Inner-shell excitation of alkali-metal atoms

Pram~na - J. Phys., Vol. 35, No. 1, July 1990, pp. 89-94. Printed in India. Inner-shell excitation of alkali-metal atoms S N TIWARY* International Centre for Theoretical Physics, Miramare, Trieste, Italy * Permanent address: Department of Physics, L. S. College, Bihar University, Muzaffarpur 842001, India MS received 23 January 1990 Abstract. Inner-shell excitation of alkali-metal atoms, which leads to auto-ionization, is investigated. Comparison is made with other available data. Basic difficulties in making accurate calculations for inner-shell excitation process are discussed. Suggestions are made for further study of inner-shell process in atoms and ions. Keywords, R-matrix; auto-ionization. PACS No. 34.80 Studies of the electron impact excitation of the inner-shell of alkali-metal atoms and alkali-like ions have been of growing interest for both theorists and experimentalists. One of the primary reasons for this interest stems from the fact that inner-shell excitation may lead to auto-ionization which plays a very important role in explaining the structure observed in the total ionization cross section curve. In recent years, a number of theoretical calculations (Tiwary 1976, 1981, 1982, 1983a-g, 1985, 1986, 1989, 1990; Tiwary et al 1983a, b, 1985, 1988; Tiwary and Rai 1973, 1975, 1976; Tiwary and Nicolaides 1983; Faisal and Tiwary I980; Hibbert et al 1982; Kingston et al 1987; Srivastava and Rai 1977; Berrington et al 1978; Peterkop 1977) and experimental observations (Nygaard (1975) and Pejcev and Ross (1977)) have been made but there are striking discrepancies between the prediction of various theoretical methods in both qualitative as well as quantitative behaviour. Consequently, the existing theoretical situations are of special interest from both scattering and structure point of view in order to resolve the discrepancies. Over the past decade, several quantum-mechanical approximations, e.g. plane wave Born approximation (PWBA), modified plane wave Born approximation (MPWBA), Vainshtein approximation (VPSA), Glauber approximation (GA), Coulomb Born approximation (CBA), asymptotic Green function approximation (AGFA), distortedwave Born approximation (DWBA), quantum defect theory (QDT), Fano method, diagonalization method, close-coupling (CC); hyperspherical coordinate method, complex-rotation method, R-matrix method etc., have been employed to study the inner-shell process in atoms and ions. Nygaard (1975) and Pejcev and Ross (1977) have measured the total ionization cross sections for the alkali-metal atoms by electron impact and have observed the structure in the results. This structure has been attributed to the excitation of the inner-shell electron which leads to auto-ionization. We have investigated extensively the inner-shell process in alkali-metal atoms and 89

90 S N Tiwary Li 1 50- ~,.I DWB lsa 2s2 S~"~'ts 2sa zs T O %{ with exchange) N'-o o 30- c o CBA. - "ow8...,. {without exchonge)... 8 \ u~ 10 - PWBA R- Matrix I 1 5 0 100 150 200 Electron impact energy {ev) Figare I. Electron impact excitation cross section for the I s2s z auto-ionizing level in lithium atomic system. alkali-like ions from both scattering and structure points of view. For the case of the simplest alkali-metal atom i.e. lithium (Li), the following scattering reaction has been studied e- + Li(ls22s)~e - + Li(ls2sZ). (1) Figure 1 displays the present R-matrix results obtained using single-configuration Hartree-Fock were function from Clementi and Roetti (1974) along with other available results. It is clear from the figure that DWB with exchange and R-matrix methods agree with each other qualitatively but differ significantly in nature from PWBA and DWB without exchange and similar other approximations (not shown in the figure). DWB with exchange and R-matrix cross sections are still rising in the close vicinity of the threshold and decrease monotonically at high impact energies, whereas the PWBA and DWB without exchange and similar other approximations show a broad and flat maximum away from threshold. Unfortunately, no direct experimental evidence is available for inner-shell excitation which leads to autoionization in the alkali-metal atoms by electron impact but from the results of Nygaard, Pejcev and Ross, it is clear that the rise is abrupt in the cross section. Figure 2 shows the angular distribution of electrons in the case of lithium in the DWB with exchange at different impact energies. At 70 ev, the differential cross section decreases with the increasing scattering angle and reaches a minimum value at about 60 and then rises in the backward scattering region. It is interesting to note that the cross section in the backward direction is considerably larger than the forward direction. At 80eV, the cross section is almost the same in the forward and backward directions. At high energies, the forward scattering dominates over the backward which is the usual behaviour. From this pattern of angular distribution as well as the abrupt rise in the integrated cross section obtained in the distorted-wave Born approximation with exchange and the R-matrix method, it is well established that

