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Valence photoelectron spectra of Na, K, Rb and Cs with s or p outermost electron: Evolution of the states from LS-coupling to jK-coupling
Journal of Electron Spectroscopy and Related Phenomena 161 (2007) 99–104
Valence photoelectron spectra of Na, K, Rb and Cs with s or p outermost electron: Evolution of the states from LS-coupling to jK-coupling M. M¨aa¨ tt¨a a,∗ , J. Schulz b,1 , S. Hein¨asm¨aki a , H. Aksela a a
Department of Physical Sciences, Box 3000, FIN-90014 University of Oulu, Finland Department of Physics, Uppsala University, Box 530, SE-75121 Uppsala, Sweden
b
Available online 1 March 2007
Abstract The development of the fine structure of the np5 (n + 1)s and np5 (n + 1)p (n = 2 in Na, n = 3 in K, n = 4 in Rb, n = 5 in Cs) configurations of alkali atoms ions from LS-coupling to jK-coupling has been studied by comparing the experimental photoelectron spectra with the results obtained with analytical LS-coupling and jK-coupling models. © 2007 Elsevier B.V. All rights reserved. PACS: 32.80.Fb; 42.62.Fi Keywords: Alkali metals; Valence photoelectron spectrum; Pure coupling schemes; LS-coupling; jK-coupling
1. Introduction Valence np (n = 2 in Na, n = 3 in K, n = 4 in Rb, n = 5 in Cs) photoelectron spectra of alkali metal atoms with one electron outside the closed shell display features typical for open shell atoms. If the ground state valence s electron is laser excited into the p state before the ionization, the effect of the character of the outermost electron to the valence photoelectron spectrum can be investigated. When going from Na to Cs the relativistic effects become stronger and the structures of the spectra split up into two groups related to the spin–orbit splitting of the np subshell (n = 2 in Na, n = 3 in K, n = 4 in Rb, n = 5 in Cs). The validity of a coupling scheme is changed accordingly. Main lines of the photoelectron spectra of alkali metal atoms have been studied earlier by S¨uzer and coworkers [1,2]. Experiments where synchrotron radiation has been combined with laser excitation have been performed for Na [3], K [4], Rb [5] and Cs [6]. In the present work we remeasured both np5 (n + 1)s and np5 (n + 1)p photoelectron spectra with the same resolution for the alkali metal atoms. This allowed us to compare
∗
Corresponding author. Tel.: +358 8 553 1326; fax: +358 8 553 1287. E-mail address: [email protected] (M. M¨aa¨ tt¨a). 1 Permanent address: MAX-lab, Lund University, Box 118, 221 00 Lund, Sweden. 0368-2048/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2007.02.031
the experimental photoelectron spectra of both the ground and laser-excited atoms as a function of Z in detail. The validity of the coupling scheme for the same atom but with s or p outermost electron has been investigated as well as the evolution of the coupling as a function of atomic number for both the np5 (n + 1)s and np5 (n + 1)p configurations. Previously only the experimental np5 (n + 1)s photoelectron spectra measured using He I and He II resonance lines were compared in [1], and calculated np5 (n + 1)p photoelectron spectra of Na, K and Rb were discussed in [4]. 2. Experiment The experiments were carried out at the undulator beamline I411 at the 1.5 GeV MAX-II storage ring at Lund, Sweden [7,8] with the setup which has been described in more details in [9]. The np5 (n + 1)s spectra were measured with photon energy of 61 eV for Na, 60 eV for K and Rb and 72 eV for Cs. np5 (n + 1)p states can be reached by a two step process including a (n + 1)s → (n + 1)p laser excitation followed by direct np photoemission (Na 3s → 3p followed by direct 2p photoemission). The laser can be tuned to excite either the 2P 2 2 3/2 state or the P1/2 state, only the P1/2 excitation has been studied in this work. The wavelength for this excitation is for Na 589.6 nm, for K 769.9 nm, for Rb 794.8 nm, for Cs 894.3 nm.
