Ordering and segregation reactions in b.c.c. binary alloys

ORDERING

AND

SEGREGATION

REACTIONS

GERHARD

IN B.C.C.

BINARY

ALLOYS*t

INDEN:

Within the Bragg-Williams-Gorsky approximation the most stable atomic configuration3 of b.c.c. solid 3olution3 will be determined in dependence on two energy parameters 11’7”’and W’“‘, the interchange The most stable atomic configuration3 of energies between nearest and next-nearest neighbors. homogeneou3 solid solution3 (configuration diagrams) are presented for a representative 3et of values out of the whole range of W1 and W (*I. From these results the most stable states of solid solutions, if heterogeneous concentration distributions are permitted (in the pha3e diagrams), are determined. REACTIOSS

D’ORDRE

ET

DE

SEGREGATIOS D-ASS DES CGBIQL-ES CESTRES

ALLIAGES

BISAIRES

Dan3 l’approximation de Bragg-William+Gorsky les configurations atomiques les plus stable3 ont et& determinees pour de3 alliages binaires cubiques cent& en fonction de deux parametres Bnerge. tiques Wr’ et W2’ repr63entant les energie3 d’echange entre premier3 et 3econds voisina. Pour un choir repre3entatif de valeurs de ces parametres les configuration3 le3 plus atables d’alliages homogenes A partir de ce3 resultat les &at3 lee plus stable3 ont et6 determin6es (diagramme de configuration). de3 alliages 3an3 re3triction d’homogeneite (diagrammes de phases) ont et& cleduit3. ORDSUSGS-

T_XD ESTMISCHUXGSRE_~KTIOSES

IS

K.R.Z.

BIS:iRE?r’

LEGIERUXGES

Die stabilsten _1ltomanordnungen in k.r.z. Mischkri3tallen werden in der Bragg-Williams-Gorsky Saherung in Abhangigkeit van zwei Energieparametern W”) und W*‘, den Austauschenergien zwi3chen nachsten und iibernachsten Sachbarn, ermittelt. Fur eine reprasentative Auswahl fiir 1V”’ und W2’ au3 dem gesamten Wertebereich werden die stabilsten dtomanordnungen fiir homogene Mischkristalle (Konflgurationsdiagramme) dargestellt. Mit Hilfe dieser Ergebni33e werden die stabilsten Yischkristallzustilnde, rrenn heterogene Konzentrationsverteilungen zugelassen 3ind (ZustandJdiagramme), bestimmt. INTRODUCTION

The descriptions of the ordering reactions in b.c.c. binary alloys by means of model calculations generally deal with the hypothetical case of homogeneous concentration distribution in the alloy.(‘-l”) The results may t.hen be summarized in a configuration diagram which indicates the most stable atomic configurations in a homogeneous alloy in dependence on temperature and composition. In addition, at each temperature and composition Dhemost stable atomic configurations are described by means of suitable order parameters. If the const,raint of homogeneous concentration distribution is removed(ll--15) it is possible to include also segregation reactions by considering the free energies of the most stable confignrations in dependence on composition. This leads to the phase diagram. The ground states of b.c.c. binary alloys’16*17’have already been analyzed for the whole range of values of the interchange energies Wci) = -ZV$& + VyL + V& i = 1,2 between nearest (nn) and nest nearest (nnn) neighbors. The results are summarized in Fig. 1. Only for positive values of WI) and W(*), i.e. for ordering tendency in nn and in nnn, homogeneous ground states are predicted. If there is any segregation tendency in either neighborhood ( Wi) < 0) heterogeneous ground states are predicted. In the following these results will be complemented tith the numerical * Received December 20, 1973. t Presented at the ‘International Sympo3ium OrderDisorder Transformation in Alloya,’ organized by Deutsche Gesellschsft fur Metallkunde, Ttibingen 1973. $ Msx-Planck-Institut fur Eisenforschung, Dusseldorf, Federal Republic Germany. ACT.1 2

JIETALLLRGICA,

VOL.

