2014 ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Сер. 4. Том 1 (59). Вып. 2 ELECTRON POSITIVE ION OF GOLD ATOM. ORBITAL ENERGIES

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1 2014 ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Сер. 4. Том 1 (59). Вып. 2 ФИЗИКА УДК I. Yu. Yurova EFFECTIVE -DEPENDENT POTENTIAL FOR THE SYSTEM: ELECTRON POSITIVE ION OF GOLD ATOM. ORBITAL ENERGIES St. Petersburg State University, , St. Petersburg, Russian Federation We present reativistic effective -dependent potentias (RELP) for eectron ion Au +. We determined parameters of ELP from Auger and optica spectroscopic data. Orbita energies of atom we cacuated by two approaches: non-reativistic with Schrödinger equation and reativistic one with Dirac equation. We cacuated bound energies of 34 orbita states of Au atom with presenty deveoped effective potentia RELP. Resuts are in good agreement with Dirac Hartree Fock cacuations and experimenta data. We appied potentia RELP for osciator strengths cacuations. Our resuts of osciator strengths vaues agree with Dirac Hartree Fock these. With the combination of RELP and core poarized dipoe correction method we received osciator strengths conforming to experimenta data. We appied the RELP to cacuation of the eectron-eectron screening constant λ in metaic God. For these purposes, we cacuated the 4f 7/2 orbita energy shift of God in metaic phase appying perturbation theory with RLEP. From the comparison of cacuated and experimenta vaues of 4f 7/2 energy shift we estimated the vaue of eectron-eectron screening constant as 0.08 < λ < 0.3 for different systems formed by soid god. Refs 22. Figs 1. Tabes 5. Keywords: effective potentia, ion, atom, spectroscopy, God, orbita energy, cacuation, osciator strength, energy shift, fine structure. 1. Introduction. Considerabe efforts are directed to experimenta and theoretica studies of God atom and God systems such as semi-infinite media, thin fims, quantum wires, nanopartices for many years unti nowadays. One can get knowedge about God and God systems from the specia edition: God Buetin (Word God Counci, London). There are basic experimenta methods in the fied of the examination of God: spectroscopic determination of high energy eves of Au atom [1, 2], X-ray photoemission spectroscopy [3] and Auger spectroscopy [4] for core eves. Above spectroscopic methods, there is another type of experiment the eastic peak eectron spectroscopy (EPES) [5]. EPES deas with the eectron scattering from the soid God and God system surfaces at intermediate and high non-reativistic impact energies. Theoretica investigation of heavy atoms eectronic structure deas with the Dirac Hartree Fock (DHF) [6, 7] cacuations. Either a eectrons of an atom or vaence eectrons ony are taken into consideration in sef-consistent fied cacuations. In atter case, a eectrons except vaence these, are substituted by effective potentia [8]. Note, that DHF resuts are not aways accurate, especiay for high energy eves (see Tabe 1). There are areadyknown 148

2 eectron God effective potentias [9, 10] however, these potentias did not provide proper vaues of Au orbita energies. Au orbita energies (a are negative), and radia expectations Tabe 1 Leve Orbita energy R n Present DHFS IPM DHF Experiment Present DHF 7s [1] re 8.67 re s [12] re 2.83 re s [2] re 0.94 re s [3] re 0.41 re s [3] re 0.18 re s [3] re re s [3] 2993 re re p [1] re 11.5 re p [1] re 4.10 re p [3] re 1.07 re p [3] re 0.44 re p [3] re 0.20 re p [3] re re d [2] re 1.84 re d [2] re 0.46 re d [2] re 0.17 re f [3] re 0.44 re 0.50 A vaues are in atomic units. Present unmarked nonreativistic cacuations; re reativistic ones. Reativistic vaues with > 0 were averaged over ( 1/2, + 1/2) configuration. The goa of the present consideration is to deveop an approach as to Au orbita states and to eectron Au scattering cacuations as we to obtain theoretica resuts corresponded to above referred spectroscopic and EPES experiments. For these purposes we deveoped the e Au + ( ) -dependent non-reativistic and reativistic effective interaction potentias and considered its appications to cacuations of orbita energies, eectron densities, dipoe osciator strengths of Au atom and orbita energy shift cacuation. 149

