Radiative Lifetimes of Rydberg States in Neutral Gallium

Transcription

1 Vol. 115 (2009) ACTA PHYSICA POLONICA A No. 3 Radiative Lifetimes of Rydberg States i Neutral Gallium M. Yildiz, G. Çelik ad H.Ş. Kiliç Departmet of Physics, Faculty of Arts ad Sciece, Selçuk Uiversity, Campus Koya, Turkey (Received March 7, 2008; revised versio December 2, 2008) Radiative lifetimes were calculated usig weakest boud electro potetial model theory for 4s 2 s 2 S 1/2 ( 7), 4s 2 p 2 P 0 1/2 ( 5), 4s 2 p 2 P 0 3/2 ( 6), 4s 2 d 2 D 0 3/2 ( 6), 4s 2 d 2 D 0 5/2 ( 6) series i eutral gallium. The use of the quatum defect theory ad Marti s expressios allowed us to supply lifetime values alog by meas of the series above. The obtaied i this work preseted good agreemet with theoretical ad experimetal values. Moreover, some lifetime values ot existig i the literature for highly excited Rydberg states i gallium atom were obtaied usig this method. PACS umbers: Cs 1. Itroductio With the advet of the better laser systems ad other optical devices, spectroscopic parameters such as trasitio probabilities ad atomic lifetimes ca be studied i highly excited atomic ad ioic systems particularly o the high-lyig Rydberg states. The determiatio of spectroscopic data for eutral ad ioized atomic systems has bee a active research area i astrophysics, plasma physics, laser physics ad thermouclear fusio research. It is importat to kow the lifetime of highly excited Rydberg atoms, which is a parameter that ca be compared to existig theories i the literature. Durig the last two decades, the most of researchers have studied the trasitio probabilities, eergy levels, oscillator stregths ad lifetimes of excited states belogig to gallium beig i group-three elemets havig a groud state cofiguratio of 4s 2 of 4p [1 5]. Especially, Zheg et al. have calculated eergy values usig fit procedure i gallium atom. They have cosidered four coefficiets i Marti s expressio [5]. Particularly, lifetimes research i high Rydberg states is importat, because the exact lifetime measure are used i astrophysics, laser physics, physical chemistry [6, 7]. The most of ivestigatios are limited to few low excitatio radiative lifetimes ( 8 12). May of the moder experimetal techiques still ecouter difficulties i the exact measuremet of spectroscopic parameters such as oscillator stregths, trasitio probabilities ad lifetimes. The choices of theoretical methods for calculatio of the spectroscopic data i may electro atoms ad heavy ios are ot obvious ad ot so easy. The problem ca be solved more or less exactly for systems havig few electros such as helium. More correspodig author; gulteki@selcuk.edu.tr geeral multi-electro systems caot be treated with such precisio. It is impossible yet to solve may electro systems without imposig severe approximatios. May theoretical methods exist for calculatio of spectroscopic parameters for atomic or ioic systems such as the Hartree Fock approximatios, cofiguratio iteractio methods, semi-empirical methods ad may-body perturbatio theories. The calculatio of some physical parameters correspodig to idividual eergy levels, especially highly lyig levels ad the Rydberg states are always difficult problems i theoretical studies, because of idistiguishability of equivalet electros ad the ecessity of takig ito accout may cofiguratios or orbital basis-set fuctios [8]. Therefore, semi-empirical methods where oe or more parameters are eeded to be adjusted accordig to the existig experimetal data ca be useful [9 11]. I the preset paper, we have studied the lifetimes of some high Rydberg levels i eutral gallium usig the weakest boud electro potetial model theory (WBEPMT). The were listed here for atoms havig the pricipal quatum umber up to = 50. Our theoretical procedure is described i Sect. 2, we preset ew spectroscopic. Coclusive remarks are give i Sect The method The WBEPMT has bee developed [12, 13] ad applied for determiatio of some physical parameters i may-electro atomic ad ioic systems by Zheg et al. [14 19]. Zheg suggested a ew model potetial to describe the electroic motio i multi-electro systems ad separated electros withi a give system ito two groups, the weakest boud electro (WBE) ad o- -weakest boud electros (NWBE). The WBE i may- (641)

