Creator:P.T. Landsberg
Date Created:August 1, 1959
Place Created:Aberdeen, Scotland
Keywords:negative temperatures,thermodynamics
Context:article reprinted from the Physical Review
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PHYSICAL RKVIKW	VUI.II H K 115, N l> M K K K \	AUGUST I. IVS9
Negative Temperatures
P. T. Landsbkrc; Mnristkal College, UmvrtUy, Aberdeen, Scotland (Received Man-h 6, I9S9)
It it shown by an example how the concept of negative temperature finds a natural place in the approach to thermodynamics which was originated by Carathfodory.
1. INTRODUCTION
' I ""HE concept of spin temperature does not so far
X seem to have been discussed from the point of view of Carathfodory's principle in thermodynamics. A brief discussion of" alternative signs for the absolute temperatures of equilibrium systems is, however, contained in an attempt to extend the approach originated by Carath&dory.1 The purpose of this note is to show explicity how a negative temperature finds a natural place in this system of ideas. The results derived here are therefore to some extent contained in, or simple corollaries from, the approach developed in I. But they are difficult to obtain from I without working through the whole paper, so that the present exposition may be of use.
Consider a spin system which is decoupled from the lattice, and whose thermodynamic equilibrium state can be described by specifying the strength B of a uniform applied magnetic field, and the temperature r* —1/7", where T is the absolute temperature. From a statistical mechanical analysis, the phase space may be taken to be the (S,r) plane. Curves of constant magnetic field have then the form shown in Fig. 1. These have been drawn for the high-temperature approximation, when
M~-CHT,	(1)
U-C(B*+B')t,	(2)
S-A-C(W+B>)^/2,	(3)
where A, B, and C are constants. M is the average value of the magnetic moment in the direction of the applied field, and U is the mean internal energy. B is determined by the interaction among the spins. The proof of Eqs. (1) to (3) is straightforward, the main formulas having been given by Abragam and Proctor" (see Appendix for additional derivations).
The above example shows incidentally that the usual definition of an ideal paramagnetic material by M '•ftB/T) is too broad, since this law can also be obtained for interacting spins. The following additional restrictions may be suitable for the ideal paramagnetic material: (i) fl-0 implies M=0, and conversely; (ii) the internal energy in zero field is independent of tem-
> P. T. Landsberg, Revs. Modem Phys. 28. 363 (1956), to be referred to aa I. A more detailed exposition is contained in a forthcoming book on thermodynamics which is being published by Intencience Publishers, Inc., New York.
■A. Abragam and W. G. Proctor, Phyi. Rev. 10», 1441 (1958).
peraturc. With this definition the system specified by Eqs. (1) to (3) is an ideal paramagnetic system only
if fl-0.
2. ANALYSIS OF THE PHASE SPACE
The thermodynamic state of any system must be specified by at least two independent variables (I, p. 365). One may chose T and S (as above), or T and another variable x, say the magnetic field. If one has only one independent variable, * and T cannot be independent. The theory of the thermodynamic equilibrium of such systems should therefore be derivable from the theory of equilibrium with respect to the variable x. For instance, the condition Ti= Tt for thermal equilibrium can then be stated by using only the variable x (i.e., the magnetic field). From this observation one obtains, as a special case, a remark by Abragam and Proctor (reference 2, p. 1444), who note with respect to systems satisfying Curie's Law that conclusions about the magnetic moment of the system can be derived "without bothering about spin temperature at all."
There is therefore little thermodynamic interest in considering a spin system which is subject to quasistatic adiabatic processes only, since this has only one independent variable. It will therefore be assumed that it is possible to link adiabatically the states represented by any two points in the phase space of Fig. 1—unless the contrary can definitely be established for specific points of this space. The concepts of I can then be applied, and one has two independent variables in T and S, or T and B. The question as to the validity of this assumption focusses attention on the search for the irreversible adiabatic processes which may be possible with spin systems. (For example, the above assumption would be invalid if no irreversible adiabatic processes were possible for spin systems.)
The set /J.—Denote by 0 the set of points which has the property that the states represented by any two
Flo. I. Thermodynamic phase space of spin system.