22, AE’GGST

1954

946

determinat.ion of the evolution of these ground states with temperature presented in form of conf@ration and phase diagams. Despite the crudeness of the approsimation the Bragg-Wlliams-Gorsky (BWG) formalism has been adopted in the numerical treatment in order to reasonably limit t,he amount of computational nork. Furthermore, much more elaborated computations based e.g. on Kikuchi’s cluster variation method give qualitatively the same results as the BJ’X treatment as has been sholv-n by Sate and KikuchP) and van Baal(ig) for t’he particular case ?W = 1.575PJ. The quant,itative discrepancies which concern t.he ratio of thermal energy at t,he transition temperature and the energy difference between the ordered and the disordered state are known.(20’ DESCRIPTION OF THE CONFIGURATIONS

ATOMIC

The various atomic configurations in the tist and second coordination sphere of b.c.c. alloys are commonly expressed in terms of occupation probabilit’ies pJL, pBL of the components g and B on four f.c.c. sublattices L = I, II, III, IV.(3-g) The lattice sites on I and II are nearest neighbors to those on III and IV whereas the sites on I or III are next nearest neighbors to those on II or IV, respectively. The nomenclature used for t.he description of the ordered configurations has been chosen in terms of face centered space groups consistent with the choice of f.c.c. sublattices as basic units (Struliturbericht-symbols in

-ACT_4

METILLTRGICI,

VOL.

-‘?,

197-1

of a hypothetical alloy which is not alio~~ed to undergo any decomposition wiII be determined b- the standard method of minimum search of the configurational part of the free energy-‘i,sJ

SkT + -y--

PIG. 1. Ground 3tates (T = OK) of b.c.c. alIoy3 in dependence on the values of the interchange energiea WC” and V*‘?’ between nearest and nest-nearest neighbors.

brackets) Fdk

:

(47) random pAI

Frn3c

zz

distribution

in jztb and in HIM

p_yr= p;y =

(B2) order in nn, random II i

p_<1c

distribution

in tltzn

III _

PA= = PA -r PA - P,Ir FmSttt. (DO,) order in WL and in nnn I_

II,

xv

zpy #PA PA --PA (DO,*)f order in IM and in ~TUE. P-41 = PA” = P;II f P&Iv F93 ~7,

order in ~b and in mn &I

= py

f

P;1

II _c

T- PA

IV

Fd3t)z (332) random

distribution in nn, order in tlnn III I PA -PA fP_1 11 zz.zP,‘v. I _

Since only three of these occupation probabilities may be chosen as independent variabIes it is useful to introduce t,hree long range order parameters, x = $(pA1 f pA1’ - pTjl - p,“) describing the order in nn, and !J = j(pf:‘: - P_~~‘), z = ;S(pp,I - pAI’) describing t,he order in rmn.(7-9) TEMPERATURE DEPENDENCE EQUILIBRIUM STATES

OF

THE

The determination of the ~emperat~e dependence of the most stable state of an alloy will be done in two steps : In a first step the most stable at,omic configuration t This particular superstructure which part.icular set of values W”J = l,5Wa) > by an asterisk in order to distinguish from DOZ. Both superstructures may be each other (see Fig. 4).

arises only for the 0 has been marked this superstructure in equilibrium with

3 (PAL h PA L f

pBL In pEL).

(1)

The first term Cko is the internal energy of bhe pure components, the second term is the energy of mixing, the t.hird t’erm gives the energy contribution due to ordering and the fourth term is the entrop- contribution of t,he configuration which may also be expressed in terms of x, y, 2 (equation -i in Ref. 9), cA and cg is the composition, k is Boltzmann’s constant. If at a given temperature and composition the necessary and sufficient equilibrium conditions (e.g. see Ref. 9, p. 630) admit. one unique solution x~~~,(c~, T), T) these values correspond to the Ymin(CB, T, , ~~~~~~~~ most stable state of the homogeneous alioy. If several solutions sat,isfy these conditions, one of them gives the lowest value of free energy. This solution then corresponds to the most stable state of the homogeneous alloy-, the other ones represent metastable states. These states obtained by numerical techniques wilt be summarized in the configuration diagrams. The second step consists in an examination, for each t.emperature, of the free energies of the most stable atomic configurations F(c,, T, rainy Jtmin, zmin) = T). The curve 1Pmin(Cg’ T = const) is then Fmin tCBt examined to see whether it exhibits inflection points or not. If there are inflection points then a decomposition in two phases given by the common tangent const~~tion kt.h different compositions and different These results wiI1 be degrees of order is predicted. summarized in phase diagrams. NUMERICAL

