Helium-like Ions Below. the N=2-9 Hydrogenic Thresholds

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1 Adv. Studies Theor. Phys., Vol. 6, 01, o. 7, The 1,3 S e Itershell Autoioizig States of Helium-like Ios Below the N=-9 Hydrogeic Thresholds M. Dieg *, A. Wagué Laboratory Atoms Lasers, Departmet of Physics Faculty of Scieces ad Techiques Uiversity Cheikh Ata Diop, Dakar, Seegal. I. Sakho Departmet of Physics, UFR Scieces ad Techologies Uiversity of Ziguichor, Ziguichor, Seegal M. Biaye Departmet of Physics ad Chemistries Faculty of Scieces ad Techologies of Formatio ad Educatio Uiversity Cheikh Ata Diop, Dakar, Seegal * topdige@yahoo.fr Abstract Doubly excited itershell 1,3 S e autoioizig states i helium below the N=-9 hydrogeic thresholds of helium-like ios are calculated with a ovel approach. We have combied the variatioal method with the o-liear parameters of Hylleraas ad the β-parameters of screeig costat by uit uclear charge. Our theoretical results up to Z=10 i helium-like series compare very well with some of the most accurate existig theoretical calculatios ad with experimetal observatios. PACS umbers: v, 31.5.Jf, Pf, bu Keywords: electroic correlatios, ios, helium-like systems, siglet states, triplet states, matrix elemets, doubly-excited, autoioizatio, semi-empirical ad empirical calculatios, Rydberg states

2 1346 M. Dieg et al 1. Itroductio Doubly or multiply excited atomic systems play a very importat role i the uder-stadig of a atomic system. I most cases they are coupled to the cotiuum via Coulomb iteractio. Thus decayig to the cotiuum is a mai mechaism for the de-excitatio of these atomic systems. Experimetally these multiples excited states appear as promiet peaks i the electro spectroscopy. Therefore the study of these states is crucial i correctly iterpretig the-observed spectra. Herrick ad Siaoglu [4] ad Li [] have itroduced a classificatio scheme, which is widely used, alog with a hyperspherical coordiate descriptio of He. This paper presets a theoretical ivestigatio of eergies o the iter-shell 1,3 (, ) A e K T N S doubly excited states of the helium-like ios below the N=-9 hydrogeic thresholds are calculated by usig variatioal method with the oliear parameter of Hylleraas [10,11] ad the semi-empirical procedure of Screeig Costat by Uit Nuclear Charge (SCUNC) [7]. Sectio presets the theoretical procedures used i this work. I sectio 3 the comparisos of our results with the other theoretical calculatios ad experimetal data are also made.. Theory ad calculatios.1. Hamiltoia ad Hylleraas-Type Wave Fuctios The Schrödiger equatio for the helium-like io ca be writte as: r r r r, (1) H Φ Nll r 1 r E Φ Nll r 1 r (, ) = (, ) H is the Hamiltoia of the helium-like ios, r 1 ad r are the coordiates of the electros with respect to the ucleus. The wave fuctios for the descriptio of the iter-shell siglet ad triplet doubly excited states are give here [10,11]: l υ = N l 1 (, ) 1 ( 1 ) ( 0 υ r r Φ Nll = r r N r λ r1r ) + υ = 0 l υ ' = l 1 ' υ ur ur j k m r r r λ ' r r r + r r r r r ( ) ( ) ( )( ) exp υ ' = 0 ( λ r + λ ' r ) 1 () N ad are the pricipal quatum umbers; l is orbital agular momets. j, k, m are Hylleraas parameters with (j, k, m 0), λ ad λ are variatio parameters. The set of parameters (j, k, m) defie the basis states (i.e. the cofiguratios). The eve values of k defie the symmetric wave fuctios describig the siglet states, while the odd values defie the atisymmetric wave fuctios for the triplet states. The fial form of the wave fuctios of the iter-shell siglet ad triplet doubly excited state icludig the correlatio effects to the mixig of cofiguratios ca be expressed as follows:

