Journal
127
ofthe LessCommon Metals, 168( 1991) 127 144
ELECTRONIC STRUCTURE, ALLOY PHASE STABILITY AND PHASE DIAGRAMS A. GONIS, M. SLUITER and P. E. A. TURCHI Chemistry and Materials Science, L268, Lawrence Livermore Na~ionaf Labora~o~, Livermore, CA 94550 (U.S.A.]
G. M. STOCKS and D. M. NICHOLSON ~e~afs and Ceramics, Oak Ridge Na~fo~afLabora~o~, Oak Ridge, TN 37831 (U.S.A.) (Received March 26,199O)
Summary We review the relevance of electronic structure calculations to the determination of alloy phase stability and alloy phase diagrams. The Co~ollyW~li~s method, the generalized perturbation method, the embedded cluster method and the method of concentration waves are presented and their main features are discussed and compared. The results of calculations of effective cluster interactions in substitutionally disordered alloys and of phase diagrams for specific alloy systems, e.g. PdRh, PdV and AlLi, are shown and work currently in progress is briefly described.
1. Introduction In this paper, we review and contrast a number of methods which have been developed for the study of the thermodynamic properties of substitutionally disordered alloys. The common characteristic of the methods discussed is that all rely on the ~rstpr~ciples calculation of the electronic structure of judiciously chosen systems to extract sets of parameters that can be used in connection with statistical mechanical models to calculate thermodynamic properties. The big prize sought in these calculations is undoubtedly the determination of the phase diagram of an alloy. We will see to what extent the various methods have succeeded toward accomplishing that goal. Present day methods for the firstprinciples calculation of alloy phase stability, chemical short and longrange order, and ultimately alloy phase diagrams are essentially based on the socalled threedimensional Ising model, in co~ection with various approximate schemes for treating the statistical mechanical aspects of Elsevier Sequoia/Printed
in The Netherlands
128
the problem. In these methods, one seeks to obtain the partition function [ 11,
(1.1) where /l = l/k, T, with k, being Boltzmami’s constant and T denoting the absolute temperature, and where Tr denotes the trace with respect to alloy configurations. As is well known, the Hamiltonian of the classical Ising model has the simple form, H= f 1 J;,aioj
(1.2)
‘.J
where J, is an exchange integral between spins on sites i and j, and the occupation numbers oi, 2m component system take the values + m, k (m  1 ), . . . , k 1, 0. From 2 one can readily obtain all the thermodynamic properties of a system. One may attempt to describe binary random, or partially ordered substitutional alloys, AB, c, where c denotes the concentration of atoms of type A in the alloy, by means of a Hamiltonian of the same form as that shown in eqn. ( 1.2), (1.3) through an appropriate redefinition of the various parameters. Here, the effective pair interaction, Vij, is defined as VV=( V$*+ V;”  2 VP”), where V;B is the pair interaction between atoms of type a and /3 (a, /? denote atoms A or B) at sites i and j, and the occupation numbers ui assume the values l(  1) for an A(B) atom at site i. More generally, the alloy Hamiltonian can be written in terms of higher order effective interactions, involving triplets, quadruplets, etc., of sites in the form, Ha”“y=:C
I/l/qu,+:
1.1
1
I/,,oiujo,+...
