First Principles Theory of Disordered Alloys and Alloy Phase Stability

Almost immediately after the discovery of quantum mechanics, physicists and metallurgist began the task of trying to understand the origins of alloy phase stability in terms of the underlying electronic structure. Hume-Rothery [1] used the chemical concepts of electronegativity and atomic size supplemented with the solid state physics notion of electron to atom ratio to formulate a set of rules that could be used to rationalize the complex array of phases that form when two elemental metals are mixed to form an alloy. The Hume-Rothery rulesand their progeny [2], [3] have proved very successful in providing metallurgists and alloy designers with a basis for understanding alloy phase stability.

curacy to allow for the calculation of the small energy differences between different alloy 1 phases; secondly, a method for implementing this theory for the types of phases encountered in alloy phase diagrams, pure elemental metals, ordered intermetallic compounds, substitutionally disordered alloys, and, ultimately, the liquid state; finally, techniques for including the statistical mechanics of compositional rearrangements that occur at finite temperature and are responsible for the most important entropic contribution to the free energy of the disordered solid solution phases.
For the first, the local density approximation (LDA) to density functional theory (DFT) [4] [5][6] provides a basis for obtaining energetics to sufficient relative precision to allow one to ask questions of metallurgical interest e.g. calculation of the small energy changes associated with allotropic transformations and intermetallic compound formation. For the second, modern electronic structure methods allow the equations of LDA-DFT to be solved to a very high level of accuracy for systems with underlying periodicity. However, these methods are still limited to situations in which the basic unit cell _ontain8 a small number of atoms, typically a few tens, ideally < 20. Thus, disordered solid solution phases present a particular problem since there in no underlying small unit cell a_d experiment measures thermodynamic averages. It was for calculating these averages that the so called coherent potential approximation (CPA) was developed [7] [8]. The Korringa-Kohn-Rostoker (KKR)-CPA, that will form the underpinning for these lectures, is the first principles implementation of the basic CPA idea [9] [10][11] [12] within LDA.
Because of the limitations imposed by modern electronic structure methods as to the size of systems that can reasonable handled, it is at the third element, d making the link between the LDA based electronic structure methods and the statistical mechanics of compositional rearrangements, that the difficulty in obtaining a first principles theory of phase stability lies. In recent years several distinct approaches have been developed for making this connection.
In one class of methods the energetics obtained from first principles calculations are mapped onto a generalized Ising model which is then used in connection with either the Monte Carlo method [13] [14] or the cluster variation method (CVM) [15] to treat the statistical mechanics of compositional rearrangements. In the Connolly-Williams method [16][17] the interactions are obtained by mapping the first principles energetics of a set of small unit cell ordered compounds to parameters of a generalized Ising model. In the generalized perturbation method (GPM) [18] [19] the Ising model interactions are obtained by expanding the electronic energy about the disordered state which is treated using the LDA-KKR-CPA method.
An alternative approach that does not require the electronically mediated interaction be mapped onto an effective Ising model is the mean field concentration functional (MF-CF) method of Gyorffy and Stocks [20]. In the MF-CF method, the electronically mediated quantum mechanical nature of the interactions is kept all the way through. Clearly this cannot be done without serious approximation. In the MF-CF method there are two major approximations.
Firstly, statistical mechanics is treated only within mean field theory. Secondly, the electronic structure of the disordered phase is treated using the LDA-KKR-CPA. Unfortunately, the application of LDA-DFT to substitutionally disordered alloys is not a straight forward matter. Although, in a substitutional disordered alloy, the atoms can still be though of as occupying an underlying periodic lattice the occupancy of those lattice sites by the various atom types is random. For a binary AcB(l_c) alloy we can specify the occupancy of a give lattice site I_ by a random variable _i that takes values _ -1 and _ = 0 according as the site is occupied by the A or B atomic species. This is pictured schematically in fig. 1. The electron density and hence the effective one electron potential entering the Kohn-Sham equations then become functions, p(F; { R/}; {_i}) and rely(F; p(_';{R/}; {_i})) of "_heset {_i} of occupation variables that define a specific alloy configuration.
There are two major reasons why these equations cannot be handled in a direct manner. Firstly, it is not possible to solve the LDA self-consistency equations for an arbitrary configuration that has no translational symmetry. Secondly, computation of Clearly, for a random alloy the ensemble of configurations is prohibitively large. In the gedanken scheme outlined above the procedure is clear, first solve the LDA equations making p(F; {/_}; {Zi}) self-consistent with v.,,y(F;p(F; {P_}; {ZI})), compute the total energy, then average. In the LDA-KKR-CPA method this direct approach is avoided by a subtle inversion of the self consistency and averaging processes that makes the partially averaged electron densities (p(F))i,o self-consistent with the partially averaged single site potential (v_(F -1_.; [p(r-')]))_,o.These partial averages are define suc._ that ith-site is occupied by aSh-species whilst all other sites are averaged over. The KKR-CPA method is then used to calculate (p(r-')),,°[231.

