CASCADES AFTER K-VACANCY PRODUCTION AND ADDITIONAL IONIZATION OR EXCITATION IN ATOMS OF LIGHT ELEMENTS

The Auger effect is the main de-excitation process in light atoms after the creation of an inner vacancy. Such two-electron transitions usually involve neighboring shells, and thus de-excitation tends to proceed gradually in several steps as a cascade of Auger and radiative transitions between various configurations. After each Auger transition, the ionization degree increases; therefore, as a result of such a cascade, various multiply charged ions are obtained. This process can play an important role in the production of ions in various astrophysical objects, such as active galactic nuclei, X-ray binaries, interstellar gas, and the winds of B-type stars (Weisheit 1974; Petrini 1992; Zsargó et al. 2003). Auger transitions are energetically constrained, and thus an Auger cascade often ends in excited states, which can be de-excited later through radiative transitions.

Consequently, cascade data are necessary for modeling astrophysical plasma under conditions where the inner vacancies can be effectively created by photoionization or particle impact. Namely, such conditions are realized in the vicinity of cosmic X-ray sources or in high-temperature plasma. The populations of excited levels along with the data for radiative transitions (Palmeri et al. 2008) can be used to interpret the registered characteristic X-ray spectra.

In order to reduce the extent of computations for cascades, averaged calculations of transitions between configurations are usually performed. Such an approximation was not only used in the first extensive calculations of cascades for light elements (Opendak 1990; Kaastra & Mewe 1993), but also in various later works (Kochur et al. 1995; Jonauskas et al. 2000; Kochur & Petrini 2004; El-Shemi & Lotfy 2005). However, the results of detailed level-by-level calculations demonstrate that the results of cascade can significantly depend on the many-electron quantum numbers of the initial and intermediate states (McGuire 1975; Hasoglu et al. 2006; Paladoux et al. 2010; Partanen et al. 2010). Thus, in Kučas et al. (2012), the results of detailed calculations of cascades after K-vacancy production for the astrophysically important light elements (Ne, Mg, Si, S, and Ar) were presented.

During photoionization or ionization by the electron impact of an inner shell, the other shells can also be ionized or excited. Their probability can be estimated using the sudden perturbation model to describe an additional ionization (shake-off) or excitation (shake-up) at the production of an initial vacancy. The total probability of single shake processes for inert gases Ne and Ar was estimated to be up to 20% (Carlson & Nestor 1973; Mukoyama et al. 1999). Moreover, multiple shake processes are possible, but their probability is approximately 10 times smaller (Kochur & Popov 2006), and thus we do not consider such initial states.

This work supplements the data for cascades after single K-vacancy production (Kučas et al. 2012) and allows us to take into account the ionization or excitation of the other shell in addition to the single ionization of the K-shell. The cascades from configurations that can be obtained through the shake-off and shake-up processes are considered. Shake-up excitations are limited by the change in the principal quantum number ${\rm{\Delta }}n$ $\leqslant$ 2. The results of detailed calculations of cascades from such two-vacancy states are presented for the same astrophysically important light elements (Ne, Mg, Si, S, and Ar) as investigated by Kučas et al. (2012).

The sudden perturbation model is valid when the energy of the incident particle exceeds by several times the single ionization threshold. In the single-configuration approximation, more general expressions were presented by Kučas et al. (2012) for double photoionization as well as for simultaneous ionization and excitation probabilities without separating these processes into two steps. The most exact calculations of double ionization or ionization with excitation can be performed using the many-body perturbation theory and other correlation methods (Amusia et al. 2012). On the other hand, the populations of levels depend on the specific conditions typical for the considered object. Thus, we present the results for the cascading decay of the initial two-vacancy states without their excitation probabilities; they should also be taken into account when modeling processes in astrophysical plasma.

Astrophysical objects usually contain various ions of the same element; therefore, we have calculated cascades from the two-vacancy states for all of the ions in which Auger transitions are possible.

In this study, as in our previous work (Kučas et al. 2012), we calculate the wave functions, energy levels, and Auger and radiative transition rates in the single-configuration quasi-relativistic approximation using the atomic structure code (Cowan 1981). For the ions of light elements, the structure of the atomic shells is mainly determined by the Coulomb interaction; thus, we do not separate the nl^N shells into nlj^N subshells and we use the LS coupling scheme for the classification of levels.

