The role of magnetism in the calculation of phase diagrams

Physica 103B (1981) 82-100 © North-Holland Publishing Company

THE ROLE OF MAGNETISM IN THE CALCULATION OF PHASE DIAGRAMS G. I N D E N l~;,,,, ~-Planck-Insatut [fir Eisenforschung GmbH, Diisseldorf, Fed. Rep. Germany In 1963 Prof. Meijering discussed the effect of magnetism on the shape of miscibility gaps in binary systems formed with a ferro- and a non-magn,tic component. In this treatment a Bragg-Williams-Gorsky approach for spin ~ was used for the calculation of the magnetic contributions to the Gibbs energy. Starting: from these ideas a more general but empirical treatment of the magnetic effects is developed which allows us to take account of any moment and of short-range order (sro) effects above the Curie or N6ei temperatures. Based on this treatment :theinfluence of magnetic ordering on the phase equilibria will be analyzed, including miscibility gaps and atomic orderSng. ~7or particula," binary systems with strong magnetic effects the results will be discussed in comparison with available experimental data.

I. Introduction T h e r e are two aspects of the influence of magnetic ordering reactions on phase stability of alloys which have to be considered in phase diagram calculations. First, magnetic ordering can be considered as a separate effect taking place without alteration of the other physical properties. In such cases a formal and global treatment of the magnetic effects would suffice to account for the magnetic stabilization cffect. This situation is approximately fulfilled in pure elements and r a n d o m alloys. The second aspect is the mutual influence of atomic ordering or segregation reactions and magnetic ordering. These effects occur in alloys. Their description necessitates a detailed knowled.;e of the atomic and magnetic state which depends on temperature and composition but which is not obtainable from ft rmal treatments. Here physical models need to be developed. This second aspect was treated by Prof. Meijering in his paper in 1963 [1] for binary alloys with one ferromagnetic component and with a segregation tendency between the atoms. The magnetic ordering was treated in the B r a g g - W i l l i a m s - G o r s k y approximation using spin -~. It was found that the ferromagnetic interaction produces either a rise or a fall of the phase boundary of the miscibility gap (as compared to the paramagnetic state), and in some cases the gap closes at a sharp critical point. This latter finding is not restricted to alloy systems with a segregation tendency nor is it related to the approximation used. This will be discussed later and an example of a magnetically induced two-phase fieM a r o u n d the Curie t e m p e r a t u r e will be shown. In the present paper both aspects of magnetic influences on phase stability will be covered. First, an empirically derived formal treatment of magnetic ordering contributions will be p r e s e n t e d which is particularly aonronriate for computer eait'ullatinrt¢ It t,k,~ ~""'~"nt of ti %. ..1 ..1.t ~. . •. .U ..i l.t ~ % , , t k . . ' l t.~,~_ ~_.~ t^, . . . . • t.lg.,l ~,~l...x lit...~] Cl.lilk,I ot short range order (sro) effects and it works with only three parameters: (1) the critical temperature, Tc, of lro (Curie or N6ei temperature); (2) the total magnetic entropy; and (3) the ratio of the short range order portion to the total a m o u n t of magnetic enthalpy. It will be illustrated with particular examples of how this method can be used in alloy systems. T h e n the effect of the mutual influence of atomic and magnetic ordering will be discussed. For this it will be necessary to apply physical models like the one used by Prof. Meijering. For practical applications, however, it is necessary to extend the treatment to higher spins (spin 1, -~, 2). With those higher m o m e n t s the magnetic energies also become higher and more important. This will be illustrated with an example. 82

G. lndenlRole of magnetism in calculating phase diagrams

S~

The most striking effect of the atomic order on the magnetic order (and vice versa) is on the criticaB temperatures which can be considerably raised or lowered. This suggests the use of the previously mentioned formal treatment also in cases of mutual influence simply by introducing the modified critical temperatures into the formalism.

2. Formalistie treatment of phase stability 2.1. General aspects The relative stability of different phase states is determined by the values of the Gibbs energies of the phases under consideration. These energies vary with temperature and composition in a complicated manner due to the numerous physical effects occurring in dependence on these variables. There are essentially two approaches to developing mathematical expressions for the Gibbs energy (or equivalently the entropy and enthalpy). One is a formal power series expansion with a sufficient number of adjustable parameters to obtain an adequate description. This technique, initiated on a large scale by Kaufman [2], is generally applied to the high temperature states of alloys since these exhibit sufficiently simple temperature and composition dependences to allow a truncation of the series expansions at low orders. In temperature ranges where ordering reactions are occurring this technique is not applicable in practice (at ieasl if more than an exact reproduction of already well known diagrams is envisaged). The power series is particularly appropriate for computer techniques; however it does not alk,w us to treat the various physical effects additively. Although exact additivity cannot be expected in reality, it seems to be approximately fulfilled for the specific heat [3]. Additivity is generally accepted for pure elements for the con!fibutions due to thermal vibrations, anharmonic vibrational effects, the free electrons, and also for the magnetic effects. The additivity need not to be destroyed if these effects are interdependent (e.g. an electronic s,",ecific heat dependent on the degree of ferromagnetic coupling, or a Debye temperature differing above and below the Curie temperature), since this mutual dependence can be accounted for by a relatively simple composition or the temperature dependences of the parameters entering the description of the individual effects. An alternative to the power series description should therefore start wilh the specific heat which allows us to determine all required thermodynamic functions by integration. This procedure will now be developed for the magnetic order-disorder effects. 2.2. Formalistic treatment of magnetic effects It is proposed to describe the magnetic specific heat by means of tile following formulae .4]: 1+7

cl,O=KtrORln~ P

r st° =

.3

1 -- 7.3,

/(sr°R

In 1, -J-' 7.-5

forT.=

T ~ -

for I" > 1.

