Review of the electronic specific heat of B. C. C. and F. C. C. solid solutions of first long period transition elements

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Review of the electronic specific heat of B. C. C. and F.

C. C. solid solutions of first long period transition elements

K.P. Gupta, C.H. Cheng, Paul A. Beck

To cite this version:

K.P. Gupta, C.H. Cheng, Paul A. Beck. Review of the electronic specific heat of B. C. C. and F. C.

C. solid solutions of first long period transition elements. J. Phys. Radium, 1962, 23 (10), pp.721-726.

�10.1051/jphysrad:019620023010072100�. �jpa-00236669�

721.

REVIEW OF THE ELECTRONIC SPECIFIC HEAT OF B. C. C. AND F. C. C. SOLID SOLUTIONS OF FIRST LONG PERIOD TRANSITION ELEMENTS

By ^{K. P.} GUPTA, ^{C. H. CHENG and PAUL A.} BECK, University ^of Illinois, Urbana, Illinois, ^{U. S. A.}

Résumé.

_{On passe} en revue les résultats relatifs aux chaleurs spécifiques des solutions solides

cubiques centrées, ^et certaines données nouvelles sur les système ^{V-Cr et} Ti-V. Les chaleurs spéci- fiques ^sont remarquablement bien décrites par ^un modèle de bandes rigides, ^même pour les mélanges’

ternaires.

En ce qui ^concerne les solutions solides cubiques ^{à faces} centrées, les valeurs de _03B3 mesurées ont

une interpretation électronique simple seulement pour les alliages ferromagnétiques ^{ou non} magné- tiques. ^{Pour les} alliages ^où coexistent des interactions ferro- et antiferromagnétiques, ^{la valeur} expérimentale ^de 03B3 parait comprendre ^{souvent une} contribution magnétique. ^{En se limitant aux} alliages ^non magnétiques ^ou ferromagnétiques ^on peut déterminer la forme de la bande d pour la structure cubique ^à faces centrées et pour des concentrations allant de ^{8 à 10} électrons d par atome.

Abstract.

A review will be given ^{of the} electronic specific ^{heat vs. electron} concentration for b. c. c. solid solutions, including ^new data for the systems V-Cr and Ti-V. The latter alloys

show a maximum of _03B3 at approximately ⁶⁰ % V. The electronic specific heat coefficients correlate well will the superconductive transition temperatures, according ^{to the} Bardeen-Cooper-Schriffer theory, ^{with an} interaction coefficient V independent ^of composition. ^{It was} found that the

CsCl-type ^ordered alloys ^{in the} ternary ^Ti Fe-TiCo and TiCo-TiNi systems possess electronic specific

heat values paralleling ^{those for the b. c. c. Cr-Fe} alloys ^{at the same} average electron ^concen- trations. The electronic specific heat values for the b. c. c. alloys of transition elements with one

another can be apparently ^{described to a} surprising ^{extent in terms of a more or less} rigid band,

the degree ^of filling up of this band with electrons being determined essentially by the average electron concentration for the alloy. This appears ^to hold even in cases where the electron con-

centration is determined by averaging ^{over three} different atomic species. ^As found also for the

alloy VFe, ^{with a} CsCl-type ^ordered structure, ordering ^causes little, ^if any, change ^{in the elec-}

tronic specific heat values of these alloys.

New low temperature specific heat measurements with f. c. c. solid solutions in the Mn-Fe, Mn-Ni, Fe-Ni, ^V-Ni systems gave coefficients for the specific ^{heat term} linear in temperature,

which are in some cases quite inconsistent with each other if interpreted ^as electronic specific

heat coefficients. Consistent _03B3 values were obtained for simple ferromagnetic alloys ^{and for non} magnetic alloys. ^For alloys ^{where the} simultaneous presence of both ferromagnetic and antiferro-

magnetic interactions is known from magnetic measurements, ^or may be considered as very likely,

the measured _03B3 values are anomalously high. ^{It has been} proposed by ^Marshall (1) ^{that a} tempe-

rature-linear contribution to the low temperature specific ^{heat can} arise in instances where a

sufficient number of spins ^are located in a near zero magnetic ^{field. It} appears from ^our results that this condition is often fulfilled in alloys having ^both ferromagnetic ^and antiferromagnetic

interactions. As a result, in such cases the 03B3 value measured at low temperatures comprises, ⁱⁿ