(1.4)
I.,.k
Quite often, one must simply assume that the expansion in eqn. (1.4) converges rapidly enough to be useful, a property that must be verified on a casebycase basis. Also, the mode of determination of the I/r/,,,k, the socalled effective cluster interactions (ECIs) may often bring into question their uniqueness and consequently the validity of calculated properties. In any case, once the ECIs have been determined for a given alloy system, one seeks to obtain the thermodynamic properties of an alloy through a calculation of the partition function, eqn. ( 1.1). At the same time, it should be pointed out that the parameters I/i/,,,k are usually both concentration and temperature dependent in contrast to the exchange integrals entering the Ising model that are constant with respect to the physical parameters of a system. That eqn. ( 1.4) is a meaningful and sensible representation of the energy of an alloy has been demonstrated to be true in the case of normal metals and their alloys [2] through the use of pseudopotential theory. In transitionmetal alloys, where disorder effects can be much stronger than in normal metals, the justification of an expansion such as the one in eqn. ( 1.4) is much more problematic. At the
129
same time, the success of phenomenological theories, based on eqn. ( 1.4), in the study of ordering processes has initiated extensive work aimed at providing a direct link between electronic structure calculations and statistical models. In this paper, we review four methods which allow the determination of ECIs, namely, the ConnollyWilliams method (CWM) [3], the generalized perturbation method (GPM) [4], the related embedded cluster method (ECM) [5], and the method of concentration waves (CWs) [6]. The last three methods are based on the coherent potential approximation (CPA) [7121 and certain of its extensions [4, 13211 and have been reviewed in previous publications [5, 22, 231, in connection with both tight binding [5] (TB) and muffintin (MT) [22,23] Hamiltonians. Once a set of ECIs has been obtained, it remains to specify the statistical model to be used to calculate thermodynamic and statistical properties of interest. In this regard, Kikuchi’s cluster variation method (CVM) [24261 and Monte Carlo simulations [27,28] can be mentioned. These techniques have evolved to the stage in which they can incorporate both concentration and temperature dependent interactions. In our discussion, we consider only the dependence on concentration, and treat explicitly the case of binary alloys. The latter restriction is used only for the sake of clarity of presentation; multicomponent alloys can be treated through essentially straightforward generalizations of the methods to be presented here.
2. The method of Connolly and Williams In the approach proposed by Connolly and Williams [3] the total energy for a given binary alloy configuration, 1, is written in the form, E”‘(a) =I
v,(a) Lj”,
(2.1)
where T/I,(a) and t”, denote, respectively, the configuration independent ECIs associated with clusters of IZsites and the nsite correlation functions, the letter a denotes the lattice parameter, and the sum runs over all cluster types, including the empty lattice, on a fixed lattice structure. The correlation functions are given explicitly by the expression,
(2.2) where N,, is the number of sites in cluster n, and the sum runs over all nth order clusters of a given type. These correlation functions can be found essentially by “inspection” for any ordered alloy configuration, while the corresponding energies, Ejj), can be obtained from a calculation of the electronic structure of the ordered material. A simple matrix inversion then allows the identification of ECIs by means of eqn. (2.1). It is possible to ascribe a concentration dependence to the ECIs, v,(a), because firstprinciples electronic structure calculations allow the selfconsistent determination of the lattice parameter at each concentration. However, in spite of
130
this desirable feature and of recent applications [29321 of the CWM, this method is beset by a number of conceptual and computational difficulties. The most important among these regards the uniqueness of the interactions. These parameters can be strongly dependent on the set of ordered structures used in the fit through eqn. (2.1). Such a “configurational” dependence of the ECIs can lead to significant differences in calculated thermodynamic properties, e.g. alloy phase diagrams. It also brings into question the meaning of convergence of eqn. (2.1), as this equation will presumably converge at different rates, if at all, for different sets of configurations and maximum clusters. The results of calculations [33] to be discussed in the following sections indicate that eqn. (2.1) in the CWM should be used with care. In particular, the condition of rapid convergence is by no means completely fulfilled, and even within the same cluster truncation the V, are not unambiguously defined. On the other hand the CWM can, under an artful choice of ordered structures, lead to reasonable results and to phase diagrams that compare favorably with those determined experimentally. The conceptual drawbacks of the CWM are alleviated to some extent by the CPAbased methods, the GPM, the ECM and that of CW, and we now turn to a discussion of those methods.
