J. Phys. B: At. Mol. Opt. Phys. 32 (1999) 3711–3725. Printed in the UK

PII: S0953-4075(99)97356-X

Simultaneous L- and M-shell ionization of a 80 Se target deduced from the analysis of energy shifts and relative intensities of K x-ray lines induced by various projectiles M Polasik†, S Raj‡, B B Dhal‡, H C Padhi‡, A K Saha§, M B Kurup§, K G Prasad§ and P N Tandon§ † Faculty of Chemistry, Nicholas Copernicus University, 87-100 Toru´n, Poland ‡ Institute of Physics, Bhubaneswar-751005, India § Tata Institute of Fundamental Research, Colaba, Mumbai-400005, India Received 9 September 1998, in final form 4 June 1999 Abstract. Average energy shifts of Kα and Kβ x-ray satellite lines and Kβ/Kα intensity ratios have been experimentally evaluated for a 80 Se target bombarded by various projectiles with atomic numbers in the region 3 6 Zp 6 16 at a given projectile energy of 3.3 MeV/u. The data have been analysed using an approach, based on the results of extensive single-configuration Dirac–Fock (DF) calculations, performed within the multiconfiguration DF method for various distributions of holes in subshells of a 80 Se ion. This provides information about the average number of holes in various subshells of 80 Se ions at the time of emission of the K x-ray lines. The primary average number of L- and M-shell holes in a 80 Se target produced at the moment of collisions with projectiles and the average ionization probabilities per electron for L and M shells have been deduced by means of a simple statistical scaling procedure which accounts for all processes that modify the number of L- and M-shell holes prior to the K x-ray emission. It has been found that the average ionization probabilities per electron for L and M shells increase considerably with Zp2 . For larger Zp the average ionization probability per electron is significantly higher for M shells than for L shells. On the other hand, for small Zp the ionization probability per electron seems to be slightly higher for L shells than for M shells. Moreover, the dominant role of the simultaneous multiple ionization of L and M shells accompanying the ionization of a K shell has been observed for projectiles with Zp > 3.

1. Introduction The study of collision processes has always played an important role in the development of modern atomic physics. At the fundamental level it provides data necessary for an evaluation of quantum mechanical models of many-body interactions. At a more practical level the experimental data are of great importance to other branches of physics such as solid state physics, laser physics, astrophysics, plasma physics and nuclear physics. Less than 30 years ago, a method was developed to determine the L-shell ionization probability in collisions from measured K x-ray spectra. Observing these spectra we are limited only to a small subset of collisions which can be considered as nearly central in the L- and M-shell scale since for a K x-ray transition an initial K-shell hole must be present. These collisions preferentially pick out a small range of impact parameters and so the results would not be expected to represent the general situation for the production of L and M holes in the collisions. On the other hand, the ionization probability determined for the small impactparameter range allows much more detailed comparison with theoretical predictions than the 0953-4075/99/153711+15$30.00

© 1999 IOP Publishing Ltd

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total cross-sections. Moreover, the ionization probabilities determined from measured K x-ray spectra are rather weakly influenced by uncertainties in the Auger and Coster–Kronig yields. This method has been applied to many studies on the L-shell ionization probability in collisions of target atoms with Z < 30 [1–24]. Since 1987 it has also been applied to studies of target atoms (with Z > 40) bombarded by various light and heavy projectiles [25–33]. Recently this method has been extended to determine the M-shell ionization probability in collisions with light projectiles [34–37] and, with the help of extensive multiconfiguration Dirac–Fock (MCDF) calculations, to determine the M-shell ionization probability in collisions with heavy projectiles [38, 39] and with 4 He2+ ions [40]. In this paper we present the results of a systematic study on the simultaneous L- and M-shell ionization of a 80 Se target bombarded by various projectiles with atomic numbers in the region 3 6 Zp 6 16 at a given projectile energy of 3.3 MeV/u (i.e. at a velocity closely matching that of the L-shell electrons of a 80 Se target). This study is based on the analysis of the K x-ray spectra of a 80 Se target. Since the S9+ ion has been the heaviest projectile (Zp = 16) used in our study, the x-ray spectrum of a 80 Se target bombarded by S9+ projectiles is the most interesting one and has been presented in figure 1. The spectrum shows 80 Se L x-ray peaks in the region of 0–2 keV and S9+ K x-ray peaks in the region of 2–4 keV, low background x-ray peaks in the region of 5–7 keV coming from the target holder and two peaks in the region of 11–15 keV corresponding to Kα and Kβ satellite lines of a 80 Se target. In the case of the collisions of energetic heavy projectiles (such as O, F, Al, and S ions) with mid-Z target atoms (such as 80 Se atoms; Z = 34), the ionization of a K shell is always accompanied by the simultaneous multiple ionization of L and M shells and leads to numerous (even hundreds) different configurations of the target atoms. For a particular configuration with more than one open subshell a large number of initial and final states may exist and between these states a

Figure 1. The x-ray spectrum of a 80 Se target bombarded by S9+ projectiles.

