The effect of two gamma states on the free energy of austenitic iron-nickel-chromium alloys

THE

EFFECT

OF TWO GAMMA STATES ON THE IRON-NICKEL-CHROMIUM

FREE ENERGY ALLOYS*

OF AUSTENITIC

A. P. MIODOWNIKj. Anomtious thermodymbmic properties in the iron-nickel~bromium system have been explained in terms of the model of two gamma states proposed by Kaufman, Clougherty and Weiss for pure iron. This model has been extended to allow an analysis of ternary alloys, and magnetic properties have been used to provide the necessary parameters for a quantitative evaluation of enthalpies and excess entropies. Chromium appears to stabilise the ferromagnetio gamma phase in the same way as nickel, but has a much more marked effect on the relative degeneraoies of the two gamma states. Satisfaotory agreement between experimental and calculated values suggests that the concept of two gamma states separated by a small energy gap should be considered in thermodynamic treatments of other iron alloys. INFLUENCE

DES

DEUX ETATS GAMMA SUR L’ENERGIE LIBRE AUSTENITIQUES FER-NICKEL-CHROME

DES ALLIAGES

Lee propri&% thermodynamiques anormales du systbme fer-nickel-chrome ont 6tB expliqut%s B l’aide du modble des deux citats gamma propose par Kaufman, Clougherty et Weiss pour le fer pur. Ce modAle a Cltbg&&ali& pour permettre une Etude des allisges ternsires, et lea proprikt& magnetiques ont et& utilides pour fournir les param&res n&essaires & une Evaluation quantitative dss enthalpies et des entropies. I1 semble que le chrome stab&e la phase ferrom~~tique gamma de la mGme fqon que le nickel, mais qu’il a une influence beaucoup plus nette sur les d$&n&escenoes des deux &ats gamma. L’aooord satisfaisant qui eat apparu entre les valeurs exp$rimcntales et les valeurs calculQeamontre qu’il faut tenir oompte du concept de deux Btats gamma d’hnergies peu diffbrentes dans les calculs thermodynamiques relatifs aux autres alliages du fer. DER

EINFLUR VON ZWEI GAMMA-ZUSTIiNDEN AUF DIE FREIE AUSTENITISCHEN EISEN-NICKEL-~H~OM-LEGIER~GEN

ENERGIE

VON

Anomale the~od~~isohe Eigensohaf~n des Eisen-nickel-prom-Sys~ms wurden mit dem von Kaufman, Clougherty und Weiss fiir reines Eisen vorgeschlagenen Model1 der zwei Gamma-ZustTulde e&l&t. Dieses Model1 wurde erweitert, urn die Analyse tern&rer Legienmgen zu ermagliohen. Zur Bestimmung der fiir die quantitative Auswertung der Enthalpien und tfberschuDentropien notwendigen Parameter wurden magnetische Eigenschaften herangezogen. Chrom stabilisiert die ferromagnetische Eamma-Phase auf dieselba Weise wie Nickel, hat jedoch einen weitaus stiirkerenEinfluB auf die relativen Gntartungen der zwei Gamma-Zustiinde. Wegen der befriedigenden nbereinstimmung zwischen experimentellen und berecbneten Werten sollte daa Konsept der zwei, duroh eine schmale Energieliike getrennten Gamma-Zust~ndo beit hermod~namischen ~ehandlungen anderer E&en-Legierungen in Betracht gezogen werden.

INTRODUCTION

The magnetic and electrical properties of pure iron, and the behaviour of certain iron compounds suggest that gamma iron can exist in two distinct electronic stages.(1$2) Kaufman et al.(‘) have assumed that the equilibrium between these two states is given by by the relation :

f/(1 - f) = kdg;,)exp (-~/~T)

(1)

where (f) is the fraction of iron atoms in the higher energy state, (AE) represents the difference in energy between the two states, and (gl, go) are the degeneracies of the upper and ground level respectively. Consider&ion of anomalous magnetic properties led Weiss@) to extend this concept to binary iron-nickel alloys, and by postulating a variation of (As) with nickel content as shown in Fig. 1, many of the unusual property changes in this system can be satisfactorily explained. The behaviour of the iron* Received July 17, 1969; revised October 6, 1969. t Department of Metallurgy and Materials Technology, University of Surrey, England. ACTA B

METALLURGICA,

VOL.