3. Methods based on the coherent potential approximationthe and CWs
GPM, the ECM
The methods to be discussed in this section overcome the difficulties associated with the uniqueness of the CWM by providing a proper treatment of the random part of the disorder characterizing a substitutional alloy. Here, the central idea is to begin by determining an appropriate reference medium which properly interpolates between the various alloy configurations, and to study statistical fluctuations by means of either perturbation theory methods, as is done in the GPM, through the exact treatment of cluster configurations in the alloy, as is prescribed by the ECM, or through the use of linearresponse theory, as in the approach underlying the method of CW. All of these methods rely on the CPA for the determination of a proper effective medium to describe the completely disordered state of an alloy. We shall begin our discussion of these methods by reviewing briefly the CPA, emphasizing those aspects that are most relevant for the introduction of the GPM, the ECM and the method of CW. 3.1. The coherent potential approximation It is now generally acknowledged that an application of the CPA within the Greenfunction formalism introduced for the treatment of ordered materials by Korringa [34], and by Kohn and Rostoker [35] (KKRCPA) provides a reliable firstprinciples description of the electronic structure and related properties of substitutionally disordered alloys. The CPA is a singlesite meanfield theory which allows the selfconsistent determination of a translationally invariant effective medium that replaces the configurational average of the disordered material. The
131
medium is determined through the condition that the scattering of an impurity atom of any of the alloy species vanishes on the average. In the KKRCPA the oneelectron random potentials, which for simplicity are taken to be of the nonoverlapping muffintin form, are characterized at energy E by the partialwave scattering amplitudes f+(E), where a refers to the species and L stands for the angular momentum indices, 1, m. The fa,J E) are taken to be configuration independent, although in a selfconsistent calculation they do depend on concentration. The reference medium is described within the CPA by an ordered array of effective scattering amplitudes, f,(E), which are obtained as the solutions of the equation
1
c”X”=O
(3.1)
n where c” is the concentration )(a = Ama( I  f()oAma)l
of the atoms of type a in the alloy and (3.2)
Here, ma[= (t”)‘1 is the inverse of the ontheenergyshell scattering matrix (transition matrix) corresponding to an atom of type a in the alloy, with ta =  f”/ ,/E, and Zoo is the sitediagonal element of the scattering path operator introduced by Gyorffy and Stott [36]. In general, the various quantities in eqns. (3.1) and (3.2) are matrices which possess nonvanishing matrix elements between different angular momentum states, as is indicated by the full matrix notation employed here. In most practical applications, and specifically in calculations based on spherically symmetric MT potentials and carried out to L < 2, the various matrices become diagonal, a feature which greatly facilitates the performance of numerical calculations. The selfconsistent nature of the KKRCPA is clearly displayed when we note that Am” is the difference Am”=fima
(3.3)
and that d3k[rirG(k)]’
(3.4)
where G(k) denotes the familiar KKR structure constants of the underlying lattice and the integral extends over the first Brillouin zone (BZ) of the reciprocal lattice. The CPA condition, eqn. (3.1), is based on the assumption that the concentration of each species is uniform throughout the alloy. In order to examine ordering and segregation tendencies in alloys, one must consider the effects of concentration fluctuations, &p, associated with site i and species a in the alloy. The forms chosen for SC: have led to two formally distinct but numerically quite similar (at least in the case of metallic alloys) methods for the study of ordering effects. The choice SC: = cy  ca, where c4 is the “instantaneous” concentration of species a at site i, is best suited for the study of fluctuations driven by long range interactions. It leads to the consideration of phase stability in terms of static concentration waves [37, 381. It also leads naturally to a definition of longrangeorder (LRO)
132
parameters and pair correlation functions, and can be shown [37] to be consistent with phenomenological ordering theories of the Landau form. On the other hand, the choice SC; =py CO, where pp = l(0) if site i is (is not) occupied by an atom of type ct, underlies the development of the GPM and the ECM and is best suited for the study of fluctuations driven by shortranged interactions, and SRO effects. Thus, the two different modes of expanding the ordering energy in terms of concentration fluctuations are complementary and, at least for binary alloys of transition metals, have been found to yield very similar numerical results. In contrast to the method of concentration waves, which is consistent with linearresponse theory and the meanfield nature of the CPA, the GPM and the ECM are perturbationlike methods which break away from the selfconsistency underlying the CPA. This, at first glance, undesirable feature is compensated by the ability to calculate nonlinear effects caused by finite concentration fluctuations. These two methods lead to rapidly convergent expansions of the configurational energy and allow the use of statistical models for the calculation of the entropy, e.g. CVM and MonteCarlo simulations. Furthermore, they allow the calculation of multisite effective interactions which in some cases are necessary in order to obtain converged, accurate expansions of the ordering energy. In fact these two methods are strongly related to one another [5], with the ECM providing the summation of certain classes of diagrams in the GPM expansions. We now turn to a somewhat more detailed discussion of these different methods. 