2 642 M. Yildiz et al. -electro systems is also the electro which ca most easily be excited or ioized. Accordig to the WBEPM theory, the WBE moves i the potetial field produced by the ucleus ad the o-weakest boud electro. This potetial field ca be divided ito two parts, oe of which is the Coulomb potetial. The secod part is the dipole potetial. This dipole momet affects the behavior of the WBE. It would make the pricipal quatum umber ad the agular mometum quatum umber l of the WBE to be the effective pricipal quatum umber ad effective agular mometum quatum umber l. The itroductio of d effectively modifies the iteger quatum umbers ad l ito o-itegers ad l. These terms differ from the usual core polarizatio potetial which behaves as 1/r 4 asymptotically, i that the effective dipole momet of the core is used as a parameter rather tha beig derived from the polarizability of the core i the electric field of the WBE. Some atomic or ioic properties i multi-electro systems such as trasitio, excitatio ad ioizatio may be referred to weakest boud electro s behavior. The treatmet of WBE gives some accurate iformatio about these properties [12 19]. I the WBEPM theory, the Schrödiger equatio of the weakest boud electro uder o-relativistic approximatio is give as [5, 17, 18]: [ 1 ] V (r i ) ψ i = ε i ψ i, (1) V (r i ) = Z d(d + 1) + 2dl +. (2) r i 2r 2 i I Eq. (2), V (r i ) is potetial fuctio produced by the o-weakest boud electros ad ucleus. Moreover, Z is the effective uclear charge, r i is the distace betwee the weakest boud electro i ad ucleus. I this theory, electroic radial wave fuctios are preseted as a fuctio of the Laguerre polyomial i terms of some parameters [5, 17, 18]: ( ) 2Z l +3/2 [ 2 Γ ( + l ] 1/2 + 1) R l (r) = exp ( Z r ) r l L 2l +1 l 1 ( ( l 1)! 2Z r ). (3) ad l parameters have bee give to be = + d ad l = l + d. (4) The eergy expressio of weakest boud electro is give as ε = Z 2 2 2, (5) where both Z ad d are ukow parameters. This complicatio could be removed makig Eq. (5) similar to quatum defect theory. The eergy expressio i the quatum defect theory is [5, 20]: ε = Z2 et 2 2, (6) where = δ ad δ is quatum defect. δ is approximately costat for a give fixed orbital quatum umber. Z et is the io core charge. For a eutral atom Z et is 1 ad for a sigly charged io Z et is 2 etc. For eutral atoms, eergy eigevalue ca be give as 1 ε() = 2( δ) 2, (7) where is the pricipal quatum umber. Also, δ quatity i Eq. (7) ca be determied by Marti s expressio [5, 21]: δ = a + b( δ 0 ) 2 + c( δ 0 ) 4 + d( δ 0 ) 6 +e( δ 0 ) 8, (8) where δ 0 is quatum defect of lowest eergy state i the give Rydberg series ad it is a costat. Coefficiets a, b, c, d, e ca be determied from the first four experimetal values by solvig Eq. (7) ad Eq. (8). The eergy levels of high Rydberg states ca be obtaied by extrapolatig with these values. Marti has bee applied to sodium atom for aalysis of spectra ad the have bee obtaied i high precisio. The lifetime of excited levels for may electro atomic systems ca be calculated usig the followig formula [5, 22]: τ = τ 0 ( ) α, (9) where the parameters τ 0 ad α are coefficiets i relevat series. Accordig to the WBEPM theory, these coefficiets will be determied usig the experimetal values of eergy ad lifetime. 3. Results ad coclusios The coefficiets a, b, c, d, e i Eq. (8) ad the values of δ 0 belogig to differet series are fitted to experimetal eergy values [23] i the gallium atom ad these are give i Table I ad Table II. I this paper, we have cosidered five coefficiets by addig oe coefficiet to the coefficiets cosidered by Zheg et al. [5] i Marti s expressio to determie eergy values. τ 0 ad α coefficiets have bee calculated usig the WBEPM theory. We have calculated the values of eergy levels i the gallium atom ad compared them with the experimetal values take from NIST. These are listed i Tables III VII. For some levels, where o documeted values exist, our calculated values have ot bee compared. Eve though i some cases the deviatios are 5 6 cm 1, it ca be see that most of the deviatios are less tha 1 cm 1 i cosiderig all data. Radiative lifetime values of six Rydberg series 4s 2 s 2 S 1/2 ( 7), 4s 2 p 2 P1/2 0 ( 5), 4s2 p 2 P3/2 0 ( 6), 4s2 d 2 D3/2 0 ( 6), 4s2 d 2 D5/2 0 ( 6) have bee calculated i Ga(I). Available experimetal ad theoretical data are quite limited for high Rydberg states. Therefore, our lifetime obtaied from the WBEPM theory have bee compared with relativistic may-body perturbatio theory () give by Safroova et al. [24] ad the of Carlsso et al. icludig both experimetal ad theoretical data [25].

