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Distorted wave approach to calculate Auger transition rates of ions in metals
Nuclear Instruments and Methods in Physics Research B 182 (2001) 8±14
www.elsevier.com/locate/nimb
Distorted wave approach to calculate Auger transition rates of ions in metals Stefan A. Deutscher
a,*
, R. Dõez Mui~ no a, A. Arnau b, A. Salin c, E. Zaremba
d
a
Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 San Sebasti an, Spain Departamento de Fõsica de Materiales, UPV/EHU, Aptdo. 1072, 20080 San Sebasti an, Spain Laboratoire de Physico-Chimie Mol eculaire, UMR 5803 CNRS-Universit e de Bordeaux, 351 Cours de la Lib eration, 33405 Talence Cedex, France d Department of Physics, Queen's University at Kingston, Ont., Canada K7L 3N6 b
c
Abstract We evaluate the role of target distortion in the determination of Auger transition rates for multicharged ions in metals. The required two electron matrix elements are calculated using numerical solutions of the Kohn±Sham equations for both the bound and continuum states. Comparisons with calculations performed using plane waves and hydrogenic orbitals are presented. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 78.70.)g; 79.20.Rf; 73.90.+f Keywords: Highly charged ions; Many-body response; Density functional
1. Introduction The neutralization and relaxation of multicharged ions in solids is a complex problem involving many dierent processes [1,2]: electronic processes like resonant electron transfer (oneelectron process) or Auger processes (two-electron), and radiative as well as many-electron processes. Perturbative approaches using the unperturbed target electron wave functions to evaluate quantum mechanical transition amplitudes
*
Corresponding author. Tel.: +34-94-30-18-174; fax: +3494-30-15-600. E-mail address: [email protected] (S.A. Deutscher).
are not valid since the slow multicharged ion strongly perturbs the target electron states. This means that the plane wave Born approximation (PWBA), or equivalently linear response theory (LRT), has to be improved upon, at least to account for the distortion of the target electron wave functions introduced by the projectile ion. The way to proceed is similar to the case of atomic physics when one improves PWBA introducing the continuum distorted wave (CDW) method [3,4] or further developments of it like CDW-EIS [5], whenever a full coupled channel calculation [6] is not feasible. In the case of solid targets, one way to account for the distortion of the valence electron wave functions is a self-consistent non-linear screening calculation within density functional
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 6 4 8 - 6
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
theory (DFT), applied to a static impurity in jellium [7,8]. Although it has some limitations, it allows to obtain a complete description of the electron states for simple free electron like metals. In this work, we focus on Auger processes involving metal valence electrons, such as KVV or LVV transitions, although intra-atomic Auger processes (KLV, LMV, KLL, etc.) will be considered as well. The role of valence electrons in the neutralization and relaxation of multicharged ions is still not well understood: experiments show [9± 18] that the time-scale for these processes is shorter than what has been predicted by dierent theoretical models based on atomic [19±22] or solidstate physics [23±27]. First steps have already been taken but none of them has accounted for the distortion of the wave functions in calculating the system response: a preliminary calculation [28] using hydrogenic states and plane waves showed the relative importance of intra-atomic and Auger processes in which valence electrons are involved. Later the role of the distortion of the wave functions was partially accounted for [29,30] in a study of L-shell ®lling rates of N and Ne ions in metals. However, a full non-linear calculation of the selfconsistent system response is still an unsolved problem that we try to answer in dierent steps: (i) evaluation of the two-electron matrix elements using a complete basis set that accounts for the presence of the ion in the metal, and (ii) self-consistent calculation of the system response function at the random phase approximation (RPA) level. In this work, we address step (i) and show that it is equivalent to a Hartree-only calculation of the system response, which is required prior to the solution of the full integral equation. The full selfconsistent response includes collective excitations of the system (plasmons) and has an intrinsic fundamental interest, although it is not expected to change signi®cantly the value of the Auger transition rates, except for low transition energies. However, such a calculation is computationally very demanding. In practice, the solution of the integral equation is better obtained using a basis set that discretizes the continuum and allows a reduction of the problem to a matrix inversion, as done, e.g., in surface physics problems [31]. One
9
possible choice for such a basis set is Sturmian functions [32] shown to be useful in surface physics problems [33].