3.2. The method of concentration waves Perhaps the most outstanding accomplishment of the CW approach to ordering phenomena is the ability to predict instabilities (spinodals) and SRO intensities. Such predictions are based on the knowledge, heretofore obtainable from experimental studies only (e.g. Xray, electron, and neutron diffraction), of a characteristic function St2)(k). The method of concentration waves allows the calculation of Stz)(k) from a knowledge of the electronic structure of an alloy. Although in principle the method of CW is exact within the context of linearresponse theory, its implementation requires the use of a singlesite meanfield approach (BraggWilliams) to ordering. Thus, the method cannot account for the effect of SRO on the configurational entropy. On the other hand, since it allows the treatment of pairwise interactions of infinite range, it may be expected to yield a not too inaccurate estimate of the transition temperature, at least for longrange pair interactions. At the same time pairwise interactions, even taken to be concentration and temperature dependent, may not be sufficient for the calculation of accurate alloy phase diagrams. A more appropriate treatment of the statistical mechanics and the inclusion of multisite interactions may be required, a task which can be accomplished within the cluster expansion schemes afforded within the GPM and the ECM. The characteristic function Sc2)(k) can be obtained through an expansion of the thermodynamic function, Q, (the Helmholtz free energy) of an alloy in terms of concentration fluctuations about the uniform medium of the CPA. If environmental effects can be considered as being small in some sense, Q can be expanded
133
c” or, equivalently
in powers of SC: =cr&, = c,  c,
Q=q,+~s:“sci+f~ I
for a binary alloy, in powers
S’,“dCisC,+~ c s~;~dc,6c,dc,+...
of
(3.5)
1.l.k
1.1
where (3.6) is the nparticle correlation function calculated for all ci = c, and Q,, is the concentrationdependent but configurationindependent contribution of the homogeneous CPA medium. The summation indices i,j, k, . . . , in eqn. (3.5) run over all sites in the material. Since in the homogeneous medium all sites are equivalent, and by definition C&,=0
(3.7)
the term linear in 6c vanishes identically, and for infinitesimal fluctuations we can write
(We note that in systems with polyatomic bases the singlesite contributions, linear in 6c, must be taken into account.) We now introduce the lattice Fourier transforms 6c, =;I
&,, exp( ik.R,,)
(3.9)
l, and Q(k)=;
1 (P)‘exp(ik*R,,,,)=~ ,r.,n#n
3(k+q)
Z(q)(Z”“)*
(3.10)
q
where all kspace summations (integrations) extend over the first Brillouin zone of the reciprocal lattice, and ?“’ denotes the site offdiagonal term of the scattering path operator determined in the CPA. Upon using these quantities and a certain amount of algebraic manipulation consistent with the CPA condition, we can write eqn (3.8) in the form, (3.11) where E, is the Fermi energy and the implicit dependence of the various quantities on the energy is to be understood. It is to be pointed out that eqn. (3.11) represents
134
the bandenergy contribution to A52 which, in most cases, can be expected to be the dominant one. Equation (3.11) is valid for general multicomponent alloys, yielding the effective pair interaction between sites i#j by means of a direct Fourier transform. In the case of binary alloys we find explicitly,
sf’=&1 Sf2)(k) BZ
exp(
(3.12)
ik.R,)
k
where V,, is the volume of the first Brillouin Zone, S~2)(k)=~Im~ dE(6X)*Q(k)[l~(lc)(6X)~Q(k)]‘1 jr m
(3.13)
and
dX=XAXB
(3.14)
with X” being given by eqn. (3.2). The term L in eqn. (3.13) can be chosen to ensure that the CowleyWarren SRO parameter, a, is properly normalized, ]a( k) d2 k = 1. This is known as the spherical approximation. In the high temperature limit, 1= S,,. In deriving these equations we have used the fact that for MT potentials and for L I 2 the X matrices are diagonal in 1, m to obtain the fairly simple result of eqn. (3.13). Further, if the term c( 1  c) ( 6x)2 Q(k), the socalled vertex correction, can be neglected as being small, we obtain (3.15) for the effective pair interaction between sites i and j. It can easily be shown that $3 = SAA+ $?B  S”B  $!A I/
‘J
11
‘I
(3.16)
11
an expression of familiar form. In the more general case in which vertex corrections must be taken into account the pair correlation function can be written as E, Sp)= ?I,( x
m
dE(GX)” @8’+ [
1 (X2)(?)2(Zk/)*+... k#t
I
(3.17)
This realspace expansion shows explicitly that S$?) contains the response of all medium sites (summation over k). Finally, the S (d)(k) can be used to obtain an expression for the Cowley SRO parameter, a(k), given by the wellknown Krivoglaz, ClappMoss formula, a(k)=[l
c(l
c)PS(~)(~)]~’
a quantity which can be extracted from experimental or neutron diffuse scattering.