3 Radiative Lifetimes of Rydberg States i Neutral Gallium 643 Spectral coefficiets of eergy level series for Ga(I). TABLE I Series a b c d e δ 0 4s 2 s 2 S 1/2 ( 7) s 2 p 2 P 0 1/2 ( 5) s 2 p 2 P 0 3/2 ( 6) s 2 d 2 D 0 3/2 ( 6) s 2 d 2 D 0 5/2 ( 6) The coefficiets of lifetime for Ga(I). TABLE II Series τ 0 α 4s 2 s 2 S 1/2 ( 7) s 2 p 2 P 0 1/2 ( 5) s 2 p 2 P 0 3/2 ( 6) s 2 d 2 D 0 3/2 ( 6) s 2 d 2 D 0 5/2 ( 6) Eergy values ad radiative lifetimes for Ga(I): 4s 2 s 2 S 1/2 ( 7) series. TABLE III τ cal [s]

4 644 M. Yildiz et al. Eergy values ad radiative lifetimes for Ga(I): 4s 2 p 2 P 0 1/2 ( 5) series. TABLE IV τ cal [s] τ exp [s] TABLE V Eergy values ad radiative lifetimes for Ga(I): 4s 2 p 2 P 0 3/2 ( 6) series. τ cal [s]

5 Radiative Lifetimes of Rydberg States i Neutral Gallium 645 Eergy values ad radiative lifetimes for Ga(I): 4s 2 d 2 D 0 3/2 ( 6) series. τ cal [s] τ exp [s] TABLE VI TABLE VII Eergy values ad radiative lifetimes for Ga(I): 4s 2 d 2 D 0 5/2 ( 6) series. τ cal [s] τ exp [s]