The KKR-CPA Method for Electronic Structure of Random Alloys
In outlining the KKR-CPA method, rather than dealing with eq. 1, it is necessary to deal with the corresponding equation for the single particle Green function G(F, P; c) o where v.(F-1_;_.) is the effective potential a_sociated with the nth-site. We shall associate this with the single site partial average potential (v,(F-1_,;[p(r-')]))i,o, more specifically with the KKR.-CPA approximation to it v%(F-1_; _,,). Implicit in this is the assumption that there are only a of these potentiais i.e. that the effective potential that an electron feels at a given site depends only on the occupancy of that site and the overall composition and does not depend on the local configuration of surrounding sites. We also assume that it is sufficient to approximate the full non-spherical potential by a muffin-tin form.
Given a set of effective single site potentials the KKR-CPA provides a direct method for calculating the partially averaged Green function (G(F, P; e))i,o [23] [24]from which the partially averaged electron densities (p(r-'))i,arequired by the self-consistency procedure can be obtained [24] '/ equation for a single muffin-tin scatterer with the vector r_ = _-l_ being measured relative the center of the nth muffin-tin sphere. In eq. 11 ¢£_L_,(e) is the scattering path matrix [25] nm tn,L(e)gnn,,(e)TL,,L(e) (12) n,m_p L u where np gLL" are the angular momentum matrix elements of the free particle propagator connecting sites n and p. The scattering path matrix is the generalization to many scatterers of the sil.tsie site t-matrix, t_,L. Whereas ta,L converts an incoming partial wave in angular momentum channel L into an outgoing scattered wave for a single muffin-tin potential, rg_,(e) converts the incoming L -th partial wave at site rn into _heoutgoing L' -th scattered partial wave at site n in the presence of all of the other scatterers.
Clearly, in eq. 11, by focusing on the site diagonal Green function for the nth site, holding the occupancy of this site at the o-th-species, and averaging over the remaining sites yields the partially averaged Green function (G(F,r';e))i,o that we require. The result is Thus, the problem of finding the partially averaged single site electron aenslty na_ oeel_ reduced to finding a theory for the partially averaged scattering path matrix (T£"_.,),,°. This is what the KKR-CPA is set up to do. " The essence of the KKR-CPA is to approximate the scattering properties of the disordered array of real scattering centers, each characterized by a single site t-matrix t°, by the scattering properties of an ordered array of e._ective scatterers characterized by some effective t-matrix, tv(e), and then to choose to(e)(e) such that it gives the best approximation to the real system. In the KKR-CPA best is defined in the sence that this approximation is the best that can be achieved whilst considering only single-site partial averages. As such the KKR-CPA specifically ignores effects associated with specific local configurations.
The scattering path matrix, Tc,"n(e), for an axray of effective scatterers is simply given by the site diagonal element of eq. 12 where all the t-matrices are tv(e)'s. Since, the underlying lattice is periodic the resulting equation for the ordered array of effective scatterers can be solved using be lattice Fourier transforms to yield _c"_(e) = / dl_:e_:_" [t_t(e)-G(_; e)]-1 Where yc,""(e), t_¢(e) and G(_¢;e) are matrices in (LL _) and the latter axe the KKRstructure constants [23]. l_,, is the vector connecting sites n and m. If we now replace the effective t-matrix at site n by the t-matrix for the a th species the corresponding site diag,:;nal scattering path matrix ya'""(e) describes the scattering from an a impurity em_bedded in the effective medium.
The CPA is then obtained by requiting that replacing a single effective scatterer at some site n by a real scatterer produces no further scattering when averaged over all of the species. Mathematically this statement reduces to the requirement o where _'°'""(e) is given by the solution to the single impurity problem Taken together eqs.15,16, and 14 form a self-consistency condition that determines the effective KKR-CPA scatterer. We refer to these equations collectively as the KKR-CPA equations.
Finally, making the associations in eq. 13 yields the KKR-CPA approximation, p,_(r-'),to the partially averaged single site electron density that we require for the LDA self-consistency step. p=_(r-') is given Before closing this section we note that a couple of other quantities of interest can be obtained straight forwardly from the partially averaged Green function. The single site density of states is given by (20) from which the configurationally averaged total density of states can be obtained by performing the final average over the species O The Fermi energy can be obtained from the charge neutrality condition a where the Ihs of eq. 22 is simply the average nuclear charge in the system, Z,. The final quantity of interest for future discussions is the Bloch spectral function, AS(_:, e), [24] ,4_(_:,_) =-ll,_'_(_-_t,)f_ _(a(_+fi,.; _+ _._; _))d_. (23) The Bloch spectral function contains a complete description of the electronic structure of a random alloy. It is the generalization to a disordered system of the band structure of pure metals and ordered compounds. Indeed, for an ordered system it reduces to A/3(k, e) = _ 6(e -e_,_) 6-function peaks of eq. 24 broaden into peaks with finite width and finite heights whose position in I_, e space trace out the band structure. 2.1.$ Self-consistency and Total Energy As indicated above, given a set of effective single site potentials the KKR-CPA provides a direct method for calculating the partia]ly averaged electron densities _a(r-')that are required in order to develop a self-consistency procedure. All that now remains is to specify how the configurationa]ly averaged single site potentia]s, 0a(r-') are related to (r-')and then to specify how the configurationa]ly average energy E is to be obtained from the self-consistent _a(r-') and 0°(r-').
Since the KKR-CPA method is itself a mean field theory of the effects of disorder on the electronic structure, we have chosen to specify the single site potentia]s in & mean field approximation. Specifically, the potentia] on an o site is taken to be the exchange potential corresponding to _'(r-') plus the solution of Poisson's ecllation for a electron density consisting of _°(r-*)on the central site and the concentration averaged electron density _(r-')= _o ca_°(r-') on all other sites. In the mu/_n-tin approximation this results in [26] i"-' and _o is the average of the interstitial electron density a=A It should he noted that the muffin-tin potential given by (25) for the disordered structure has the same form as that for ordered structures [27], except that the interstitial electron density po is replaced by _0. The LDA-KKR-CPA algorithm is now straightforward. Initial guesses of the partially averaged electron densities F' associated with each of the alloying species are used to compute initial partially averaged potential functions _-a(__ }1,; _",p, _o) which are to be used in the KKR-CPA calculation of new paxtially averaged electron densities. The input and output F' are compared, if they are not equal within some prescribed tolerance then a new guess of the input _ is made according to some appropriate mixing prescription and the process is .repeated until self-consistency is obtained. Once the self-consistent F'(r-') have been obtained within the KKR-CPA all that remains is to calculate the configurationally averaged total energy /_. , In order to establish a relationship between _'(r-') and/_ we begin with the observation [12] that the grand potential N(T, V, p) is in general related to the configurationally averaged density of states through the thermodynamic identity da(T, where/_ is the electron chemical potential and N(#) is the number of electrons F Equation 28 can be integrated to yield an expression for fl that is linear in fi that can be configurationaJly averaged within the CPA. Thus, the above procedure circumvents the need to form the troublesome configurational averages of the square of the charge density that occurs in the standard expression for the total energy. The result of this procedure is where N is the configuration averaged integrated density of states, The first term in eq. 30 is the familiar contribution of the eigenvalue sum plus the electron hole entropy. The remaining term is the so called double counting term written in an unfamiliar form. The important point is, once it has been decided to use the CPA, that the relationship between the electron density and integrated density of states is fixed because they are derived from the CPA Green function. Using single site potentials specified by eq. 25 the second term on the rha ot eq. _0 can 0e integrate, to obtain [12] an expression for f_. Taking T = 0 yields a configl,.rationally averaged energy of the form or where Ej is the expression derived by Janak [27] for ordered systems for a crystal potential in the muffln-tin form, excepting that the pure metal charge densities of Janak are replaced by the configurationally averaged ones defined above. A remarkable feature of eq. 32 is, thanks to the use of CPA, that E retains the variational properties characteristic of E[p] for pure systems. Namely a"-f = 0 (33) Furthermore, as is the case for normal LDA-DFT, taking the variation of the potential energy U[p] with respect to the single site densities yields the effective potential that enters the Schr_linger equation These latter are clearly some of the reasons for the success of the KKR-CPA theory for the total energy [12].

Electronic Structure and Properties of Binary Alloys
By now there are several KKR-CPA codes in existence. They have been used to understand a large body of experimental measurements of the electronic properties of disordered solid solutions. Here we mention just few that have come out of our work. We do this to establish the overall correctness of the KKR-CPA description of the electronic structure and, more importantly, to make the point that the electronic structure provides an interpretation of experimental probes such as X-ray photoelectron spectroscopy and residual resistivity, and further, provides a basis for understanding the driving mechanisms behind ordering phenomena and alloy phase stability. In fig. 4 we show calculated total and component densities of states and XPS spectra for two concentrations of CucPdl-c that were calculated by Winter et ci. [28]. For each composition, the lower left of the four frames compares the calculated XPS spectrum with experiment. It should be noted that the calculated spectra take proper account of the optical matrix elements and that this is a importjfeature in obtaining the pl, _ing agreement between theory and experiment.
"L.a_.'b-The residual resistivity is a particularly sensitive measure of the electronic structure of a disordered alloy since it not only depends on the topology of the Fermi surface but also on the extent to which it is smeared out by disorder. In fig. 5 we show the calculated concentration dependence of the residual resistivity of Ag_Pdl_= alloys obtained by Butler and Stocks   For the Cu-based alloys the calculated results are in excellent agreement with the measured values for the cold worked samples. This is suggestive that the departure for the annealed samples results from the presence of chemical short range order, the effects of which are not considered in the calculations, and which is broken up by cold work.
Having pointed to some of the successes of the KKR-CPA in understanding the electronic structure of disordered alloys we now turn to the energetics.

