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Chapter10 Thermodynamic Aspects of Magnetism

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Chapter10 Thermodynamic Aspects of Magnetism CHAPTER10 THERMODYNAMIC ASPECTS OF MAGNETISM (this chapter deals with fundamentals, and may be skimmed over during a first reading) This chapter applies thermodynamic methods to the analysis of magnetism in solids, and provides a brief introduction to critica/ phenomena. 1. BASIC THERMODYNAMICS: OUTSIDE MAGNETISM We first recall some results on equilibrium thermodynamics in simple systems, in particular in the absence of an applied magnetic field. The presentation is based on the work by Callen [1], to which the reader is invited to refer both for a refresher and for derivations that are not presented here. Classical thermodynamics deals with macroscopic systems at equilibrium, and describes the changes in characteristic quantities between equilibrium states. The simplest example is that of an isolated system made up of N particles. The interna! energy U of this system, given by the sum of the kinetic and potential energies of the component particles, is then a constant, and its equilibrium state is described by the condition of maximum entropy S. A small change around this equilibrium state leads, for a single component fluid, to a change in the interna! energy: dU= TdS -pdV + ~dN (10.1) For a mixture, it is necessary to introduce the sum ~j dNj over the various chemical species j. In a solid, the scalar quantities p and V have tobe replaced by the stress and strain, integrated over the volume of the system. The equilibrium state is quantitatively characterized by a small number of macroscopic variables, called the state variables. The pressure p, the temperature (absolute) T, and the chemi cal potentialjlj are intensive· state variables. Their uniformity characterizes mechanical, thermal and diffusion equilibria, respectively. These variables appear, in nearly ali thermodynamic equations, together with "conjugate" extensive variables: the 322 MAGNETISM - FUNDAMENTALS volume V, the entropy S and the number of particles or moles Nj (depending on the the definition chosen for the chemical potential). The fact that the differential of U involves the differentials of S, V a.nd Nj implies that these are the natural variables for U. One can also formally consider the condition of equilibrium of an isolated system as a minimum of U, given the entropy, volume and number of particles. Thus U plays the role of a thermodynamic potential, just as the condition for equilibrium in a mechanical system is described by the minimum in its potential energy. A system forced to remain at constant temperature T is no longer isolated. Its equilibrium state is then given by the minimum, consistent with the imposed constraints, of the free energy F = U - TS, the differential of which can be written: dF = -SdT- pdV + LJljdN j (10.2) j The natural variables for F are T, V and Nj- When temperature and pressure are the experimentally controlled variables, free enthalpy, or the Gibbs energy G =U- TS + p V, is involved. This bas the differential: dG = -SdT + Vdp + LJljdNj (10.3) j The natural variables are then T, p and Nj- Changing from one thermodynamic potential to another one is a simple procedure. The Legendre transformations, as from U to For to G, correspond to the change in the physical variables that are experimentally controlled. As U(S, V, Nj), F(T, V, Nj) and G(T, p, Nj) are state functions, their mixed second derivatives are necessarily equal, i.e. (10.4) from which: = _ S Or (dF) and ( ~~ )T N. = - p, dT V • N·J ' • J which leads to: (10.5) These thermodynamic identities are called Maxwell relations. We note that the variables involved (T, V, Nj in this example) appear explicitly at the bottom of these equations. They immediately indicate which thermodynamic function can lead to useful relations. The useful thermodynamic potential is that for which the natural 1O - THERMODYNAMIC ASPECTS OF MAGNETISM 323 variables are the relevant, i.e. the controlled variables, in the problem. In the above example, it is therefore F(T, V, Nj ). Analysis of the stability conditions of a phase at equilibrium leads to the derivation of various inequalities between the response coefficients, such as: Cp C (as 1 aT) p' N j = T or (as 1 aT) V' N j = Tv Cp and Cv are the specific heats, or beat capacities, at constant volume and pressure, respectively. We note that these coefficients concern pairs of conjugate variables. It can be shown that Cp and Cv are always positive, and that Cp > Cv. 2. THERMODYNAMIC POTENTIALS FOR AN UNDEFORMABLE MAGNETIC SYSTEM {1, 2} In the presence of magnetic fields Ho, H, Bo, B, and of magnetization M, the situation becomes more complicated. Furthermore, we are usually interested in solids, hence p and V