First-principles calculations of spin and angle-resolved resonant photoemission spectra of Cr(110) surfaces at the 2 p - 3 d resonance
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b First-principles calculations of spin and angle-resolved resonant photoemission spectraof Cr(110) surfaces at the 2 p - 3 d resonance F. Da Pieve and P. Kr¨uger ALGC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium. ICB, UMR 6303 CNRS - Universit´e de Bourgogne, F-21078 Dijon, France. (Dated: January 30, 2018)A first principles approach for spin and angle resolved resonant photoemission is developed withinmultiple scattering theory and applied to a Cr(110) surface at the 2 p -3 d resonance. The resonantphotocurrent from this non ferromagnetic system is found to be strongly spin polarized by circularlypolarized light, in agreement with experiments on antiferromagnetic and magnetically disorderedsystems. By comparing the antiferromagnetic and Pauli-paramagnetic phases of Cr, we explicitlyshow that the spin polarization of the photocurrent is independent of the existence of local magneticmoments, solving a long-standing debate on the origin of such polarization. New spin polarizationeffects are predicted for the paramagnetic phase even with unpolarized light, opening new directionsfor full mapping of spin interactions in macroscopically non magnetic or nanostructured systems. PACS numbers: 78.20.Bh,78.70.-g,75.20.Ls,79.60.-i
In recent years, the theoretical description of absorp-tion/photoemission spectroscopy in the X-ray region hasbeen boosted by the merge of density functional theory(DFT) with many body approaches such as dynamicalmean field theory [1, 2], many body perturbation theory[3–5] and by the development of time-dependent DFT[6]. However, second order processes, like resonant inelas-tic X-ray scattering (RIXS) and resonant photoemission(RPES), remain a major challenge for theory. For RPES,existing approaches are semiempirical [7–10], based on awell defined two-holes final state and on small clusters,and thus do not take into account the delocalization ofintermediate states, the bandstructure of the system andmultiple scattering effects in the propagation of photo-electrons.The huge experimental output from RPES on corre-lated materials [7, 11–16] and the intriguing quest for adetermination of local magnetic properties put forwardby pioneering experiments [14–16] call for advancementsin the theoretical description of this spectroscopy. In ex-periments on CuO and Ni, it was shown that the RPESphotocurrent with circular polarized light is spin polar-ized in antiferromagnets [14, 15] and Curie paramag-nets [16]. It was claimed that a specific combination ofspin resolved spectra provides a direct measure of thelocal magnetic moments [14–16]. The issue is of funda-mental importance in the search for a tool to access thelocal magnetic properties in antiferromagnetic, magneti-cally disordered and/or nanostructured systems at theircrossover with the transition temperature. The inter-pretation was however rejected on the basis of symmetryanalysis [17], but explicit calculations predicting the line-shape and intensity of such fundamental signal are stilllacking and remain highly desirable.In this letter, we present the first ab-initio methodfor RPES in solids, based on a combined formulationwithin the real space multiple scattering (RSMS) ap- proach [18, 19] and DFT, and its application to Cr(110)at the 2 p -3 d resonance. By comparing the antiferromag-netic (AFM) and Pauli-paramagnetic (PM) phase of Cr,we solve the long-standing debate about the possibilityto determine local magnetic moments in macroscopicallynon magnetic systems by means of spin resolved RPESwith circular polarized light. New interesting effects inthe PM phase by unpolarized light suggest that othermechanisms are active and could be exploited for map-ping the origin of the different spin polarization (SP)components in paramagnets and magnetically disorderedsystems. Theoretical formulation.
