Effective interactions in the binary metallic systems

The basic approach in the microscopic theory of binary metal systems has been developed. The electron-ion model Hamiltonian with nonlocal manyparticle interactions was obtained using the statistical operator averaging for the electron-nuclear model over the localized electron states. The role of ort...

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Дата:	2004
Автори:	Koval, S.P., Solovyan, V.B.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2004
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/118891
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Effective interactions in the binary metallic systems / S.P. Koval, V.B. Solovyan // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 101-110. — Бібліогр.: 5 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1188912017-06-02T03:04:31Z Effective interactions in the binary metallic systems Koval, S.P. Solovyan, V.B. The basic approach in the microscopic theory of binary metal systems has been developed. The electron-ion model Hamiltonian with nonlocal manyparticle interactions was obtained using the statistical operator averaging for the electron-nuclear model over the localized electron states. The role of orthogonalization and exchange effects on the formation of electron-ion interactions is investigated. The transition to the ion metal model with effective manyparticle interactions was performed. Розроблено базисний підхід у мікроскопічній теорії бінарних металічних систем. Шляхом статистичного засереднення статистичного оператора електрон-ядерної моделі за станами підсистеми локалізованих електронів побудовано гамільтоніан електрон-іонної моделі з нелокальними багаточастинковими взаємодіями. Досліджено роль ефектів ортогоналізації та обміну у формуванні електроніонних взаємодій. Здійснено перехід до іонної моделі з ефективними багаточастинковими взаємодіями. 2004 Article Effective interactions in the binary metallic systems / S.P. Koval, V.B. Solovyan // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 101-110. — Бібліогр.: 5 назв. — англ. 1607-324X PACS: 71.55.Ak DOI:10.5488/CMP.7.1.101 http://dspace.nbuv.gov.ua/handle/123456789/118891 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The basic approach in the microscopic theory of binary metal systems has been developed. The electron-ion model Hamiltonian with nonlocal manyparticle interactions was obtained using the statistical operator averaging for the electron-nuclear model over the localized electron states. The role of orthogonalization and exchange effects on the formation of electron-ion interactions is investigated. The transition to the ion metal model with effective manyparticle interactions was performed.
format	Article
author	Koval, S.P. Solovyan, V.B.
spellingShingle	Koval, S.P. Solovyan, V.B. Effective interactions in the binary metallic systems Condensed Matter Physics
author_facet	Koval, S.P. Solovyan, V.B.
author_sort	Koval, S.P.
title	Effective interactions in the binary metallic systems
title_short	Effective interactions in the binary metallic systems
title_full	Effective interactions in the binary metallic systems
title_fullStr	Effective interactions in the binary metallic systems
title_full_unstemmed	Effective interactions in the binary metallic systems
title_sort	effective interactions in the binary metallic systems
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2004
url	http://dspace.nbuv.gov.ua/handle/123456789/118891
citation_txt	Effective interactions in the binary metallic systems / S.P. Koval, V.B. Solovyan // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 101-110. — Бібліогр.: 5 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kovalsp effectiveinteractionsinthebinarymetallicsystems AT solovyanvb effectiveinteractionsinthebinarymetallicsystems
first_indexed	2025-07-08T14:50:50Z
last_indexed	2025-07-08T14:50:50Z
