Model pseudopotential of the electron – negative ion interaction

Generalization of the Anderson model to describe the states of electronegative impurities in liquid-metal alloys is the main aim of the present paper. The effects of the random inner field on the charge impurity states is accounted for selfconsistently. Qualitative and quantitative estimation of...

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Datum:	2003
Hauptverfasser:	Rudavskii, Yu., Ponedilok, G., Klapchuk, M.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2003
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/120763
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spelling	irk-123456789-1207632017-06-13T03:03:43Z Model pseudopotential of the electron – negative ion interaction Rudavskii, Yu. Ponedilok, G. Klapchuk, M. Generalization of the Anderson model to describe the states of electronegative impurities in liquid-metal alloys is the main aim of the present paper. The effects of the random inner field on the charge impurity states is accounted for selfconsistently. Qualitative and quantitative estimation of hamiltonian parameters has been carried out. The limits of the proposed model applicability to a description of real systems are considered. Especially, the case of the oxygen impurity in liquid sodium is studied. The modelling of the proper electron-ionic interaction potential is the main goal of the paper. The parameters of the proposed pseudopotential are analyzed in detail. The comparison with other model potentials have been carried out. Resistivity of liquid sodium containing the oxygen impurities is calculated with utilizing the form-factor of the proposed model potential. Dependence of the resistivity on impurity concentration and on the charge states is received. Узагальнюється мікроскопічна модель Андерсона з метою опису станів електронегативних домішок у розплавах рідких металів. Самоузгоджено враховується вплив випадкового внутрішнього поля на зарядові стани домішки. Проведена якісна і кількісна оцінка параметрів гамільтоніана, досліджуються межі застосовності мікроско- пічної моделі до опису конкретних систем. Розглянуто конкретний випадок домішки кисню в рідкому натрії. У такій системі важливим є вибір потенціала взаємодії електронів з домішками. Детально проаналізовано параметри запропонованого псевдопотенціала, проведено порівняння з іншими модельними псевдопотенціалами. Використавши форм-фактор запропонованого потенціала, пораховано питомий опір рідкого натрію з домішками кисню. Отримано залежність домішкового питомого опору від зарядового стану домішок та їх концентрації. 2003 Article Model pseudopotential of the electron – negative ion interaction / Yu. Rudavskii, G. Ponedilok, M. Klapchuk // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 611-628. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 71.23.-k, 71.23.An, 72.15.Rn DOI:10.5488/CMP.6.4.611 http://dspace.nbuv.gov.ua/handle/123456789/120763 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	Generalization of the Anderson model to describe the states of electronegative impurities in liquid-metal alloys is the main aim of the present paper. The effects of the random inner field on the charge impurity states is accounted for selfconsistently. Qualitative and quantitative estimation of hamiltonian parameters has been carried out. The limits of the proposed model applicability to a description of real systems are considered. Especially, the case of the oxygen impurity in liquid sodium is studied. The modelling of the proper electron-ionic interaction potential is the main goal of the paper. The parameters of the proposed pseudopotential are analyzed in detail. The comparison with other model potentials have been carried out. Resistivity of liquid sodium containing the oxygen impurities is calculated with utilizing the form-factor of the proposed model potential. Dependence of the resistivity on impurity concentration and on the charge states is received.
format	Article
author	Rudavskii, Yu. Ponedilok, G. Klapchuk, M.
spellingShingle	Rudavskii, Yu. Ponedilok, G. Klapchuk, M. Model pseudopotential of the electron – negative ion interaction Condensed Matter Physics
author_facet	Rudavskii, Yu. Ponedilok, G. Klapchuk, M.
author_sort	Rudavskii, Yu.