The cross section for valenceband photoemission to a final state | v, k i , where v de-notes a valence band hole and k a photoelectron state, isgiven by I ( ω, q, k ) = X v | T kv ( ω, q ) | δ ( ǫ k − ǫ v − ~ ω )where ~ ω and q are the photon energy and polariza-tion. Here the independent particle approximation hasbeen assumed (i.e., all many-electron eigenstates aresingle Slater determinants corresponding to the sameeffective one-electron hamiltonian). According to theHeisenberg-Kramers formula [20], the transition matrixelement T kv ( ω, q ) is the sum of a direct and a resonantterm. In the latter, photon absorption leads to an inter-mediate state | c, u i , with a core hole ( c ) and an electron ina formerly unoccupied state | u i , which decays to the finalstate | v, k i through a participator Auger process [20, 21].To lowest order in the autoionization process, the transi-tion matrix element is given by T kv ( ω, q ) = h k | D q | v i + X cu h kc | V ( | vu i − | uv i ) ~ ω + ǫ c − ǫ u − i Γ h u | D q | c i (1)where D q is the dipole operator, V the Coulomb oper-ator and Γ the width of the intermediate state. Spec-tator Auger decay leads to different, namely two-hole fi-nal states and is not considered here. Participator andspectator channels can in principle be separated experi-mentally by using a photon bandwidth smaller than thecore-hole lifetime, as they show different photon energydependence (linear for the participator, and no photonenergy dependence for the spectator). Here we focus onthe physical effects at the origin of spin polarization anddichroism as well as their directional-dependence in the“pure” participator channel.The RPES intensity can be written in a compact formas I ( ω, q, k ) = X ijLL ′ σ M ω,qiLσ ( k ) I ijLL ′ ( ǫ v , σ ) M ω,qjL ′ σ ( k ) ∗ Here, i, j label atomic sites, L ≡ ( lm ) angular momentumand σ spin quantum numbers. ǫ v = ǫ k − ~ ω is the energyof the valence hole. The quantity I ijLL ′ ≡ − iπ ( τ − τ † ) ijLL ′ is the essentially imaginary part of the scattering pathoperator. It comes from the simplification of the sum overdelocalized valence states through the so called opticaltheorem in RSMS [22] and it contains the bandstructureinformation. The matrix elements M ω,qiLσ ( k ) are given by M ω,qiLσ ( k ) = X jL ′ B ∗ jL ′ ( k ) A jL ′ ,iL ( ǫ k σ k , ǫ v σ )The B jL ′ ( k ) are the key quantities in the RSMS approachand represent the multiple scattering amplitudes of thecontinuum state k ≡ ( k σ k ) [22]. The matrix elements A jL ′ ,iL ( ǫ k σ k , ǫ v σ ) are given by the sum of the direct ra-diative process ( A D ), the resonant process with directCoulomb decay ( A C ) and the resonant process with theexchange decay ( A X ), see Eq. (1). A D and A C are site-and spin-diagonal ( ∼ δ ij δ σ k σ ). We have A D = h iǫ k L ′ σ | D q | iǫ v Lσ i A C = − X j ′ cL u L ′ u σ u Z E F dǫ u I j ′ j ′ L u L ′ u ( ǫ u σ u ) ~ ω + ǫ c − ǫ u − i Γ ×h iǫ k L ′ σ, j ′ c | V | iǫ v Lσ, j ′ ǫ u L u σ u ih j ′ ǫ u L ′ u σ u | D | j ′ c i A X = X cL u L ′ u Z E F dǫ u I jiL u L ′ u ( ǫ u σ k ) ~ ω + ǫ c − ǫ u − i Γ ×h jǫ k L k σ k , ic | V | jǫ u L u σ k , iǫ v Lσ ih iǫ u L ′ u σ k | D | ic i The sums over unoccupied states u have been againsimplified through the optical theorem. The exchangeterm A X is not strictly site-diagonal because of the non-locality of the exchange interaction together with the de-localized nature of the states u . In the RSMS approachthe Coulomb matrix elements h kc | V | vu i and h kc | V | uv i can be exactly developed in one- and two-center terms. Inmetallic Cr, the Coulomb interaction is strongly screened.As a result, two-center terms are by at least one orderof magnitude smaller than the one-center terms [23] and have been neglected here. In general, the 2 p -3 d excitedintermediate states might display excitonic effects, whichcould be taken account for with a Bethe-Salpeter descrip-tion [3, 5]. For Cr metal, these effects are quite smallbecause of the large 3 d band width ( ∼ sp electrons, and thus neglected here.Photoemission spectra from Cr(110) are calculated inRSMS with a cluster of 151 atoms (see Fig. 1a) and self-consistent spin polarized potentials, obtained by a scalarrelativistic LMTO [24] calculation for bulk Cr in the lo-cal spin density approximation. Except for the 2p corelevel, all states entering the RPES calculation are de-veloped in RSMS. The 2p orbital is obtained by solv-ing the scalar relativistic Schr¨odinger equation with self-consistent spin-polarized LMTO potentials. The 2p / spin-orbit coupled states are then constructed using stan-dard angular momentum algebra and the spin-orbit cou-pling constant is taken from an atomic calculation [25].We consider the AFM order of CsCl-type which is agood approximation to the true spin density wave (SDW)ground state of Cr. The calculated magnetic momentis 0.74 µ B in reasonable agreement with experiment(0.62 µ B ). At the (110) surface, the transverse SDWpropagates along [100] or [010] [26]. Therefore, we take e z = [001] as magnetization and spin-quantization axisthroughout this paper. We also consider the Pauli PMstate, corresponding to a non-magnetic calculation. Spinorbit (SO) coupling of the valence and continuum statesis neglected (it is as small as 0.03 eV for Cr-3 d [27]). Results.