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[11]:
3. Calculations For comparison with the experimental np5 (n + 1)s and 5 np (n + 1)p photoelectron spectra we have calculated the relative intensities using analytical LS-coupling [10] and jKcoupling models [11]. The starting point in the analysis is the general expression for the photoionization intensity from polarized atoms [12]. The dynamical information for the angle-averaged total intensity is encoded in the coefficients Bk0 kkγ for k0 = k = kγ = 0 which in the jj coupling scheme read: √ B000 = 3 |αf Jf , lj : JDα0 J0 |2 (1) ljJ
(jK) B000
√ ˆ f2 Jˆ f2ˆl−2 = 3Jˆ 02 K 0
J0
L0
S0
Kf
Jf
j0
2
|dl |2
(4)
l
where we have reduced the one-particle matrix elements to LS-coupling thus removing the j dependence of the continuum waves. Within our geometrical model, the relative intensities of lines in the photoelectron spectra, belonging to same configuration, are independent of one-particle amplitudes and mixing coefficients of the final ionic states. Therefore, we do not need to calculate these quantities. 4. Results
where angular momenta with the subscript 0 refer to the initial atomic state, those with subscript f refer to the final ionic state and quantities without subscripts to the total values obtained by summing the final ion and continuum quantum numbers. The “geometrical” models discussed in this paper amount to various simplifications of the matrix elements in (1) so that the photoelectron intensities become dependent only on the angular momentum quantum numbers and not on the ionization amplitudes. This is often a good approximation if the contributions of various partial waves stay constant in the energy region considered. We work within the assumption that there is at most a single open shell in the ground state of the atom and that the ionization takes place from a closed shell. We also assume that apart from the ionized electron the remaining configuration of the atom remains unperturbed. With these simplifications the many-electron ionization matrix elements can be simply factorized into one-particle ionization matrix elements and coupling coefficients. We have for the pure jj coupling: √ (jj) B000 = 3Jˆ f2 jˆ 02 |dj |2 (2) lj
where √dj = ljdn0 l0 j0 and where we have used the notation xˆ ≡ 2x + 1. In the pure LSJ coupling case the matrix element in (1) becomes, after recoupling and reduction to one-particle matrix elements, to [10]: ⎧ ⎫2 1/2 l0 j0 ⎪ ⎪ ⎨ ⎬ √ (LS) ˆ 2fˆl−2 × ˆ 02 S0 L0 J0 B000 = 3Jˆ 02 Sˆ 0−2 Sˆ f2 Jˆ f2 L j 0 ⎪ ⎪ ⎩ ⎭ j0 Sf L f J f × |dl |2 (3) l
where dl = ldn0 l0 . The last interesting case is the so-called jK-coupling, which amounts to coupling the total angular momentum of the valence hole state to the orbital angular momentum of the valence electron to give an intermediate quantum number Kf . Coupling this to the spin of the valence electron results in the total final ionic state angular momentum Jf . Within the jK-coupling (1) becomes
Figs. 1 and 2 show the experimental np5 (n + 1)s and np5 (n + 1)p photoelectron spectra for Na, K, Rb and Cs, respectively. In normalization of the spectra the sum of intensities for the manifolds is taken to be 100%. The energies and energy splittings of the photoelectron lines were obtained from optical data [13,14]. For Na and K we have used the LS-coupling based notation 2S+1 LJ , which gives four main lines 3 P2,1,0 and 1 P1 in the np5 (n + 1)s spectra. The structure of np5 (n + 1)p photoelectron spectra is more complicated and 10 final states 3 S1 , 3 D3,2,1 , 1 D2 , 3P 1 1 3 2,1,0 , P1 , S0 are generated. The final state D3 is not populated as the transition to that state is forbidden from 2 P1/2 initial state. The order of the LS-coupled final states usually obeys the Hund rules, which predict the order of 3 P2 , 3 P1 , 3 P0 and 1 P1 from lower to higher energy states for the np5 (n + 1)s photoelectron spectra, and of 3 D3 , 3 D2 , 3 D1 , 3 P2 , 3 P1 , 3 P0 , 3 S1 , 1 D2 , 1 P , 1 S for the np5 (n + 1)p photoelectron spectra. In the latter 1 0 case the correct order is , however, 3 S1 , 3 D2 , 3 D1 1 D2 , 1 P1 , 3 P2 , 3P , 3P , 1S . 1 0 0 For Rb and Cs we have used the jK-coupling based energylevel notation of [K]Jf . Configuration np5 (n + 1)s has four states, one with Jf = 0, two with Jf = 1 and one with Jf = 2 and configuration 5p5 6p 10 states, two with Jf = 0, four with Jf = 1 and three with the Jf = 2, the transition to Jf = 3 state being forbidden. The splitting of final states into two groups related to spin–orbit splitting of the np subshell increases with increasing Z. The purities of the two final states of np the photoionization process in LS- and jK-coupling schemes calculated with the Cowan code [15] for ground and laser excited alkali metal atoms are plotted in Fig. 3. The purities of np5 (n + 1)s1 P1 /(2 P1/2 )2 [1/2]1 final state in the first panel show dramatic change as a function of Z, LS-coupling is valid for Na but K already deviates from LS-coupling still being not pure in jK-coupling. Rb and Cs can be well described with a pure jK-coupling. The second panel shows the purity of the np5 (n + 1)p1 P1 /2 P1/2 2 [3/2]1 state which is completely different from the purities plotted in the first panel. Pure LS-coupling fails in describing the states even for Na, opposite to the finding of the first panel. K is not well described either in LS-coupling or jK-coupling. jK-coupling is the best for Rb and shows increasing trend just like in the first panel. Similar trends as shown in the first panel were obtained for the np5 (n + 1)s configuration
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Fig. 1. Experimental np6 (n + 1)s → np5 (n + 1)s photoelectron spectra of Na, K, Rb and Cs. First and lowest panels show the calculated photoelectron spectra for Na and Cs.
