First-principles study of short range order and instabilities in AuCu, AuAg and AuPd alloys

Acta m¢tall, mater. Vol. 39, No. 4, pp. 493-501, 1991

0956-7151/91 $3.00+ 0.00 Copyright © 1991PergamonPress pie

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FIRST-PRINCIPLES STUDY OF SHORT RANGE ORDER A N D INSTABILITIES IN Au-Cu, Au-Ag A N D Au-Pd ALLOYS T. M O H R I t, K. TERAKURA 2, S. TAKIZAWA 2 and J. M. S A N C H E Z 3 1Department of Metallurgical Engineering, Hokkaido University, Sapporo 060, Japan, 2Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan and 3Department of Mechanical Engineering, Center for Materials Science and Engineering, University of Texas, Austin, TX 78712, U.S.A. (Received 10 April 1990; in revised form 24 August 1990)

Abstract--Interaction energiesderived from first-principlestotal energy calculation are used together with the Cluster Variation Method in order to study phase diagrams, ordering instabilities and the Fourier spectrum of the Warren-Cowley short range order parameters for Au4:3u, Au Ag and Au-Pd alloys. The present study is carried out using both the tetrahedron and the tetrahedron-octahedron approximations of the CVM. It is shown that the tetrahedron approximation predicts the wrong Fourier spectrum for the SRO parameters although it gives qualitatively correct phase diagrams. In particular, the tetrahedron approximation fails to reproduce the expected maxima in the fluctuation spectrum at the -type reciprocal space vectors, characteristic of L 10and L 12ordering systems. The correct features of the k-space fluctuation spectrum is obtained in the tetrahedron-octahedron approximation. Rrsumr--43n utilise les 6nergies d'interaction drrivres du calcul d'rnergie totale ~ partir des premiers principes, ainsi que la mrthode variationnelle des amas (MVA) afin d'rtudier les diagrammes de phases, les instabilitrs d'ordre, et le spectre de Fourier des param&res d'ordre ~ courte distance (OCD) de Warren et Cowley dans les alliages Au~Cu, Au-Ag et Au-Pd. La prgsente 6tude est effecture en utilisant ~ la fois les approximations t&rardrique et tgtrardrique-octardrique de la MVA. On montre que l'approximation tgtrardrique prrvoit un spectre de Fourier faux pour les paramrtres d'OCD bien qu'il donne qualitativement des diagrammes de phases corrects. En particulier, l'approximation, trtrardrique est en drfaut pour reproduire les maximums attendus du spectre de fluctuation sur les vecteurs du type (100) du rgseau rrciproque, maximums qui sont caractrristiques des systrmes ordonnrs LI 0 et L12. L'aspect correct du spectre de fluctuation dans l'espace des k est obtenudans l'approximation trtrardrique-octardrique. Zusammenfassung--Mit Wechselwirkungsenergien, die aus Grundberechnungen der Gesamtenergie abgeleitet sind, werden mittels der Cluster-Variationsmethode (CVM) Phasendiagramme, Ordnungsinstabilit/iten und das Fourierspektrum der Warren-Cowley-Nahordnungsparameter fiir Au~2u-, Au-Ag- und Au-Pd-Legierungen behandelt. Die vorliegende Arbeit nutzt hierbei sowohl die Tetraeder- als auch die Tetraeder~)ktaeder-N/iherung der CVM. Die tetraedern~iherungergibt das falsche Fourierspektrum fiir die Nahordnungsparameter, obwohl sic qualitativ richtige Phasendiagramme liefert. Insbesondere versagt die Tetraedern/iherung bei der Darstellung der erwarteten Maxima im Fluktuationsspektrum bei den (100)-Vektoren des reziproken Raums. Die richtige Struktur des Fluktuationsspektrums im k-Raum ergeben sich mit der Tetraeder~)ktaeder-N/iherung.

