Gi ant Res o nance in 4d Photo-Ionization and -Ex ci ta tion of High Z Atoms and Ions; Sys tem atic Study of Dy namic Elec tron-correlation

Jour nal of the Chi nese Chem i cal So ci ety, 2001, 48, 499-504 499 Gi ant Res o nance in 4d Photo-Ionization and -Ex ci ta tion of High Z Atoms and Ions; Sys tem atic Study of Dy namic Elec tron-correlation Tsutomu Watanabe ac * ( ), Xiao-Min Tong b ( ), Masashi Sato a ( ) and Shunsuke Ohtani bc ( ) a Di vi sion of Nat u ral Sci ences, In ter na tional Chris tian Uni ver sity, Ohsawa, Mitaka-shi, To kyo 118-8585, Ja pan b Cold Trapped Ions Pro ject, ICORP, Ja pan Sci ence & Tech nol ogy Cor po ra tion (JST), Axis Chofu 3F, 1-40-2 Fuda, Chofu, To kyo 182-0024, Ja pan c In sti tute of La ser Sci ence, Uni ver sity of Elec tro-communications, Chofu-shi, To kyo 182-8577, Ja pan For the study of so-called 4d gi ant res o nance of an Xe atom and its iso-electronic se quence, the lin ear re - sponse method with the den sity func tional the ory is ap plied. Par tic u lar at ten tion is made for the 4d photoexcitation pro cess of Xe to gether with the con tin uum spec trum and the ex ci ta tion peaks in the ex per i ment are iden ti fied for the first time. Sys tem atic study for iso-electronic se quence of Xe-like ions and at oms, i.e., I -, Xe, Cs +, Ba ++ and La 3+ are re ported in or der to see the role of the dy namic elec tron cor re la tion to 4d-photoexcitation/ion iza tion pro cesses. I. INTRODUCTION For the photo-absorption mea sure ment of Xe at oms in the pho ton en ergy range of 70-140 ev, a big broad spec trum was ob served, and this spec trum could not ex plained in terms of the first or der per tur ba tion treat ment. This can be ex - plained by many-body per tur ba tion the ory or ran dom phase ap prox i ma tion in many body treat ment, and it can be at trib - uted to the elec tron-electron dy namic cor re la tion dur ing photo-excitation/ion iza tion pro cesses. 1 This spec trum be - came fa mous be cause the shift of en ergy is as large as the or - der of sev eral tenth ev, and the spec trum is com pletely re ar - ranged by dy namic cor re la tion. In this sense, the so-called 4d-giant res o nance in photo-ionization in heavy at oms has been the sub ject of study in atomic phys ics and now a days many ex per i men tal as well as the o ret i cal ap proaches have been pub lished. One of the typ i cal ex am ples is ob served in photo-ionization of 4d elec tron in a neu tral Xe atom and these phe nom ena have been in ves ti gated for a long time. 1,2 Whereas the gi ant res o nance in a neu tral atomic sys tem has been ex - ten sively stud ied, some 4d photo-excitation peaks in Xe at - oms have been mea sured and these ex ci ta tion peaks have not yet been ex plained the o ret i cally. Fur ther more, a sys tem atic study for the iso-electronic se quence of Xe-like ions has not been made the o ret i cally yet. In this pa per we are go ing to em ploy the lin ear re sponse method based on the den sity func tional the ory for the study of 4d photo-excitation/ion iza tion pro cesses. A de tailed de - scrip tion will be made for the cal cu la tion of the Xe photoexcitation/ion iza tion pro cess. The re sults of cal cu la tion will be com pared with ex per i men tal data. We have made the sys - tem atic study of iso-electronic se quence of Xe-like ions from I -, Xe, Cs +, Ba ++ and to La 3+. The in tro duc tion of this sys tem - atic study will be made, be cause the whole sys tem atic ap - proach is go ing to be pre sented some where else. 3 The lin ear re sponse the ory based on the den sity func - tional the ory is the ex pres sion for the elec tron den sity change by an ex ter nal field and its cor re spond ing susceptibility. Usually this susceptibility by ex ter nal dis tur bance should in - clude a dy namic cor re la tion ef fect. This elec tron den sity change