<1

,

(i) (2)

where R Is the gas constant. There are three parameters in the description: T¢. K ~r'~and K .... . The latter two will be expressed in terms of generally known quantities. From (1) and (2) the magnetic enthalpy is immediately obtained by integration [,~]. One obtains: for 7.= T/Tc < I T -"

I .!

o

Cp~° d T

84

G. lnden/Role of magnetism in calculating phase diagrams

= K~ORT~[(l_r)ln(l_~.)+zln(. 1 +¢1 + f C ) + ln~,.X/r~+ 1 j 1+I" , fi + \ +X/~ a r c t g ( ~ 3 1 ) - X/3 arct g /2¢ [ , ~ J+ 1\ + 2X/3 a r c t g ( ~ ) ] ,

(3)

T for ¢ =-~-> 1" T

•.~ ~. ~,~

. r~f

c~dT

=KS'°RT~ ( r + ~ ) l n ( r + l ) - ( r - 1 ) l n ( r - 1 ) + r l n

+¢+r2+d+

~

- cos(-5)ln(C - 2r 2 cos(-~)+ 1 ) - cos(~-~)ln(,'+ 2¢2 c o s ( 5 ) + 1) +2 sin{ " for ~ ) [arctg('r - c°s(cr/5)'~ arctg(~- + cos(w/5)~] sin(or/5) ] _ sin(or/5) ]J +2 sin(~-ff)[arctg( r - cos(3cr/5)~ arctg('r+ cos(3cr/5)~] sin(3cr/5) ] sin(3rr/5) ]J + ln(-54)+ cos(5)ln(4 sin2(5)) + ~ s in(5)+ cos(-~)ln(4 s i n : ( ~ ) ) - ~-s,n~T,].t. or. [3¢r\1

(4)

The pure long range and short range order contributions are then: AH~,O(Tc) = 0.645K~ORTc, A!l,,~o(oo)= 0.598K, rORT¢. In a similar way as for the enthalpy, the entropy can be derived from (1) and (2) by integration. Here again the entropies of Iro and sro will be evaluated separately: T

AS~(T) =

dT, for T < Tc, 0 T

f -~-t" dT, SrO

A,.c;S~(T) =

for T > Tc.

rc

These integrals can immediately be tra~sformed to the series [4]: A$ (~) - KI¢°R - --3--

AS(T)~r"_ K ~°R_g,IF(l)- F(z-5)),

(5) (6)

where

2(:,. For 0 -< y _
G. Inden/Role of magnetism iv~ calculating phase diagrams

85

term. In order to get the same accuracy with six terms in the range V ' 2 - 1 -< y _< 1, one should make use of the following relation which F ( y ) fulfills: ,/r 2

The total entropies of lro and sro turn out as: AS~°(Tc) = S ( T c ) - S(O K) =

aS=o(oo) = S(oo)- S(Tc)=

KlrOqr2R 12 '

K~ro,/r2R

20

"

The parameters K ~° and K ~ can be expressed in terms of A S lr°'*di' - A S ~ c ) + AS(%r~ and

A H k°'*dis = A~r-l~ + AH(==°).

Instead of the latter quantit~ it is preferable to introduce the fraction f defined as

f=

÷,tu,,o,

which is the ratio of the magnetic enthalpy due to sro to the total amount of magnetic enthalpy. One obtains: (7)

K "~°- 0"784fAS°~-'di~/R 0.598 - 0.211f " KirO = A$°'d"eiS/R _ 0.493K =o O.822

(8)

The fraction f is expected to vary in the range 0 . 5 < f < 0 . 9 , depending on the crystal structure (coordination number) and cn the magnetic moment [5, 6]. The magneuc Gibbs energy " ~ G is generally expressed with reference to the completely random (i.e. paramagnetic at T = 0o) state. With this reference state any magnetic ordering produces a stabilization effect. It follows that for T < Tc:

masG =

AH~

,~o + A H ~ ) - (AH(To

T a S ~,o tr),

(%)

and for ,'- > T~: " n G = A N N ) - AH~®~)- T[AS~¢,) + ASN)].

(9b)

2.3. Determination of the parameters A,~°r't''di' a n d f

In the =implest case of a pure component with an integral number/3 of Bohr magnetons, the t(~tal magnetic entropy is directly given by ASO,a-,~

~ ~ = R In03 + 1). -- S(T=•)-S(T=0K)

(10)

In most practical cases, however, non-integral numbers are found experimentally. In this case it has been proposed [7] to use (10) and to take for fl the experimental value. This procedure can directly be extended to alloys if the atoms are assumed to contribute individually

,%

G. lndenlRole of magnetism in calculating phase diagrams

to the magnetic entropy. Eq. (10) shou',:! then read for alloys: AS °~-'e~' = R ~ x , In(r, + l),

(11)

where/3~ are the individual moments (in/~B) of the component atoms i with mole fraction xi. If the individual moments ]3~ are not known, A S °'~-'~s can be approximated by the expression AS °'n-'~' = R in((,8) + 1), with (fl:~ representing the average magnetic moment per atom which can be deduced from magnetizath,J ~ a s u r e m e n t s . This expression cannot be used for elements or alloys witlh anti-ferromagnetic ordering tendency since there (/3) will be small or zero whereas the individual fl~ are different from zero. Th~ fraction f is a formal parameter which should be determined empirically. At present the values f = 0.4 ~or bcc and f = 0.2L for fcc alloys are suggested. ?i~ey have been derived by fitting formulae (1) and (2) to the magnetic specific heats of Fe, Co and Ni [4]. In figs. 1 and 2 these fits are shown for Fe and Ni. The calculated curves were obtained with the following parameters: Fe:

T~ = 1043 K;

f = 0.4;

A S °rd'~dis - -

R In(3.2);

Ni:

Tc = 631 K;

f = 0.28;

AS °~-'~ = 0.6R In(2).

The magnetic entropy for Ni was taken from the literature (see e.g. [7]). The agreement between calculation and experiment is satisfactory in both cases. We should remind the reader that the magnetic part of cm which has been separated from the experimental cp-values ([7, 8] for Fe, [8] for Ni) depends on the amount attributed to the remaining contributions. The fractions f suggested here rely on the accuracy of this separation and they might be subject to slight modifications. In order to illustrate the application of the present treatment to alloys, the specific heat of bcc Fe~,79Cr,, 2~ will be considered for which experimental data are available [9]. They are represented in fig. 3. The broken line ir, dicates the non-magnetic specific heat contribution c?~m which has been derived from the corresponding values for bcc Fe [8] and Cr [10]: c~,m(Fe, xCr~)= (1 - x)c~m(Fe)+ xc~m(Cr).