are no longer sufficient to describe the elastic effects. W e therefore introduce, in section 4, the stress and strain tensors, each one involving up to 6 independent components. For the moment, let us consider the case of an undeformable solid, i.e. the effects of the thermal variables (S, T) and of magnetic variables (M, Ho) only. We begin with the expression for the work associated with an elementary change. Legendre transformations will then provide the whole range of thermodynamic potentials. The equation for the magnetic work that we will use as a start is: (10.6) where J.1<J represents the magnetic permeability of vacuum, not a chemi cal potential. The first term corresponds to the magnetostatic energy of the applied field stored over ali space. It can often be omitted, which amounts to choosing a reference state where the applied field would exist, but where the sample would initially be in the state corresponding to zero field. The second term describes the effect of the magnetic field on a sample with magnetization M. The integration is automatically limited to the sample's volume because there is no magnetization outside the sample. This expression may be simplified if the applied field is uniform. The second term then becomes: Ilo Ho dm, where m is the total magnetic moment of the sample: m = JMdV, or again: f>W mag = Ilo Ho dm11. Here, m11 depicts the component of the total magnetic moment parallel to the applied field. 324 MAGNETISM - FUNDAMENTALS Introducing this expression into the interna! energy leads to a variety of thermo­ dynamic functions, with various pairs of natural variables: • with the natural variables S and m11, a thermodynamic potential U(S, m11) such that: dU(S, m11) = T dS + Ilo Ho dm11 (10.7) • with T and m11, a thermodynamic potential F(T, m11) such that: dF(T, m11) = - S dT + Jlo H0 dm11 (10.8) Use of a Legendre transformation then leads to functions: • of S and H0, thus a thermodynamic potential U(S, Ho) such that: dU(S, H0) = T dS- Jlo m11 dHo (10.9) • or ofT and Ho. a thermodynamic potential F(T, Ho) such that: dF(T, Ho) = - S dT-Ilo m11 dHo (10.10) The relation F ( T, H 0 ) = F ( T, M) - Jl o JH oMdV describes the conversion from F(T, M) to F(T, H 0). This is in fact exactly the same procedure as that used in section 5.1.1 of chapter 5 when analysing the most favourable situation in terms of magnetic domains. The free energy F(T, M) contains anisotropy, exchange and demagnetising field terms; the term -Jlo JHoMdV is the Zeeman term for the interaction with the applied field (symbolised by EH in chap. 5, § 7.1.2). Equilibrium corresponds to the minimum of F(T, Ho), with respect to the interna! degrees of freedom. Here this corresponds to the distribution within the sample of M, which in practice is its orientation. We have chosen not to give a name to the many thermodynamic functions that can be devised. The important piece of information is the indication of the natural variables which follows the symbol U or F, i.e. their arguments. While the above thermodynamic functions involving Ho are of particular interest because the applied field is frequently an experimentally controlled variable, they are not the only ones of interest. The variable that is physically most meaningful is the macroscopic mean field, or induction, B. It can be shown that the elementary work can also be expressed in terms of B, with the total field: H = Ho + Hdem• as the conjugate variable. Here Hdem designates the field created, both inside the sample, and outside (we sometimes refer to the dispersion field, or stray field), by the distribution of magnetization in the specimen: dWmag = JHdBdV = jHdB11dV) (10.11) Neither B nor H are extensive variables, and an integration over ali space is necessary as they are not limited to the interior of the sample. 1O - THERMODYNAMIC ASPECTS OF MAGNETISM 325 Tbese same procedures can lead to tbe faur state functions: U (S, B11), F (T, B11), U (S, H) and F (T, H). Their respective differentials are written: dU( S, B11) = TdS + f HdB11dV (10.12) dF(T,B11) = -SdT + JHctB11dV (10.13) dU(S, H) = TdS- f B11dHdV (10.14) dF ( T, H) = - SdT - f B 11 dHdV (10.15) 3. MAXWELL RELATIONS AND INEQUALITIES The eigbt functions defined above provide, via the Maxwell relations wbicb generalise equation (10.4), pbysically useful expressions for the beat involved, or the variation in temperature, associated with the application of a magnetic field. Thus, the iso-entropic cbange in temperature can be described in terms of a magnetotbermal coefficient, wbicb can be deduced from equation (10.9) associated with U(S, H0): _QI_) =-J.to(am/1) =-J.to(am/1) (aT) =_J.toT(am11) 00.16) (aHo s as Ho aT Ho as Ho CHo aT Ho because (aT 1aS)Ho = T 1CHo• wbere CHo = T(aS 1aT)Ho is tbe beat capacity at constant applied field*.
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