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very large number of transitions (with different energies) is possible. As a consequence the observed two peaks in the region of 11–15 keV, the first one corresponding to Kα satellite lines of a 80 Se target and the second one to Kβ satellite lines, are made up of an enormous number of components which are strongly overlapped and which are not resolved because of the relatively low detector resolution. Due to this fact our study on the simultaneous L- and M-shell ionization of a 80 Se target is based only on the average energy shifts of Kα and Kβ satellite lines and Kβ/Kα intensity ratios evaluated from the measured x-ray spectra for a 80 Se target bombarded by various projectiles. We thus propose an approach for the reliable analysis and interpretation of the experimentally evaluated average energy shifts of Kα and Kβ satellite lines and Kβ/Kα intensity ratios, which is based on the results of extensive singleconfiguration Dirac–Fock (DF) calculations, performed within the MCDF method for various distributions of holes in subshells of a 80 Se ion. This approach consists of two stages. In the first stage very detailed information about the dependence of the Kβ/Kα intensity ratios and Kα and Kβ transition energies on the distributions of holes in various subshells and qualitative arguments have enabled us to choose those electronic configurations which are most important at the time of emission of the K x-ray lines. In the second stage, in order to evaluate the average number of holes in various subshells of 80 Se ions we select for particular projectiles only a few (2–3) configurations, a combination of which gives a good reproduction of experimentally obtained K x-ray intensity ratios and average energy shifts. The primary average number of L- and M-shell holes in a 80 Se target produced simultaneously with the K-shell vacancy at the moment of collisions with projectiles (and the average ionization probabilities per electron for L and M shells) have been deduced by means of a simple statistical scaling procedure which accounts for all processes that modify the number of L- and M-shell holes prior to the K x-ray emission. The results of this paper evidently depend on the reliability of the MCDF method applied to the case of multiply ionized 80 Se atoms. Therefore, it is worth mentioning that several theoretical models for reliable descriptions of very complex x-ray spectra accompanying the ionization in collision processes based on the MCDF method have been tested and successfully applied to various many-electron atoms [41–45]. The results of this series of papers have been effectively implemented in the theoretical analysis of Kα and Kβ x-ray spectra of molybdenum [38], palladium and lanthanum [39] generated in collisions with oxygen ions. Moreover we have succeeded [46] in attributing the measured positions of various L and K x-ray peaks and the measured Kβ/Kα intensity ratios for highly ionized swift projectiles of 75 As and 80 Se passing through a thin carbon foil to certain types of electronic configurations based on the extensive MCDF calculations of a similar kind as presented in this paper. 2. Experimental details and data analysis procedure Various projectile ions used in the experiment were obtained from the BARC-TIFR 14 MV tandem accelerator at Mumbai. The post accelerated ions are energy selected by the 90◦ analysing magnet and finally directed into the experimental chamber situated at the 30◦ beam port with the help of a switching magnet. The projectile energy used for exciting the target atoms is given in column 3 of table 1. Prior to entering the experimental chamber the beam was collimated to a diameter of 1.5–2.0 mm. A Se target of thickness 25 µg cm−2 (on 15 µg cm−2 carbon backing) was used for carrying out the measurements. The target was placed with its face perpendicular to the incident beam. Typically, beam currents of the order of a few nA were used for the experiment and were monitored by the charge collection on the insulated target chamber acting as a Faraday cup. Details of the experimental setup can be found elsewhere [47].

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M Polasik et al Table 1. Experimental Kβ/Kα intensity ratios and average Kα and Kβ x-ray transition energies and energy shifts for a 80 Se target bombarded by various projectiles. The errors in the reported energies vary between 10 and 20 eV. ‘0’ corresponds to photoionization data. Transition energies Energy shifts (keV) (eV) Projectile atomic number (Zp )

Zp2

EP (MeV/u)

Kβ/Kα intensity ratio

Kα

Kβ

Kα

Kβ

0 3 6 8 9 13 16

0 9 36 64 81 169 256

— 3.28 3.31 3.31 3.31 3.30 3.12

0.161 ± 0.001 0.164 ± 0.006 0.186 ± 0.003 0.198 ± 0.006 0.201 ± 0.006 0.209 ± 0.009 0.204 ± 0.007