3 1,3 S e itershell autoioizig states of helium-like ios 1347 Ψ ur ur (, ) = r Nll 1 r a jkm jkm Φ Nll, (3) a jkm are the eigevectors which ca be determied by solvig the Schrödiger equatio: ur ur urur ΗΨ ( r1, r) = ( r1, r) Nll ΕΨ, (4) Nll I what follows, for the sake of brevity we shall deote the triad of Hylleraas parameter (j, k, m) by q. ( H ) 0 Nllqq ' E N Nllqq' a = (5) q ' q ' with: H N = Nllqq' ΦNllq Φ, (6) Nllq' = Φ H Φ, (7) Nllqq' Nllq Nllq' Where N Nllqq are the ormalisatio factor ad H Nllqq the matrix elemets of Hamilto operator. The iter-shell siglet ad triplet doubly excited wave fuctios were foud i the basis cotaiig the cofiguratios with the followig coditio for the Hylleraas parameters j + k + m 3, correspodig to the basis dimesio D = 13 or 7. I order to obtai the miimum eigevalue i which we are iterested, the calculatios are carried out for various values of the parameters λ ad λ. The eigevalues E obtaied i the preset calculatios follow the Hylleraas- Udheim theorem [5] ad do ot iclude the Feshbach shifts because of the use of the icomplete basis sets of the wave fuctios. These calculatios have bee carried out i the framework of the variatioal method usig iteractio basis states with has real Hamiltoia. Accordig to the Hylleraas-Udheim theorem, a good approximatio for the eigevalues is obtaied whe the miima of the fuctios (d (H(λ,λ )/dλdλ =0) coverge with icreasig values of the dimesio D ad whe the fuctios exhibit a plateau. λ 0 ad λ 0 deotes the values of the λ ad λ -parameter correspodig to the miima of the fuctio ad the miimum eigevalue... Geeral formalism of the SCUNC method The screeig costat by uit uclear charge formalism (SCUNC) to calculate the eergy resoaces of the helium-isoelectroic sequece covergig to the N=- 9 hydrogeic thresholds. I the framework of the SCUNC-method, total eergy of, A S KT ; + L π excited states is expressed i the form (i Ry) [7,8]: ( ) 1 N A S A S 1 E + π ( K, T) ; L ; Z + π = Z + 1 β ( K, T) ; L ; Z, N N N (8) Furthermore, i the framework of the screeig costat by uit uclear charge formalism, the β-screeig costat is expressed i terms of the variatioal λ 0 ad

4 1348 M. Dieg et al λ 0 who deotes the values of the λ ad λ -parameter of Hylleraas correspodig to the miima of the fuctio ad the miimum eigevalue (7) as follows. ' A S 1 π ' 1 N λ0 λ0 β + ( KT, ) ; L; Z, λ0, λ + N 0 = Z N + L+ 1 L N + L+ S( S+ 1) + L+ S( S+ 1) (9) 1 Usig (9), total eergy of (, ) A S K T N L π doubly excited states i the helium-lie ios the expressed as follows: ' A S 1 N 0 0 E + π λ + λ ( K, T) ; L ; Z = Z + 1 N N Z N + L+ 1 L , N + L+ S( S+ 1) + L+ S( S+ 1) (10) The simple expressio (10) is used to calculate resoace parameters for itershell 1,3 S e autoioizig states i helium (the two electros occupy the differet shells) without a complex calculatio program. For example the 3(1, 0) S level i Ry is: Z E λ λ' Z E λ λ' Results ad discussios 1,3 The results obtaied i the preset work for total eergies of the (, ) A e K T N S states of helium-like ios below the N=-9 hydrogeic thresholds ad Z 10 are reported i Table I ad II. I table III we preset a compariso of the results obtaied presetly for + 1,3 e 1,3 3(1, 0) S, 4(1, 0) + e + S, 1,3 e + 1,3 e 4(,0) 3 S ad 5(,0) 3 S autoioizig states of the helium atom below the N=,3 hydrogeic thresholds ad Z 10 with the screeig costat by uit uclear charge method of Sakho [7], the complex-coordiate rotatio method of Ho [17], the computig double sum over the complete hydroge spectrum of Ivaov ad Safroova [1], the Fechbach projectio formalism method of Bachau et al [6],,the time idepedet variatio perturbatio theory of D. Ray et al [3] ad the trucated diagoalizatio method of Lipsky et al [9]. The agreemet betwee the preset results ad the theoretical data for all these states of He i Table III are see to be satisfactory.