Energies of Mixing
Given a theory of the total energy of the disordered state it is a relatively straightforward matter to calculate the energy of mixing AE _z. For a binary substitutional A_BI_= alloy A.£_" is given by where E _ and E B are the ground state energies of the pure A and pure B metals. The way the calculations proceed is entirely standard; for a given underlying crystal structure, the energy is calculated as a function of lattice spacing for both pure metals and disordered alloys. The ground state energy and equilibrium lattice spacing are given, in plots of energy ¢s lattice constant, by the minimum energy and corresponding lattice constant. The ground state energies are then used in eq. 35 to obtain the energy of mixing. It is perhaps worth a c_mment on the significance of the energy of mixing defined in this way since, ideally, the compositionally homogeneously disordered state to which the LDA-KKR-CPA energies correspond is only realizable at infinitely high temperature but we are evaluating it at T -OK. Certainly, this energy does not have to correspond to measured enthalpies of mixing since these measurements are often made at moderate temperature. However, excepting in systems where short range order effects are particularly pronounced one would not expect them to be wildly different. Fig. 7 shows the results of our calculations of AE _x for the CucZnl_, alloy system [12] [32] for both fcc and bcc structures. Since, structural energy differences cannot be J  fig. 7 are referenced to the fcc by taking taking the structural energy differences for the elemental metals from full-potential FLAPW calculations.
The negative sign of AE miximplies an ordering tendency which, indeed, is a feature of the rather complicated Cu¢Znl_¢ phase diagram [33]. Interestingly AE _x is not strictly parabolic and this suggests a complex phase diagram. Comparison of the calculated energies of mixing in the disordered phase with available experimental measurements [34] and with those obtained from assessed phase diagrams [35] suggests that the calculated values that are approximately a half of the ezperirnental ones. Since, as we shall see later, the calculated values produce a quite good description of the phase diagram of Cu-rich CucZnl_, alloys, this discrepancy may be more apparent than real. However, recent calculations [36] of the total energy of the disordered state that attempt to go beyond mean field theory in the potential reconstruction step and include some account of charge correlations between neighboring sites suggest that this discrepancy may be due to the use of the mean field potential reconstruction and the fact that there is some small charge transfer.
In fig. 8 we show calculated energies of mixing and equilibrium lattice spacing for AlcAgl_c alloys [37]. The shape of the calculated energy of mixing curve deviates strongly from the parabolic c(1 -c) shape assumed in regular solution theory and is AI concentration AI concentration, again indicative of s complex phase diagram. The positive energies of mixing for AIrich .alloy suggests that these, alloys would want to phase separate, whilst the neg&tive values of AE _= for Ag-rich ahoy suggests that these alloys would want to order at low temperature. We shall return to this observation l&ter. The calculated concen_ trstion dependence of the equilibrium lattice spacing deviates slightly from Vegard's rule with the sign of the deviation being positive for Al-rich alloys and negative _or Ag-rich. Comparison with the concentration c]ependence observed experimentally is made difficult by the fact that for pure Ag and pure AI the calculated equilibrium lattice constants deviate from experiment with opposite signs, however, it is the case that the sense of the deviation from Vegard's rule for the experimented results is in agreement with the calculation.

LDA-KKR-CPA Based Theories
Having developed a theory of the electronic structure and energetics of the ideal random solid solution we can now turn to the task of building a theory of ordering and phase stability on it. Thus, we have to address the dit_cult task of accounting for the statistical mechanics of concentration fluctuations within a first principles theory. In the following subsections we will briefly describe and show results for two rather different approaches, a concentration functional (CF) theory implemented within mean field theory, and the generalized perturbation method (GPM). Whilst both methods are based on the underlying LDA-KKR-CPA theory of the disordered state, the emphasis is rather different. The mean field theory places stress on retaining the electronic interactions in their full generality. The consequences of this approach is that it is only possible to treat the statistical mechanics of concentration fluctuation within mean field theory. Thus potentially important cluster effects within the configurational entropy are neglected. However, as with most mean field theories the results of this approach not only provide important insights into the physical mechanism that drive particular orderings but also provide a reference from which to judge the need for mcluchng a OeLLertream_en_ oi the configurational entropy such as that provided by the Monte Carlo method or th_ " cluster variation method (CVM).
In the GPM, stress is placed on obtaining a better description on the configurational entropy than that provided by the mean field or point approximation. Consequently, the electronic interactions are mapped onto a generalized Ising model, and the electronic energy of specific configurations is partitioned between site, pairs, triplets, etc., nearest neighbors, next nearest neighbors, etc.. However, when coupled with the LDA-KKR-CPA the GPM provides a first principles means by which these interactions can be calculated. Once, calculated questions of short range order and phase stability can be explored using the CVM and Monte Carlo method.

Mean Field Concentration Functional Theory
The first pripciples concentration functional tL_ory introduced by Gyorfry and Stocks [20] is based on an adaptation of the classical density functional theory of liquids [39] to a lattice gas model of substitutionady disordered alloys. Though a powerful device that produces a number of interesting results, the theory is much like the density functional theory of the electron gas in that without the local density approximation, which converts the theory into a practical computational method, little can be done with it. For the concentration functional method, the approximation that makes the method tractable is the mean field (MF) approximation. The MF-CF method was set out in the original paper and has been restated several times since, both in applications to alloy ordering [40], [41] and in connection with the disordered local moment theory of ferromagnetic ordering in the 3d itinerant ferromagnets [42]. Thus,we will only state the principle results.
The central result of the mean-field CF theory is that there is a mean field grand potential, flMF ( = keT In 1 c_ whose solution defines the equilibrium set {C/}o of concentrations. The interpretation of ficP^({c/}) in eq. 36 is that it is the CPA grand potential corresponding to an inhomogeneous set of local concentrations {._ ) i.e. the probability that a site, say n, is occupied by an A atom, namely c_, differs from site to site. We refer to this as the inhomogeneous CPA [18], [20]. The reason for the CPA electronic . grand potential appearing in eq. 36 is that we are reqmred Oy tne mean ]lei_ u_r) to take averages with respect to the inhomogeneous product distribution i where p_(_), which now varies from site to site, and is given by As remarked earlier, it is for performing averages with respect to a probability function that is a product of independent site probability functions that the CPA was developed. Although, this inhomogeneous KKR-CPA recipe implicit in eq. 36 can not be implemented numerically, it is a very useful formal device because it can be expanded about the homogeneous limit, i.e. c_ ffi _ Vi, and it allows the calculation of the derivatives sl" -- and where _ is the uniform concentration of the high temperature disordered phase. The significance of SO) is provided by eq. 38. As was pointed out by Gy_ffry and S_._ck_ [20] the second derivative is the mean-field approximation to_the Orstein-Zerni,_ke direct correlation function [43] and it plays a variety of important roles in the _heory of compositional order. In particular .