C;'°qIR} F._.~e

~ I,

..... o ~

CTgIR] cotc

exp

Ni

15

calc

i

-~

exp

o

0

2(3

3

0

05

1000

20f9 T[K]

Fig. 1. Magnetic specific heat of Fe. Full line: present calculation. Symbols: values derived from experiment by subtraction of all non-magnetic contributions (O ref. 7; • ref. 8).

0

0

/.00

800

1200

T(Ki Fig. 2. Magnetic specific heat of Ni. Full line: present calculation. Symbols: values derived from experiment by subtraction of all non-magnetic contributions [8].

G. Inder~Role of magnetism in calculating phase diagrams

$7

C~[J-tool-1.K-~I Fe~ Crz,. - -'

60

~

.

40

i

.....

_- _--" Tc=~J23K

20 I

t

i

[

[ ~t

_

500

.

.

.

.

.

.

.

1000 T[K]

Fig. 3. Specific heat of F0.79Cr0.2t. Dashed fine: non-magnetic specific heat derived by' linear interpolation of the values [or bcc Fe [8] and Cr [10]. Full line: present calculation. The magt ;ic part has been added to the non-magnetic values. Symbols: experimental values [9].

The magnetic contribution could be calculated using the following values:

Feo.79Cr0.21: Tc = 923 K; The value (fl> (fl)/Pe

=

=

f = 0.4;

AS °r*'~ = R In(1.61 + 1).

1.61 is obtained from the variation of the magnetization with the Cr-content [I1]"

2.11 - 2.36Xc,.

In fig. 3 the full line is obtained by adding the calculated magnetic specific heat to the non-magnetic values. The agreement between experiment and calculation is satisfactory. The measurement started with a quenched specimen and shows successively negative and positive deviations from a smooth curve in the range 700 K < T < 830 K. They can be attributed to segregation in the quenched bcc phase.

2.4. Application to C o - V : Prediction of unknown phase relations and their experimental confirmation

The most recent phase diagram determination of the Co-rich C o - V alloys [12] is showr, in fig 4 1400

TIK1

/ /f/'7;

1200

"\ 1000

/i x

[ t \

/ / 800

t Co

t /

. . . . . . . . . . . . .

' ~

!

Lt2

\

I .... a .....

\,,, \

_

_

01

L

02

'1

Xv Fig. 4. Phase diagram of Co-rich Co--V alloys as quoted in the literature [12]. The dots indicate composition and a~nealing temperature of alloys to be discussed in figs. 7-9.

88

G. Inden/Role of magnetism in calculating phase diagrams

According to this diagram homogeneous random fcc solid solutions (type A1) should exist at Xv-< 0.1 and for T > 950 K. At Xv > 0.1 heterogeneous equilibria between A1 and ordered phases are observed, AI + LI,, and A,1 + LI* (.LI~ differs from LI2 by the occurrence of satellite reflections [13]). The s'ery steep decrease in the Curie temperature, on the other hand, indicates a strong magnetic destablizafion of the C o - V alloys with increasing V-content. If this destabilization is not compensated by the chemical (i.e. non-magnetic) part of the Gibbs energy of formation, a magnetically induced two-phase field should occur around the Curie temperature. It will now be shown that this two-phase field is oredicted in this instance and is observed. Th~ .nagnetic Gibbs energy can be calculated according to eqs. (9a) and (9b) taking for the Curie tempera:ure Tc = 1377-4040Xv [12]. The composition dependence of the magnetic moment per atom is not ~no~.qn. Therefore the simple dilution has to be postulated, i.e.
(12)

G ~ o # ' ( T ) = ".~coo~.,,.f::,,'rx ,-,co ~ j, ~+ xvOG~,,,m~(T) + F,'._,,m.AW.rx ,-,c.,v ~-,,.

since only completely random atomic distributions are considered (AI), FG~movAI takes the simple form: I:G~:~A~ = VH~o#" + R T ( x c o In Xco + Xv In Xv).

(13)

,,

be treated as being temperature independent in the present approximation. Furthermore, a parabolic concentration dependence typical of a pairwise interaction between the atoms can be presumed. Experimental enthalpies of formation determined at T = 1473 K (i.e. at sufficiently high temperatures that randomness is fulfilled) are available [14]. The results are shown in fig. 5. The experimental values refer to Co and V in their physical states at T = 1473 K. At this temperature Co is

FH~-~vAI c a n

/ *5

//

?\

x V,

o

_~

6Hbv c ~ c c

~

CoY

Fig. 5. Enthalpy of formation of C o - V alloys at T = 1473 K. O Experimental values of random fcc alloys with reference to the natural states of the pure components at 1473 K [14]. C) Transformed values: for pure Co and Co09V01 transformation to the state of complete magnetic disorder (cmd), by the present calculation, for pure V transformation to the fcc state of V according to [ 16]. Full line: enthalpy of formation of paramagne:ic fcc C o - V alloy" fitted to the experiments with a parabolic function.

G. lndentRole o[ magnetism in calculating pha.se diagrams

~,;

fcc and magnetically short range ordered, V is bcc and nrn, whereas Punm.a~ • ~¢oV in (13) refers to non-magnetic and fcc standard states. Therefore one must write FLI'nm.AI ~I CoV

unm,AI -- ~rl C o V

~

_~ OEJ'nm,fcc - C o Jr-Jr(7,o

v OLrnm,fcc -- - ~ V ~ V

= H ~ , # ' ( 1 4 7 3 K)- xco°H~(1473 K)- xv°H~(1473 K) +xco[°H~(1473 K ) - 0tr"m'ee{14"3 • ~Co' ~ " K)] + Xv[°Hv~(1473 K ) - ° H ~ ( 1 4 7 3 K)]. The first three terms together form the enthalpy of formation, at T = 1473 K, of the random fcc C o - V alloys, with reference to the natural states of the pure components at this temperature. The fi~mt bracketed term represents the magnetic energy of pure Co at T = 1473 K due to short range order, i.e. AH~o(oo) - A H ~ ( 1 4 7 3 K). These expressions can directly be calculated using eq. (4) and the parameter:; for pure fcc Co: Tc = 1377 K, f = 0.28,/3 = 1.7/za [15]. One obtains: A H ~ ( o o ) - A H ~ ( 1 4 7 3 K) = - 1750 J/mole.