addition to the true electronic specific ^heat coefficient, ^{also a} magnetic component. Using only ^03B3

values obtained for simple ferromagnetic ^{and for} nonmagnetic alloys, ^the shape ^{of the d-band}

for the f. c. c. solid solutions has been nevertheless successfully determined in the electron concen-

tration range from approximately ^{8 to 10.}

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM TOME} 23, ^OCTOBRE 1962,

In a previous ^paper [1] electronic specific ^heat

vs. electron concentration data were published ^for

b. c. c. solid solutions of first long period ^tran-

sition elements. Measurements made recently [2]

with b. c. c. solid solutions in the Ti-V System

confirm the peak ^{in the} electronic specific ^heat predicted [1] ^{for these} alloys (2). Figure ^{1 shows}

(1) ^MARSHALL (W.), Specific heat of dilute alloys. Phys.

.Rev., 1960,118,1519.

(2) ^{It was noted} [2] that the electronic specific ^heat

coefficients of these alloys correlate well with their super- conductive transition temperatures [2, 3] ^{in accordance} with the B. C. S. theory, ^{and with a value} of the interaction

coefficient ^V independent ^of composition. This situation is similar to that found by ^{Goodman et} al for the b. ^{c. c.}

U-Mo alloys [4].

the combined electronic specific ^{heat data for the}

b. c. c. alloys in the entire electron concentration range where solid solutions with this ^{structure are} formed by ^the transition elements of the first long period.

The second high peak ^of y corresponds roughly

to the onset of ferromagnetism ^{in the Cr-Fe} alloys ;

the question ^{arises as to} whether this peak may be of magnetic origin. ^The high temperature specific

heat data of Schrôder [5] clearly ^indicate (fig. 2),

that the large peak ^{in the term} of the low tempe-

rature specific heat linear in temperature, ^which

occurs at around Cr + ¹⁹ % ^{Fe is not of} magnetic origin, but that it is indeed an electronic specific

heat coefficient.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010072100

722

FIG. 1.

_Y vs. average electron concentration for b. ^{c. c.}

Ti-V, V-Cr, ^Cr-Fe and Fe-Co alloys. Filled squares

represent data for close packed structures [2].

FIG. 2. - Specific ^{heat vs.} composition ^{of Cr-Fe} alloys

at various temperatures [5].

It was recently ^found [6] ^{that the} CsCI-type

ordered alloys ^{in the} ternary ^{TiFe-TiCo and TiCo-} TiNi systems have electronic specific heat values

paralleling very closely those for the b. c. c. Cr-Fe

alloys ^{at the same} average electron concentrations

(fig. 3). These , results represent further confir-

.

mation of the electronic specific ^heat nature of the measured _y values. They also underline the sur-

prising ^{extent to which the} electronic specific ^heat

of the body ^{centered cubic} alloys of transition elements with one another can be interpreted ⁱⁿ

terms of a more or less rigid d-band, ^the degree ^of filling up of this band with electrons being ^deter-

mined by the average electron concentration of the

alloy. ^{This is now shown for} ternary alloys, ^where

the electron concentration is determined by ^avera- ging ^{over three} different atomic species’. ^Orde- ring apparently ^makes little, if any, différence in the electronic specific ^{heat. This is not} very ^sur-

prising in ^view of the work of Rayne [7] ^with AuCU3 Also, ^newer results for VFe with a CsCI-type ^orde-

red structure [8] show the electronic specific ^heat

FIG. 3.

y ^vs. average electron concentration for ternary Ti-alloys ^{with CsCI} type ordered structure. 0 repre- sents TiFe-TiCo alloys. ^{stands for} alloy

Dashed line gives y values for b.. ^{c. c. Cr-Fe} alloys [1].

to be nearly ^{the same as that for the disordered} alloy ^{of the same} composition.

The density ^{of states vs.} energy, calculated from the data given ⁱⁿ figure ¹ according ^{to the method}

of Hoare [9], ^is given ⁱⁿ figure 4, ^{where the} density

FIG. 4.

Density of states for each spin orientation per eV

vs. energy in eV for the d-electrons. Zero of energy

scale corresponds ^{to Cr.}

72

of states for the two spin directions is plotted sepa-

rately. ^The interpretation presented ^{in the} figure

makes use of magnetic saturation moment data for these alloys ^{as shown in the} Slater-Pauling

curve [7], ^{and of neutron} diffraction data [12].

