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Correlation structure in nondipole photoionization M. Ya. Amusia,1,2 A. S. Baltenkov,3 L. V. Chernysheva,2 Z. Felfli,4 S. T. Manson,5 and A. Z. Msezane4 3 0 1Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel 0 2A. F. Ioffe Physical- Technical Institute, St. Petersburg 194021, Russia 2 3U. A. Arifov Institute of Electronics, Tashkent 700187, Uzbekistan 4Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314 n 5Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303 a J (Dated: February 2, 2008) 2 The nondipole parameters that characterize the angular disribution of the photoelectrons from the 3d subshell of Cs are found to be altered qualitatively by the inclusion of correlation in the h] form of interchannel coupling between the 3d3/2 and 3d5/2 photoionization channels. A prominent p characteristic maximum is predicted only in the parameters for 3d5/2 photoionization, while the - effect for 3d3/2 is rather weak. The results are obtained within the framework of the Generalized m Random Phase Approximation with Exchange (GRPAE), which in addition to the RPAE effects takesinto account therearrangement of all atomic electrons dueto thecreation of a 3d vacancy. o t a PACSnumbers: 31.20Tz, 31.50.+b,32.80.Dz,32.80.Fb,32.80.Hd . s c i Nondipole effects in the photoionization of atoms and that it was demonstrated recently[21] that the interac- s y molecules were thought to be of any importance only tion between photoionization channels belonging to dif- h at multi-keV photon energies[1, 2] despite indications to ferent components of the spin-orbit doublets 3d3/2 and p the contrary[3, 4, 5]. However, in an upsurge of work 3d in Xe, Cs and Ba affect dramatically the partial 5/2 [ on the subject, it was found both theoretically and ex- photoionizationcrosssections. Specifically, itwas shown perimentally that nondipole effects are often of signifi- that due to this interchannel coupling interaction, the 1 v cance at photon energies of hundreds and even tens of partial 3d5/2 cross section acquires an additional promi- 4 eV[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. It has been nent maximum. This result gave the explanation for the 0 determined that photoelectron angular distributions are recent experimental observation of this effect in Xe[22] 0 particularly affected by these nondipole photoionization and also predicted similar, even more dramatic effects 1 channels. In lowest order, the effect of the non-dipole for 3d photoionization in Cs and Ba. 0 3 channels is to add two new terms to the well-known In this communication we present the results of our 0 dipole expression for the photoelectron angular distri- studies, based on the approach developed earlier[21], of / bution, along with their associated dynamical parame- the nondipole angular distribution parameters for 3d s c ters, γ and δ, the nondipole parameters[5, 7, 8, 17, 18]. photoionization in Cs to illustrate a new phenomenon i These parameters arise from the interference of dipole where impressive manifestations of intra-doublet in- s y and quadrupole (E1 and E2) channels in much the same terchannel interactions producing new structures in h manner that the β parameter, which characterizes the nondipole parameters are predicted, thereby also con- p dipole photoelectron angular distribution, arises from tradicting the notion that correlation does not affect v: dipole-dipole interference[19]. nondipole parameters significantly. Xi Studies of these nondipole parameters have found γ The theoretical results were obtained using a spe- and δ to be essentially structureless, except in regions cially modified random-phase-approximation with ex- r a of dipole or quadrupole resonances[11, 16] or Cooper change (RPAE) approach previously developed for half- minima[13]. It has also been averred that correlation filled subshells[23, 24]. This calculation showed clearly effects are unimportant for nondipole effects[20]. In this the physics of the phenomenon observed[22] and led to communication, we show that assertion to be incorrect. results[21] that are in both qaulitative and quantita- In fact, we report on a new phenomenon, a new kind of tive agreement with experimental 3d Xe data. Explic- structure in nondipole parameters