Journal of Magnetism and Magnetic Materials 260 (2003) 146–150

Thermodynamic potentials for simple magnetic systems

Gustavo Castellano* Facultad de Matematica,! Astronom!ıayF!ısica, Universidad Nacional de Cordoba,! Ciudad Universitaria, 5000 Cordoba,! Argentina

Received 29 April 2002; received in revised form 1 July 2002

Abstract

A strong discrepancy of criteria appears in texts on Statistical Mechanics when associating the partition function of simple magnetic systems with thermodynamical potentials. The aim of this work is to provide an adequate description, starting from the maximization of entropy. Finally, a discussion about some thermodynamic properties is given. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 05.70.a

Keywords: Entropy; Free energy; Magnetic moment

1. Introduction relationship between entropy and the probability density operator r# becomes There are many possibilities of expressing the entropy S of a system in thermodynamic equili- S ¼k Trðr# ln r#Þ: brium in terms of the corresponding probability density r: The expression first introduced by Gibbs When dealing with simple magnetic systems, [1] provides adequate results for the thermody- the r# operator acts on spin coordinates; if the namic properties of most systems in equilibrium, eigenfunctions are used for constructing the and has therefore been used almost universally. operators in the previous expression, a simple For a classical system of N particles described by sum has to be assessed. An example for magnetic generalized coordinates and momenta q and p; this systems is given by the well-known one-dimen- expression takes the well-known form sional chain of N nuclei, whose spin coordinates Z can take the eigenvalues si (as long as a suitable S ¼k r lnðCN rÞ d3N p d3N q: basis is chosen). If there is no interaction among the nuclei, but only with an external magnetic The constant CN is introduced in order to obtain induction of magnitude B; the system will be the correct units. For quantum systems, the governed by the following Hamiltonian eigen- values: *Corresponding author. Tel.: +54-351-433-4051; fax: +54- 351-433-4054. XN E-mail address: [email protected] H ¼m Bsi; ð1Þ (G. Castellano). i¼1