6 646 M. Yildiz et al. Carlsso et al. employed experimetal laser spectroscopic techiques. It is kow that the measurig of radiative lifetimes is ormally performed at room temperature. I this case, there will be a backgroud field due to preset black-body i the measurig volume [25]. This field is very weak at optical frequecies, but as oe goes ito the ifrared regio it will icrease i stregth. Sice the frequecies for trasitios betwee differet high-lyig Rydberg states are ofte i the ifrared regio, trasitios will be iduced by this black- -body backgroud field. The measured values will be shorter tha the values measured without the black-body iduced trasitios. Therefore, some estimates ad correctios were doe for this effect by Carlsso et al. These values are preseted i our tables. At the same paper, Carlsso et al. used the multi-cofiguratioal Hartree Fock () wave fuctios to calculate lifetime values i a froze-core approximatio [25]. Their theoretical are give as a comparable data i Tables III VII. While the calculatio procedure for the systems with a few electros ca be carried out easily, the calculatios have become more difficult ad complex i the case of icreasig umber of electros. Especially, for the excited states ad Rydberg states of may-electro systems, more cofiguratios must be cosidered. Therefore, calculatios become more complicated. Because of the difficulties metioed above, the theoretical ad experimetal studies geerally cosiders low lyig states rather tha highly excited states. It ca be see from tables that our are very close to the correspodig theoretical ad experimetal. These prove that Marti s expressio is coveiet for the Rydberg series of Ga(I). The WBEPM theory has a simple calculatio procedure. It ca be used to calculate the lifetimes for both highly excited states ad low lyig states without ay icrease i complexity i calculatio process. Previously, we have employed the WBEPM theory for several physical parameters such as trasitio probabilities, oscillator stregths ad ioizatio potetials i may-electro atoms. We have obtaied very satisfactory [26 30]. I this study, by courtesy of this method, we have calculated higher lifetimes tha published i the literature i Ga(I). Ackowledgmets The authors gratefully ackowledge the Selçuk Uiversity Scietific Research Projects (BAP) Coordiatig Office for support. Project No: Refereces [1] G. Jösso, H. Ludberg, S. Svaberg, Phys. Rev. A 27, 2930 (1983). [2] G. Jösso, H. Ludberg, Z. Phys. A Atoms Nuclei 31, 151 (1983). [3] G. Jösso, C. Leviso, S. Svaberg, C.G. Wahlström, Phys. Lett. A 93, 121 (1983). [4] G. Jösso, C. Leviso, I. Lidgre, S. Svaberg, C.G. Wahlström, Z. Phys. A Atoms Nuclei 322, 351 (1983). [5] N.W. Zheg, Z.G. Li, D.X. Ma, T. Zhou, J. Fa, Ca. J. Phys. 82, 523 (2004). [6] E. Biemot, P. Palmeri, P. Quiet, Z. Dai, S. Swaberg, H.L. Xu, J. Phys. B, At. Mol. Opt. Phys. 38, 3547 (2005). [7] W.O. Youis, S.H. Allam, Th.M.E. Sherbii, At. Data Nucl. Data Tables 92, 187 (2006). [8] L.L. Shimo, N.M. Erdevdi, Opt. Spectrosc. 42, 137 (1977). [9] C. Zhou, L. Liag, L. Zhag, Chi. Opt. Lett. 6, 161 (2008). [10] C. Zhou, L. Liag, L. Zhag, Chi. Opt. Lett. 5, 438 (2007). [11] N.W. Zheg, Z.Q. Li, J. Fa, D.X. Ma, T. Zhou, J. Phys. Soc. Japa 71, 2677 (2002). [12] N.W. Zheg, A New Outlie of Atomic Theory, Jiag Su Educatio Press, Najig [13] N.W. Zheg, Chi. Sci. Bull. 22, 531 (1977); 33, 916 (1988). [14] N.W. Zheg, T. Wag, D.X. Ma, T. Zhou, J. Fa, It. J. Quat. Chem. 98, 281 (2004). [15] N.W. Zheg, T. Wag, Chem. Phys. 282, 31 (2002). [16] N.W. Zheg, T. Wag, R.Y. Yag, J. Chem. Phys. 113, 6169 (2000). [17] N.W. Zheg, T. Wag, T. Zhou, D.X. Ma, J. Phys. Soc. Japa 71, 1672 (2002). [18] N.W. Zheg, Y.J. Su, Sci. Chia Ser. B 30, 15 (2000). [19] N.W. Zheg, T. Wag, Astrophys. J. Suppl. Ser. 143, 231 (2002). [20] D.R. Bates, A. Damgaard, Philos. Tras. A 242, 101 (1949). [21] W.C. Marti, J. Opt. Soc. Am. 70, 784 (1980). [22] O.V. Rykova, Y.F. Veroaliee, Opt. Spectrosc. 1, 23 (1994). [23] NIST Atomic Spectra Database, Physical Referece Data. [24] U.I. Safroova, T.E. Cowa, M.S. Safroova J. Phys. B, At. Mol. Opt. Phys. 39, 749 (2006). [25] J. Carlsso, H. Ludberg, W.X. Peg, A. Persso, C.G. Wahlström, Z. Phys. D, At. Mol. Clust. 3, 345 (1986). [26] G. Çelik, E. Akı, H.Ş. Kılıç, Eur. Phys. J. D 40, 325 (2006). [27] G. Çelik, H.Ş. Kılıç, E. Akı, Tur. J. Phys. 30, 165 (2006). [28] G. Çelik, J. Quat. Spectrosc. Radiat. Trasfer 103, 578 (2007). [29] G. Çelik, E. Akı, H.Ş. Kılıç, It. J. Quat. Chem. 107, 495 (2007). [30] G. Çelik, M. Yıldız, H.Ş. Kılıç, Acta Pol. Phys. A 112, 485 (2007).