title	Model pseudopotential of the electron – negative ion interaction
title_short	Model pseudopotential of the electron – negative ion interaction
title_full	Model pseudopotential of the electron – negative ion interaction
title_fullStr	Model pseudopotential of the electron – negative ion interaction
title_full_unstemmed	Model pseudopotential of the electron – negative ion interaction
title_sort	model pseudopotential of the electron – negative ion interaction
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/120763
citation_txt	Model pseudopotential of the electron – negative ion interaction / Yu. Rudavskii, G. Ponedilok, M. Klapchuk // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 611-628. — Бібліогр.: 23 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT rudavskiiyu modelpseudopotentialoftheelectronnegativeioninteraction AT ponedilokg modelpseudopotentialoftheelectronnegativeioninteraction AT klapchukm modelpseudopotentialoftheelectronnegativeioninteraction
first_indexed	2025-07-08T18:32:25Z
last_indexed	2025-07-08T18:32:25Z
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fulltext	Condensed Matter Physics, 2003, Vol. 6, No. 4(36), pp. 611–628 Model pseudopotential of the electron – negative ion interaction∗ Yu.Rudavskii, G.Ponedilok, M.Klapchuk National University “Lvivska Politechnika” 12 S.Bandera Str., 79013 Lviv, Ukraine Received December 15, 2002 Generalization of the Anderson model to describe the states of electroneg- ative impurities in liquid-metal alloys is the main aim of the present pa- per. The effects of the random inner field on the charge impurity states is accounted for selfconsistently. Qualitative and quantitative estimation of hamiltonian parameters has been carried out. The limits of the proposed model applicability to a description of real systems are considered. Espe- cially, the case of the oxygen impurity in liquid sodium is studied. The mod- elling of the proper electron-ionic interaction potential is the main goal of the paper. The parameters of the proposed pseudopotential are analyzed in detail. The comparison with other model potentials have been carried out. Resistivity of liquid sodium containing the oxygen impurities is calcu- lated with utilizing the form-factor of the proposed model potential. Depen- dence of the resistivity on impurity concentration and on the charge states is received. Key words: liquid metals, electronegative impurity, Anderson model PACS: 71.23.-k, 71.23.An, 72.15.Rn 1. Introduction The study of the properties of electronegative impurities in condensed matter has been provided for a long time and it is still actual. The basic results of this topic are gathered in monographs [1,2]. Let us consider the system of liquid-metal alloy containing electronegative im- purities of such elements as H, O, Cl, N, C, F. It has practical application to the problems of corrosion resistance of materials. Liquid metals Li, K, Na, Pb and their mixtures LixPb1−x, LixNa1−x, NaxK1−x are used as the heat-carriers in nuclear re- actors due to their large thermal capacity, thermal conduction and low melting temperatures [3–5].The utilization of these metal alloys in thermonuclear reactors is prospective as well [6]. ∗This paper is dedicated to Professor Myroslav Holovko on the occasion of his 60th birthday. c© Yu.Rudavskii, G.Ponedilok, M.Klapchuk 611 Yu.Rudavskii, G.Ponedilok, M.Klapchuk It still remains urgent to solve an important technological problem, caused by aggressive corrosive effect of liquid-metal phase on structural materials (steels). The dissolution of the constituent atoms of steel (Fe, Ni, Co, Mo, Cr and other) in a liquid phase takes place during the direct contact of the heat-carrier with structural material. It gives rise to violation of stoichiometric constitution of the surface layer of hard alloys. This predetermines the loss of valuable physical-chemical properties of this layer and its subsequent destruction. The dissolution of hard alloys in al- kaline melts is a rather complicated physical and chemical process, which depends on numerous factors: chemical structure of a structural material and liquid-metal alloy, temperature, radioactivity, etc. Detailed discussion of experimental data for processes of structural materials interaction with alkaline metal alloys and their in- terpretation within the framework of simple kinetic and thermodynamic models can be found in [3,4]. Solubility kinetics of structural material in a metallic alloy essentially depends on the availability of electronegative impurities in the structure of the alloy. These impurities are always present in larger or smaller amounts in alkaline metal alloys. It was experimentally established that temperature dependence of the equilibrium solubility of gaseous impurities in liquid metals is approximately described by the expression lnC = A− BT−1, where C is concentration of impurities, A and B are certain constants, which