1. INTRODUCTION Recently, a systematic first-principles study of phase stability was carried out in noble metals alloys [1-3] and noble-transition metal alloys [4] by combining total energy electronic structure calculations with the Cluster Variation Method (CVM) [5]. Such calculations are motivated by the interest in deriving temperature-composition phase diagrams strictly from first-principles using, as input, only the atomic numbers of the constituent elements. The method has also been used to study Ni-A1 [6] and semiconductor alloys [7]. In particular, for the noble metals and the noble-transition metal alloys investigated here, the calculated heats of formation predict the correct ground states and successfully distinguish between the different segregating and ordering tendencies among these systems [1-4].

An important component of the first-principles approach to the study of alloy phase stability is the CVM itself. In particular, the CVM provides an efficient and accurate framework in which all relevant configurational thermodynamic properties for a given system, including phase equilbrium, can be systematically obtained from a unique free energy formula. This was demonstrated in a previous publication [8] in which transition temperatures, ordering spinodals, cluster densities and Short-Range Order Diffuse Intensities (SROI) were all derived self-consistently for an Ising system. The main objective of the present study is to extend this approach, previously used for Ising systems, to a realistic alloy model in which the free energy is derived from first-principles electronic structure calculations. We focus on the calculation and analysis of the SROI since this quantity gives a very detailed picture of the state of order in the alloy.

493

494

MOHRI et al.: SHORT RANGE ORDER IN Au42u, Au-Ag AND Au-Pd ALLOYS

Furthermore, the comparison of the calculated SROI with experimental X-ray and/or neutron diffuse scattering experiments also provides an important test of the accuracy of the theoretical models. The present study is carried out for three Au-based systems, namely Au-Cu, A u - A g and Au-Pd, with the emphasis placed on ordering reactions involving the L10 phase. Of these alloy systems, A u - C u has been extensively investigated and, at the present time, phase diagram [9], ordering instabilities [10] and SROI [11] are well established experimentally. The characterization is less complete for the A u - P d and A u - A g systems. For example, although A u - A g alloys are known to form solid solutions for the entire composition range [9], SROI has not been investigated in detail. Thus, the theoretical framework presented here can be put to test more readily in the case of the Au--Cu system, whereas some of the predictions for both A u - A g [12] and A u - P d [20] await experimental confirmation. In the next section we briefly review the essential theoretical aspects involved in the development of a first-principles configurational free energy for alloys. Our main results are presented and discussed in Section 3. The last section is devoted to concluding remarks. 2. THEORETICAL BACKGROUND

2.1. First-principles free energy expression A key aspect in the present first-principles approach is the assumption that the alloy's ordering energy can be expressed in terms of relatively shortranged pair and many-body interactions. Consequently, the configurational energy for a fixed molar volume or, equivalently, a fixed Wigner-Seitz cell radius r, can be written as

E(r) = ~ v,(r)¢,

(1)

(¢~'~))-t, are known a priori. On the other hand, the compound energies in equation (2) can be determined from first-principles total energy calculations. This procedure was first implemented by Connolly and Williams [15] who used Augumented Spherical Wave (ASW) [16] and Local Density Functional (LDF) approximations to calculate the self-consistent electronic structure and total energies of several ordered compounds. Here we investigate three alloy systems of the type Au-B, where B is one of Cu, Ag and Pd. For each of these systems, total energy calculations were carried out using the ASW and L D F approximations for five crystal structures at several volumes: pure Au and pure B in the f.c.c, structure, Au3 B and AuB3 in the L12 structure, and AuB in the L10 structure. The relativistic effect, except the spin-orbit interaction, was included [27]. A set of five effective interactions, corresponding to the "empty" duster (n = 0), the point (n = 1), the nearest-neighbor (nn) pair (n = 2), the nn triangle (n = 3) and the nn tetrahedron (n = 4) clusters, are obtained from equation (2) for different values of r. Table 1 gives the effective interactions v,(r) for the Au-B alloy systems, at their low temperature equilibrium volume for 1:1 stoichiometry. The volume (or r) dependence of effective interactions can be found in Refs [1] and [2]. The configurational entropy for the ordered and disordered phases is calculated using the Cluster Variation Method (CVM). Thus far, all the first-principles studies of alloy phase equilibrium mentioned in the introduction have been carried out using the tetrahedron approximation of the CVM. In this approximation, the entropy in the disordered f.c.c. phase takes the form [5] (~-IY.~" ; ! ) 6N' y Stetr a ~-

--

ka

In .