The electronic structure of Cr(110) is well ac-counted for in the RSMS approach as can be seen fromthe comparison between the local density of states (DOS)of a Cr atom in the cluster and of bulk Cr (Fig. 1b). Non-resonant angle-resolved photoemission spectra (ARPES)are shown in Fig. 1c. Differences with respect to exper-iments [28] are expected as our approach does not con-tain local many-body interactions and layer-dependentpotentials, which could play a role for a quantitative de-scription of the peak renormalization and dispersion be-haviour of the energetic structures [29]. However, themain features of the experimental spectra are reproducedin the calculation, confirming that RSMS provides a rea-sonably good description of valence band photoemissionfrom metals as previously shown for Cu(111) [22].Spin resolved, angle integrated PES and RPES spec-tra are shown in Fig. 2 for the AFM phase and severalphoton energies across the L -edge absorption threshold.Left circular polarized light incident along the magne-tization axis [001] is considered. In this “parallel” ge-ometry the spectra, right polarized light produces thesame spectra but with up and down spin exchanged. Themaximum peak intensity as a function of photon energyis plotted in Fig. 2b and shows the expected Fano pro-file. The first photon energy (551.0 eV) is too low to ex-cite the core electron and so only direct PES is possible. (cid:1) (cid:2)(cid:3) (cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:9) -4 -2 0 2energy - E F (eV) l o ca l DO S ( a r b . un it s ) bulkcluster AFM (b) -4 -3 -2 -1 0 1binding energy (eV) pho t o e m i ss i on i n t e n s it y ( a r b . un it s ) θ (c) ARPES ν=21.2 PM h eV FIG. 1: (a) Cr(110) cluster used in the RSMS calculations.The two magnetic sublattices of the AFM state are in red andblue. (b) DOS in the AFM phase for a bulk atom (LMTO)and a central atom in the cluster (RSMS). (c) ARPES spec-tra from Cr(110) along the h i azimuth for different polarangles θ with respect to the surface normal. Unpolarized lightalong the [001] axis was considered. Experimental data from[28]. i n t en s i t y ( a r b . un i t s ) RPES upRPES downPES -4 -3 -2 -1 0binding energy (eV)010002000 -4 -3 -2 -1 0 01 552 555 558photon energy (eV)0.11101001000 peak intensityh ν=551.0 h ν=552.4 h ν=554.4 h ν=585.1 Angle-integrated PES
AFM left pol.light (a) (b) eV eV eVeV
FIG. 2: a) Spin-resolved, angle integrated RPES and PESspectra of AFM Cr(110) with circular polarized light incom-ing along the spin quantization axis [001] and photon ener-gies across the L -edge resonance. A gaussian broadening of0.27 eV FWHM was applied. Note the different intensity scalefor hν = 554 . When the photon energy is raised to 552.4 eV, just belowthe absorption edge, direct and resonant process interfere destructively, giving rise to the dip in the Fano profile.Strong resonant enhancement is observed between 552and 554.5 eV (see e.g. the spectrum for 554.4 eV), whichcorresponds to transitions from the 2 p / level into theunoccupied Cr 3 d band. At hν = 585 . p / -3 d resonance is spin-polarized when circular polarized light is used.We now turn to angle and spin resolved spectra atmaximum resonance (h ν =554.4 eV), focusing on theirfour “fundamental” combinations (and their relationto local magnetic properties), constructed by differentchoices of photoelectron spin ( ↑ , ↓ ) and light helicity(+ , − ) ≡ (left,right):tot ≡ ( ↑ +) + ( ↑ − ) + ( ↓ +) + ( ↓ − ) (total)spr ≡ ( ↑ +) + ( ↑ − ) − ( ↓ +) − ( ↓ − ) (spin-resolved)dic ≡ ( ↑ +) − ( ↑ − ) + ( ↓ +) − ( ↓ − ) (dichroic)mix ≡ ( ↑ +) − ( ↑ − ) − ( ↓ +) + ( ↓ − ) (mixed)The “mixed” spectrum was the one considered inRefs [14, 