Thermodynamic aspects of itinerant electron magnetism

305 THERMODYNAMIC ASPECTS OF ITINERANT ELECTRON MAGNETISM

E.P. W O H L F A R T H Department of Mathematics, Imperial College, London SW7, U.K.

The magnetic, thermal and magnetoelastic properties of strong and weak ferromagnets are discussed from the point of view of elementary thermodynamics. This is shown to provide a valuable link between the empirical evidence and quantum mechanics. It is concluded on this basis that strong itinerant ferromagnets, in particular nickel, have properties below the Currie point governed mainly by spin waves. For weak itinerant ferromagnets the evidence for the dominance of single particle excitations is very much stronger. A critical index ~- relating T~ with the Stoner factor as ( I - 1 y is shown from the evidence to be close to '.

1. Introduction

The classical work of Herring [1] is concerned with exchange interactions a m o n g itinerant electrons but, although this work is one of over 400 pages, it deliberately avoids detailed discussion of finite t e m p e r a t u r e effects nor is there any but passing reference to the experimental basis of the subject. No c o m p a r a b l y weighty discussions of these matters have yet been given although there have been a few brief reviews [2]. On the occasion of the first international conference on itinerant electron magnetism it appears appropriate to review these matters again. It will be shown that the t h e r m o d y n a m i c s and the empiricism of the subject are very closely interrelated. T h e r m o d y n a m i c s and the concomitant statistical mechanics deal with real life variables such as temperature, pressure, shear stress, alloy concentration, magnetic field strength etc. and with their influence on magnetization, susceptibility, specific heat, thermal expansion, elastic constants etc. At worst an intelligent application of the elementary principles of t h e r m o d y n a m i c s leads to a check of the consistency of sometimes quite disparate experimental data. Those who apply this perfectly reasonable procedure to the analysis of observations are often malevolently accused of merely proving that t h e r m o d y n a m i c s works. Where else is it even conceivable that the continuing verification of a complex philosophy on the basis of empiricism serves to convict the philosopher of the sin of naivet6? At best, the results of a t h e r m o d y n a m i c approach are model dependent and this serves as a link between the observations and a clear decision as to which rival model is applicable to a Physica 91B (1977) 305-314 © North-Holland

real material. This a c h i e v e m e n t is clearly one of immense value and no one can reasonably regard it with malevolence. H o w e v e r , the number of occasions where this decision is fairly clear cut is quite small in the field of itinerant electron magnetism. Among a n u m b e r of reasons for this is the unfortunate fact that fitting experiments by means of t e m p e r a t u r e laws can very frequently be misleading. For example, the low t e m p e r a t u r e magnetization of an itinerant metal is very often fitted by two terms going a s T 312 and T 2, where it may be clear that either higher order terms may also be important or that a fit with a few terms is unreliable anyway. The most diabolical example of a frequently misleading t e m p e r a t u r e fit is the slough of despond known as the C u r i e - W e i s s law [3]. A third example concerns the T 3 In T specific heat of paramagnons which was quoted ex cathedra as providing the explanation of data for N i - R h alloys only for the o b s e r v e d anomalies to be subsequently ascribed to clustering effects [4]. H e n c e it is clear that cases where a fairly unambiguous decision as to the appropriate model can be made on the basis of t h e r m o d y n a m i c s are rather rare but are thus all the more precious where they do occur. Such cases, it is claimed here, do exist and will be discussed. H o w e v e r , t h e r m o d y n a m i c s and statistical mechanics stand in their own right as supplementing in a fully satisfying description of the natural p h e n o m e n a concerning itinerant electron magnetism not only the empiricism but also the quantum mechanics of the single particle and m a n y body effects, the elementary excitations, the phase transitions etc. The richness of the subject and the continuing a p p e a r a n c e of new ideas is clear from the great interest which this