N K G A T I V K T K M I' K K A T IJ K K S
519
points of 0 can be linked adiabatically, and thai no points which can be in the set arc exduded (I, p. 371). Note that points such as R are outside 0. For at infinite absolute temperature the entropy has the value A for all finite fields. R can be approached indefinitely, even in a quasistktic adiabalic manner, bv raising the field, but it cannot be attained with finite fields. Thus no point on the r=0 axis can lie in 0, excepting the point for which S—.4.
The set F(0).—This is the set of points which constitutes the "frontier" of S (I, p. 376), i.e., it contains all boundary points of 0. F(0) consists of the curve H=0, part of the axis r = 0, and we shall regard it as open in the direction of the axis 5=0, since the theory leading to Eqs. (1) to (3) ceases to apply in any case as the low entropy values are approached. The part AR of the r=0 axis, including the point 5—A, belongs to F(0). The assertion that a point lies in F{0) does not mean, of course, that the point belongs to 0. Which of the points of F(0) belong to 0 must be decided by a separate argument. The criterion to be used is that, if a point of F(P) belongs to 0, then the stale which this point represents must be adiabatically linked with the rest of 0. If it does not belong to 0, it Is not adiabatically linked with the rest of 0.
The set y.—One can lake one further step without deciding which points of F(fi) belong to 0. Consider the set of those points of F(fi) which do in fact lie in 0, and remove them from 0. One is then left with two disconnected sets, to be denoted by 71 and 71, which lie respectively to the left and to the right of the axis r™0. Each set is an open set, i.e., it consists of internal points only, because it contains no boundary points. As in I, p.376, if a set 7 is referred to in the sequel, either one of these open connected subsets of 0 is meant. Each point of 71 possesses a neighborhood, every point of which lies also in 71, and the same holds for 72.
J. EXISTENCE OF TWO SUBSETS v: GENERALIZATIONS
The fact that the set 0 contains two sets 7 holds clearly for all systems having a finite number of energy levels [and is not a property of the specific model leading to Eqs. (1) to (3)].1 For such systems it is a fact of statistical mechanics that the entropies at 7' = fl and r —0 are independent of external parameters. However, critical values of such parameters may exist for which the entropy jumps abruptly from one value to another, for example, if a degeneracy is split by a magnetic field at absolute zero. It follows that on a plot such as that of Fig. 1, the set 0 can have only Isolated points [which lie in F(fi) and therefore outside 7] on the t = 0 axis. In an analogous plot of entropy against absolute temperature, the set 0 can have only isolated points on the T=() axis. It follows that whenever positive and negative tern-
' It holds alao for ail infinity uf levels, provided there exist I Kith an upper and a lower hound to the cnergv. See alao N. F. Raniaev, Phya. Rev. 103, 20 (1956).
Flo. 2. Alternative forma of Ihc phue apace. "Reduced" rcpreaenu-tion which doea not dia-tirvuiah between positive and negative temperatures. [ym/f+B', zm
(W+W-1].
peratures have been defined for a set 0, which refers to a system having a finite number of energy levels, that set contains (at least) two disconnected sets 7.
An even more general conclusion of this kind can be derived from thermodynamics. That an integrating factor of d'Q is the absolute temperature T, holds in general only for a set 7, since no troubles can arise in these sets from the existence of boundary points. In such a set it may be possible to approach the hypersurface 7*—0 and t=0 as closely as desired (and it may also be possible to define these sets by a limiting process, as discussed in I, p. 379). However, they cannot be reached in 7, since the mathematical processes wfiich enabled one to define the absolute temperature T as an integrating fact of d'Q, presume that T is finite and nonzero. Hence one can conclude that the sets 7 cannot actually attain or cross the hypersurfaces T=0 and r-0.
i. THERMODYNAMIC CONSIDERATIONS
In the preceding sections we have started with statistical mechanics and hence introduced the thermodynamic phase space together with its simplest topological properties. It is instructive to invert this procedure. Given the thermodynamic properties of the system in the form of Eqs. (1) to (3), what can one say about the limitations of the treatment which gave rise to them?