16] and claimed to be sensitive to local mag-netic moments in non-ferromagnetic samples.The normal emission RPES spectra (Fig 3a,b) (totalspectra) for parallel geometry consist of a single peak at0.8-0.9 eV binding energy, very similar to the low energynon resonant spectrum in Fig.1c ( θ = 0 o ). AFM and PMspectra are almost identical except for a small shift of ∼ . d bands. The dichroic (dic) and spin-resolved(spr) signals vanish for both PM and AFM phase, asexpected since the system is globally non-magnetic inboth cases, and the set up is non chiral.However, the mixed signal is non-zero with a large am-plitude ( ∼ / T C [16]) with disordered and/orfluctuating magnetic moments, but a Pauli PM state,where the magnetization is strictly zero in all points ofspace. Therefore, our finding that the mixed signal isessentially unchanged when going from the AFM to thePM state unambiguously proves that it is unrelated to lo-cal magnetic moments, in contrast to the interpretationin Refs [14, 16].Rather than being of magnetic origin, the non-zeromixed signal is in fact induced by angular momentumtransfer from the light helicity to the electron spin viaSO in the core shell together with a strong exchange -5 -4 -3 -2 -1 0binding energy (eV) totdicsprmix f unda m en t a l s pe c t r a ( a r b . un i t s ) -4 -3 -2 -1 0 (a) (c)(b) (d) AFM PMPM PM off-normalparallelnormalparallelnormalparallel normalperpendicularemission emissionemission emissiongeometry geometrygeometry geometry
FIG. 3: Angle-resolved fundamental spectra of Cr(110).RPES as thick lines for hν = 544 . × o (vector e in Fig.1 a).Light incidence parallel (a,b,d) or perpendicular (c) to spin-quantization axis e z . Light vector in (c) is shown as p inFig.1a. effect in the decay process. To see this, consider lightwith left (+) helicity and a non-magnetic ground state.The 2 p / -3 d optical transition has a larger amplitude forspin-up than for spin down electrons because of the dom-inantly parallel alignment of spin and orbit in 2p / . Forexample, for an empty or spherically symmetric 3 d shellthe intensity ratio is 5:3. Consider now a spin-up elec-tron transition. The RPES intermediate state has oneextra spin-up electron in the 3 d -shell (denoted u ↑ ) and a2 p -hole of dominant spin-up character. This state decaysthrough Coulomb interaction to the photoemission finalstate with one 3 d -hole and the photoelectron. The directCoulomb matrix elements is of the form h kσ, c ↑| V | vσ, u ↑i which is independent of the photoelectron spin σ . So thedirect decay alone would lead to a spin-balanced pho-tocurrent. For the exchange decay, the matrix elementis h kσ, c ↑| V | u ↑ , vσ i ∼ δ ( σ, ↑ ). This is roughly as a largeas the direct Coulomb term for spin-up electrons (the ra-dial matrix elements are exactly the same) but it is zerofor spin-down electrons. Since the exchange matrix ele-ments are substracted from the direct terms in Eq. (1),the transition probability for spin-up electron emission isstrongly reduced by the exchange process. This showsthat a core-valence transition of a spin up electron leads,through autoionization, to a strongly spin polarized pho-tocurrent with a majority of spin down electrons. Asmentioned before, left circular polarized light promotesdominantly spin-up electrons in the 2 p / -3 d transition. Therefore it produces a majority of spin-down photoelec-trons. Under the assumption of complete cancellation be-tween direct Coulomb and exchange matrix elements forparallel spins and by neglecting the direct valence photoe-mission, the ratio of spin-down to spin-up photoelectronsis 5:3, which corresponds to a spin-polarization (ratio ofmixed over total signal) of − /