have experimentally defined values for different alloys and for different impurities [3–5]. The mechanism of catalytic activity of gaseous impurities in alkaline alloys has not been fully investigated so far. Thermodynamic research methods in condensed metallic systems permit to define integrated (macroscopic) characteristics of alloys, to construct the diagrams of state. However, the microscopic reasons, which deter- mine the impurity behaviour in alloys in such studies cannot be revealed. Therefore, the effect of different gaseous impurities on the properties of metals and their alloys cannot be predicted. Thus, from a technological point of view, undesirable effects cannot be deliberately excluded when liquid metals and structural materials are contacted. Detailed research of the impurity behaviour at a microscopic level is important for a deeper understanding of interaction mechanisms of liquid-metal phase with structural alloys and for explaining the corrosive phenomena on a medium interface. A computation of the following physical quantities is important in order to un- derstand the microscopic processes: - chemical potential or the coefficient of the activity of the impurities; - maximum solubility of the impurities in liquid metals; - charge impurity state; - spin-polarized (magnetic) state; - structure of the neighbouring environment of the impurity and some others. 612 Model pseudopotential. . . Let us consider a monocomponent metallic alloy of alkaline metal, in which gaseous impurities are included. The microscopic approach to the description of liquid alkaline metals Li, K or Na is identical. The only quantitative difference is due to microscopic parameters, which characterize these metals. Let us give a qualitative estimate of impurity states in liquid alkaline metals. The experiments testify that the oxygen is the most dangerous in liquid sodium or in sodium based liquid alloys. At least eight experiments provided by authors [7] show that oxygen exists in ion- ized form in liquid sodium due to its high electronegativity. The following chemical elements H, O, Cl, N, C, F belong to the class of electronegatives. Electronegativity is characterized by the capability of the atoms, which are included into the struc- ture of a molecule and other compounds or a solvent, to join electrons. The electron affinity energy (E0) is an important parameter of electronegativity. An electronic structure of oxygen (1s22s22p4) shows that a free atom can be in the state O2− with two electrons additionally localized on the p-shell . However, the effective atomic charge state of oxygen in a condensed metallic medium can be changed from neutral to O2−, taking any intermediate values, depending on the parameters of a system. For the sake of simplicity we shall model atomic charge state of oxygen as local- ization process of electron from conductivity band to non-degenerate local s-level. The basic supposition in explaining the formation mechanism of a negative oxygen ion is the electron localization under the action of an effective one-particle potential. Twofold filling of a local level corresponds to the charge state O2−. The formation of an effective charge of oxygen impurity in liquid metal can be presented as a process of hybridization of local level with the states from conductivity band. This quali- tative interpretation of the formation of an effective impurity charge is explained within the framework of Anderson model [8]. The paper is organized as follows. In section 2 we generalize the Anderson model to describe the states of electronegative impurities in liquid-metal alloys. In sec- tion 3 we perform the analysis of the proposed effective pseudopotential of electron – negative ion interaction. The best method for testing the applicability of model pseudopotential in a real system is to calculate some material characteristics and compare them with the experimental data. Section 4 is devoted to the calculation of the change of impurity resistivity of liquid sodium containing the oxygen impurities. 2. Microscopic model of the system “metallic alloy – gaseous impurity” Let us consider the separate gaseous impurity dissolved in liquid alkaline metal. The liquid-metal phase will be described within the framework of electron-ionic model, which for such metals gives satisfactory computational results of electronic and structural properties. Let R1, . . . ,RN be the coordinates of atoms of metallic alloy, which accept ar- bitrary values in volume V . The impurity has a coordinate R0. We selected the 613 Yu.Rudavskii, G.Ponedilok, M.Klapchuk following full model Hamiltonian in coordinate representation: Ĥ = Hcl + Ĥel−i + Ĥel−el . (2.1) Energy operator of electron-ion interaction is written as follows: Ĥel−i = − ~ 2 2m ∑ 16i6N ∆i + ∑ 16i6N ∑ 16j6N V (\| ri − Rj \|) + ∑ 16i6Ne V0(\| ri − R0 \|). (2.2) In this formula r1, . . . , rN are electron coordinates of a metallic