306 C o n f e r e n c e has aroused, and t h e r m o d y n a m i c s has helped to create this richness and these ideas. 2. Effects of temperature magnetic transition metals

and

field:

n* = ~n(1 + ~') = £~ N ( E ) d E [ e x p { ( E - ~ + s--n l1r . I

(l)

where n is the n u m b e r of particles per atom, n ~ the n u m b e r with + spin, ~ the relative magnetization, N(E) the density of single particle states per atom per spin, # the chemical potential and I the effective interaction between the itinerant electrons which may be short range intraatomic Coulomb corrected for correlation. Even these physically simple equations are difficult computationally as sr occurs on both sides, but they have been solved for a number of N(E) curves [6]. It is found that curves of ,~(T) are rather insensitive to N(E) and have a close r e s e m b l a n c e to the o b s e r v e d curves of the ferromagnetic transition metals. The Curie temperature, given by ~--I

d E = 1,

X

2"~f[N(E)]

af(T)aE dE/

Ferro-

The most straightforward model for describing the effects of t e m p e r a t u r e on the properties of itinerant electrons is the Stoner model [5] in its archetypal form i.e. where only the single particle excitations are involved. Later developments are concerned with the temperature effects arising from collective excitations such as spin waves. To some approximation and in some t e m p e r a t u r e regimes these two effects are additive. Hence the two types of excitations may usefully be considered separately. The Stoner equations read

++-tx~H)/kT}+ 1]

ceptibility above T~, given by

(2)

where f is the Fermi function, has been found to have the tendency of being larger than observed for these metals when using calculated values of I [7]. This continues to point to the possible importance of spin waves in determining T~ for strong ferromagnets. The qualitative agreement derived from (1) for ~'(T) below T~ is thus also inconclusive. The paramagnetic sus-

although again in qualitative agreement with observation, can equally not be used to decide on the best description of the ferromagnetic transition metals, since a straight Curie-Weiss law does equally well [81. Such a Curie-Weiss law may arise for several reasons, e.g. from strictly localized electrons or the spin fluctuation effects of Moriya and K a w a b a t a and others [9]. One important result emerges if the Curie t e m p e r a t u r e is not determined by the exchange splitting nl~'(0), i.e. by the saturation magnetization ~(0) itself. Under pressure, which we will describe later as a very relewmt therm o d y n a m i c variable indeed, T~. would thus tend to vary differently than ((0), thus providing some evidence for the origin of T,. itself. Spin waves excitations lead to a low temperature d e p e n d e n c e of the relative magnetization of the form

= 1 - c(kT/D) ~l'~,

(4)

where c is a constant, and D is the spin wave stiffness constant calculated on the basis of the itinerant electron model. Whereas this dependence has been clearly observed at low temperatures [10] there is no clear experimental evidence regarding the t e m p e r a t u r e dependence of the correction terms which may arise, for example, from an application of the Stoner equations (T 2,T~e A/kr...) or from the temperature d e p e n d e n c e of D(T'-, T~I~...). The low temperature specific heat of the ferromagnetic transition metals [11] has three distinct contributions going as T, T 3;2 and T ~. The first follows from single particle excitations but it is not clear what part of this contribution is of normal non-ferromagnetic origins and what part is " m a g n e t i c " , in the sense of Stoner [5]. One analysis, for nickel-rich N i - P t alloys [12], seems to point to such a magnetic contribution for pure nickel. If this is substantiated it would be valuable t h e r m o d y n a m i c evidence for the importance of single particle excitations in a strong ferromagnet at low temperatures. The spin wave term, going a s T 312, is intrinsically

307 small and not easy to observe [13]; as already stated, there is no doubt of the presence of the analoguous term in the magnetization. The thermodynamic evidence regarding the properties of the ferromagnetic transition metals at finite temperatures is thus unhappily not very conclusive. The basic properties, the magnetization and susceptibility as a function of temperature, follow in their general behaviour the predictions coming from either single particle or spin wave excitations. The last are clearly present as evidenced by the T 312 term in the low temperature magnetization, while a linear magnetic term in the specific heat would point to a single particle effect. The relative importance of these excitations near the Curie temperature is still not clear so that the validity of (2) in determining Tc itself is also debateable for the ferromagnetic transition metals. For weak itinerant ferromagnets the situation seems rather clearer, although even this has been regarded as arguable.

