Atomic Static Displacements and Their Effect on the Short Range Order in Alloys

is obtained by the collective variables method. The theoretical results are illustrated by numerical calculations performed for disordered alloys of the K-Cs and Ca-Ba systems. A drastic effect of the ASD on the -function behaviour in the rst Brillouin zone is observed. The ASD smooth the dispersion of the -function. Negative values of the short-range order parameter on the rst coordination sphere indicate a trend to the ordering in alloys of the systems investigated. The ASD are shown to favour the ordering tendency. The theoretical conclusions concerning the temperature inuence on the short range order parameter perfectly agree with the experimental data from the treatment of the X-ray diffuse scattering in binary alloys.


Introduction
It is well known that the short-range order (SRO) in binary alloys is caused by the di erence in e ective interactions between ions of two kinds, namely, see 1,2] h k ?k i V 2 (k) ; (1) where h k ?k i and V 2 (k) are the Fourier components of the binary correlation function and the ordering potential V 2 (R) = V AA (R) + V BB (R) ?2V AB (R) ; (2) respectively, index j = A; B denotes a sort of an alloy component.The SRO in uences di erent alloy characteristics, such as electrical conductivity, magnetic c Z.Gurskii, Yu.Khokhlov and galvanomagnetic properties 1,3].A close correlation between the SRO and mechanical properties is also observed 3].That is why, investigation of the factors which alter the SRO in alloys is an urgent problem.Atomic static displacements (ASD) could be regarded as one of such factors.
Formation of metal solid solutions is accompanied by the arising of local lattice distortions.The latter are characterised by the ASD with respect to the ideal mean lattice sites.The ASD have a drastic e ect on the X-ray (neutron) di use scattering 1].They determine a lattice parameter dependence on alloy concentration.However, the mutual in uence of the ASD and the SRO on each other has not been investigated yet within the microscopic theory.
Study of the ASD e ect on the SRO formation is the purpose of the present paper.It is organized as follows.Derivation of explicit expressions for the alloy free energy by the collective variables method is given in section 2. Special attention is paid to the original moments of the considered approach.Behaviour of the binary correlation function Fourier components in the Brillouin zone principal directions is analysed in section 3. The theory is illustrated by numerical calculations carried out for the alloys of K?Cs and Ca?Ba systems.Calculations have been performed for two cases: 1) with the ASD taken into account; 2) in the rigid lattice approximation, that is without the ASD.The in uence of the ASD on the SRO parameter R values is also considered in section 3. The dependence of R on temperature and alloy concentration is presented.Conclusions in section 4 complete the paper.