In order to obtain the populations of different ion states, the FORTRAN computer code package Cascades has been developed. The algorithm was constructed using the following procedures, applied repeatedly for each step of the ionization. First, all of the possible radiative electric dipole decay channels for the initial ion configurations are determined and the electric dipole transition rates are calculated. Then, all of the Auger decay channels for each configuration, formed during the radiative transitions, are established and the Auger transition rates are calculated. All of the levels arising during cascades are sorted by decreasing energy and all of the transition rates between the levels are collected in the corresponding transition rate matrix. Using such a matrix, the populations of the levels P(Cγ) are calculated directly following the evolution of the cascade, according to the formula

$$\begin{eqnarray}&&P(C\gamma )=\displaystyle \sum _{{C}^{\prime }{\gamma }^{\prime }}P(C^{\prime} \gamma ^{\prime} )\displaystyle \frac{A(C^{\prime} \gamma ^{\prime} -C\gamma )}{A(C^{\prime} \gamma ^{\prime} )},\end{eqnarray} \tag{ 1 }$$

where γ is the level of configuration C, A(C'γ'–Cγ) is the Auger or radiative transition rate, and A(C'γ') is the total de-excitation rate of level γ'. The summation over C' is performed over all of the configurations from which the radiative or Auger transitions to configuration C are possible. The same radial orbital for the Auger electron has been used for all of the transitions between the initial and final configurations. The average energy of the Auger electron is determined through an iterative process, averaging the transition energy with a weight equal to its rate.

Only those ion levels with a population exceeding 0.005% are included in the next step of the cascade calculations. The inaccuracy introduced by such a simplification is insignificant for most results, however, the omitted parts of the populations are accumulated for the final ions and may constitute 1%–2% of the population of initial configuration. Comparatively small relativistic corrections for light elements are rather effectively taken into account by the quasi-relativistic approximation. Thus, the accuracy of the results presented in this work mainly depends on the correlation effects that are not taken into account by the single-configuration approximation. The main correlation types for configurations with inner vacancies were discussed by Karazija (1996); among the configurations considered in our work, only 3s3p^N (N > 1) belongs to the class of strongly interacting configurations (due to its mixing with the 3s²3p${}^{N-2}$3d configuration). On the other hand, the correlation corrections tend to be partly compensated during a cascade. The experimental data are available only for the relative distribution of ions in different stages of ionization after the integral cascade, which corresponds to the production of a single K-vacancy with additional ionization and excitation. The results for Ne and Ar presented in Kučas et al. (2012) show that such a distribution essentially differs from the experimental data for the cascade from a single K-vacancy state, but we obtain fairly good agreement by taking into account the contribution from the initial two-vacancy states. This indicates that the calculation of cascades from such states for light elements in the single-configuration quasi-relativistic approximation enables us to describe cascades fairly accurately.

3.1. Populations of Excited Configurations and Final Ions Distribution

Two main types of results for cascade evolution are of interest for astrophysical applications. The populations and other characteristics of the excited levels from which the radiative transitions take place are necessary for modeling the emission spectra. The final ions distribution according to the ionization degree, configuration, and level is useful for investigating the ionization equilibrium in the plasma. All of these calculated data are presented in the form of extensive tables (Tables 1 and 2). A small portion of every table is devoted to one initial state of one particular element as an example and the complete electronic tables are also provided.

Table 1.  Populations of Levels of Excited Configurations Produced During the Cascade of Auger and Radiative Transitions for Ne, Mg, Si, S, and Ar Ions