Here the Curie temperature is now taken to be given by (2) which becomes T~

= T ~ ( [ - 1),

(6)

= IN(EF)

(7)

where [

and

r/

I ". 2 [ [ N ( I ) ( E F ) \ 2

N(2)(EF)]

(8)

Also Xo

= I . ~ 2 N ( E F ) / ( [ - 1)

(9)

is the high field susceptibility at 0 K. The saturation magnetization ~r(0) is given by ~(0) 2 = 3 ( • - l)/c, where N(2)(EF)" 8\N(EF)/

l_ \ N ( E F ) /

N(EF)

3o) 3. Effect of temperature itinerant ferromagnets.

and

field:

Weak

If the relative magnetization at 0 K is small compared to I then relations (1) may be expanded in powers of ~" and only a few terms in this expansion regarded as sufficient. The resulting equation for M ( H , T), the magnetization as a function of temperature and field [14[, may be related to experiments with a view to assessing whether the single particle excitations are dominant in determining the magnetic and thermal properties. There are indeed on the face of it good reasons for expecting such dominance since the low relative magnetization leads to a low exchange splitting and thus a smaller phase space for spin waves. On the other hand, arguments continue to be put forward that even for weak itinerant ferromagnets spin fluctuation effects are important [15]. The paper of Moriya given at this Conference summarizes the theoretical background and the evidence [16]. The relationship determining M ( H , T ) on the basis of (l) is [14]

[,M,., T)I M(O,O) J

-

?,(..

r)]i-1 _ 1 LM-~,~JL ~]

2XoH M(O,O)"

(5)

The spin wave stiffness D, defined by (4), is also proportional to ( [ - 1) in, so that there is a very simple relation between the 4 observables ,~(0), Tc, X0 and D to which we will return below. The implications of (5)-(10) are so simple that it is in principle very easy to assess the validity of the model on the basis of the relevant experimental data. The challenge is then to make a similarly detailed experimental comparison with the recent more complicated models [9, 15, 16]. One such comparison is concerned with the relation between T~ and ( [ - 1). Writing this as T¢ - ( [ - 1)~, relation (6) gives 7 = ~, Moriya and Kawabata [9] give r = 43 and Murata and Doniach and Yamada [9] give T = I . If Tc is not determined by [ at all but, for example, by independent spin wave effects one would have, formally, ¢ = 0. Another comparison will have to be concerned with the susceptibility above T¢ which (5) shows to be given by x ( T ) = 2Xo/(T2/T~ - 1)

(11)

or an extension thereof if higher order terms ( T 4. . . . ) are important. As discussed in [3] this law is not always obeyed although not too much should be made of this. Examples confronting each other are ZrZn2 where [3] )¢(T) obeys a

308 Curie-Weiss law (though with a constant given by (1 I) at T,., i.e. xoT~,!) and Ni3A1 [181 where the quadratic law applies well. More generally, the results (5) shows that Arrott plots, giving M ( H , T ) 2 as a function of H / M ( H , T ) , are straight lines with slopes F =I/B

= 2x, M(0,0) 2

(12)

and intercepts G = -A/B

= M(O, 0)2(1

T2/T~).

