Calculations of Energy Levels Using the Weakest Bound Electron Potential Model Theory

Transcription

1 Vol. 115 (2009) ACTA PHYSICA POLONICA A No. 3 Calculatons of Energy Levels Usng the Weakest Bound Electron Potental Model Theory T.-Y. hang and N.-W. heng Department of Chemstry, Unversty of Scence and Technology of Chna Hefe, Anhu , P.R. Chna (Receved November 29, 2008) In ths paper, we ntroduce a new method for calculaton of energy levels n detal and gve our for several so-spectrum-level seres as examples: [He] 2s2p 1 P 1, [He] 2s2p 3 P 0, [He] 2s2p 3 P 2, and [He] 2s3s 3 S 1 seres of Be-lke sequence; [Ne] 3s 2 3d 2 D 3/2 seres and [Ne] 3s 2 3d 2 D 5/2 seres of Al-lke sequences; [Ne] 4p 2 P 1/2 seres, [Ne] 5d 2 D 5/2 seres, and [Ne] 6f 2 F 7/2 seres of Na-lke sequences. In the method I() = T lm () T (, n), where I(), T lm (), and T (, n) denote onzaton potental, seres lmt, and energy level of a gven member, respectvely. The expresson of non-relatvstc part of I() s derved from weakest bound electon potental model theory and relatvstc effects of I() are ncluded by usng a sx-order polynomal n. Our are compared wth the expermental data and wth those obtaned by other theoretcal method. PACS numbers: b, p 1. Introducton The energy levels are wdely appled n many felds, such as plasma dagnoss, astrophyscs, laser development, and analytcal chemstry. Therefore, more and more attenton has been pad on calculatons of hgh- -precson energy levels of atoms and ons. Many theoretcal methods have been developed n ths feld: multconfguraton Hartree Fock (MCHF) method [1 3], multconfguraton Drac Hartree Fock (MCDHF) method [1 3], confguraton nteracton (CI) method [4 7], relatvstc many-body perturbaton theory (R) [8 14], and weakest bound electon potental model (WBEPM) theory [15 23], etc. In recent years, many excellent works have been done by theoretcal researchers. The MCHF method has been wdely used to calculate propertes of atoms and ons ncludng energy levels. Irma and Fscher reported energy levels and transton probabltes of neutral argon usng both MCHF method and MCDHF method [1]. Fscher and Tachev also appled MCHF method to the study of energy levels and lfetmes for the Be-lke to Ne-lke sequences [2]. Two years later, the same group reported an extenson of ths work to the sequences wth electrons [3]. In ther calculatons, the MCHF approach was used for obtanng the best radal functons for the nteractng terms and the relatvstc effects were ncluded through the Bret Paul approxmaton. Dzuba et al. used both Hartree Fock (HF) and CI method to study the energy levels and lfetmes of Nd(IV), Pm(IV), Sm(IV), and Eu(IV) [4]. They also reported energy levels of Ge, Sn, Pb [5] and barum and radum [6, 7]. They started ther calculatons from the relatvstc Hatree correspondng author; e-mal: nwzheng@ustc.edu.cn Fock method and used CI method to treat the nteracton between valence electrons. Another powerful method n ths feld s R. Yb-, Al-, Ga-lke ons have been nvestgated by ths method [8 10]. Energy levels of slver, beryllum, magnesum, francum and znc soelectronc sequences were also studed by ths method [11 14]. The core valence correlatons are ncluded beyond the secondor thrd-order of R. Snce the WBEPM theory was proposed [15], many studes have been performed to calculate energy levels for atoms and ons [16 24]. In these works, the authors used the concept of the spectrum- -level-lke seres and a new formula takng perturbaton nto account for calculaton. Excellent are obtaned n those works. Devatons between the theoretcal and expermental values are generally smaller than 1 cm 1. The purpose of ths paper s to ntroduce another new method to calculate energy levels proposed by one of authors (N.W.), recently [25]. After ntroducng the concept of so-spectrum-level seres, we gve the of several so-spectrum-level seres ncludng Be-, Al-, and Na-lke sequence as examples. Accordng to the WBEPM theory, the expresson of nonrelatvstc onzaton potental s wrtten as a functon wth two-order n. The relatvstc effects are ncluded by usng a sx-order polynomal n. We compare our wth the expermental data and wth those obtaned by other theoretcal method. 