for the two final state with same Jf . The trend shown in the second panel is more typical for np5 (n + 1)p configuration due to strong mixing of LS states. Figs. 1 and 2 show experimental np5 (n + 1)s and np5 (n + 1)p photoelectron spectra for alkali metal atoms. The second panel of Fig. 1 shows the experimental 2p5 3s photoelectron spectrum of Na and first panel its intensity distribution predicted by the analytical LS-coupling scheme. The calculated LS-coupling model predicts the 2p photoelectron spectrum of ground state of Na very well. In the 2p5 3p spectrum (second panel of Fig. 2) of Na only six lines are resolved, the 3 P2,1,0 and 1 P1 lines overlap heavily. Calculated 2p5 3p photoelectron spectrum of laser excited
Na clearly deviates from the experimental one. The distribution of the intensity between these final states is not correctly predicted by the pure LS-coupling model, which do not take the mixing of the states into account. Due to the mixing, the 1 P1 and 3 P1 lines gain some intensity from the lines 3 D1 and 3 S1 . Also the states with Jf = 2 mix strongly, and 1 D2 loses part of its intensity to the 3 P2 and 3 D2 lines. Cubaynes et al. [3] calculated the relative intensities taking into account the mixing of states and obtained a better agreement between experiment and calculations. 3p5 4s spectrum of K shows the same main lines 3 P2,1,0 and 1 P1 as the 2p5 3s spectrum of Na, but due to configura-
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Fig. 2. Experimental np6 (n + 1)p2 P1/2 → np5 (n + 1)p photoelectron spectra of Na, K, Rb and Cs. First and lowest panel shows the calculated photoelectron spectra for Na and Cs.
tion interaction between the 3p5 4s 3 P1 and 3p5 3d 1 P1 states the 3p5 3d 1 P1 line appears in the spectrum and steals intensity from the 3p5 4s 3 P1 line. In the 3p5 4p spectrum of K of the lines 3 D1 and 1 P1 start to separate, which shows that the transition toward jK-coupling is already visible because the inner-shell spin–orbit interaction is stronger than in the case of Na. np5 (n + 1)s and np5 (n + 1)p photoelectron spectra of Rb and Cs split up into two groups of lines for the np-hole total angular momenta j = 1/2 and 3/2. The mixing of np5 (n + 1)s and np5 nd configurations in Rb and Cs creates the 4p5 4d 3 P2,1,0 lines
in the 4p5 5s spectrum of Rb and the 5p5 5d lines in the 5p5 6s spectrum of Cs. In np5 (n + 1)p photoelectron spectra of Rb and Cs the np5 nf states were found not to be populated, because configuration interaction between the configurations np5 nf and np5 (n + 1)p is very weak. The splitting of the np shell is bigger in the 5p5 6p spectrum of Cs than in 4p5 5p spectrum of Rb, the final states 5p5 (2 P3/2 )6p2 [3/2]1 and 5p5 (2 P1/2 )6p2 [3/2]1 are even more separated than the corresponding lines in the Rb spectrum. Theoretical 5p5 6s and 5p5 6p photoelectron spectra of Cs calculated by analytical jK approximation are seen in the lowest
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out, however, are the effects of electron correlation which manifest themselves as the interference of the pure-coupling levels, and also energy-dependent effects of the spin–orbit interaction in the one-particle ionization matrix elements. The methods used here are suitable for identifying the lines and general structures of the spectra in the absence of strong correlation. Multiconfiguration effects of the one-particle ionization matrix elements are, on the other hand, negligible as long as we are away from Cooper minima where the individual amplitudes change drastically. The 2p5 3s spectrum of Na predicted with the pure LS-coupling scheme is in good agreement with the experimental spectrum but the model fails in describing the 2p5 3p photoelectron spectrum of Na. When comparing the calculated 5p5 6s and 5p5 6p spectra of Cs, the pure jK-coupling model is seen to describe better the np5 (n + 1)s spectrum, but also for the 5p5 6p spectrum the resemblance between theory and experiment is fairly good. The coupling scheme for Rb and Cs deviates strongly from Na and K because the spin–orbit interaction of the np hole (n = 4 in Rb and n = 5 in Cs) is of the same magnitude or larger than the Coulomb interaction between the hole created and the outermost s electron. Acknowledgments
Fig. 3. Purities of given states. For details, see the text.