The second bracketed term represents the lattice stability z.~°E[[?'c-'bc¢(1473K) which is a temperature independent value according to Kaufman (16]: A°H~5¢'b¢¢ = -9000 J/mole. Thus, one obtains: ~H~m'Al = FHcov(1473 K ) - 1750xco- 9000Xv

in J/mole.

The enthalpy of formation with reference to the hypothetical nm and fcc states of Co and V can be deduced from the experimental values in fig. 5, taking as reference the dashed line. The parabolic interpolation of the scarce experimental data yields FLI'nm.A1 rzCoV -- --65000Xc,,Xv Inserting

in J/mole.

(I.I.)

(14) into (13) and taking account of the magnetic contribution, one obtains for the Gibb~

energy: Gcov(Xv, T) AI

._

magi-,Ai t lr C . o V ~,~ . & V , T ) - 65000(1 - Xv)Xv + RT[(1

-

xv)ln(l - xv)-~ xv in xv].

(15)

The numerical analysis of eq. (15) yields the phase relations of C o - V alloys ~ormed with fcc Co and hypothetical fcc V. They are shown in fig. 6. A b o v e about 1280 K the magnetic transformation occurs continuously (second-order transformation indicated bv the hachure) whereas beh;w 12S0 K a mi~cibili~x

15ooj TIK]

// fcc,porarnagn

1000

(All

500

Co

xv

Fig. 6. Calculated hypothetical phase diagram of fcc C o - V alloys formed with fcc Co and hvpothetical fcc V. For the sake ,f simplicity only states of complete atomic disorder (A1) have been considered in the calculation Heterogc,~eous state's (A1, fm) ÷ (A1, pm) are predicted around the Curie temperature at T < 1280 K.

90

G. lnderdRole of magnetism in calculatingphase diagrams

1~°°I/'r ~' TIK]

I'~

1200

•

Al.pm

Q

'

A1. fm÷ Al.pm.,

10 •

/

80t "1" Co

I 0.1

, I 02

Xv

Fig. 7. Phase diagram of C o - V alloys as obtained by combining figs. 4 and 6. The hachure of the Curie temperature indicates a continuous (i.e. second-order) transformation.

gap is predicted separating the ferromagnetic and the paramagnetic phase states. This finding is the analogue for a system with negative ~:nthalpy of formation to the sharply closing miscibility gap found by Meijering [1] for a system with positive enthalpy of formation. Putting the calculated diagram (fig. 6) and the experimental one (fig. 4) together we obtain the predicted diagram, fig. 7. This prediction will now be compared with experimental data. The best method for checking the existence of the magnetically induced miscibility gap is M6ssbauer spectroscopy since this method is particularly sensitive to ferromagnetic and paramagnetic states. If 57Fe isotopes are used, a ferromagnetic state produces a six-line spectrum; a paramagnetic state produces a single-line spectrum. ~ e amount of ~plitting depends on the degree of magnetic long range order (i.e. proportional to the difference between Tc and temperature of the measurement, here room temperature) and on the atomic environment of the STFe-atoms. In bcc F e - V alloys the hypedine field splitting decreases with the number of nearest neighbour V-atoms [17]. The same tendency can be expected for the fcc structure. Two alloys were prepared: Co0.94V0.0:TFe0.01 and Co0.89V0.10S7Fe0.0~, from the starting materials Co (0.999), V (0.997) and S7Fe (0.999) by arc melting in an Ar atmosphere. The alloys were remelted several times to obtain homogeneity and additionally homogenized in evacuated quartz ampoules at T = 1100°C for 12 d. Then, filings were produced and afterwards heat treated: 1300 K 17 h/H20 all alloys, additional treatments 1123 K 11 d/H20 for Xv = 0.05 and 963 K 11 d/HeO both compositions. The compositions and temperatures are marked in figs. 4 and 7 by dots. According to the literature (fig. 4) all alloys should be homogeneous after the applied heat treatments. According to the predicted phase equilibria heterogeneous states should be observed, at least for Xv = 0.1. In figs. 8 and 9 the M6ssbauer spectra of the two alloys are shown for each heat treatment. These spectra will now be analyzed qualitatively. A quantitative analysis is not required at this stage and for the present purpose.

2.4.1. Alloy Coo.94Vo.o557Feo.ol In the homogeneous random atomic solution (annealing at 1300 K) essentially two spectra are expected in an intensity ratio 1:0.6 which originate from 57Fe-atoms having 12 Co-atoms or 11 Co- and 1 V-atom in their nearest neighbourhood. These two spectra can be detected in fig. 8a.

G. lnden/Role off magnetism in calculating phase diagrams 0

-Vmm( t"

91

~Vma~ |

K 'rlhtH=0

••

11d/H20

C°o~ Voos Feom Fig. 8. M~issbauer spectra of Co0(~V0.0557Fe00~ after different treatments: 1373 K 12d/H20+filing+ (a) 1300 K 17h/H:O. (b) 1123 K 11 d/l-120.