2. The concepton of the so-spectrum-level seres In our prevous works, the concepton of the so- -spectrum-level seres has already been proposed and used to calculate the onzaton potental [26 29]. In order to descrbe our method clearly, we ntroduce the concepton of the so-spectrum-level seres frst. The concepton of so-electronc seres s usually used n the study (629)

2 630 T.-Y. hang, N.-W. heng for the onzaton energes of ground state. All members (atom and ons) n an soelectronc seres have the same electron confguraton. As a result, the concepton of soelectronc seres s assocated wth the electron confguraton and cannot provde the nformaton relatng to the terms or energy levels. However, for the study of ground state, all systems le n the lowest energy state. Therefore, the concepton of soelectronc seres s convenent to nvestgate the regulartes of onzaton energes of ground state. But for the atoms or ons wth excted electrons, each electron confguraton usually gves rse to several terms and each term splts nto several spectral levels further. Under ths stuaton, the concepton of so-electronc seres s too rough to study the energes. Therefore, we ntroduce the concepton of the so- -spectrum-level seres study of the general regulatons of excted states of atoms. An so-spectrum-level seres s a seres of energy levels that s composed of energy levels wth the same spectral level symbol n a gven so- -electronc seres. For example, Be(I) ([He] 2s3p 3 P 1 ), B(II) ([He] 2s3p 3 P 1 ),..., O(V) ([He] 2s3p 3 P 1 ),..., make up an so-spectrum-level seres named Be(I) ([He] 2s3p 3 P 1 ). From the defnton, n a gven so-spectrum- -level seres, not only the electron confguraton but also the spectrum energy levels s defned. The only varable parameter s the nuclear charge. Therefore, the onzaton energy of WBE n a gven so-spectrum-level seres could be approxmate to a functon of nuclear charge. 3. Theory and method WBEPM theory s based on the consderaton of the dynamc successve onzaton, the choce of zero energy n quantum mechancs and the separaton of the weakest bound electron (WBE) and non-weakest bound electrons (NWBE) [15, 16]. A free partcle wth N electrons and a nucleus of charge +e n ts ground state can gve rse to N stages of onzaton. The onzed speces n N successve onzaton stages are, respectvely, neutral atom, unpostve on,..., +( 1) on. The concepton of WBE s referred to the defnton of the onzaton of a free partcle: the onzaton potental for a free partcle s defned as the energy requred completely to move the WBE from the partcle. Therefore, we classfed the electrons n an atom or on system nto two types, WBE and NWBE. The WBE s the electron whch s most weakly bound to the system and excted or onzed frst, the rest electrons are called NWBE. In terms of exctaton or onzaton, the WBE n a gven system dffers from the rest of the electrons n behavor. We can separate the WBE and the NWBE, and the problem of the WBE can be treated as a one-electron problem. Each electron n the N-electrons system acts sooner or later as a WBE n the onzaton procedure. By removng the frst, the second,... N-th WBE, an N-electrons atom can gve N stages of onzaton. Each stage