ex pres sion can be re writ ten in terms of the suscep - tibility with in de pend ent par ti cle ex pres sion and the self - consistent in ter ac tion which in cludes dy namic cor re la tion ef - fects. In the pres ent case, an ex ter nal dis tur bance is taken to be the di pole pho ton-electron in ter ac tion, and it has been suc - cess fully de vel oped by one of the pres ent au thors and oth - ers. 4,5 The elec tronic tran si tion due to a di pole in ter ac tion of pho tons with an atomic sys tem, and the dy namic elec tron cor - re la tion un der some per turb ing po ten tial is taken into ac count by lo cal field cor rec tions. In Sec. II, we de scribe the the o ret i cal method em ployed in the pres ent prob lem. In Sec. III, we in tro duce the way of prac ti cal cal cu la tion as nu mer i cal pro ce dure and the steps for each ex e cu tion. The re sults of nu mer i cal cal cu la tion for Xe Spe cial is sue for the Fourth Asian In ter na tional Sem i nars on Atomic and Mo lec u lar Phys ics

500 J. Chin. Chem. Soc., Vol. 48, No. 3, 2001 Watanabe et al. and the com par i son with ex per i ments and also the re sults of cal cu la tion for ions of Xe iso-electronic se quence cal cu la - tions are in tro duced. The com par i son with ex per i men tal ob - ser va tion is given. In IV the phys ics of 4d-giant res o nance, i.e. the role of dy namic cor re la tion and its nu clear charge de - pend ence are dis cussed. II. THEORETICAL METHOD When an ex ter nal time de pend ent in ter ac tion V ext (r, t) is ap plied to charged par ti cle sys tem such as an atom, mol e - cule, solid ma te rial and so on the charge den sity change due to this ex ter nal in ter ac tion mul ti plied by the re sponse func - tion (r, r ; t t ) cor re spond ing to this ex ter nal in ter ac tion and in te grated over in ter ac tion time is t ext ( r, t) dt ' dr ' ( r, r '; t t ') V ( r ', t '). If we take the Fou rier trans form of this charge den sity change, we ob tain the fre quency de pend ent form of the charge den sity charge us ing fre quency de pend ent sus cep ti bil ity (r, r ; ) as, ext ( r, ) ( r, r'; ) V ( r ', ) dr '. Usually this sus cep ti bil ity cor re spond ing to the ex ter nal in - ter ac tion in cludes the ef fects of dy namic cor re la tion due the re sponse of ex ter nal in ter ac tion. In the den sity func tional the ory, the fre quency de pend - ent charge den sity of elec trons in the atom in the pres ent case is de fined by the fol low ing form in terms of one elec tron wavefunction i(r, ); N 2 ( r, ) i ( r, ), i where i is an or bital la bel num ber and N is the num ber of elec - trons in the atom or the ions. Equa tion 2 can be re writ ten in an alternative way in terms of in de pend ent par ti cle ex pres sion of susceptibility IPA (r, r ; ) and a self con sis tent in ter ac tion V SCF (r, ) as IPA SCF ( r, ) ( r, r'; ) V ( r, ) dr'. Here, IPA (r, r ; ) is ex pressed in terms of un per turbed one elec tron eigenfunction i(r) and its eigenvalue i as IPA ( r) ( r) ( r ) ( r ) ( r, r '; ) ( ), * * i j j i ni n j i, j ( j i ) i (1) (2) (3) (4) (5) where i is an imag i nary in fin i tes i mal used to en sure the out - go ing wave bound ary con di tions. The self-consistent field V SCF (r, ) can be ob tained by den sity func tional the ory with an op ti mized ef fec tive po ten - tial and a self-interaction cor rec tion. 4,6 It should sat isfy the self-consistent equa tion, SCF ex t ( r, ) V ( r, ) V ( r, ) dr ' r - r V ( r) xc ( r, ), 0 ( r ) (r) where 0(r) is the ground-state elec tron den sity and Vxc(r) is the so-called ex change-correlation po ten tial and ap prox i - mated by lo cal den sity ap prox i ma tion (LDA). 