q(?) is related to the Warren-Cowley
--IJ short-range order parameter, _0, that can be measured using X-ray, neutron and electron diffuse scattering experiments [44]. Specifically _(_)is given by where a(_) and S{_)(_) are the lattice Fourier transforms of a_ and _{2} respectively. -0 The important point about eq. 44 is that S{2)(_) car be evaluated from information available at the end of a KKR-CPA calculation. For the case where only the band structure contribution to the grand potential is considered, the first term on the right of eq. 30, explicit expressions are given in reference [40].
Before showing the results of calculations of S(2}(_) for a number of alloy systems. a few preliminary comments are in order. Firstly, the configurational entropy that appears in eq. 36 is simply the ideal entropy of mixing. Thus, effects that that explicitly depend on higher order entropic contributions are neglected. However, by taking this tack we are able to retain the electronic interaction in there full generality. This allows us to identify the specific electronic mechanisms that are responsible for the particular form of the SRO of clustering seen in any particular system. Secondly, since the LDA-KKR-CPA method is a mean field theory of the effects of disorder on the electronic structure the overall theory has a high degree of internal consistency.
In what follows we shall begin by considering alloy systems were it is sufficient to approximate the electronic grand potential by the band structure contribution alone, i.e. systems were charge transfer effects contained in the double counting terms axe small. This need n_,t be an arbitrary approximation since the LDA-KKR-CPA charges " a_aociated with the x,arious species are outputs of calculations. Subsequently, we shall comment on a system NicCr__c were it is necessary to include these effects.
Finally, as regards subsequent calculations of transition temperatures for ordering and for phase separation the theory of the diffuse scattering intensity a(_), as developed above, does not satisfy the conservation rule a,, = 1/nsz/dq(q) = I. (45) This failure, results from the fact that MF-CF theory does not satisfy the fluctuationdissipation theorem. This generic failure of mean field theories has been known for a long time, together with a method, introduced by Onsager [ where Ac i_ a normalization constant that enforces the conserv_.tion rule 45. In what follows, we _hall refer to this variously, as the Onsager cavity field corrected theory or the mesa spherical approximation. This corr¢ct';,m makes a significant improvement to phase transition temperature at little extra computational cost. Thus we will use it as a matter of course.

Fermi-surface nesting
The first application of the mean field concentration functional theory outlined above was to Cu-rich CucPdl-c alloys [20]. At low temperature Cu-rich CucPdl-c alloys order into a series ot one anQ _,, dimensional LPOS [50], [51]. Recently, Ceder et. al [51] have used the calculated MF--CF values of S(2)(_) to calculate the phase diagram of the LPOS. In this work they make the assumption that S(2)(q) can be interpreted as a Fourier trans'._rmed pair potential, t_(_), and then use it in a mean field free energy expression to _lculate the range of stability of the LPOS. Their results are summarized in fig. 10 and in table 1 [20], [40], [41]. The notation for the LPOS is that of Fisher and Selke [52]. As can be seen from table 1 there is excellent agreement between the list of structure that are calculated to exist and those that have actually been observed.

Concentration Dependent Interactions
As we saw in the previous subsection, the rapid concentration dependence of a(_) for Cu-rich CucPdl_c alloys implies that the underlying interactions are very concentration dependent. However, Fermi surface nesting is not the only way such extreme concentration dependence can arise.
As we remarked earlier 2.3 Ag_All_c is a particularly interesting alloy system in that the calculations of the energies of mixing, AE _=, are positive for Al-rich alloy suggesting that these alloys want to phase separate, and negative for Ag-rich alloy suggesting that these alloys want to order at low temperature. This conjecture is supported by explicit calculation of a(_). In fig. 11 Table 1: Calculated and observed LPOS in CucPdl-c alloys. The structures that whose stability was analyzed are listed in column 1.

Column 2 (3) indicates whether the structure is found to be stable in the calculations (experiments)
at low temperature. It is very satisfying that the results of calculations of S(2)(q_are bear out whist was exl_.-cted on the basis of the energy calculations. The experimental phase di_tgram is quite complex. For Ag-rich alloys phase_ that are based on Ecc and hcp lattices dip down to low temperatures. These are phases that we have not considered since, our calculations were performed for an underlying lattice that is fcc. Thus, the _xistence of the fcc based CuPt structure is masked. For Al-rich alloys the situttion is more interesting. The experimental equilibrium phase diagram shows, at room temperature, a two phase field between an fcc o-phase disordered solid solution of Ag in AI and a hcp related phase, based on the chemical composition Ag2AI, which is the equilibrium phase for Ag content > 40 atomic percent [33], [53]. However, it is known [53] that within the two phase field that there is a metastable miscibility gap and that this is responsible for driving the formation of Guinier-Preston(GP) zones [54] [55], [56]. GP-zones provide a fundamental strengthening mechanism in many commercial Al-based alloys, for exar._ple alloys based on AI with a few percent Cu find, amongst a wide set of uses, application in aircraft skins. In AIAg alloys, the GP-zones are coherent precipitates that form under appropriate annealing conditions and are comprised of essentially pure Ag even though the parent phase is very AI rich [53]. Presumably, the miscibility gap found in our calculations for fcc phase is the one that is responsible for providing the driving mechanism for the formation of the GP-zones.

Band Filling
There are fairly general arguments based on the tight binding model that suggest alloys between late transition metals with roughly half filled d-bands should order whilst those with nearly empty or nearly full d-bands should cluster in the disordered phase and, therefore, phase separate at low temperatures [57, 58, 59, 60, 61]. The exact positions of the crossover from cl,_steringto SP.O depends on the details of the Hamiltonian, but the trend is robust. Further, experimental binary alloy phase diagrams involving transition metals from the same series show ample evidence of this general trend [33]. The Pdclth_l_c)alloy system is a particularly good exampled this band filling effect. These late 4d-transition metals are of almost equal size, and there arc only small strain and charge transfer effects. In such a system it is expected that the eigenvalue sum should dominate the energetics and the clustering predicted by tight binding models should not be obscured by some other mechanism. The phase diagram of PdcRho_c) is very simple, below melting it is a solid solution until the _emperature drops into the miscibility gap when it phase separates into Pd-rich and Kh-rich phases.
The CPA energy of mixing is positive indicating that Pd and Rh would prefer not to mix but to form clusters. Figure 12 shows the calculated short range order diffuse scattering pattern obtained on the basis eq. 44. Clearly, the peaks in the diffuse whe:e GCeA(q,z) is the lattice Fourier transform of the CPA t.,reen sunc_Jo_ a_ _,,_ complex energy z = _r + _, the imaginary part of which is the Bloch spectral function A(_, e) defined in eq.23. Even though we evaluate S(2)(_, T) using the full KKR-CPA formula [40] the approximate form of eq. 47 it instructive because it relates the diffuse scattering to the spectral function, features of which, as we have pointed out earlier, are measurable using photoelectron spectroscopies and determine such properties as the residual resistivity. Thus a link is made between physics related measurements of the underlying electronic structure and the metallurgical interesting phase stability of the system. A schematic depiction of how this occurs is shown in fig. 14. For the case of a band (top-frame) where # is small (right frames) A(_; er) is 6-function like and is contained within the BZ (middle-frame) consequently the integral over the BZ of its derivative (lower-frame) is zero. For a band where "ris large on the scale of the dispersion (left frames) A(k; er) may be asymmetrical and intersect with the BZ boundary. In this case the integral over the BZ of its derivative is large giving rise to large (clustering) contributions to S(_)(_, T). Clearly, as can be seen in fig. 13 this is the case for PdcRh1_©. Quite generally, we can assert that large clustering contributions to S(2)(_, T) will appear when er is just below an extremum of a band for an appreciable part of the BZ. Obviously, this situation occurs most readily when cF is close to the top of the d-band.
At high temperatures S(2)(_ = 0,T) goes to zero and the system is random as dictated by entropy. However, as the temperature is reduced S{2)(k -0,T) grows and clustering type SRO becomes stronger. Eventually, the denominator in eq. 44 will go to zero indicating that the system is unstable to phase separation; infinitesimal fluctuations _n concentration reduce the free energy. This is the spinodal temperature,