(13)

Here A and B are the Landau coefficients which will be discussed below. Higher order terms in F, i.e. in B, have been obtained [17, 18] and used to discuss the temperature dependence of G. The occurrence of such terms is completely natural and their inclusion gives good agreement with experiment for ZrZn2 [19] and N i - P t [20]. The temperature dependence of the saturation magnetization M(0, T), in particular, is given by (13) and contains the characteristic T 2 term to which may again be added a T 4 term if higher orders have to be included. For Ni3AI, another weak itinerant ferromagnet, a very good fit to a T 2 law alone was observed over a wide range of temperatures [18]. However, these same authors also found a good fit of the observed data for Ni3Al by initially fixing the coefficient of T 3 from the value of TF, defined by (9), obtained from paramagnetic Ni3A1 alloys with lower nickel contents and supplementing this T e term with the spin wave contributions going as T 3/-" and T v2. The coefficient of (TITs) 3/2 so obtained was found to be close to that for iron and nickel, and the coefficient of ( T I T s ) ~/2 actually considerably higher. In view of the warning given in the Introduction not too much significance should be read into these results. However, yet another alloy system, the F e - N i invar alloys, has also been investigated from the point of view of low temperature fits of the magnetization [21]. Again, the data were found to fit a T 2 + T 3/2 term, the coefficient of the former being both important [21] and reasonable from the present point of view [22]. Hence, it must be concluded that there is a fundamental difference between strong or almost strong itinerant ferromagnets like nickel and iron, and weak itinerant ferromagnets. The latter seem, on the basis of considerable experimental evidence, to have clear cut T 2 terms in the sa-

turation magnetization while such evidence is absent for the former materials. Other influences on Arrott plots have also been considered. It has to be stressed that this type of representation of magnetic isotherms is particularly sensitive to even slight perturbations. For example, the magnetization may not be uniform in space and the resulting dependence of M ( r ) has been found to influence the shape of Arrott plots in a characteristic manner [231. These enable metallurgical and other fluctuations of M to be in principle assessed from the deviations of Arrott plots from the linear dependence given by (5). One particularly important type of fluctuation of M is that which arises near the Curie point where it may indeed be bigger than M itself. Where this Ginzburg criterion applies the mean field basis of the itinerant model breaks down and the magnetic properties are determined on the basis of models where these fluctuations are the main feature. The limits of applicability of the mean field model have been calculated for weak itinerant ferromagnets [24]. In the M2, H / M plane the straight line (14)

2 B M 2 = X,, i _ H / M

and the coordinate axes define a triangle within which the model breaks down. Here B is given by (12) and X~ is a critical susceptibility related to the Ginzburg criterion e, which is determined in terms of the band structure, through the relation X,, = Xo/2E.

( 15 )

The proper results within this triangle must be obtained on the basis of the modern theory of phase transitions. However, for the weak itinerant ferromagnets normally investigated, T~ and thus the temperature range of breakdown eT,, are usually so small that there has not yet appeared any clear evidence for the expected critical phenomena such as a susceptibility, temperature relation involving a critical index y > I. When the mean field model continues to apply it is natural to use (5) to calculate the free energy, thus forming the basis of the Landau theory of phase transitions. It is found that 12,141. (16)

F = ½AM2+~BM 4- HM,

where

M=M(H,T)

and

the

Landau

co-

309 efficients are given by (12) and (13). The free energy is measured from a zero at M = 0 (paramagnetic state) and the spatially dependent terms discussed in [23] are omitted, corresponding to a uniform M. Although (16) is excessively simple this is partly deceptive. This expression is usually taken to be applicable near the Curie point where there is, however, the danger of breakdown due to critical phenomena. For weak itinerant ferromagnets the magnetization M is always small (compared to nNIxB, where N is the number of atoms per unit volume and n the number of particles per atom); equivalently, Tc'~TF, given by (8). Hence the Landau theory holds at all temperatures (outside ETc about To) as long as T/TF'~ 1, including the lowest. A system of real life materials whose free energy is as simple as (16) is not c o m m o n and should be hence all the more cherished. Deductions from (16) are firstly those relating to the magnetic properties already summarized, and secondly those relating to thermal properties. The last were fully discussed in [14] so that only a summary need be given here. One result concerns the discontinuity of the specific heat at Tc, recalling the result stated earlier that the expected critical phenomena have not yet been seen for these materials. It follows that M(0, 0) 2 A C - - -

st(0)3 - T 3,

(17)