subsystem, the amount of which coincides with the number of metal atoms due to one-valence of alkaline elements. It is assumed that the electrons of valence impurity shell remain localized on the impurity. Pseudopotentials V (\|ri − Rj\|) and V0(\|ri − R0\|) describe electron scattering on the ions of metal and impurity, accordingly. The first term in formula (2.2) is the operator of a kinetic energy of a free electron subsystem. The last term in (2.1) describes the energy of the pair electron-electron interac- tion Ĥel−el = 1 2 ∑ 16i6=j6N Φ(\| ri − rj \|) = 1 2 ∑ 16i6=j6N e2 \| ri − rj \| . (2.3) Non-operator part Hcl describes the energy of classical ion-ionic interaction. In order to represent the secondary quantization, as a base we shall use flat waves to decompose the field electronic operators ϕk(r) = 1√ V exp (ikr) (2.4) and s-shell localized on the impurity ψ0(r) = 1√ πr3 p exp ( −\|r − R0\| rp ) . (2.5) Wave vector k in (2.4) goes through the specified values in impulse quasi-continuous space Λ: Λ = { k : k = ∑ 16α63 2π V 1/3nαeα, nα ∈ Z, (eα, eβ) = δαβ } . Let us remark that ψ0(r) is not orthogonal to flat waves (2.4). Apart from this, its inclusion into the basis causes overfilling of the last. However, the errors introduced by such an approximate procedure will not affect the regularity of a qualitative picture. In the representation of the secondary quantization operator (2.1) with 614 Model pseudopotential. . . allowance for only a certain class of Coulomb electron-electron interactions we have the following: Ĥ = Hcl + ∑ k∈Λ ∑ σ=±1 Ek a + kσ akσ + ∑ σ=±1 E0 d + 0σ d0σ + ∑ k∈Λ ∑ q∈Λ ∑ σ=±1 ( Vq a + kσ ak−q, σ + V0,q a + kσ ak−q, σ ) + ∑ σ=±1 U0 n̂0σn̂0,−σ + ∑ k∈Λ ∑ σ=±1 ( Wk a + kσ d0σ +W ∗ k d + 0σ akσ ) + ∑ k∈Λ ∑ q∈Λ ∑ σ,σ′=±1 Pq a + kσ ak−q, σ n̂σ′ + ∑ k∈Λ ∑ σ 6=σ′ ( Uk n̂σ′ a+ kσ d0, σ + U∗ k d + 0, σ akσ n̂σ′ ) . (2.6) Here, akσ(a+ kσ) and d0, σ(d+ 0, σ) are the annihilation (creation) Fermi-type operators for electrons in the states {k, σ} and {R0, σ}, where σ = ±1 is quantum spin number, which accepts two values according with two possible orientations of an electronic spin relatively to the quantization axis. Ek = ~ 2k2/2m is energy spectrum of the electrons in the states ϕk(r), and E0 is the energy of the localized electronic state ψ0(r). n̂σ = d+ σ dσ is the spin-dependent occupation number operator for the localized state. The matrix elements Vq and V0,q characterize the processes of elastic scattering of electrons on the ions of metal and impurity. Their explicit analytical forms are as follows: Vq = 1 N ∑ 16j6N e−iqRj v(q), V0, q = e−iqR0 v0(q). (2.7) Formfactors of scattering pseudopotentials v(q) = ∫ V V (\|r\|) e−iqr dr, v0(q) = 1 V ∫ V V0(\|r\|) e−iqr dr (2.8) depend only on the module of momentum transfer q due to the locality of pseu- dopotentials V (\|r\|) and V0(\|r\|). The processes of nonelastic scattering of electrons caused by their transition from the state localized on the impurity into conduction band and on the contrary, are characterized by a matrix element Wk = 1 V ∫ V e−ikr ( −~ 2∆r 2m + VLF(r) ) ψ0(r) dr. (2.9) Here, VLF(r) = ∑ 16j6N V (\|r− Rj\|) + V0(\|r − R0\|) (2.10) is the potential of a local field of metal ions and the impurity, which acts on the electron at a point r ∈ V . 615 Yu.Rudavskii, G.Ponedilok, M.Klapchuk The term ∑ σ U0 n̂σn̂−σ in Hamiltonian (2.6) descends from the operator of Coulombic electron interaction and describes Hubbard repulsion of electrons lo- calized on the impurity, with the intensity U0. U0 = ∫ dr1 ∫ dr2\|ψ0(r1)\|2 e2 \|r1 − r2\| \|ψ0(r2)\|2 = 5 8 e2 rp , (2.11) that for the atom of oxygen can be approximately about 1–5 eV. The process of elastic scattering of electrons on the charged impurity is described by a matrix element Pq = ∫ V e−iqr Φ̃(r) dr. (2.12) Here, the value Φ̃(r) = ∫ V Φ(\|r − r′\|) \|ψ0(r ′)\|2 dr′, (2.13) has the sense of potential energy of the electron in a field, which is generated by the electron localized on the shell ψ0(r). Matrix elements can be written down in the other form, structural multipliers being separated explicitly Wk = e−ikR0 w(k), Uk = e−ikR0 u(k), Pk = e−ikR0 p(k). (2.14) Coefficients w(k) = 1√ V ∫ V e−ikr ( −~ 2∆r 2m + VLF(r) ) ψ0(r) dr, u(k) = 1√ V ∫ V e−ikr Φ̃(r)ψ0(r) dr, p(k) = 1 V ∫ V e−ikr Φ̃(r) dr (2.15) ),( 0 s-R),( 0 sR )',( 0 sR ),( 0 sR ),( 1 sk )',( 2 sk )',( 0 sR ),( 1 sk ),( 2 s-k ),( 1 sk ),( 2 s¢k ),( 3 sk Figure 1. Feynman diagrams 616 Model pseudopotential. . . do not depend here on the nodal index and are considered in the coordinate system related to the impurity. Hamiltonian (2.6) does not take into account all processes with participation of two electrons. Specifically, processes represented by Feynman diagrams (figure 1) are neglected. On the diagrams, as usually, the line, which exits from the top accords with the electron creation processes in the states indicated in the diagram, while the lines, which enter the top accord with annihilation processes, respectively. A double dashed line denotes the matrix element of Coulomb electron interaction operator. Actually, in Hamiltonian (2.6) only the electrostatic effects including two electrons are taken into account and the processes of exchange character are not considered. 