0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0304-8853(02)01286-6 G. Castellano / Journal of Magnetism and Magnetic Materials 260 (2003) 146–150 147 where m is the magnetic moment for each nucleus. However, this last association seems to be borne in Thermodynamic properties can be derived from mind when looking for the total magnetization of the corresponding partition function the system, even when f is referred to as the X Helmholtz potential, since the expression always Q ¼ e bH; used is  qf where b ¼ 1=kT; k being the Boltzmann constant M ¼ ; qB and T the temperature of the system; the summa- T;N tion runs over all the possible states of the system. For the present discussion, attention will be which produces the correct result focused on the case of spin-1 particles. The 2 Nm mB partition function will then read M ¼ tanh : ð6Þ 2 2kT  mbB N Q ¼ 2 cosh : ð2Þ 2 On the other hand, most texts state that the internal energy may be computed as the mean value of Eq. (1). This may be written as This typical result appears in many text books on  elementary Statistical Mechanics. Since nuclei are / S qðln QÞ independent, the overall partition function was U  H ¼ ; ð7Þ qb B;N expected to read as the product of single summa- tions q for each spin (Q ¼ qN ). Usually, the partition function is associated with some thermo- which for the present example gives dynamic potential f by means of NmB mB U ¼ tanh : ð8Þ 2 2kT f ¼kT ln Q: ð3Þ In addition, if the association of Eq. (3) with the Although the example chosen appears to be very Gibbs potential is always borne in mind, an simple, no agreement is shown in the literature expression for the entropy can be derived about which thermodynamic potential corre-  sponds to Eq. (3). For instance, some authors [2– qf S ¼ ; 4] suggest that f is the Helmholtz free energy, i.e., qT B;N the Legendre transformation in the internal energy U which changes S by T: obtaining for this system f ¼ U TS; ð4Þ  mB NmB mB S ¼ kN ln 2 cosh tanh : ð9Þ whereas others [5,6] take it as the Gibbs free 2kT 2T 2kT energy, which means a further transformation At this point, an apparent contradiction arises, involving the substitution of the total magnetiza- since by simply adding U TS BM the Gibbs tion M by the external field B: potential provided by Eq. (3) is not reobtained. Although it may be thought that the error is the f ¼ U TS BM: ð5Þ acceptance of f as the Gibbs free energy, the reader may verify that this and even more contra- Obviously, the discrepancy is not just a matter dictions arise if f from Eq. (3) is taken as the of ‘‘names’’ for the potential in Eq. (3), since the Helmholtz free energy. description of the thermodynamical properties of Unfortunately, text books on Statistical Me- our system will not be the same in the two cases. chanics pay little attention to this controversy, 148 G. Castellano / Journal of Magnetism and Magnetic Materials 260 (2003) 146–150 disencouraging an adequate comprehension of the of thermodynamics, fundamentals of the subject. This is not a minor dU ¼ T dS þ oW þ z dN; ð10Þ issue, since the identification of the partition function with a thermodynamic potential must where oW must represent the external work done hold not only for the case of non-interacting spins. on the system, z being the chemical potential (in In the next section, some basic steps are reviewed this expression we assume there are no further in order to achieve the correct associations with thermodynamical parameters to describe our thermodynamical properties. system). The external work of an external field B which produces a variation dM on a system of total magnetization M is given by B dM—the 2. Maximization of entropy and connection with energy for creating the field may be excluded from thermodynamics our system, since it has nothing to do with the thermodynamic properties in which we are inter- In order to avoid misinterpretations, the expres- ested (a very good explanation is detailed in sions for the probability distribution of the system Appendix B of Ref. [2]). Although agreement chosen will be rederived by means of extremizing about this point is very well accepted by all the the entropy function, adequately imposing the literature (see e.g., Ref. [7]), oW is not always corresponding constraints. adequately written when analyzing thermody- Since the nuclei in our system are independent, namic properties of a magnetic system. However, attention will be focused on one single spin. The once the system has clearly been defined—in our joint probability for the whole system will be example, the set of nuclei—the expression for dW obtained by simply multiplying the probabilities is not arbitrary at all. (It is worth noting that the for all separate nuclei. If the eigenvectors of the induction field includes in fact ‘‘internal’’ effects, spin operator are taken as the basis for describing since it accounts both for the external magnetic spin states, PðiÞ represents the probability that our field H and the field due to magnetization of the 71 system under consideration. The expression ex- spin be with quantum number si ð¼ 2Þ: The normalization condition for these prob- ternal induction field refers to the exclusion of this abilities provides the first constraint that must be magnetization term. Referring to the constitutive used relation between induction and magnetic fields, the X external induction is m H; m being the magnetic PðiÞ¼1: 0 0 permeability of vacuum, and not H as often The mean value of the interaction Hamiltonian of considered. In addition, if oW is taken as H dM; Eq. (1) might also be chosen as a constraint. Or inconsistencies in units may arise). The Euler equivalently, since the external magnetic field relation for a magnetic system thus reads remains constant, the total magnetization M must U ¼ TS þ BM þ zN: ð11Þ be regarded as a mean value. In the present case, the individual magnetization Mi will be taken as When a system is under the action of an external the second constraint field which is held constant, it is useful to introduce the enthalpy, defined as the difference M ¼ m /s S: i i of the internal energy and a term which accounts Now attention will be focused on the internal for the coupling with the outside world. In the case energy of our system. At first sight, it looks natural of magnetic systems, it usually referred to as to have the internal energy as BM; i.e., to accept ‘‘magnetic enthalpy’’ the association of Eq. (7) as valid. As mentioned E ¼ U BM: above, many texts take this association as obvious, although it is in disagreement with first principles. This is precisely the magnitude associated with the In order to avoid misleading ideas, it is worth mean value of the Hamiltonian of Eq. (1): no starting from the combined first and second laws matter the configuration of the system, the field B G. Castellano / Journal of Magnetism and Magnetic Materials 260 (2003) 146–150 149 will remain constant (see e.g., Ref. [8]). Although related to the enthalpy BM if the identification the magnetic enthalpy may not always be regarded aM ¼B=T is allowed. Then, for the whole as a Legendre transform of the internal energy (the system this may be written as necessary conditions are detailed in Ref. [2, p. kT ln Q ¼ U TS BM; ð13Þ 142]), its physical meaning is clear: it represents the internal energy of a new system, which also includes confirming the identification with the Gibbs the sources of the field B: Now, if we make the potential, as suggested in Eq. (5), is the correct adequate identification of the mean value given by one. As stated above, when some kind of interac- Eq. (7), we have E ¼BM; and therefore the tion among spins is present, U will not vanish, but internal energy will vanish for all temperatures and Eq. (13) will still be valid. magnitudes of B: We will return to this point later. Before carrying on, it is worth noting that if any interaction among the spins is included in the 3. Discussion Hamiltonian of Eq. (1), all the arguments stated above will hold: the only difference in the analyisis Now that the partition function has adequately is that the mean value of this new Hamiltonian will been associated with Gibbs free energy, we should not look like BM; and an additional constraint be able to verify some thermodynamic properties should appear in the next steps. for our system. First, we will return to Eq. (13): the The set of probabilities which extremize the term U in the right-hand side has been deliberately entropy restricted by the constraints mentioned introduced here in order to emphasize the fact that above can now be obtained through the usual the internal energy vanishes for this system. Is this procedure, using Lagrange multipliers. These a reasonable idea? Unlike other physical situa- multipliers will be called a0 and aM for the tions, this system is built up by elements which normalization condition and the individual mag- interact with an external field but without ‘‘ab- netization, respectively; the extremum condition sorbing’’ this interaction energy for themselves. A then reads comparison may be done with an ideal gas, i.e., a no X system of non-interacting point mass particles ½ð Þ ð Þ ð Þ ð Þ ¼ d a0P i kP i ln P i aM P i si 0: within a rigid cylinder with a free piston at one Now, since all infinitesimal variations dPðiÞ must end: the external field is the pressure which be independent, this condition implies performs mechanical work on the system when volume changes occur. The particles absorb this ½a0 k k ln PðiÞaM si dPðiÞ¼0: work, converting it into kinetic energy, which will Since this equation must hold for any dPðiÞ; the not disappear when the external pressure vanishes. bracket on the left-hand member must vanish: If the reader is not acquainted with this example, he is invited to carry out the maximization of the a k k ln PðiÞa s ¼ 0: ð12Þ 0 M i entropy bearing in mind that instead of the If this equation is multiplied by PðiÞ and summa- volume, it is the pressure that is held constant, tion over i is performed, the identification of the which again implies the correspondence of the partition function per nucleus as partition function and the Gibbs free energy. The X usual form of the Hamiltonian for free particles q ¼ eaM msi=k ¼ e1a0=k takes into account the energy which is effectively arises naturally. Eq. (12) thus becomes absorbed by the particles, which allows a better comprehension of the association of kT ln Q with kT ln q ¼ a TM TS ; M i i the Gibbs free energy. In our spin system, however, Si being the entropy per nucleus. Following the when the magnetic field disappears, no energy can usual arguments (see e.g., Ref. [5]), in this be maintained in the nuclei. Thus, the fact that the equation, the right member can be associated with internal energy vanishes is indeed a reasonable idea; a thermodynamic potential, since the first term is however, Eq. (11) may not be considered as a 150 G. Castellano / Journal of Magnetism and Magnetic Materials 260 (2003) 146–150

