Chapter 12

1 The essential idea is that in theN limit of largesystems (on our own macroscopic scale) it is not onlyconvenient but essential to realize that matter will undergomathematically sharp, singular phase transitions to states inwhich the microscopic symmetries, and even the microscopicequations of motion, are in a sense violated. The symmetryleaves behind as its expression only certain characteristicbehaviors. P W Anderson, More is Different, Science177, 393(1972). Chapter 12 Thermodynamics in SolidsLodestones fragments of magnetic1 FeO+Fe2O3(Fe3O4) although known tothe ancients were, according to Pliny the Elder, first formally described in Greek6thcentury writings. By that time they were already the stuff of myth,superstition and amazing curative claims, some of which survive to this day.

2 TheChinese used lodestones in navigation as early as 200 and are credited withinventing the magnetic compass in the 12th century in the modern era has magnetism become well understood, inspiring countlesspapers, books2and more than a dozen Nobel prizes in both fundamental and appliedresearch. Among the forms of macroscopic magnetism : In an external magnetic fieldBthe spin-state degeneracy1 Named, as one story goes, for Magnus, the Greek shepherd who reported a field of stones thatdrew the nails from his Stephen Blundell, Magnetism in Condensed Matter Oxford Maser Series in Con-densed Matter Physics (2002); Daniel C Mattis, Theory of Magnetism Made Simple, World Sci-entific, London (2006); Robert M. White, Quantum Theory of Magnetism: Magnetic Propertiesof Materials, 3rd rev. ed.

3 , Springer-Verlag, Berlin (2007).12 Chapter AND MAGNETISMof local (atomic) or itinerant (conduction) electronic states is lifted (Zeemaneffect). At low temperature this results in an induced macroscopic magneticmoment whose vector direction lies parallel to the external field. This is referredto most materials removing the external field restores spin-state degeneracy,returning the net moment to : Macroscopic magnetization may also be induced with a mag-netization vectoranti-parallelto the external field, an effect called diamag-netism. In conductors diamagnetism arises from the highly degenerate quan-tum eigen-energies and eigenstates (referred to asLandau levels.)4formedby interaction between mobile electrons and magnetic fields. Diamagnetismis also found in insulators, but largely from surface quantum orbitals ratherthan interior bulk cases are purely quantum phenomena, leavingmacroscopic diamagnetism without an elementary explanatory ,7 Allsolids show some diamagnetic response, but it is usually dominated by anyparamagnetism that may be high magnetic fields and low temperatures very pure metals exhibit anoscillatory diamagnetismcalled the de Haas-van Alfen effect whose source isexclusively the Landau Magnetism Ferromagnetism:Ferromagnetism is an ordered state of matter in whichlocal paramagnetic moments interact to produce aneffective internal mag-netic fieldwith collective alignment of moments throughout distinct regionscalled domains.

4 Due to these internal interactions, domainscan remainalignedeven after the external field is alignment abruptly disappears above a material specic tem-perature called the Curie temperatureTc, at which point ordinary local para-3 Itinerant (conduction) electron paramagnetism is referred to as spin or Pauli Landau Diamagnetism of Metals, Z. , 629 (1930).5D. Ceresoli, et. al. Orbital magnetization in crystalline solids, Phys. Rev. B74, 24408(2006).6 Niels Bohr, Studier over Metallernes Elektrontheori, Kbenhavns Universitet (1911).7 Hendrika Johanna van Leeuwen, Problmes de la thorie lectronique du magntisme, Journalde Physique et le Radium,2361 (1921).8D. Shoenberg, Magnetic Oscillations in Metals, Cambridge University Press, Cambridge(1984). IN SOLIDS3magnetism reasserts.

5 Antiferromagnetism:At low temperatures, interactions between adja-centidenticalparamagnetic atoms, ions or sub-lattices can induce anti-alignment of adjacent paramagnets, resulting in a netzeromagnetic mo-ment. Ferrimagnetism:At low temperatures interactions betweenunequivalentparamagnetic atoms, ions or sub-lattices can produce anti-alignment ofmoments, resulting in asmallresidual both ferrimagnetism and antiferromagnetism increasing temperature weak-ens anti-alignment with the collectiveinducedmoments approaching amaximum. Then, at a material specic temperature called theN eeltempera-tureTN, anti-alignment disappears and the materials becomes this Chapter general concepts in the thermodynamics of magnetism and mag-netic fields are discussed as well as models of local paramagnetism and WorkCentral to integrating magnetic fields and magnetizable systems into theFirst Lawof Thermodynamicsis a formulation of magnetic work.