After annealing at 1123 K, i.e. within the two-phase field, this alloy should produce almost the same spectra as in fig. 8a plus additional ones for the V-rich phase which, at room-temperature, also becomes ferromagnetic but with a lower Tc and therefore with a lower splitting. These additio~al spectra cannot be resolved in fig. 8b, either owing to the small amount present or to an insufficient driving force for the segregation reaction to overcome the strain energy of the coherent precipitation. However, these spectra will be very useful for interpretating the spectra of the eecond alloy.t

2.4.2. Alloy CoosgVoAoSTFeo.o~ The homogeneous solid solution obtained after annealing and quenching from 1300 K shows a much smaller hyperfine field splitting than previously (fig. 9a). This is due to the considerably lower T. and to -Vma x C

0 I

÷Vrnax i

.3~OOK17h/H20

COoB9Vmo Feom Fig. 9. M6ssbauer spectra of Co0.s9V0.1oSTFe.0.Olafter different heat treatments: 1373 K 12 d/H20 + filing + (a) 1300 K 17 hfl-l:O; (bj 963 K 11 d/l-120.

t Note added in proof: the existence of the two-phase states for xv = 0.05 could meanwhile be confirmed directly by high-temperatun' X-ray diffraction [31].

q2

G. lnden/Role of magnetism in calculating phase diagrams

different atomic environments of the 57Fe-atoms: here essentially three spectra with the intensity ratios 0.75:1:0.6'are expected from STFe-atoms with 0 V-atoms, 1 V-atom, and 2V-atoms as nearest neighbouts. These three spectra produce the considerably broadened lines. Annealing at 963 K produces a clearly detectable change (fig. 9b). The spectra of fig. 9a are no longer present. Instead of these a central line (or very weakly split spectrum) appears which can be attributed to a V-rich L12-phase. The rest is very similar to the spectrum with Xv = 0.05 and it should therefor,~: be due to a Co-rich phase. The splitting is higher in fig. 9b than in fig. 8 suggesting a higher T~ or eqW~wdently Xv < 0.05. This is corroborated by the reduction in intensity of the spectrum of 57Fe with 1 V :~earT~, neighbour as compared to fig. 8. ~ i s second experiment confirms the calculated phase diagram in fig. 7 with magnetically induced heterogeneous equilibria More erperiments at other temperatures and compositions are presently runni,g. 2.5. Extension to atomic ordering

"13acre is no problem in applying the q,-formalism to atomic ordering too. The critical temperature of iro is presumed to be known (either by experiment or by model calculations, see section 3). The total entropy of the atomic ordering process is known if the long range ordered structure is known. The latter is generally described in terms of sublattice occupations p~ which give the fractions of atoms i on a sublattice position L. For bee alloys four sublattices and for fcc alloys at least eight sublattices need to be considered (see fig. 10). The entropy of any atomic distribution on these sublattices (binary alloy without vacancies) is given by S-

R ~

(Pk In p~, + p~ In p~).

(16)

r/ L = !

Thus. for :he evaluation of the entropy of the ordering reaction one needs the initial and finai values of p t. An example for this will be giver~ for the alloy CuZn (transition B2--, A2).

[I

I~I....I-~I I

LLI'~'

lI"

2nn

I

2nn Inn

2°°,'

J °.-Iv"'i

/ f/

•

/I

Fig. 10. Subdivisionof the bcc and fcc lattice positionsin terms of sublattices characterized by first and second nearest-neighbour relationships. In bcc alloys four sublattices and in fcc alloys at least eight sublattices are needed to describe ordered atomic arrangements produced by ordering forces between first and second nearest neighbours.

G. Inden/Role of magnetism in calculating phase diagrams Cu Zn

8C

% IJIKlmole of CuasZn(~sl

• exp

-~[~(Cul+CplZn)]

: 6[

0,3

....

colc. f=02 calc.f=0.3

t

f . _L__ 20

'

t~O .

.600.

.

.B00.

TIKI

Fig. 11. Comparison of the calculated specfic heat for transition B2--, A2 in Cu05Zn0..~ with experiments [19].

The fraction f for atomic ordering reactions is still uncertain owing to the lack of reliable experimental data. There is only one classica? example, CuZn [18], shown in fig. 11. The, dotted line represents the non-configurational part of Cp which has been calculated by the weighted sum of ce for Cu and Zn [19]. At temperatures T < 500 K the experimental values follow this line since changes in the atomic configurations cannot take place. Two calculated curves are shown in fig. 11 with f = 0.2 and f = 0.3. The entropy has been evaluated in the following way: initial state B2: p[u

..m = p N = ezn = P ~ = 1,

p ~ . = p~n = pc. ..m = p~V = O;

final state A2: p & = . ~ , ~ : 0.5,

as'2-A2 = - R In(0.5).

The critical temperature is 741 K. The agreement between the calculated and experimental values is satisfactory in the range 600 K < T < 7 5 0 K. At T < 6 0 0 K the experimental values are lower than the calculated ones as is expected if the equilibrium states of order could not be reached during the experiments. No decision is possible as to the value of f. At present it is therefore recommended to use the values f = 0.4 for bcc and f = 0.28 for fcc as they have been obtained for magnetic transformations. 3. Model ~,alculations of phase stability

3.1. General viewpoints De.~-itc the success of the formalistic treatment outlined above for magnetic and atomic ordering, it is not possible to abstain from any physical model, particularly if the mutual influence of atomic and magnetic order is to be considered. Such models should be sufficiently simple to be applicable to practical cases. This means: (1) :l'hc atomic and magnetic interactions are treated as pairwise interactions (V}k) for kth neighbours i and A J!~) for kth neighbouring spins Si and Sj). (2) The interchange energies W~ )= -2V~,~)+ v~,k)+ V}k) are independent of temperature or composition. The same assumption is made for the magnetic interchange energies J!~).