18, MAY

1970

541

nickel alloys confirms that the two gamma states are characterised by markedly different magnetic configur&tions. The ground state in pure iron appears to be &ntiferrom~gnetic with a low magnetic moment (mO.5 pB), whereas the higher energy state is ferromagnetic with a large magnetic moment (~2.8 ,+,). In pure iron the number of atoms in the ferromagnetic state is never large enough to allow Iong-range coupling, and produce mscroscopic fe~oma~etism ; however the addition of nickel reverses the relative stability of the two electronic states, so that with nickel contents in excess of 30 per cent, the alloys begin to exhibit & Curie temperature. Although this model hes undoubtedly been successful in explaining anomalous changes in the ela,stic modulus and coefficient of expansion as well as the magnetic properties of these alloys, the model does not seem to have been extended to other systems. It is the purpose of the present paper first to apply the two gamma state model to account for anomalies found recently in the thermodynamic properties of iron-nickel alloys at 1200°C,(4) and subsequently t.o

542

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TABLE 1. Individual magnetic moments on iron and nickel atoms in f.c.c. Fe-Ni 8110y&~J XNi

LLB(F~)

0.74 0.60 0.50

I

,

0.20

I

0.40

1

0.60

0.80

XNL FIQ. 1. Variation of the energy difference between the two gamma states with nickel content (after Weiss’3’).

extend the analysis to anomalies in iron-nickelchromium alloys.(5*13) The analysis is of general application, and it is likely that similar effects will exist in most austenitic iron alloys. ENTROPY CONTRIBUTIONS ARISING FROM EXISTENCE OF TWO GAMMA STATES IN BINARY IRON-NICKEL SYSTEM

THE THE

Kaufman et uZ.(~)have derived the total free energy contribution due to two gamma states in pure iron as AF_

= E,, -

RT [I + (s&J exp

(-AEIJWI

(2)

Differentiating with respect to temperature yields the following expression for the magnetic entropy/g-mole of iron atoms (ASMFe): ASMFe = R In (1 + a) -

__ (a) AEIRT (1 + a)

where a = (gl/gO) exp -

1

(3)

AEIRT

(4) For the binary iron-nickel system, the necessary value of (AE) has already been calculated by Weiss,t3) and in order to estimate AS, it therefore only remains to determine the variation of (gJg,,) with nickel content. In general, the value of (gJg,,) can be related to the relative magnitudes of the Bohr magneton numbers appropriate to the two states, assuming that these are predominantly due to spin contributions,(3) and thus : k71/%))5%WI

+ 1)/W,

+ 1) w (B, + lMrs0 + 1) (5)

where (J) is the number of spin states, and (t!?)is the number of Bohr magnetons. Since equation (3) refers to the entropy contribution/ g-mole of iron atoms, the value of j? required is the magnetic moment associated with the iron atoms in the alloy. Neutron diffraction measurements indicate that this moment remains essentially unchanged over a range of nickel contents,(s) Table 1. The value of ,u&F) associated with the ferromagnetic gamma state (yF) can therefore be taken as remaining relatively unchanged over the whole system. Not

2.91 2.72 2.60

&(Ni) 0.62 0.66 0.67

much is known about the effect of nickel on the antiferromagnetic state (yA F) ; the Neel temperature of an 18 ‘A Ni 9 % Cr steel is reduced to 40°K according to Kondorsky and Sedov,(‘) and it is likely that ~1~ (YAP) must be small compared to the ferromagnetic ,+,(XF) in this system. For the purposes of evaluating (AS) it has therefore been assumed that : = constant

,uB(F)/,u,(AF)