panels of Figs. 1 and 2, respectively. Mixing of np5 (n + 1)s and np5 nd configurations was not included in calculations, which is why the lines assigned as 5p5 5d are missing from theoretical predictions. For 5p5 6s configuration, calculations overestimate the intensity of 5p5 (2 P1/2 ) 6s 2 [3/2]1 line in binding energy around 17 eV because mixing of the states with Jf = 1 was omitted. The calculated 5p5 6p photoelectron spectrum of Cs reproduces the experimental spectrum quite well as was pointed out also in Ref. [6]. Appearance of two additional lines in theoretical spectrum can be explained by the mixing of the 5p5 (2 P1/2 )6p2 [3/2]2 and2 [5/2]2 states. Part of the intensity of the line 5p5 (2 P3/2 )6p2 [5/2]2 is going to 5p5 (2 P3/2 )6p2 [3/2]2 line. Because of this mixing, 1 D /(2 P 2 5 2 3/2 ) [3/2]2 line vanishes also in the np (n + 1)p photoelectron spectra of K and Rb but the line can clearly be seen in the 2p5 3p spectrum of Na. As seen from Fig. 3, mixing takes place also between final states with Jf = 1, the line 5p5 (2 P1/2 )6p2 [1/2]1 gets some intensity from the line 5p5 (2 P1/2 )6p2 [3/2]1 . 5. Conclusions The pure coupling schemes investigated in this work are in principle a tool for mapping the strength of the spin–orbit coupling in the different ionic states of the atom. What are left
The authors thank The Swedish Research Council (VR), and the Research Council for Natural Sciences and Engineering of the Academy of Finland for economic support. This work was supported by the European Community-Research Infrastructure Action under the FP6 “Structuring the European Research Area” Programme. The authors thank the staff of MAX-lab and M. Tchaplyguine, T. Rander, H. Bergensen, A. Lindblad, G. ¨ Ohrwall, S. Svensson, R. Sankari, S. Osmekhin, M. Huttula and E. Kukk for support. References [1] S. S¨uzer, B. Breuckmann, W. Menzel, C.E. Theodosiou, W. Melhorn, J. Phys. B 13 (1980) 2061. [2] S. S¨uzer, Chem. Phys. Lett. 72 (1980) 256. [3] D. Cubaynes, M. Meyer, A.N. Grum-Grzhimailo, J.-M. Bizau, E.T. Kennedy, J. Bozek, M. Martins, S. Canton, B. Rude, N. Berrah, F.J. Wuilleumier, Phys. Rev. Lett. 92 (2004) 233002. [4] M. Meyer, D. Cubaynes, F.J. Wuilleumier, E. Heinecke, T. Richter, P. Zimmermann, S.I. Strakhova, A.N. Grum-Grzhimailo, J. Phys. B 39 (2006) L153. [5] J. Schulz, M. Tchaplyguine, T. Rander, H. Bergersen, A. Lindblad, G. ¨ Ohrwall, S. Svensson, S. Hein¨asm¨aki, R. Sankari, S. Osmekhin, S. Aksela, H. Aksela, Phys. Rev. A 72 (2005) 032718. [6] J. Schulz, M. M¨aa¨ tt¨a, S. Hein¨asm¨aki, M. Huttula, R. Sankari, E. Kukk, T. Rander, S. Svensson, S. Aksela, H. Aksela, Phys. Rev. A 73 (2006) 062721. [7] M. B¨assler, J.-O. Forsell, O. Bj¨orneholm, R. Feifel, M. Jurvansuu, S. Aksela, S. Sundin, S.L. Sorensen, R. Nyholm, A. Ausmees, S. Svensson, J. Electron Spectrosc. Relat. Phenom. 101–103 (1999) 953. [8] M. B¨assler, A. Ausmees, M. Jurvansuu, R. Feifel, J.-O. Forsell, P. de Tarso Fonseca, A. Kivim¨aki, S. Sundin, S.L. Sorensen, R. Nyholm, O. Bj¨orneholm, S. Aksela, S. Svensson, Nucl. Instrum. Methods A 469 (2001) 382. [9] M. Huttula, M. Harkoma, E. Nømmiste, S. Aksela, Nucl. Instrum. Methods Phys. Res. A 467–468 (2001) 1514.