2. Theory We have calculated the Auger rates evaluating the two-electron matrix elements of the Coulomb interaction v r r0 1=jr r0 j between electron states that are described at dierent levels of approximation (see Section 3). In the following, atomic units are used throughout, unless otherwise stated. The rate Cca of Auger capture to the bound state jai is given by X vjk1 k2 ij2 Cca 2p jhak3 j^ k1 ;k2 ;k3
d ek1 ek2
ek3 ea ;
1
where jk1 i and jk2 i are occupied valence electron states in the Fermi sea, jk3 i is an unoccupied state above the Fermi level, and jai is a bound state of the embedded ion. ek1 , ek2 , ek3 and ea are the energies of the states, respectively. In Fig. 1, we
Fig. 1. Scheme of Auger transitions in the screened Coulomb potential V r of an ion embedded in jellium; the wave functions are sketched for the bound (jai) and continuum (jk1;2;3 i) states. kF and r denote the Fermi momentum and radial coordinate, respectively.
10
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
sketch a typical Auger transition involving two valence electrons (KVV, for instance). The rates for other Auger processes in which one or more intra-atomic decay channels are involved (such as the KLV or KLL transitions) can be calculated in a similar way, substituting the corresponding continuum state by the appropriate bound state. When the distortion of the metal states due to the presence of the ion is neglected (i.e., in LRT), the jk2;3 i states can be described by plane waves and the two-electron matrix elements appearing in Eq. (1) can be factorized as the product of a oneelectron matrix element times the imaginary part of the response function [34]. In general, this simpli®cation is not possible. However, an expression more general than Eq. (1) can be obtained using a self-energy formalism [31] in terms of the imaginary part of the screened interaction W r; r0 ; x that is related to the response function v r; r0 ; x according to Z XZ dr dr0 /a r/k1 r0 Im W r; r0 ; x Cca 2 k1
/a r0 /k1 r; where x ek1 W r; r0 ; x is 0
2
ea , the screened interaction Z
0
W r; r ; x v r r Z dr2 v r
dr1 r1 v r1 ; r2 ; xv r2
r0 ; 3
and v r;r0 ;x v0 r;r0 ;x v r1
Z
Z dr1
dr2 v0 r;r1 ;x
r2 K xc r1 ;r2 ; xv r2 ; r0 ; x : 4
The self-consistent response function v r; r0 ; x can be calculated at dierent levels of approximation by solving Eq. (4) in which the zero-order response v0 r; r0 ; x and the exchange and correlation kernel K xc r1 ; r2 ; x appear in the general case. Neglecting K xc gives the RPA result, while
v v0 is the Hartree approximation. The imaginary part of v0 is given by Im v0 r; r0 ; x X p /k2 r/k3 r/k3 r0 /k2 r0 d x ek3 ek2 : k2 ;k3
5
A direct substitution of Eq. (5) in Eq. (3), and then of ImW r; r0 ; x in Eq. (2) gives Eq. (1). 3. Results and discussion The evaluation of Eq. (1) for the Auger capture rate Cca to the bound state jai requires the calculation of four wave functions: one for the jai bound state, and three for the jki i continuum states i 1; 2; 3. Each of them can be calculated at dierent levels of approximation. The simplest approach is to use a hydrogenic wave function for the bound state, and plane waves for the continuum states. A more sophisticated description of any of these wave functions, including the target distortion by the ion, is achieved by numerically solving the Kohn±Sham equations self-consistently. In the latter description, the bound state wave function is screened and shifted up in energy, and the continuum wave functions are strongly modi®ed due to the charge piling up in the vicinity of the ion. In Fig. 2 we plot the Auger capture rate per spin state CKVV to the bound 1s state (K-shell) as a function of the ion charge Z and for rs 2 (rs is de®ned in terms of the background electronic 1=3 density n0 , as rs 3= 4pn0 ). Only one K-shell hole is considered in the initial state electronic con®guration. In the ®gure, results are shown for several combinations of wavefunctions calculated at dierent levels of approximation: the sequence of four letters labels the approximation in which the respective wave functions of the jai, jk1 i, jk2 i and jk3 i states are calculated. Hydrogenic wave functions are labelled with `h', plane-wave continuum wave functions with `b' (Bessel functions), and bound or continuum wave functions calculated self-consistently with `j' (after the jellium in which the ion is embedded).
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
Fig. 2. KVV Auger rate at dierent levels of approximation as a function of the nuclear charge Z of the embedded ion. Note the appearance of kinks when continuum states are described with jellium wave functions.