that has nothing to itly relativistic calculations[25] were subsequently per- do with resonances or Cooper minima. This structure, formed using the relativistic-random-phase approxima- as shallbe shownbelow, is due solely to many-body cor- tion (RRPA)[26] which confirmed the qualitative and relations; it has no analogue in a single-particle picture. quantitative accuracy of the results in [21]. Furthermore, the structure is induced by relativistic in- To summarize the theoretical approach, we consider teractions; it does not arise in a non-relativistic calcula- the 3d and 3d subshells each as half-filled atomic 5/2 3/2 tion. Thus,thestudyofthisstructureisadirectmeasure subshells. Thispermitsustoapplystraightforwardlythe of relativistic correlation effects. RPAE methodology to include many-body correlations To understand the basic origin of this structure, note (includinginterchannelcoupling)forhalf-filledsubshells. 2 Exchange is neglected between these two sorts of elec- quadrupole, ql±2,0, are defined as trons, the six that form the 3d (called ”up”), and the 5/2 ∞ four forming 3d (called ”down”) electrons. However, 3/2 intherealhalf-filled3dsubshellonewouldhavefiveelec- dl±1 ≡dnl,ǫl±1 =Z φnl(r)rφǫl±1(r)dr, trons. But these corrections, 6/5 and 4/5, respectively 0 can be introduced easily into the calculational scheme. ∞ 1 encTehoenf ”wuep”coanncden”tdraotweno”netlehcetrionnvsesutipgoantioeancohfotthheerinaflnud- ql±2,0 ≡qnl,ǫl±2,0 = 2Z φnl(r)r2φǫl±2,0(r)dr. (2) 0 demonstrate that the effect of the 3d (”down”) elec- 3/2 trons upon the 3d5/2 (”up”) manifests itself not only in Hereφnl(r)andφǫl′(r)aretheradialpartsoftheHartree- the partial cross sections σ (ω), σ (ω)[21] and some- Fock (HF) one-electron wave functions. In the uncor- 5/2 3/2 what in the respective angular anisotropy parameters related (HF) approximation, the formulae for γ (ω) nl β (ω), β (ω)[25], but also in the nondipole parame- and δ (ω) can be used directly with the dipole and 5/2 3/2 nl tersofthe photoelectronangulardistributions, γ(ω)and quadrupole amplitudes. δ(ω). In all these cases the effect of the 3d photoion- An effect of correlation, however, is to render these 3/2 ization channels upon the 3d leads to the creation of transition matrix elements complex; the correlated 5/2 an additional maximum, while the action of the 3d5/2 dipole amplitudes are termed Dl±1 and the correlated electrons upon 3d3/2 proved to be generally negligible. quadrupole amplitudes Ql±2,0. To obtain the corre- Note that since the matrix elements for the photoioniza- sponding expressionsfor γ and δ , one has to perform nl nl tion process are strongly modified by correlation in the the following substitutions[5, 17] form of interchannel coupling, it is evident that charac- teristics of the process other than the integrated cross section; for example, the photoelectronspin polarization |dl±1 |2→|ReDl±1 |2 +|ImDl±1 |2 (see [27] and references therein) are also modified. |ql±2,0 |2→|ReQl±2,0 |2 +|ImQl±2,0 |2 The angular distribution of photoelectrons from an nl dl±1ql±2,0cos(δl±2,0−δl±1)→[ReDl±1ReQl±2,0 subshellcreatedbylinearlypolarizedlight,includingthe lowestordernondipoleparameters[5,7,8,17,18]isgiven +ImDl±1ImQl±2,0)]cos(δl±2,0−δl±1)− by −[ReDl±1ImQl±2,0−ImDl±1ReQl±2,0)] sin(δl±2,0−δl±1) (3) dσnl(ω) = σnl(ω)[1+β (ω)P (cosθ) wherethe δl′ arethe single-particle(HF) phase shifts for dΩ 4π nl 2 the designated channel, and Re and Im stand for real + [δ (ω)+γ (ω)cos2θ]sinθcosφ], (1) and imaginary parts, respectively. nl nl The method that we use here, just as in [21], is where σ (ω) is the partial cross section, β (ω) is the thespin-polarizedrandom-phaseapproximationwithex- nl nl dipole angular anisotropy parameter, γ (ω) and δ (ω) change(SPRPAE).However,fortheintermediate3dsub- nl nl are nondipole parameters, P (cosθ) is a Legendre poly- shell SPRPAE is not sufficient; the effects of core rear- 2 nomial,θ isthe anglebetweenthephotoelectronandthe rangement(relaxation)mustbetakenintoaccount. This photon polarization directions, and φ is the angle be- isdonebygoingfromRPAEtothegeneralized(GRPAE) tween the photon momentum and the projection of the or,inourcase,fromSPRPAEtoSPGRPAEwhichtakes photoelectron momentum in the plane perpendicular to into account that while a slow photoelectron