Band Filling, Off-diagonal Randomness and Relativity
One of the strengths of first principles approaches to studying phase stability is that they have the possibility to allow us to understand ordering/clustering in alloy systems where different underlying mechanisms are competing, i ne 1_1_ alJoysystem pro_Ju_ " a nice example of this where the competition is between band filling, off-diagonal randomness and relativity. Despite the fact that the NicPtl-c alloy system is comprised of two elemental metals from the end of there respective transition metal series (3d for Ni and 5d for Pt) these alloys exhibit SRO in the rJisorderedphase [63] and form ordered structures (L12, Ll0) upon slow cooling to low temperatures [33]rather than clustering and phase separations as would be expected on the basis of band filling arguments. Aware of this exception to the rule, Treglia and Ducastelle [64] studied NicPtl_c using a tight-binding model Hamiltonian and concluded that there was no simple way of avoiding the prediction that NicPt(___)should cluster in the disordered phase, and phase separate at low temperatures. In the end, they suggested that spin-orbit coupling, which was_neglected in their non-relativistic treatment of the problem, might give rise to a repulsion between the Pt atoms and hence override the usual band filling argument. To substantiate or reject this interesting hypothesis was the purpose of two recent reconsiderations of the problem by Pinski et al. [65]and Lu et -1. [17], [66]. Unfortunately, these calculations resulted in apparently conflicting results. The first suggested that the ordering tendency in the Ni_Pt(l_c} system is due to a size effect that results from the very different d-band widths of Ni and Pt (o._-diagomzl rundom,ess in the language of tight binding models), whilst Lu et al. [17]concluded that a relativistic effect but not specifically to do with the spin-orbit coupling was responsible.
In fig. 16 where V. is the equilibrium volume of the system indicated by the suffix o = L l0, Ni and Pt respectively. The calculations were performed using state-of-the-art selfconsistent, scalar-relativistic and non-relativistic FLAPW calculations [17], [66]. They found that non-relativistically _E is positive implying phase separation at low temperatures, but scalar-relativistically it is negative and, therefore, consistent with the observed ordering tendency. They therefore concluded that relativity, specifically the mass-velocity and Darwin terms (since they also neglected spin-orbit coupling) was responsible for the ordering tendency.

Charge Transfer Effects
So far we have ignored all contributions to S(2)(_) save those resulting from the band structure contribution to the grand potential.
In systems that exhibit significant charge transfer this is not sufficient. Recently, the basic MF-CF theory of Gyortfy and Stocks [20]  For those alloys characterized by both small effective 'charge transfers' AQ and small densities of states at the Fermi level, nA(tF) and ns(e_.), the compositional correlations are dominated by the band-filling term _(q), such is the case for CuPd alloys where SRO is determined by Fermi surface nesting [40]. In systems which also have small effective charge transfers, AQ _. 0, but differ in that they possess both a sizable fi(e_.) and An(er), it turns out that the charge arrangement is still sensitive to the compositional environment and there is an important contribution to the interchange energy S(_)(q) from charge re-arrangement and screening contributions. This can be considered loosely as an effect coming from 'local' Fermi energy adjustments as the number of electronic states available varies with atomic composition.
Foralloys in which the atomic species have differing electronegativities and therefore non-negligible AQ, once again we find that this category can be further subdivided according to the nature of the electronic structure in the vicinity of the Fermi energy. For systems with large AQ but small An(eF), the dominant functional form in these cases is roughly a sum of the band-filling energy _(q) and a screened electrostatic interaction between charges +AQ and -AQ with a screening length I,_,, i.e.

+ (51)
If long-range order develops, it is evident that the Coulomb term becomes the Madelung energy associat_:dwith ordered charges on a Bravais l_.ttice, in orderea alloys, _lie Madelung ener_ can, of course, be quite substantial. It should be evident from the above equations that the electrostatics can also be important for the SRO effects in the high-temperature alloy. Indeed, as found by Johnson, St,mnton, and Pinski, the SBO in NiCr alloys results from a competition between a c_ustering, band-filling _,_(q) and an ordering, electrostatic contribution in the nickel-r[_ alloys [48]. The sum of those contributions yield an a(q) which has peaks at (1, ½,0). This rather surprising result comes from the band-filling, q-0 peaks in $= being canceled from the tails of the electrostatic q = (1,-_,0) peak, and the electrostatic peak being diminished, not canceled, (reducing the transition temperature) from the tails of the band-filling contribution to S(2}. In NiCr, then, there is a subtle cancellation between different electronic contributions giving rise to a robust result for the SRO.

Magnetic Efl'e_.ts: Ordering in Fe©V1_c
Before closing this section we make a few remarks regarding alloy phase stability in alloys that are magnetic. This is a large subject that we will not attempt to cover here. The pro_ess that had been made within extensions of the KKR-CPA and CF approaches was the subject of the lectures of Prof. Staunton at the AHoy Phase St=biliq/NATO ASI held in 1989 [42]. Thus, we will only review some results on the Fe_VI-= alloy system that have been obtained since that time and serve to highlight the quantitative nature of the theory and the importance of the magnetic effects in influencing the phase stability.
The generalization of the LDA-KKR-CPA method to alloy systems that are ferromagnetic using spin-polarized local density theory is quite straight forward and has produced results for the ground state total and partial magnetic moments that are in good quantitative agreement with magnetization and neutron scattering measurements [711, [72], [42], [T3], [74].Similarly, the concentration functional theory of ordering and phase stability can be generalized to the case of ferromagnetic systems. Here we assume that we are addressing the problem of chemical ordering in a system that is magnetically ordered i.e. the ordering/phase separation temperature is below the Curie temperature. In which case we have, at l_ast formally, to consider the electronic grand potential to depend not only on the set of local concentrations {c./} but also the a set of local magnetic moments, {/=i}, [42]. As a consequence, S_ ) consists of two terms [42], [75], [74] -,, + "_" .