11.202 11.216 11.247 11.267 11.277 11.303 11.323

12.496 12.531 12.635 12.710 12.751 12.860 12.957

0 14 45 65 75 101 121

0 35 139 214 255 364 461

The x-rays emitted in the collision process were detected in a Canberra Si(Li) detector of a good resolution (160 eV FWHM at 5.9 keV) placed at an angle of 135◦ with respect to the beam direction. The gain of the x-ray spectrometer was set at ∼10 eV/ch. No noticeable changes in the peak energies (610 eV) from standard sources were seen throughout the experiment. The energy calibration of the spectrometer was made with standard sources of 55 Fe, 57 Co and 241 Am. The emitted x-rays pass through a 1.15 mg cm−2 thick Mylar (chamber window), 1 cm air path and 1.15 mg cm−2 thick Mylar foil fixed on the Si(Li) detector before reaching the 25 µm beryllium window of the detector. The terminal voltage and charge state of different projectiles have been adjusted to get a projectile energy of approximately 3.3 MeV/u. The peak areas and centroid positions of the K x-ray lines of 80 Se are determined with the help of a multi Gaussian least-square fitting programme incorporating a nonlinear background substraction. The measured x-ray line intensities were corrected for detector efficiency, absorption in the Mylar chamber window and a small air path and self absorption in the target. The efficiency of the detector was determined theoretically as described elsewhere [48]. 3. MCDF method The MCDF method applied in the present study has been mainly developed by Grant and coworkers and is described in detail in several papers [49–52]. In this method the Hamiltonian for the N-electron atom is taken in the form N N X X hD (i) + Cij (1) H = i=1

j >i=1

where hD (i) is the Dirac operator for the ith electron and the terms Cij account for electron– electron interactions and come from one-photon exchange process. The latter are a sum of the Coulomb interaction operator and the transverse Breit operator. The atomic state functions with total angular momentum J and parity p are represented in the multiconfigurational form X cm (s)8(γm J p ), (2) 9s (J p ) = m

where 8(γm J p ) are configuration state functions (CSFs) built from one-electron spinors, cm (s) are the configuration mixing coefficients for state s, γm represents all information required to define a certain CSF uniquely.

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The information about the average number of holes in various subshells of 80 Se ions at the time of emission of the K x-ray lines have been conducted using a special average-level version of MCDF calculations (see Jankowski and Polasik [53]) which gives values of the Kβ/Kα intensity ratios for Ti, Cr, Fe, Ni, Cu, Zn, and Ge in a significantly better agreement with highly accurate experimental data of Perujo et al [54] than the theoretical predictions of Scofield [55] and the results of standard average-level and extended average-level versions of MCDF calculations (see Grant et al [49]).

4. Results and discussion The experimentally evaluated Kβ/Kα intensity ratios and average Kα and Kβ x-ray transition energies and energy shifts for a 80 Se target bombarded by various projectiles with atomic numbers in the region 3 6 Zp 6 16 at a given projectile energy of about 3.3 MeV/u are presented in table 1. The experimental Kβ/Kα intensity ratio, as a function of Zp , is shown in figure 2. Moreover, the experimental average Kα and Kβ energy shifts for a 80 Se target, as functions of Zp2 , are presented in figure 3. It can be found from table 1 and figure 2 that the measured Kβ/Kα intensity ratio increases in the beginning and then achieves a value of 0.209 ± 0.009 at Zp = 13. The average energy shifts of both Kα and Kβ satellite lines, as functions of Zp2 (see figure 3), show a fast increase for projectiles with Zp 6 8 after which the change becomes slow.

Figure 2. The experimental Kβ/Kα intensity ratio, as a function of Zp for a 80 Se target.

Figure 3. Experimental average energy shifts for a 80 Se target as functions of Zp2 : (a) Kα energy shifts, (b) Kβ energy shifts.