3. The structure of the effective electron-ionic interaction potential It is still an extremely difficult problem to calculate the total effective electron- ion interaction from the first principles with any degree of precision. Consequently, the potential is generally presented in a model form which includes in a simple parametric way all the features dictated by the physics of the situation. Ashcroft, Heine-Abarenkov, Cohen, Animalu model potentials are widely applicable in liquid metal physics. Parameters of these potentials have been investigated and approved completely enough ([11–13,20]). We have evaluated in chapter 5 the resistivity of the liquid sodium with Ashcroft’s potential (including screening by the conduction electrons) [11]. v(r) = { 0, r 6 rc, −Ze2/r, r > rc. (3.1) where rc is core radius. The Fourier transform of the potential (3.1) is v(q) = −4πZe2 Ωq2 cos(qrc). (3.2) Parameters for liquid sodium are rNa c = 1.66 a.u. = 0.0878 nm, Ω = 270 a.u. – atomic volume of liquid Na at 100◦C. Screened function by the conduction electrons in Heldart-Vosko approximation is as follows [11]: ε(q) = 1 + 4πZ Ωq2 ( 2 3 EF)−1λ( q 2kF )[1 − f(q)], λ(y) = 1 2 + 1 − y2 4y ln ∣∣∣∣ 1 + y 1 − y ∣∣∣∣ , f(q) = 1/2q2 q2 + 2kF/(1 + 0.01574(Ω/Z)1/3) , (3.3) where kF = (3π2Z/Ω)1/3 = 0.4786 a.u.−1. 617 Yu.Rudavskii, G.Ponedilok, M.Klapchuk Now let us consider the interaction between the electron and the negative ion. Besides Coulomb interaction −(Z/r) there appears the term −α/r4 (α is ion polar- izability). At large distances the pseudopotential has an asymptotic form at r → ∞ [2] U(r) ≈ −α/r4. (3.4) Such an asymptotic behaviour of the potential is allowed to adsorb the effects of polarization since they affect the attraction electron and its localization at the impurity. The potential of polarizative electron-ion interaction is selected in [14] as follows: U(r) = −α/2(r + rp) 4, (3.5) where parameter rp is the cutting radius, found for hydrogen rp = 0.74aB, polariz- ability α = 9/2a3 B. The effective potential chosen in [15] includes the observed polarizability in the following way: U(r) = V (r) − αe2 2(r2 + r2 p) 2 , (3.6) where V (r) is the central potential, parameter rp is taken, somewhat arbitrary, to be the average distance from the nucleus of the outer electrons of the neutral atom. α serves as an eigenvalue once the binding energy E0 is specified. The values of α reasonably agree with experimental values for ions O−, C−, F−. Similar form of the potential is adopted in [16]: U(r) = UHS(r) + 2(1 − e−r/r0) r − α(1 − e−r/rp) (r2 + r2 p) 2 , (3.7) where VHS is the Hartree-Fock-Slater potential for the neutral atom. The parameter rp was arbitrarily chosen to be 1.5aB, 2.5aB, 3.5aB, 4.5aB, respectively, for all atoms in each of the successive rows of the periodic table. The atomic polarizability α is chosen as the best possible from the experimental and theoretical literature. Theoretically, the value of ion polarizability can be found from the following formula α = 2e2 ∑ n6=0 \|Z0n\|2 En − E0 , Z is the operator of electron dipole moment, n characterizes the system state, En is the energy of this state. E0 is electron energy of affinity. Short distance interaction permits to apply the zeroth radius potential method widely used in atomic physics. Wave function in the coupled s-state in δ-potential ψ(r) = B √ γ/2π exp(−γr)/r. Then, polarizability α = B2/2γ4. Parameters B, γ are in [10]. Assuming all the above mentioned, we will be modelling the potential of the interaction between the electrons and the negative ions. But following [15], we must note that the method of model potential has the following disadvantages. The semiempirical parameters α and rp do not arise naturally from the formalism. Thus, the only criterium we have 618 Model pseudopotential. . . for the accuracy of the method is its agreement with experimental results. So, we use the values of α [16] and rp [15], taken from the experiment. We shall discuss herein the form of the electron wave function in the negative ion. Since the electron binding energy in the negative ion is considerably smaller than the electron binding energy in the atom, the size of negative ion is greater than the atomic size. The attraction of outer electrons takes place in the valence electron region. Thus, we can use one-electron approximation for a weakly bounding electron out of the atom. Radial electron wave function at large distances is as follows: ψl(r) = C√ r Kl+1/2(γr), where l is the orbital electron moment, K is Macdonald’s function. Wave function in such a form is used for these characteristics of a negative ion, which