6 Using Maxwell s fields,9theenergy generated within a volumeVin a time t, by an electric fieldEacting ontrue charge currentsJ Joule Heat is10 WM= tSVJ EdV .( )Therefore thequasi-static and reversible11magnetic work donebythe system is WMQS= tSVJ EdV .( )9 Maxwell fields in matter and free space are the local averages that appear in his equations heat and work are not state functions they do not have true differentials, so the wiggly s are used instead to represent incremental work in an interval of time specifying reversibility non-reversible hysteresis effects are AND MAGNETISMU sing Maxwell s Equation (in cgs-Gaussian units)12 H=4 cJ+1c D t( )the work donebythe system is WMQS= t c4 SV( H) EdV 14 SV D t EdV .( )Using the vector identityU V= (V U)+V U( )this becomes WMQS= t c4 SV (H E)dV+SVH EdV 14 SV D t EdV.

7 ( )The first integral on the right can be transformed by Gauss theorem into a surfaceintegral. But since the fields are static (non-radiative), they fall off faster than1r2sothat for a very distant surface the surface integral can be neglected. Then, with theMaxwell Equation (Faraday s Law) E= 1c B t ( )incremental work donebythe system is WMQS= t 14 SVH B tdV+14 SV D t EdV ( )= 14 SVH BdV+14 SV D EdV ,( )where the integrals are over the volume of the sampleandsurrounding free : The fields arefunctionsofthecoordinatesxand are not just simple variables,12 Although cgs units have fallen out of pedagogical favor in newer text books, they offer unrivaledclarity in presenting the subtle issues involved in thermodynamics of magnetic and electric IN SOLIDS5so that here the wiggly deltas in B(x)and H(x)representfunctionalchanges, changes in thefieldsnot the the discussion to magnetic phenomena, the magnetic contribution toquasi-staticwork donebythe system is therefore WMQS= 14 SVH BdV ,( )so that the thermodynamic identity becomes U=T S pdV+14 SVH BdV.

8 ( )From the Helmholtz potential, defined asF=U TS, F= U T S SdT ,( )which when combined with gives the change F F= SdT pdV+14 SVH BdV .( )Defining a magnetic enthalpyHasH=U+pV 14 SVB HdV ,( )gives, using , an enthalpy change H H=T S+Vdp 14 SVB HdV .( )Finally, a magnetic Gibbs potentialGis defined asG=F+pV 14 SVB HdV ,( )6 Chapter AND MAGNETISM which with gives the Gibbs potential change G G= SdT+Vdp 14 SVB HdV .( )Magnetization density13 Mand polarization density14 Pare introduced by the linearconstitutive relationsH=B 4 M( )andD=E+4 P( )in which casequasi-staticmagnetic work may be written WM1QS= 14 SVB BdV SV M BdV ( )or WM2QS= 14 SVH HdV+SV H MdV ,( )whilequasi-staticelectric work may be written WP1QS= 14 SVD DdV+SV P DdV ( )or WP2QS= 14 SVE EdV SV E PdV.

9 ( )Both alternatives have, as the first term, total field energies integrals over all space,both inside and outside matter. The second terms are integrals onlyoverV which13 Total magnetic moment per unit electric dipole moment per unit IN SOLIDS7includes just the volume of magnetized (polarized) matter. Since magnetic (electric)thermodynamics is primarily concerned with magnetized (polarized) matter, onepractice is to bravely ignore the total field energies completely. Another is to absorbthe field energies into the internal energyU. But since neither option is entirelysatisfactory a third way is discussed below, in Subsection , these results in terms oflocal average fields are general and thermo-dynamically they are not convenient to apply. Nor are they the fieldsthat appear in a microscopic magnetic (electric) quantum hamiltonian.

10 In quantummagnetic (electric) models the hamiltonians for individual magnetic (electric) mo-ments depend only on thelocalB(localE) fields in which theindividual particlesmove. In the absence of internal currents or inter-particle interactions, this is thesame as theexternal (applied) fieldB0(E0) the field before the sample is the sample is introduced,internal fieldscan additionally result from:a. Interactions between induced moments which are accounted for by additionalterms in the hamiltonian. These interactions may be reducible to effectiveinternal fields [see, for example, Subsection , below],b. Internal demagnetizing fields arising from fictitious surface poles inducedbyB0,16c. Internal currents induced by the applied field (especially in conductors). Models and Uniform FieldsTherefore, in microscopic models magnetic and electric hamiltonians are expressedin terms ofuniform applied fields(B0,E0) presentbeforematter is introduced.