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4.1. Average number of holes in various subshells of L and M shells at the time of emission of the K x-ray lines To evaluate the average number of holes in various subshells of 80 Se ions at the time of emission of the K x-ray lines, the experimental data have been analysed with the help of extensive singleconfiguration DF calculations, performed within the MCDF method for various distributions of these holes. Because the x-rays are emitted by 80 Se ions which exist in states corresponding to even hundreds (for heavy projectiles) of different electronic configurations it is not possible to point out in a reliable and unambiguous way, to what exact combination of configurations the measured average energy shifts and Kβ/Kα intensity ratios correspond. Therefore, in present studies we have proposed a well-defined approach based on our many years’ experience in analysing heavy-ion-induced x-ray spectra, which consists of two stages. Results of the first (preliminary) stage together with simple qualitative arguments enabled us to eliminate electronic configurations, the role of which at the time of emission of the K x-ray lines is negligible and select those which are most important. At the second stage replacement of many substantial configurations by a combination of only a few of them is done to calculate the average number of holes in various subshells of the target atom so that a good fit to experimentally obtained intensity ratios and average energy shifts for particular projectiles is achieved. 4.1.1. Role of various electronic configurations at the time of emission of the K x-ray lines. Results of the part of very detailed single-configuration DF calculations performed within the MCDF method (i.e., the effect of removing electrons from given subshells on the Kβ/Kα intensity ratio and Kα and Kβ transition energies) are presented in table 2. It is very important to note that the effects of the removal of electrons from the subshells are nonadditive for the Kβ/Kα intensity ratio and Kα and Kβ transition energies. In general, it can be seen from table 2 that the Kβ/Kα intensity ratio is influenced, first of all, by the ratio of the number of holes in 3p and 2p subshells. Moreover, the ionization of the 2p subshell causes the largest shifts of Kα and Kβ energies. In particular, since there are plenty of electrons in the 3d subshell of a neutral atom, its ionization (which seems to be substantial at the time of emission of the K x-ray lines) can modify both the Kβ/Kα intensity ratio and Kβ energy to a great extent. On the other hand, the ionization of 2s and 3s subshells plays rather a minor role at the time of emission of the K x-ray lines. It is because the number of electrons in 2s subshell is three times smaller than in a 2p subshell. Moreover, the very fast L1 Li X Coster–Kronig transitions [56] cause the transfer of holes from the 2s to 2p subshell before the emission of the K x-ray lines. The probability that a hole is transferred from the 2s to 2p subshell prior to the K x-ray emission is given by 0LCK 7.8 1 = = 0.71, (3) 0K + 0L1 2.5 + 8.5 where 0K is the total (radiative plus Auger) width for the K shell [57], 0L1 is the total (radiative is the width of plus Coster–Kronig plus Auger) width for the L1 (2s) subshell [58] and 0LCK 1 L1 Li X Coster–Kronig transitions [58]. Using this scaling procedure we can estimate that the relative population of 2s holes (i.e. the number of holes in a 2s subshell relative to the total number of holes in an L shell) at the time of emission of the K x-ray lines is " # 0LCK 1 1 1 1.00 − (4) = (1.00 − 0.71) = 0.07. 4 0K + 0L1 4 We see that the number of holes in a 2s subshell is approximately 7% of the total number of holes in an L shell.

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Table 2. Effect of removing electrons from given subshells on the Kβ/Kα intensity ratio and average Kα and Kβ transition energies (eV) for 80 Se. Initial configuration

Kβ/Kα intensity ratio

Kα energy shift

Kβ energy shift

1s−1 1s−1 2s−2 1s−1 3s−2 1s−1 4s−2 1s−1 2p−1 1s−1 2p−2 1s−1 2p−3 1s−1 2p−4 1s−1 2p−5 1s−1 3p−1 1s−1 3p−2 1s−1 3p−3 1s−1 3p−4 1s−1 3p−5 1s−1 3d−10

0.158 0.178 0.167 0.160 0.187 0.241 0.331 0.512 1.053 0.129 0.106 0.082 0.056 0.029 0.230

0.0 72.7 5.5 0.4 43.4 88.0 133.9 181.1 229.6 4.2 8.6 13.4 18.4 23.7 −9.9

0.0 155.6 29.1 3.8 95.6 193.6 294.2 397.0 502.3 10.9 22.8 35.6 49.4 64.3 127.5

Similarly, the effect of ionization of a 3s subshell is also small because it also contains just two electrons (a small number when compared with 18 electrons in the whole M shell). Taking into account the results obtained within the semiclassical approximation model using Dirac– Hartree–Fock wavefunctions (SCA-DHF method) for Pd [39], we may assert that the ionization probability per electron is much smaller for a 3s subshell than for 3p and 3d subshells. The very fast M1 Mi X Coster–Kronig and M1 Mi Mj super Coster–Kronig transitions [59] cause the transfer of holes from 3s to 3p and 3d subshells and make the importance of the 3s subshell even smaller at the time of emission of the K x-ray lines. The probability that a hole is transferred from a 3s subshell to 3p and 3d subshells prior to the K x-ray emission is given by CK + 0MsCK 0M 1 1

0K + 0M1

=

5.3 = 0.68, 2.5 + 5.3

(5)

where 0K and 0M1 are the total widths for a K shell [57] and M1 (3s) subshell [59], respectively; CK sCK and 0M 0M are the widths of M1 Mi X Coster–Kronig and M1 Mi Mj super Coster–Kronig 1 1 transitions, respectively [59]. Using this scaling procedure we can estimate that the number of holes in a 3s subshell is less than 4% of the total number of holes in an M shell. Therefore, in the theoretical analysis of the experimental results we neglect holes in both 2s and 3s subshells. On the other hand, the 3d subshell becomes very rich in holes. It is because besides the Mi Mj X Coster–Kronig transitions which strongly favour the presence of holes in a 3d subshell at the time of emission of the K x-ray lines, there are two other arguments which show that the number of holes in 3d subshell is large. First, the number of electrons in a 3d subshell at the moment of collision is large. Secondly, according to the results of SCA-DHF calculations for Pd [39], the ionization probability per electron for a 3d subshell can be two times higher than for 3s and 1.6 times higher than for 3p. Because of the two reasons mentioned above—the nonadditivity of the effects of the removal of electrons from various subshells (see table 2) and the dominating role of the simultaneous multiple ionization of L and M shells (see the introduction and table 7 for projectiles with Zp > 3)—it turned out to be necessary to perform additional very extensive single-configuration DF calculations performed within the MCDF method. The results of