(3.18) measurements
such as Xray
13s
3.3. The generalized perturbation method We continue to consider only the bandstructure contribution dynamic potential of an alloy, which can be written in the form E, S(E,)=
to the thermo
 j. dE N(E) rn
(3.19)
in terms of the integrated electronic density of states. Within the local density approximation to the total energy of an alloy [39], it can be shown that Q(E,) can be written as a sum of two terms: (i) the concentration dependent but configuration S$ c, E, ) of the CPA reference medium; and (ii) a configurationdependent energy Q’({p:], E, ).
independent
energy
This last energy term can be expressed as EJ,
Q’({pp},E,)=
$NIrn,
dETrln(lXi) mm
(3.20)
where the trace is taken over both site and angular momentum indices, N denotes the number of sites in the system and i is the strictly site offdiagonal part of 2. On expanding the logarithm in eqn. (3.20) we obtain the result, Q’({p~},E,)=$Nlm
f k=O
1 dETrq
(3.21)
_m
Through the use of CPA algebra, and for the case of binary alloys, this last equation can be expressed in the form (3.22) where 6~,,~= pz  c, and V,$,.,,., defines the kth order effective action involving sites 11,)n2, . . . , nk,
cluster inter
(3.23) the trace being taken over angular momentum indices and the prefactor of 2 accounting for electron spin. Thus, to the lowest order in the perturbation, ]SXa, the quantity 9’ which is commonly called the ordering energy is given by Q’=f 2 v;;~sc,,&,
(3.24)
where V& is the effective pair interaction between sites 0 and II, vb’,’= I/AA+ OII T/BB III1 2 t,‘Ae OII
(3.25)
136
Therefore, as far as the second order terms of the expansion are the predominant ones, a positive (negative) sign of V,,, indicates a tendency toward ordering (phase separation). In closing this subsection, we point out the formal similarity between Syj and V$, although the expressions (3.17) and (3.23) correspond to different expansions of the thermodynamic potential. The expression for V”$ can be obtained from the Sj/2),eqn. (3.23) by ignoring all terms outside the square brackets but the first one. Since these terms tend to be small it is not surprising that numerical calculations reveal very similar results. 3.4. The embedded cluster method We consider eqn. (3.20) in connection with a system consisting of a cluster of n sites, C,,, embedded in a disordered material. An exact treatment of this system would require a complete configurational average over all configurations of the material surrounding the cluster. Clearly such an exact treatment of configurational averaging is not computationally feasible and approximations must be made. At the first level of approximation, all sites outside the cluster are taken as being occupied by effective medium scatterers, or “CPA atoms”. Calling this the coherent potential approximation for the configurational energy, we can write, I, I, i.j
i.j,
k
where E,, is the contribution of the CPA medium and the T/;E:,,,;, are the ECIs as defined [51 in the embedded cluster method (ECM) I$ I/(“’  [ JJ’~~,~,~,,~_,]~ =  ‘, Im 1 dE Tr In[IIjQj(cve”)] [IT,Q”oJd)] I f,lp..lti=[ J$‘~,~,~,,._J,~ m (3.27)
Here, j( even) and j( odd) denote cluster configurations with ev$n and odd numbers of minority (B) atoms, respectively, 0 is the matrix (1  XZ) occurring in eqn. (3.20), and the double prime on the summations denotes sums over distinct sets of sites. The symbol [V,:,; _,,,,_,] z denotes an effective multisite interaction among the n sites of cluster C,, under the restriction that site n is occupied by an atom of type GI. Upon expanding the quantity Tr In Q in eqn. (3.27), we obtain [5] the GPM expression, eqn. (3.23). Thus, the ECM and the GPM are rather similar approximations, with the former corresponding to a summation of sets of diagrams (terms) of the latter. In fact, numerical studies [5] indicate that at least in the case of metallic systems the first term in the GPM expansion is essentially numerically indistinguishable from the complete summation provided by the ECM. 4. Numerical