ANALYSIS

The analysis of the most stable atomic configurations of alloys with homogeneous concentration distribution reveals both types of transitior@+ between the different ordered states : Continuous transitions characterized by a continuous variation of the internal energy during the transition (frequently called ZsD order transitions(21)). The transition temperatures separate regions with different symmetry ofthe configurations genera& characterized by a temperature dependent variation of a typical order parameter below the transition temperature and a constant value of this order parameter above the

ISDES:

ORDERISG

AXD

SEGREG_1TIOS

transition temperature. These transition temperatures are indicated by hatched lines in the cotiguration and phase diagrams, t>hehachure indicating the region of variation of the Qpical order parameter. Discontinuous transitions characterized by a discontinuous variation of the internal energy during the transition. At the transition temperature two different, atomic configurat.ions with different degree of order but with hypothetically Used and equal composition are in equilibrium with each other. These cotigurations continue to exist as me&table states in a narrow range beyond the transition temperature. The temperature of equal free energies of the two atomic structures is indicated by solid lines in the configuration diagrams. In most cases the discontinuous transitions could be analyzed at a given temperature and composition due to the obtaining of two solutions, one stable and the other metastable. The reversion of the characteristics of stability of either solution indicates the transition. A well discernible discontinuity of the internal ener,7 has been obtained in these cases. In some dubious cases, when the temperakre range of the coexisting stable and metast,able solutions was too small to be detected numerically, the discontinuity of the int,ernal energy at the transition temperature has been rerealed by eraluating the stable solution in temperature steps down to 4.10-s T/T,(0.5), i.e. units of the maximum transition temperature TK which in all cases arises at cg = 0.5. In principle, the continuity of a transition cannot be proved by this stepwise analysis. Nevertheless, for continuous transitions a decision could be obtained from the stepwise analysis due to the different variation of the internal energy: in this case the values of the internal energy from above and below the transition temperature tended to a common value at the transition temperature which could be determined in this may with a numerical accuracy of about 4.10V8 T/T&0.5). _tb example of such an analysis is given in Fig. 2. Obviously at cB = 0.1 the transition DO, % B2 is a discontinuous transition whereas at cg = 0.2 this transition is continuous. For the particular values of W(l) and W2’ chosen in this instance the nature of the transition can also be obtained in a rigorous way by analytical methods which leads to the same results.“’ .A discontinuous transition of the homogeneous alloy always implies t,hat heterogeneous t.wo phase transitions do occur in the phase diagram if heterogeneous concentrations are permitted. In t,his case the transformation is of first order and the two atomic structures are separated by a tno phase region. The trro

REACTIOSS

-50

IX

___9

B.C.C.

.f‘___

--TGtemperaiu:a

BIS_kRY

ALLOTS

917

_- _/ 5’:

___

s37907: in units of TK I S,

FIG. 2. Temperature dependence of the configurational portion of the internal energy UI during 8 discontinuous tr8nsition B2 + DO, at cB = 0.1 and & continuous trensition B2 Z+ DO, at c, = 0.2 for values of the interchange energies WI” = 2W”*’ > 0.

phases now differ in composition as well as in atomic configurations. The phase limits of the two phase regions are indicated b>- solid lines in the phase diagrams. Heterogeneous two phase transitions also occur in some cases where the corresponding transformations in the con@auration diagram are continuous, but in most cases continuous t.ransitions of the hypothetically homogeneous alloy remain single phase transitions, when the constraint of hom0geneit.y is removed. These transitions are again represented by hatched lines in the phase diagrams. RESULTS OF NUMERICAL CALCULATIONS