Mg5+ 1s2s2p5 Initial Configuration
Excited State						Initial Levels
q	Configuration	No	Level	$E(\mathrm{eV})$	fy	1	2	3	4	5	6	7
5	1s2s2p5	1	[3S]4P2.5	1308.142	4.08·10−2	100	⋯	⋯	⋯	⋯	⋯	⋯
5	⋯	2	[3S]4P1.5	1308.388	4.08·10−2	⋯	100	⋯	⋯	⋯	⋯	⋯
5	⋯	3	[3S]4P0.5	1308.54	4.08·10−2	⋯	⋯	100	⋯	⋯	⋯	⋯
5	⋯	4	[1S]2P1.5	1320.394	5.81·10−2	⋯	⋯	⋯	100	⋯	⋯	⋯
5	⋯	5	[1S]2P0.5	1320.577	5.95·10−2	⋯	⋯	⋯	⋯	100	⋯	⋯
5	⋯	6	[3S]2P1.5	1329.444	3.33·10−2	⋯	⋯	⋯	⋯	⋯	100	⋯
5	⋯	7	[3S]2P0.5	1329.558	3.22·10−2	⋯	⋯	⋯	⋯	⋯	⋯	100
5	1s22s2p4	⋯	(3P)4P2.5	30.780	1	2.85	1.84	⋯	⋯	⋯	⋯	⋯
5	⋯	⋯	(3P)4P1.5	30.967	1	1.22	0.54	3.40	⋯	⋯	⋯	⋯
5	⋯	⋯	(3P)4P0.5	31.076	1	⋯	1.70	0.68	⋯	⋯	⋯	⋯
5	⋯	⋯	(1D)2D1.5	43.836	1	⋯	⋯	⋯	0.47	4.74	0.05	0.54
5	⋯	⋯	(1D)2D2.5	43.840	1	⋯	⋯	⋯	4.14	⋯	0.59	⋯
5	⋯	⋯	(1S)2S0.5	51.081	1	⋯	⋯	⋯	0.92	0.90	0.11	0.15
5	⋯	⋯	(3P)2P1.5	55.159	1	⋯	⋯	⋯	0.26	0.11	2.61	1.03
5	⋯	⋯	(3P)2P0.5	55.387	1	⋯	⋯	⋯	0.04	0.22	0.54	2.04
5	1s2s22p4	⋯	(1D)2D1.5	1285.120	3.30·10−2	⋯	⋯	⋯	⋯	⋯	⋯	0.01
5	⋯	⋯	(1D)2D2.5	1285.158	3.18·10−2	⋯	⋯	⋯	⋯	⋯	0.01	⋯
6	1s22s2p3	⋯	(2D)3D2	215.697	1	24.07	8.69	42.83	19.02	23.02	2.34	3.36
6	⋯	⋯	(2D)3D1	215.699	1	6.36	24.90	10.47	9.19	18.39	1.08	2.77
6	⋯	⋯	(2D)3D3	215.706	1	37.35	31.66	10.35	33.84	17.36	4.50	2.04
6	⋯	⋯	(2P)3P0	220.812	1	4.29	0.85	4.90	3.02	2.70	0.29	0.50
6	⋯	⋯	(2P)3P1	220.815	1	9.27	12.84	4.84	8.83	8.58	0.98	1.31
6	⋯	⋯	(2P)3P2	220.821	1	14.06	16.46	22.01	13.66	16.48	1.88	1.61
6	⋯	⋯	(2D)1D2	232.411	1	0.01	⋯	⋯	3.61	4.02	70.78	70.16
6	⋯	⋯	(4S)3S1	233.093	1	0.51	0.52	0.53	2.04	1.99	0.79	0.80
6	⋯	⋯	(2P)1P1	237.526	1	⋯	⋯	⋯	0.95	1.49	13.46	13.69
6	1s22p4	⋯	3P2	254.217	1	4.45	2.83	0.85	16.32	6.04	6.28	2.16
6	⋯	⋯	3P1	254.466	1	1.48	1.48	4.41	5.84	11.41	2.16	4.73
6	⋯	⋯	3P0	254.577	1	⋯	1.62	0.68	1.37	4.94	0.55	1.95
6	⋯	⋯	1D2	259.406	1	⋯	⋯	⋯	2.24	2.18	22.29	23.15
6	⋯	⋯	1S0	266.994	1	⋯	⋯	⋯	0.43	0.47	4.43	4.70

(Data presented in this table are available in the tar.gz file.)

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Table 2.  Populations of Levels of Final Configurations After the Entire Cascade for Ne, Mg, Si, S, and Ar Ions

S4+ 1s2s22p63s24p Initial Configuration
Initial Levels: 1–3P0, 2–3P1, 3–3P2, 4–1P1
Final State				Initial Levels
q	Configuration	Level	E(eV)	1	2	3	4
4	1s22s22p63s2	1S0	0.000	0.09	0.09	0.05	0.25
4	1s22s22p63s3p	3P0	8.966	0.03	0.02	0.01	0.01
4	⋯	3P2	9.089	0.03	0.04	0.10	0.05
	Population of ion stage	⋯	⋯	0.15	0.14	0.16	0.31
5	1s22s22p63s	2${{\rm{S}}}_{0.5}$	71.689	10.15	10.33	10.56	10.39
5	1s22s22p53s3p	[3P]4${{\rm{D}}}_{3.5}$	245.859	0.01	0.02	0.14	⋯
	Population of ion stage	⋯	⋯	10.16	10.35	10.69	10.39
6	1s22s22p6	1S0	159.961	37.60	36.72	34.45	35.76
6	1s22s22p53s	3P2	329.455	25.78	26.90	30.67	28.56
6	⋯	3P0	330.693	6.88	6.27	4.57	5.33
	Population of ion stage	⋯	⋯	70.26	69.89	69.70	69.64
7	1s22s22p5	2${{\rm{P}}}_{1.5}$	440.457	11.64	12.13	12.94	12.71
7	⋯	2${{\rm{P}}}_{0.5}$	441.704	7.77	7.47	6.49	6.92
	Population of ion stage	⋯	⋯	19.42	19.60	19.43	19.64