The combination of these data with the electronic

specific ^{heat results} clearly indicates that the first

sub-band, lying in the electron concentration _range between 4 and 6 ( fig. 1) ^{or in the} energy range between -1.2 and 0 eV (fig. 4), represents ^an equal number of electrons with positive ^{and with} negative spin direction, ^{since the} alloys ^{in this}

range have ^{no net} permanent ^{moment. The}

second sub-band, which extends over the electron concentration _range from about 5.8 to 8.4 ( fig.1)

or in the energy range from - ^{0. 4 to} + 1. 35 ^eV (fig. 4), clearly represents electrons in one spin

orientation only. The atomic moments, ^{as mea-}

sured by saturation magnetization ^or by ^neutron diffraction, correspond ^{to the number} of electrons added to this sub-band. The fractional saturation moment of Fe (2.2 yB) ^{can now be} understood ^{as a} result of the fact that the second sub-band begins

at an electron concentration of approximately 5.8,

and that all the electrons added to this sub-band contribue to the moment.

Although the above model is consistent with the known expérimental facts, ^it certainly ^{does not fit}

the usual concept ^{of the structure of the d-band}

of Fe. In the usual picture, Fe is considered to

have both + ^and - spin orientations at the Fermi surface [13] while the above discussed results

suggest [1] that in Fe there should be d-electrons at the Fermi surface with ^one spin orientation only.

Recent measurements by Walmsley [14] ^{of the} change ^{in the contact} potential between Fe and Cu

on applying ^{an external} magnetic field gave ^a direct confirmation of this concept. Walmsley ^{also esta-}

blished that the sign ^of spin ^{of the} d-electrons at the Fermi surface in Fe is positive.

Finally, ^{the third} sub-band, extending ^from approximately 8.2 electrons per ^{atom to} the end

of the range of stability ^{of the b. c. c.} structure at around 8.7 electrons per ^atom ( fcg. 1) ^{or from} + ^1.06 to + 1. 64 ^eV (fig. 4), represents ^electrons

with the opposite spin ^direction only, ^{as shown} by

the decreasing saturation moment with increasing

number of electrons per ^atom (Slater-Pauling curve).

The separation ^{of the} positive (" ^second ") ^and negative (" ^third ") ^{sub-bands here is} approxi- mately 1. 5 eV. This value is much larger ^{than the} exchange energy associated with the Curie tempe-

rature. Since Tc 1000 oK for all of the alloys concerned, the latter exchange energy

The observed separation, ^{which is at least an order}

of magnitude larger, ^must correspond ^{instead to}

the much larger intra-atomic exchange energy which gives ^{rise to} (unaligned) ^{atomic moments}

well above the Curie température.

It is interesting to compare ^the electronic spécifie

heat results with magnetic susceptibility ^{data for}

some of the same alloy systems, published recently by Taniguchi, ^{Tebble and Williams} [15]. Compa-

rison of figure ^{5 with} figure 1 shows the expected

FIG. 5.

Magnetic susceptibility ^{vs. electron concen-}

tration for b. c. c. alloys ^{of first long} period ^transition

elements [15].

parallelism ^{at least} in the electron concentration range between 5 and 6.5. (For ^{the V-Cr} alloys

this was already ^{found earlier} by Childs, ^Gardner

and Penfold [16].) However, the X ^{and the} y ^vs.

composition ^{curves differ from} each other _very

significantly for the Ti-V alloys. ^The magnetic susceptibility ^{of Ti-Nb} alloys ^{also varies in a}

manner similar ^to that of the Ti-V alloys ^{in the}

same electron concéntration _range, as shown by Jones, Pessall and McQuillan [17]. ^The reality ^of

the différence between ^the variation of x ^{and of} y

can not be doubted, ^{but its} explanation is, ^at present, unknown.

Low température specific heat measurements have been made with f. c. c. solid solutions in the

Mn-Fe, Mn-Ni, Fe-Ni, ^{and V-Ni} systems [20].

