amorphous Fe88Zr7B4Cu1 O’Handley, R. C. 2000. Modern Magnetic Materials, Principles alloys. J. Appl. Phys. 85:5130–5132. and Applications. John-Wiley and Sons, New York. Ginzburg, V. L. and Landau, L. D. 1950. Zh. Eksp. Teor. Fiz. Pauling, L. 1938. The nature of the interatomic forces in metals. 20:1044. Phys. Rev. 54:899. Handrich, K. 1969. A Simple Model for Amorphous and Liquid Ruderman, M. A. and Kittel, C. 1954. Indirect exchange Ferromagnets. Phys. Status Solidi. 32:K55. coupling of nuclear magnetic moments by conduction Harrison, W. A. 1989. Electronic Structure and the Properties electrons. Phys. Rev. 96:99. of Solids: The Physics of the Chemical Bond. Dover Russell, H. N. and Saunders, F. A. 1925. New regularities in the Publications, New York. spectra of the alkaline earths. Astrophys. J. 61:38. Heisenberg, W. 1928. On the theory of ferromagnetism. Z. Phys. Slater, J. C. 1930. Atomic shielding constants. Phys. Rev. 49:619. 36:57–64. Hund, F. 1927. Linienspektren und periodische system der Slater, J. C. 1937. Electronic structure of alloys. J. Appl. Phys. elemente. Springer, Berlin. 8:385. Kammerlingh-Onnes, H. and Tuyn, W. 1929. Leiden Comm. 198. Van Vleck, J. H. 1932. The Theory of Electric and Magnetic Kaneyoshi, T. 1984. Amorphous Magnetism Chemical Rubber Susceptibilities. Oxford University Press, New York. Corp., Boca Raton, FL. Weiss, P. 1907. L’hypothese du Champ Moleculaire et la Kasuya, T. 1956. Electrical resistance of ferromagnetic metals. Propriete Ferromagnetique. J. Phys. 6:661. Prog. Theor. Phys. 16:45. Yafet, Y. and Kittel, C. 1952. Antiferromagnetic arrangements Kobe, S. 1969. Phys. Status Solidi. 32:K55. in ferrites. Phys. Rev. 87 (2):290–294. Kobe, S. 1970. ibid. 41:K13. Yosida, K. 1957. Magnetic properties of Cu–Mn alloys. Phys. Kobe, S., Handrich, K., Tverd, Fiz. and Leningrad, Tela. 1971. Rev. 106:893. 13, 887 and Sov. Phys. Solid State. 13:734. Langevin, P. 1907. Magnetisme et Theorie des Electrons. Ann. KEY REFERENCES Chem. Phys. 5:70. London, F. 1950. Superfluids, Vol. 1. John Wiley and Sons, Solid State Physics Texts—Introductory (Undergraduate) New York. Level MacLaren, J. M., McHenry, M. E., Eberhart, M. E., and Eisberg, R. and Resnick, R. 1974. Quantum Physics of Atoms, Crampin, S. 1990. Magnetic and electronic properties of Molecules, Solids, Nuclei and Particles. John Wiley and Sons, Fe/Au multilayers and interfaces. J. Appl. Phys. 61:5406. New York. MAGNETIC MOMENT AND MAGNETIZATION 25

Rosenberg, H. M. 1990. The Solid State. Oxford Science Publica- Berkowitz, A. E. and Kneller, E. 1969. Magnetism and tions, Oxford. Metallurgy, Vol. 1. Academic Press, San Diego, CA. Solymar, L. and Walsh, D. 1998. Lectures on the Electrical Bozorth, R. M. 1951. Ferromagnetism. Van Nostrand, Properties of Materials. Oxford University Press, New York. New York. Solid State Physics Texts—Intermediate to Graduate Level Chen, C.-W. 1986. Metallurgy of Soft Magnetic Materials. Dover, Ashcroft, N. W. and Mermin, N. D. 1976. Solid State Physics. Holt, New York. Rinehart and Winston, New York. Coey, J. M. D. 2010. Magnetism and Magnetic Materials, Blakemore, J. S. 1981. Solid State Physics. Cambridge Univer- Cambridge University Press, Cambridge. sity Press, Cambridge, UK. Cullity, B. D. 2009. Introduction to Magnetic Materials, 2nd ed. Burns, G. 1985. Solid State Physics. Academic Press, San Diego, IEEE Press, Hoboken, NJ. CA. Kouvel, J. S. 1969. Magnetism and Metallurgy 2 (A. E. Berkowitz Kittel, C. 1976. Introduction to Solid State Physics. John Wiley and E. Kneller, eds.), p. 523. Academic Press, New York. and Sons, New York. McCurrie, R. A. 1994. Ferromagnetic Materials: Structure and Properties. Academic Press, London. Physics of Magnetism Morrish, A. H. 2001. The Physical Principles of Magnetism. IEEE Chikazumi, S. 1978. Physics of Magnetism. Kreiger, Malabar, FL. Press, New York. Keefer, F. 1982. Helimagnetism. In McGraw-Hill Encyclopedia O’Handley, 2000. See above. of Physics (S. P. Parker, ed.). McGraw-Hill, New York. Morrish, A. H. 1965. The Physical Principles of Magnetism. Wiley, Fine Particle and Nanomagnetism New York. Bean, C. P. and Livingston, J. D. 1959. J. Appl. Phys. Mydosh, J. A. 1993. Spin Glasses, An Experimental Introduction. 30:120S–129S. Taylor and Francis, London, Washington, DC. Bitter, F. 1932. Phys. Rev. 38:1903; ibid. 41:507 (1932). White, R. M. 1983. Quantum Theory of Magnetism, Springer Jacobs, I. S. and Bean, C. P. 1963. Magnetism, Vol. 3 (G. T. Rado Series in Solid State Physics 32. Springer-Verlag, Berlin. and H. Suhl, eds.). Academic Press, New York. Luborsky, F. E. 1962. J. Appl. Phys. 32:171S–183S. Magnetic Materials