(6) The values of (gJg,,) used in the determination of (AS) are shown in Table 2. The value of 1.6 for O-30 % nickel alloys is based on the figure of 1.6 derived by Kaufman et al. for pure iron(l) which, according to TABLE 2. Values of the degeneracies of the two gamma states estimated from Bohr magneton numbers and the calculated value for pure iron

-50 O-O.3 0.3 0.3-1.0

91/90

911%

Cr = 0 Cr = 0.05 1.6 1.0 0.6

1.5 1.0 0.65

9hcl

91/90

91180

Cr = 0.10

Cr = 0.20

Cr = 0.30

1.4

1.2 1.0 0.85

i-8 l:o

:::

equations (5) and (6) should remain unchanged providing there is no change in the ground state. At higher nickel contents there is a reversal of ground states, so that the appropriate value of (gJg,) is the reciprocal of 1.6 or ~50.6. At 30% nickel, (AE) = 0 and it seems appropriate to use a value of (gJg,,) M 1.0. The value of ASzmN’ in iron-nickel alloys may now be evaluated since the effect is directly proportional to the number of iron atoms in the alloy and therefore : ASF,“-Ni = (1 - X&ASg& (7) where (XNi) is the mole fraction of nickel. Fig. 2 shows the calculated values of ASgmNi for a series of temperature between 1000°C and 1400°C. The marked curvature of the plots leads to an appreciable excess entropy term. Figure 3 allows a comparison of this excess entropy term with the experimental values available for 12OO”C,which are based on a single gamma state model with a normal contribution from the entropy of mixing.f4) Smoothing the stepwise transition in gl/g,, values (Table 2) with nickel content does not alter the general relationship but, produces a better fit in the vicinity of 30 % Ni. The calculated value of AS, appears to account for a considerable portion of the observed excess entropy in the system.

FREE

MIODOWNIK:

ENERGY

OF

AUSTENITIC

AH

0.601 0.40

c

0.20

t

Fe-Ni-Cr

0 Calculated

-1000

-

-600

-

0’

v Experimental (5)

0.20

0.40

x

-0.20 I-

I

I

0.20

I

I

0.40

0 60

X

0.80

AH~-Ni = (l-X,i)(fyF)AE

(3)

TABLE 3. Calculation of AH= arising from the presence of two gamma states in binary Fe-Ni alloys at 1323°K

A::

0.55 0.55

0.80

NI

NL

AHgPNi in Table 3 is obtained by multiplying AE by the fraction of iron atoms in the alloy (l-X,i) and the proportion of ferromagnetic gamma f(yF)* as obtained from equation (1).

f (YP)

0.60

Fm. 4. Comparison of experimental heats of formation and the calculated enthalpy contribution from two gamma states for binary iron-nickel alloys.

-2

Fm. 2. Calculated values of the entropy contributed by two gamma states.

I--XNi

543

ALLOYS

AE

q Experimental

AHr(calc)

AH,

+820 +820

+450 +405

Figure 4 shows a comparison of the excess heat of formation obtained by Kubaschewski and StuarP) at 105O’C with the values calculated by means of equation (8). The agreement is satisfactory and reinforces the conclusion that anomalous thermodynamic values in gamma iron-nickel alloys are associated with the presence of two gamma states. This hypothesis can now be tested by attempting to calculate the excess entropy and enthalpy values observed in ternary systems. THE EFFECT OF CHROMIUM ON ENERGY DIFFERENCE BETWEEN TWO GAMMA STATES

-;5

THE THE

Following the procedure used by Weiss in the binary system,(3) it seems appropriate to use the magnetic properties of the ternary alloys to estimate (AEPe-cr-Ni). According to Weiss and Tauer(*) the Curie temperature T, is related to the Bohr magneton number (/?*) and the number of nearest neighbours with spin parallel to the central reference atom (IZt - 211) through the empirical expression:

(4)

T, = 113.5 I.Zt -

ZJ.1In (/?* + 1)

(3)

Table 4 shows that this relationship is held remarkably well in a variety of alloys. In the case of binary nickel alloys p* and 1Zt - .ZJl are related to the ratio TABLE 4. Comparison of experimental and calculated Curie temperatures’5l Alloy 0.20

0.40

0 60

O-80

XNL

Fro. 3. Comparison of calculated and experimental excess entropies in binary iron-nickel alloys. * Note that f(yF) this concentration.