Strain Fluctuations
In all of the forgoing it has been assumed, teven in the disordered state, that the atoms occupy the sites of well defined underlying periodic lattice. Clearly, this will only be a good approximation for alloys for which in the language of Hume-Rothery the size difference is small. In alloys where this is not the case the average lattice is still periodic, as is made manifest by the fact that disordered alloys still have well defined X-ray diffraction Bragg scattering peaks, however, local relaxations occur. This situation is pictured schematically in fig 19 where we have envisioned the local lattice relations around large and small atoms. The lattice surrounding large atoms expands locally whilst the lattice surrounding small atoms responds by contracting.
As we have mentioned above [69], and, as has been discussed by others [78], [79], [80], [81] these local relaxation, or strain fluctuations, can give rise to important effects in alloy phase stability. Furthermore, they can now be measured very accurately using synchrotron radiation and anomalous scattering techniques [82], [83].
In a recent publication Gyorffy et. al. considered the problem of includin_ strain fluctuation within the concentration functional theory outlined above [69]. Although no calculations have yet been performed that include strain fluctuations, for the sake of completeness we reproduce the major theoretical points here.
The theory is based on a straightforward generalization of the mean field theory where the static force constants _oo(i,j) are given by From this point of view, the first principles mean-field theory method advocated here is particularly promising since it requires calculations in _-space and treats the small and large _ limits on an equal footing. In this respect, it is quite different from supercell or finite cluster based methods [84]. These censiderations are also relevant to the applicability of the ConnoUy-Williams type of approaches.

Clearly, to account for large elastic interactions, the effective Hamiltonian must include long-ranged forces for which both the CVM and the Monte Carlo method become difficult to implement. Fortunately, as was recently demonstrated by Marius et al.
[ 85], under these circumstances the mean-field theory becomes a better and better approximation.

Generalized Perturbation Method
The  We will not repeat the derivations here but will simply outline the method, quote the important formulae, and show some illustrative results. As in previous section we will assume that for the electron system it is sufficient to work at T=0 K. Whence, ffP,t = ECP,t _/_N where E cPA is the LDA-KKR-CPA energy of the random alloy eq. 32, p is the electron chemical potential an_l N ffi N(eF) is the integrated density of states at the Fermi energy p ffi eF. The configuration dependent contribution of the energy can be written as an expansion in concentration In fig. 21 we illustrate schematically the content of eq. 66. For example the 2"d order pair wise interaction Vo{_ } consists of an interaction at site 0 propagating in the CPA medium to site 1, interaction at site 1, and returning to site 0. It should be noted that nowhere do we build in any explicit approximation to either the concentration dependence, the complexity (pair, triplet,...), 6r the range of the interactions other than that implicit in the CPA. Thus these factors have to be investigated for each alloy system considered. In fig. 22 solid solution. However, a BI9 structure has been reported [87]. Interestingly, this . structure can be thought of as a monoclinic distortion of the Lio structure. For most of the systems that have been investigated the triplet and higher order interactions are generally small. However there are some notable exceptions. For example in fcc Ni0.TsAlo.zs the nearest neighbor triplet interaction is 2.6 mRy/atom. Whilst, thi_ is small compared with the nearest neighbor pair interaction (14.2 toRy/atom) it is large compared with the second (-I.0 mRy/atom) and third (-1.7 toRy/atom) neighbor pairwise interactions.
In the remainder of this subsection we will show the results for calculated phase diagrams for two model systems Cu(1.c)Znc [32] and Al{1_c)Nic [89] These systems serve to illustrate some of the successes and some of the remaining difficulties with the current implementation of the GPM method that neglects the double counting contributions.

S.3.1 Phase Stability of Cu-rlch CuZn Alloys
The Cutn-e)Znc alloy system has been one of the most studied and is an example of one of the Hume-Rothery electron compounds. Hume-Rothery related the position of the a-_ (fcc-bcc solid solution) two phase boundary to the electron to atom ratio _. Figure 23 shows the effective pairwise interactions for fcc and bcc alloys. These interactions correspond to the energies of mixing shown in fig. 7. For both bcc and fcc, the interactions show a strong ordering tendency at all concentrations. Interactions more distant than second neighbor are an order of magnitude smaller than those shown. A ground state analysis including pairwise interactions out to fourth (fifth) neighbor fcc (bcc) indicate that the L12 (DO_) structure are stable at c -0.25 and 0.75 and the Ll0 (B2) structure is stable at c=0.5. Figure 24 shows the calc_Ilated [32] and assessed [33] Cuo__)Znc phase diagrams. The calculated phase diagram was obtained using the tetrahedron-octahedron (irregular tetrahedron) approximation CVM for fcc .

Phase Stability of Ni-rich NiAI Alloys
During the last few years NiAI alloys have been at the centex of a great deal of alloy development work based on modifications to two of the ordered compounds Llo structure NisAl and B2 structure NiAI. Although single crystals of Ni3AI are quite ductile as cast polycrystalline material is extremely brittle but can be ductilized by the addition of a few hundred part_/mill_on of boron. B2 structure NiAI, however, has so far resisted all attempts to make it ductile in say useful way. In addition to being a candidate structural material, B2 NiAI is also interesting as a possible shape memory alloy. For Ni concentrations around 62.5 percent _-phase NiA! transforms Martensitically to the 7R structure. It is observed that the transition temperature is very composition dependent ranging from 0 K for c ,_ 0.60 to well above room temperature for c --0.66. Consequently, these a!.loys are interesting candidates as controllable transition temperature shape memory ._]loys. In addition to being interesting engineering materials these alloys exhibit a humor of interesting pre-Martensitic phenomena. As a result, they have become one of the model systems for studying Martensitic transformation. We shall have more to say on this in 4.2.
In figure 25 we show the calculated results for the LDA-KKR-CPA mixing energies and the corresponding GPM first sad second neighbor pairwise interactions for both fcc and bcc NiAI alloys [89]. In addition the formation energies of ordered structures IO.O " " " ' " ' " """""" ' "'""''"" ' " " Since the muffin-tin approximation is notoriously poor when used to calculate structural energy differences we opted for the semi-empirical values. We note that use of the calculated values results in a phase diagram that is much worse than that shown below.
For Ni-rich alloys, the experimentally observed structures [33] are orderings on fcc or bcc lattices or, as in NisAls, a closely related structure. A careful analysis of the ground state energies obtained using the GPM interactions (see fig.25 )reveals that at c=0.5 bcc based B2 is stable with respect to fcc based LI, whilst at c=0.75 fcc based LI2 is stable with respect to bcc based DO3 [92], [89]. At c-0.625 the most stable phase is, in the notation of Finel and Ducastelle [93], phase 9. However, closer examination reveals that this structure is closely related to the experimentally observed structure of NisAl3 namely PtsGas. The PtsGa3 structure is a bct structure with a c/a-ratio intermediate (c/a --1.24) between that of bcc (c/a -1.0) and fcc (c/a -v/_) when the latter is viewed as bct.
In fig. 26 we reproduce the phase diagram calculated by Sluiter et al. [89] using the cluster variation method (CVM) [94] with calculated GPM fcc and bcc interchange energies and KKR-CPA heats of mixing together with _per/menta/structural energy differences [95]. The calculated phase diagram closely resembles the experimental one [33]. The B2-Llo two phase region is in good agreement with experiment, the calculated order-disorder temperature of the Llo phase is 1770°C which is just abo_e the melting point as was surmised by Cahn et al. NisAl3 phase are both missing from the calculated phase diagram since neither can be represented by the current GPM-CVM methods. It should be noted that the calculated order/disorder temperature for the B2/bcc phases of 5345°C is much greater than the melting temperature of the B2 phase.