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fulltext	Condensed Matter Physics, 2004, Vol. 7, No. 1(37), pp. 101–110 Effective interactions in the binary metallic systems S.P.Koval’ 1 , V.B.Solovyan 2 1 The Ivan Franko National University of Lviv, 8 Kyrylo and Mephodii Str., Lviv, 79005, Ukraine 2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011, Lviv, Ukraine Received February 24, 2004 The basic approach in the microscopic theory of binary metal systems has been developed. The electron-ion model Hamiltonian with nonlocal many- particle interactions was obtained using the statistical operator averaging for the electron-nuclear model over the localized electron states. The role of orthogonalization and exchange effects on the formation of electron-ion interactions is investigated. The transition to the ion metal model with ef- fective manyparticle interactions was performed. Key words: basic approach, electron-nuclei model, electron-ion model, ion model, manyparticle interactions PACS: 71.55.Ak The investigation of manyparticle effective interactions in the binary metallic systems is very interesting from different viewpoints. In particular, it is important in interpreting thermodynamic and structural characteristics of alloys and melts, in describing the quantum states of impurities in metals, structure phase transitions etc. The basic approach, developed in papers [1,2], makes it possible to perform pre- cise calculations without employing the model isights or any adjusted parameters. It gives the advantage of coupling the analytical and numerical analysis which becomes very substantial for the subsequent employment of the results produced. In accordance with the basic approach let us consider the electron-nuclei model for the binary metallic system which consists of two types (c = a, b) of nuclei (Nc is the number of nuclei of c type with the charge Qc · e) and Ne = ∑ c NcQc electrons, −e is the electron charge. To describe the electron subsystem we use the compound basis of one particle wave functions {Ψσ} = {Ψla} ⊕ {Ψlb} ⊕ {Ψk}, (1) c© S.P.Koval’, V.B.Solovyan 101 S.P.Koval’, V.B.Solovyan which consists of two orthogonal subspaces – localized {Ψlc(r)}(c = a, b) and delo- calized functions {Ψk(r)}. The functions of localized electron states are formed from the atomic orbitals ϕµc,ic(r) ≡ ϕµc (r−Ric) according to the Bogoliubov method [3], for example Ψλa,ja (r) = ϕλa,ja (r) − 1 2 ∑ µa ∑ ia(6=ja) (ϕµa,ia, ϕλa,ja )ϕµa,ia(r) − 1 2 ∑ µb ∑ ib (ϕµb,ib, ϕλa,ja )ϕµb,ib(r) + · · · , (2) where λa, µa are the quantum states numbers of localized electrons that form the ionic cores, and Rja ,Ria are nuclei coordinates. The subspace of Ψk(r)-functions can be constructed using different methods, for instance with the help of the technique described in paper [4] and grounded on the plane wave basis {ϕk}. Let us calculate the partition function of the above described model over the electron variables in grand canonical ensemble Zµ = Spe { exp [ −β ( Ĥ − µN̂e )]} , (3) which plays a role of the effective statistical operator for nuclei (ionic) subsystem. The Hamiltonian of an initial model in the second quantization representation takes the form Ĥ = Ĥn(R) + ∑ σ1,σ2 ∑ s εσ1,σ2 a+ σ1,saσ2,s + 1 2 ∑ σ1,...,σ4 ∑ s1,s2 Vσ1,σ2,σ3,σ4 a+ σ1,s1 a+ σ2,s2 aσ3,s2 aσ4,s1 , (4) where the component Ĥn(R) is the operator sum of the nuclei kinetic energy and their mutual Coulomb interaction. The second term in the right part of (4) presents the electrons’ kinetic energy and attraction with nuclei, the third term presents electron-electron interaction. Here aσ,s are Fermi operators, which correspond to Ψσ(r)-functions (with the spin projection s), {σ} = {λa, ja}, {λb, jb}, {k}′. The “stroke” means that some number of wave vectors, which is equal to the number of localized functions in {Ψla}, {Ψlb} subspaces are excluded from k-spectrum. Matrix elements in (4) are defined as follows: εσ1,σ2 = Tσ1,σ2 + ∑ c Nc ∑ jc=1 vjc σ1,σ2 , vjc σ1,σ2 = − Qc V ∑ q VqRσ1,σ2 (q) exp(−iqRjc ), Vσ1,σ2,σ3,σ4 = 1 V ∑ q6=0 