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 À 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 À1 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 À À 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 :  À24 2 ð = Þ; defined formally as mB 9 27 10 Am J T ð À m ¼ m ¼ 9:27  10 21 erg=G ð9Þ ~ ¼ ~ ¼ ~ ~ ð Þ B m IAu~r ÂJ~ 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 ÂJ~ 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 Á 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, À1 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 ÁÁÁ ÁÁÁ 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.