For ion charges Z P 6 successive improvement of the wave functions for the jai, jk1 i and jk2 i states from a lower level of approximation to the corresponding self-consistent one (h; b ! j) increases the Auger rate each time by about one
11
order of magnitude. For smaller ion charges, the dierence among the various descriptions can be smaller, but it remains sizable. It can be seen from the nearly coinciding curves for the jjjb and jjjj case that only the high-energy jk3 i continuum state is relatively unaected by the presence of the ion and can be described by a plane wave. The most sophisticated calculation (jjjj) of the Auger rate CcK shown in Fig. 2 exhibits kinks at Z 8 and Z 16. They are related to the appearance of bound p states (2p and 3p, respectively) in the initial con®guration of the ion. For Z 6 8 (16), the 2p (3p) state of the ion merges into the continuum and a large amount of continuum electronic charge surrounds the atom, increasing the Auger capture rate. For larger Z, on the other hand, the respective p state becomes bound and the screening electronic charge has a pronounced bound character, decreasing CcK . The appearance of bound states opens new, intra-atomic, channels. This is shown in Fig. 3, where we plot the Auger rate per spin state calculated using the jjjj approach for rs 2, for all processes signi®cant in the ®lling of the K-shell. In the ®gure, KVV (capture from the continuum),
Fig. 3. Contributions to the total K-shell ®lling rate (solid line) in the jjjj approximation as a function of the nuclear charge Z of the embedded ion: rates per spin state for the KVV (dashed line) and intra-atomic (symbols) transitions. The arrows on top of the graph mark the onset of the dierent bound states, the horizontal dashes their binding energies Eb Z.
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S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
KLV, KLL, and some of the insigni®cant KLM rates are shown. For reference, we also plot the energy levels of the bound states. To allow comparison with experimental data [35], the total Auger K-shell ®lling rate CK is drawn with a solid line, as the sum of all the individual rates after inclusion of the necessary statistical factors [29]. Whenever a bound state appears a new channel opens, and its inclusion results in the disappearance of the kinks. Therefore, the total rate CK shows a smooth dependence on Z. So far, we have shown results in which the initial electronic con®guration of the ion is ®xed and the nuclear charge is varied. Let us now turn to a dierent situation in which we follow the ®lling of the L-shell of an ion of ®xed charge (e.g., Ne), as a function of the number nL of electrons (or number nL of holes) already present in the L-shell. In Fig. 4, we show the Auger capture rate per spin state, Cc2s , to the bound 2s state of a Ne ion, as a function of the number of L-shell holes in the ®nal state. Three dierent electronic densities (corresponding to dierent values of rs ) are considered. The initial electronic con®guration of the ion does not include any holes in shells other than the L-shell. It can be seen from the ®gure that only the total number of initial L-shell holes matters. The partly overlapping symbols show, on the other hand, that dierent 2s and 2p sub-shell distribu-
Fig. 4. Comparison of CLVV rates as a function of the number of L-shell holes in jjbb and jjjj approximation for dierent rs values.
tions of the holes hardly change the calculated Cc2s rate. Results obtained at the jjbb and jjjj levels are compared. The dierence between these two levels of approximation is specially signi®cant for high number of holes in the L-shell, for which the medium is strongly perturbed. In [29], the necessity of accounting for the ion perturbation in the calculation of the captured electron initial wave function was shown. Fig. 4 shows that the perturbation of the initial wave function of the emitted electron due to the presence of the ion also plays a role in the calculation of the Auger rate. The role of more weakly bound, M-shell, states in the ®lling of the 2s bound state of Ne is shown in Fig. 5, in which we plot the Auger capture rate Cc2s per spin state, and the C2s ®lling rate per spin state, calculated as the sum of the LVV (Cc2s ), LMV (CLMV ), and LMM (CLMM ) processes. All calculations are performed at the jjjj level of approximation. The LMV and LMM processes are possible only when the 3s state of the ion is bound. The small arrows in Fig. 5 mark the onset of 3s binding for the dierent electron densities: for rs 2 (3) at least nL 4 (2) holes are needed, while for rs 1:5 the 3s state is not bound. It can be seen that for nL 0 there is a strong background density dependence and the intra-atomic channel is
Fig. 5. Ne L-shell ®lling rate CLVV in the jjjj approximation as a function of the number of L-shell holes (solid line) for dierent rs values. The arrows mark the onset of 3s binding, and the dashed lines the total rates including intra-atomic transitions involving the M-shell.