VqRσ1,σ4 (q)Rσ2,σ3 (−q), Tσ1,σ2 = ( Ψσ1 ∣ ∣ ∣−h̄2∇2/2m ∣ ∣ ∣ Ψσ2 ) , Rσ1,σ2 (q) = (Ψσ1 \|exp(iqr)\|Ψσ2 ) , Vq = 4e2 π q2. (5) 102 Effective interactions According to the papers [1,2], we will split the calculation of Ze(µ) into two stages. In the first stage we will calculate the track of statistical operator over the localized electron states λc(c = a, b). In the second one – over the collectivized electron states. After the first stage computation we obtain the effective statistical operator of the collectivized electron subsystem in the ionic field, which determines the Hamiltonian of electron-ion model on {Ψk}-subspace: Splc { exp[−β(Ĥ − µN̂e)] } = exp[−β(Ĥef − µN̂k)], Ĥef = Ĥi(R) + ∑ k1,k2 ′ ∑ s v2(k1,k2\|R)a+ k1,sak2,s + 1 2 ∑ k1,k2,k3,k4 ′ ∑ s1,s2 v4(k1,k2,k3,k4\|R)a+ k1,s1 a+ k2,s2 ak3,s2 ak4,s1 + · · · , N̂k = ∑ k,s ′a+ k,sak,s. (6) Here Ĥi(R) = ∑ c NcE 0 c + ∑ c [ −h̄2/(2Mc) ] ∑ jc ∇2 jc + 1 2 ∑ c1,c2 ∑ jc1 ,jc2 Vc1,c2(Rjc1 − Rjc2 ) (7) is the Hamiltonian of isolated ion subsystem, E0 c is the energy of an isolated ion and Vc1,c2(Rjc1 − Rjc2 ) is the pair ion-ion potential in the Hartree-Fock approximation (jc1 6= jc2 at c1 = c2), Vc1,c2(R) = 1 V ∑ q Vc1,c2(q) exp(iqR), Vc1,c2(q) = VqZc1(q)Zc1(q), (8) where Zc(q) = Qc − 2 ∑ λc Rλc,λc (q) is the effective ion valency function, which at q → 0 tends to its true value – the ion valency Zc. The second term in Ĥef describes the one particle electron-ion interaction and the third describes the two electron interaction in the ion field. To simplify the second stage computations (the averaging over the conductive electron states ) we will turn from operators ak,s to Ck,s, which conforms with plane waves basis {ϕk(r)} = {V −1/2 exp(ikr)} according to the expressions: ak,s = ∑ q (Ψk, ϕq)Cq,s. (9) 103 S.P.Koval’, V.B.Solovyan This relationship, simultaneously with the completeness condition for {Ψσ}-basis, written in the momentum space ∑ l (Ψl, ϕq1 )(ϕq2 , Ψl) + ∑ k ′ (Ψk, ϕq1 )(ϕq2 , Ψk) = δq1,q2 , (10) permits to recalculate the Hamiltonian Ĥef matrix elements. The sum calculation over the wave vectors k can be done in the general form without concrete definition of functions Ψk(r) . As a result, the general form of Hamiltonian Ĥef remains, but instead of matrix elements Tσ1,σ2 , Rσ1,σ2 (q) (see (5)), calculated on the basic functions Ψσ(r), we obtain matrix elements T 0 σ1 ,σ2 , R0 σ1,σ2 (q) which are calculated on the functions of OPW -system: {Ψla(r)} ⊕ {Ψlb(r)} ⊕ {χk(r)}, (11) where χk(r) = ϕk(r) − ∑ c ∑ lc (Ψlc, ϕk)Ψlc(r) (12) – orthogonalized plane wave. Thus, we obtained {σ} = {la} ⊕ {lb} ⊕ {k} without any restrictions for the wave vector k. Further, we extract from operator Ĥef the electron liquid Hamiltonian ĤEL and the electron-ion interaction operator V̂ei: Ĥef = Ĥi(R) + Ĥ EL + V̂ei Ĥ EL = ∑ k,s εka + k,sak,s + 1 2V ∑ q Vq ∑ k1,k2 ∑ s1,s2 a+ k1+q,s1 a+ k2−q,s2 ak2,s2 ak1,s1 V̂ei = ∑ k1,k2 ∑ s A2(k1,k2\|R)a+ k1,sak2,s + 1 2 ∑ k1,...,k4 ∑ s1,s2 A4(k1,k2,k3,k4\|R)a+ k1,s1 a+ k2,s2 ak3,s2 ak4,s1 + · · · , (13) where εk = h̄2k2/2m. We present here the main terms of matrix elements A2(k1,k2\|R), A4(k1,k2,k3,k4\|R), disregarding the components caused by gibridiza- tion effects between localized and delocalized electrons : A2(k1,k2\|R) = T 0 k1,k2 − εk1 δk1,k2 − 1 V ∑ c Qc ∑ jc R0 k1,k2 (q) exp(−iqRjc ) + 1 V ∑ c ∑ lc,s ∑ q6=0 Vq { R0 k1,k2 (q)R0 lc,lc(−q) − 1 2 R0 k1,lc(q)R0 lc,k2 (−q) } + A4(k1,k2,k3,k4\|R) = = 1 V ∑ q6=0 Vq { R0 k1,k4 (q)R0 k2,k3 (−q) − δk1,k4+qδk2,k3−q } . (14) We pass on to the second stage of computation, i.e., the partition function cal- culation over the collectivized electron states. As in the reference system we use the 104 Effective interactions electron liquid model with Hamiltonian Ĥ EL . We will take into account the electron- ion interactions in terms of perturbation theory method. As one can see from (2), (14), the operator V̂ei describes nonlocal one- and two-electron interactions, which have manyparticle character relatively