Atomic static displacements and their effect on the short range order in alloys

The influence of local atomic static displacements (ASD) on the short range order formation in binary alloys is investigated within the microscopic theory. Explicit expression for the binary correlation function Fourier components ‹ρkρ−k› is obtained by the collective variables method. The theor...

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Published in:Condensed Matter Physics
Date:1998
Main Authors: Gurskii, Z., Khokhlov, Yu.
Format: Article
Language:English
Published: Інститут фізики конденсованих систем НАН України 1998
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/118933
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Cite this:Atomic static displacements and their effect on the short range order in alloys / Z. Gurskii, Yu. Khokhlov // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 389-400. — Бібліогр.: 11 назв. — англ.

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Khokhlov, Yu.
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Khokhlov, Yu.
citation_txt Atomic static displacements and their effect on the short range order in alloys / Z. Gurskii, Yu. Khokhlov // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 389-400. — Бібліогр.: 11 назв. — англ.
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container_title Condensed Matter Physics
description The influence of local atomic static displacements (ASD) on the short range order formation in binary alloys is investigated within the microscopic theory. Explicit expression for the binary correlation function Fourier components ‹ρkρ−k› 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 ‹ρkρ−k›-function behaviour in the first Brillouin zone is observed. The ASD smooth the dispersion of the ‹ρkρ−k›-function. Negative values of the short-range order parameter on the first 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 influence on the short range order parameter perfectly agree with the experimental data from the treatment of the X-ray diffuse scattering in binary alloys. В рамках мікроскопічної теорії досліджується вплив локальних статичних зміщень атомів (СЗА) на формування близького порядку в бінарних сплавах. Методом колективних змінних отримано явний вираз для Фур’є-компонент бінарної кореляційної функції ‹ρkρ−k›. Результати теорії ілюструються числовими розрахунками, виконаними для невпорядкованих сплавів систем K-Cs та Ca-Ba. Спостерігається сильний ефект СЗА на поведінку ‹ρkρ−k›–функції у першій зоні Бриллюена. СЗА згладжують дисперсію ‹ρkρ−k›-функції. Від’ємні значення параметра близького порядку на першій координаційній сфері вказують на тенденцію до впорядкування у сплавах досліджуваних систем. Показано, що СЗА сприяють цій тенденції. Висновки теорії стосовно впливу температури на параметр близького порядку дуже добре узгоджуються із експериментальними даними, отриманими із обробки дифузного розсіяння рентгенівських променів у бінарних сплавах.
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fulltext Condensed Matter Physics, 1998, Vol. 1, No 2(14), p. 389–400 Atomic static displacements and their effect on the short range order in alloys Z.Gurskii, Yu.Khokhlov Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii St., UA–290011 Lviv, Ukraine Received January 26, 1998 The influence of local atomic static displacements (ASD) on the short range order formation in binary alloys is investigated within the microscopic the- ory. Explicit expression for the binary correlation function Fourier compo- nents 〈ρkρ−k〉 is obtained by the collective variables method. The theo- retical results are illustrated by numerical calculations performed for disor- dered alloys of the K-Cs and Ca-Ba systems. A drastic effect of the ASD on the 〈ρkρ−k〉 -function behaviour in the first Brillouin zone is observed. The ASD smooth the dispersion of the 〈ρkρ−k〉 -function. Negative values of the short-range order parameter on the first coordination sphere indi- cate 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 influence on the short range order parameter perfectly agree with the experimental data from the treatment of the X-ray diffuse scattering in binary alloys. Key words: atomic static displacements, short-range order, disordered alloy, binary correlation function, free energy PACS: 05.70.Ce; 65.50.