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
not opened yet. The latter means that the continuum electrons are weakly perturbed, as can be seen in Fig. 4 by comparing the jjjj and jjbb descriptions. However, when nL is large, the density dependence of the rates is much weaker and the intra-atomic channel is opened. In this case, the target electronic states are strongly perturbed by the presence of the impurity and the screening becomes atomic-like, i.e., the wavefunctions have Coulombic character close to the embedded ion. 4. Conclusions We have found that a distorted-wave description of the one-electron wave functions, both for the continuum and the bound states, is essential in the calculation of Auger rates: they change several orders of magnitude when the plane wave states (b) are replaced by the proper continuum states (j). The relative importance of KVV processes in the ®lling of the K-shell decreases with growing nuclear charge Z, as new intra-atomic channels become available. When the contributions from all the channels are added up, the total K-shell ®lling rate CK saturates for Z > 10 at about 8 10 3 a.u. We have also shown that the dependence of the C2s ®lling rate on the background electronic density is much stronger when the L-shell of the ion is almost ®lled. In other words, the dependence of the L-shell ®lling rate on the electronic density of the target is more important in the last steps of the neutralization sequence. In summary, non-linear eects in the response of the solid have to be included when calculating Auger rates of processes in which valence electrons are involved. Acknowledgements We wish to thank P.M. Echenique for many useful discussions. Help and support by Euskal Herriko Unibertsitatea, Eusko Jaurlaritza, Gipuzkoako Foru Aldundia and the Spanish D.G.I.C.Y.T. is gratefully acknowledged. One of us (E.Z.) acknowledges help and support by DIPC.
13
References [1] A. Arnau et al., Surf. Sci. Rep. 27 (1997) 113. [2] J. Burgd orfer, Review of Fundamental Processes and Applications of Atoms and Ions: Atomic Collisions with Surfaces, World Scienti®c, Singapore, 1993, p. 517. [3] I.M. Cheshire, Proc. Phys. Soc. 84 (1964) 89. [4] Dz. Belkic, R. Gayet, A. Salin, Phys. Rep. 56 (1979) 279. [5] P.D. Fainstein, V.H. Ponce, A.E. Martinez, Phys. Rev. A 47 (1993) 3055. [6] G. Schiwietz, Phys. Rev. A 42 (1990) 296. [7] C.O. Almbladh, U. von Barth, Z.D. Popovic, M.J. Stott, Phys. Rev. B 14 (1976) 2250. [8] E. Zaremba, L.M. Sander, H.B. Shore, J.H. Shore, J. Phys. F 7 (1977) 1763. [9] J.-P. Briand, L. de Billy, P. Charles, S. Essabaa, P. Briand, R. Geller, J.P. Desclaux, S. Bliman, C. Ristori, Phys. Rev. Lett. 65 (1990) 159. [10] F.W. Meyer, S.H. Overbury, C.C. Havener, P.A. Zeijlmans van Emmichoven, D.M. Zehner, Phys. Rev. Lett. 67 (1991) 723. [11] J. Limburg, S. Schippers, R. Hoekstra, R. Morgenstern, H. Kurz, F. Aumayr, HP. Winter, Phys. Rev. Lett. 75 (1995) 217. [12] H.J. Andra, A. Simionovici, T. Lamy, A. Brenac, G. Lamboley, J.J. Bonnet, A. Fleury, M. Bonnefoy, M. Chassevent, S. Andriamonje, A. Pesnelle, Z. Phys. D 21 (1991) 135. [13] J. Thomaschewski, J. Bleck-Neuhaus, M. Grether, A. Spieler, N. Stolterfoht, Phys. Rev. A 57 (1998) 3665. [14] P.A. Zeijlmans van Emmichoven, C.C. Havener, F.W. Meyer, Phys. Rev. A 43 (1991) 1405. [15] H. Kurz, F. Aumayr, C. Lemell, K. T oglhofer, HP. Winter, Phys. Rev. A 48 (1993) 2192. [16] D. Schneider, M.A. Briere, J.W. McDonald, J. Biersack, Radiat. E. Def. Solids 127 (1993) 