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[10] A. Verweyen, A.N. Grum-Grzhimailo, N.M. Kabachnik, Phys. Rev. A 60 (1999) 2076. [11] Ph. Wernet, J. Schulz, B. Sonntag, K. Godehusen, P. Zimmermann, A.N. Grum-Grzhimailo, N.M. Kabachnik, M. Martins, Phys. Rev. A 64 (2001) 042707. [12] S. Baier, A.N. Grum-Grzhimailo, N.M. Kabachnik, J. Phys. B 27 (1994) 3363.
[13] NIST Atomic Spectra Database. http://physics.nist.gov/PhysRefData/ ASD/. [14] C. Moore, Natl. Bur. Stand. (U.S.) Circ. No. 467 (U.S. GPO, Washington, DC, 1971), Vol III. [15] R.D. Cowan, The Theory of the Atomic Structure and Spectra, University of California Press, Berkeley, 1981.
Valence photoelectron spectra of Na, K, Rb and Cs with s or p outermost electron: Evolution of the states from LS-coupling to jK-coupling M. M¨aa¨ tt¨a a,∗ , J. Schulz b,1 , S. Hein¨asm¨aki a , H. Aksela a a
Department of Physical Sciences, Box 3000, FIN-90014 University of Oulu, Finland Department of Physics, Uppsala University, Box 530, SE-75121 Uppsala, Sweden
b
Available online 1 March 2007
Abstract The development of the fine structure of the np5 (n + 1)s and np5 (n + 1)p (n = 2 in Na, n = 3 in K, n = 4 in Rb, n = 5 in Cs) configurations of alkali atoms ions from LS-coupling to jK-coupling has been studied by comparing the experimental photoelectron spectra with the results obtained with analytical LS-coupling and jK-coupling models. © 2007 Elsevier B.V. All rights reserved. PACS: 32.80.Fb; 42.62.Fi Keywords: Alkali metals; Valence photoelectron spectrum; Pure coupling schemes; LS-coupling; jK-coupling
1. Introduction Valence np (n = 2 in Na, n = 3 in K, n = 4 in Rb, n = 5 in Cs) photoelectron spectra of alkali metal atoms with one electron outside the closed shell display features typical for open shell atoms. If the ground state valence s electron is laser excited into the p state before the ionization, the effect of the character of the outermost electron to the valence photoelectron spectrum can be investigated. When going from Na to Cs the relativistic effects become stronger and the structures of the spectra split up into two groups related to the spin–orbit splitting of the np subshell (n = 2 in Na, n = 3 in K, n = 4 in Rb, n = 5 in Cs). The validity of a coupling scheme is changed accordingly. Main lines of the photoelectron spectra of alkali metal atoms have been studied earlier by S¨uzer and coworkers [1,2]. Experiments where synchrotron radiation has been combined with laser excitation have been performed for Na [3], K [4], Rb [5] and Cs [6]. In the present work we remeasured both np5 (n + 1)s and np5 (n + 1)p photoelectron spectra with the same resolution for the alkali metal atoms. This allowed us to compare
∗
Corresponding author. Tel.: +358 8 553 1326; fax: +358 8 553 1287. E-mail address: [email protected] (M. M¨aa¨ tt¨a). 1 Permanent address: MAX-lab, Lund University, Box 118, 221 00 Lund, Sweden. 0368-2048/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2007.02.031
the experimental photoelectron spectra of both the ground and laser-excited atoms as a function of Z in detail. The validity of the coupling scheme for the same atom but with s or p outermost electron has been investigated as well as the evolution of the coupling as a function of atomic number for both the np5 (n + 1)s and np5 (n + 1)p configurations. Previously only the experimental np5 (n + 1)s photoelectron spectra measured using He I and He II resonance lines were compared in [1], and calculated np5 (n + 1)p photoelectron spectra of Na, K and Rb were discussed in [4]. 2. Experiment The experiments were carried out at the undulator beamline I411 at the 1.5 GeV MAX-II storage ring at Lund, Sweden [7,8] with the setup which has been described in more details in [9]. The np5 (n + 1)s spectra were measured with photon energy of 61 eV for Na, 60 eV for K and Rb and 72 eV for Cs. np5 (n + 1)p states can be reached by a two step process including a (n + 1)s → (n + 1)p laser excitation followed by direct np photoemission (Na 3s → 3p followed by direct 2p photoemission). The laser can be tuned to excite either the 2P 2 2 3/2 state or the P1/2 state, only the P1/2 excitation has been studied in this work. The wavelength for this excitation is for Na 589.6 nm, for K 769.9 nm, for Rb 794.8 nm, for Cs 894.3 nm.