The binary alloy free energy
Consider a substitutional binary alloy.Atoms of two kinds A and B are placed arbitrarily on N crystal lattice sites.Their con guration is given by the set f R g of numbers R which equal +1 if the site R is occupied by the A-kind atom and equal -1 otherwise.The alloy Hamiltonian within the pair interatomic interaction approximation, after summing over electron states 2,4] has the form: Here V ij (q); i; j = A; B is the Fourier transform of the e ective interaction between ions of i and j kinds, V AB (q) = V BA (q).The explicit expressions for V ij (q) are given in 2,4,5].Let us take into account the fact that the local ASD are present in an alloy.Then, the coordinates of the lattice sites are the following ones: R = R 0 + R (4) where R are the ASD with respect to the sites R 0 of the ideal mean lattice.Assume that R does not depend on the kind of an atom and perform the Fourier transformation of R: R = 1 Here is the k-th Fourier component of the occupation numbers, 0;k , the Kronecker symbol.Equations ( 6) and (7) with allowance for (4) to (7) in the harmonic approximation 5] takes the form H( R ) = H 0 (^ ) where H 0 (^ ) = NV 0 + p is the Hamiltonian of an ideal mean lattice without displacements.The explicit expressions for potentials V 0 ; V 1 and V 2 (k) are presented in 2,5].They have the following physical meaning: V 0 is the part of alloy energy which does not depend on atomic con guration, V 1 indicates the di erence between alloy component atomic characteristics 5] and V 2 (k) is the Fourier transform of the ordering potential.
The addends H 1 (k; A k ; ^ k ) and H 2 (k; A k ; ^ k ) are linear and quadratic in A k amplitudes, respectively.The explicit equations for them see in 5].
We proceed from the grand partition sum to nd the free energy The following notations are introduced in (10): = (k B T) ?1 is the inverse temperature, i { the chemical potentials of the alloy components.Symbol Tr f R g in (10) means summing over all the possible values of the occupation numbers f R g.
One can rewrite equation ( 10) with a view of ( 8) and using the rigid ideal lattice of an alloy as a reference system, as follows: are the addends of the alloy Hamiltonian (8) renormalized by the ASD and The next notations are accepted in ( 12)-( 15): G are the reciprocal lattice vectors, and (0) { the force constant matrix of the reference system.The correlated average crystal (CAC) in the rigid lattice approximation is used as a reference system.One can get familiarized with the CAC term value in 2,6].The expression for (0) is given in 6], also see appendix 2 in 5].
is the transition Jacobian to the CV space and with , the Dirac delta function.
The general ideas of the CV method are presented in 2,7].We omit them here and pay attention to the original moments of the given paper.Including potential Ṽ1 ( ) (13) into the transition Jacobian ( 17) is an important feature of the approach considered here.It allows one to achieve an adequate description of the alloy physical properties within the simplest Gaussian approximation of the CV method and the rigid lattice approximation 8].
Calculation of the grand partition sum (16) can be performed analytically in the Gaussian approximation.Details of the consideration are omitted because they are similar to those, given in 5,8].Then the grand potential per one atom equals F(T; ) = ?kB TN ?1 ln Z = Ṽ0 ( ) ? ?1 (ln 2 Here M n = @ n @x n ln cosh x  20) and ( 13) that M n (n = 0; 1; 2) are complex functions of temperature, potential Ṽ1 and alloy component chemical potentials.Equation  " k : (26) Here " k and ! 2 k are eigenvectors and eigenvalues of the force constant matrix (0) , respectively, = 1; 2; 3 { the polarization index and m is the average ion mass, see 2,6] for details.Analyse result (26).One can conclude from ( 15) and ( 26) that the ASD amplitudes A k are small if the pair interatomic potentials V AA and V BB Fourier components are similar: V AA (q) V BB (q).Really, P k 0 at V AA (q) = V BB (q) and then A k = 0.This conclusion allows one to clear up the nature of the well-known phenomenological Hume-Rothery rules 9] on the microscopic level.Using equations ( 13), (24) and condition (25) one can prove that Ṽ1 = V 1 : (28) It means that the potential Ṽ1 as well as the cumulants Mn (20) do not depend explicitly on the ASD amplitudes A k .This result simpli es very much the calculation of the alloy free energy (23).Let us analyse more carefully equation ( 23) for the alloy free energy.The third term in (23) proportional to ?1 is entropy (S), while the rest of the terms de ne the alloy internal energy (E).One can get the next formulae for E and S considering equations ( 23) and (21) in the high temperature limit: V 2 (k) 1.
Equation ( 29) determines the energy of an average crystal: all the lattice sites are occupied by mean ions which interact via the mean potential.v = v A C A + v B C B with v i { the potential of an i-kind ion.Equation (30) de nes the con gurational entropy of an ideal binary solution.Thus, the high temperature limit of the CV method Gaussian approximation is equivalent to the well-known W.Bragg { E.Williams theory which ignores the pair atomic correlations.By the way, the di erence F = F(T; C) ?E id + TS id with F(T; C) (23) indicates contribution of the SRO e ects to the alloy free energy.