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M Polasik et al Table 3. Numbers of levels (for the initial configuration), the calculated Kβ/Kα intensity ratios and average Kα and Kβ energy shifts (eV) for electronic configurations of 80 Se ions corresponding to various distributions of holes in 2p, 3p and 3d subshells (the ratio of the number of 3d to 3p holes is asumed to be three). Initial configuration

Number of levels for the initial configuration

Kβ/Kα intensity ratio

Kα energy shift

Kβ energy shift

1s−1 2p−1 3p−1 3d−3 1s−1 2p−2 3p−1 3d−3 1s−1 2p−2 3p−2 3d−6 1s−1 2p−2 3p−3 3d−9 1s−1 2p−3 3p−3 3d−9

1 226 2 925 11 923 951 1 272

0.173 0.228 0.211 0.159 0.276

47.3 93.4 98.1 110.5 151.5

153.4 254.5 325.6 380.4 508.8

Table 4. The electronic configurations of 80 Se ions (and their weights), the combination of which correctly reproduces the experimentally obtained Kβ/Kα intensity ratio and average Kα and Kβ energy shifts (eV) for a projectile with atomic number Zp = 13. Initial configuration

Weight

1s−1 2p−2 3p−2 3d−6 1s−1 2p−2 3p−3 3d−9 1s−1 2p−3 3p−3 3d−9

0.75 0.15 0.10

Experimental values

Kβ/Kα intensity ratio

Kα energy shift

Kβ energy shift

0.209

105.2

354.9

0.209 ± 0.009

101 ± 10

364 ± 20

these calculations have given us very detailed information about the dependence of the Kβ/Kα intensity ratios and Kα and Kβ transition energies on the distributions of holes in 2p, 3p and 3d subshells. The preliminary estimations of the average number of holes in various subshells of 80 Se ions (at the time of emission of the K x-ray lines) have shown that a good reproduction of measured results is possible only if the ratio of the number of 3d to 3p holes is about three. Such a ratio is consistent with qualitative arguments presented above (after equation (7)) which indicate the relative richness of 3d holes when compared with the number of 3p (and 3s) holes. Therefore, in the theoretical analysis of the experimental results this ratio has been assumed to hold (see tables 3 and 4). For the clarity of the discussion and to give the reader a better idea of our approach, we have decided to present only a small part of the results of our studies. In table 3 we have listed those electronic configurations (chosen among hundreds of possible configurations) which are dominating and indispensable to determine the avarage number of holes in L and M shells for various projectile atomic numbers Zp . As mentioned above, the single-configuration DF calculations performed within the MCDF method are very extensive. These calculations are of a large scale because there exist thousands of initial (see the numbers of levels for the initial configurations in the second column of table 3) and final levels corresponding to a certain distribution of holes in 2p, 3p and 3d subshells. Between these levels hundreds of thousands of transitions are possible. 4.1.2. Selection for particular projectiles of only a few configurations, a combination of which gives a good reproduction of experimentally obtained data. In the second stage, using results sampled in table 3, we obtained values of the average number of holes in L and M shells of 80 Se at the time of emission of the K x-ray. This has been done by finding such a combination of contributions of only a few (2–3) electronic configurations which do experimentally reproduce obtained intensity ratios and average energy shifts for particular projectiles. As an example, in

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Table 5. The estimation of the average number of holes in 2p, 3p and 3d subshells of 80 Se ions at the time of emission of the K x-ray lines induced by various projectiles. Average number of holes Projectile atomic number (Zp )