94

G. lndenl Role of magnetism in calculating phase diagrams

(3) The treatraent of atomic ordering should include first and second nearest-neighbour interactions. This is ,~lirectly evident for bcc alloys since ordering reactions between first and second nearest neighbours are observed experimentally (e.g. in Fe--Si [20] and Fe-AI [2I]). It has been shown [22] that this also holds "for fee alloys, although superstructures of type L10 and L12 occurring in many fee alloy systems are ordered atomic arrangements between first nearest neighbours only. In many systems these superstructures are stabilized by an ordering tendency between first (Wo - 0) > 0) and a strong segregation t e n ~ , c y between second nearest neighbours (W~) < 0) with considerable modifications in the phase d i a g , , ~.,as compared to the situation W~ I = 0 [22]. (4) ~+e treatment of magnetic ordering can be restricted to nearest neighbour interactions in many c~+es. ~i,:cond nearest-neighbour interactions, if considered, turn out much smaller than those for nea¢est neighbours. They can therefore be neglected in many cases. (5) The magnetic o:dering should be considered for spin -12, 1, 3, 2. This is important for the determin~+~.tion of the correct value of the magnetic ordering energy from Curie temperatures since this value varies sensibly with spin number for the same fixed Tc-value. A model which fulfills all these requirements is the Bragg-Williams-Gorsky model (BWG model). It has been shown that this model yields critical temperatures which are in good agreement with experience in binary [23, 24] and ternary alloys [25, 26]. This model does not account for effects of sro. It could be shown that this deficiency can be overcome, as far as the critical temperature is concerned, by a global sro correction [23]. It cannot be denied that this model is the lowest level of sophistication. Its capabality is its simplicity and surprisingly good description of reality as far as critical temperatures of lro and phase relations are concerned. More sophisticated treatments like the cluster variation method do not yield the same agreement, e.g. for oi'dering systems like Fe-Si the B2-structure is not predicted to occur for Xs~< 0.25 [27], in contradiction with the experiment [20]. For fcc alloys this method is still restricted to nearest neighbours only. A simultaneous treatment of atomic and magnetic ordering has not yet been formulated. Therefore, at this stage the BWG-formalism remains the best suited formalism if practical applications are envisaged.

3.2. Magnetic ordering model Four the sake of simplicity the following presentation will be restricted to alloys with atoms having all the s~Lme spin. It will be shown that other situations, e.g. binary alloys with one non-magnetic component, can directly be derived therefrom. The enthalpy of a magnetic element with pairwise interactions is written in the BWG (or mean field) approximation as (e.g. [27]):

"~agH= ½N~

z(k)j(k)or 2,

(17)

where the summation goes over the different neighbouring shells (k, = 1, 2 . . . . ), Z (k) is the coordination number, j
~ H = -~ :~ z'k~S~t~k>,2. Since So is known and constant it is eq,.dvalently possible to define S~J
G. lndenlRole of magnetism in calculating phase diagrams

W h e n written in the form of e q (17), ferromagnetism occurs for JCk)<0 in analogy to the sign convention for the atomic interchange energies W (k) which are n~gative for segregation t e n d e n c y This tendency stabi~zes |ike a t o m p a ~ agmnst unlike pairs, T h e magnetic entropies for different spin values have been derived by the m e t h o d of steepest descent [28]:

spin !2:

C18)

m'S = - R [ ( ~ - 0.)ln(~- or) + (~ + 0.)In(1+ 0.)]; spin 1 [29]: ' ~ ' $ = R [ I n ( 8 - 6o- + 2 ~ / 4 - 30.2) - (1 - 0.)1n(2(1 - 0.))- (1 + 0.)In(or + X/4 - 3o2)];

(19)

spin 2a: (20)

m,sS = R[In(Z-3/2 + Z ,/2 + Ztr2 + Z 3 a ) _ or In Z], with 1 [20. - I + ~3/2(~VG + H + ~ O - H)], Z = 3(3- 20.) O = 4 + ( 3 - 2o-)(57- 20o'2), H = 3 V ~ ( 3 - 20.)x/(4or 2 - 11)2 + 4; spin 2:

(21)

ma~S = R[In(Z -2 + Z - ' + 1 + Z + Z 2) - o" In Z ] , with

z=U~

[U2+ v~-uv+P(U+ v+e)]

u+2 v P + 2l \ / u + v - p

33/ U = 3(2-0")

X/0.(50.2 - 27) + 3x/[3H], 27)- 3[3x//rgi,

V = 3 ( 2 - 0")

H = - 50. 6 + 70o .4 - 333o .2 + 540, p = 50. 2 - 10o- - 3 12(o-- 2) z The magnetic Gibbs energy is then given by ma,G(T, 0-)= masH -- Tmags. The Carie temperature can be obtained from the condition d~G]

= 0,

which is fulfilled for continuous transformations. This yields R T c = - $o(So3+ 1) N g Z*k)J (k).

]cr-1 40--2

96

G. lnden/Role of magnetism in calculating phase diagrams

This equation is generally written in the form T~ = - So(So + 1) ~ Zo') to,) "-, k~ 3

where ks = Boltzmann's constant. This suggests that we express j(k) in units of ks 1 K = lk-unit.t In binary alloys one has to replace J(~) by the weighted average of J ~ , J ~ , J ~ , where the weights are $i :en by the fractions of A - A , A - B , B - B pairs in the alloy: "" '--, p ~ J ~

+ pl~d.lt~/; + ~ $

+ pl~Z~r~,

So(So + 1)_.__. z k ~ j ~ T¢ = - 3ka(aXA + ~3xs) ~

"

(22)

+ pgr~J~d+ ( p ~ + p~),,)J~l,

where a and/3 arv generally 1 except if A or B is a non-magnetic component, then ~ = 0 or/3 = 0, respectively. This factor stems from the weight which is to be attributed to the magnetic entropy, since only the magnetic components contribute to it. For a disordered atomic solution one obtains: p~

= xL

t , ~ = x~,

p ~ = p ~ = x~x~,

which yields T~,~=

So(So + 1) 3ka

1 O~XA +

#xs -k'-

Z(k)[XAJ~ + xsJt~d- XAXS~'I •-(k)l,,

(23)

with M (k) = - 2 J ~

+ J(~, + J~r~.