‘‘fundamental equation’’, since not all partial ðqU=qTÞM;N ; enhancing the conclusions drawn derivatives of U make sense and therefore, it is above. An alternative way to see that CM should not possible to draw all thermodynamic properties vanish is to recall that heat capacities must be of the system from this equation alone [9]. related by  A further comment can be stated about this null qM qB C ¼ C þ T internal energy. From Eq. (9), the heat capacities can M H qT qT be assessed; they provide an idea about the B;N M;N qM=qT 2 capability of our spin system for absorbing energy B;N ¼ CH T : by heat transfer. In the case of the heat capacity at qM=qB constant field H (or constant external induction B), a T;N simple derivative produces the well-known result The right-hand member of this expression can be  2 computed using Eqs. (14) and (6), reobtaining a qS ðmBÞ N 2 mB null value for CM : CH ¼ T ¼ 2 sech : ð14Þ qT B;N 4kT 2kT The capability of absorbing heat from a thermal 4. Conclusion reservoir when the field is held constant obviously depends on the source of B; which must provide Care has been taken in the derivation of energy so that the system changes its magnetiza- thermodynamical properties of a non-interacting tion when changes of temperature occur. This 1 spin-2 system from basic Statistical Mechanics. A would suggest that no energy is necessarily correct association of the partition function with absorbed by the system when heat interchanges Gibbs free energy has been achieved, reobtaining occur, but it may be transferred to the sources of well-known formulae for the entropy S and the B: In fact, this can be checked by simply applying total magnetization M: On the other hand, the Eq. (10) for a process in which B and N are improper identification of the internal energy U maintained constant; the variation of U with T with the mean value of the interaction Hamilto- must therefore obey the well-known identity   nian has been overcome, replacing it by the qU qM magnetic enthalpy. It has been shown that in fact, ¼ CH þ B : these associations of thermodynamic potentials are qT B;N qT B;N valid also for magnetic systems with non-null The right-hand member of this equation vanishes, interaction among spins. as can be verified easily by combining Eqs. (6) and (14). This precisely confirms the fact that, although CH is not zero, the system does not absorb energy References when heat interchanges occur at constant B: On the other hand, an expression for the heat capacity [1] J. Gibbs, Elementary Principles in Statistical Mechanics, CM at constant magnetization may also be Dover, New York, 1960. derived. With M given by Eq. (6), the expression [2] H. Callen, Thermodynamics and an Introduction to for S of Eq. (9) may be rewritten in terms of T; M; Thermostatistics, 2nd Edition, Wiley, New York, 1985. [3] K. Huang, Statistical Mechanics, 2nd Edition, Wiley, New and N; obtaining "# York, 1987. kN 4 [4] R. Kubo, Statistical Mechanics, North-Holland, SðT; M; NÞ¼ ln Amsterdam, 1981. 2 1 ð2M=mNÞ2 [5] L. Reichl, A Modern Course in Statistical Physics, University of Texas Press, Austin, 1980. 2kM 2M arctanh : [6] H. Stanley, Introduction to Phase Transitions and Critical m mN Phenomena, Oxford University Press, New York, 1971. [7] J. Jackson, Classical Electrodynamics, 2nd Edition, Wiley, Since S is not explicitly dependent on T; the heat New York, 1975. capacity at constant magnetization will be identi- [8] N. Ramsey, Phys. Rev. 103 (1956) 27. cally zero: this coincides with the fact that CM ¼ [9] Y. Zimmels, Phys. Rev. E 52 (1995) 1452.