In order to get a representation of t.he diagrams independent of the numerical values of the interchange energies all temperatures are presented in a relative scale. The reference temperature is gi-ien b- the maximum order-disorder t,ransit,ion temperature T, which always occurs at cB = 0.5: T,(O.S) = (2W(l) - 1.5P’)/k for W(l) > 0, V(l) > 1.5 FFF”’and T,(0.5) = 1.5W”J/k for JV2’ > 0, W1’ Q 1.5JV2). Since all diagrams are s_ymmetrical with respect to cB = 0.5, only one half of t.he diagrams is represented in Figs. 3-5. Figure 3 shows the resuks for the range of interchange energies WI) > 1.5 JVc2)> 0. In t,his range the ordering energy in nn prevails over that in nnn and two types of ordered atomic configurations, B2 and DO,, are predicted. In particular Fig. 3(a) gives an example of entirely cont.inuous transitions in the confi,aration diagram and a two phase region BZ + DO, in the phase diagram. For laxer values of W(l) (Fig. 3b) the extent of this two phase region increases and the transition temperature B;, + DO, is raised. The transition t.emperature A2 + B2 therefore

3(a).tt'rl' = ?.?Tv'"' > 0.

FIG.

Fro. 3(b). iPi

= 2.OW’f’ > 0.

&gram

ccnf+prakoo

G

.5

.25 froc!lcn

FIG.

strikes the two phase region giving rise to a two phase region A2 + DO,. A further reduction of the ordering tendency in nn extends these two phase regions to higher temperatures (Fig. 3~). For values It’(r) only slightly above 1.5fW in addition to the two phase region Ai” f DO,: a further two phase region DO3 -i_ B2 occurs near the equiatomic composition which leads to a miscibility gap when the transition B” + DO, strikes the tlvo phase region DO, + B2 (Fig. 3d). For the particular value V(l) = 1.51V2) > 0 the free energies of both ordered configurations B2 and 332 become equal and both atomic structures are equally stable. The same holds for DO, and FZ3m. The resulting diagrams are represented in Fig. 4. It is to be not.iced that, with the above values of the int,erchange energies the mathematical formalism becomes formally analogous to that of t,he f.c.c. lattice.(3*12*22)t If the ordering energy in nnn prevails over that in w, 0 < G’(l) < 1.5Ws), three different ordered configurations do occur, DO,, FZ3m and B32 (see Figs. 5a-c). With decreasing values for V(l) the extent of the two phase regions decreases. For ‘II”(i) o 0.8W(2) only cont.inuous transitions do occur in both diagrams (e.g. Fig. 5~). For negative rakes of one of the energy parameters the numerical results are similar in the two cases -l.5W2) Q IV) < 0, FPr) > OorflW) > 0, ?V*) < 0] and they may be represented simultaneously in one diagram (Fig. 6a-cl). Therefore the inscriptions in the diagrams are labelled by two sets of interchange energies one of them being inserted in brackets. The same indication of both sets has been used in the text.