of successve onzaton processes corresponds to the removal of a WBE from the related subsystem. Accurate treatment of the WBE can provde accurate knowledge of atomc and onc propertes. As the onzaton energy s defned as the energy requred completely removng the weakest bound electron from an atom or an on n ts ground or excted states, the energy of a level n spectrum-level-lke seres [17 24] can be wrtten as T (n) = T lm I exp, (1) where T lm s the onzaton lmt for a spectrum-level-lke seres. I exp s the onzaton energy of WBE. In order to get the value of T (n) the value of I exp s needed. In ths work, we use the theoretcal value of onzaton energy, I cal, to replace the I exp. Therefore, we get T (n) = T lm I cal. (2) We dvded approxmately the onzaton energy nto nonrelatvstc part and relatvstc part [30] I cal = I nr + I r, (3) where I nr represents the nonrelatvstc energy and I r represents relatvstc energy. Now we employ the WBEPM theory to calculate the nonrelatvstc energy I nr. Accordng to WBEPM theory [15, 16], the Schrödnger equaton of WBE s [ 1 ] V (r ) ϕ = ε ϕ. (4) In WBEPM theory the potental functon V (r ) n Eq. (4) may be wrtten as (n atomc unts) V (r ) = d(d + 1) + 2dl +, (5) r 2r 2 where s the effectve nuclear charge, l s the angular quantum number of WBE, and r s the dstance between the WBE and the nucleus. Parameter d s ntroduced to modfy the ntegral quantum number n and angular quantum number l nto nonntegral n and l. Substtutng Eq. (5) nto Eq. (4) and solvng the Schrödnger equaton of the WBE, we can obtan the followng expressons of energy egenvalue and the radal functon: ε = 2 (6) 2n 2 and R = C exp ( ) r r l L 2l +1 n l 1 n ( 2 r n ), (7) where n s the effectve prncpal quantum number wth n = n + d, l s the effectve angular quantum number wth l = l + d, C s the normalzaton factor, and L 2l +1 n l 1 ( 2 r n ) s the generalzed Laguerre polynomal. Because Eq. (4) s the non-relatvstc one-electron Schrödnger equaton, the energy egenvalue of WBE obtaned from Eq. (4) negatve value of the non-relatvstc part of onzaton energy, I nr : I nr = ε = 2. (8) 2n 2 For an so-spectrum-level seres we can wrte Eq. (1)

3 Calculatons of Energy Levels Usng the Weakest Bound Electron nto T (, n) = T lm () I exp (), (9) and the effectve nuclear charge was proposed as a functon concernng nuclear charge [15], that s = ( σ) 2 + g( 0 ), (10) where 0 s the nuclear charge of the frst member n an so-spectrum-level seres, for example, 0 s the nuclear charge of the Be atom for so-spectrum-level seres Be(I) ([He] 2s3p 3 P 1 ), σ s the screenng constant of the frst member, and relatvely ncrease factor g s a parameter that ndcates the effect on the effectve nuclear charge due to the ncrease n the nuclear charge n seres. The non-relatvstc onzaton energy can be wrtten as I nr () = ( σ)2 + g( 0 ) 2n 2. (11) In order to obtan the values of parameters n, σ, and g, we consdered the frst dfference of the non-relatvstc onzaton potental. In an so-spectrum-level seres, the plot of the frst dfference of the onzaton potental, I nr = I nr ( + 1) I nr (), vs. nuclear charge, would be a straght lne. Because the relatvstc part of onzaton potental s qute small, the plot of the frst dfferences of the expermental onzaton potental I exp = I exp ( + 1) I exp () vs. nuclear charge could be approxmated to the plot of the frst dfference of the onzaton potental, I nr = I nr ( + 1) I nr (), vs. nuclear charge. In ths work, the expermental data are taken from Ref. [31]. Therefore, the effectve prncple quantum number n can be treated as a constant approxmately, and can be obtaned from the plot of I exp and nuclear charge. The screenng constant of the frst