is deter- mined by a weakly bounding electron. Other model wave functions one can find in a monograph [2]. We have to take the s-state wave function as: ψ0(r) = 1√ πr3 p e−r/rp, (3.8) rp is taken to be the average distance from the nucleus of the outer nl-electrons of the neutral atom. 3.1. Electron-neutral impurity interaction potential. The potential parameters We shall select the effective potential of the interaction of electrons with neutral impurities as follows: V0(r) = Ae−r/rp r2 − α (r2 + r2 p) 2 , (3.9) A > 0, α, rp > 0 are potential parameters. The Fourier transform of the potential (3.9) v0(q) = ∫ V V0(r) e−iqr dr can be written as follows: v0(q) = 4πAe2 q arctg(qrp) − απ2e2 rp e−qrp. (3.10) The first term of expression (3.9) is some analytical approximation of repulsive interaction between electrons and neutral impurity. The parameter A can be defined from physical reasons. It is a well known fact that the bound states spectrum of potential falling more rapidly than Coulomb potential, consists of finite number of energy levels or cannot contain them whatsoever. The energy of ground state level is supposed to coincide with the affinity energy. Then, 619 Yu.Rudavskii, G.Ponedilok, M.Klapchuk �� Figure 2. Pseudopotential of interaction between electrons and neutral impurities O, C, F, Cl. the value of parameter A is chosen such that the resulting potential will support an s-state with the binding energy of the negative ion. E0 = 4π ∞∫ 0 r2 [ ~ 2 2m ( dψ0 dr )2 + V0(r)ψ 2 0(r) ] dr The equation for the parameter A is obtained as follows: E0 = ~ 2 2mr2 p + 4Ae2 3r2 p − 4αIe2 r4 p , where I = ∞∫ 0 dx x2 (1 + x2)2 e−2x, x = r/rp. We have provided all calculations in the atomic system of units: 1 a.u. of energy = 1 Ry = e2/2aB = 13.6 eV, 1 a.u. of length = 0.0529 nm = aB, ~ = m = e = 1. Then, the potential (3.9) in the atomic system of units transforms into V0(r) = 2Ae−r/rp r2 − 2α (r2 + r2 p) 2 , where A = 3/8r2 pE0 + 3αI/r2 p − 3/8. The pseudopotential of interaction between electrons and neutral impurities O, C, F, Cl is plotted in figure 2. All potential parameters are given in table 1. 620 Model pseudopotential. . . Table 1. Parameters of model potential V0(r) of interaction between electrons and neutral impurities O, C, F, Cl. Experimental values are taken from [16] () and [17] (). Elements rp (aB) [16] −E0 (Ry) α (a3 B) A (aB) O 1.2 0.1077 5.19 0.278 C 1.71 0.09188 14.2* 0.527 F 1.16 0.2534 4.05* 0.249 Cl 2.046 0.2656 23.5** 0.968 3.2. Potential of interaction between electrons and charged impurity We now turn to the matrix element (2.12). We calculate the effective pseudopo- tential of interaction between electrons and charged impurity in the Hartree-Fock approximation (see [18]) Ṽ0(r) = V0(r) + 〈n̂〉 ∫ V Φ(\|r − r′\|) \|ψ0(r ′)\|2 dr′, (3.11) where 〈n̂〉 is the spin-dependent occupation number operator for localized state. This value must be evaluated self-consistently. The system of equations for self-consistent calculation of charge impurity state is obtained in [18]. We can take here 0 6 〈n̂〉 6 2 (according to the Pauli principle). Now, the expression Φ̃(r) can be written as follows: Φ̃(r) = 4πe2 q2V ∑ q 1 q2 eiqrρ1s(q), here, we denote electron density Fourier coefficient ρ1s(q) = 1 πr3 p ∫ V dr′e−2r′/rpe−iqr′ = q4 0 (q2 + q2 0) 2 , q0 = 2/rp. Then, Φ̃(r) = q4 0e 2 πr Im +∞∫ −∞ eiqrdq q(q2 + q2 0) 2 , Finally, we obtain Φ̃(r) = e2 r [ 1 − e−2r/rp − r rp e−2r/rp ] . The effective pseudopotential of interaction between electrons and charged im- purity (3.11) becomes as follows: Ṽ0(r) = V0(r) + 〈n̂〉e 2 r [ 1 − e−2r/rp − r rp e−2r/rp ] . (3.12) 621 Yu.Rudavskii, G.Ponedilok, M.Klapchuk �!�"�#�$ #�% &'�( !� �% )+-,�.0/214355 55 6 )+-,�.0/55 55 6 )+-,�.8755 55 6 )+-,�.879143:::: ; Figure 3. The pseudopotential of interaction between electrons and charged oxy- gen impurity at 0 6 〈n̂〉 6 2. The pseudopotential of interaction between electrons and oxygen impurity at 0 6 〈n̂〉 6 2 is plotted in figure 3. The contribution from the Hartree-Fock potential increases when local level is occupied by one electron. It concerns the charge state O−. When 〈n̂〉 = 1.5, the bound state of the electron with the ion is absent. The form-factor of the effective impurity pseudopotential ṽ0(q) = v0(q) + Pq〈n̂〉 = ∫ dre−iqr [ V0(r) + 〈n̂〉 ∫ dr′\|ψ0(r ′)\|2Φ(\|r − r′\|) ] contains the Hartree-Fock potential caused by the charge of impurities. Pq = 4πe2 q2 [ 1 − q2r2 p (4 + q2r2 p) ( 1 + 4 4 + q2r2 p )] . (3.13) It can be rewritten in the atomic system of units as follows: ṽ0(q) = 8πA q arctan(qrp) − 4απ2 rp e−qrp + 〈n̂〉8π q2 [ 1 − q2r2 p (4 + q2r2 p) ( 1 + 4 4 + q2r2 p )] . In the limit q → 0, the form-factor v0(q) is as follows: v0(q) = v0(0) + βq + pq2, q → 0, v0(0) = (8πArp − 4π2α)/rp, β = 4απ2, p = −2απ2rp/2. (3.14) The form-factor v∗0(q) = v0(q)/v0(0), q = qaB of the oxygen impurity is plotted in figure 4. 