XoTc from (6), (9) and (10), so that a very small discontinuity indeed is predicted [5]. For a homogeneous specimen of ZrZn2 it was found [25] that ~ C is as tiny as 420 mJ/grm, tool. K, compared to the value 280 derived from (17). "Considering the extreme sensitivity of X0 to sample condition, the agreement is considered to be good" [25]. To generalize, the magnetic contribution to the specific heat over the whole temperature range 0 < T < To, still defined relative to the state M = 0, is given by

Cm

=

- -

M(O, O)2T(T~ - 3TZ)/2XoT~,

(18)

so that the corresponding entropy change from 0 to T~ is zero. In [25] the measured entropy change for ZrZn2 is given as "0.02R ln2". Finally, using (18) again, the magnetic specific heat at

low temperatures is given by Cm = ~mT,

where M(0, 0) 2 Ym-

2X07~

~r(0)2 - 1~,

(19)

giving "Ym 6.4mJ/grm. mol. K 2 for the above ZrZn2 specimen. Measurements for a single compound like this are inconclusive since the negative Ym is of course supplemented by a stabilising positive linear electronic specific heat Ye which it is difficult to disentangle. For a whole alloy system the negative effect is, however, more easily discernable and the measurements on N i - P t alloys [26] which give a peak in the total y = Ye + Ym at the critical concentration for ferromagnetism were analysed using (19) [12]. A similar analysis should hold for some other alloy systems (Ni-Rh, NiCr . . . . ). All results obtained for weak itinerant ferromagnets vary characteristically with composition in an alloy series. These simple though powerful results are again amenable to experimental verification. As far as M(0,0) [or equivalently ~'(0)], Tc and X0 are concerned, the relations (6), (9) and (10) imply =

-

T~ - M(0, 0) 2 - Xo' - [C - C0l,

(20)

where Co is the critical concentration for ferromagnetism. Experimental plots of these quantities are called Mathon plots [27], and have been used to assess the applicability of the basic model. If, for example, Tc and M(0, 0) are determined by different effects, for example by spin wave and single particle exchange splitting effects, respectively, then this would in principle show up on measured plots of this type. Mathon plots for Ni3AI compounds off stoichiometry [18] are shown in fig. 1. The results are not plentiful but fit the relationships (20) with values of Co = 74.51_+0.06% Ni [Note that here X refers to the paramagnetic susceptibility]. For N i - P t alloys near the critical concentration [20] C0=42.0% Ni, the values of M(O, O)/T~ and of xoM(O, 0) z vary little between C = Co and 50.2%. Within the errors of the experiments it appears, therefore, that (20) applies reasonably well to a number of alloy systems. Recalling that Moriya and Kawabata [9]

310

~

50~

4X, I 10 e m u l

'- 5000

90 T2(K 2) i

40~

8 0 ; 4 , ," }'

70

M2 e m u 6 0 ! ~00!,

I

5o 4

I

4 0 i 2 0 c'-)

30 20, i !0C~, !

10 1 I 73 5

74

74 5 °/o N i

76~

75

Fig. I. M a t h o n plot,; for Ni,AI 1181.

had a critical index r for T~ equal to ~ compared to r = / from (20) it may be that even more accurate magnetic measurements are needed to distinguish these two models. In principle, at least, a recipe for this choice of models is provided in this way. Further, the two examples show clearly that T~ is certainly not independent of M(0,0) (r = 0) as would be the case if T~ is determined by effects completely unrelated to the exchange splitting. It was claimed earlier that the strong and weak ferromagnets may show this significant difference. Mathon plots for the spin wave stiffness D are less plentiful. For N i - P t Beille et al. [281 obtained roughly a parabolic relationship for D as a function of [ C - C01. For F e - N i invar alloys Kohgi et al. [29] gave a plot of D versus C for several investigations and found those which are reliable [30] to again follow this parabolic relation. Similar results for invar were also obtained by Hennion et al. [31] from whose data a plot of D 2 v e r s u s C gives almost a perfect straight line between Co= 28.5% Ni and about 65% Ni, close to the maximum of the SlaterPauling curve! Here there is thus no question but that the relevant critical index has a value very close to ~. Further neutron measurements of D for related alloys are awaited with interest.