member σ and the relatvely ncrease factor g can be calculated later. Therefore, the energy levels can be obtaned from the followng equaton: T (, n) T lm () ( σ)2 + g( 0 ) 2n 2. (12) As 2 s the hghest power of Eq. (12), the relatvstc effects such as mass velocty, the Darwn term, and the spn orbt term cannot be ncluded completely n Eq. (12). In order to get more accurate energes, the relatvstc effects must be taken nto account. As mentoned above, n a gven so-spectrum-level seres, the only varable parameter s the nuclear charge. Therefore, n ths work, the relatvstc energes are presented as a unvarant functon of a sx-order polynomal n nuclear charge 6 I r () = a. (13) 0 We consdered the devatons between expermental onzaton potental I exp () and the onzaton potental I nr () calculated from Eq. (11) are equal to the relatvstc part I r (). Here expermental values of onzaton potental are taken from NIST data base [31]. By fttng devatons I exp () I nr () to Eq. (13), we can obtan the values of coeffcents a ( = 0 6). The theoretcal value of onzaton energy can be expressed as the followng equaton: I cal () = ( σ)2 + g( 0 ) 2n a. (14) Then for an so-spectrum-level seres, we obtan T (n) = T lm [ ] ( σ) 2 + g( 0 ) 6 2n 2 + a. (15) As the s determned for a gven member, Eq. (14) reduces as Eq. (2) and one can calculate T (n) through I cal and T lm. 4. Results and dscusson We studed the energy levels of so-spectrum-level seres along Be-, Al-, and Na-lke sequences usng the method mentoned above. The parameters needed n Eq. (15) are lsted n Tables I II, and the of ths work are lsted n Tables III IX. 0 0 TABLE I Parameters of Eq. (14) for [He] 2s2p 1 P 1; [He] 2s2p 3 P 0; [He] 2s2p 3 P 2; and [He] 2s3s 3 S 1 Parameters [He] 2s2p 1 P 1 [He] 2s2p 3 P 0 [He] 2s2p 3 P 2 [He] 2s3s 3 S 1 σ g n a a a a a a a

4 632 T.-Y. hang, N.-W. heng TABLE II Parameters of Eq. (14) for [Ne] 3s 2 3d 2 D 3/2 seres, [Ne] 3s 2 3d 2 D 5/2 seres, and [Ne] 3s 2 3d 2 D 5/2 seres of Al-lke sequences and [Ne] 4p 2 P 1/2 seres, [Ne] 5d 2 D 5/2 seres, and [Ne] 6f 2 F 7/2 seres of Na-lke sequences. Parameters Al-lke sequences Na-lke sequences [Ne] 3s 2 3d 2 D 3/2 [Ne] 3s 2 3d 2 D 5/2 [Ne] 4p 2 P 1/2 [Ne] 5d 2 D 5/2 [Ne] 6f 2 F 7/2 σ g n a a a a a a a TABLE III Energy levels [cm 1 ] for the [He] 2s2p 1 P 1 method [32] TABLE IV Energy levels [cm 1 ] for the [He] 2s2p 3 P 0 method [32]

5 Calculatons of Energy Levels Usng the Weakest Bound Electron TABLE V Energy levels [cm 1 ] for the [He] 2s2p 3 P 2 method [32] Table I presents the parameters for the Be-lke seres ncludng [He] 2s2p 1 P 1 ; [He] 2s2p 3 P 0 ; [He] 2s2p 3 P 2 ; and [He] 2s3s 3 S 1. For these seres, the expermental data from = 4 to = 16 are used to obtan the parameters n, σ, and g. The expermental data are taken from Ref. [31]. The effectve prncple quantum number n s obtaned from the frst dfferences of I exp (). σ s the screenng constant of the frst member of a gven so- -spectrum-level seres, and we calculated t from Eq. (12). g s called relatve ncrease factor whch ndcates the effect on the effectve nuclear charge. Each member n a so-spectrum-level seres s used to obtan the relatve g, and g s the arthmetcal average of the g. When obtaned the parameters a ( = 0 6) from the least- -squares fttng, = 20, 22, 24 are added for [He] 2s2p 1 P 1 seres; = 22, 24 are added for [He] 2s2p 3 P 0 seres; = 22, 23 are added for [He] 2s2p 3 P 2 seres; and = 22, 24 are added for [He] 2s3s 3 S 1 seres. Results of these Be-lke seres are lsted n Tables III VI. All the energes are gven n cm 1 unt. We compared our wth the expermental data and those obtaned by Safronova et al. [32, 33]. In each table, column 1 s the nuclear charge of every member of a gven so-spectrumlevel seres. Column 2 lsts the expermental data taken from the NIST data base [31]. The NIST collected accepted data were obtaned by dfferent authors all around the world and crtcally evaluated the relablty of those TABLE VI Energy levels [cm 1 ] for the [He] 2s3s 3 S method [33] TABLE VII Energy levels [cm 1 ] for the [Ne] 3s 2 3d 2 D 3/2 seres of Al-lke sequences method [9]