Inner-shell excitation of alkali-metal atoms 91 '0 X I t. U') NO O C O,4- "1.6 1.2 0.8 I\ Li Z 1 s22s 2 S ---~+s 2s" 2S 1 DWB 7(~ t'\i / t. U C 0.4 120 (ev) ~ 121 o I I I I 40 80 120 Scattering angle (de<j) Figure 2. Angular distribution of inelastically scattered electrons at incident energies 70, 80, 100 and 120eV. 7 - x % 5 - C.o O.4-- o 1 I I I 30 50 70 90 110 EleCtron impact energy tev) Figure 3. Total electron impact excitation cross section for the lowest-lying auto-ionizing level in sodium. Curve A, R-matrix results; Curve B, first Born approximation results; Curve C, Glauber approximation results. the exchange is significantly important for the resonance type character in the cross section. It also reflects that inclusion of exchange is essential. Figures 3 and 4 exhibit the low-energy electron impact integrated cross sections of the lowest-lying auto-ionizing levels generated due to the inner-shell excitation 1 s 2 2s 2 2p 6 3s -~ 1 s 2 2s 2 2p ~ 3s 2 and 1 s 2 2S 2 2p 6 3S 23p 6 45 --~ 1 s 2 2s 2 2p ~ 3s2 3p5 4s 2 complex transition in sodium and potassium respectively, obtained using single configuration

92 S N Tiwary N lo -- 5 X ~0 o 83,q I I I I 20 O0 IO0 Electron impact energy (ev) Figure 4. Integrated cross section for the excitation of the auto-ionizing level in potassium. Curve A, R-matrix results; Curve B, first Born approximation results; Curve C, Glauber approximation results. Hartree-Fock wave functions for both initial and final states within the R-matrix. The present R-matrix results are compared with other available PWBA and GA results which have been also obtained using Hartree-Fock functions. The general features of these results are similar to those of lithium. From the above considerations, we conclude that PWBA, GA, VPSA and similar approximations are not useful for inner-shell processes in atoms. Only the DWB with exchange, R-matrix or similar approximations are reliable for such study. It is interesting to note that DWB without exchange supports the PWBA behaviour, whereas with exchange it follows the R-matrix predictions. It indicates that the exchange plays an extremely important role for the inner-sheu process in atoms. We propose a new mechanism for the exchange operation as follows: e- + ls22s~ ls22s2 ~ ls2s 2 +e-. (2) The incident electron penetrates the lithium electron charge cloud configuration ls 22s and the R-matrix sees the scattering system ls22s 2 within the boundary and finally the one core electron is ejected. In recent years, we have performed extensive calculations for the theoretical excitation energies and optical oscillator strengths of elements for the inner-shell excitation transitions in the potassium iso-electronic sequence using the Hartree-Fock (HF) and configuration interaction (CI) wave functions. Our investigation shows that the Hartree-Fock length and velocity (fl and f~ respectively) forms of the oscillator strengths may differ by a factor of two and the effect of correlation decreases with the increasing atomic number (Z). The CI fl and f~ are in satisfactory agreement in all systems (see table 1). A large number of the configurations with all sorts of correlations e.g. external correlations, semi-internal and internal correlations may be included. The general form of our configuration interaction wave functions is as follows: M ~P(LS) = ~ aidpi(~ils) (3) i=1 where all quantities are defined elsewhere (see our Tiwary et al 1983).