113. [17] S. Ninoyima, Y. Yamazaki, K. Sawatari, M. Iruko, K. Komaki, T. Azuma, K. Kuroki, M. Sekiguchi, Nucl. Instr. and Meth. B 115 (1996) 177. [18] S. Winecki, C.L. Cocke, D. Fry, M.P. St ockli, Phys. Rev. A 53 (1996) 4228. [19] C.P. Bhalla, Phys. Rev. A 8 (1973) 2877. [20] N. Vaeck, J. Hansen, J. Phys. B28 (1995) 3523. [21] S. Schippers, J. Limburg, J. Das, R. Hoekstra, R. Morgenstern, Phys. Rev. A 50 (1994) 540. [22] S. Schippers, J. Limburg, J. Das, R. Hoekstra, R. Morgenstern, Phys. Rev. A 50 (1994) 4429(E). [23] J. Burgd orfer, P. Lerner, F.W. Meyer, Phys. Rev. A 44 (1991) 5674. [24] U. Wille, Nucl. Instr. and Meth. B 125 (1997) 310. [25] N. Stolterfoht, A. Arnau, M. Grether, R. K ohrbr uck, A. Spieler, R. Page, A. Saal, J. Thomaschewski, J. BleckNeuhaus, Phys. Rev. A 52 (1995) 445. [26] A. Arnau, P.A. Zeijlmans van Emmichoven, J.I. Juaristi, E. Zaremba, Nucl. Instr. and Meth. B 100 (1995) 279. [27] A. Arnau, R. K ohrbr uck, M. Grether, A. Spieler, N. Stolterfoht, Phys. Rev. A 51 (1995) 3399.
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[28] R. Dõez Mui~ no, A. Arnau, P.M. Echenique, Nucl. Instr. and Meth. B 98 (1995) 420. [29] R. Dõez Mui~ no, N. Stolterfoht, A. Arnau, A. Salin, P.M. Echenique, Phys. Rev. Lett. 76 (1996) 4636. [30] R. Dõez Mui~ no, A. Salin, N. Stolterfoht, A. Arnau, P.M. Echenique, Phys. Rev. A 57 (1998) 1126. [31] P.M. Echenique, J.M. Pitarke, E.V. Chulkov, A. Rubio, Chem. Phys. 251 (2000) 1. [32] M. Rotenberg, Adv. At. Mol. Phys. 6 (1970) 233.
[33] S.A. Deutscher, X. Yang, J. Burgd orfer, Phys. Rev. A 55 (1997) 466. [34] P.M. Echenique, F. Flores, R.H. Ritchie, in: H. Ehrenreich, D. Turnbull (Eds.), Solid State Physics: Advances in Research and Applications, Vol. 49, Academic Press, New York, 1990, p. 49. [35] M. Grether, A. Arnau, R. K ohrbr uck, A. Spieler, N. Stolterfoht, Nucl. Instr. and Meth. B 115 (1996) 157.
www.elsevier.com/locate/nimb
Distorted wave approach to calculate Auger transition rates of ions in metals Stefan A. Deutscher
a,*
, R. Dõez Mui~ no a, A. Arnau b, A. Salin c, E. Zaremba
d
a
Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 San Sebasti an, Spain Departamento de Fõsica de Materiales, UPV/EHU, Aptdo. 1072, 20080 San Sebasti an, Spain Laboratoire de Physico-Chimie Mol eculaire, UMR 5803 CNRS-Universit e de Bordeaux, 351 Cours de la Lib eration, 33405 Talence Cedex, France d Department of Physics, Queen's University at Kingston, Ont., Canada K7L 3N6 b
c
Abstract We evaluate the role of target distortion in the determination of Auger transition rates for multicharged ions in metals. The required two electron matrix elements are calculated using numerical solutions of the Kohn±Sham equations for both the bound and continuum states. Comparisons with calculations performed using plane waves and hydrogenic orbitals are presented. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 78.70.)g; 79.20.Rf; 73.90.+f Keywords: Highly charged ions; Many-body response; Density functional
1. Introduction The neutralization and relaxation of multicharged ions in solids is a complex problem involving many dierent processes [1,2]: electronic processes like resonant electron transfer (oneelectron process) or Auger processes (two-electron), and radiative as well as many-electron processes. Perturbative approaches using the unperturbed target electron wave functions to evaluate quantum mechanical transition amplitudes
*
Corresponding author. Tel.: +34-94-30-18-174; fax: +3494-30-15-600. E-mail address: [email protected] (S.A. Deutscher).