4. In angle integratedRPES at maximum resonance (Fig. 2a, hν =554.4 eV) wefind a SP of − .
21, in good agreement with such modelestimation. These values agree also well with the mea-sured spin-polarization in CuO [14] and Ni [16], whichis 10–40% depending on binding energy. Our findingsclarify the physical mechanism inducing the presence ofthe mixed signal in both phases, and point to a criticalre-examination of experimental observations.Interestingly, we find that, contrary to the previousset up, it is possible to have a net spin polarization sig-nal on the PM phase. This is possible under appropriategeometrical conditions, and even with unpolarized light.Such SP can be of opposite sign and be due to differentactive mechanisms. In Fig. 3c, normal emission spectraare shown for light incident along [1¯10], i.e. perpendic-ular to the spin-quantization axis s = e z (perpendiculargeometry). As before, the dichroic signal is zero, as lightincidence ( p ) and electron emission vector ( n ) lie in amirror plane of the surface (see Fig. 1 a). However, theset up (including spin resolution) is chiral, since the threevectors p , n and s form a right-handed frame. ThusSO-induced SP cannot be ruled out by symmetry and asmall, positive SP (in this case transverse to the scatter-ing plane) is indeed observed in RPES, even for unpolar-ized light. A similar SP from PM surfaces for unpolar-ized light was theoretically predicted in direct PES [30]in a relativistic approach and confirmed by experiments[31, 32]. It was ascribed to broken symmetry due to theoff-normal light incidence together with SO in the ini-tial states and phase shift differences. We do not observethis effect in non-resonant PES since the SO coupling inthe Cr 3 d valence states is very weak and neglected here.However, for RPES, such SP has to be related to the dy-namical SP studied in atomic physics, which is known tobe related to phase shift differences in the final outgoingwaves, and to be generally small [33, 34]. Our result con-firms that such SP exists for an atom embedded in a solidand that it survives to the multiple scattering effects.A SP signal in the PM phase is also present for paral-lel geometry with off-normal emission (Fig. 3d). In thiscase, the system composed by the surface, light incidence(along e z ) and electron emission vector, is chiral. There-fore a dichroic signal is observed even in non-resonantPES, known as circular dichroism in angular distribu-tion [35]. In RPES, the angular momentum of the photonis partly transferred to the electron spin through the SOcoupling in the 2 p shell, leading to non-zero intensity alsofor spin resolved and mixed signals. The spin polariza-tion is negative, i.e. photoelectrons are mainly polarizedantiparallel to their emission direction, because of the ex-change process in the autoionization decay. This findingsuggests a Fano-like effect in resonant processes for offnormal emission directions, which could be well studiedalong the same lines as direct PES on paramagnets [36].In conclusion, we have presented a first-principles ap-proach for RPES in solids and its application to Cr(110).By comparing Pauli PM and AFM states, we have shownthat the mixed signal is essentially independent of localmagnetic properties and we have clarified its origin: con-trary to previous interpretations, this effect is induced byan angular momentum transfer from the photon to theelectron spin, through SO coupling in the core level andthe exchange process in the autoionization decay. Our re-sults show that caution must be taken in linking the spinpolarized or mixed signal to local magnetic moments, allthe more so as the photoelectron spin may have compo-nents along and across the light helicity. New effects inthe SP suggest that a mapping of spin interactions inparamagnets and disordered magnetic structures couldbe obtained via full tomography experiments at the coreresonances even with unpolarized light. [1] O. ˇSipr, J. Min´ar, A. Scherz, H. Wende, and H. Ebert,Phys. Rev. B , 115102 (2011).