Inner-shell excitation of alkali-metal atoms 93 Table 1. Comparison of excitation energies (AE in atomic unit) and optical oscillator strengths of length (f~) and velocity (f~) forms for transitions 3p63d 2D--*3p53d2 2pO, 2Do ' 2FO obtained using Hartree-Foek (HF) and configuration-interaction (C1) wave function. Sc 2 + Ti3 + V 4 + Wave Transition function AE fl f~ AE fl f~ AE fl f~ 2D~2p HF 1.586 1"18 0.63 1"855 1"12 0-59 2-098 1-03 0.54 CI 1"493 0.71 0.78 1"762 0-65 0-72 2"023 0-78 0-86 2D-'}2D HF 1-630 2.46 1.24 1-904 2"31 1'15 2.151 2"13 1-06 CI 1-531 1.40 1'16 1,846 0.56 0.44 2-086 I'82 1-55 2D-'}2F HF 1"455 1"80 1"14 1"695 1"69 1"06 1"917 1"56 0"98 CI 1-396 0.64-0.644 0-646 1-21 1-16 1-847 0-46 0-44 These results demonstrate that the inclusion of correlations is indispensable to obtain reliable results. They also indicate that if HF wave functions are used for the calculations of cross sections the result may be off by as much as a factor of two. There are numerous practical difficulties associated with accurate scattering as well as structure calculations for the inner-shell excitation process in many electron atoms or ions. For the scattering calculation reliable wave functions are needed for both the initial and final target states. HF wave functions are not adequate for heavy atoms or ions because of complexity, for multi-dectron atoms, or ions, a huge number of configurations are required which make the computational problem cumbersome and calculations become very expensive and in many cases not feasible. For example, Tiwary et al (1983b) have found that a single configuration yielded 68 states which are very close to each other and interact strongly. We have to solve 68 coupled second order integro-differential equations. Another challenging situation is how to handle double continuum electrons which interact via the Coulomb operator 1/r12 which play an important role in the threshold behaviour. This situation occurs in the excitation of the inner-shell followed by auto-ionization. Experimental study of such process is very limited since the third generation (e, 2e) coincidence experiment must be used. Finally, we would like to make some constructive and fruitful suggestions for obtaining reliable results for auto-ionization. 1. PWBA, GA, VPSA and similar approximations are not appropriate where DWB with exchange, R-matrix and similar methods can yield reliable results. 2. Hartree-Fock description for the atomic states involved in the transitions is not adequate especially for complex systems. Configuration interaction wave functions, which include all types of correlations, are suitable for accurate calculations. Correlations must be included same amount in both states otherwise imbalance of correlations can destroy the results. Our experience also suggests that the inclusion of the pseudo-orbitals in the basis set is essential for obtaining reliable results. 3. Interactions between the auto-ionizing state and the associated continuum, radiative and non-radiative processes between the scattered and the ejected electrons should also be incorporated in elaborate calculations.

94 S N Tiwary Acknowledgements The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. He would also like to thank Professors J Macek, D H Madison, B Crasemann, G Dujardin, H Le Rouzo, L Hellner and M Gaillard for their encouragement and fruitful suggestions. He is also grateful to referees for valuable suggestions and finally to Bihar University, India, for leave. References Berrington K A, Burke P G, Dourneuf M Le, Robb W D, Taylor K T and Ky Lan Vo 1978 Comput. Phys. Commun. 14 367 Clementi E and Roetti C 1974 At. Data Nucl. Data Table 14 177 Faisal F H M and Tiwary S N 1980 I/erh. Dtsh Ges. 29 597 Hibbert A, Kingston A E and Tiwary S N 1982 J. Phys. BI5 L643 Kingston A E, Hibbert A and Tiwary S N 1987 J. Phys. 1120 3907 Nygaard K J 1975 Phys. Rev. All 1475 Pejcev V and Ross K J 1977 J. Phys. BI0 L291 Peterkop R 1977 Theory of ionization of atoms by electron impact (Trans.) Ed. D G Hummer (JILA, Colorado Associated University Press) Srivastava R and Rai D K 1977 J. Phys. BI0 269 Tiwary S N 1976 Studies of excitation, auto-ionization and ionization of atoms Ph.D. Thesis, Banaras Hindu University Tiwary S N 1981 J. Phys. BI4 2951 Tiwary S N 1982 Chem. Phys. Lett. 93 47 Tiwary S N 1983a Chem. Phys. Lett. 97 222 Tiwary S N 1983b J. Phys. BI6 L459 Tiwary S N '1983c Phys. Reo. A28 751 Tiwary S N 1983d d. Phys. B16 L247 Tiwary S N 1983 Chem. Phys. Lett. 96 333 Tiwary S N 1983f Astrophys. J. 269 803 Tiwary S N 1983g Astrophys. J. 272 781 Tiwary S N 1985 Phys. Rev. A32 627 Tiwary S N 1986 National workshop, Banaras Hindu University Tiwary S N 1989 Structure and scattering calculations of atoms and ions, Ph.D. thesis, Bihar University Tiwary S N 1990 Can. J. Phys. (to be published) Tiwary S N, Burke P G and Kingston A E 1983a XIII ICPEAC, Berlin, West Germany, p. 20 Tiwary S N, Kingston A E and Hibbert A 1983b J. Phys. BI6 2457 Tiwary S N, Macgk J and Madison D H 1985 Phys. Rev. A32 254 Tiwary S N and Nicolaides C A 1983 Che~ Phys. Lett. 97 283 Tiwary S N and Rai D K 1973 Phys. Lett. A43 411 Tiwary S N and Rai D K 1975 J. Phys.!!8 1109 Tiwary S N and Rai D K 1976 J. Phys.!19 631 Tiwary S N, Singh A P, Singh D D and Sharma R J 1988 Can. J. Phys. 66 405