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There are two types of initial configurations with a K-vacancy: one with an additional excitation and the other with an additional ionization. However, all of the notations in both tables are similar, and thus only the data for one configuration type are presented as an example in Tables 1 and 2.

In Table 1, for every excited level produced during the cascade, we give the ionization degree of the ion ($q)$, the configuration and level notation, the energy ($E)$, the fluorescence yield (fy), and the level population. These data are also presented for the initial configuration, the levels (initial levels) of which are numbered at the beginning of the table. The population is given as a percentage of the initial level population, which is taken to be equal to 100. However, the transitions from some excited configurations to others are also possible. Thus, some part of these populations is taken into account repeatedly and for this reason the summary population of all excited configurations usually exceeds 100. If the population of the level is smaller than 0.005, then the corresponding line is omitted from the table (e.g., for 3s3p ³P₁). The energies of all of the levels are given with respect to the ground level of the initial ion (Mg⁵⁺ in the given portion of the table). The levels of the same configuration are ranged in order of increasing energy. The configurations for the same ion are given in order of increasing total population. We use the LS coupling scheme for the classification of levels, and their notations correspond to those adopted in the Cowan code. The angular momenta, which couple the momenta of the shells, are placed in angle brackets. The self-explanatory single term of a shell with one electron or single vacancy is omitted. The angular momenta of the open shells are coupled gradually beginning from the inner shells.

The structure of the emission spectrum generated during the cascade is mainly determined by radiative transitions from the excited states whose decay through more probable Auger transitions is forbidden. Auger transitions end in various configurations because they are only possible from the highly excited autoionizing levels. In the case of a cascade from 1s2s2p⁵ for ${\mathrm{Mg}}^{5+}$, such configurations are 1s²2s2p³, 1s²2s2p⁴, 1s²2s2p⁵, ${}^{}\mathrm{and}$ 1s²2p⁴. The more intense radiative transitions are possible from the levels of configuration 1s²2s2p³. We must note that the populations of its levels differ significantly for cascades from various initial levels. In order to present the complete data, the excited levels whose decay is possible through both radiative and Auger transitions are also included in Table 1. Such a configuration for the considered cascade is 1s2s²2p⁴.

The role of various radiative transitions generated during the cascade from the initial single K-vacancy and two-vacancy configurations can be illustrated by the corresponding emission spectrum for neon (Figure 1). The probabilities of two-electron processes at the initial ionization of the K-shell were calculated using the expressions presented in the appendix of Kučas et al. (2012). The most intense maximum of this spectrum at 849.8 eV corresponds to the radiative transitions from the single-vacancy 1s2s²2p⁶ state, which can also be de-excited in a nonradiative manner. It accounts for the rather large natural width of this state, equal to 0.266 eV, which is mainly determined by more probable Auger transitions dominating in light elements. Consequently, both lines 1s2s²2p⁶–1s²2s²2p^{5 2}${{\rm{P}}}_{1/2,3/2}$, separated by a distance of only 0.096 eV, coalesce into one wide maximum. The radiative 2p–2s transitions from the excited configurations after the Auger cascade have an essentially smaller probability. However, the impossibility of decaying in a nonradiative manner also essentially diminishes its natural width. This property can compensate for smaller radiative transition probabilities and the less probable population of two-vacancy states during the initial ionization of atoms. Most of the lines in the other energy interval are again generated by the same cascade after the production of a single K-vacancy. They correspond to the transitions from the 1s²2s2p⁵ configuration populated due to the radiative 2p–1s transition. The levels of this configuration have a very small natural radiative width. Line 4 at 35.43 eV corresponds to the 1s²2s2p^{5 1}P₁–1s²2s²2p^{4 1}D₂ transition with a width of only $2.54\cdot {10}^{-5}$ eV. Several other transitions between the terms of these configurations give rise to line 3 at 30.31 eV with the same width and a group of six lines in the interval 26.54–26.68 eV with an even smaller width of $5.7\cdot {10}^{-6}$ eV. In this rather simple spectrum, only one line of comparable intensity belongs to the cascade from the initial two-vacancy state. This is line 1 at 23.95 eV, corresponding to the transition 1s²2p^{6 1}S₀–1s²2s2p^{5 1}P₁ and its 1s²2p⁶ state is obtained through the Auger transition from the initial configuration 1s2s2p⁶3s.