The next section is concerned with magnetoelastic effects including pressure effects. Clearly a set of relations analogous to (20) but involving pressure should apply, giving a further series of relevant experimental investigations.

4. Magnetoelastic effects For strong itinerant ferromagnets the observed magnetoelastic effects are in general small. Thus the change of the Curie temperature with hydrostatic pressure P is essentially zero for iron and cobalt while OTJOP = + 0.37 K kbar t [32] for nickel. The pressure dependence of the saturation magnetization was obtained [33] from measurements of the small volume magnetostriction, as ao~

1 aM

OH

V aP

121)

where o~ is the volume strain, giving a small negative pressure derivative of M(0,0). Other magnetoelastic effects for the ferromagnetic transition metals are summarized by Carr [341. The most primitive calculation of aTJOP takes T, to be determined by a single characteristic energy E of any origin (exchange, band

311 width . . . . ), giving

oTc

aP

- FKTc, (22)

where r = [a ln___.EE[ I

&0

I

is a Griineisen constant. For a band model where this energy is the width of the d-band, F - 3 5 and this result suffices to explain the sign and qualitatively the magnitude of the observed pressure derivative [35]. Correction terms to (22) have also been considered and the whole simple formalism applied to nickel rich binary alloys [35, 36]. For the pressure dependence of M(0,0) Mathon [37] showed that it has contributions both from a change of the d-band structure under pressure and from s-d transfer under pressure. The last effect is important for nickel at low temperatures, where it determines the magnitude and sign completely, but the first effect begins to outweigh it near the Curie temperature where aM(O, Tc)/aP reaches large positive values. Since this shift of the importance of Mathon's two effects happens when the magnetization b e c o m e s small, the s-d transfer effect will be ignored when discussing weak itinerant f e r r o m a g n e t s below. The importance of s-d transfer for such materials remains to be assessed. The situation regarding the ferromagnetic transition metals and their dilute alloys is thus fairly clear but not unusually exciting. The Lang-Ehrenreich formula (22) gives a reasonable account of the small observed pressure derivatives of T~ but the physical origin of the characteristic energy E is not revealed. The situation is different for weak itinerant ferromagnets. It has recently been summarized [38] and hence only some salient features need be given here. Consider relation (6) for the Curie t e m p e r a t u r e corresponding to single particle excitations. The resulting pressure derivative of T~ is given by

OT~_ 5KT c_ ~ __ ]

OP

3

Tc'

where a - 6_

5KIT: I. F-

(23)

J

Here the first term arises as a result of the

pressure d e p e n d e n c e of the characteristic temperature TF and was already given as relation (22) above. The second term arises as a result of the pressure d e p e n d e n c e of [ and has the form given here on the basis of K a n a m o r i ' s theory of correlation [39] with I the effective and Ib the bare intraatomic coulomb interaction. The occurrence of this large negative term transforms the ennui of this subject when applied to the pure metals to one of much greater interest when applied to weak itinerant ferromagnets. The whole equation (23) was applied by Brouha et al. [40] to some rare e a r t h - - i r o n a n d - cobalt intermetallic c o m p o u n d s and found to fit very well over a range of Curie t e m p e r a t u r e s about 200-1200 K. It was found that F is close to ~5 and that the values of a for the cobalt c o m p o u n d s were all close to 2200 K: kbar i and for the iron c o m p o u n d s close to 1000 K 2 kbar -l. Note that a d e p e n d e n c e of T c on I - - 1 different from (6) would of course also lead to a different pressure dependence, the negative term in (23) going as - T c " - ~ / L The same holds for the two other examples which now follow. For all three classical invar alloy systems F e - N i , F e Pd, and F e - P t relation (23) was found to hold [41] over a wide range of Curie temperatures, about 300-600 K, with a single p a r a m e t e r ~ = (2000 + 100) K 2 kbar -~. Using a reasonable value of the characteristic t e m p e r a t u r e TF [27] it follows that I/Ib ~--0.9, thus giving an estimate, yet to be evaluated, of the correlation effect. M e a s u r e m e n t s of Tc at higher pressures should follow the relationship, obtained f r o m the second term in (23), T ~ ( e ) = T~(0)(l - P / P c ) ,