Zp2

2p

3p

3d

3 6 8 9 13 16

9 36 64 81 169 256

0.3 ± 0.1 1.0 ± 0.2 1.5 ± 0.2 1.65 ± 0.2 2.1 ± 0.2 2.4 ± 0.2

0.2 ± 0.1 0.7 ± 0.2 1.2 ± 0.3 1.45 ± 0.3 2.25 ± 0.4 2.95 ± 0.5

0.6 ± 0.3 2.1 ± 0.6 3.6 ± 0.8 4.35 ± 0.9 6.75 ± 1.2 8.85 ± 1.4

table 4 the electronic configurations of 80 Se ions (and their weights), the combination of which correctly reproduces the experimentally obtained Kβ/Kα intensity ratio and average Kα and Kβ energy shifts for projectile with Zp = 13 are presented. We have chosen these electronic configurations from table 3 in the following way. Firstly, we have picked out from table 3 the electronic configuration, for which the Kβ/Kα intensity ratio and average Kα and Kβ energy shifts approximate to experimental values to the highest degree. This electronic configuration has dominated, i.e. its corresponding weight has been the highest. Then, we have found two (in this example) additional electronic configurations, the admixture of which has enabled us to get (by optimalization of weights) the values of the Kβ/Kα intensity ratio and average Kα and Kβ energy shifts giving a satisfactory agreement with their experimental counterparts. These three configurations represent those hundreds of configurations which are responsible for the spectrum but the weights of which we cannot determine. Weights associated with these three configurations according to table 4 enable one to estimate the average number of holes in 2p, 3p and 3d subshells of 80 Se ions at the time of emission of the K x-ray lines. The estimation of the average number of holes in 2p, 3p and 3d subshells of 80 Se ions at the time of emission of the K x-ray lines (for various projectile atomic numbers Zp ) are given in table 5. It can be seen from this table that the average number of holes in 2p and 3p subshells of 80 Se ions increases considerably with Zp . For small Zp the ionization of the 2p subshell seems to be slightly higher than that of 3p while for larger Zp the opposite is true. The ionization of the 3d subshell seems to be very large in all cases (see table 5). The obtained results enable one to explain the maximum for the Kβ/Kα intensity ratio which can be seen in figure 2. Generally the ionization of all the subshells of 80 Se, besides the 3p subshell, causes an increase of the Kβ/Kα intensity ratio (see table 2). As mentioned above this ratio is especially sensitive for the ionization of 2p subshell (the increase of the ratio) and 3p subhell (the decrease of the ratio), because 2p electrons are directly responsible for the Kα transitions and 3p electrons for the Kβ transitions. For the projectiles with 3 6 Zp 6 9 the average number of holes in 2p and 3p subshells increases considerably with Zp and seems to be higher for 2p than 3p (see table 5). Therefore, the Kβ/Kα intensity ratio increases rapidly in this region. For Zp = 13 the average number of holes in 3p subshell is rather slightly higher than in 2p subshell. That is why the increase of the Kβ/Kα intensity ratio becomes less significant in the region between Zp = 9 and Zp = 13. For Zp = 16 the average number of holes in 3p subshell is much higher than in 2p subshell, which causes the decrease of the Kβ/Kα intensity ratio in the region between Zp = 13 and Zp = 16. According to this explanation the Kβ/Kα intensity ratio achieves the maximum for the projectile with Zp = 13.

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4.2. Average L- and M-shell ionizations at the time of the ion–atom collisions Experimentally observed Kβ/Kα x-ray intensity ratios and average Kα and Kβ x-ray energy shifts for a 80 Se target (table 1) reflect the average number of holes in L and M shells at the moment of the K x-ray emission shown in table 5. The determination of the average ionization probabilities per electron for L and M shells requires, however, the knowledge of the initial average number of holes produced in L and M shells just at the moment of collision. The primary average number of holes in L and M shells can be deduced by means of a simple statistical scaling procedure which accounts for all processes that modify the number of Land M-shell holes prior to the K x-ray emission. Various processes like Coster–Kronig, super Coster–Kronig, Auger, L and M radiative transitions [56, 59] are involved. Let us briefly discuss their influence on the L- and M-shell holes. The LMM Auger, LMX Auger, LXY Auger, L radiative transitions diminish the number of L holes by one. The LMM Auger transitions produce two holes in the M shell. The Li Lj M Coster–Kronig, LMX Auger, L–M radiative and Mi Mj Mk super Coster–Kronig transitions produce one hole in the M shell. The MXY Auger and M radiative transitions diminish the number of M holes by one. It has to be noted that Li Lj M Coster–Kronig and Li Lj X Coster– Kronig transitions change the hole distribution within the L subshells but they do not modify the total number of L holes. Furthermore, Li Lj Lk super Coster–Kronig transitions that would increase the number of L vacancies by one are energetically forbidden for 80 Se. The Mi Mj X Coster–Kronig transitions change the hole distribution within the M subshells but they do not modify the total number of M holes. In the case of 80 Se the most important are the fast Li MM Auger transitions [56, 58] which diminish the number of L holes by one and produce two holes in the M shell. The probability that one L hole is filled up prior to the K x-ray emission (and two holes in the M shell are produced) is given by 1 0LA2 MM 1 0LA3 MM 1 0LA1 MM + + = 0.23 (6) 4 0K + 0L1 4 0K + 0L2 2 0K + 0L3 where 0K and 0Li are the total widths for the K shell [57] and Li subshell [58], respectively; 0LAi MM are the widths for Li MM Auger transitions [58]. The other processes are negligible [56–59]. The primary average number of L- and M-shell holes in a 80 Se target produced at the moment of collisions with projectiles (deduced by means of a simple statistical scaling procedure) and the average ionization probabilities per electron for the L shell (pL ) and M shell (pM ) have been presented in table 6. Results sampled in table 6 are also displayed in figure 4, where the ionization probabilities per electron for L and M shells are plotted versus the square of atomic number of a projectile. It can be seen that the average ionization probabilities per electron for L and M shells increase considerably with Zp2 . The increase of the ionization probability with Zp2 for the M shell seems to be almost linear for both small and large projectile atomic number. These two regions of linear growth are separated by a nonlinear one with a weakly visible inflexion point. The variation of the L-shell ionization probability with Zp2 is more complex. The growth seems to be linear for large Zp only. In this region pL becomes smaller than pM and it increases more slowly. It seems to be quite natural that the larger charge of the projectile influences more strongly the weakly bound M-shell electrons of the target atom. On the other hand, one might wonder that for small Zp , pL seems to be slightly higher than pM . It is possible because we consider here only the near-central collisions in which the projectile comes close to the nucleus of the target atom. Moreover, in all our experiments the velocity of the projectile closely matches the velocity of the L-shell electrons of a 80 Se target which implies that the ionization of the L shell approaches maximum for given atomic