6 634 T.-Y. hang, N.-W. heng TABLE VIII Energy levels [cm 1 ] for the [Ne] 3s 2 3d 2 D 5/2 seres of Al-lke sequences method [9] TABLE IX Energy levels [cm 1 ] for the [Ne] 4p 2 P 1/2 seres; [Ne] 5d 2 D 5/2 seres; and [Ne] 6f 2 F 7/2 seres of Na-lke sequences. [Ne] 4p 2 P 1/2 [Ne] 5d 2 D 5/2 [Ne] 6f 2 F 7/2 T exp [31] T cal T exp [31] T cal T exp [31] T cal data. Therefore, we took the data taken from the NIST data base for comparson. Column 3 lsts the obtaned usng WBEPM theory. The obtaned by Safronova et al. are contaned n column 4. Usng the R, Ref. [32] reported the energy levels of n = 2 states of berylum-lke ons wth nuclear charges rangng from = [32]. In ther calculatons, both the Coulomb nteracton and the Bret Coulomb nteracton are carred out to the second order. In the next year, the same group extended the theory to study the 2l3l states of berylum-lke ons. 16 even-party (2s3s, 2p3p, 2s3d) excted states and 20 odd-party (2s3p, 2p3s, 2p3d) excted states were studed n ther work [33]. In general, ther are n good agreement wth expermental data. Most devatons are smaller than 1000 cm 1, when several of them are cm 1 and only a few are more than 4000 cm 1. The comparson shows that our are at the same level wth Safronova s ones. Table II gves the parameters for the two Al-lke seres and three Na-lke seres. The energy levels of these fve seres are lsted n Tables VII IX. For two Al-lke seres, the expermental data from = 13 to = 22 are used to obtan the parameters n, σ, and g. When we obtaned the parameters a ( = 0 6) from the least-squares fttng, = 23, 26, 29 are added for [Ne] 3s 2 3d 2 D 3/2 seres and = 24, 27, 28 are added for [Ne] 3s 2 3d 2 D 5/2 seres. The taken from Ref. [9] are gven for comparson. For three Na-lke seres, the expermental data from = 11 to = 17 are used to obtan the parameters n, σ, and g. When obtaned the parameters a ( = 0 6) from the least-squares fttng, = 21, 27, 30 are added for [Ne] 4p 2 P 1/2 seres; = 22, 26, 29 are added for [Ne] 5d 2 D 5/2 seres; and = 22, 27, 29 are added for [Ne] 6f 2 F 7/2 seres. The man source of error n the present method s the ncomplete treatment of relatvstc effects. We fnd that the devatons between nonrelatvstc energes and the expermental data show an ncreasng trend wth the ncrease n along the so-spectrum-level seres. That s because the relatvstc effects on the radal wave functons become more evdent for hgh. Although the relatve devatons caused by relatvstc effects are very small, the absolute devatons wll be lttle large ones. In present work, n order to mprove the accuracy of our, the relatvstc correctons of an so-spectrum-level seres are ncluded by a sx-order polynomal n. By ncludng the relatvstc effects, devatons between our and the expermental data become much smaller. However, ths treatment s successful at present calculaton. But for ons wth much hgher nuclear charge, the more precse consderaton of relatvstc effects s necessary. In concluson, employng the WBEPM theory, we calculated the energes for Be-, Al-, and Na-lke sequence. Equaton (11) s derved to calculate the non-relatvstc energes, and the relatvstc correctons are taken nto account by a sx-order polynomal n nuclear charge.

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