" 2 "

-5

(3)

n

where v, and ¢, are, respectively, the effective interaction energy and correlation function for a subcluster specified by a subscript n. The energy expression of equation (1) is based on the description of the state of order in terms of cluster correlation functions, as developed in Ref. [13] and extended to multicomponent systems in Ref. [14]. In particular, equation (1) holds for any stoichiometric compound m with energy E ~m) and cluster correlations ¢("m). Proper choice of a set of ordered compounds allows the inverson of equation (1) [14], from which one obtains the effective interactions Vn(r) for a given value of r in terms of the set of compound energies E~m)(r) and the corresponding cluster correlations ~~,")

v,(r) = ~ E(m)(r) {¢(.,.)}-1

(2)

m

Since the atomic arrangement of pure metals and of ordered compounds are uniquely determined, the matrix of cluster correlations ¢(.m), and its inverse

\

~kl

/

,,

i

/

where ka is the Boltzmann constant, N is the total number of lattice points and xi, y~ and w~kt are, respectively, the cluster densities or probabilities for the point, nearest neighbor (n.n.) pairs and n.n. tetrahedron dusters in the configurations i,/j and ijkl. The subscripts i, j . . . of the cluster densities take values + 1 for a Au atom and - 1 for a B atom (either Cu, Ag or Pd). The duster densities can be written in terms of the correlation functions for the point (¢1), n.n. pair (¢2), Table 1. Effectiveinteraction energies (in Ryd/4-atoms)for three kinds of Au based alloys at their low temperature equilibrium volume for 1:1 stoichiometry AuCu AuAg AuPd v0 0.00551 - 0.01311 - 0.02071 vI 0.01812 0.00025 -0.00333 o2 0.08044 0.01371 0.02990 v3 0.00552 0.00048 0.00579 v4 - 0.00012 - 0.00010 0.00038

MOHRI et al.: SHORT RANGE ORDER IN Au~2u, Au Ag AND Au-Pd ALLOYS n.n. triangle (~3) and n.n. tetrahedron (~4) clusters as [13, 14] xi = 1{1 + i~l }

(4)

Z

1

y0 = ~i{1 + (i q-J)~l-q- tJ~2}

(5)

and

wokt = ~4{1 + (i + j + k + 1)~1 + (ij + ik + il + j k +jl + kl)~: + (ijk + ijl + ikl +jkl)~3 + ijkl~4}.

(6)

In the tetrahedron-octahedron approximation, the entropy for the disordered state takes the form [13] St_0

( I~uukt,..N')(1-IW~k,N')Z(I~IY~N]) 6 = - kB In \ijklmn

J ~,~kt

/ \ 'J

/

g = E -- TS + #~

ized only up to the largest cluster employed in the entropy expression, in the present case the octahedron and/or the tetrahedron clusters, and no information is obtained for longer range correlations. This shortcoming is circumvented by studying the Fourier spectrum of the correlation functions, as first proposed in the context of the CVM by one of the present authors [17]. In particular, the Fourier spectrum of the fluctuations in the concentration variable xi, or the point correlation function ~1 is equal to the short-range order diffuse intensity which, furthermore, can be directly measured by means of X-ray or neutron diffraction experiments. In order to obtain the SROI, one begins by formally expanding the CVM free energy in terms of deviations A~, (p) of the correlation functions at each lattice site Rp from their equilibrium value ~,. Up to second order in the deviations A~,(p), the change in free energy relative to the equilibrium value is given by

(7)

where zu~ and uuum. are the triangle and octahedron cluster densities, which are related to the correlation functions by relations similar to equations (4)-(6) [131. Given the effective interactions v.(r), the configurational energy E, and the configurational entropy S in either the tetrahedron or tetrahedron-octahedron approximation, the grand potential is defined as

495

AF = ~p~, ~,F,~.(p,p')A~.'(p')

(11)

where

F..,(p,p') =

ei.(p)e~.,(p')

(12)