to ions. In this paper, in order to avoid inconveniences, we restrict ourselves to a simple metal case, neglecting the atomic function ϕµa (r−Rja ) overlap integrals. To simplify the high order diagram calculation for perturbation theory series we will restrict ourselves to the main terms in A2(k1,k2\|R) and A4(k1,k2,k3,k4\|R), which are additive over ion partial structure factors Sc q = Nc ∑ jc=1 exp(iqRjc ). (15) But in the first order of perturbation theory series we will also take into account the terms proportional to the product of two partial structure factors. The statistical operator track for the electron-ion model over the electron vari- ables defines the statistical operator for the ion model of the metal with effective manyparticle interactions: exp { βµ∗Ne } · Spe ( exp { −β [ Ĥ − µ∗N̂e ]}) = = exp { −β [ F EL + ∑ c Ncωc + Ĥef(R) ]} , (16) where µ∗ ≡ µ∗(β, N/V ) is the electron chemical potential, F EL is free energy for electron liquid model, ωc is the single ion energy in the electron liquid surroundings in the pair correlation approximation relatively to the reference system, Ĥef(R) is the ion subsystem Hamiltonian where the interionic energy is written in terms of structure factors: Ĥef(R) = − ∑ c ∑ jc [ h̄2 2Mc ∇2 jc ] + 1 2V ∑ c1,c2 ∑ q6=0 V (2) c1,c2 (q) [ Sc1 q Sc2 −q − δc1,c2Nc1 ] + ∑ n≥3 (−1)n−1 n!V n−1 ∑ c1,...,cn ∑ q1,...,qn V (n) c1,...,cn (q1, . . . ,qn)Sc1 q1 · · ·Scn qn δq1+···+qn,0. (17) Here, we obtain the following expressions for the interionic potentials: V (2) c1,c2 (q) = VqZc1(q)Zc2(−q) − 1 V ∑ k ac1 2 (k + q,k)ac2 2 (k,k + q)µ2(q,−q\|k), V (n) c1...cn (q1, . . . ,qn) = 1 V ∑ k µn(q1, . . . ,qn\|k)ac1 2 (k,k − q1)a c2 2 (k − q1,k − q1 − q2) × · · ·acn 2 (k − q1 − · · · − qn−1,k − q1 − · · · − qn), (18) 105 S.P.Koval’, V.B.Solovyan where µn(q1, . . . ,qn\|k)−k is the component n-particle static semi-invariant correla- tion function of the reference system [5]. Particulary, in the local field approximation µ2(q,−q\|k) = µ0 2(q,−q\|k) { 1 + VqV −1µ0 2(q,−q)[1 − Gq] }−1 , (19) where Gq is static local field correction function and µ0 2(q,−q) = ∑ k µ0 2(q,−q\|k) (20) is the pair correlation function of noninteracting electron subsystem. The effective ion valence function is determined by expressions: Zc(q) = Qe − 2 ∑ λc Rλcλc (q) − 1 V ∑ k,s nk,sP c k,k(q), P c k,k(q) = ∑ λc { ∑ µc Rλc,µc (q)ϕλc (k)ϕ∗ µc (k) − ϕλc (k − q)ϕ∗ λc (k) − ϕλc (k)ϕ∗ λc (k + q) ∑ µc } . (21) The contributions to effective interionic potentials which appear in the second and higher orders in perturbation theory series are obtained in the following approxima- tion for V̂ei: V̂ei =⇒ V̂ (0) ei ≡ ∑ c 1 V ∑ q6=0 S−q ∑ k a2(k + q,k)C+ k+q,sCk,s . (22) Here, the main term of two-electron interaction is taken in Hartree-Fock approxi- mation. 1 2 3 4 5 6 q a0 -10 0 10 20 30 40 50 60 70 +1 0 -1 k = kF t = -1, 0, 1 a0 3 Ry Figure 1. Depending of the kinetic energy contribution on wave vector q (k = kF, t = −1, 0, 1). 106 Effective interactions In accordance with (18), n-particle interionic potentials are of the screening char- acter, particularly V (2) c1,c2(q) have the following asymptotics: lim q→0 (V (2) c1,c2 (q)) = VqZc1(q)Zc2(q) 1 − L(q)G(q) 1 + L(q)[1 − G(q)] , lim q→∞ (V (2) c1,c2 (q)) ≈ VqQc1Qc2, (23) where L(q) = VqV −1µ0 2(q,−q). As we can see, the nonlocal matrix elements of electron-ion interaction, simultaneously with electron liquid model correlation func- tions, completely define the character of n-particle interionic potentials. 1 2 3 4 5 6 -20 -15 -10 -5 0 5 q a0 t = -1 k = kF a0 3 Ry Figure 2. The exchange interaction between localized and conductive electrons in plane waves (solid curve) and ortogonalized plane waves (dashed curve). 