Complex Lattice Alloys
Up to this point we have only considered alloy systems that can be treated as solid solutions on an underlying simple lattice (fcc, bcc). In this final section we turn our attention to alloys where it is necessary to consider systems as being composed of several underlying sub-lattices. Some of the many situations where this is appropriate are illustrated schematically in fig. 27. These include states of partial long range order, ofl'-stoichiometric compounds, and binary compounds with ternary additions. In the following we will refer to these as complex lattice or multi-sublattice alloys. The extension of the LDA-KKR-CPA method described in section 2.1 to complex lattices is atraight forward [97],

4.1
Ordering Energies Although the energy difference between the ideal disordered phase of an alloy and a related ordered phase is not generally experimentally measurable it does provide an important measure of the strength of the ordering interactions present in the system. The order-disorder energy AE°-_ for a binary AcB(I-e} may be defined as where z refers to the structure of the disordered lattice and X refers to some ordered structure based on this lattice and Ed(z,c) and E°(X) are the corresponding ground state (T = 0°K) energies. Clearly, when AE°'d is large the alloy is strongly ordering •and the order-disorder temperature is high. As is pictured schematically in fig 27 we treat the ideal stoichiometric NiAI B2 (CsCI) structure ordered phase as two interpenetrating simple cubic lattices having sub-lattice compositions Nit.0Alo_ and Nio_All.o. In order to minimize the relative i_ error when subtracting the large energies of the individual phases we also treat the disordered phase Ni0.sAlo.s as comprising two interpenetrating simple cubic lattices each with composition Nio.sAlo.s.
In fig. 28 we show the energies of bcc disordered Nio.sAlo.s and ordered B2 structure NiAI. In fig. 28 the minima of the energy versus lattice spacing curves give the ground state energies, E°(B2) and E_(bcc, c) respectively, the difference, E°(B2)-Ed(bcc, c), then gives the order-disorder energy AE°-d(B2, bcc) = 47.7 mRy/atom. The minima in the two curves occur at different lattice spacings. The predicted equilibrium lattice spacings are 5.45 a.u. and 5.40 a.u. for the disordered and ordered phases respectively, corresponding to a volume expansion of approximately 3% on disordering. The experimentally determined lattice spacing for stoichiometric NiAI is 5.455 Lu. [101].
The LDA-KKR-CPA not only allows one to treat the fully disordered and fully ordered states but also opens up the possibility to study the evolution of the energy as a function of the long range order (LRO) parameter. If we consider the disordering of NiAI by random exchanges of Ni and AI atoms between the Ni and AI sub-lattices, as pictured schematically in fig. 27, we may model the LRO state by two interpenetrating simple cubic sub-lattices having compositions Ni0_,}AI ¢ and NicAl(l_c} for concentrations c in the range 0 _<c _ 0.5. If we define the LRO parameter as t/= (2c-1) then ---1, c = 1 corresponds to the ordered B2 structure and q = 0, c = 0.5 corresponds to the homogeneously disordered state.
In fig. 29 we show the energy of Ni0.sAlo.s plotted as a function of the LRO parameter. The zero of energy is taken to be the energy of the disordered state. It is interesting that the T/dependence of the ordering energy departs strongly from the quadratic dependence that would be expected if the interactions were entirely pairwise. throughout the whole range of hand energies, the DOS curve for the disordered alloy is relatively structureless. Clearly, upon ordering the weight in the DOS close to _.,-is reduced and the weight at low energies is increased resulting in stabilization of the B2 structure.
The changes in the densities of state__o_ fig. 30 result in a significaat rearrangement of charge. In fig. 31 we show'ho--w the charge within equal volume Voronoi polyhedra associated with each sublattice evolve as a function of LRO. In the disordered state _-0.3 electrons/atom are traasferred from AI to Ni sites. As the LRO builds up the atoms on their correct sub-lattice transfer even more charge whilst antisite atoms become more neutral. The net effect is to increase the Madelung energy over and above thst associated with simple ordering of the charges a_ociated with the Ni and AI sites in the disordered alloy. This effect is also stabilizing.
Before closing this section, we note that the order-disorder energy obtained from this direct calculation, AE°-_(B2,bcc)-46.7 mRy/atom, is larger by a factor of two Using the values of section 3.3 yields AE°-_(B2,bcc) = 20.3 mRy/atom. Clearly if the GPM interaction and disordered state energies are internally consistent the two methods should yield the same order/disorder energy. However, the GPM interactions were calculated on the basis of the band structure contribution _s to the total energy alone, neglecting double _unting corrections (contriL,ttions that were included in the energy calculations), thus it is not n,_essarily surprisi_g that the results differ. That the two vah,_ di_er so much is presumably due to the relatively large charge transfer f_,_ i,-, _is _!;_ oy_tc:_. ',iVh_,t is surprisiing is thP._ an internally consistent picture eme,'ges fro.,r_the GPM _ il,,_ _. charge transfer terms are neglected and they are used i,'L_ conju_ctl, on with the K,_ ,., A disordered state energy defined above. A resolution ,_:'_'_' this dilemma may i__y_'_; _ a c_ge self-consistency scheme that goes beyond th..-.. _,imple metro field theo_', _ so far. Work is in progress on this matter [ O.16(21r/a). For Nio.s2sAlozTs the dip becomes more pronounced as the temperature is lowered towards Ts4. Since the phase transformationis first order, the phonon frequencydoes not go to zero at Ts4 _ 80K as it wouldin an ideal soft phonon transformation. However, it is clear that this anomaly is closely connected with the phase transformation. Indeed, for Nio.s_Alo_Tsthe reciprocalof qo is close to the wavelength of the Martensitic 7R.structure into which it transforms. Thus, understanding the mechanism that gives rise to the pre-Martensitic phonon softening can be expected to provide clues to the mechanism that drives the Martensitic transformation. and Harmon together with the Fermi surface that they obtained. Zhao and Harmon identified a Fermi surface nesting vector (marked by the arrow) that was responsible for the dip that they obtained in there calculations. For pure metals and for ordered systems the methodology followed by Zhao and Harmon is quite sound since it is based on standard band theory technology. Unfortunately, for disordered alloys and off-stoichiometric compounds it is much less well founded. In order to be able to treat the off-stoichiometric Ni0.s_sAlo._Tscompound, Zhao and Harmon made use of the rigid band model. They based their calculations on the e'_tronic structure of stoichiometric NiAI assuming that the only effect on the electronic structure of adding excess Ni is to change the position of the Fermi energy. They choose valences of 1 and 3 for Ni and AI respectively (arguing that, in Ni, the other nine d-electrons that are normally considered valence electrons are in a fixed do core-like configuration), modify eF accordingly, and use this modified ordered compound band s':ucture to perform the calculation of the phonon spectra. The problem with this approach is that the effect of off-stoichiometry, i.e. the creation of anti-site defects, is not only to modify the position of eF, but also to smear out the electronic structure in a complex energy and _ dependent manner. Thus the whole question of the existence of a band structure and Fermi surface is thrown into question.