[2] J. Braun, J. Min´ar, H. Ebert, M. I. Katsnelson, and A.I. Lichtenstein, Phys. Rev. Lett. , 227601 (2006) .[3] W. Olovsson, I. Tanaka, T. Mizoguchi, G. Radtke, P.Puschnig and C. Ambrosch-Draxl, Phys. Rev. B ,195206 (2011).[4] J. Vinson, J. J. Rehr, J. J. Kas, and E. L. Shirley, Phys.Rev. B , 115106 (2011)[5] R. Laskowski and P. Blaha, Phys. Rev. B , 205104(2010)[6] O. Bunau and Y. Joly, Phys. Rev. B , 155121 (2012)[7] S. R. Mishra, T.R. Cummins, G.D. Waddill, W.J. Gam-mon, G. van der Laan, K.W. Goodman and J.G. Tobin,Phys. Rev. Lett. , 052407 (2005).[9] O. Tjernberg, G. Chiaia, U. O. Karlsson and F. M. F. deGroot, J. Phys.: Condens. Matter (1997) 9863[10] H. Ogasawara, A. Kotani, P. Le F`evre, D. Chandresrisand H. Magnan, Phys. Rev. B , 7970 (2000).[11] M. Morscher, F. Nolting, T. Brugger and T. Greber,Phys. Rev. B , 140406(R) (2011).[12] M.C. Richter, J.-M.Mariot, O.Heckmann, L. Kjeldgaard,B.S. Mun, C.S. Fadley, U. L¨uders, J.-F. Bobo, P. DePadova, A. Taleb-Ibrahimi and K. Hricovini, Eur. Phys.J. Special Topics , 175 (2009).[13] T. Ohtsuki, A. Chainani, R. Eguchi, M. Matsunami,Y. Takata, M. Taguchi, Y. Nishino, K. Tamasaku, M. Yabashi, T. Ishikawa, M. Oura, Y. Senba, H. Ohashi,and S. Shin, Phys. Rev. Lett. , 047602 (2011).[14] L. H. Tjeng, B. Sinkovic, N. B. Brookes, J. B. Goedkoop,R. Hesper, E. Pellegrin, F. M. F. de Groot, S. Altieri, S.L. Hulbert, E. Shekel, and G. A. Sawatzky, Phys. Rev.Lett. , 1126 (1997).[15] L. H. Tjeng, N. B. Brookes, B. Sinkovic, J. Electron Spec-trosc. Relat. Phenom. , 189 (2001).[16] B. Sinkovic, L. H. Tjeng, N. B. Brookes, J. B. Goedkoop,R. Hesper, E. Pellegrin, F. M. F. de Groot, S. Altieri, S.L. Hulbert, E. Shekel, and G. A. Sawatzky, Phys. Rev.Lett. , 3510 (1997).[17] G. van der Laan, Phys. Rev. Lett. , 733 (1998).[18] J. J. Rehr and R. C. Albers, Rev. Mod. Phys. , 621(2000).[19] D. S´ebilleau, R. Gunnella, Z.-Y. Wu, S. Di Matteo andC. R. Natoli, J. Phys.: Condens. Matter , 175 (2006).[20] A. Tanaka and T. Jo, J. Phys. Soc. Jpn. , 2788 (1994)[21] C. Janowitz, R. Manzke, M. Skibowski, Y. Takeda, Y.Miyamoto, and K. Cho, Surf. Sci. Letters L669(1992).[22] P. Kr¨uger, F. Da Pieve, and J. Osterwalder, Phys. Rev.B , 115437 (2011).[23] For nearest-neighbor (NN) two-center Coulomb termsthe average electron distance equals the NN distance d .For the one-center terms the average distance is about d/
2. Here it is even smaller because of the strong local-ization of the core-orbital near the nucleus. The screenedCoulomb interaction is exp( − r/λ ) /r , where λ is thescreening length. The ratio between NN and on-siteterms is therefore about χ = exp( − d/ /λ ) /
2. UsingThomas-Fermi theory and taking 1 nearly free electron(4s) for Cr, we get λ = 0 . χ = 0 . , 3060 (1975).[25] R. D. Cowan, The Theory of Atomic Structure and Spec-tra, University of California Press, Berkeley, 1981.[26] K.-F. Braun, S. F¨olsch, G. Meyer, and K.-H. Rieder,Phys. Rev. Lett. , 3500 (2000)[27] G. van der Laan and B. T. Thole, Phys. Rev. B , 13401(1991).[28] P. E. S. Persson and L. I. Johansson, Phys. Rev. B ,2284 (1986).[29] J.S`anchez-Barriga et al. , Phys.Rev. B , 205109 (2012)[30] E. Tamura and R. Feder, Europhys. Lett. , 695 (1991).[31] J. Kirschner, Appl. Phys. A , 3 (1987).[32] N. Irmer, R. David, B. Schmiedeskamp, and U. Heinz-mann, Phys. Rev. B , 3849 (1992).[33] U. Hergenhahn, U. Becker, J. Electron Spectrosc. Relat.Phenom. , 225 (1995).[34] B. Lohmann, J.Phys.B:At.Mol.Opt.Phys. , L643(1999)[35] J. Henk, A. M. N. Niklasson, and B. Johansson, Phys.Rev. B , 13986 (1999).[36] J. Min´ar, H. Ebert, G. Ghiringhelli, O. Tjernberg, N.B. Brookes and L. H. Tjeng, Phys. Rev. B63