7 The nor mal pro ce dure is to solve equa tions (4) and (6) un til con ver gence is reached. In prac tice, these can be solved by trans form ing them into an in te gral equa tion for V SCF (r, ) and it can be solved by discretization of the r-space. Sub sti tuting the re - sults of V SCF (r, ) into equa tion (4), we can ob tain the in - duced den sity (r, ). In ad di tion, our com puter code is treated sep a rately for up-spin state and for down-spin state con sid er ing the case of open-shell at oms in gen eral. De tailed de scrip tion was given in Ref. 5. The dy namic polarizability un der an ex ter nal field V ext (r, ) is given in terms of sus cep ti bil ity in in de pend ent par ti cle ap prox i ma tion IPA (r, r ; ) and a self-consistent field V SCF (r, ) as ( ) V ext ( r, ) IPA ( r, r '; ) V SCF ( r, ) drdr '. The self con sis tent field V SCF (r, ) in cludes the ex ter nal field and an in duced one due to the re dis tri bu tion of the elec tron den sity per turbed by the ex ter nal field. Af ter ob tain ing V SCF (r, ), the photo-ionization and ex ci ta tion cross sec tions are cal cu lated by us ing the for mula: 4 ( ) [ ( )]. c and com pared with the ex per i men tal data. III. CALCULATION FOR THE XE ATOM AND ITS ISO-ELECTRONIC SEQUENCE We de scribe the way to make a prac ti cal cal cu la tion for photo-excitation/ion iza tion cross sec tion in the case of an Xe atom. In the first stage, we made a cal cu la tion for the ground state as well as the ex cited states of Xe at oms us ing the den - sity func tional en ergy cal cu la tion code. This is the in de pend - (6) (7) (8)

Gi ant Res o nance in Photo-Absorption of High Z Ions J. Chin. Chem. Soc., Vol. 48, No. 3, 2001 501 ent par ti cle ap prox i ma tion and it does not take the elec tron cor re la tion in ter ac tion into ac count. We call this ap prox i ma - tion the lo cal spin den sity ap prox i ma tion (LSDA). 4,5 This means that we treat the elec tron den sity of the up-spin part and that of down-spin part sep a rately in our com puter code, con sid er ing treat ment of open shell at oms such as Eu. The photo-ionization cross sec tions are es ti mated by the first or - der ap prox i ma tion; i.e. V SCF (r, ) in Equa tion (7) is re placed by the un per turbed field V ext (r, ). At the sec ond stage, af ter ob tain ing the photo-excitation/ion iza tion cross sec tion by LSDA, we ob tain a self-consistent in ter ac tion V SCF (r, ) by solv ing equa tion (6) iteratively in a self-consistent way. We call this pro ce dure the time de pend ent lo cal ized spin den sity ap prox i ma tion (TDLSDA). The mean ing of the ter mi nol ogy is that we have al ready taken into ac count the time de pend ent self-consistent in ter ac tion through its fre quency de pend ence. We show the re sults of cal cu la tion by LSDA and those by TDLSDA in Fig. 1. The up per part of the fig ure in di cates that the cal cu lated cross sec tion of 4d photo-excitation/ion - iza tion of Xe by LSDA as func tions of pho ton en ergy. A dot - ted line in di cates the to tal ex ci ta tion/ion iza tion cross sec tion, but a solid line shows the f-state (nf as well as kf) only. The dif fer ence be tween the two curves shows the tran si tion to p state. We can see that al most all the ex ci ta tion peaks are iden - ti fied as 4d-np ex ci ta tions. The lower part of the fig ure in di - cates the cal cu lated cross sec tion by TDLSDA. Again the dot ted line in di cates all pos si ble fi nal and in ter me di ate states (p as well as f states) are taken into ac count, and the solid line in di cates only f fi nal as well as f in ter me di ate states are taken into ac count. The dif fer ence in di cates that not only p state ex - citation and ion iza tion, but also con tri bu tion of p-inter mediate state is not so small in the 90-145 ev pho ton en ergy re gion. From the com par i son of photo-excitations of TDLSDA with those of LSDA, these two p-state ex ci ta tion spec tra are al - most iden ti cal. This shows that the p-state ex ci ta tion peaks are not in flu enced