The second law will be used in the following form: All points ol 7 are i-points. By an t-point is meant a point y of 7 such that every 7-neighborhood of it contains a point which is adiabatically inaccessible from y [I, (All)]. Let this law be applied to Fig. 2(a), assuming B=0. It is clear that the point 0 cannot be adiabatically linked with the points of the set 0. For if it were, one could pass from any point in 7 to all neighboring points, by choosing paths along adiabatics and going through 0. Hence the points of 7 would not be t-points. This argument, which applies to ideal paramagnetic systems, shows that at low temperatures and low field* such systems cannot exist. If Bs*0 it is, of course, clear from Fig. 2fa) that the system cannot exist at low tern-
520	P. T. LANDSBKRG
peratures. A similar argument, applied to the origin of Fig. 2(b), shows that this point is adiabatically unlinked with the rest of the points of y (whether B—0 or Br*0). The removal of one point from the set 0 must mean that physically also the immediate neighborhood of this point does not correspond to true states of the system. The origin of Fig. 2(b) corresponds to infinitely high temperatures and infinitely high fields. Thus thermodynamics suggests that the theoiy of Sec. 1 is either a high-temperature low-field theory, or else a low-temperature high-field theory. The first possibility is in fact the correct one.
It may be argued that at points such as 0 one has physically no means available of chosing one adiabatic rather than another on which to leave 0. In the present method of presenting thermodynamics, however, such questions do not arise, since any curve which lies in a set 0 and consists of adiabatics represents a quasistatic adiabatic linkage of two states. The possibilities of such linkages depends exclusively on the topology of the thermodynamic phase space.
Toahigh-temperature theory the third law of thermodynamics in its conventional form cannot be applied. In a more general formulation, the third law is a statement which asserts which of the boundary points F(Jj) of a set y may be regarded as adiabatically linked with the rest of y, and are therefore to be regarded as also belonging to the set 0. In general, all boundary points F(y) which do not correspond to a state at the absolute zero of temperature can be included in a set 0. By this principle all points of the curve H=0 of Fig. 1 are to be regarded as part of the set 0. This applies also to the state of maximum entropy A. This conclusion is in agreement with the result obtained from the assumption stated at the beginning of Sec. 2.
It is only for a set y that the second law asserts the existence of an entropy and an absolute temperature. In the present example, therefore, it tells us that there exist an entropy Si and an absolute temperature T, for the set Ti, and similarly that there exist functions 5>, Tt for >1. Since the equation i'Q— TdS remains valid under the transformation T-*CT,5-'C-'S, however, there is still some arbitrariness in the definitions. If one set y is considered by itself, for example, the choice of sign is still available, and this gives rise to various types of thermodynamics, as considered in I.
In the present example the signs chosen for T and 5 in ?1 and in yt are also arbitrary from the thermodynamic point of view. It is only if one considers the relation between the two sets that additional restrictions can be brought into play. For instance, one can use the statistical mechanical result that for a given field one can have two physically distinct states of the same entropy. It then becomes very convenient to choose opposite signs for the absolute temperatures in 71 and 71, and to choose the additive constants in the entropy in such a way that the entropies are positive in both y, and y,.
ACKNOWLEDGMENT
I am indebted to Professor M. Fierx for correspondence.
APPENDIX
Using the notation of Abragam and Proctor,' the Hamiltonian of the system is W-Bt+B 1, where
Tr(£r,)-Tr(fl,)-Tr(F^,)-0.
The mean internal energy in the high-temperature ap-- proximation is, with 0m 1/kT,
O-Tt(We-*w)/Tr(e-"r)SSri(W-0W)/TTl.
The mean magnetic moment in the direction of the field is
a=Tr(Ue-")/Ti{f-")srTi(,lf-0tfW)/TTl. Since B,= -MB, where B is the nlagnetic field,
Cat-C(B'+B,)/T, flgzCB/T, (Al)
where
C«Tr(Jf)/*Tr/, fl*-Tr(B,>)/Tr(V).
In the text U and U have been written for 0 and B.
The relation TdS-iU+MdB becomes, upon using B and T as variables,
iS-CT~l(B'+B,)dT-CHT-'dH
Hence the entropy can be taken in the form
S-^-JCr-«(fl»+fl').	(A2)
This yields Eqs. (1) to (3) of the text.