622 Model pseudopotential. . . q v∗0(q) 1 Figure 4. The form-factor v∗0(q) of the oxygen impurity. The proposed pseudopotential will be helpful in calculating the charge and spin- polarized oxygen impurity states in liquid sodium. The system of equations for self-consistent calculation of charge impurity state is obtained in [18]. Calculating some material characteristics and comparing them with the exper- imental data is the best method for testing the applicability of model potential in a real system. We have to calculate the resistivity of liquid sodium with small concentration of the oxygen impurities in the next section. 4. Resistivity of liquid sodium containing the oxygen impurities Liquid-metal resistivity is caused by the scattering of electron waves on the atom- ic thermal oscillations (i.e., phonon scattering). This is a thermal part of resistivity ρT. Besides this, an electron scattering on the impurities took place. This part is sometimes called the impurity or the residual resistivity ∆ρC. ρ = ρT + ∆ρC, where ∆ρC ∼ ∆C. The latter relation is used in the case of small concentration of impurities C [21]. Let us first discuss the pure liquid metal. In the liquid state all scattering is supposed to be confined to a spherical Fermi-energy shell. In the relaxation time 623 Yu.Rudavskii, G.Ponedilok, M.Klapchuk approximation the solution of the Boltzmann equation for the relaxation time yields 1/τF = 2πnvF π∫ 0 dθ sin θ(1 − cos θ)σ(qF, θ), where σ(qF, θ) is the differential scattering cross section per scattering center and vF is the carrier velocity at the Fermi surface. Born approximation is used to determine σ. This, in turn, requires the square of the matrix element of the model potential that scatters the electrons \|v(q)\|2: σ(qF, θ) = 1 4π2 ( m ~2 )2\|v(q)\|2. Since the scattering occurs on the spherical Fermi surface, we have q = 2qF sin θ/2, and hence 1/τF = 1 2π m2 ~4 Ω 2q4 F ∫ 2qF 0 \|v(q)\|2S(q)q3dq. S(q) is liquid structure factor for the model of hard spheres. The theoretical structure factor is compared with the experimental data of Gingrich and Heaton [19] for the alkali metals by authors of [20]. It is apparent that up to and including the major diffraction peak, the structure factor is well reproduced by the model fluid. S(q) = {1 − nC(q)}−1, (4.1) where the direct correlation function in momentum space is given by C(q) = −4πd3 1∫ 0 s2 sin qs qs (α + βs+ γs2)ds. (4.2) The parameters α, β, γ are functions of a packing-density parameter η: η = (π/6)nd3, α = (1 + 2η)2/(1 − η)4, β = −6η(1 + η/2)2/(1 − η)4, γ = (1/2)η(1 + 2η)2/(1 − η)4. For a long wavelength (i.e., small momentum transfer) the limit for the structure factor is known thermodynamically: S(0) = nkTχT, where χT is the isothermal compressibility. For sodium nkTχT = 0.024 at 100◦C. The theoretical hard-sphere value is S(0) = 1/α = (1−η)4/(1+2η)2. For η = 0.45, which is our fit to the sodium data, S(0) = 0.025. The resistivity ρT may now be calculated from ρT = mvFΩ Ze2 1 τF . (4.3) 624 Model pseudopotential. . . Table 2. Comparison of liquid-metal resistivities for liquid Na, K at 100◦C. Metal ρexp ρH−A [20] ρA [11] ρA Na 9.6 9.5 6.0 9.64 K 13 16.2 9.4 13.2 The resistivity for liquid metals is evaluated in [20] with Heine-Abarenkov poten- tials and the Animalu-Heine modification of this potential. The results are close to the experiment. We have calculated the sodium resistivity with Ashcroft’s potential (3.1) (including screened function in Heldart-Vosko approximation (3.3)). In [11] the screened function has been used in Hartree approximation (without including elec- tron exchange and correlation effects). The experimental value is ρexp = 9.6 µΩcm (see table 2). So, we are convinced of the fact that the resulting value of resistivity is sensitive to the screened function and to small changes in the potential. Now let us discuss the effect of the impurities on resistivity. When adding the impurity, the resistivity of liquid metal increases. The increment of resistivity is given as follows (the relation is obtained for crystals) ∆ρ ∆C = mvF e2 σ, (4.4) where σ is total scattering cross section of conductivity electrons per impurity, ∆C is atomic part of the impurities. σ = 2π ∫ π 0 (1 − cos(θ))σ(qF, θ) sin(θ)dθ. We have calculated the scattering cross section per pseudopotential ṽ0(q) for a charged impurity. ∆ρ ∆C = mvF πe2 ( 2m ~2 )2 ∫ 1 0 \|ṽ(2qFz)\|2z3dz. (4.5) Concentration dependencies of the residual resistivity for several values 〈n〉 = 0, 1, 2 are given in figure 5. The dash-dot line accords to experimental data [23]. Experimentally obvious relation is as follows: ρ350◦C = ρ0 350 + ∆ρ = 18.9(1 + ∆C)µΩ cm, where ρ350◦C is full resistivity of liquid Na at temperature 350◦C, ∆ρ is the increment of resistivity caused by oxygen concentration rising, ∆C is the change of oxygen concentration (O2 in at. %). The description of some experiments on controlling the oxygen impurity in Na is given in [22,23]. Dependence of the residual resistivity on the impurities charge states represented in figure 6 can be useful for impurity identification in liquid-metal heat-carriers. 