The resulting coefficients of the ^low temperature specific ^{heat term linear in} temperature, ^{if inter-} preted ^as electronic spécifie ^heat coefficients, ^{are in}

several cases quite inconsistent with the rigid ^band

model. As seen in figure 6, ^the y values obtained by Walling ^{and Bunn} [18] ^{for f. c. c.} Co-Ni solid

solutions, ^{which are} strongly ferromagnetic, ^{are in} reasonably good agreement with the values obtai- ned by Keesom and Kurrelmeyer [19] ^{and in this} laboratory [20] ^{for f. c. c. Fe-Ni} alloys ^{in the same}

electron concentration range. These Fe-Ni alloys

are also strongly ferromagnetic. ^{The value of y}

for ordered MnNi3, ^as reported by ^Goldman [21],

724

is also fairly ^{close to} that of the Co-Ni solid solution at the same eleetron concentration (fig. 6). ^{It is}

known that ordered MnNi3 ^{is also} ferromagnetic,

with a Curie temperature ^{of 475°C} [22].

FIG. 6.

_y vs. electron concentration for f. c. c. alloys

in the Mn-Fe, Mn-Ni, Fe-Ni, ^{Co-Ni and V-Ni} systems.

For MnN’3 y is given ^{fôr the} following conditions : disor- dered (symbol ), short-range ^ordered (symbol e-), and long-range ^ordered (symbol -C). ^For Fe,N’3 y ^was determined after cooling ^{in a} magnetic ^field (symbol 0),

as well as without magnetic ^field (symbol U) [20].

However, ^the y value reported by ^Goldman [21]

for disordered MnNi3, is very much higher. ^This

result was recently ^confirmed [20]. (See ^{the two} nearly coinciding ^c symbols ⁱⁿ figure 6). ^Dreesen

concluded from his study of the effect of order on

the Hall constant of MnNi3 [23] ^{that the observed}

différence in the _y values between ordered and disordered MnN’3 ^{is not a result of a} change ^{in the}

band structure upon ordering. ^This conclusion is

supported by ^{all results obtained to date on the}

effect of ordering ^on the electronic specific ^{heat of} alloys having ^no magnetic ^moments [7, 8]. ^This

raises the question why ^the coefficient of the low

temperature specific ^{heat term} linear in tempe-

rature increases _upon ordering ^{in the case of} MnNi3.

Overhauser first proposed ^a theory [24] ^{for a low} temperature specific ^{heat term of} magnetic origin,

which is linear in temperature. ^This theory ^pos-

tulates that ^a large ^{number of} spins ^{are located in}

a near zero field as a result of the presence of

stationary spin ^{waves in an} antiferromagnetic

structure. In view of the fact that disordered

MnNi3 ^is f erromagnetic [22], it appears that Over- hauser’s theory ^{could not be} applied ^{to it. On}

the other hand,, Marshall’s theory [25]. ^{wich is}

also based on spins located in nerr zero field, ^does

not require antiferromagnetic structure. It appears very probable ^{that the increment} of the obser-

ved _y value for disordered MnN’3 ^{over that}

of ordered MnN’3 ^{is a} result of the mechanism described by ^Marshall [25] and that the condi- tion of spins ^{located in} near zero average field is satisfied in the disordered MnN’3 alloy ^{as a}

result of its peculiar magnetic structure. It is known from the work of Kouvel and Graham [26]

that this alloy shows " exchange anisotropy " ^when

cooled through the magnetic transformation tempe

rature range in ^a magnetic ^{field. The observed}

unusual magnetic effects result from the presence in this material with an overall ferromagnetic

structure of locally antiferromagnetic spin ^arran- gements. ^{It has been} suggested by ^Kouvel [27]

that in the Cu-Mn and Ag-Mn alloys, ^{which also}

show " exchange anisotropy ", ^the simultaneous presence of ferromagnetic ^and antiferromagnetic

interactions on an atomic scale is a result of the statistical variations in the separation ^of pairs ^of

near Mn atoms, and of the strong dependence ^of

the type ^of exchange coupling between Mn pairs ^on

the distance of separation. ^{It is} extremely likely

that the same mechanism accounts also ^{for the} exchange anisotropy observed in disordered MnNi3.

Consequently, it is reasonable to assume that many atomic moments in this alloy ^are located in a near zero average local field resulting ^{from the near com-} pensation in these local areas of the ferroma-

gnetic ^and antiferromagnetic exchange ^interac-

tions. It may be expected, therefore, ^{that the}

condition required by Marshall’s theory is satisfied.

In order to test the above hypothesis, ^an MnNi3 alloy specimen, which had been disordered by annealing ^at 1 000 OC and preserved in this condi- tion by quenching, ^was re-annealed for two hours at 485 OC, ^{in order to achieve} short range ordering

and thus to decrease the number of nearest neighbor

Mn pairs, ^which presumably give ^{rise to local anti-} ferromagnetic coupling. This heat treatment was

selected on the basis of results reported by ^Marcin-

kowski and Brown [22]. ^{As seen in} figure 6, ^the

measured _y value after this heat treatment

(symbol e-) ^was considerably ^{lower than} that in the disordered condition. In the case of disordered

MnNi. ^the magnetic component ^of y approximately

doubles the appearent " electronic specific heat ".