3 2. Effective eectron ion Au + potentia (ELP) Anaytic effective potentias for the eectron heavy ion interaction. In spite of great variety of effective potentias, ony few of them have been deveoped for the modeing the interaction between an eectron and heavy atom with nucear charge Z nuc > 70. For exampe, it is the Independent Partice Mode (IPM) effective potentia: Z nuc η Ω(r) + η V IPM (r) =, Ω(r) =H(exp(r/d) 1) + 1, (1) r where η = 1 for eectron one charged positive ion interaction; η = 0 for eectron neutra atom interaction; H = d(z nuc η) 0.4, the parameter d is tabuated for Z nuc =2 103; IPM = V IPM η=1 and V (a) IPM = V IPM η=0. For Au and Au + the parameter d = The second effective potentia is the Dirac Hartree Fock Sater (DHFS) one [10]: V DHFS (r) = Z 3 nuc A i exp( α i r), (2) r i=1 V (a) DHFS DHFS = V DHFS, r r 0 ; r, r > r 0, at 0 r <. The vaues of parameters in the potentia (2) are tabuated for Z nuc = For Au and Au + ion they are equa to: A 1 = , A 2 = , A 3 =1 A 1 A 2, α 1 = , α 2 =3.6914, α 3 = We cacuated orbita energies of Au atom with potentias V IPM and DHFS, Tabe 1. One coud concude, that those potentias especiay, did not provide accurate vaues of orbita energies (see Tabe 1). That is why we deveoped new e Au + effective potentia provided proper vaues of orbita energies. Moreover, it occurred, that our e neutra atom Au effective potentia provided accurate eastic e Au + cross sections (see beow) Eectron Au + -dependent effective potentias. According to the theory of -dependent effective potentias [8, 16], we used the expression: ( r) = (r), (3) where is the projection operator in Hibert space onto the subspace with fixed orbita momentum. The exchange and Couomb potentias are usuay incuded in the ( r) [8, 11]. In present consideration we did not incude the core poarization, exchange and correation in the ( r) separatey. However, as indicated in Tabe 1, the successfu fit to orbita energy vaues suggests that the present effective potentia incudes effects of exchange and correation. Radia potentias (r) in (3) are determined as: (r) =V h + 1 i= 1 ( C (+) i r exp q () i ) r 1 r, (4) ( ) exp q () V h = C 2 2 r. (5) r(r + h) From the expansion of (r) atsmar, one can obtain the equaity: 150 C (+) 2 h + C (+) 1 1= Z nuc. (6)

4 We assume that at >2 potentias coincide with the potentia 2. Parameters of (4), (5) we find from the fitting of cacuated bound atom energies to experimenta data anaogized [9, 11]. Note, that we coud not reach the success in fitting unti the additiona term V h (5) was incuded in expression (4). The fitting way aows anybody to use the spectroscopic data bank with accurate vaues of atom bound energies known for the most of atoms of periodic tabe. To cacuate orbita energies we soved numericay radia Schrodinger and Dirac equations with potentias (4), (5) and obtained parameters of effective -dependent non-reativistic potentia (ELP) and reativistic one (RELP) correspondingy. For fitting we have appied reference states 4s, 5s, 6s, 4pj, 5pj, 6pj and 4d j, 5d j, 4f j,with averaging over j for ELP, and with one vaue of j from two possibe these (j = ± 1/2) for RELP; experimenta data had been taken from [1 4, 12]. Note, that the parameter C (+) 1 one can find from the condition (6). For cacuations we appied our own computer code in non-reativistic case, for Dirac equation we appied the DHF computer code presented by professor I. I. Tupitsyn. In Tabe 2 parameters of ELP and RELP radia potentias (4), (5) are presented. Both V ELP and V RELP potentias are given by same expressions (3) (6) with same vaues of a parameters except the parameter h. Parameters of the potentia (2) Tabe 2 =0 h =0.100 h =0.068 (re) =1 h =0.072 h = (re) >1 h = h = (re) A vaues are in atomic units. i q () i C (+) i We present resuts for orbita energies and radia expectations in Tabe 1. We inserted in Tabe 1 vaues, obtained with DHS a eectrons method, and with effective potentias (1) and (2) as we. There is good agreement of present orbita energies with experimenta data and DHF resuts in both non-reativistic ELP and reativistic RELP cacuations. We present deviations in percents of cacuated orbita energies from experimenta vaues in Tabe 3. ELP and RELP 3s, 3p, 7s, 6p, 7p energies agree experimenta data and DHF resuts without fitting. Note, that RELP 1s, 2s, 2p energies agree DHF and experimenta data better than non-reativistic ones. The potentia DHFS appeared poor, and the potentia V IPM seems sighty better for cacuations of orbita energies of Au atom (Tabe 3). Energies of spinorbit spitting of Au for n =2 5 are presented in Tabe 4. The accuracy of RELP fine structure resuts is about %. 3. Dipoe osciator strengths. We empoy radia potentias (2) in the cacuations of Au dipoe osciator strength f γj,γ,j. According to [13], it is the function of matrix eements from components of the eectron dipoe moment D i and energies of bound states E γj, E γ j 151