by dy namic cor re la tion. The de tails of the ex ci ta tion part of the cal cu lated re - sults are shown in Fig. 2. The up per part in di cates again the re sults by LSDA and the mean ings of dot ted line and solid line are the same as be fore. Con tri bu tion of f-state ex ci ta tion is neg li gi bly small. In com par i son of the up per fig ure with the lower one, 6p to 9p peaks are al most iden ti cal with each other. This means that p-state ex ci ta tion is well ex plained by the in de pend ent par ti cle model. Fig. 3 shows the com par i son of the re sults of pres ent cal cu la tion by TDLSDA with the re - sult of ex per i ment. 8 The up per fig ure in di cates the global spec trum and the lower one in di cates the de tails of the ex ci ta tion part. As for the ex ci ta tion pro file, a Lorentzian pro file is as sumed and the width is set as in equa tion 5 is equal to 0.05 ev con sid er ing usual the res o lu tion of soft X-ray spec trom e ter. We em ploy this value for through out this pa per. Speaking of the up per fig ure, some dis crep ancy of the con tin uum spec trum with that of ex per i ments ap pears, and the rea son of this dis crep - ancy is not known. Prob a bly the dy namic cor re la tion for the ion iza tion pro cess is so sen si tive that the de vi a tion of the or - der of 5 ev may eas ily oc cur. In the lower fig ure, TDLSDA re sults are shifted a lit tle up ward in or der to com pare eas ily our re sults with the ex per i ment. The fine struc ture sep a ra tion of the ex ci ta tion peaks from j = 5/2 and j = 3/2 from Xe 4d or bit was taken into ac - count (we adopt the fine struc ture sep a ra tion E = 2.1 ev at ion iza tion po ten tial IP (J = 5/2) and IP (J = 3/2)). 9 The peak of 6p and 6p in di cate the photo-excitation peaks from 4d 5/2 to 6p and that from 4d3/2 to 6p and other peaks have a sim i lar mean ing. We can iden tify the ex ci ta tion peaks of Xe for the Fig. 1. Cal cu lated photo-excitation/ion iza tion spec tra of Xe atom as a func tion of pho ton en ergy: The up per fig ure in di cates the spec trum by LSDA (lo cal spin den sity ap prox i ma tion) and the lower fig ure shows the cal cu la tion by TDLSDA (time de pend ent lo cal spin den sity ap prox i ma - tion). The solid line in di cates the spec trum con - cern ing the f-state only (nf-discrete states and kf- continuum state, and the dot ted line in di - cates the spec trum which con cerns all states (all p- and f-states).

502 J. Chin. Chem. Soc., Vol. 48, No. 3, 2001 Watanabe et al. first time. We have no ex ci ta tion peaks for the f-state; around the 8p and 8p peaks we might have 4f and 5f peaks or 4f and 5f peaks but they are too small to be de tected. We have made a sim i lar cal cu la tion for 4d photoexcitation/ion iza tion for neg a tive I -, Cs +, Ba ++ and La 3+. These ions have the same elec tronic struc tures (5p closed shell) and dif fer ent cen tral nu clear charges. This re search is go ing to be sub mit ted some where else. 3 Here we will de - scribe the char ac ter is tic fea ture in the sys tem atic ap proach over this iso-electronic se quence. For an I - neg a tive ion, there ex ists cal cu la tion by two kinds of ran dom phase ap prox i ma - tions RPA and RPAR (RPA with re lax ation). The cal cu la tions were made by Amusia et al. 10 and Radojevic et al. 11 I - has no ex cited bound state. Gi ant res o nance con tin uum spec tra by TDLSDA is largely shifted from the spec tra by LSDA and TDSDA spec tra agrees with those by RPA 10 and with RPAR. 11 The con tin uum spec trum is spread from 60 ev to 140 ev and the char ac ter is tic fea ture of the spec trum is al most the same as that of Xe. At the pres ent time, there is no ex per i men tal data yet. Since we have three kinds of data which in clude dy - namic elec tron cor re la tion ef fect, the ex per i men tal data will not be so far