(24) where Pc =

TZc(O)/2a"

Note the analogy of (24) with (20). Relation (24) has been tested for ZrZn2 [42] and an F e - N i - M n invar alloy [43], with respective values of Pc 8.5 (or 18) and 27 kbar. The parabolic dependence given by (24) is not always obeyed, there being a tendency towards a more linear decrease in some cases (critical index ~" greater than ½), or a more curved d e p e n d e n c e in others (T<~). H o w e v e r , it may be claimed that m e a s u r e m e n t s on a range of alloys with differing Tc values at low pressures are more reliable than for one alloy over a wide high pressure

range. As already stated, the former type of experiment continues to support (23). The final example refers back to relation (20). Measurements on N i - P I alloys [20] were carried out on the pressure dependence of all the quantities contained in this relation. A single curve was found to represent well the logarithmic derivatives of T~, M(O,0)~', X(,' and tC C.I as functions of ] C - C o l , with a single parameter dCJdP I x 10 ~kbar ~ which was itself also calculated satisfactorily from elementary thermodynamics [20]. It thus appears from presstire experiments that (20) and its theoretical basis are fully verified. In particular the Curie temperature is determined by the single particle excitations as it is proportional to M(0,0). The critical index r continues to be close to '~. The results of relation (23) may be formally represented by including in the free energy expression (16) additional terms, thus extending the Landau theory [44]. These terms :ire, to lowest order, I AF(w) =-w:2K

CtoM:,

(25)

where K is the compressibility, to the volume strain and C the magneloelastic coupling constant related to a, given by (23), by [44] (26)

a = 2 K C I . t ~ N ( E v ) T ~.

Using (16)+(25) it is easy to deduce the volume magnetostriclion (the change of volume arising from a change of M) and the magnetic contribution to the thermal expansion. It is found that to = K C [ M 2 ( H ,

T)

M~(O, T)],

(27)

this being a convenient modification of (25) which makes to = 0 when H = 0. Measurements of this effect were reported for ZrZne [45] and Ni3AI and N i - P t [46]. The dependence of to on M e was verified in all cases over a wide range of temperatures below and above T,. This is clearly a model independent result but serves to show the applicability of the equations in a wide temperature range "~ TF. The wdues of the coupling constant derived from these measurements agree well with those obtained from the pressure dependence of T,. and the thermal expansion [47, 48]. Does this merely constitute a proof thai thermodynamics works'? Even a

small temperature dependence of K ( " 1461, 'a,, T -~, agrees well with an obvious extension of the model [441. Finally, using the measured values of K(" and reh/tions (16)+(25), values of I/1~, may be obtained thus continuing to provide experimental evidence for a fundamental Irealment. For Ni~AI and N i - P I [46] I/I~, w a x found to have the respective vahles 0.5 and 0.85, c)f the same order as for Fe-Ni invar alloys obtained earlier. The second result obtained from (25) is concerned with the temperature dependence of the magnetic volume thermal expansion coeflicienl /3~1. It is found that [481 toM(T) :

K(" M(O, O ) : T ' / T [ .

(28)

i.e.