In ordered solutions the corresponding values for p!~) have to be introduced in (22). 3.3. Atomic ordering model

Long range ordered atomic configurations can best be described in terms of sublattice occupations. Since first and second nearest-neighbour interactions have to be considered, four sublattices for bcc and at least eight sublattices for fee alloys must be introduced as shown in fig. 10. (The superstructures of type Ni,Mo o r P t 2 M o cannot be described in terms of these sublattices.) The enthalpies are immediately obtained by simple counting of the number of bonds formed out of the pure components, i.e. they are calculated with reference to (sometimes hypothetical) pure nonmagnetic components with the same crystal structure as the alloy formed: Vltb~ = _N[(4W(r~ + 3W(2))XAXs + lt6_~aw(n_ 3 W(2))(p~, + pU_ p r o _ pW)2

+8W {(pA- pA) ~ +

- p~):}l,

(24) IV

IV

FH:-'~ = - N [ ( 6 W (')- 8XAW (:) + 3 W(:))xA + 8W (I) ~--t--(p~' + p~÷W)2 _ ~W(2) L=, ~'~ p~p~÷r.].

(25)

The Gibbs energy is then obtained by combining (24), (25) and (16). The equilibrium state of order is obtained by solving the problem of minimization of Gibbs energy with respect to the variables ptA under the constraint Z~_-i p~ = nxa.

,+ l k - u n i t = 13.8 x 10 TM J = 8.6 × 10 -5 e V .

G. lnden/Role of magnetism in calculatingphase diagrams

~;.7

3.4. Simultaneous consideration of magnetic and atomic ordering In the magnetic treatment the reference state ( " ~ G = 0 ) was given by the hypothetical state of complete magnetic d i s o ~ e r a t T = 0, In the atomic ordering treatment the reference states of the alloy were the pure components in the same crystal structure as the alloy under consideration at T = 0. In the combined treatment of magnetic a n d atomic ordering the chosen reference states are the pure components in the crystal structure of the alloy and in the completely disordered! magnetic state at T = 0. These reference states are generally hypothetical states. The Gibbs energy of an alloy at a given temperature and compo.dtion is then set up with the following terms. (1) The magnetic Gibbs energy of the pure components

xl"'G(.r!t ), So,). (2) The sum of the magnetic and atomic enthalpy of ordering. Both terms together can be written in the identical form as the pure atomic part (eq. (24) or (25)) if the substitution W(k)-~ W °:) + o-:M (~) is performed with M (k) = -2J(kAt~+ J ~ + J ~ . The case of a non-magnetic component i is inc]luded setting simply J ~ ) = 0. (3) The sum of the atomic entropy of ordering (16) and of the magnetic entropy (18)-(2I). In the case of a non-magnetic component the magnetic entropy has to be weighted by the fraction of magnetic atoms in the alloy.

3.5. Short range order correction The entities J!~), M ~k) and W ~k~ are to be interpreted as real physical energies. It is only if their numerical values are determined from experimental critical temperatures of lro that the BWG model interferes. For a given energy the BWG treatment overestimates the critical temperature since be~ ond T, complete disorder exists in this model instead of sro configurations. This has been discussed in [23] for atomic and magnetic ordering in bcc alloys and in [22] fol ~fcc alloys. The final result of this discussion was to account for this effect by a correction factor K for atomic and/~ for magnetic ordering in the following way: (T°'d-'~')~, = K(T°rd"dis)BWG. (T~)~., = ft (Tc)BWG. The values of K depend on the ratio W(2)/W (~). They should be taken from [22.23]. For #. the value ft = 0 . 8 has been obtained in [23].

3.6. Application to a hypothetical binary system The following calculations have an illustrative character only. Therefore the whole treatment ~ill be made without consideration of the previously mentioned sro correction. The W ¢k), J~) and M ~:~ values are thus pure model parameters ("BWG-energies"). and vice In order to demonstrate the effect of the mutual influence of atomic on magnetic or,.,er, ,~ versa, a hypothetical bcc binary system will be discussed which is shown in fig. 12. It is presumed that (a) A is ferromagnetic with Tc = 1400 K; (b) B is a non-magnetic component; (c) J ~ = J[~ = 0 ==>M °) = J~/~,; (d) the ordering tendency between nearest neighbours is W a) = 550k, W a) = 0. Fig. 12 shows the critical temperature of atomic and magnetic lro as they would vary width

G. lndenlRole of magnetism in calculating phase diagrams

A2, pm T(K) 1000

500

0

I¢: .

.

.

.

.

-

t

•

....,.

0.2

A

t

. ......

. ._

0k

,:..i::i~ . . . . . . . . . .

'.......,..:..:i:...,,,t;~,,.

0.6

S

0.8

xB

Fig.. 12. Hypothetical bcc binary alloy system with the presumed properties. (a) Ferromagnetic component A, non-magnetic component •, J ~ <0, Jl~ = 0, J,~a = 0 (linear variation of Tc with composition). (b) Ordering tendency between nearest nei,ghbours (W °) = 550k, W t2) = 0). The critical reg. peratures of atomic and of magnetic Iro have been calculated separately with the BWG model according to (a) and (b). Outside the shaded area these calculations remain correct if magnetic and atomic ordering is simultaneously considered. Wilthin the shaded area they will be modified owing to mutual interactions. The hachure indicates continuous transformations.

composition if no mutual influence occurs. In the regions where only one kind of order exists the~e critical temperatures are the final ones. In the shaded area, where both kinds of order simultaneously exist, the mutual influence on both critical temperatures will take place. This effect will now be calculated assuming spin -~ and spin 1 for the component A.

3.6. ~. Spin ½ assumed T~ ( A ) = 1400 K ~ ] ~ - - -700k-units according to (23). Fig. 13 shows the result of the aumerical calculation. In the ranges where no mutual influence occurs, fig. 12, Tc and T Bz-'A2 remain unchanged. In the case of mutual influence drastic changes occur:

Sph112 T[K]

~

A2,pm

100G

A

02

04

0,6

0.8

B

X8

Fig. 13. BW..~-calculation for the hypothetical system in fig. 12 now with simultaneous consideration of magnetic and atomic ordering reactions. Spin 1/2 is presumed for the A-atoms. The variation of the critical temperatures for hypothetical homogeneous alloys is showr= by dashed lines inside the two-phase field.

G. lnden/ Role of magnetism in calculating phase diagrams

T

[gl ~

~

,ooo+

~

soin I

+ -,N +-

........... ....................