5 1,3 S e itershell autoioizig states of helium-like ios 1349 I table IV, we preset a compariso of the curretly calculated eergy resoaces of the (1, 0) S ad (,0) 3 S Rydberg series of the atom with the results obtaied by Sakho [7], Bachau et al [6], Lipsky et al [9], D. Ray et al [3] ad with the saddle-poit complex-rotatio method of Che [1]. Compariso idicates a good agreemet betwee the preset calculatios ad the others ab iitio oes. 3 e 3 e I table V, the compariso of excitatio eergy of (1, 0) S ad (,0) 3 S levels for Z= is made with available theoretical literature values. 1 The results for the excitatio eergies of the ( K, T e ) + S level of He I, Li II, Be III ad B IV are listed i table VI ad compared with the photoabsorptio experimets of Madde ad Codlig [14], with the electro-impact experimet of Hicks ad Comer [13], with advaced light source experimets of Scully et al [16] ad with the photoioisatio experimetal results of Diel et al [15]. Here we ote a good agreemet betwee the preset theoretical data ad the experimetal observatios.

6 1350 M. Dieg et al 4. Coclusio 1,3 We have carried out extesive calculatios for iter-shell (, ) A e K T N S doubly excited states eergies of helium-like ios below the N=-9 hydrogeic thresholds. The calculatios have doe by usig the variatioal method with the o-liear parameters of Hylleraas ad the β-parameters of screeig costat by uit uclear charge. The simple expressio (10) is used to calculate resoace parameters for itershell 1,3 S e autoioizig states i helium (the two electros occupy the differet shells) without a complex calculatio program. We ote here geerally satisfactory agreemet betwee the preset results ad the other available calculatios. Our results should be useful refereces for theoretical ad experimetal works. Ackowledgmets. The authors would like to thak the Swedish Iteratioal Developmet Agecy (SIDA), the Iteratioal Cetre for Theoretical Physics (ICTP) for support. Refereces [1] A. I. Ivaov, I. U. Safroova, Opt. Spectrosc. 75, 516 (1993) [] C. D. Li, Commets At. Mol. Phys. 19, 89 (1987) [3] D. Ray, B. Kudu, P. K. Mukherjee, K. Ohtsuki ad K. Oho, Phys. Lett. A. 136, 43 (1989) [4] D. R. Herrick ad O. Siaoglu, Phys. Rev. A 11, 97 (1975) [5] E. Hylleraas ad B. Udheim, Z. Phys. 65, 759 (1930) [6] H. Bachau ad al., Atomic Data ad Nuclear Data Tables, 48, 168 (1991) [7] I. Sakho, Euro. Phys. J. D 61, 67 (011) [8] I. Sakho, J. At. Mol. Sci., 0 (011) [9] L. Lipsky, R. Aaia ad M. J. Coeely, Atomic data ad uclear data tables. 0, 17 (1977) [10] M. Biaye, M. Dieg, I. Sakho ad A. Wagué, Chi. J. Phys. 47, 166 (009) [11] M. Dieg, I. Sakho, M. Biaye ad A. Wagué, Chi. J. Phys. 48, 38 (010) [1] M. K. Che, Phys. Rev. A 56, 4537 (1997) [13] P. J. Hicks ad J. Comer, J. Phys. B. At. Mol. Phys. 8, 1866 (1975) [14] R. P. Madde ad K. Codlig, Phys. Rev. Lett. 10, 516 (1963) [15] S. Diehl, D.Cubayes, J. P. Bizau, F. J. Wuilleumier, E. P. Keedy, J. P. Mosier ad T. J. Morga, J. Phys. B: At. Mol. Opt. Phys. 3, 503 (1999) [16] S. W. J. Scully, I. Ālvarez, C. Ciseros, E. D. Emmos, M. F. Gharaibeh, D. Leiter, M. S. Lubell, A. Müller, R. A. Phaeuf, R. Pütter, A. S. Schlachter, S. Schippers, W. Shi, C. P. Balace ad B. M. McLaughli, J. Phys. B: At. Mol. Phys. 39, 3957 (006) [17] Y. K. Ho, Phys. Rev. A. 3, 137 (1981)