are not valid since the slow multicharged ion strongly perturbs the target electron states. This means that the plane wave Born approximation (PWBA), or equivalently linear response theory (LRT), has to be improved upon, at least to account for the distortion of the target electron wave functions introduced by the projectile ion. The way to proceed is similar to the case of atomic physics when one improves PWBA introducing the continuum distorted wave (CDW) method [3,4] or further developments of it like CDW-EIS [5], whenever a full coupled channel calculation [6] is not feasible. In the case of solid targets, one way to account for the distortion of the valence electron wave functions is a self-consistent non-linear screening calculation within density functional
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 6 4 8 - 6
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
theory (DFT), applied to a static impurity in jellium [7,8]. Although it has some limitations, it allows to obtain a complete description of the electron states for simple free electron like metals. In this work, we focus on Auger processes involving metal valence electrons, such as KVV or LVV transitions, although intra-atomic Auger processes (KLV, LMV, KLL, etc.) will be considered as well. The role of valence electrons in the neutralization and relaxation of multicharged ions is still not well understood: experiments show [9± 18] that the time-scale for these processes is shorter than what has been predicted by dierent theoretical models based on atomic [19±22] or solidstate physics [23±27]. First steps have already been taken but none of them has accounted for the distortion of the wave functions in calculating the system response: a preliminary calculation [28] using hydrogenic states and plane waves showed the relative importance of intra-atomic and Auger processes in which valence electrons are involved. Later the role of the distortion of the wave functions was partially accounted for [29,30] in a study of L-shell ®lling rates of N and Ne ions in metals. However, a full non-linear calculation of the selfconsistent system response is still an unsolved problem that we try to answer in dierent steps: (i) evaluation of the two-electron matrix elements using a complete basis set that accounts for the presence of the ion in the metal, and (ii) self-consistent calculation of the system response function at the random phase approximation (RPA) level. In this work, we address step (i) and show that it is equivalent to a Hartree-only calculation of the system response, which is required prior to the solution of the full integral equation. The full selfconsistent response includes collective excitations of the system (plasmons) and has an intrinsic fundamental interest, although it is not expected to change signi®cantly the value of the Auger transition rates, except for low transition energies. However, such a calculation is computationally very demanding. In practice, the solution of the integral equation is better obtained using a basis set that discretizes the continuum and allows a reduction of the problem to a matrix inversion, as done, e.g., in surface physics problems [31]. One
9
possible choice for such a basis set is Sturmian functions [32] shown to be useful in surface physics problems [33].
2. Theory We have calculated the Auger rates evaluating the two-electron matrix elements of the Coulomb interaction v r r0 1=jr r0 j between electron states that are described at dierent levels of approximation (see Section 3). In the following, atomic units are used throughout, unless otherwise stated. The rate Cca of Auger capture to the bound state jai is given by X vjk1 k2 ij2 Cca 2p jhak3 j^ k1 ;k2 ;k3
d ek1 ek2
ek3 ea ;
1
where jk1 i and jk2 i are occupied valence electron states in the Fermi sea, jk3 i is an unoccupied state above the Fermi level, and jai is a bound state of the embedded ion. ek1 , ek2 , ek3 and ea are the energies of the states, respectively. In Fig. 1, we
Fig. 1. Scheme of Auger transitions in the screened Coulomb potential V r of an ion embedded in jellium; the wave functions are sketched for the bound (jai) and continuum (jk1;2;3 i) states. kF and r denote the Fermi momentum and radial coordinate, respectively.