+m 1. Introduction It is well known that the short-range order (SRO) in binary alloys is caused by the difference in effective interactions between ions of two kinds, namely, see [1,2] 〈ρkρ−k〉 ∼ V2(k) , (1) where 〈ρkρ−k〉 and V2(k) are the Fourier components of the binary correlation function and the ordering potential V2(R) = VAA(R) + VBB(R)− 2VAB(R) , (2) respectively, index j = A,B denotes a sort of an alloy component. The SRO influences different alloy characteristics, such as electrical conductivity, magnetic c© Z.Gurskii, Yu.Khokhlov 389 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 lat- tice distortions. The latter are characterised by the ASD with respect to the ideal mean lattice sites. The ASD have a drastic effect on the X-ray (neutron) diffuse scattering [1]. They determine a lattice parameter dependence on alloy concentra- tion. However, the mutual influence of the ASD and the SRO on each other has not been investigated yet within the microscopic theory. Study of the ASD effect 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 ofK−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 influence of the ASD on the SRO parameter αR values is also considered in section 3. The dependence of αR on temperature and alloy concentration is presented. Conclusions in section 4 complete the paper. 2. 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 configuration is given by the set {σR} of numbers σR 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: H(σR) = 1 2N ∑ Ri,Rj {∑q[(VAA(q) 1+σRi 2 1+σRj 2 + VAB(q) 1+σRi 2 1−σRj 2 + +VBA(q) 1−σRi 2 1+σRj 2 + VBB(q) 1−σRi 2 1−σRj 2 )eiq(Ri−Rj)]}. (3) Here Vij(q), i, j = A,B is the Fourier transform of the effective interaction between ions of i and j kinds, VAB(q) = VBA(q). The explicit expressions for Vij(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 = R0 + δR (4) where δR are the ASD with respect to the sites R0 of the ideal mean lattice. Assume that δR does not depend on the kind of an atom and perform the Fourier 390 Atomic static displacements in alloys transformation of δR: δR = 1√ N ∑ k∈BZ [ δRk exp(ikR 0) + δR−k exp(−ikR0 j) ] , (5) Symbol k ∈ BZ means that the wave vector k takes N values in the first Brillouin zone (BZ). The ASD (δRj ; as well as δRk) are random quantities in a disordered alloy. Let us separate Ak, a configurationally independent part of δRk, by means of the following relationship [5] δRk = i Ak 2 ( ρ̂k − √ Nδ0,k ) A−k = −Ak. (6) Here ρ̂k = 1√ N ∑ R σR exp(−ikRi 0). (7) is the k-th Fourier component of the occupation numbers, δ0,k, the Kronecker symbol. Equations (6) and (7) indicate that the Fourier components of the local lattice distortions are caused by fluctuations of the impurity concentration waves with respect to the average value CB = NB/N where Ni, i = A,B is the number of i-kind atoms. Component B is regarded as an ”impurity”. One should emphasize that the approximation (6) works very well in the whole region of CB values (0 < CB < 1) in such alloys where dependence of the mean lattice parameter on CB is close to the linear one [1,5]. Let us expand the factor exp(iqR) in (3) in power series of the static displace- ments δR, restricting ourselves to the square of δR. The alloy Hamiltonian H(σ) (3) with