7 1,3 S e itershell autoioizig states of helium-like ios 1351 Table I: Eergy resoaces of doubly excited ( KT, ) N S The results are expressed i Rydberg: 1a.u= Ry= eV (N=-9 ad =3-10) states of helium-like ios (Z=-10). (K,T) A N 3(1,0) + 4(1,0) + 5(1,0) + 6(1,0) + 7(1,0) + 8(1,0) + 9(1,0) + 10(1,0) + Nll S+1 L π s3s 1 S e s4s 1 S e s5s 1 S e s6s 1 S e s7s 1 S e s8s 1 S e s9s 1 S e s10s 1 S e (K,T) A N 4(,0) + 3 5(,0) + 3 6(,0) + 3 7(,0) + 3 8(,0) + 3 9(,0) (,0) + 3 5(3,0) + 4 Nll S+1 L π 3s4s 1 S e 3s5s 1 S e 3s6s 1 S e 3s7s 1 S e 3s8s 1 S e 3s9s 1 S e 3s10s 1 S e 4s5s 1 S e (K,T) A N 6(3,0) + 4 7(3,0) + 4 8(3,0) + 4 9(3,0) (3,0) + 4 6(4,0) + 5 7(4,0) + 5 8(4,0) + 5 Nll S+1 L π 4s6s 1 S e 4s7s 1 S e 4s8s 1 S e 4s9s 1 S e 4s10s 1 S e 5s6s 1 S e 5s7s 1 S e 5s8s 1 S e (K,T) A N 9(4,0) (4,0) + 5 7(5,0) + 6 8(5,0) + 6 9(5,0) (5,0) + 6 8(6,0) + 7 9(6,0) + 7 Nll S+1 L π 5s9s 1 S e 5s10s 1 S e 6s7s 1 S e 6s8s 1 S e 6s9s 1 S e 6s10s 1 S e 7s8s 1 S e 7s9s 1 S e (K,T) A N 10(6,0) + 7 9(7,0) (7,0) (8,0) + 9 Nll S+1 L π 7s10s 1 S e 8s9s 1 S e 8s10s 1 S e 9s10s 1 S e Z -E -E -E -E

8 135 M. Dieg et al Table II: Eergy resoaces of doubly excited 1 e ( KT, ) N S The results are expressed i Rydberg: 1a.u= Ry= eV (N=-9 ad =3-10) states of helium-like ios (Z=-10). (K,T) A N 3(1,0) - 4(1,0) - 5(1,0) - 6(1,0) - 7(1,0) - 8(1,0) - 9(1,0) - 10(1,0) - Nll S+1 L π s3s 3 S e s4s 3 S e s5s 3 S e s6s 3 S e s7s 3 S e s8s 3 S e s9s 3 S e s10s 3 S e (K,T) A N 4(,0) - 3 5(,0) - 3 6(,0) - 3 7(,0) - 3 8(,0-3 9(,0) (,0) - 3 5(3,0) - 4 Nll S+1 L π 3s4s 3 S e 3s5s 3 S e 3s6s 3 S e 3s7s 3 S e 3s8s 3 S e 3s9s 3 S e 3s10s 3 S e 4s5s 3 S e (K,T) A N 6(3,0) - 4 7(3,0) - 4 8(3,0) - 4 9(3,0) (3,0) - 4 6(4,0) - 5 7(4,0) - 5 8(4,0) - 5 Nll S+1 L π 4s6s 3 S e 4s7s 3 S e 4s8s 3 S e 4s9s 3 S e 4s10s 3 S e 5s6s 3 S e 5s7s 3 S e 5s8s 3 S e (K,T) A N 9(4,0) (4,0) - 5 7(5,0) - 6 8(5,0) - 6 9(5,0) (5,0) - 6 8(6,0) - 7 9(6,0) - 7 Nll S+1 L π 5s9s 3 S e 5s10s 3 S e 6s7s 3 S e 6s8s 3 S e 6s9s 3 S e 6s10s 3 S e 7s8s 3 S e 7s9s 3 S e (K,T) A N 10(6,0) - 7 9(7,0) (7,0) (8,0) 3 9 Nll S+1 L π 7s10s 3 S e 8s9s 3 S e 8s10s 3 S e 9s10s 3 S e Z -E -E -E -E