5 Tabe 3 Deviations of cacuated orbita energy from experimenta vaues (in percents) 1s 6s 2p 6p 3d 5d 4f V IPM V DHFS V present Reativistic vaues with > 0 were averaged over ( 1/2, + 1/2) configuration. Tabe 4 Reativistic nj orbita energies (a are negative) and radia expectations Leve 7p 6p 5p 4p 3p 2p 5d 4d 3d Orbita energy R nj RELP DHF Experiment RELP DHF / / /2 [1] / / / / / / / / / /2 [1] / / / / / / / / / /2 [1] / / / / / / / / / /2 [3] / / / / / / / / / /2 [3] / / / / / / / / / /2 [3] / / / / / / / / / /2 [2] / / / / / / / / / /2 [2] / / / / / / / / / /2 [2] / / / / / / /2 as: 4f A vaues are in atomic units / / /2 [3] / / / / / / /2 f γj,γ,j = 2(E γ j E γj) D γj,γ,j 3(2j +1) 2, (7) where γ, γ are a quantum numbers except j, j ones, D γj,γ,j 2 = γjm D i γ J M 2, i M,M index i designsx, y, z. For transitions 6s 1/2 np j, j =1/2, 3/2, we have: D 6s1/2,np j 2 = a j ( P 6s1/2 r Pnpj 2 + Q 6s1/2 r Qnpj 2 ), a 1/2 =2,a 3/2 =4, (8)

6 P ni, Q ni are arge and sma components of Dirac wave functions. We cacuated dipoe osciator strengths for 6s np 1/2 and 6s np 3/2 transitions for n =6, 7inRELP,we appied dipoe correction method with core poarization, as it was done in [14], and in DHF approximations. We present resuts for dipoe osciator strengths in Tabe 5. Osciator strengths of 6s j np j transitions of atom Au, n =6, 7 Tabe 5 Transition Present Present CPDC DHF Other theories Experiment 6s 1/2 6p 3/ [16] (16) [17] s 1/2 6p 1/ [16] (7) [17] s 1/2 7p 3/2 3.16( 2) 4.07( 3) 5.48( 2) 6s 1/2 7p 1/2 2.36( 3) 1.08( 3) 1.69( 2) 7s 1/2 7p 3/ s 1/2 7p 1/ Present reativistic effective -dependent potentia (RELP), CPDC core poarization dipoe corrected method. Energy vaues and wave functions were cacuated by professor I. I. Tupitsyn. DHF [15]. Many configuration DHF [15]. CPDC DHF [14]. One can see in Tabe 5, that DHF method [15] and presenty appied DHF with RELP approximation give osciator strengths approximatey twice higher experimenta ones, however our dc cacuations and the theory [14] gives vaues being in the good agreement with experimenta data [16, 17]. One coud expain this agreement by the incusion in DHF the CPDC effect opposite other cacuations. 4. Orbita energy shifts in soid god and estimation of eectron eectron screening constant. The other appication of RELP is the cacuation of orbita energy shifts in soids. 4f orbita of Au atom is the most suitabe one for the investigation orbita energy shift due to various perturbation effects in soid state [4]. One can write the eectron eectron interaction potentia in soids in accordance with [18, formua (8)]: V ee (r, r )=C exp( r r /λ) r r, (9) C =0forfreeatomandC = ±1 for buk metas, custers with different impurities and nanopartices with different sizes and pretreatment conditions. We consider V ee as the perturbation causes of atom orbita energy shift. In order to cacuate the correspondent energy shift we averaged the interaction V ee (r, r ) over Wigner Zeitz ce with pane wave approximation for Boch functions [19]: V ee (r, r )= 1 V ee (r, r )dr, (10) Ω WZ WZ where Ω WZ the voume of Wigner Zeitz ce; R 0 =2.99a 0 for God [20]. In order to cacuate integra in expression (10) we appied the expansion of V ee (r, r ) onto a sequence 153