Since in the equilibrium state F.., (p, p') is translationally invariant, this matrix of second derivatives can be diagonalized with respect to the real space variable p and p ' by a Fourier transform. Thus, equation (11) becomes [17]

(8)

where the Lagrange multiplier # is usually referred to as the effective chemical or diffusion potential. The equilibrium of the grand potential go, is obtained by minimization with respect to the correlation functions ¢ and the Wigner-Seitz cell radius r at a fixed temperature T, external pressure P~ and effective chemical potential/~

where

F,n,(K)= ~ F,,,(p)exp(iK" Rp)

(14)

P

A¢,(K) =~A~,(p)exp(iK.Rp)

(15)

P

0___g_=g 0

(9)

and

~g --

OV

=

-eox

(10)

where V is the volume. Equilibrium between any two phases m 1 and rn2 is determined by the equality of grand potentials g(0m~) (T,/~, Pex) = g(m:) (T, #, Pox). This relation is used throughout in this work in order to compute the equilibrium phase diagrams. 2.2. Stability and short range order diffuse intensity The CVM formulation given in the previous section provides a detailed picture of short range order in the alloy in terms of cluster densities. However, from the minimization of the free energy and/or grand potential, short-range order can be character-

Equation (13) provides the basis for the instability analysis in reciprocal space as well as the computation of the SROI. In particular, the eigenvalues of the Fourier transformed matrix of second order derivatives F,~,(K) reveal the stability of the system with respect to correlation and/or concentration waves of wave vector K. Vanishing of one of the eigenvalues of F,,,(Ko) for a particular wave vector K0 gives rise to the instability or spinodal. If the highest temperature instability occurs for K0 at the Brillouin zone center (0,0,0), the phenomenon is commonly known as a clustering spinodal, whereby the system becomes unstable with respect to infinitely long wave length concentration waves. On the other hand, ordering spinodals occur when the instability takes place for a concentration wave with K0 located at the Brillouin zone boundaries [18, 19]. In the case of the f.c.c, structure, symmetry considerations place

496

MOHRI et al.: SHORT RANGE ORDER IN Au-Cu, Au-Ag AND Au-Pd ALLOYS

the possible ordering spinodals at special points in K-space where two or more point group symmetry elements intersect: (1,0,0) (X-point), (1,1/2,0) (Wpoint) and (1/2,1/2,1/2) (L-point) [18, 19]. ((LJ, K ) represents all the equivalent (I,J,K) family in the reciprocal space in 2n/a unit, where a is the lattice constant.) It is pointed out that, for second order or continuous transitions, the locus of the highest ordering spinodal temperature coincides with the transition temperature. On the other hand, for first order transitions, the ordering spinodal provides a metastability limit for a system that has necessarily been quenched or supercooled below the equilibrium transition temperature. In the so-called Gaussian approximation, the expectation values ( ~ . ( K ) ~ . . ( - K ) ) of the Fourier transform of the correlation functions are given by the inverse of the matrix F..,(K). In particular, for n = n ' = 1, corresponding to the point correlation function, the SROI is given by [17]

2

i

2'F-

_

/0,o,o>

= 0,½,o>

/ t/

~o~ I /1000

I 1500

I 2000

Temperoture ( K )

ISRO(K) = (41 (K)~l ( - - K ) ) -- (~1 (K))(~1 ( -- K)) = mk B T F ~ 1(K)

(16)

The SROI in equation (16) provides short-range order information for any arbitrary length or distance in real space, and it is the result of using the CVM free energy functional to determine the fluctuation spectrum of the correlation functions in the Gaussian approximation. Furthermore, since the SROI diverges at the ordering spinodals, the instability temperature for any wave vector K m a y be obtained from the vanishing of the T/IsRo(K ). 3. RESULTS AND DISCUSSION A plot of the T/IsR o vs temperature, obtained in the tetrahedron approximation, is shown in Fig. 1 for the AuqSu system at 1:1 stoichiometry and for the three types of f.c.c, special point ordering waves. As it can be inferred from Fig. 1, the tetrahedron approximation predicts the same value of SROI for both the (1,0,0) and (1,1/2,0) reciprocal space vectors, whereas the SROI is significantly different at the (1/2,1/2,1/2) position. Furthermore, ISRO diverges simultaneously for both the (1,0,0) and (1,1/2,0) waves at 845 K, indicating the onset of an ordering instability or spinodal. The instability temperatures Ts determined by the divergence of the SROI for the (1,0,0) wave vector are shown as a function of concentration for Au42u, Au-Ag and Au-Pd alloys in Figs 2, 3 and 4, respectively. Included in the figures are the equilibrium phase boundaries also calculated in the tetrahedron approximation. We note from Figs 2-4 that the first-principles calculations give phase diagrams which are qualitatively correct. However, quantitatitive disagreement with experiments are also apparent. For example, in the case of Au-Cu alloys, the calculated transition temperatures are approximately 30% higher than