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Table 6. The primary average number of L- and M-shell holes in 80 Se target produced at the moment of collisions with various projectiles (deduced by means of a simple statistical scaling procedure) and the average ionization probabilities per electron for L and M shells. Average number of holes

Average ionization probability per electron

Projectile atomic number (Zp )

Zp2

L shell

M shell

L shell

M shell

3 6 8 9 13 16

9 36 64 81 169 256

0.4 ± 0.1 1.3 ± 0.2 1.9 ± 0.3 2.1 ± 0.3 2.7 ± 0.4 3.1 ± 0.5

0.6 ± 0.4 2.2 ± 0.8 3.9 ± 1.3 4.8 ± 1.5 7.7 ± 1.9 10.4 ± 2.2

0.05 ± 0.01 0.16 ± 0.02 0.24 ± 0.03 0.27 ± 0.04 0.34 ± 0.05 0.39 ± 0.06

0.03 ± 0.02 0.12 ± 0.04 0.22 ± 0.07 0.27 ± 0.08 0.43 ± 0.10 0.58 ± 0.12

Figure 4. The average ionization probabilities per electron for the L shell (pL ) and M shell (pM ) of a 80 Se target as functions of Zp2 .

number of the projectile. Thus, it is possible that in the case of the near-central collisions for small Zp the velocity matching effect which favours the ionization of the L shell prevails over the stripping of the M shell. Let us notice, that for all projectiles the ionization probability per electron for the K shell (pK ) is very small (much smaller than pL and pM ) also in the case of the near-central collisions. However, as has been mentioned above observing the K x-ray lines we are sure that one electron from the K shell has been removed in all collisions we have considered. 4.3. Role of the simultaneous multiple ionization of L and M shells accompanying the ionization of the K shell in collisions with various projectiles Assuming no correlation between the ionized electrons, the multiple ionization of M and L shells with the simultaneous ionization of the K shell can be expressed by a binomial

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distribution. In this case, the cross section σKLn M r for removing one K-shell electron, n L-shell electrons and r M-shell electrons is given by     Z +∞ 2 8 σKLn M r = 2πb × pK (b)(1 − pK (b)) × (pL (b))n (1 − pL (b))8−n 1 n 0    18 r 18−r × (pM (b)) (1 − pM (b)) db, (7) r where pK (b), pL (b) and pM (b) are the impact-parameter-dependent average ionization probabilities per K-shell, L-shell and M-shell electron, respectively, 0 6 n 6 8, and 0 6 r 6 18. In the impact parameter region where pK (b) is different from zero, pL (b) and pM (b) are almost constant and it is reasonable to approximate pL (b) by pL (beff ) and pM (b) by pM (beff ), where beff is the value of b for which the contribution to the K-shell ionization cross section is nearly maximum. Then the ionization cross section can be rewritten as   8 σKLn M r ∼ (pL (beff ))n [1 − pL (beff )]8−n = n   Z +∞ 18 × (pM (beff ))r [1 − pM (beff )]18−r 4π bpK (b)(1 − pK (b)) db r 0   8 = (pL (beff ))n [1 − pL (beff )]8−n n   18 × (pM (beff ))r [1 − pM (beff )]18−r × σK , (8) r where σK represents the single K-shell ionization cross section. The estimation of the contributions of the single (KL0 M0 ), double (KL0 M1 and KL1 M0 ), and multiple ionization (the simultaneous two shell ionization (KL0 Mr and KLn M0 ) and the simultaneous three-shell ionization (KLn Mr ) for n 6= 0 and r 6= 0) in a 80 Se target produced at the moment of collisions with various projectiles are included in table 7. The sum of the contributions of KL0 M0 , KL0 Mr , KLn M0 and KLn Mr (for n 6= 0 and r 6= 0) is to be equal to one for all projectiles due to the property of the binomial distribution. As we expected, one can observe the dominant role of the simultaneous multiple ionization of L and M shells accompanying the ionization of the K shell in collision processes for projectiles with Zp > 3 Table 7. The estimation of the contributions of the single (KL0 M0 ), double (KL0 M1 and KL1 M0 ), and multiple ionization (the simultaneous two shell ionization (KL0 Mr and KLn M0 ) and the simultaneous three-shell ionization (KLn Mr ) for n 6= 0 and r 6= 0) in a 80 Se target produced at the moment of collisions with various projectiles. The sum of the contributions of KL0 M0 , KL0 Mr , KLn M0 and KLn Mr is equal to one for all projectiles. Projectile atomic number (Zp ) 3 6 8 9 13 16