allowance for (4) to (7) in the harmonic approximation [5] takes the form H(σR) = H0(ρ̂) + ∑ k∈BZ [H1(k, δAk, ρ̂k) +H2(k,Ak, ρ̂k)], (8) where H0(ρ̂) = NV0 + √ NV1ρ̂0 + 1 2 ∑ k∈BZ V2(k)ρ̂kρ̂−k (9) is the Hamiltonian of an ideal mean lattice without displacements. The explicit expressions for potentials V0, V1 and V2(k) are presented in [2,5]. They have the following physical meaning: V0 is the part of alloy energy which does not depend on atomic configuration, V1 indicates the difference between alloy component atomic characteristics [5] and V2(k) is the Fourier transform of the ordering potential. The addends H1(k,Ak, ρ̂k) and H2(k,Ak, ρ̂k) are linear and quadratic in Ak amplitudes, respectively. The explicit equations for them see in [5]. We proceed from the grand partition sum to find the free energy Z̃ = Tr{σR} exp    −β  H(σ)− ∑ i=A,B µiNi      . (10) The following notations are introduced in (10): β = (kBT ) −1 is the inverse tem- perature, µi – the chemical potentials of the alloy components. Symbol Tr{σR} in 391 Z.Gurskii, Yu.Khokhlov (10) means summing over all the possible values of the occupation numbers {σR}. 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: Z = exp [ −NβṼ0(µ) ] Tr{σR} exp  −βṼ1(µ) ∑ R σR − 1 2 ∑ k∈BZ βṼ2(k,Ak)ρ̂kρ̂−k   . (11) Details are given in [5]. Here Ṽ0(µ) = V0 − 1 2 (µA + µB) + 1 4 A0Φ (0)A0 , (12) Ṽ1(µ) = V1 − 1 2 (µA − µB) + 1 2 P 0A0 − 1 2 A0Φ (0)A0 , (13) Ṽ2(k,Ak) = V2(k)− P kAk + 1 2 AkΦ (0)Ak (14) are the addends of the alloy Hamiltonian (8) renormalized by the ASD and P k = 1 4 ∑ G {(G− k)[VAA(G− k)− VBB(G− k)]− −(G+ k)[VAA(G+ k)− VBB(G+ k)]} . (15) The next notations are accepted in (12)-(15): G are the reciprocal lattice vectors, and Φ(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 Φ(0) is given in [6], also see appendix 2 in [5]. The grand partition sum (11) is calculated by the collective variables (CV) method [2,7]. Equation (11) is rewritten in the following way within the CV method [2,7,8]. Z̃ = exp [ −NβṼ0(µ) ] ∫ . . . ∫ exp  −1 2 β ∑ k∈BZ Ṽ2(k,Ak)ρkρ−k   J(ρ) ∏ k∈BZ dρk (16) where J(ρ) = Tr{σR}J(ρk, σR) exp [ −βṼ1(µ) ∑ R σR ] (17) is the transition Jacobian to the CV space and J(ρ, σR) = ∏ k∈BZ δ ( ρk − 1√ N ∑ R σR exp(−ikR) ) (18) with δ, 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(µ) (13) into the transition Jacobian (17) is an important feature of 392 Atomic static displacements in alloys 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 F̃ (T, µ) = −kBTN −1 ln Z̃ = Ṽ0(µ)− β−1(ln 2 +M0) + 1 2 Ṽ2(0)M 2 1 1 + βṼ2(0)M2 + +(2Nβ)−1 ∑ k∈BZ ln[1 + βṼ2(k,Ak)M2] . (19) Here Mn = ∂n ∂xn ln cosh x ∣ ∣ ∣ ∣ ∣ x=βṼ1(µ) n = 0, 1, 2 . . . (20) are cumulants [2,7]. It is seen from (20) and (13) that Mn (n = 0, 1, 2) are complex functions of temperature, potential Ṽ1 and alloy component chemical potentials. Equation CA − CB = ∂F̃ ∂µB − ∂F̃ ∂µA (21) determines the difference 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 F (T, C) = F̃ (T, µ) + ∑ i=A,B µiCi (22) and solve equation (21) to find the alloy free energy F (T, C) as a function of temperature and component concentration. Then, F (T, C) = Ṽ0 + 1 2 Ṽ2(0)M 2 1 1 + βṼ2(0)M2 + β−1    ln 2 +M0 − 1 2N ∑ k∈BZ ln[1+ +βṼ2(k,Ak)M2] } + ( β−1x− Ṽ1 ) { M1 [ 1 1 + βṼ2(0)M2 + + 1 N ∑ k∈BZ βṼ2(k,Ak) 1 + βṼ2(k,Ak)M2  +M1M2 [ βṼ2(0)M1 1 + βṼ2(0)M2 ]2    , (23) where Ṽ0 = Ṽ0(µ) + 1 2 (µA + µB), Ṽ1 = Ṽ1(µ) + 1 2 (µA − µB), (24) 393 Z.Gurskii, Yu.Khokhlov and x = βṼ1(µ), see (13), is the solution of the system of equations ∂F (T, C) ∂Ak = 0, (25) and (21). Solution of equation (25) is given in [5]. We present the final result omitting details Akλ = ∑ λ (Pkǫkλ) mω2 kλ εkλ . (26) Here εkλ and ω2 kλ are eigenvectors and eigenvalues of the force constant matrix Φ(0), respectively, λ = 1, 2, 3 – the polarization index and m = ∑ i=A,B miCi (27) is the average ion mass, see [2,6] for details. Analyse result (26). One can conclude from (15) and (26) that the ASD amplitudes Ak are small if the pair interatomic potentials VAA and VBB Fourier components are similar: VAA(q) ≈ VBB(q). Really, P k ≡ 0 at VAA(q) = VBB(q) and then Ak = 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 = V1 . (28) It means that the potential Ṽ1 as well as the cumulants Mn (20) do not depend explicitly on the ASD amplitudes Ak. This result simplifies very much the calcu- lation 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 β−1 is entropy (S), while the rest of the terms define 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: βV2(k) ≪ 1. Eid = V0 + V1(CA − CB) + 1 2 V2(0)(CA − CB) 2 , (29) Sid = −kB ∑ i=A,B Ci lnCi . (30) Equation (29) determines the energy of an average crystal: all the lattice sites are occupied by mean ions which interact via the mean potential. v̄ = vACA + vBCB with vi – the potential of an i-kind ion. Equation (30) defines the configura- tional 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 difference ∆F = F (T, C)− Eid + TSid with F (T, C) (23) indicates contribution of the SRO effects to the alloy free energy. 394 Atomic static displacements in alloys 3. 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) diffuse scattering intensity [1,3]. Besides, they are related to the SRO parameter αR = 1 4CACB ∑ k∈BZ 〈ρkρ−k〉 exp(ikR) . (31) Here αR is the value of the SRO parameter on the R-coordination sphere, 〈ρkρ−k〉 – the Fourier components of the binary correlation function. Calculation of 〈ρkρ−k〉 does not face any difficulties within the Gaussian approximation of the CV method [2,7] 〈ρkρ−k〉 = − ∂ ln Z̃ ∂ ( 1 2 βṼ2(k) ) = [1 + βṼ2(k,Ak)M2] −1 . (32) Potential Ṽ2(k,Ak) (14), renormalized by the ASD, takes the form [10] Ṽ2(k,Ak) = V2(k)− 1 2 ∑ λ (P kεkλ) 2 mω2 kλ (33) Figure 1. Behaviour of the ordering po- tential Fourier transform V2(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 ac- count. Curves 1 refer to alloy K0.7Cs0.3 while the curves 2 correspond to alloy K0.1Cs0.9. which is Ṽ2 < V2 in the whole first Bril- louin zone except for the points of high symmetry where vector P k = 0 [5]. One can notice from (32), (33), (20) and (26) that the binary correlation function Fourier components directly depend on the ordering potential, tem- perature and the ASD. Besides, they are complicated functions of potential V1 and the alloy concentration via cu- mulant M2 and the equilibrium atomic volume. In the present paper the theoreti- cal results are illustrated by numeri- cal calculations performed for the al- loys of K − Cs and Ca − Ba systems. Solid solutions of the body centred cu- bic (bcc) structure exist in wide ranges of temperature and alloy concentration in the both systems [11]. The renormal- ized potential Ṽ2(k,Ak) (33) forK−Cs and Ca−Ba alloys are shown by dashed lines in figures 1 and 2, respectively. 395 Z.Gurskii, Yu.Khokhlov Figure 2. Behaviour of the ordering po- tential Fourier transform V2(k) in the [111] direction in alloys of Ca−Ba sys- tem at T = 750K. Notations are the same as in figure 1. Curves 1 refer to alloy Ca0.5Ba0.5 and the curves 2 cor- respond to alloy Ca0.2Ba0.8. The bare ordering potentials Ṽ2(k) are depicted by full lines. Details of the cal- culations are omitted because they are the same as those in [2,5,8]. The po- tential Ṽ2(k,Ak) has the absolute min- imum in the [111] direction in the al- loys of the systems investigated. The ASD smooth the dispersion of the or- dering potential V2(k) in the first Bril- louin zone, especially in the [100] di- rection. It is seen from figures 1 and 2 that an additional minimum appears owing to the ASD in the [111] direction in the alloys of K − Cs and Ca − Ba systems. Potential Ṽ2(k,Ak) (33) is