7 4f 7/2 binding energy, ev 84.0 a b , Å , Å The dependence of 4f 7/2 orbita energy on eectron eectron interaction constant λ in metaic god system: a C =+1;b C = 1 of modified Besse functions with variabes r/λ,r /λ (see [18, formua (13)]). In resut, we obtained the expression for averaged eectron eectron interaction potentia: V ee (r, r ) [ ( C 3λ2 = R R ) ( exp R )( ( 0 r exp λ λ λ) λ ( 2r exp r ) )], r < R 0 λ 0, r R 0. (11) In the first order of perturbation, the energy shift is equa to the sum of matrix eements: P4f3±1/2 Q4f3±1/2 D 4f3±1/2 = P 4f3±1/2 V ee + Q 4f3±1/2 V ee, (12) where P 4f3±1/2 and Q 4f3±1/2 are arge and sma components of RELP orbita wave function 4f 3±1/2. Appying expression (11), we obtained the 4f 7/2 orbita energy as the function of eectron-eectron screening constant λ for C = ±1, see Figure. One can appy the experimenta vaues of 4f 7/2 binding energy shift in number of nanopartices produced by different pretreatments [21] and aoyed systems with soid Au [22] for estimation the eectron eectron screening constant vaues; in resut we obtained the interva 0.08 < λ < Concusion. We presented new reativistic effective -dependent potentia RELP for eectron God interaction. It was been successfuy appied to the God atom orbita energies and eectron densities cacuations opposite to other known effective potentias. We appied RELP to the cacuation of dipoe osciator strengths of Au atom. The incusion in RELP of dipoe correction due to the core poarization effect aowed us to get resuts we agreed with experimenta data. The new way of estimations of eectron eectron screening interaction constant λ in metas due to the cacuation of orbita energy shift with RELP and fit theoretica resuts to experimenta data was been proposed. Cacuations of λ were made for different systems formed by metaic God. Aknowegements. The author is peased to acknowedge Professor I. I. Tupitsyn for giving an opportunity for appication of the computer code for the soution of Dirac equation with effective potentia. The author is thankfu to Professor I. I. Tupitsyn for the presentation of a eectrons DHF orbita energies, wave functions and radia expectations of Au atom. 154

8 References 1. Ehrhardt J. C., Davis S. P. Precision waveengths and energy eves in God // J. Opt. Soc. Am Vo. 61. P Brown C. M., Ginter M. L. Absorption spectrum of Au // J. Opt. Soc. Am Vo. 68. P Lotz W. Binding energies in free atoms // J. Opt. Soc. Am Vo. 60. P Carson T. A. Photoeectron and Auger Spectroscopy. New York: Penum Press, Gergey G. Eastic peak eectron spectroscopy // Scanning Vo. 8. P Descaux J. P. Reativistic Dirac Fock expectation vaues for atoms with Z =1toZ = 120 // At. Data Nuc. Data Tabes Vo. 12. P Bratsev V. F, Deyneka G. B., Tupitsyn I. I. // Bu. Acad. Sci. USSR (Russian), Phys. Ser Vo. 41. P Laughin C., Victor G. A. Mode potentia methods // Adv. At. Mo. Phys Vo. 25. P Green A. E. S., Sein D. L., Zacher A. S. Anaytica independent partica mode for atoms // Phys. Rev Vo P Savat F., Martinez J. D., Mayo R., Pareada J. Anaytica Dirac Hartree Fock Sater screening function for atoms (Z =1 92) // Phys. Rev. (A) Vo. 36. P Chu X., Chu S.-I., Laughin C. Spectra and tempora structures of high-order harmonic generation of Na in intense mid-ir aser fieds // Phys. Rev. (A) Vo Fugge J. C., Martensson N. Core-eve binding energies in metas // J. Eectron Spectroscopy and Reated Phenomena Vo. 21. P Sobe man I. I. Introduction to the theory of atomic spectra. Oxford: Pergamon, Migdaek J., Bayis W. E. Reativistic Hartree Fock osciator strengths for the owest s p and p d transitions in the first few members of the Ag(I) and Au(I) isoeectronic sequences, with aowance for core poarization // J. Quant. Spectrosc. Radiat. Transfer Vo. 22. P Descaux J. P., Kim Y.-K. Reativistic effects in outer shes of heavy atoms // J. Phys. (B) Vo. 8. P Gruen D. M., Gaudioso S. L., McBeth R. L., Lerner J. L. Appication of matrix isoation spectroscopy to quantitative sputtering studies. I. Energies and osciator strengths of the resonance transitions of god atoms isoated in nobe gas matrices // J. Chem. Phys Vo. 60. P Gaarde M. B., Zerne R., Caiyan L. et a. Determination of radiative ifetimes of excited states in neutra god using time-resoved vacuum-utravioet aser spectroscopy // Phys. Rev. (A) Vo. 50. P Feder R., Goisch H., Meinert D. et a. Low-energy e, 2e. spectroscopy from the W(001) surface: Experiment and theory // Phys. Rev. (B) Vo. 58. P Caaway J. Energy band theory. New York; London: Academic Press, Kambe K. Cohesive energy of nobe metas // Phys. Rev Vo. 99. P Radnik J., Mohr C., Caus P. On the origin of binding energy shifts of core eves of supported god nanopartices and dependence of pretreatment and materia synthesis // Phys. Chem. Chem. Phys Vo. 5. P Rousset J. L., Cadete Santos Aires F. J., Sekhar B. R. et a. Comparative X-ray photoemission spectroscopy study of Au, Ni, and AuNi custers produced by aser vaporization of buk metas // J. Phys. Chem. (B) Vo P Статья поступила в редакцию 20 ноября 2013 г. Контактная информация Yurova Inna Yurievna Dr. Sci., Professor; e-mai: inna-yurova@ramber.ru 155