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[11]:
3. Calculations For comparison with the experimental np5 (n + 1)s and 5 np (n + 1)p photoelectron spectra we have calculated the relative intensities using analytical LS-coupling [10] and jKcoupling models [11]. The starting point in the analysis is the general expression for the photoionization intensity from polarized atoms [12]. The dynamical information for the angle-averaged total intensity is encoded in the coefficients Bk0 kkγ for k0 = k = kγ = 0 which in the jj coupling scheme read: √ B000 = 3 |αf Jf , lj : JDα0 J0 |2 (1) ljJ
(jK) B000
√ ˆ f2 Jˆ f2ˆl−2 = 3Jˆ 02 K 0
J0
L0
S0
Kf
Jf
j0
2
|dl |2
(4)
l
where we have reduced the one-particle matrix elements to LS-coupling thus removing the j dependence of the continuum waves. Within our geometrical model, the relative intensities of lines in the photoelectron spectra, belonging to same configuration, are independent of one-particle amplitudes and mixing coefficients of the final ionic states. Therefore, we do not need to calculate these quantities. 4. Results
where angular momenta with the subscript 0 refer to the initial atomic state, those with subscript f refer to the final ionic state and quantities without subscripts to the total values obtained by summing the final ion and continuum quantum numbers. The “geometrical” models discussed in this paper amount to various simplifications of the matrix elements in (1) so that the photoelectron intensities become dependent only on the angular momentum quantum numbers and not on the ionization amplitudes. This is often a good approximation if the contributions of various partial waves stay constant in the energy region considered. We work within the assumption that there is at most a single open shell in the ground state of the atom and that the ionization takes place from a closed shell. We also assume that apart from the ionized electron the remaining configuration of the atom remains unperturbed. With these simplifications the many-electron ionization matrix elements can be simply factorized into one-particle ionization matrix elements and coupling coefficients. We have for the pure jj coupling: √ (jj) B000 = 3Jˆ f2 jˆ 02 |dj |2 (2) lj
where √dj = ljdn0 l0 j0 and where we have used the notation xˆ ≡ 2x + 1. In the pure LSJ coupling case the matrix element in (1) becomes, after recoupling and reduction to one-particle matrix elements, to [10]: ⎧ ⎫2 1/2 l0 j0 ⎪ ⎪ ⎨ ⎬ √ (LS) ˆ 2fˆl−2 × ˆ 02 S0 L0 J0 B000 = 3Jˆ 02 Sˆ 0−2 Sˆ f2 Jˆ f2 L j 0 ⎪ ⎪ ⎩ ⎭ j0 Sf L f J f × |dl |2 (3) l
where dl = ldn0 l0 . The last interesting case is the so-called jK-coupling, which amounts to coupling the total angular momentum of the valence hole state to the orbital angular momentum of the valence electron to give an intermediate quantum number Kf . Coupling this to the spin of the valence electron results in the total final ionic state angular momentum Jf . Within the jK-coupling (1) becomes
Figs. 1 and 2 show the experimental np5 (n + 1)s and np5 (n + 1)p photoelectron spectra for Na, K, Rb and Cs, respectively. In normalization of the spectra the sum of intensities for the manifolds is taken to be 100%. The energies and energy splittings of the photoelectron lines were obtained from optical data [13,14]. For Na and K we have used the LS-coupling based notation 2S+1 LJ , which gives four main lines 3 P2,1,0 and 1 P1 in the np5 (n + 1)s spectra. The structure of np5 (n + 1)p photoelectron spectra is more complicated and 10 final states 3 S1 , 3 D3,2,1 , 1 D2 , 3P 1 1 3 2,1,0 , P1 , S0 are generated. The final state D3 is not populated as the transition to that state is forbidden from 2 P1/2 initial state. The order of the LS-coupled final states usually obeys the Hund rules, which predict the order of 3 P2 , 3 P1 , 3 P0 and 1 P1 from lower to higher energy states for the np5 (n + 1)s photoelectron spectra, and of 3 D3 , 3 D2 , 3 D1 , 3 P2 , 3 P1 , 3 P0 , 3 S1 , 1 D2 , 1 P , 1 S for the np5 (n + 1)p photoelectron spectra. In the latter 1 0 case the correct order is , however, 3 S1 , 3 D2 , 3 D1 1 D2 , 1 P1 , 3 P2 , 3P , 3P , 1S . 