= f below (X = 0.30) and (1 -f)

above

Ni CiFe @e lJ1Fe Fe,.&r,., Cr

I% 0.6 2.22 2.8 NO.5 0.9 0.4

IZt -

215-1 T,(“K)calo. 12 8 12

640 1061 1820

4 8 8

-153 582 306

T,(“K)obs. 636 1040 ~1800 (extrap) -80 580 312

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METALLURGICA,

ofthe two gamma phases by the following equations :t3)

[,.6X+ 2.;;1;a;’ + “y+6’“] (10)

/I* = and

(11)

For ternary additions of chromium these relationships become : 2.8(1 - x - y)

P =

[0.6X + +

0.5(1 -

(1

+

X -

(1 +

a)

Y)a -

a)

pPY

and I.???-

211 = 12 [l x

1

(12)

1

(13)

(l ;l”,;,ay)]

1 _ 2(1 -

X + KY)

(I + a)

TABLE 5. The reduction in the average magnetic moment produced by chromium f.c.0. in alloys

Ni-Cr (11) Co-Cr (12) yFe-Cr(os10) Fe 45%Ni-Cr

PB@)

-4.4 -6.6 -6.4 -5.7

(9)

The calculated value for Fe-Cr is based on a relationship developed by Goodenough’2) pp

= -(Vi TABLE

+ M) + 2n,l

18,

(14)

1970

where V1 is the number of outer electrons in the gaseous solute atom, M is related to the number of holes in the d and s shells of the solvent and equals 2 for iron, and ndl is a measure localised d electrons taken as 0.8 from the value in the nickel-chromium system.(2) The average effect of Cr in gamma iron-chromium appears to be ~-6.4pu, whereas the value in the nickel-chromium system is ~-4.4 ,~n. A weighted value between these two limits has therefore been used according to the Fe/Ni ratio in the alloy. Experimental values of the variation of T, and the saturation magnetisation at room temperature BRT for iron-nickel alloys containing up to 20% Cr have been determined by Jackson and Russell.(s*lo’ Because the Curie temperatures of many alloys is near room temperature, the values of Is,, must be corrected to yield the saturation magnetisation PO at O°K.(lO) PROCEDURE

where the symbols remain as defined previously with the addition that y equals the percentage of chromium and psc’ is the reduction in the average Bohr magneton number produced/mole of chromium atoms. Table 5 shows that it may not be possible to assign a constant value to pBC’ in the ternary system.

Alloy system

VOL.

FOR

THE

EVALUATION

OF

AE

In principle AE can now be obtained by evaluating the parameter a from equations (9-13) and then substituting this value into equation (4). Examination of equation (10) shows that an iterative method must be used, since the value of @* to be substituted into equation (9) already requires prior knowledge of the value of a, and information on the value of 1Zt - .ZJ[ is not independently available. Consequently the following procedure has been adopted : 1. Experimental values of Do and T, from Jackson and Russell have been entered into equation (9) to yield a value of [Zf - ZJl. 2. This value is then entered into equation (11) to yield a value of a. 3. This value of a is entered into equation (10) to yield a value of p* appropriate to T,. 4. The process is repeated with the new value of /3* so obtained, until the value of p* remains essentially unchanged. 5. The value of a which leads to a constant ,!I* is

6. Equilibrium between two gamma states for iron-nickel-chromium alloys

%Cr

%Ni

T,(“K)

ART

DO

,8*

5 5

30 40 50

335 555 675

0.40 0.95 1.10

0.90 1.03 1.13

1.22 1.14 1.11

5 5

60 ::

725 690 590

1.05 0.85 0.62

1.06 0.87 0.65

1.06 0.87 0.65

10

30

265

0.20

0.40

0.85

::