Recent'_y,Zhao and
As we have pointed out earlier a much better way to calculate the electronic structure of off-stoichiometric compounds is, at the outset, to face up to the fact that they • " are disordered and to use the LDA-KKR-CPA method. At the present time, there is no first principles theory of the dynamical matrix for disordered alloys similar to that of Varma and Weber. However, since the mechanism identified by Zhao and Harmon rests critically upon the picture of the electronic structure an(t Ferma surlace o_ta_nea using the rigid band model, we can inquire into its validity. Specifically, if the effect of adding disorder into the problem is to smear out those parts of the Fermi surface " r_ponsibh for the nesting, this would negate the mechanism identified by Zhao and Harmon. We have calculated the electronic structure of Nio.s2sAlo_Tsusing the complex lattice LDA-KKR-CPA method. We assume that the structure is B2 and that the excess Ni goes onto the Al-sublattice. Thus the system is describe by two interpenetrating simple cubic lattices, one is occupied entirely by Ni, i.e. has the composition Nil.oAIo.0, the second has Ni and AI randomly distributed with composition Nio.2sAle.n. Once the self-consistent crystal potentials have been obtained the Aa(_, e) was calculated using a generalizs_ on of eq. 23. AAs(_, e) along a number of directions in the _t,_ plane for a number of values of _, is displayed down the left hand side of fig. 34. For _¢z = 0, AS(_c,e) is quite sharp in all directions, although, it is sharpest along the F-X direction and less sharp along DM. As _, is increased AS(_,e) remains sharp along (100)-directions for i_, < 0.3(2_r/a) whilst along (ll0)-directions it is rapidly smeared out by the disorder on the Al-sublattice. If we interpret the loci of the peaks in AS(l_, e) as the Fermi surface and the width as the inverse mean free path of the electrons, clearly the Fermi surface is well-define around the X-points in the Brillouin zone and much less so elsewhere.
On the right of fig. 34 we show a summary plot of the loci of the peaks in As(_, e). What is clear from this plot is that the Fermi surface is quite well-defined over a' substantial region of the Brillouin zone around the (100)-direction. Since it is the Fermi surface in the neighborhood of this direction that, according to Zhao and Harmon, is responsible for the nesting, this is an important observation. Furthermore, in this Legion the Fermi surface is quite fiat normal to the _., --constanZ plane for values of k, around 0.15(21r/a).
In fig. 35 we plot the Fermi surface in the _c, -0.15 plane in the extended zone scheme. We have also indicated the part of the Fermi surface that is welldefined and a spanning vector that can give rise to nesting. The spanning vector is -_ 0.18(110)(27r/o), this is identical to that of Zhao and Harmon. Both are somewhat larger than the actual lock-in vector for the 7R structure of 1/7(llO)(21r/a).
Even though we do not have a first principles theory of the phonon spectra in disordered off-stoichiometric compounds our results do lend strong support to the conclusions of Zhao and Harmon that the pre-Martensitic phonon softening observed in B2 phase Nio.e2sAIo.3n is Fermi surface driven. Such an observation has important consequences for future theories in that is implies that the interactions giving rise to the pre-Martensitic behavior are long ranged. Thus we can conjecture that it will not be possible to describe this behavior on the basis of simple pair and short ranged interatomic potentials without the explicit inclusion of these subtle band structure effects.
Since the effects of disorder on the electronic structure are extremely energy and dependent, it is also clear that its effects will have to be considered at the outset. The extent to which such conclusions apply to the Martensitic phase transformation itself remains to be seen. That disorder is importan_ in understanding the nature of the atomic displacements that precede the Martensitic transformation has already been provided by molecular dynamics studies based on embedded atom potentials [112]. In this work it was demonstrated that the characteristic "tweed" diffraction pattern is obtained for random positioning of the Ni and AI atoms on the disordered sublattice but it is not if the NiAI atom__ are placed in some ordered array. Thus, it appears that a theory that includes both compositional and displacement fluctuations will be  required in order to fully understand these complex phenomena.

Conclusions
We have reviewed the local density approximation, first principles Korringa, Kohn, Rostoker coherent potential approximation theory of the electronic structure and energetics of substitutionally disordered alloys. We have also reviewed the mean field concentration functional and generalized perturbation method theories of ordering and alloy phase stability that have been built on the LDA-KKR-CPA description of the disordered state. Results of calculations of short range order parameters and phase diagrams for a number of alloy systems have been presented that illustrate different ordering and clustering behavior. We have placed emphasis on understanding the driving mechanisms of ordering phenomena in terms of the underlying electronic structure. A number of mechanisms have been identified (Fermi surface nesting, band filling, offdiagonal randomness, relativity, charge transfer, magnetism, strain fluctuations, etc.) that provide a modern, first principles, basis for understanding ordering/clustering and alloy phase stability and for the interpretation of the Hume-Rothery rules and ideas based on tight-binding models. An important ingredient in this understanding is that features in the electronic structure of the disordered state tl_at drive the ordering processes are also amenable to verification by experimental probes of the electronic structure of the disordered state. Thus, for example, the topology and smearing of an alloys Fermi surface that, on the one hand, determines its residual resistivity, can be responsible for the development of LPOS on the other. To date, we have not made any numerical applications of the strain fluctuation theory presented in these notes. With the development of the anomalous X-ray scattering technique [ti2jt_i_jtaat can now measure 1oca4atoalic aisp_acellJc_,L_ ,. u,_', _, _ _,,_: to a high degree of precision this promises to be a fruitful area. This is nowhere more likely than in the INVAR problem where concentration, magnetic, and displacement fluctuations are intricately coupled sad in _-phase NiAI where concentration and disas placement fluctuations appear as precursorsto the Martensitic phase transformation. Faulkner a. al. [113] have developed a new direct Monte Carlo approach that is based on the calculation of the energetics of specific local configurations embedded in an otherwise homogeneously disordered solid solution. The cluster energies are obtained from the embedded cluster method [19] which, in turn, is based on the LDA-KKR-CPA treatment of the disordered phase. In many respects the new method can be viewed as a next step beyond the MF-CF theory, in that, there is no need to partition the electronic energy between sites, however, by going directly to the Monte Carlo method the statistical mechanics is treated essentially exactly, rather than the point approximation of MF-CF theory. Although only in the early stages of development the results obtained are very encouraging. m m I I