leaves the the photon polarization. The expressions for γ (ω) and atom,the fieldseenis modifieddue to the alteration(re- nl δ (ω)intermsofdipoleandquadrupolematrixelements laxation) of the wave functions of all other atomic elec- nl were first obtained in a slightly different form in [5] and trons as a result of the creation of the inner-subshell va- later in [7, 8, 13, 28]. Most of the attention in the past cancy. GRPAE is discussed at length in [29]. Its exten- was given to l = 0 and l = 1, where the expressions for sion for a system with two types of electrons ”up” and γ and δ are relatively simple. Here we are interested ”down”, the transition from SPRPAE to SPGRPAE, is nl nl in the more complex case of l = 2, and the explicit ex- straightforward. pressions are given elsewhere[8]. For present purposes, Theresultsofourcalculationsforγforphotoionization however,itisnecessarytopointoutthatthe expressions ofCs3d and3d areshowninFig. 1;boththecorre- 5/2 3/2 involve ratios of quadrupole to dipole matrix elements, latedandtheuncorrelatedHartree-Fock(HF)resultsare along with cosines of phase shift differences. shown to emphasize the features brought about by cor- In the present calculation, the 3d → ǫp,ǫf dipole relation. As seen from the figure, the outstanding differ- amplitudes and 3d → ǫs,ǫd,ǫg quadrupole amplitudes encebetweencorrelatedanduncorrelatedresultsappears are taken into account for both 3d and 3d pho- in the value of γ for 3d photoionization in the region 5/2 3/2 5/2 toionization. The matrix elements for dipole, dl±1, and just above a photon energy of 740eV. The fact that this 3 significant structure in γ is not seen in the uncorrelated, rameter. A significant correlation structure in the 3d 5/2 HF, calculation is proof that its existence is due to cor- channel, seen just above the 3d threshold is evident, 3/2 relation. Specifically, we have found that it is due to along with changes in δ in the 3d channel just at its 3/2 interchannel coupling among the 3d and 3d pho- threshold. These effects of correlation occur for exactly 5/2 3/2 toionization channels; by interchannel coupling we mean the same reason as discussed above for γ. It is apparent simply configurationinteractioninthe continnum. Since from comparison of Figs. 1 and 2 that the changes in the ”real”wave function for the final continuum state in δ due to correlation are qualitatively larger than for γ. aphotoionizationprocessis multichannel,thereismixed This is probably due to the fact that the values of δ are inwiththe finalstate wavefunctions for3d photoion- roughly an order of magnitude smaller than γ. 5/2 ization a small amount of 3d channels. Then, due to Owing to the geometry of a number of experimental 3/2 the3d→ǫf shaperesonanceintheregionjustabovethe setups to measure nondipole photoelectron angular dis- 3d threshold, the 3d dipole cross section is much tributions,γ andδareoftennotmeasuredindividually[9, 3/2 3/2 larger than the 3d . Thus mixing a small amount of 10]; instead, ζ = γ + 3δ is investigated. To connect 5/2 the wave function of the 3d → ǫf channel with the with such experimental investigations, ζ for the 3d 3/2 5/2 wave function of the much smaller 3d channel alters and 3d channels of Cs are shown in Fig. 3. Since ζ is 5/2 3/2 the 3d dipole matrix elements significantly; the 3d simply a linear combination of γ and δ, the explanation 5/2 5/2 cross section maximizes with the 3d → ǫf just above forthe correlationstructureheresimplyfollowsfromthe 5/2 its threshold,anddropsoff considerablyatthe energyof previous discussion for γ and δ individually. the 3d →ǫf shape resonance. Inconclusionthen,ithasbeenshownthat,duetospin- 3/2 Butthe γ parameter,asdiscussedabove,is essentially orbit induced interchannel coupling between the dipole aratioofquadrupoletodipolematrixelements. Thereis photoionization channels arising from the two members also interchannel coupling among the quadrupole chan- ofthe3dspin-orbitdoubletinCs,anewstructureresults nels, 3d → ǫs,d,f and 3d → ǫs,d,f. But there in the nondipole photoelectron angular distribution pa- 5/2 3/2 are no significant resonances in any of these channels in rameters γ and δ (and, of course, ζ). Similar results are the energy range of interest, and all of the channels are foundforXe3dandBa3d;thesewillbepresentedinafu- of