Fig. 1. T/IsRo-T curves for Au-Cu at 1: 1 stoichiometry. The triangle indicates (1/2,1/2,1/2) wave, while (1,1/2,0) and (1,0,0) waves degenerate and are plotted by open circles. Note that the right and left hand axes correspond to (1/2,1/2,1/2) and (1,0,0) ((1,1/2,0)) waves, respectively. actually observed. As discussed in previous publications [1-4], the primary source of this discrepancy is the fact that the energy associated with local atomic relaxations is not included in the calculation of the energy of formation except for the ordered stoichiometric L10 and L12 compounds. Furthermore, the phase diagrams in Figs 2 to 4 are obtained using the configurational free energy and, consequently, relatively important vibrational contributions to the free energy are neglected. Despite these shortcomings, the results are nevertheless encouraging considering the fact that the only input to the calculations are the atomic numbers of the constituents elements and crystal structures of some ordered phases. With regard to the Au-Ag alloys we note that, although they are known to form solid solutions for the entire composition range [9], the calculated phase diagram shown in Fig. 3 indicates the possibility of L10 and L12 ordering below 200 K [12]. Since kinetics is very sluggish at this low temperature, there is no experimental evidence at the present time to confirm or deny the first-principles predictions. The calculation of phase boundaries display the largest discrepancy for the Au-Pd system and a detailed analysis for this alloy is reported in a separate publication [20]. As suggested in the previous section, the SROI provides a more stringent test of the theoretical model. As a general rule, the calculation of the SROI using a CVM free energy functional represents a significant improvement [17, 21] over the most commonly used Krivograz-Clapp-Moss [22, 23] formula

MOHRI e t al.:

II00

SHORT RANGE ORDER IN Au-Cu, Au-Ag AND Au-Pd ALLOYS

/

1000

497

x.

/ I

900

P ..3 =

\. s°

I:: 1900

\

°~-q~~b;°~'o -~.o % °

\\

~; ~c~

\ \

700

\

~o

-xt, '

° ,

N

\ :'" 6,o,o)

I

I

10

20

I.....

I,

50

40

I

I

I

I

I

50

60

70

80

90

Au

Fig. 2. The phase diagram of Au-Cu obtained by the first-principles calculation [12] and the <1,0,0> instability locus.

which is based on the Bragg-Williams [24] approximation for the free energy. Plots of the SROI predicted by the tetrahedron approximation on a (100) reciprocal space plane for a Au--Cu alloy at 1:1 stoichiometry, are shown in Fig. 5(a) and (b) for, respectively, reduced temperatures T I T o of 3.2 and 1.03, where To is the order~tisorder transition temperature for a stoichiometric AuCu. Note that the calculated order190 180

Disorder

-

~

disorder transition temperature is approximately 1000 K (see Fig. 2) whereas the experimental value is 683 K. Therefore, comparison of calculated and experimental SROI will be carried out using a reduced temperature scale. Comparison of Fig. 5(b) with, for example, the diffuse intensity measurements by Metcalfe and Leake [11] carried out at the same reduced temperature of 1.03 (700 K for the experimental data), underscores the intrinsic deficiency of the tetrahedron approximation. In particular, we note that the approximation fails to reproduce the expected intensity

~

170-

Disorder

350

160

140

LI°

E~" 130

" " - . 0,o,o)>

E

~25C

/

120


l 10

I 20

I ~0

I 40

1 50

I 60

t 70

t 80

L 90

I 100

AU (at. %)

Fig. 3. The phase diagram of Au-Ag obtained by the first-principles calculation [12] and the {1,0,0> instability locus.