results
In this section we report on the results of calculations of ECIs and of alloy phase stability and phase diagrams based on the methods discussed in the previous section. Results associated with applications of the method to firstprinciples MT
137 TABLE 1 Tetrahedron effective clusters interactions for WV alloys calculated in the CWM for various sets of orderedstructures:(a)(A,L1~,L1,,,B},(b)~A,D0~~A,B,L1,,,LI,AB,,B/and(c)(A,D0~,,40, B}. (40 designates an AzB2 phase in Kanamori’s notation [47]) V
u
h
‘
2 3 4
0.09455 0.00118 0.00299
0.09455 0.00694 0.00587
0.08647 0.00956 0.0084 1
Interactions are designated as follows: 2, nearest neighbor pair; 3, equilateral triangle between nearest neighbors; 4, tetrahedron formed by nearest neighbors.
and more general potentials, as well as to tight binding (TB) Hamiltonians are exhibited. Table 1 shows the nearestneighbor pair, triangle and tetrahedron cluster interactions, Vl, VT and Vi, respectively, calculated within the CWM in a TB description of &the Hamiltonian. The superscript T refers to the tetrahedron truncation of eqn. (2.1) which takes into account the [ 100] family of ordered structures, namely, LI,, (A,B and AB,), and LI,, (AB), in addition to the pure elements A and B. We see that especially the triangle and tetrahedron parameters can vary significantly. (Somewhat surprisingly, the [ 1001 family gives the smallest values of VT and VT ). It is, therefore, obvious that (i) the V,,even within the same cluster truncation are not unambiguously defined, and (ii) the rapid convergence criterion of eqn. (2.1) is not completely fulfilled within the CWM. On the other hand, a judicious choice of ordered structures can indeed lead to alloy phase diagrams that compare favorably with experimental ones. Figure 1 shows the solid state portion of the AlLi phase diagram, calculated with the CWM in the tetrahdron approximation on both the f.c.c. and b.c.c. lattices. Total energies were computed with the full potential linearized augmented plane wave (FLAPW) [40] method, and the configurational aspects of alloying were treated with the Cluster Variation Method [24, 251 in the tetrahedron approximation [41]. Total energies of only 5 (6) ordered structures were required for a description of phase stability on the f.c.c. (b.c.c.) lattice. The CWM result in Fig. 1 compares favorably with experimental findings. It correctly yields an aluminumrich f.c.c. solid solution, with a high solubility for lithium, and a virtually aluminumfree f.c.c. lithium phase. Lithium is calculated to be more stable in the f.c.c. than the b.c.c. structure because of the neglect of vibrational entropy. A very stable B32 phase is predicted at about equiatomic composition, in agreement with the observation that the B32Al Li phase melts before disordering sets in. The calculation yields that a metastable LI, phase with composition Al,Li can form from the aluminumrich f.c.c. solid solution, in agreement with experimental evidence [42]. In addition to providing a phase diagram and elucidating underlying metastable phase equilibria, the CWM allows the evaluation of thermodynamic potentials and activities, and of variations of lattice parameter and bulk modulus with composition in various phases.
138 2500
2000
1500 8 k 1000
500
0
’ 0
I 0.2
0.4 CLi
0.6
0.8
1
(at. %)
Fig. 1. The solid state part of the AILi phase diagram. Both f.c.c. and b.c.c. based phases have been included. The f.c.c. based ordered phases have been repressed because of the greater stability of the b.c.c. based superstructures. The metastable f.c.c. based equilibria are indicated with broken lines. The [ 1001 f.c.c. ordering spinodal is denoted with a dotted line.