625 Yu.Rudavskii, G.Ponedilok, M.Klapchuk Figure 5. Concentration dependencies of the residual resistivity calculated for charged impurities 〈n〉 = 0, 1, 2. Figure 6. Dependence of the residual resistivity on the charge states of the impurities. 5. Conclusions We have constructed the generalization of microscopic Anderson model to de- scribe the states of electronegative impurities in liquid metal alloy. The case of oxygen impurity has been considered in detail. The proposed potential of interac- tion between conducting electrons and negative ions has been used in calculating the increment resistivity of liquid sodium containing oxygen impurities. The results are close to experimental data. The observed dependence of impurity resistivity on the concentration of the impurities and the charge states can serve as an effective method of controlling the impurities in liquid-metal heat-carriers. The form-factor of the potential will be used for self-consistent calculation of the charge and spin- polarized states of the oxygen impurity. References 1. Messi G. Negative Ions. Moscow, Mir, 1979. 2. Smirnow B.M. Negative Ions. Moscow. Atomizdat, 1978. 3. Beskorovainyj N.M., Ioltukhovskii A.G. Constructive Materials and Liquid-metals Heat-carriers. Moscow, Energoatomizdat, 1983. 4. Greznov G.M., Evtikhin V.A. et.al. Material Science of Liquid-metal System of Ther- monuclear Reactors. Moscow, Energoizdat, 1989. 5. Linchevskii B.V. Thermodynamic and Kinetic Interaction between Gaseous and Liquid Metals. Moscow, Metallurgia, 1986. 6. Mihailov V.N., Evtikhin V.A. et.al. Lithium in Thermonuclear and Cosmic Energetics of XXI Century. Moscow, Energoatomizdat, 1999. 626 Model pseudopotential. . . 7. Greenwood D.A., Ratti V.K. – In: Proc. Intern. Conf. Organized by the BNES, Not- tingham, April 1973, London, p. 43–45. 8. Anderson P. Localized magnetic states in metals. // Phys. Rev., 1961, vol. 124, No. 1, p. 41. 9. Batsanov B.M. Electronegativity of elements and chemical bond. Novosibirsk, AN US, 1962. 10. Smirnov B.M. Physic of Atom and Ion. Moscow, 1986. 11. Ostrovskii O.I., Grygorian V.A., Vishkariov A.F. Properties of Metallic Alloys. Moscow, Metallurgia, 1988. 12. Harrison U. Pseudopotentials in Theory of Metals. Moscow, Mir, 1968. 13. Yuhnovskii I.R., Gurskii Z.A. Quantum Statistic Theory of Disordered System. Kiev, Naukova dumka,1991. 14. Golovinskii P.A., Zon B.A. // Izvestia of AN US, Ser. Phis. 1981, vol. 45, No. 12, p. 2305. 15. Cooper J.W., Martin J.B. Electron photodetachment from ions and elastic collision cross section for O, C, Cl and F. // Phys. Rev., 1962, vol. 126, p. 1482. 16. Robinson E.J., Geltman S. Single- and double- quantum photodetachment of negative ions. // Phys. Rev., 1967, vol. 153, p. 4. 17. Dalgarno A. // Advan. Phys., 1962, vol. 11, p. 281. 18. Rudavskii Yu.K., Ponedilok G.V., Mikitiouk O.A., Klapchuk M.I. Electronegative im- purity states in structural disordered system. Preprint of the Institute for Condensed Matter Physics, ICMP-02-13U, Lviv, 2002, 26 p. 19. Gingrich N.S., LeRoy Heaton // J. Chem. Phys. 1961, vol. 34, p. 873. 20. Aschkroft N., Lekner J. Structure and resistivities of liquid metal // Phys. Rev., 1966, vol. 145, No. 1, p. 83–90. 21. Shpilrain E.E. Thermal Physics Properties of Alkali Metals. Moscow, Atomizdat, 1970. 22. Subbotin V.I. Physical-chemical Bases of the Liquid-metal Heat-carriers Applicability. Moscow, Atomizdat, 1970. 23. Subbotin V.I. et al. // Teplof. vysokih temper., 1965, vol. 3, p. 145 (in Russian). 627 Yu.Rudavskii, G.Ponedilok, M.Klapchuk Модельний потенціал взаємодії електронів з негативними іонами Ю.Рудавський, Г.Понеділок, М.Клапчук Національний університет “Львівська політехніка”, 79013 Львів, вул. С.Бандери, 12 Отримано 15 грудня 2002 р. Узагальнюється мікроскопічна модель Андерсона з метою опису станів електронегативних домішок у розплавах рідких металів. Са- моузгоджено враховується вплив випадкового внутрішнього поля на зарядові стани домішки. Проведена якісна і кількісна оцінка пара- метрів гамільтоніана, досліджуються межі застосовності мікроско- пічної моделі до опису конкретних систем. Розглянуто конкретний випадок домішки кисню в рідкому натрії. У такій системі важливим є вибір потенціала взаємодії електронів з домішками. Детально про- аналізовано параметри запропонованого псевдопотенціала, прове- дено порівняння з іншими модельними псевдопотенціалами. Вико- риставши форм-фактор запропонованого потенціала, пораховано питомий опір рідкого натрію з домішками кисню. Отримано залеж- ність домішкового питомого опору від зарядового стану домішок та їх концентрації. Ключові слова: негативні іони, рідкі метали, модель Андерсона PACS: 71.23.-k, 71.23.An, 72.15.Rn 628

Model pseudopotential of the electron – negative ion interaction

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