from the pres ent cal cu lated data. The cal cu lated photo-excitation/ion iza tion spec trum in the case of Cs + ion is spread from around 80 ev to 150 ev or more and the max i mum po si tion is cal cu lated at around 102 ev. From 82 to 93 ev, there ex ist ex ci ta tion peaks. The cor re - spond ing ex per i men tal spec trum is re ported by Koizumi et al. 12,13 Whereas the con tin uum part of the ex per i men tal spec - trum is not so smooth com pared with the cal cu la tion, the gen - eral fea ture is in rea son able agree ment with the cal cu la tion. By com par ing cal cu lated ex ci ta tion peaks with those of ex - per i ments, we can see that there is very good cor re la tion be - tween them. We suc ceeded in iden ti fy ing all peaks for the first time. The dy namic cor re la tion ef fects ap pear in the 4d to f-continuum most strongly. The f-state ex ci ta tion peaks are also shifted by 3.0 ev and are re mark ably de creased by tak - ing the dy namic cor re la tions into ac count from LSDA to TDLSDA. The f-state con tin uum in creases re mark ably in the Fig. 2. Cal cu lated photo-excitation spec tra of Xe atom as a func tion of pho ton en ergy.the up per fig ure in di cates the spec trum by LSDA (lo cal spin den sity ap prox i ma tion) and the lower fig ure shows the cal cu la tion by TDLSDA (time de - pend ent lo cal spin den sity ap prox i ma tion). The solid line in di cates the spec trum con cern ing f-state only (nf-discrete states) and the dot ted line in di cates the spec trum which con cerns all states (all p- and f-states). Fig. 3. Com par i son with ex per i men tal data of Xe, 8 up per fig ure: whole photo-excitation/ion iza - tion spec tra of Xe as func tions of pho ton en - ergy, lower fig ure: the en large ment of photo - excitation part. np and nf in di cate the ex ci ta - tion peaks from (4d) 5/2 and np and nf in di - cate the ex ci ta tion peaks from (4d) 3/2.

Gi ant Res o nance in Photo-Absorption of High Z Ions J. Chin. Chem. Soc., Vol. 48, No. 3, 2001 503 same way as that in the case of Xe. The photo-excitation/ion iza tion spec trum of Ba ++ in cal cu la tion is again spread over 100 ev to 150 ev. The main part of the spec trum by LSDA is again largely shifted by tak - ing TDLSDA as be fore. The ef fect of dy namic cor re la tion is sim i lar to pre vi ous cases. The height of the con tin uum part (photo-ionization) is the same or der (of the or der of 10 Mb). The ma jor ity of the photo-absorption pro cess is moved to the photo-excitation of 4d to f-states. The por tion of photoionization de creases down to about 1/3 whereas that in the case of Cs + is about 3/4. The mag ni tude of shift en ergy of dy - namic cor re la tion and the amount of de crease or in crease of ex ci ta tion to f-state or ion iza tion to f-continuum is the same or der as in the case of Xe and Cs +. This spec trum can be com - pared with an ex per i men tal one by Lucatorto et al. 14 with good agree ment. The cor re spon dence of each cal cu lated ex - ci ta tion peaks with those of the ex per i men tal one is not as com plete as be fore. We can, how ever, as sign al most all the ex per i men tal peaks. In the case of La 3+, the ion iza tion limit moves to a nearly 140 ev ma jor ity of pho ton-induced tran si tions that be - come ex ci ta tions. The amount of 4f peak at 103 ev in LSDA moves to around 118 ev and ab so lute mag ni tude is en hanced about 10 times. The peak po si tions of 4d-np ex ci ta tion do not change so much, and the ab so lute mag ni tudes in TDLSDA are greatly en hanced about ten-times as much as those by LSDA. We also made the iden ti fi ca tion of each ex ci ta tion peak and com pared them with ex per i ments by Koble et al. 15 The agree ment is sat is fac tory. IV. DISCUSSION Com paring the photo-excitation/ion iza tion spec tra of Xe-like iso-electronic se quences ion iza tion