/3M(T) = - ( 2 K ( ' M(0, 0):/I~i)'1, where wM(T) is the volume change belween 0 and T. At the Curie poinl T the thermal expansion coefficient has a discontinuity (again in the absence of any critical phenomena for which there is no evidence here) A[3 M - -- 2 K C M(0, 0 ) : / T

~'(0) ~ 1 .

(29)

Hence, comparing with the specific heal discontinuity given by (17), :l very strong prediction can be made that A/3~t is much more easily obserwlble than At'. Measurements of thermal expansion for ZrZne [491. N i - P t and Ni~AI [481 give a complete verification of all these predictions. A very close verification, in particular, i~ given of the T e and corresponding T laws in (28) with no apparent evidence for spin wave effects. The discontinuities of the thermal expansion :is a function of temperature [491 or concentration [48] tire also easily seen. l'he balance between the negative contribution to the thermal expansion coming from magnetism and the positive contributions from the phonons and conduction electrons can give low net thermal expansions over a range of temperatures, i.e. invar behaviour [38]. Finally reference only need to be made to the magnetic contributions to the elastic constanls of itinerant ferromagnels since an account of this matter was recently given [501. The bulk modulus B ( M ) = K t varies on passing the Curie point since measurements above I'~ are ideally made in a stale of constant M and below at constant H. Hence for the discontinuity of

313 B, A B = BH--BM, where Bm is taken to be extrapolated from above to below To. The result of an elementary thermodynamic analysis gives [50, 51] AB

=

4x0M(0, 0)2C 2,

(30)

where C is the magnetoelastic coupling constant defined in (25). This discontinuity contributes to that of Young's modulus, the AE-effect. A further magnetic contribution to B, and thus to E, arises from higher order terms in the free energy additional to (25) and going as ~2M2. These have been calculated [50] in terms of 1/lb and the volume magnetostriction and found to be in general small. There is as yet no satisfactory agreement between experiment and this theoretical analysis of the elastic constants of weak itinerant ferromagnets. The reasons for this failure are not clear but are felt to reside in phonon effects and thus not to provide any relevant evidence for the present purposes. 5. Conclusion

Elementary thermodynamics has been shown to provide a rich background to the empirical and quantum-mechanical description of the properties of itinerant electron ferromagnets. By providing power laws in the temperature and other thermodynamic variables for each of these properties it is in principle often possible to distinguish different theoretical models on the basis of measurements. The dangers of fitting such laws have been stressed in the Introduction. Nevertheless, a number of examples has been given where the procedure is sufficient to justify the claims of one or other theoretical model with some confidence. It is thus claimed: (1) For the ferromagnetic transition metals, in particular nickel, the evidence for single particle excitations below the Curie temperature is not strong although their effect can not be wholly excluded. It is thus not as clear as was thought at one time that the Curie temperature itself is closely related to the exchange splitting and thus the saturation magnetization. If further evidence continues to strengthen this belief then T~ must be considered as a critical temperature where spin waves become soft. Edwards [52] has proposed as an exciting possibility the softening of a very low lying spin wave mode previously observed in nickel by neutrons. The effects are

critically dependent on the d-band structure and thus models of fluctuations not including band structure effects as fully as this are rather suspect. (2) For weak itinerant ferromagnets the evidence is much more that single particle excitations below T~ are dominant, there being only weak experimental evidence for spin wave effects. The Curie temperature itself is much more clearly demonstrated as arising from single particle excitations, magnetoelastic measurements being particularly valuable in this respect. A comparison between the simple Stoner model and more advanced models including fluctuation effects is made on the basis of the value of the critical index ~', given by T ~ - ( f - 1y. Although the difference between the value ~ of the simple 3 Stoner model and the values i, 1. . . . from the later models is not large, it is claimed that the experimental evidence points to the first of 1 these, ~. The final conclusion, which may well be completely reversed at the second E P S I E M conference, is thus that weak itinerant ferromagnets continue to behave in a particularly simple manner and provide the closest experimental evidence for the simple Stoner model.

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