.25

G Cf subs!aoce

.5

B

~~~~

3(c).WC"'= I.BOTV?'> 0.

0

.ii

.5 fraction

.25

C of substance

.5

8

FIG. 4. Configuration and phase diagram for W1’ = 1.6W*’ > 0, T,(O.S) = 1.5W*‘Jk. The ordered states of twypeB7 and B32 or DO, and FY3, exhibit equal Vah~8sof free energy for these vatues of the interchange energies. .5 frcctlon

FIG. 3(d). W”’

.25

G Cf substance

= 1.58W”

.s

a

> 0.

FIG. 3. Configuration and phase diagrams for ordering tendencies between nearest and next-nearest neighbors: W”’ > l.ZiW* > 0. The maximum transition temperature T,(0.5}is given by T,(0.3) = [2W(11 - 1.3W(a1)jk

t This may be seen if in equation (1) the order parametem r, y, z are substituted by the probabilities PAL and ?V” is substituted by WxL and then equation (1) is compared e.g. with equations (?) and (3) in Ref. (22). In this instance the ratio of the interchange energiesjust compensates thedifference in coordination number in ran and in NUL In this formal tlnalogy the coniigurations A?-_4 1, DO,-LL,, {BZ,B32f-Ll, correspond to each other.

ORDERISG

ISDES:

c

_kXD

~~CC!:CP

.25

‘J

.j

.25

SEGREGATIOS

3’ subs!arce

3

REACTIOSS

IS

If”’

= l.OV’”

configuration phase

BISARY

_%LLOTS

949

The diagrams represented in Figs. 6(b-d) confirm the previous results(16*17) that no homogeneous ground states do occur if one of the interchange energies is negative. For t,he particular values Wt-cl)= - 1.5 W(f), W-ra)> 0 or [IV) = 0, W2) < 0] (see Fig. 6d) the exceptional situation arises that the free energy values of three phases, namely disordered A-rich, ordered B31 stoechiometric [or B2 stoechiometric] and disordered B-rich are equal and lie on a commdn tangent to the curve Fmi,(ca) at all temperatures T < T,(0.5). This unusual situation only holds for a hypothetical alloy which fulfills the particular energy requirements indicated above and the assumptions of t-he BWG: treatment as well. In real alloys this situation will generally not be encountered, in agreement with the phase rule. conflgurotion

FIG. 5(b).

B.C.C.

Dhcae diagram

diagram

> 0.

diagram

0

.5

diagram

0

1. fraction

o!

substance

.5

?.

3

04

@I

FIG. 6(a). Con&gunition diagr8m for the range of values -1.5w*, < WI’ < 0, W”’ > 0 or [WI > 0, wz < 01. For the values V”’ = 0, W”’ > 0 8nd [ W1’ > 0, W’“’ = 0] this con6gumt~ion diagram is identical to the corresponding phase diagram. FIG. 6(b). Phasediagram for WV”’ =: -0.643 JV*‘, W*) > 0 or [yYll’ = - W*), Ivc2’ < 01. The corresponding con.@ur8tion diagram is given in Fig. 6(a). phase dicgrom

0 fraction FIG. 5(c).

.25 of substance

tV”I = 0.5W?

phase

3grom

.5 6 > 0.

FIG. 5. Configuration and phase diagrams for ordering tendencies between nearest and next-nearest neighbors: 0 < IV” < 1,6W”‘, r,(o.J) = l.sw’“‘;k.

0

.5

1. frcction

Figure 6(a) represents the co@mtion diagram for all the cases with W(l) Q 0, W(*) > 0 or [W(l) > 0, W(2)4;01,The corresponding phase diagrams depend on the interchange energies (see Figs. 6b-d). In addit,ion the particular cases W(l)= 0,W@) > 0 and [W(l) > 0, IF(*) = 0] shall be mentioned where the phase diagram is identical to the coloration diagram in Fig. 6(a). Only continuous and homogeneous transitions are predicted in this inst,ance.

fc)

0 3f sirbstance

.s

1.

8

(df

FIQ. 6(c). Phase diagram for W(l) = -W?J, Wa) > 0 or [WI” = -0.373W”‘, W*’ < 01. The corresponding configuration diagrsm is given in Fig. 6(a). FIG. 6(d). Phase diagram for Wl’ = --l.5Wz), WC* > 0 or[W’) = 0, W2) < O]. The corresponding con@uretion diagmm is given in Fig. B(8). Fro. 6. Conf&uration and phase dkgrams for segregation [or ordering] tendency in ~8 and ordering [or segregation] tendency in RW?. The maximum tmnsition temper8ture is given either by TJO.5) = 1..5W~*‘/kor by [T,(O.5) = (2W”’ - 1*5W(~‘)/k].