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The populations of levels, configurations, and ion stages after the entire cascade of Auger and radiative electric dipole transitions are given in Table 2. The portion for ${{\rm{S}}}^{4+}$ 1s2s²2p⁶3s²4p is shown as an example in the text. The populations are given as a percentage of the initial level population that is taken to be equal to 100. The notations for all of the quantities are the same as in Table 1; again, the levels with populations for all of the initial levels smaller than 0.005 are omitted. For this reason, and due to an approximation made during the calculation of the cascade (levels with a population smaller than 0.005% are excluded), the summary population of all of the final configurations can be slightly lower than 100%. Table 2 contains ions not only in the ground configurations, but also in some levels of the excited configurations 3s3p, 2p⁵3s, and 2p⁵3s3p because radiative electric dipole transitions from such levels are forbidden by the selection rule for the total angular momentum quantum number J.

The detailed data for the final levels given in Table 2 enable the calculation of the total ion yields (Table 3). They are obtained by the summation of the populations for all of the levels of considered configuration and by averaging over the initial levels (their statistical populations are assumed). However, such a table for all of the initial ions and configurations of one element is rather large, and thus, as an example, only part of such a table for the first four ions of Ar is presented here in the text. In Table 3, the ionization degree of the final ion is indicated with respect to a degree q of the initial ion. The sum of the ion yields for the given initial configuration is usually slightly lower than 100% due to the elimination of levels with a population smaller than 0.005% in the calculation of cascade. The data for the single K-vacancy cascade are taken from Kučas et al. (2012).

Table 3.  Averaged Ion Yields After the Entire Cascades from the Initial Configuration with a K-vacancy and Additional Ionization or Excitation of the Other Shell for Ne, Mg, Si, S, and Ar Ions

${\mathrm{Ar}}^{q+}$ (q = 1–4)
Initial Configuration		${\rm{\Delta }}q$ a
q		0	1	2	3	4	5	6	7
1	1s2s2p63s23p64s	⋯	0.02	1.07	10.18	16.61	51.64	19.03	0.04
1	1s2s22p63s3p64s	⋯	1.94	13.71	24.16	45.98	13.87	⋯	⋯
1	1s2s22p63s3p65s	⋯	1.01	12.19	22.90	49.00	14.03	0.44	⋯
1	1s2s22p63s23p54p	0.86	8.53	10.21	52.92	21.43	5.16	⋯	⋯
1	1s2s22p63s23p55p	1.57	16.35	7.94	14.74	40.83	17.62	0.13	⋯
1	1s2s22p63s23p6	0.91	11.87	9.72	53.45	18.88	5.09	⋯	⋯
2	1s2s2p63s23p54s	⋯	0.05	1.93	12.93	26.38	52.04	6.23	⋯
2	1s2s22p63s3p54s	0.46	5.95	13.83	47.34	29.44	2.92	⋯	⋯
2	1s2s22p63s3p55s	⋯	⋯	⋯	29.16	49.33	5.62	⋯	⋯
2	1s2s22p63s23p44p	0.79	10.62	10.31	57.48	19.06	1.60	⋯	⋯
2	2s22p63s23p6	0.01	0.27	2.27	4.24	17.97	40.58	33.08	⋯
2	1s2s2p63s23p6	⋯	0.05	2.01	12.81	28.45	55.6	0.72	⋯
2	1s2s22p53s23p6	⋯	1.26	12.40	12.37	59.43	14.43	⋯	⋯
2	1s2s22p63s3p6	1.01	11.86	9.66	53.23	20.91	3.22	⋯	⋯
2	1s2s22p63s23p5	0.86	11.85	9.33	54.57	23.35	⋯	⋯	⋯
3	1s2s2p63s23p44s	⋯	0.19	5.67	17.55	52.43	24.03	⋯	⋯
3	1s2s22p63s3p44s	0.81	11.01	10.54	57.98	19.54	0.02	⋯	⋯
3	1s2s22p63s23p34p	0.70	11.94	10.01	64.89	12.39	⋯	⋯	⋯
3	2s22p63s23p5	0.01	0.25	2.24	6.07	28.42	62.24	⋯	⋯
3	1s2s2p63s23p5	⋯	0.08	2.56	18.10	54.20	25.00	⋯	⋯
3	1s2s22p53s23p5	⋯	1.20	12.41	19.19	66.05	1.01	⋯	⋯
3	1s2s22p63s3p5	0.94	11.90	9.99	64.41	12.72	⋯	⋯	⋯
3	1s2s22p63s23p4	0.79	12.01	9.62	66.13	11.44	⋯	⋯	⋯
4	1s2s2p63s23p34s	⋯	0.45	7.50	28.60	63.41	⋯	⋯	⋯
4	1s2s22p63s3p34s	0.71	12.25	16.75	61.73	8.54	⋯	⋯	⋯
4	1s2s22p63s3p35s	0.52	9.62	20.95	57.05	11.85	⋯	⋯	⋯
4	1s2s22p63s23p24p	0.62	12.19	13.70	65.48	7.99	⋯	⋯	⋯
4	2s22p63s23p4	0.01	0.24	2.49	13.69	42.58	40.60	⋯	⋯
4	1s2s2p63s23p4	⋯	0.52	14.29	47.81	37.28	⋯	⋯	⋯
4	1s2s22p53s23p4	0.01	1.14	13.33	46.50	39.02	⋯	⋯	⋯
4	1s2s22p63s3p4	0.83	12.05	18.56	68.49	0.05	⋯	⋯	⋯
4	1s2s22p63s23p3	0.69	12.04	15.48	71.78	⋯	⋯	⋯