one tenth of Ṽ2(k,Ak) in K − Cs alloys, compare figures 1 and 2. Dependence of Ṽ2(k,Ak) on the atomic concentration is more pronounceed in K − Cs alloys than in Ca−Ba ones. Behaviour of the binary correlation function Fourier components 〈ρkρ−k〉 in Figure 3. Temperature effects on the bi- nary correlation function Fourier compo- nents in the K0.7Cs0.3 alloy. Dashed and full curves show results obtained, respec- tively, with and without the ASD taken into account. Curves 1 refer to T = 300K and curves 2 correspond to T = 200K. some principal symmetry directions has been investigated according to equation (32) forK−Cs and Ca−Ba alloys. Cal- culations have been performed with the ASD taken into consideration (dashed curves), and without them: Ak ≡ 0 for k ∈ BZ (full curves), see figures 3-6. Drastic effect of the ASD on the 〈ρkρ−k〉 = f(k) behaviour is observed, especially for Ca−Ba alloys, see figures 3 – 6. The ASD encourage the gain- ing of the 〈ρkρ−k〉 values in the whole first Brillouin zone. They smooth dis- persion of 〈ρkρ−k〉 in the alloys stud- ied. Thus, the alloys become more sim- ilar to the ideal solutions owing to the ASD, especially at high temperatures. The 〈ρkρ−k〉 = f(k) functions strongly depend upon the atomic concentration and temperature in alloys of the both systems investigated, see figures 3-6. Dispersion of 〈ρkρ−k〉 becomes more 396 Atomic static displacements in alloys visible at the decrease of temperature, see figures 3-5. The 〈ρkρ−k〉 = f(k) func- tions display the most interesting behaviour in the [111] direction. The largest effect of the ASD on 〈ρkρ−k〉 is observed in the [110] direction of the first Brillouin zone, see figures 3 – 6. Figure 4. The same for the K0.1Cs0.9 al- loy. The values of the SRO parameter αR on the first coordination sphere R1 have been calculated according to (31). Tables 1 and 2 demonstrate depen- dence of αR1 upon temperature and alloy concentration for K − Cs and Ca−Ba systems. The αR1 negative val- ues indicate a trend to ordering in the alloys of the both systems. The ASD favour this tendency: the values of αR1 are smaller within the rigid lattice ap- proximation, see tables 1 and 2. Tem- perature has a stable effect on αR1 . The SRO parameter αR1 increases with the decrease of temperature. This ten- dency is most pronounced in alloys with x = 0.65÷ 0.7 and x = 0.5 in KxCs1−x and CaxBa1−x systems, respectively, see tables 1 and 2. The obtained results the- oretically agree with the conclusions about the temperature effect on the SRO parameter in alloys, drawn in [3] and based on the experimental investigations of the X-ray diffuse scattering. Figure 5. The same for the Ca0.5Ba0.5 alloy. Curves 1 refer to 850K while curves 2 correspond to T = 650K. 397 Z.Gurskii, Yu.Khokhlov 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 figure 3. Curves 1 refer to the Ca0.2Ba0.8 alloy and curves 2 correspond to the Ca0.8Ba0.2 alloy. Table 1. Dependence of the short-range order parameter values on the first coor- dination sphere upon temperature and alloy concentration in KxCs1−x system. Abreviations RLA and ASD denote respectively that calculations have been per- formed within the rigid lattice approximation or with the atomic static displace- ments taken into account. Concentration of potassium, x Temperature 0.1 0.3 0.5 0.7 0.9 T , K RLA ASD RLA ASD RLA ASD RLA ASD RLA ASD 300 -.216 -.228 -.279 -.294 -.307 -.336 -.298 -.328 -.292 -.281 250 -.239 -.252 -.298 -.320 -.348 -.379 -.346 -.385 -.337 -.366 200 -.258 -.272 -.324 -.345 -.378 -.432 -.382 -.453 -.368 -.421 Table 2. Values of the short-range order parameter on the first coordination sphere in CaxBa1−x system and their dependence upon temperature and alloy concentration. Concentration of calcium, x Temperature 0.1 0.3 0.5 0.7 0.9 T , K RLA ASD RLA ASD RLA ASD RLA ASD RLA ASD 850 -.203 -.213 -.269 -.281 -.307 -.331 -.298 -.321 -.225 -.226 750 -.219 -.237 -.287 -.309 -.348 -.376 -.295 -.344 -.238 -.254 650 -.244 -.264 -.316 -.341 -.378 -.413 -.322 -.352 -.259 -.286 398 Atomic static displacements in alloys 4. Conclusions The given results can be summarized in the following statements. 