40 50

435 540

0.55 0.70

0.63 0.76

0.86 0.77

IZf -211

a-l

%?JPM(TA

6.5 3.8 8.1

:.: 5:7

85

9.0 9.8 10.2

10.7 :z

is 97

2.4

71

12.5 5.5

85 92

63:: 7.7

MIODOWNIK:

FREE

ENERGY

OF

AUSTENITIC

Fe-Ni-Cr

545

ALLOYS

requires a combination of the new values of AZ with the appropriate value of (g,/gs) and the total fraction of ferromagnetm gamma :

AE

AS~-Ni-C*(alloy) = R(1 x

[l/(1 + cr)][ln(l + a) -

XNi -

yo,)

(a/l + a)AE/RT]

(15)

These values of AS@e-Ni-Cr can then be converted to excess entropies AS~-N’-C’ by the same method used for the binary alloys (Fig. 2). For this purpose it is merely necessary to calculate the effect of chromium on the terminal value of AL~‘~-~‘, as the value of ASg-c’ will remain zero (Table 8). The values of (gr/gc) used for the ternary alloys are based on the calculated value for pure iron adjusted in proportion

2000

IO00

TABLE 8. The effect of chromium on the magnetic entropy of gamma iron Cr%

AE

0 5

x

Boo

Ni

Fro. 5. Variation of the energy difference between the two gamma, states with chromium content.

then entered into equation (4) setting the temperature equal to the Curie temperature T,.* Although chromium markedly reduces the saturation magnetisation of these alloys (Tables 5 and 6), it is clear from Fig. 5 and Table 7 that chromium increases the stability of the ferromagnetic gamma state. This unexpected result nonetheless yields the correct form of the dependence of thermodynamic quantities in the ternary alloys as described below. TABLE 7. Calculated values of the Energy difference between two gamma states as a function of chromium content %Ni 30 :o”

0 %Cr

AE ccl/mole 5 %Cr

lO%Cr

0 1180 1890

382 1430 2360

460 1530 2740

EXCESS ENTROPY CONTRIBUTION ARISING FROM TWO GAMMA STATES IN IRON-NICKELCHROMIUM ALLOYS

Calculation of the entropy contribution follows the same lines as for the binary system, and merely * Weiss’s) has pointed out that the appropriate value of (gl/gO) under these circumstances tends towards 1.0, because the degeneracy of the ferromagnetic state will drop sharply when spins become ordered at T,. A value of gr/g,, of 1.0 was therefore used for the binary Fe-Ni system, although strictly speaking this leaves out of consideration changes in the value of go. For purposes of comparison a value of g,/g, of 1.00 has also been used in the present calculations, which means that the results represent an upper limit to the derived values of AE.

820 460 -&zl

81/%

ASdl@

1.60 1.50 1.00 0.85

-1.32 -1.33 -1.24 -0.64

to the saturation magnetisation of the alloys. The large effect of chromium on this parameter is due to the high value of pBCr(Table 5). AE values for alloys containing more than 10% chromium have been estimated by extrapolating the rate of change with chromium content in Fig. 5 which is -750 Cal/mole for a 10% chromium addition. Table 9 shows the relative effect of the various parameters concerned on the value of AS, and (Fig. 6) shows the correspondence between calculated and experimental values. TABLE 9. Parameters contributing to the excess entropy of alloys containing 60% nickel and O-20% chromium Cr%

AE

91/%

0 5 10 20

2360 2735 3110 3390

0.60 0.65 0.70 0.85

ASar -0.05 -0.02 -0.01 -0.004

ASrcalc

ASrobs

-0.47 -0.64 -0.49 -0.30

-0.40 -0.48 -0.44 -0.33

ENTHALPY EFFECTS ARISING FROM TWO GAMMA STATES IN IRON-NICKEL-CHROMIUM ALLOYS

Since the effect of chromium is to markedly decrease the heat of formation of iron-nickel allays,(5) it is at first sight difficult to relate this to an increase in the value of AE on adding chromium : AHg-Ni-Cr = (1 -