In spite of these desirable features and several successful applications [29321, as mentioned earlier, the CWM suffers from a number of conceptual difficulties. These arise as a consequence of (i) the rather arbitrary choice for the alloy Hamiltonian, which bears no fundamental relation to the description of the electronic structure of the alloy; and (ii) the truncation of eqn. (2.1), a requirement for inversion. The absence of a fund~ental relationship between the electronic structure and the CWM ECIs leaves no room for obtaining interactions other than essentially taking differences between energies of ordered structures. In ideal cases two structures can be found that differ only with respect to one particular correlation, and a number of correlations pertaining to very large clusters, whose interactions are supposed to be negligible. However, there is no guarantee of any kind for convergence of an expression like (2.1). That makes the second issue, the one of truncating the infinite sum, all the more questionable. In practice, interactions corresponding to larger clusters have been found to be much less important than those corresponding to small clusters, but considering that an infinite number of contributions from clusters larger than some preassigned size, nmax, has been neglected, this is a rather weak suggestion of convergence. The neglected higher order ECIs are “mapped” onto the ECI’s within nmaxby the matrix inversion in an indiscriminate manner. Errors in the total energies are distributed likewise over the ECI’s. As a result, the ECIs obtained are not unique. They depend on the particular set of interactions considered, and the set of ordered structures selected
[43]. The “ambiguity” in the CWM ECIs was found to be of the order of 30 meV in the usual tetrahedron approximation on the f.c.c. lattice. For ECIs with large absolute values such ambiguities are relatively small, but multisite ECIs with characteristically small absolute values can change by orders of magnitude or even change sign. These serious drawbacks of the CWM may to some extent be reduced, for example, by considering large numbers of ordered configurations to obtain only a small set of interactions. The large number of configurations could help “average out” the effect of clusters larger than ~l,~~and of inaccuracies in the total energies. In order to predict accurately the ordering energy, a necessity for computing a realistic phase diagram, the energies of both ordered and random configurations should be represented. This means that the set of ordered structures must include not only those actually observed in the alloy, but also other configurations which are far removed from actually occurring configurations. Research to assess the merits of these extensions is currently in progress. The results of calculations based on the method of CW are shown in Figs. 2 through 4. Figures 2 and 3 depict the SRO parameter, a(k), defined in eqn. (3.18), for alloys exhibiting phase separation, Fig. 2, or ordering behavior, Fig. 3. From eqn. (3.18), we see that at high temperatures the term proportional to #I will be small leading to an a(k) that is sensibly flat. As the temperature is reduced, a will peak at those values of k where S Q) is largest and eventually will become singular at the spinodal temperature, T,, and wavevedtor k,. This progression from flat to singular behavior is illustrated by the plots of a for Pdo,,,,Rh,~,,, in Fig. 2. The singularity indicates the instability of the concentration wave wavevector, k,, to the formation of longrange order. For example, a peak at k = (0, 0,O) indicates phase separation, while a peak at k = ( 1, 0,O) in a f.c.c. material suggests ordering in the Cu,Au or CuAu ordered structure. Peaks away from the special points may correspond to the formation of long period superstructures. A well known example is provided by the CuPd alloy, which tends to form long period superstructures. Here the CW wavevector is determined by Fermisurface nesting and is not necessarily commensurate with the lattice. The calculated diffuse scattering intensity of Cu,.,,Pd,,.zs is shown in Fig. 3. The PdRh alloy provides a convenient test case for the application of the CW method because of its particularly simple phase diagram, which consists of only liquid and f.c.c. solid solution phases. In Fig. 4, the triangles represent T, as computed at five concentrations. The triangles outline a roughly parabolic curve, the spinodal, below which the alloy is unstable and separates into palladiumrich and rhodiumrich solid solutions. In the region just outside the spinodal, the alloy is metastable; given sufficient diffusion, the alloy will phase separate. The region of metastability continues to the phase boundary beyond which the solid solution exhibits clusteringtype SRO but will not phase separate. The calculated phase boundary, indicated by dots in Fig. 4, was found by the common tangent construction using freeenergy of mixing curves based on S(‘) at k = 0. Because SC’)(k = 0) is the second derivative of the free energy of the solid solution, the free energy of mixing can be found by integrating 9’) (k = 0) twice over concentration. In these calculations, the effect of phonon broadening of the electronic states is roughly approximated by adding an imaginary part, proportional to the electronphonon
Fig. 2. Calculated SRO diffuse scattering map for Pd,, z5Rh,, 75alloy at 1350 K (top), 1300 K (middle) and 1250 K (bottom) panel.