20C 040

J

0.45

1

0.50 at. % Pd

I

0.55

,

I

0.60

Fig. 4, The phase diagram of Au-Pd near 1: 1 stoichiometry obtained by the first-principles calculation [20] and the <1,0,0> instability locus. Note the convergence of the phase boundary and the instability locus near 40 at.% of Pd.

498

MOHRI et al.: SHORT RANGE ORDER IN Au-Cu, Au-Ag AND Au-Pd ALLOYS

maxima at the (1,0,0) special points in reciprocal space and, in fact, it is constant along the ( l k 0 ) type directions. The failure of the tetrahedron approximation can be traced to the fact that it only includes nearest-neighbor correlations in both the energy and the entropy. In particular, it is well known from the study of ground states at T = 0 K that with only n.n. interactions the L10 structure is degenerate in energy with the I41/amd structure of the (1,1/2,0) special point family [25]. The SROI results shown in Fig 5(a) and (b) demonstrate that this degeneracy is not properly lifted by the entropy in the tetrahedron approximation as it is usually assumed. We emphasize, however, that the approximation predicts reasonably accurate phase diagrams as well as nearest-neighbor correlations. Significant improvement in the overall description of short-range order can be obtained using the tetrahedron-octahedron approximation of the CVM, which naturally includes correlations for both nearest- and next-nearest neighbors. A plot of the SROI for stoichiometric AuCu at a reduced temperature of T / T o = 1.10 is shown in Fig. 6. Note that the transition temperature for the tetrahedronoctahedron approximation T0(tetra.-octa. ) is esti-

mated based on the ratio T0(tetra.-octra.)/ T0(tetra.)=0.952 which is predicted by the Ising model [21]. This can be rationallized by the argument at the end of this section. Although the fine structure of the experimental SROI is not present in the first-principles calculations, the most relevant features, such as the maxima at the (1,0,0) special points, are correctly reproduced. This example shows that unlike the tetrahedron approximation, the tetrahedron-octahedron entropy successfully lifts the degeneracy in the correlation spectrum between the (1,0,0) and (1,1/2,0) special point waves. Figure 7 displays a plot of the SROI integrated over the first Brillouin zone, for both the tetrahedron and the tetrahedron-octahedron approximations, as a function of the ratio TITs, where Ts is the ordering spinodal temperature. Although the integrated intensity is a strictly conserved quantity [17,21], the approximate nature of the free energy functional used to study the fluctuation spectrum results in a divergence at the instability temperature. The extent to which the integrated intensity changes with temperature is, therefore, an additional test on the level of approximation used. The weaker temperature dependence of the integrated intensity in the tetrahedron-

,J// "/' iiiI / /

..".., ,'////

::-;;_:-;Cd/ ----~---~ Ky 1

•

..........--,.-,,\\ \ I

"'",.

",

\\

~/

//

// III

.~._ .....

/iiif/ ,,,.'"

\ \

1

Kx Fig. 5(a). Caption on facing page.

2

MOHRI et al.: SHORT RANGE ORDER IN Au-Cu, Au-Ag AND Au-Pd ALLOYS :ll

499

II

/ ;;It \ :, iIiI I[ I IiI'

/

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

K~,

: .....

/ I lJlll~llk ',, x

..-'" .." //J / / i ~\~?..\V,,'-. "'--. ....

--

:

---

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

.....