930

ACT.1

METALLURGICd,

VOL.

22,

1971

For values IF(l) < - 1.6 P), ?F(‘) > 0 or K(i) < 0, N”(g)< 0 the rrell-knon-n decomposition in disordered A-rich and B-rich phases represents t,he most stable state of the al10y.(~i) DISCUSSION

The results obtained give a survey of the phase relations in b.c.c. binary alloys with ordering or segregation tendency. They may be applied to real ahoy-s either with the aim to determine the phase relations of the alloy if the interchange energies W(i) and W(*) are known, or with the aim to get the numerical values of t,he interchange energies if the configuration or t,he phase diagram is known. Both procedures are affected in different respects by the fact that the short range order is not treated as an independent variable in the BWG formalism. The consequences will briefly be discussed in the following. In the BWG treatment the degree of short range order which may generally be expressed in terms of the total number of SEpairs, is treated as a function of long range order. The disappearance of long range order at the transit,ion temperature therefore implies, in this model, the simultaneous disappearance of short range order, too. However, in reality short range order persists beyond the transition t.emperature at which the long range order vanishes, It is for this reason that the transition temperature obtained in a BWG treatment with interchange energies taken e.g. from calorimetric data, is overestimated as compared to reality. This has been show-ne.g. for the transition -42 + B2 at the composition cg = 0.5, where the temperature TK(O.S) obtained from calculations which account for short range order(1s-20~23~2J) is about 70-80 per cent of that predicted by the BWG model. This suggests to remove the discrepancy by a correction factor. In the particular case W(l) = 1.875W(*~ > 0 for which the whole phase diagram has been obtained by van Baal(lB) by means of Kikuchi’s cluster variation method, this has been done successfuliy.(~~) For a given set of interchange energies the conflguration diagrams are derived from the known equilibrium conditions.(j*i*9) Generally there is particular interest in either the transition temperatures of ordering reactions and/or in the temperatures where decompositions occur. The transition temperatures may be derived analytically or numerically with moderate effort. The det,ermination of the two phase regions is generally more laborious. Therefore, in Fig. ifa, b), at least the upper temperature limits of the various two phase regions have been plotted against the ratio of t.he energy parameters. It is seen from Fig. ‘i(a) that for positive rakes of the interchange energies and a rat.io

FIG. i(a). Upper temperature limits of the two phase regions for ordering tendency in mn and in nnn. Dashed line: -42 A DO,with TJO.5) = (flW’ - l.uv”),% Solid line: ’ 32 -i_ DO, wit.h T,(O,j) = (2Wll - l.;ltp(~~)‘x: Dash-dot line: Al i DO, aithTh(0.5) = 1.5TV’?‘JL. FIG. i(b). ‘L’ppertemperature limits of the two phase regions for segregation tendency in nn [or in nnnf: A9 + B32 with T,(0.6) = 1.5W*‘/k andz, = tVl)/lf(“, W*) > 0 (or [AS + B1 with TJO.5) = (3W*) - I.5tV2’)/k and tin