more or less the same size. Thus, interchannel cou- ture publication. Thisis the firstcaseofstructure inthe pling does not introduce any important changes in the energy dependence of nondipole parameters discovered quadrupole photoionization matrix elements like it does that is not related to resonances (dipole or quadrupole) inthedipolecase. Sinceonlyoneparticipantintheratio or Cooper minima. And, it was shown explicitly that thatyieldsγ changessignificantlybecauseofcorrelation, correlation in the form of interchannel coupling can in- then it is evident that the ratio itself must be altered deedbeimportantforthenondipolephotoelectronangu- as well. This is the reason for the significant structure lar distribution parameters, in contradistinction to the seen in Fig. 1 for γ of 3d in the vicinity of the 3d conventional wisdom. Experimental scrutiny into this 5/2 3/2 threshold. prediction is highly desirable. The only other real effect of correlation on the γ pa- Research at Clark Atlanta University is supported by rameter is for 3d right at threshold, where the HF DoE, Division of Chemical Sciences, Office of Basic En- 3/2 resultis noticeablylowerthanthe correlatedvalue. This ergy Sciences, Office of Energy Research (AZM) and againisduetointerchannelcoupling,butinthiscasethe NSF. The work of MYaA was supported by the Hebrew effect of the 3d photoionizationchannels on the 3d University Intramural Fund. MYaA and LVC acknowl- 5/2 3/2 causes the modification. In particular, the 3d → ǫf edge the support of the International Science and Tech- 3/2 amplitude, which is extremely small at the 3d thresh- nology Center (grant No 1358). The work of ASB and 3/2 old, is strongly modified by its interchannel interaction STM was supported by the US Civilian Research & De- with the much larger 3d amplitudes. velopment Foundation for the Independent States of the 5/2 The situation for the δ parameter is shown in Fig. 2 FormerSovietUnion,AwardN0. ZP1-2449-TA-02. STM and we find here a story similar to that of the γ pa- also acknowledges the support of NSF and NASA. [1] S.T.MansonandD.Dill,inElectronSpectroscopy: The- [4] J. W. Cooper and S. T. Manson, Phys. Rev. 177, 157 ory, Techniques, and Applications, v. 2, edited by C. R. 1969. Brundle and A. D Baker (Academic Press, New York, [5] M.Ya.Amusia,P.U.Arifov,A.S.Baltenkov,A.A.Grin- 1978). berg, and S. G. Shapiro, Phys. Lett. 47A, 66 (1974). [2] A. F. Starace, in em Handbuch de Physik, v. 31, edited [6] D. W. Lindle and O. Hemmers, J. Electron Spectrosc. byW. Mehlhorn (Springer-Verlag, Belin, 1982). 100, 297 (1999). [3] M. O. Krause, Phys.Rev. A 177, 151 (1969). [7] A. 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FIG. 1 Calculated values of the nondipole photoelectron [15] B.Krässig,E.PKanter,S.H.Southworth,R.Guillemin, angular distribution parameter γ for Cs 3d5/2 and O. Hemmers, D. W. Lindle, R. Wehlitz, and N. L. S. 3d subshells in correlated (SRPAE) and uncor- 3/2 Martin, Phys. Rev.Lett. 88, 203002 (2002). related (HF) approximations. [16] N.ACherepkovandS.K.Semenov,J.Phys.B34,L495 (2001). FIG. 2 Calculated values of the nondipole photoelectron [17] M. Ya. Amusia and V. K. Dolmatov, Sov. Phys. JETP angular distribution parameter δ for Cs 3d and 5/2 52, 840 (1980). 3d subshells in correlated (SRPAE) and uncor- [18] M. Peshkin, Adv.Chem. Phys. 18, 1 (1970). 3/2 related (HF) approximations. [19] S. T. Manson and A. F. Starace, Rev. Mod. Phys. 54, 389 (1982). FIG. 3 Calculated values of the nondipole photoelectron [20] A. Derevianko and W. R. Johnson, At. Data Nuc. Data angular distribution parameter ζ for Cs 3d and Tables 73, 153 (1999). 5/2 3d subshells in correlated (SRPAE) and uncor- [21] M.Ya.Amusia,L.V.Chernysheva,S.T. Manson, A.Z. 3/2 related (HF) approximations. Msezane and V. Radojević, Phys. Rev.Lett. 88, 093002 (2002). 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 Cs 3d g -0.06 -0.08 -0.10 HF 3/2 SRPAE 3/2 -0.12 HF 5/2 SRPAE 5/2 -0.14 -0.16 720 730 740 750 760 Photon energy (eV) 0.014 HF 3/2 0.012 SRPAE 3/2 HF 5/2 0.010 SRPAE 5/2 0.008 0.006 Cs 3d 0.004 d 0.002 0.000 -0.002 -0.004 720 730 740 750 760 Photon energy (eV) 0.08 0.06 Cs 3d 0.04 0.02 0.00 -0.02 z -0.04 -0.06 HF 3/2 -0.08 SRPAE 3/2 HF 5/2 -0.10 SRPAE 5/2 -0.12 720 730 740 750 760 Photon energy (eV)