1 0 0 For Rb and Cs we have used the jK-coupling based energylevel notation of [K]Jf . Configuration np5 (n + 1)s has four states, one with Jf = 0, two with Jf = 1 and one with Jf = 2 and configuration 5p5 6p 10 states, two with Jf = 0, four with Jf = 1 and three with the Jf = 2, the transition to Jf = 3 state being forbidden. The splitting of final states into two groups related to spin–orbit splitting of the np subshell increases with increasing Z. The purities of the two final states of np the photoionization process in LS- and jK-coupling schemes calculated with the Cowan code [15] for ground and laser excited alkali metal atoms are plotted in Fig. 3. The purities of np5 (n + 1)s1 P1 /(2 P1/2 )2 [1/2]1 final state in the first panel show dramatic change as a function of Z, LS-coupling is valid for Na but K already deviates from LS-coupling still being not pure in jK-coupling. Rb and Cs can be well described with a pure jK-coupling. The second panel shows the purity of the np5 (n + 1)p1 P1 /2 P1/2 2 [3/2]1 state which is completely different from the purities plotted in the first panel. Pure LS-coupling fails in describing the states even for Na, opposite to the finding of the first panel. K is not well described either in LS-coupling or jK-coupling. jK-coupling is the best for Rb and shows increasing trend just like in the first panel. Similar trends as shown in the first panel were obtained for the np5 (n + 1)s configuration
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Fig. 1. Experimental np6 (n + 1)s → np5 (n + 1)s photoelectron spectra of Na, K, Rb and Cs. First and lowest panels show the calculated photoelectron spectra for Na and Cs.
for the two final state with same Jf . The trend shown in the second panel is more typical for np5 (n + 1)p configuration due to strong mixing of LS states. Figs. 1 and 2 show experimental np5 (n + 1)s and np5 (n + 1)p photoelectron spectra for alkali metal atoms. The second panel of Fig. 1 shows the experimental 2p5 3s photoelectron spectrum of Na and first panel its intensity distribution predicted by the analytical LS-coupling scheme. The calculated LS-coupling model predicts the 2p photoelectron spectrum of ground state of Na very well. In the 2p5 3p spectrum (second panel of Fig. 2) of Na only six lines are resolved, the 3 P2,1,0 and 1 P1 lines overlap heavily. Calculated 2p5 3p photoelectron spectrum of laser excited
Na clearly deviates from the experimental one. The distribution of the intensity between these final states is not correctly predicted by the pure LS-coupling model, which do not take the mixing of the states into account. Due to the mixing, the 1 P1 and 3 P1 lines gain some intensity from the lines 3 D1 and 3 S1 . Also the states with Jf = 2 mix strongly, and 1 D2 loses part of its intensity to the 3 P2 and 3 D2 lines. Cubaynes et al. [3] calculated the relative intensities taking into account the mixing of states and obtained a better agreement between experiment and calculations. 3p5 4s spectrum of K shows the same main lines 3 P2,1,0 and 1 P1 as the 2p5 3s spectrum of Na, but due to configura-
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Fig. 2. Experimental np6 (n + 1)p2 P1/2 → np5 (n + 1)p photoelectron spectra of Na, K, Rb and Cs. First and lowest panel shows the calculated photoelectron spectra for Na and Cs.