X,,

-

yo,) [l/l

+ a]AEFe-Ni-cr

(16) The marked effect of chromium on the ratio (g,/gs) affects the value of a and offsets the change in AE to some extent, as do changes in the ratio of iron and

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0 calculated

“I. Cr Fm. 6. Comparison of calculated and experimental excess entropies in alloys containing 60%Ni and O-20%Cr.

nickel atoms, the enthalpy

but it is clear that the contribution

to

of the two gamma states is by no means

the only factor involved. It is nevertheless instructive to see what proportion of the overall change due to chromium additions can be attributed to changes in the relative proportions of the two gamma states. Values

of AH,

are given

in Table

10 and (Fig. 7)

shows how this variation compares with the overall change in a series of alloys with constant nickel content.

“I, Cr Fm. 7. Comparison of experimental heats of formation and the calculated enthalpy contribution from two gamma states in alloys containing 40 %Fe and O-40 %Cr.

Using the value for the binary iron-nickel

0 constant A constant

TABLE 10. Enthalpy changes in ternary ironnickel-chromium alloys Composition

AE

AH,w talc

%YF

AHu obs

0 %Cr 10 %Cr 20 %Cr 30 %Cr

40%Fe 40%Fe 40%Fe 40%Fe

2350 2650 2700 2300

32 31 30 27

- 740 -825 -805 -601

- 1000 -510 -30 t-770

10 %Cr 20%Cr 30 %Cr 40 %Cr

6O%Ni 60y’Ni 60 %Ni 60 %Ni

3100 3400 3900 4600

24 16 8 0

-755 - 535 - 320 0

- 740 -80 + 340 t1050

40°/,,Fe 60”/oNi P

alloy as a baseline, the rise in AH with the addition of chromium

can be corrected

with chromium

content.

tant curve is practically with

constant

indicating

nickel

for the change in AH,

(Fig. 8) shows that the resula straight line, both for alloys and

constant

that the curvature

iron

at low chromium

con-

tents can be attributed to the two gamma state equilibrium. Kaufman (l’) has made the interesting suggestion

that

a

significant

proportion

20

contents,

of

the

30

%Cr FIQ. 8. Variation of the heat of formation with chromium content in ternary iron-nickel-chromium alloys after correction for the contribution from two gamma states.

remaining linear heat of mixing term can be attributed to the enthalpy difference between b.c.c. and f.c.c. forms of chromium. This is an attractive possibility, since chromium is one of the most stable b.c.c. elements, and there is a considerable increase in

compared with the effect of chromium in (Fig. S), which is equivalent to 3000-4500 Cal/mole.

internal energy if it is forced to assume an f.c.c. configuration. Kaufman has nreviouslv estimated

The number of extrapolations involved in the analysis preclude

I

Y

AH&Y$Cr

to be ~2500

DISCUSSION

cal/mole,@)

AND

which

may

be

CONCLUSION

and interpolations a high accuracy in

MIODOWNIK:

FREE

ENERGY

OF

AUSTENITIC

Fe-Ni-Cr

ALLOYS

547

extending the results too far away from the alloys for M, values. This effect has so far been rationalised by Kaufman for the Fe-Cr system by invoking a regular which accurate magnetic information is available, solution model.(“) but there is no doubt that the concept of two gamma The difference in energy between the two gamma states provides a consistent explanation of the states in pure iron is relatively small +420 cals/mole) thermodynamic anomalies found in the iron-nickel and iron-nickel-chromium systems, and deserves to although this is considerably larger than the changes in energy which accompany the y-fa transformation be extended to other alloys.* The basic reason why (200-250 cals/mole). The effect of different solute there is a large anomaly in the iron-nickel system additions on the energy gap and the simultaneous lies in the fact that the difference in energy between the two gamma states changes sign, coupled with the effect on (gi/g& via the effect of solutes on the Bohr fact that the ratio (gr/g,,) is large and reverses simul- magneton number can quite easily produce marked changes in relative stability with very small solute taneously with the energy change. The iron-cobalt system offers the most likely alloys in which a similar additions, and must clearly be taken into considerasituation may arise, since it is known that there is a tion in any thermodynamic treatment of iron alloys. region in which the Curie temperature and the ACKNOWLEDGEMENTS saturation magnetisation vary in the opposite way The author wishes to thank Dr. Kubaschewski of with iron content@) in the same way as in the vicinity the National Physical Laboratory and his co-workers of 30-60% iron-nickel alloys. It is proposed to for providing access to unpublished information on extend the analysis to iron-nickel-cobalt, iron-nickelthe iron-nickel-chromium system, and for valuable manganese and iron-nickel-carbon alloys, since these discussions which cleared the way towards the treatexhibit different effects of the third element on the ment outlined in this paper. Thanks are also due saturation magnetisation. to Mr. D. Barrow of the University of Surrey for Of particular interest is the implication that values assistance in preparing a computer programme to of AE may be derived from magnetic data from check preliminary calculations. ferromagnetic gamma alloys, and subsequently applied REFERENCES to the properties of gamma at the iron rich end of the 1. L. KAXJFMAN, E. CLOU~HER~Yand R. J. WEISS, Actu Met. systems concerned. In this way it should be possible 11,323 (1963). 2. J. B. GOODENOUQH,Magnetism and the Chemical Bond. to see whether the effect of alloying temperature on John Wiley (1963). the equilibrium of the two gamma states plays a 3. R. J. WEISS. Proc. chue. Sot. 82.281 (1963). 4. 0. K~BASCX~WSKIand W. SLC%I, Piog&& in Mate&G significant part in the properties of systems exhibiting Science 14 (1)3 (1969). a gamma loop. These systems which include chrom5. 0. KUBASCEEWSKIand L. E. H. STUART,J. them. Engng Data 12 (3) 418 (1967). ium, are known to show anomalies in the sense that 6. C. G. SH~~.L and M.’ K. WILKINSON, Phye. Rev. 97, the solute appears to stabilise the u phase at high 304 (1955). 7. E. I: KO~DORSKY and V. L. SEDOV, J. appl. Phys. 81, temperatures, but act as gamma stabilisers at low (6) 331 S (1960). temperatures on the basis of evidence supplied from 8. R. J. WEISS and K. J. TAUER, Phys. Rev. 102 (6) 1490 * After completion of this paper, the authors attention has

been drawn to a calculation of the twospin state entropy contribution to the excess entropy of Fe-Mn alloys.o6 The equation used, S = R (y AE(RT)-i - In (A - y)) transposes to equation (3) of the present paper with y = a/l + a. Excess entropies calculated for this system also oompare favorably with experimental values:-

(%

Calc ASr(15)

Obs ASr(l6) (cel/mol “K)

20

-0.7 -1.0 -1.0

-0.6 -0.8 -1.0

::

(1956). 9. R. M.‘BOZORTH, Ferromagnetisna.Van Nostrand (1961). 10. L. R. JACKSONand H. W. RUSSELL, Instruments 11, 280 (193X). 11. V. MARIAN, Ann. Phye. 7, 459 (1937). 12. J. CRANGLE. Phil. Mau. 2. 659 (1957). 13. W. S~ouan, P. J. SPENCER and ‘0. KUBASCEEWSKI, J. Chem. Thermodynamics 2, 117 (1970). 14. L. KAUFMAN, Trans.metalZ.Soc. A.I.M.E. 215,218 (1969). 16. L. KAUFMAN,Condensed State Reactiona at High Pressures, p. 70 (edited by W. M. MUELLER). Gordon and Breach (1967). 16. P. ROY and R. HULTOREN,Trans. metall. Sot. A.I.M.E. 238, 1811 (1965). 17. L. KAUFMAN, Private communication. 18. L. KAUFMAN. Ph&seStabi&y in Metals and A&ye p. 140 (edited by P. S. RUDMAN et al). McGraw-Hill (1967). \----I.