Double ionization

Multiple two-shell ionization

Zp2

Single ionization KL0 M0

KL0 M1

KL1 M0

KL0 Mr

KLn M0

Three-shell ionization KLn Mr

9 36 64 81 169 256

0.383 0.025 0.001 0.000 0.000 0.000

0.213 0.061 0.006 0.002 0.000 0.000

0.161 0.038 0.003 0.001 0.000 0.000

0.280 0.223 0.110 0.080 0.036 0.019

0.195 0.075 0.010 0.003 0.000 0.000

0.142 0.677 0.879 0.916 0.964 0.981

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(see the contributions of KLn Mr (n 6= 0 and r 6= 0) in table 7). Of course, the simultaneous two- (K and M) shell ionization events without the L-shell ionization do play some role (very little for large Zp and rather important for Zp = 3). For a projectile with Zp = 3 single (KL0 M0 ) ionization is also very important. 5. Summary and conclusions With the objective of investigating the dependence of the average ionization probabilities per electron for L and M shells of a 80 Se target on the projectile atomic number, the average energy shifts of Kα and Kβ satellite lines and Kβ/Kα x-ray intensity ratios have been experimentally evaluated for a 80 Se target bombarded by various projectiles with atomic numbers in the region 3 6 Zp 6 16 at a given projectile energy of about 3.3 MeV/u. The experimental data have been analysed using an approach, based on the reliable (see the introduction and section 3) results of extensive single-configuration DF calculations, performed within the MCDF method for various distributions of holes in subshells of a 80 Se ion. This provides information about the average number of holes in various subshells of 80 Se ions at the time of emission of the K x-ray lines. The primary average number of L- and M-shell holes produced at the moment of collisions with projectiles and the average ionization probabilities per electron for L and M shells have been deduced by means of a simple statistical scaling procedure which accounts for all processes that modify the number of L- and M-shell holes prior to the K x-ray emission. On the basis of the present study some general conclusions can be drawn. First, the average ionization probabilities per electron for L and M shells increase considerably with Zp2 . Second, for larger Zp the average ionization probability per electron is significantly higher for the M shell than for the L shell because the larger charge of the projectile influences more strongly the weakly bound M-shell electrons of the target atom. Third, for small Zp the ionization probability per electron seems to be slightly higher for the L shell than for the M shell. Probably in the case of the near-central collisions for small Zp the velocity matching effect which favours the ionization of the L shell prevails over the stripping of the M shell. Fourth, it has been observed that the role of the simultaneous multiple ionization of L and M shells accompanying the ionization of the K shell is dominant in collision processes for all projectiles with Zp > 3. It should be mentioned that our results for the average ionization probability per electron for the L and M shells are limited only to a small subset of collisions with a small range of impact parameters (i.e. to the collisions in which a K hole is made). Therefore, our conclusions on the average ionization probability have been drawn for the near-central collisions only and they would not be expected to represent the general situation. The authors believe that the results of this study will be helpful for better understanding of the collision processes and provide valuable information about the dependence of the average ionization probabilities per electron for L and M shells of a 80 Se target on the projectile atomic number. The basic theoretical apparatus in the paper—the MCDF method—provides a very useful tool to interpret the experimental data, especially within our approach in its application to x-ray spectra. However, it is obvious that to gain deeper insight into underlying scattering processes, one should apply theoretical methods and models which are independent of experiments. It is the authors’ hope that the present paper will encourage theoreticians to apply, e.g. SCA [60, 61] and CTMC [62–64] methods to obtain theoretical results for the ionization probabilities which can be directly compared with the results presented in this paper. Thus the role of various processes like direct Coulomb ionization or electron capture could be estimated and explained. We believe that our results can stimulate further development of

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theoretical methods for determination of the ionization probabilities per electron for L and M shells as well as experimental and theoretical research with other projectiles and targets. Acknowledgments The authours gratefully acknowledge the help of the accelerator staff of the 14 MV tandem accelerator at Mumbai. They are also grateful to the members of the target preparation group for their help in preparing the targets for the experiment. MP would like to thank Dr M Janowicz, Dr P Rymuza, M Lewandowska, and F Pawłowski for helpful discussions. This work was supported in part by the Polish Committee for Scientific Research (KBN), grant no 2 P03B 019 16, and partly by the Council of Scientific and Industrial Research, India. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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