tion interaction between the 3p5 4s 3 P1 and 3p5 3d 1 P1 states the 3p5 3d 1 P1 line appears in the spectrum and steals intensity from the 3p5 4s 3 P1 line. In the 3p5 4p spectrum of K of the lines 3 D1 and 1 P1 start to separate, which shows that the transition toward jK-coupling is already visible because the inner-shell spin–orbit interaction is stronger than in the case of Na. np5 (n + 1)s and np5 (n + 1)p photoelectron spectra of Rb and Cs split up into two groups of lines for the np-hole total angular momenta j = 1/2 and 3/2. The mixing of np5 (n + 1)s and np5 nd configurations in Rb and Cs creates the 4p5 4d 3 P2,1,0 lines
in the 4p5 5s spectrum of Rb and the 5p5 5d lines in the 5p5 6s spectrum of Cs. In np5 (n + 1)p photoelectron spectra of Rb and Cs the np5 nf states were found not to be populated, because configuration interaction between the configurations np5 nf and np5 (n + 1)p is very weak. The splitting of the np shell is bigger in the 5p5 6p spectrum of Cs than in 4p5 5p spectrum of Rb, the final states 5p5 (2 P3/2 )6p2 [3/2]1 and 5p5 (2 P1/2 )6p2 [3/2]1 are even more separated than the corresponding lines in the Rb spectrum. Theoretical 5p5 6s and 5p5 6p photoelectron spectra of Cs calculated by analytical jK approximation are seen in the lowest
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out, however, are the effects of electron correlation which manifest themselves as the interference of the pure-coupling levels, and also energy-dependent effects of the spin–orbit interaction in the one-particle ionization matrix elements. The methods used here are suitable for identifying the lines and general structures of the spectra in the absence of strong correlation. Multiconfiguration effects of the one-particle ionization matrix elements are, on the other hand, negligible as long as we are away from Cooper minima where the individual amplitudes change drastically. The 2p5 3s spectrum of Na predicted with the pure LS-coupling scheme is in good agreement with the experimental spectrum but the model fails in describing the 2p5 3p photoelectron spectrum of Na. When comparing the calculated 5p5 6s and 5p5 6p spectra of Cs, the pure jK-coupling model is seen to describe better the np5 (n + 1)s spectrum, but also for the 5p5 6p spectrum the resemblance between theory and experiment is fairly good. The coupling scheme for Rb and Cs deviates strongly from Na and K because the spin–orbit interaction of the np hole (n = 4 in Rb and n = 5 in Cs) is of the same magnitude or larger than the Coulomb interaction between the hole created and the outermost s electron. Acknowledgments
Fig. 3. Purities of given states. For details, see the text.
panels of Figs. 1 and 2, respectively. Mixing of np5 (n + 1)s and np5 nd configurations was not included in calculations, which is why the lines assigned as 5p5 5d are missing from theoretical predictions. For 5p5 6s configuration, calculations overestimate the intensity of 5p5 (2 P1/2 ) 6s 2 [3/2]1 line in binding energy around 17 eV because mixing of the states with Jf = 1 was omitted. The calculated 5p5 6p photoelectron spectrum of Cs reproduces the experimental spectrum quite well as was pointed out also in Ref. [6]. Appearance of two additional lines in theoretical spectrum can be explained by the mixing of the 5p5 (2 P1/2 )6p2 [3/2]2 and2 [5/2]2 states. Part of the intensity of the line 5p5 (2 P3/2 )6p2 [5/2]2 is going to 5p5 (2 P3/2 )6p2 [3/2]2 line. Because of this mixing, 1 D /(2 P 2 5 2 3/2 ) [3/2]2 line vanishes also in the np (n + 1)p photoelectron spectra of K and Rb but the line can clearly be seen in the 2p5 3p spectrum of Na. As seen from Fig. 3, mixing takes place also between final states with Jf = 1, the line 5p5 (2 P1/2 )6p2 [1/2]1 gets some intensity from the line 5p5 (2 P1/2 )6p2 [3/2]1 . 5. Conclusions The pure coupling schemes investigated in this work are in principle a tool for mapping the strength of the spin–orbit coupling in the different ionic states of the atom. What are left
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