coupling constant [44] to the electronic selfenergy. The resulting miscibility gap corresponds fairly closely to the experimental one [45,46]. Effective pair interactions obtained in the GPM/ECM formalisms for PdRh and PdV alloys are shown in Figs. 5 and 6. As is seen in Fig. 5, the pair interactions in the Pd,,.&h,,,S,, alloy decay very rapidly as a function of neighbor distance, and are always negative, consistent with the phase separation tendency of this alloy. This is in agreement with the existence of a miscibility gap shown in Fig. 4. On the other hand, the alternating sign of the interaction in the WV alloy leads to much richer behavior. Taking account of pair interactions up to and including fourth neighbors, we predict a stable ground state in the DO,, ordered structure at
141
Fig. 3. Calculated SRO diffuse scattering for C~,,_~~pd~, 25.
PALLADIUMRHODIUM ALLOY PHASE DIAGRAM 2mm
1600

0 RI
20
40 Rh CONCENTRATION
60
60 (al.%)
100 Rh
Fig. 4. Comparison of calculated spinodal temperatures and phase boundary with the ex~e~rnent~il~ determined phase diagram: triangles, calculated spinodal temperatures including correction for the electronphonon interaction; dotted lines, calculated phase boundary, chain curve, experimental phase boundary according to Elliot 1461; futl curve, experimental phase boundary according to Shield and Williams [45].
142
4.c
Pd Rh
I , ,
T/V2
IO

4.0
; ,
GPM
_
20 % K
q...,
ECM
5
>
2.0
Vl
>
0.0
: f I .a B ,
‘b \ ‘,\ \ \‘\ \ ‘;3._ x,’ ;c \. &m.4
‘; K
.c
I
n ECM
6.0
q
I
I
I
oGPM
0.0
d,
2.0
//’
 2.0 1
2
3
I
I
I
4
5
6
7
0
n
Fig. 5. Variation of nth neighbor effective pair interactions c = 0.5, calculated in the CPM.
20
40
60
80
100
At % Pd
in Pd,V, ~( and Pd,.Rh, ~I alloys, for
Fig. 6. The concentration dependence of the first, VI, and second, V2, neighbor interactions and the ratio,  V, / V2, in Pd,.V, ~alloys as obtained in the GPM and the ECM.
c= 0.75,as is observed experimentally. Considering only first and second neighbors yields an (incorrect) LI, ordered structure (see Fig. 6). It follows that the convergence of the energy expansions in terms of longrange pair interactions is an important ingredient in the calculation of alloy phase diagrams. The variation of first and secondneighbor effective interactions in Pd,V,_. as function of concentration c (shown in Fig. 6) indicates the excellent agreement between GPM and ECM effective pair interactions, as discussed in Section 3.4. 5. Conclusion Modern firstprinciples methods for the calculation of alloy phase diagrams rely on accurate determination of electronic structure, and associated total energies of disordered or ordered alloys. Such calculations yield immediately information on energies of mixing. More detailed information on effective cluster interactions can also be obtained through a number of schemes, such as the CWM, the method of CW, the GPM and the ECM. These interactions, along with the energies of mixing, can then be used to obtain fairly accurate predictions of phase stability at T=O K, and also at TZ 0 K when appropriate statistical thermodynamics is applied.
143
As was discussed in the body of the paper, fairly accurate phase diagrams can be obtained through judicious use of the CWM. This method, however, suffers from the conceptual drawback that it yields nonunique prescriptions for obtaining the various ECIs. Methods such as that of CW, the GPM and the ECM do yield uniquely defined ECIs, but are only recently being carried out to the calculation of phase diagrams. Work currently in progress is designed to test the feasibility of obtaining phase diagrams entirely from firstprinciples electronic structure calculations, in a unique manner. The results of these investigations are nearing completion and will be reported in a future communication.
Acknowledgments
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