=:

1

Kx

Fig. 5(b) Fig. 5. (a) The first-principles calculation of the short range order diffuse intensity around (1,1,0) for Au~Zu at 1:1 stoichiometry at T/T o = 3.2 based on the tetrahedron approximation. The demonstrated pattern is the (100) section. (b) Short range order diffuse intensity around (1,1,0) for Au-Cu at 1:1 stoichiometry at T/T o = 1.03. octahedron approximation points, once again, to the improved description of SRO relative to that obtained in the tetrahedron approximation. A final point of interest is that the first-principles calculation for Au-Cu and Au-Ag alloys predict relatively small three- and four-body interactions, with the effective pair interactions being weakly dependent in concentration. For the Au-Pd system, on the other hand, the calculated three-body effective interaction is significant, amounting to zpproximately 20% of that for the n.n. pair [20]. Consequently, one expects that both Au~2u and Au-Ag alloys should behave essentially as simple Ising models, whereas Au-Pd should display significant deviations. This is confirmed botn by the phase diagrams shown in Figs 2-4 and by the comparison shown in Fig. 8 between T/IsRo(IO0 ) vS kBT/v2, where v2 is the n.n. effective pair interaction, for an Ising model and the three Au-B systems. The largest deviation from the Ising model behavior is observed for Au-Pd alloys which, in fact, are predicted by the calculations to transform to the L10 structure via

weakly second order transition (see Fig. 4) [8]. Since for a model strictly based on pair interactions the disorder-L10 transition is first order [26], the behavior predicted by the first-principles calculations for Au-Pd can be attributed to the strong three-body interaction in this system. 4. CONCLUSIONS A complete phase stability analysis was carried out for Au-Cu, Au-Ag and Au-Pd alloys starting from first-principles electronic structure calculations for a set of ordered compounds. The finite temperature behavior of each system was investigated using both the tetrahedron and the tetrahedron-octahedron approximation of the CVM. The disorder-L10 transition is confirmed to be strongly first order for Au-Cu, whereas it is predicted to be weakly second order for Au-Pd when vibrational and local elastic energy effects are neglected. Furthermore, L10 and L12 low temperature ordering reactions are predicted for Au-Ag alloys. Analysis of the SROI revealed

MOHRI et al.:

500

SHORT RANGE ORDER IN Au-q2u, Au-Ag AND Au-Pd ALLOYS

tlI t____JI,I

/J

\

~

\\

\

i\

"

\ ................... ; ; _--'"".. . ; A ? )I/"--h /

\,.'.'""'--

=_. ._. .=. ._. .-. ~ . - - ; ~ _ t J~ J

~ - . . . _ ~ _-~-- -- ~__ - ~ - - = _~-_- -~_-=

//

..~

"',

///

", \ \

~

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

', ! i C---11

,,"

~

K, Fig. 6. The first-principles calculation of short range order diffuse intensity around (1,1,0) for Au-Cu at l:l stoichiometry at T/T 0 = 1.10. This is based on the tetrahedron-octabedron approximation. 4o

Ising

0.2

A

Au - Pd

,~/

,.J

e 30

A/Au_C u ~ 20

o 0.1 ill

2

/

% 10

"'°-~'-~-;°~.-31.0

1.2

1.4

/

°1.6

1.8

T

Fig. 7. Temperature dependence of the integrated intensity in the first Brillouin zone for Au~Eu at 1:1 stoichiometry. The open circles and the solid circles indicate, respectively, the tetrahedron approximation and the tetrahedronoctahedron approximation. The temperature axis is normalized by the instability temperature Ts obtained for each approximation.

I

0.0

' ~'

I 2.0

k 8T

Fig. 8. T/IsRo()-TIv2 relation for Au-Cu (open triangle), Au-Ag (solid triangle), Au-Pd (open circle) and Ising model (solid circle). The temperature axis is normalized by the nearest neighbor pair interaction energy v2.

MOHRI et al.: SHORT RANGE ORDER IN Au-Cu, Au-Ag AND Au-Pd ALLOYS ifitrinsic deficiencies in t h e tetrahedron approximation which, in the past, has been exclusively relied upon for first-principles phase diagram calculations. Considerable improvement in the overall description of short-range order and phase equilibrium is obtained in the tetrahedron-octahedron approximation. Acknowledgements--This work was carried out under the Visiting Researcher's Program of the Institute for Solid State Physics, the University of Tokyo. We acknowledge the support by a Grant-in-Aid for Cooperative Research from the Japanese Ministry of Education, Science and Culture. One of us (J.M.S.) wishes to acknowledge support by the National Science Foundation under Grants No. DMR-8996244 and No. INT-89-96252. REFERENCES

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