thresh old en er - gies of 4d-orbitals (av er age ion iza tion en er gies of 4d 5/2 and 4d3/2) are moved from 54.79 ev (I - ), through 72.74 ev (Xe), 93.67 ev (Cs + ), 117.12 ev (Ba ++ ) and 142.90 ev (La 3+ ), the por tion of photo-excitation con tri bu tion in creased from 0% (I - ), less than 5% (Xe), ~ 25% (Cs + ), more than 50% (Ba ++ ) and more than 95% (La 3+ ). As for mag ni tudes of dy namic cor re la tion for 4d-np tran si tion, en ergy shifts due to dy namic cor re la tion are very small through out the cases from I - to La 3+, but the ab so lute mag ni tude of ex ci ta tion peaks are also in dif fer ent in the cases of Xe and Cs +, but in the case of Ba ++ 6p peak height de creases by a half and in the case of La 3+ 6p peak height in creases by a fac tor of 10. For the con tri bu tion of 4d-to-p-state con tin uum (ion iza tion) the spec tra in the high en ergy side of the con tin uum peaks are en hanced by dy - namic cor re la tion in the cases of I -, Xe and Cs + by a small amount. For Ba ++ p-continuum con tri bu tion it self is not so large. As for the 4d-nf ex ci ta tion pro cesses, the pro cess it self does not ex ist or is not large in the case of Xe. In the case of Cs +, the en ergy shift of ex ci ta tion is neg li gi bly small but the peak height de creases by a fac tor of 1/20 for 4d-4f tran si tion. In the cases Ba ++ and La 3+ the shifts of ex ci ta tion en ergy by dy namic cor re la tion are 8 ev and 15 ev, re spec tively, and the peak heights of 4f ex ci ta tion are in creased by a fac tor of 2.5 and 10, re spec tively. For I -, Xe, Cs + the larg est dy namic cor - re la tion ef fects ap pear in the 4d-continuum f tran si tions. Com paring the f-continuum spec tra by LSDA with those of TDLSDA, con tin uum max i mum-shifts move to the high en - ergy side about 20 ev and the spec trum shapes be come broad - ened and max i mum heights de crease by a fac tor of 1/3. In the case of Ba ++, the broad en ing ef fect on the f-continuum spec - trum is ob served; the con tin uum height at 120 ev in creases by a fac tor of 5 or so. In La 3+, the f-continuum spec trum it self is neg li gi bly small. Through out this sys tem atic ap proach, np-orbital en - ergy is not in flu enced by the cor re la tion ef fects and is only in - flu enced by the tran si tion am pli tude for Ba ++ and La 3+. This can be in ter preted that np orbitals ex ist near the core and are not so much in flu enced by the photo-excitation pro cesses and these ef fects ap pear for a high nu clear charge ion, be cause outer higher an gu lar mo men tum orbitals come to the in side by strong Cou lomb forces or kp (con tin uum) tran si tion. In con clu sion, the dy namic cor re la tion ap pears mainly in the tran si tion to f-states ex cept for La 3+ ; re gard less of ex ci ta tion or ion iza tion pro cess, the dy namic cor re la tion causes a large amount of en ergy shift to the high en ergy side of the or der of 5 ev to 20 ev. The amount of cor re la tion is the same or der through out this se quence. The ef fect of nu clear charge ap - pears in the form of the gi ant con tin uum res o nance to the form of the large ex ci ta tion en ergy shift and the large peak height change of ex ci ta tion pro cess. It should be men tioned that in the case of I - and Xe the con tin uum spec tra in di cate the char ac ter is tic fea ture of typ i - cal bound-fee tran si tion spec trum. In the case of La 3+, the ex - ci ta tion peak dis tri bu tion shows, how ever, a typ i cal Cou lomb spec trum dis tri bu tion (hy dro gen-like spec trum dis tri bu tion). This can be in ter preted that due to a strong Cou lomb po ten tial with a high nu clear charge, the elec tronic level struc ture be - comes Coulombic whereas the dy namic cor re la tion still works strongly. Re ceived De cem ber 31, 2000.

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