10
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
sketch a typical Auger transition involving two valence electrons (KVV, for instance). The rates for other Auger processes in which one or more intra-atomic decay channels are involved (such as the KLV or KLL transitions) can be calculated in a similar way, substituting the corresponding continuum state by the appropriate bound state. When the distortion of the metal states due to the presence of the ion is neglected (i.e., in LRT), the jk2;3 i states can be described by plane waves and the two-electron matrix elements appearing in Eq. (1) can be factorized as the product of a oneelectron matrix element times the imaginary part of the response function [34]. In general, this simpli®cation is not possible. However, an expression more general than Eq. (1) can be obtained using a self-energy formalism [31] in terms of the imaginary part of the screened interaction W r; r0 ; x that is related to the response function v r; r0 ; x according to Z XZ dr dr0 /a r/k1 r0 Im W r; r0 ; x Cca 2 k1
/a r0 /k1 r; where x ek1 W r; r0 ; x is 0
2
ea , the screened interaction Z
0
W r; r ; x v r r Z dr2 v r
dr1 r1 v r1 ; r2 ; xv r2
r0 ; 3
and v r;r0 ;x v0 r;r0 ;x v r1
Z
Z dr1
dr2 v0 r;r1 ;x
r2 K xc r1 ;r2 ; xv r2 ; r0 ; x : 4
The self-consistent response function v r; r0 ; x can be calculated at dierent levels of approximation by solving Eq. (4) in which the zero-order response v0 r; r0 ; x and the exchange and correlation kernel K xc r1 ; r2 ; x appear in the general case. Neglecting K xc gives the RPA result, while
v v0 is the Hartree approximation. The imaginary part of v0 is given by Im v0 r; r0 ; x X p /k2 r/k3 r/k3 r0 /k2 r0 d x ek3 ek2 : k2 ;k3
5
A direct substitution of Eq. (5) in Eq. (3), and then of ImW r; r0 ; x in Eq. (2) gives Eq. (1). 3. Results and discussion The evaluation of Eq. (1) for the Auger capture rate Cca to the bound state jai requires the calculation of four wave functions: one for the jai bound state, and three for the jki i continuum states i 1; 2; 3. Each of them can be calculated at dierent levels of approximation. The simplest approach is to use a hydrogenic wave function for the bound state, and plane waves for the continuum states. A more sophisticated description of any of these wave functions, including the target distortion by the ion, is achieved by numerically solving the Kohn±Sham equations self-consistently. In the latter description, the bound state wave function is screened and shifted up in energy, and the continuum wave functions are strongly modi®ed due to the charge piling up in the vicinity of the ion. In Fig. 2 we plot the Auger capture rate per spin state CKVV to the bound 1s state (K-shell) as a function of the ion charge Z and for rs 2 (rs is de®ned in terms of the background electronic 1=3 density n0 , as rs 3= 4pn0 ). Only one K-shell hole is considered in the initial state electronic con®guration. In the ®gure, results are shown for several combinations of wavefunctions calculated at dierent levels of approximation: the sequence of four letters labels the approximation in which the respective wave functions of the jai, jk1 i, jk2 i and jk3 i states are calculated. Hydrogenic wave functions are labelled with `h', plane-wave continuum wave functions with `b' (Bessel functions), and bound or continuum wave functions calculated self-consistently with `j' (after the jellium in which the ion is embedded).
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
Fig. 2. KVV Auger rate at dierent levels of approximation as a function of the nuclear charge Z of the embedded ion. Note the appearance of kinks when continuum states are described with jellium wave functions.
For ion charges Z P 6 successive improvement of the wave functions for the jai, jk1 i and jk2 i states from a lower level of approximation to the corresponding self-consistent one (h; b ! j) increases the Auger rate each time by about one
11
order of magnitude. For smaller ion charges, the dierence among the various descriptions can be smaller, but it remains sizable. It can be seen from the nearly coinciding curves for the jjjb and jjjj case that only the high-energy jk3 i continuum state is relatively unaected by the presence of the ion and can be described by a plane wave. The most sophisticated calculation (jjjj) of the Auger rate CcK shown in Fig. 2 exhibits kinks at Z 8 and Z 16. They are related to the appearance of bound p states (2p and 3p, respectively) in the initial con®guration of the ion. For Z 6 8 (16), the 2p (3p) state of the ion merges into the continuum and a large amount of continuum electronic charge surrounds the atom, increasing the Auger capture rate. For larger Z, on the other hand, the respective p state becomes bound and the screening electronic charge has a pronounced bound character, decreasing CcK . The appearance of bound states opens new, intra-atomic, channels. This is shown in Fig. 3, where we plot the Auger rate per spin state calculated using the jjjj approach for rs 2, for all processes signi®cant in the ®lling of the K-shell. In the ®gure, KVV (capture from the continuum),
Fig. 3. Contributions to the total K-shell ®lling rate (solid line) in the jjjj approximation as a function of the nuclear charge Z of the embedded ion: rates per spin state for the KVV (dashed line) and intra-atomic (symbols) transitions. The arrows on top of the graph mark the onset of the dierent bound states, the horizontal dashes their binding energies Eb Z.