However, ^the magnitude ^{of this extra} specific ^heat

decreases rapidly ^with increasing Mn-content.

It may be ^seen in figure 6 that the curve corres-

ponding ^{to Mn-Fe} alloys is crossed by ^{the corres-} ponding ^{curve for the Ni-rich Fe-Ni} alloys. ^The’

measured _y values for the latter alloys ^at high ^Fe-

contents become extremely large. ^The question

should be considered next whether or not these very high ^{y values} may also comprise large ^ma- gnetic contributions. The magnetic properties ^of

the Fe-rich f. c. c. Fe-Ni alloys ^{have been inves-}

tigated by Kondorskii and Fedotov [28] ^{who found}

that, superimposed ^{on the overall} ferromagnetic

725

structure, ^{a latent} antiferromagnetism ^{must be}

assumed ^{in order} to account for the measured

macroscopic magnetic properties. Consequently,

it _may be assumed that, ^{for these} disordered alloys,

as well ^as for the disordered f. c. c. Mn-Ni alloys,

the condition required by Marshall’s theory may ^be satisfied in the manner here proposed. ^{In order}

to test this model, ^the alloy containing ³⁰ % ^Ni (and ^{also 5} atomic per ^cent C to stabilize the f. c. c.

structure) ^{was cooled in a} magnetic ^{field of} approxi- mately 10 kG from 400 °C ^{to room} temperature.

After this heat treatment the specimen ^was again introduced into the calorimeter and its low tempe-

rature specific ^{heat was} measured. As seen in

figure ⁶ (symbol c) ^{the heat treatment increased}

the measured _y value. The observed effect of ma-

gnetic cooling suggests strongly ^{that this} alloy ^also

exhibits exchange anisotropy, and that the mea-

sured _y value comprises ^a magnetic contribution.

The crossing ^{of the curves for} Mn-Ni and Fe-Ni

alloys ⁱⁿ figure ^{6 thus} turns out to be of no parti- cular-significance, since there is no reason to expect

that thé peculiar spin arrangement giving ^{rise to} the large magnetic contribution _{to y} should occur

in différent alloy systems ^{at the same} average elec- tron concentration.

A similar lack of consistency may be observed in figure 6 between the _y values for Mn-rich Mn-Ni

alloys and Fe-rich Mn-Fe alloys. ^{The fact that} y is pnusually large ^{for a f. c. c. Fe +11} % ^Mn alloy

has been reported ^earlier by ^Goldman [29], ^and

the much lower _y value for Fe + ⁴⁵ % ^{Mn was also}

known [30]. ^{Corliss and} Hastings ^found [31] ^that

f. c. c. Mn-Fe alloys possess long range antiferro-

magnetism, ^and Kimball, ^Gerber and Arrott found [32] that the observed magnetic ^moments

must be largely associated with the Mn-atoms.

It seems very probable ^{that, here too, the} exchange

interaction between Mn-atoms with a somewhat

larger distance of separation than that of nearest

neighbors ^is ferromagnetic. Thus, in these disor-

dered alloys ^we may ^be dealing ^with long range

antiferromagnetism ^{with a} superimposed " latent "

or short _range ferromagnetism. Again, it may be

.

assumed that the average local field in which many atomic moments are located is near zero, and that the extra large y values observed in the Fe-rich

alloys ^{have a} substantial component ^{due to the}

effect described above. It should be pointed ^out

however that, ^{in this} case, ^a magnetic contribution

to y could be also accounted for on the basis of the mechanism proposed by Overhauser, ^{since the} f. ^{c. c. Mn-Fe} alloys apparently ^{do have a} long range antiferromagnetic ^structure.