1 2 3 4 5 6 qa0 -30 -20 -10 0 10 20 k = kF t = -1, 0, 1 +1 0 -1 VH a0 3 Ry Figure 3. Dependence of direct electron-ion Coulomb interaction on wave vector q at k = kF. The lovest curve coresponds to plane waves approsimation, other ones to ortogonalization corrections at t = −1, 0, 1. For example, let us consider the properties of these matrix elements for the metal- lic lithium case, using a simple 1s-function ϕ1s(r) = π−1/2(α/a0) 3/2 exp(−αr/a0), 107 S.P.Koval’, V.B.Solovyan 1 2 3 4 5 6 qa0 -20 -10 0 10 20 k = 0 1 2 VH a2(k,q), a0 3 Ry Figure 4. The total electron-ion formfactor a2(k+q,k), averaging over the angle between k and q at k = 0, kF, 2kF. where a0 is Bohr radius and α is a variational parameter (for an isolated ion Li+ parameter α takes the value 43/16 ). This form being sufficiently precise for 1s electron description, makes it possible to perform the calculations for ac 2(k + q,k) in analytical form, which permits to perform the precise analysis of the obtained results. Figures 1–3 show the main components of electron-ion formfactor ac 2(k + q,k) for the case k = kF and a different mutual orientation of vectors k and q(t ≡ cos(k,q) = −1, 0, 1) . Figure 1 illustrates the contribution of electron kinetic en- ergy change caused by orthogonalization effects. Figure 2 shows the contribution caused by exchange interaction between localized and conductive electrons in differ- ent approximations: dashed curve corresponds to plane wave approximation and the solid one corresponds to orthogonalized plane wave. Figure 3 illustrates the direct electron-ion Coulomb interaction. Lower curve corresponds to plane waves approxi- mation, higher curves are orthogonalization corrections. Figure 4 illustrates the total 1 2 3 r/a0 -3 -2 -1 0 1 2 3 -2a0 /r 0 1 2 k / kF = 0, 1, 2 a2(r,k), Ry Figure 5. The function a2(r, k) for Li at k = 0, kF, 2kF. 108 Effective interactions formfactor dependence over k, q being the average over the angle between k and q. All figures correspond to α = 43/16. As one can see in figures 1–4, the formfactor ac 2(k + q,k) speciality is its high nonlocality (dependence on modulus and orientation of vector k), caused, firstly, by orthogonalization effects. The most important corrective term is the kinetic energy contribution (see figure 1). It generates the maximum on ac 2(k+q,k) curve (figure 4). The height of this maximum decreases with the electron kinetic energy increasing. We have calculated the Fourier transformation of formfactor ac 2(k + q,k), averaged over the k and q angle (figure 5) at k = 0, kF, 2kF. The a2(r, k) dependence has the Coulomb character (r−1) for small and for large values of r ( lim r→∞ (a2(r, k)) = −Z∗e2/r). References 1. Vavrukh M.V., Muliava Y.V. // J. Phys. Stud., 1997, vol. 1, No. 2, p. 257 (in Ukraini- an) 2. Vavrukh M.V., Yakibchuk P.M., Koval’ S.P. // Metalofizyka i novitni tehnologii, 2000, vol. 22, No. 5 (in Russian) 3. Bogolyubov N.N. Izbrannye Trudy, vol. 2. Kyiv, Naukova dumka, 1970 (in Ukrainian) 4. Vavrukh M.V., Solovyan V.B. // Ukr. Jorn. Phys., 1991, vol. 36, No. 4, p. 630 (in Ukrainian) 5. Vavrukh M.V., Vavrukh N.M., Solovyan V.B. // Phys. Stat. Sol. (b), 1993, vol. 177, No. 2, p. 361 109 S.P.Koval’, V.B.Solovyan Ефективні взаємодії у бінарних металічних системах С.П.Коваль 1 , В.Б.Солов’ян 2 1 Львівський національний університет імені Івана Франка, 79005 Львів, вул. Кирила і Мефодія, 8 2 Інститут фізики конденсованих систем НАН України, 79011 Львів, вул. Свєнціцького, 1 Отримано 24 лютого, 2004 р. Розроблено базисний підхід у мікроскопічній теорії бінарних мета- лічних систем. Шляхом статистичного засереднення статистично- го оператора електрон-ядерної моделі за станами підсистеми лока- лізованих електронів побудовано гамільтоніан електрон-іонної мо- делі з нелокальними багаточастинковими взаємодіями. Дослідже- но роль ефектів ортогоналізації та обміну у формуванні електрон- іонних взаємодій. Здійснено перехід до іонної моделі з ефективни- ми багаточастинковими взаємодіями. Ключові слова: базисний підхід, електрон-ядерна модель, електрон-іонна модель, іонна модель, багаточастинкові взаємодії PACS: 71.55.Ak 110

Effective interactions in the binary metallic systems

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