1. The ASD have a drastic effect on the binary correlation function Fourier components 〈ρkρ−k〉 behaviour in the first Brillouin zone. They smooth the dispersion of the 〈ρkρ−k〉 = 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 diffuse scat- tering experiments. Acknowledgements The financial support given by the National Academy of Sciences of Ukraine is gratefully acknowledged. The authors are grateful to Professor I.Stasyuk for interest in this activity and stimulating discussions. References 1. Krivoglaz M.A. Theory of X-ray and thermal neutron scattering by real crystals. New York, Plenum, 1989. 2. Yukhnovskii I.R., Gurskii Z.A. Quantum-statistical theory of disordered systems. Kiev, Naukova Dumka, 1991 (in Russian). 3. Iveronova V.I., Katsnelson A.A. Short-range order in solid solutions. Moscow, Nauka, 1977 (in Russian). 4. Hafner J. From Hamiltonians to phase diagrams. Berlin, Springer-Verlag, 1987. 5. Gurskii Z., Khokhlov Yu. Microscopic theory of atomic static displacements in sub- stitutional binary alloys. // J. Phys.: Condens. Matter, 1994, vol. 6, p. 8711-8724. 6. Gurskii Z.A., Chushak Ya. G. Lattice dynamics of binary alloys // Phys. Stat. Sol. (b), 1990, vol. 157, No 2, p. 557-566. 7. Yukhnovskii I.R. Phase transition of the second order. Collective variables method. Singapore, World. Sci. Publ. Co., 1987. 8. Khokhlov Yu., Gurskii Z. A new approach for thermodynamic property investigations of binary alloys. // Metallofizika i Nov. Tekhnol., 1996, vol. 18, No 5, p. 3-12. 9. Hume-Rothery W., Raynor G.V. The structure of metals and alloys. London, Istitute of Metals, 1954. 10. Gurskii Z. Microscopic theory of binary substitional alloys with atomic, static displace- ments taken into account. // Ukr. Phys. Journ, 1990, vol. 35, No 11, p. 1738-1744 (in Ukrainian). 11. Hansen P.M., Anderko K. Constitution of binary alloys. New York, McGraw-Hill, 1958. 399 Z.Gurskii, Yu.Khokhlov Статичні зміщення атомів та їхній вплив на близький порядок у сплавах З.Гурський, Ю.Хохлов Інститут фізики конденсованих систем НАН України, 290011 м. Львів, вул. Свєнціцького, 1 Отримано 26 січня 1998 р. В рамках мікроскопічної теорії досліджується вплив локальних ста- тичних зміщень атомів (СЗА) на формування близького порядку в бі- нарних сплавах. Методом колективних змінних отримано явний ви- раз для Фур’є-компонент бінарної кореляційної функції 〈ρkρ−k〉 . Ре- зультати теорії ілюструються числовими розрахунками, виконаними для невпорядкованих сплавів систем K-Cs та Ca-Ba. Спостерігається сильний ефект СЗА на поведінку 〈ρkρ−k〉 –функції у першій зоні Брил- люена. СЗА згладжують дисперсію 〈ρkρ−k〉 -функції. Від’ємні значен- ня параметра близького порядку на першій координаційній сфері вказують на тенденцію до впорядкування у сплавах досліджуваних систем. Показано, що СЗА сприяють цій тенденції. Висновки теорії стосовно впливу температури на параметр близького порядку дуже добре узгоджуються із експериментальними даними, отриманими із обробки дифузного розсіяння рентгенівських променів у бінарних сплавах. Ключові слова: атомні статичні зміщення, близький порядок, невпорядкований сплав, бінарна кореляційна функція, вільна енергія PACS: 05.70.Ce; 65.50.+m 400
id nasplib_isofts_kiev_ua-123456789-118933
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1607-324X
language English
last_indexed 2025-11-29T07:04:00Z
publishDate 1998
publisher Інститут фізики конденсованих систем НАН України
record_format dspace
spelling Gurskii, Z.