A

99

0.2

\

~

0/.

0.6

0.8

B

XB

Fig. 14. Similarcalculationas in fig. 13but for A atomswithspin1. in the ferromagnetic state T B2"~ is strongly depressed. In the low temperature range, where magnetic order is nearly perfect, T m-'~ approximates the parabolic shape which corresponds to an "effective" atomic interchange energy W ~ v c = 550k + or2J~2~= ( 5 5 0 - ~ - - ) k - u n i t s

375k-units.

This interchange energy would produce a maximum critical temperature of only 750 K. However. approaching XB= 0.5 also implies approaching the Curie temperature TP 2 and therefore a dec,ease of or. i.e. an increase of w(ekfl)ective.This effect causes the steep increase of T B2+A2 in the range 0.3 ~ 9:B:-=0.4. For similar reasons the Curie temperature of the ordered states is depressed i~a the range 0.3 -< XB--<0.5: with increasing XBthe number of A - A pairs decreases reaching the value zero in the ordered state at xB = 0.5. According to eq. (22) T~a2 decreases and finally vanishes in the same way.

3.6.2. Spin 1 assumed To(A) = 1400 K f f J ~ = -263k-units. Fig. 14 shows the result of the numerical calculation in this case. The same comments as for spin apply here. The "effective" atomic ordering tendency in the ferromagnetic regime is only W~2d~,.~= 287k-units. The stronger effect in the spin 1 case as compared to spin ~ for the same Tc curve in both cases is clearly visible. 4. Conclusions

Both methods outlined here have their advantages and disadvantages. The formalistic q,-treatment is simple, easily applicable to computer calculations of phase diagrams. rand it act'2_ou_n_L$Wj'__thsufficient accuracy for Iro and sro effects. Its deficiency is that it cannot account for effect~ of mutual influence of magnetic and atomic ordering. Model treatments can describe the mutual influence of magnetic and atomic ordering. Its deficiencies are that sro effects cannot be described (these are globally accounted for in the energy determinations from critical temperatures) and the numerical calculation of the equilibrium states necessitates a search for minima in the multidimensional space of the occupation probabilities. The convergence of the routines generally depends on the starting values. Therefore this technique is inappropriate for automatic computer programs.

I00

G. lndenlRole of magnetism in calculating phase diagrams

]'here is the possibility of combining both methods. The phase diagrams should be calculated with the cp-formalism; the input data, particularly t!he critical temperatures, can and should be calculated with the model treatment. Acknowledgements The author wishes to thank Prof. W. Pepperhoff, Mannesmann Research Inst., Duisburg, for stimu%ting discussions and particularly for making available unpublished experimental data. Thanks are due ~ ~ Dr. W. Wepner, MPI Dfisseldorf, for his assistance in the entropy calculations. The continuous suptx, rt 3f this work by Prof. W. Pitsch, IVEPIDfisseldoff, is gratefully acknowledged. Re|erences [1] J.L. Meijering, Philips Res. Rep. 18 (1963) 318. [2] L. Kaufman and H. Bernstein, Computer Calculations of Phase Diagrams (Academic Press, New York, 1970). [3] G. Grimvall and I. Ebbsj6, Phys. Scripta 12 (1975) 168; G. Grimvall, Phys. Scripta 12 (1975) 173; 13 (1976) 59. [4] G. Inden, Report of the Project Meeting C A L P H A D V, Diisseldorf (1976) p. III.4-1. [5] C. Domb, Advan. Phys. 9 (1960) 149-361, esp. p. 288. [6] A. Paskin, J. Phys. Chem. Solids 2 (1957) 232. [7] J.A. Hofman, A. Paskin, K J . Tauer and R.J. Weiss, J. Phys. Chem. Solids I (1956) 45. [8] M. Braun, Doctor Thesis, K61v (1964). [9] W. Pegperhoff, Mannesmann-Forschung, Duisburg/Germany, private communication. [10] R. Kohlhaas, M. Braun and O. Vollmer, Z. Naturforschg. 20a (1965) 1077. [11] M.V. Nevitt and A.T. Aldred, J. Appl. Phys. 34 (1963) 463. [12] Y. Aoki, Y. Obi and H. Komatsu, Z. Metailk. 70 (1979) 436. [13] Y. Aoki and S. Hashimoto, Z. Metallk. 71 (1980)32. [14] PJ. Spencer and F.H. Putland, J. Chem. Thermodyn. 8 (1976) 551. [15] J Crangle, Phil. Mag. 46 (1955) 499. [lCq Ref. 2, p. 184. [171 ~ Shiga and Y. Nakamura, J. Phys. F: Metal Phys. 8 (1978) 177. [181 C. Sykes and H. Wilkinson, J. Inst. Metals 61 (1940) 223 [!~ I R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser and K . K Kelley, Selected values of the Thermodynamic Properties of Elements (American So¢. Metals, Ohio. 1973), p. 154 for Cu and p. 570 for Zn. t201 G. laden and W. Pitsch, Z. Metallk. 63 (1972) 253. [211 H. Okamoto and P.A. Beck, Metall. Trans. 2 (1971) 569. P.R. Swann, W.R. Duff and R.M. Fisher, Metall. Trans. 3 (1972) 409. [221 (3. lnden, J. de Phys. 38 Suppl. to no. 12, (1977) C7-373. [231 G. Inden, Z. Metallk. 66 (1975) 577. [24] G. Schlatte and W. Pitsch, Z. Metallk. 67 (1976) 462. [25] G. lnden, Z. Metallk, 66 (1975) 648. [26] G. Inden, Phys. Status Solidi (a) 56 (1979) 177. [27] J.S. Smart, Effective Field Theories of Magnetism (Saunders, Philadelphia, 1966). [28] R.H. Fowler, Statistical Mechanics (Cambridge Univ. Press, Cambridge, 1966). [29] S.V. Semenovskaja, Phys. Status Solidi (b) 64 (1974) 291. [~)] G. Schlatte, G. Inden and W. Pitsch, Z. Metallk. 65 (1974) 94. [31] G. inden, unpublished.