As a further test of the assumption ^{that in}

several of the f. c. c. solid solutions the measured _y values have a large magnetic component, ^{a series} of non-magnetic V-Ni solid solution alloys ^were

also measured. As seen in figure 6, the measured _y values for the non-magnetic alloys represent ^{a rea-} sonable extension of the curve for the " purely "

ferromagnetic Co-Ni and Fe-Ni alloys (no " ^latent

antiferromagnetism ^{"). The magnetic contri-}

bution to Y ^discussed above, ^{which occurs in} alloys having ^both ferromagnetic and antiferro-

magnetic interactions, ^{is absent} for both of these

cases. Consequently, ^{it seems} justifiable ^to pro- pose that the ^true electronic specific heat of the f. c. c. alloys in the electron concentration range between 8 and 10 may be represented approxi- mately by ^the heavy ^{line in} figure ^{6. The} points

for the two Mn-Ni alloys ^{near the} equiatomic ^com- position ^{are also near this curve.}

Acknowledgements. ^{- This} review is largely

based on experimental ^work performed during ^the

last six years ^{with the} following colleagues ^and graduate students : C. T. Wei, ^K. Shroeder, ^{E. A.}

Starke Jr. and E. C, ^van Reuth. The work is

supported by ^a grant from the National Science Foundation. In various earlier phases ^{it was} supported by the U. S. Air Force and by ^{the U. S.}

Army ^{Research Office} (Durham).

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Phil. Mag., 1960, 5, ^155.

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FERROMAGNETIC PROPERTIES OF Fe AND Ni IN RELATION TO THEIR BAND STRUCTURE

By R. GERSDORF,

Natuurkundig Laboratorium der Universiteit van Amsterdam.

Résumé. 2014 ^De nombreuses propriétés de Fe et Ni peuvent être reliées les unes aux autres au

moyen d’une théorie empirique supposant que le moment atomique magnétique n’est pas constant dans les métaux de transition mais est supposé dépendre ^{dans ces cas de la} température, ^du

volume et du champ magnétique ^externe.

Les propriétés ^en question sont, parmi d’autres, ^la magnétostriction ^{en volume} dépendant ^du champ, la variation de magnétisation ^{et du} point ^{de Curie avec la} pression ^et les déviations par

rapport ^aux courbes de Brillouin de la magnétisation ^à saturation en fonction de la température.

Les variations du moment atomique magnétique ^{sont estimées en} incorporant les moments

magnétiques localisés dans la théorie des bandes pour les électrons 3d.

Abstract.

Several properties of the elements Fe and Ni can be correlated with each other

by ^{means of an} empirical ^{theory, which assumes} that the atomic magnetic ^moment does not have

a constant value in transition metals, ^{but is} thought ^to depend ^{in these cases on} temperature, volume, and external magnetic ^field.

The properties ⁱⁿ question are, among others, ^the field-dependent ^volume magnetostriction,

the change ^of magnetization ^and Curie point with pressure, ^and the deviations of the temperature dependence of the saturation magnetization ^{from the} appropriate ^{Brillouin curves.}

Variations of the atomic magnetic ^{moment are estimated} by incorporating ^localized magnetic

moments into the electron band theory of the 3d-electrons.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM TOME} 23, ^OCTOBRE 1962,

Some of the physical properties of iron metal

display unexplained anomalies. We found [1] ^that

the field-dependent ^volume magnetostriction, h’O,

does not disappear ^{at low} temperatures, ^{but re-}

mains approximately ^constant. Moreover, ^the temperature dependence ^{of the} saturation _magne- tization deviates more than 10 % from the theore-

tically expected ^{Brillouin curve} [2], while the value for the exchange energy ^as found from spin ^wave theory, ^differs considerably ^{from the} value, ^calcula-

ted from the experirnentally determined Curie point [3]. ^{In other} ferromagnetics ^these properties give

a better fit with theory.

These difficulties ^can be removed by considering

the localized atomic magnetic moment, 6, of the atoms in a magnetic ^{metal as a} variable. This is in contradistinction ^{to the moments in} magnetic salts, ^{which have} fixed values that have been calcu- lated. We assume that a is a function of tempe-

rature, volume, ^{etc. For a} certain value of _a, the energy will be a minimum ; we assume a qua- dratic dependence ^{of the} energy upon ^a in the

neighbourhood ^{of this} equilibrium ^value.

If we define a as the magnetic ^moment per ^atom

expressed ^{in Bohr} magnetons, ^{and m as a relative} change ^{in volume of the} lâttice, ^{then the terms of}

the free energy per atom, F, ^which depend ^{on a}

or w, ^can be written as :

The eff ects of lattice vibrations, causing ^the

thermal expansion ^are omitted here. The first

term gives ^the quadratic dependence of F on a,

which we have already mentioned ; ^{the second}