9 1,3 S e itershell autoioizig states of helium-like ios 1353 Table III: Compariso of doubly excited +, 1,3 e 3(1, 0) S, +, 1,3 e 4(1, 0) S like ios (Z=-10) with other results. Eergies E are i Ry., +, 1,3 e 4(,0) 3 S ad +, 1,3 e 5(,0) 3 S states of helium- (K,T) A N 3(1,0) + 4(1,0) + Nll S+1 L π s3s 1 S e s4s 1 S e Preset Sakho Ho Ivaov ad Preset Sakho Ivaov ad results Safroova results Safroova Z -E -E -E -E -E -E -E (K,T) A N 4(,0) + 3 5(,0) + 3 Nll S+1 L π 3s4s 1 S e 3s5s 1 S e Preset Bachau D. Ray Preset Bachau D. Ray results ad al. ad al. results ad al. ad al. Z -E -E -E -E -E -E (K,T) A N 3(1,0) - 4(1,0) - Nll S+1 L π s3s 3 S e s4s 3 S e Preset Sakho Ivaov ad Preset Sakho Ivaov ad results Safroova results Safroova Z -E -E -E -E -E -E (K,T) A N 4(,0) - 3 5(,0) - 3 Nll S+1 L π 3s4s 3 S e 3s5s 3 S e Preset Bachau Lipsky Preset Bachau Lipsky results ad al. ad al. results ad al. ad al. Z -E -E -E -E -E -E

10 1354 M. Dieg et al Table IV: Compariso of doubly excited (1, 0) S ad (,0) 3 S (=3-10) states of helium-like ios (Z=) with other theoretical results. Eergies E are i Ry. Preset Sakho Bachau Lipsky D. Ray Che (K,T) A N Nll S+1 L π results ad al. ad al. ad al. -E -E -E -E -E -E 3(1,0) + s3s 1 S e , (1,0) + s4s 1 S e (1,0) + s5s 1 S e (1,0) + s6s 1 S e (1,0) + s7s 1 S e (1,0) + s8s 1 S e (1,0) + s9s 1 S e (1,0) + s10s 1 S e (,0) + 3 3s4s 1 S e (,0) + 3 3s5s 1 S e (,0) + 3 3s6s 1 S e (,0) + 3 3s7s 1 S e (,0) + 3 3s8s 1 S e (,0) + 3 3s9s 1 S e (,0) + 3 3s10s 1 S e e 3 e Table V: Compariso of doubly excited (1, 0) S ad (,0) 3 S (=3-10) states of helium-like ios (Z=) with other theoretical results. Eergies E are i Ry. Preset Sakho Bachau Ivaov ad Lipsky Che (K,T) A N Nll S+1 L π results ad al. Safroova ad al. -E -E -E -E -E -E 3(1,0) - s3s 3 S e (1,0) - s4s 3 S e (1,0) - s5s 3 S e (1,0) - s6s 3 S e (1,0) - s7s 3 S e (1,0) - s8s 3 S e (1,0) - s9s 3 S e (1,0) - s10s 3 S e (,0) - 3 3s4s 3 S e (,0) - 3 3s5s 3 S e (,0) - 3 3s6s 3 S e (,0) - 3 3s7s 3 S e (,0) - 3 3s8s 3 S e (,0) - 3 3s9s 3 S e (,0) - 3 3s10s 3 S e Table VI: Compariso of excitatio eergy of some states of He I, Li II, Be III ad B IV with some experimet observatios. Both resoace positios are expressed i ev ad the positios are measured from the groud state of He I (E = eV), Li II (E = eV), Be III (E = eV) ad B IV (E = eV).The ifiite Rydberg uit (1Ry = eV) is used for eergy coversio. (K,T) A N 3(1,0) + 4(1,0) + 5(1,0) + 6(1,0) + Nll S+1 L π s3s 1 S e s4s 1 S e s5s 1 S e s6s 1 S e Received: September, 01 He I E[ preset work] E [14] E [13] Li II E[ preset work] E [16] E [15] Be III E[ preset work] B IV E[ preset work]