Khokhlov, Yu.
2017-06-01T14:31:21Z
2017-06-01T14:31:21Z
1998
Atomic static displacements and their effect on the short range order in alloys / Z. Gurskii, Yu. Khokhlov // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 389-400. — Бібліогр.: 11 назв. — англ.
1607-324X
DOI:10.5488/CMP.1.2.389
PACS: 05.70.Ce; 65.50.+m
https://nasplib.isofts.kiev.ua/handle/123456789/118933
The influence of local atomic static displacements (ASD) on the short range order formation in binary alloys is investigated within the microscopic theory. Explicit expression for the binary correlation function Fourier components ‹ρkρ−k› 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 ‹ρkρ−k›-function behaviour in the first Brillouin zone is observed. The ASD smooth the dispersion of the ‹ρkρ−k›-function. Negative values of the short-range order parameter on the first 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 influence on the short range order parameter perfectly agree with the experimental data from the treatment of the X-ray diffuse scattering in binary alloys.
В рамках мікроскопічної теорії досліджується вплив локальних статичних зміщень атомів (СЗА) на формування близького порядку в бінарних сплавах. Методом колективних змінних отримано явний вираз для Фур’є-компонент бінарної кореляційної функції ‹ρkρ−k›. Результати теорії ілюструються числовими розрахунками, виконаними для невпорядкованих сплавів систем K-Cs та Ca-Ba. Спостерігається сильний ефект СЗА на поведінку ‹ρkρ−k›–функції у першій зоні Бриллюена. СЗА згладжують дисперсію ‹ρkρ−k›-функції. Від’ємні значення параметра близького порядку на першій координаційній сфері вказують на тенденцію до впорядкування у сплавах досліджуваних систем. Показано, що СЗА сприяють цій тенденції. Висновки теорії стосовно впливу температури на параметр близького порядку дуже добре узгоджуються із експериментальними даними, отриманими із обробки дифузного розсіяння рентгенівських променів у бінарних сплавах.
The financial support given by the National Academy of Sciences of Ukraine is gratefully acknowledged. The authors are grateful to Professor I.Stasyuk for interest in this activity and stimulating discussions.
en
Інститут фізики конденсованих систем НАН України
Condensed Matter Physics
Atomic static displacements and their effect on the short range order in alloys
Статичні зміщення атомів та їхній вплив на близький порядок у сплавах
Article
published earlier
spellingShingle Atomic static displacements and their effect on the short range order in alloys
Gurskii, Z.
Khokhlov, Yu.
title Atomic static displacements and their effect on the short range order in alloys
title_alt Статичні зміщення атомів та їхній вплив на близький порядок у сплавах
title_full Atomic static displacements and their effect on the short range order in alloys
title_fullStr Atomic static displacements and their effect on the short range order in alloys
title_full_unstemmed Atomic static displacements and their effect on the short range order in alloys
title_short Atomic static displacements and their effect on the short range order in alloys
title_sort atomic static displacements and their effect on the short range order in alloys
url https://nasplib.isofts.kiev.ua/handle/123456789/118933
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