Pair correlation functions and short-range order in alloys of K-Cs and Ca-Ba systems
The Fourier components of the binary correlation function are important alloy characteristics.They are needed for the calculation of the X-ray (neutron) di use scattering intensity 1,3].Besides, they are related to the SRO parameter h k ?k i exp(ikR) : (31 Here R is the value of the SRO parameter on the R-coordination sphere, h k ?k i { the Fourier components of the binary correlation function.Calculation of h k ?k i does not face any di culties within the Gaussian approximation of the CV method 2,7] 14), renormalized by the ASD, takes the form 10] Ṽ2 (k    The values of the SRO parameter R on the rst coordination sphere R 1 have been calculated according to (31).Tables 1 and 2 demonstrate dependence of R 1 upon temperature and alloy concentration for K ?Cs and Ca?Ba systems.The R 1 negative values indicate a trend to ordering in the alloys of the both systems.The ASD favour this tendency: the values of R 1 are smaller within the rigid lattice approximation, see tables 1 and 2. Temperature has a stable e ect on R 1 .The SRO parameter R 1 increases with the decrease of temperature.This tendency is most pronounced in alloys with x = 0:65 0:7 and x = 0:5 in K x Cs 1?x and Ca x Ba 1?x systems, respectively, see tables 1 and 2. The obtained results theoretically agree with the conclusions about the temperature e ect on the SRO parameter in alloys, drawn in 3] and based on the experimental investigations of the X-ray di use scattering.Table 1.Dependence of the short-range order parameter values on the rst coordination sphere upon temperature and alloy concentration in K x Cs 1?x system.Abreviations RLA and ASD denote respectively that calculations have been performed within the rigid lattice approximation or with the atomic static displacements taken into account.

Conclusions
The given results can be summarized in the following statements.
1.The ASD have a drastic e ect on the binary correlation function Fourier components h k ?k i behaviour in the rst Brillouin zone.They smooth the dispersion of the h k ?k i = f(k)-function.
2. Tendency to ordering becomes more pronounced owing to the ASD in the alloys of K ?Cs and Ca ?Ba systems.
3. Dependence of the SRO parameter on temperature obtained theoretically agrees with the conclusions drawn from the treatment of X-ray di use scattering experiments.

F
)determines the di erence of alloy components chemical potentials at the given alloy concentration.The explicit form for equation (21) is presented in 8].One has to perform the Legandre transformation (21) to nd the alloy free energy F(T; C) as a function of temperature and component concentration.Then, Ṽ1 ( ), see (13), is the solution of the system of equations @).Solution of equation (25) is given in 5].We present the nal result omitting details A k = X (P k k ) m! 2 k

Figure 1 .
Figure 1.Behaviour of the ordering potential Fourier transform V 2 (k) in the 111] direction in alloys of K?Cs system at T = 250K.Dashed and full curves show results obtained, respectively, with and without the ASD taken into account.Curves 1 refer to alloy K 0:7 Cs 0:3 while the curves 2 correspond to alloy K 0:1 Cs 0:9 .

Figure 2 .
Figure 2. Behaviour of the ordering potential Fourier transform V 2 (k) in the 111] direction in alloys of Ca ?Ba system at T = 750K.Notations are the same as in gure 1. Curves 1 refer to alloy Ca 0:5 Ba 0:5 and the curves 2 correspond to alloy Ca 0:2 Ba 0:8 .

Figure 3 .
Figure 3. Temperature e on the binary correlation function Fourier components in the K 0:7 Cs 0:3 alloy.Dashed and full curves show results obtained, respectively, with and without the ASD taken into account.Curves 1 refer to T = 300K and curves 2 correspond to T = 200K.

Figure 6 .
Figure 6.Dependence of the binary correlation function on atomic concentration in the Ca ?Ba system alloys.Direction 111] of the Brillouin zone.Notations are the same as in gure 3. Curves 1 refer to the Ca 0:2 Ba 0:8 alloy and curves 2 correspond to the Ca 0:8 Ba 0:2 alloy.

Table 2 .
Values of the short-range order parameter on the rst coordination sphere in Ca x Ba 1?x system and their dependence upon temperature and alloy concentration.