out, of t,he interval 0.8 Q lP/lV(2) g 3.6 no heterogeneous two phase regions are predicted and therefore the phase diagrams are identical to t.he corresponding configuration diagrams. It is to be noticed that the phase diagrams obtained represent the phase relations only within the b.c.c. crystal structure which in this instance is assumed to be stable at a11temperatures and all compositions. In practice the stability of ot.her phases (e.g. liquid) has to be considered, too, and the above results immediately apply only for those temperatures and compositions at which the b.c.c. structure is found to be the st,able one. However, the above results may generally be used for an analysis of the relative stability vvith respect to other phases since they give the phase relations of h~otheticall~ b.c.c. solid solutions also in those temperature and composition ranges lvhere t.hey may not be investigated e~perimenta~y.(26*2i) The numerical results obtained above only concern the ordering reactions of the comp0nent.s due to “chemical” interactions. The influence of further contributions to the int.eraction energies, e.g. “magnetic” contributions, may appreciably alter the ordering reactions predicted above and have to be incorporated into the model, too. This is to be realized if experimental phase diagrams are compared with those presented above. In the particular instance of b.c.c. Fe-S alloys the magnetic contributions have been eraluated(i*) and t,he f&al results agree semiquantitat,ively n-&h the experimenta observations.(28)

ISDES:

ORDERISG

_iSD

SEGREGATIOS

ACKSOWLEDGEXENTS

The author wishes to express his gratitude to Prof. W. Pitseh for many t-aluabie discussions and a critical reriem of t.he manuscript. Thanks are due to Dr. C. ran Baal, Technische Hogeschool Delft/ Setherlands, for kindly sending the phase diagram presented at Tiibingen.(la) The financial support of the Yerein Deutscher Eisenhiittenleute and the Kommisaion der Europaischen Gemeinschaften, Dire&ion Stahl, is gatefuIly acknowledged.

RE,lCTIOSS 11. 12. 13. 14. 15. 16. 17. IS. IQ. “0.

REFERENCES 1. I-. S. GORSKE-,Z. Phya. 50, 64 (19%). 2. W. L. BRAGG and E. J. WILLIAX~, Proc. Roy. Sot. A145, 699 (1931); A151, 340 (1935); A152 231 (1935). 3. H. &TO, Sci. Rep. RITU A3, 2-L (1951). 4. S. JIxrsrr~.~, J. Phya. Sot. Japan 6, 131 (1931). 5. K. &MCHI, Sci. Reps. Tohoku Cniu. 35, 30 (1931). 6. P. S. R~Dx_%~, Beta Net. 8, 321 (1960). 7. V. V. GEICHE~KO V. 31. D~SILESKO and -1.. -4. SNRSOV, Fiz. metal. metalloved. 13, So. 3, 321 (1962). 6. D. J~EISEAFIDT sncl0. KRISEMEST, Arch. Eisenhiittentcea. 38, 293 (1963). 3. G. I~DES and W. PITSCE, 2. Xerallk. 62, 627 (1951). 10. G. I~DES and W. Pr~scrr, Z. -IfetuZZft.63, 18 (1979).

Ql. “2. 23. ‘74.

25. 26. _‘i.

2s.

IS

B.C.C.

BETdRT

_4LLOTS

9.51

R. BECSER, 2. MetaE& 29, 245 (1937). W. S~OCICLEY. J. Chem. Phua. 6. 130 (lQf8). L. &T.EKAB, bokl. dkadem.“Xauk SS$R 13i), 56? (1960). G. SCEL_*~E, G. ISDE~ and IV. PITSCR. Z. XetaU. 65, 94 (1974). G. I~DES and W. PITSCH, Chemical metallurgy of iron and steel, Proc. Int. Symp. -Metal. Chem. ShefFeld (1971) p. 314. The Iron and Steel Institute, London, (1973). M. J. RICELLRDS and J. W. Ckrs, dcta Met. 19, IQ63 ( 197 I). S. >I. _iLLES and J. w. CaaS, _-l&z _\fet. 20, 4”:~ (1972). If. Sara and R. KIKicCHI, presented at the international symposium order-disorder transformation in alloys, Tubingen (lQT3). C. Ji. vss B_uL, presented at the international symposium order-disorder transformation in alloys, Tubingen (1973). L. GCTT>L+X, Solid &We Phyaica, Vol. 3, p. 143. &xsdemic Press (1956). L. D. L%sD.~c and E. Y. LIESHITZ, StatiaticuE Phyaica. Pergamon Press (1959). V’. GMIS. 2. _~fetnZZk.84, 268 (1973). -4. MUSTER, Statistiache Thermodyrbamik. Springer (1956). G. FOCRSET, in Phase stability in mrtala a& alloys, edited by P. S. RCDU~, J. SPRISGER and R. I. J.IFFEE. McGraw-Hill (1967). G. Isn~s, unpublished article. L. K~cFM_~~ and H. BER~STEIS, Computer calculations of phaae diagrams. .kcademic Press (1970). _X. P. JIro~o~f?irhl, Metallurgy of Ironand Steel, Proc. Int. Symp. Met. Chem. Sheffield (1971). The Iron and Steel Institute, London (iQf3). G. SCHUTTE, submitted to phys. status. aolidi. (a).