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S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
KLV, KLL, and some of the insigni®cant KLM rates are shown. For reference, we also plot the energy levels of the bound states. To allow comparison with experimental data [35], the total Auger K-shell ®lling rate CK is drawn with a solid line, as the sum of all the individual rates after inclusion of the necessary statistical factors [29]. Whenever a bound state appears a new channel opens, and its inclusion results in the disappearance of the kinks. Therefore, the total rate CK shows a smooth dependence on Z. So far, we have shown results in which the initial electronic con®guration of the ion is ®xed and the nuclear charge is varied. Let us now turn to a dierent situation in which we follow the ®lling of the L-shell of an ion of ®xed charge (e.g., Ne), as a function of the number nL of electrons (or number nL of holes) already present in the L-shell. In Fig. 4, we show the Auger capture rate per spin state, Cc2s , to the bound 2s state of a Ne ion, as a function of the number of L-shell holes in the ®nal state. Three dierent electronic densities (corresponding to dierent values of rs ) are considered. The initial electronic con®guration of the ion does not include any holes in shells other than the L-shell. It can be seen from the ®gure that only the total number of initial L-shell holes matters. The partly overlapping symbols show, on the other hand, that dierent 2s and 2p sub-shell distribu-
Fig. 4. Comparison of CLVV rates as a function of the number of L-shell holes in jjbb and jjjj approximation for dierent rs values.
tions of the holes hardly change the calculated Cc2s rate. Results obtained at the jjbb and jjjj levels are compared. The dierence between these two levels of approximation is specially signi®cant for high number of holes in the L-shell, for which the medium is strongly perturbed. In [29], the necessity of accounting for the ion perturbation in the calculation of the captured electron initial wave function was shown. Fig. 4 shows that the perturbation of the initial wave function of the emitted electron due to the presence of the ion also plays a role in the calculation of the Auger rate. The role of more weakly bound, M-shell, states in the ®lling of the 2s bound state of Ne is shown in Fig. 5, in which we plot the Auger capture rate Cc2s per spin state, and the C2s ®lling rate per spin state, calculated as the sum of the LVV (Cc2s ), LMV (CLMV ), and LMM (CLMM ) processes. All calculations are performed at the jjjj level of approximation. The LMV and LMM processes are possible only when the 3s state of the ion is bound. The small arrows in Fig. 5 mark the onset of 3s binding for the dierent electron densities: for rs 2 (3) at least nL 4 (2) holes are needed, while for rs 1:5 the 3s state is not bound. It can be seen that for nL 0 there is a strong background density dependence and the intra-atomic channel is
Fig. 5. Ne L-shell ®lling rate CLVV in the jjjj approximation as a function of the number of L-shell holes (solid line) for dierent rs values. The arrows mark the onset of 3s binding, and the dashed lines the total rates including intra-atomic transitions involving the M-shell.
S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 182 (2001) 8±14
not opened yet. The latter means that the continuum electrons are weakly perturbed, as can be seen in Fig. 4 by comparing the jjjj and jjbb descriptions. However, when nL is large, the density dependence of the rates is much weaker and the intra-atomic channel is opened. In this case, the target electronic states are strongly perturbed by the presence of the impurity and the screening becomes atomic-like, i.e., the wavefunctions have Coulombic character close to the embedded ion. 4. Conclusions We have found that a distorted-wave description of the one-electron wave functions, both for the continuum and the bound states, is essential in the calculation of Auger rates: they change several orders of magnitude when the plane wave states (b) are replaced by the proper continuum states (j). The relative importance of KVV processes in the ®lling of the K-shell decreases with growing nuclear charge Z, as new intra-atomic channels become available. When the contributions from all the channels are added up, the total K-shell ®lling rate CK saturates for Z > 10 at about 8 10 3 a.u. We have also shown that the dependence of the C2s ®lling rate on the background electronic density is much stronger when the L-shell of the ion is almost ®lled. In other words, the dependence of the L-shell ®lling rate on the electronic density of the target is more important in the last steps of the neutralization sequence. In summary, non-linear eects in the response of the solid have to be included when calculating Auger rates of processes in which valence electrons are involved. Acknowledgements We wish to thank P.M. Echenique for many useful discussions. Help and support by Euskal Herriko Unibertsitatea, Eusko Jaurlaritza, Gipuzkoako Foru Aldundia and the Spanish D.G.I.C.Y.T. is gratefully acknowledged. One of us (E.Z.) acknowledges help and support by DIPC.
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