Note.

^a ${\rm{\Delta }}q$ is the change of ionization degree with respect to the initial one. (Data presented in this table are available in the tar.gz file.)

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The distribution of ions according to their ionization degree after the entire cascade mainly depends on Auger transitions because, during each such transition, the ionization degree increases by 1 and the Auger transitions are more probable than radiative transitions in light elements. The wider distribution corresponds to the cascades from configurations with one excited electron. The cascades from two-vacancy configurations, similar to the cascades after single K-vacancy production, give rise to the ion distribution, whose maximum usually corresponds to the next-to-last or last ionization degree. The second case more often takes place for simpler initial configurations with a smaller number of shells.

The comparison of ion distributions from the same initial configuration for various elements confirms the conclusion formulated by Kučas et al. (2012) that the yield of lower-charged ions increases while the yield of higher-charged ions decreases in such an isoelectronic sequence, and thus a narrower distribution corresponds to a higher initial ion. A higher ionization degree is obtained for the larger number of electrons in the outer shell or with the presence of an excited electron.

3.2. Supplementary Electronic Tables

During the photoionization or ionization of an inner shell, the other shells can also be ionized or excited. Such two-electron processes are less probable than single ionization, but they must be taken into account to obtain agreement with the experimental data for emission spectra or ion yields. The results of this work supplement our previous data for cascades after single K-vacancy production (Kučas et al. 2012).

We present the results of Auger and radiative cascades after the production of a vacancy in the K-shell and the additional ionization or excitation of the other shell for the same astrophysically important elements, Ne, Mg, Si, S, and Ar. Detailed level-by-level calculations are performed for all of the ions of these elements using the single-configuration quasi-relativistic approximation. The following data are presented:

• 1.  
  population, energy, and fluorescence yield for the excited levels, from which the radiative transitions take place;
• 2.  
  final ions distribution according to ionization degree, configuration, and level;
• 3.  
  distribution of ion yields averaged over initial states.

The investigation of cascades from the initial two-vacancy states confirms and supplements the regularities of cascades formulated in our work for K-vacancy states.

The many-electron quantum numbers of the initial level influence the population of excited levels more strongly compared to the final levels after the cascade.

The structure of the emission spectrum generated during the cascade is mainly determined by radiative transitions from the excited states whose decay through more probable Auger transitions is forbidden. In addition, some excited levels, whose de-excitation is possible through both radiative and Auger transitions, must be taken into account because their small fluorescence yield can be compensated for by the large value of the transition probability.

After the cascade of Auger and radiative transitions, the ions are obtained not only in the ground configurations, but also in some levels of excited configurations, whereas radiative electric dipole transitions from such levels are forbidden by the selection rule for the total angular momentum quantum number J. The cascades from two-vacancy configurations, similar to the cascades after the single K-vacancy production, give rise to the ion distribution whose maximum usually corresponds to the next-to-last or last ionization degree. The second case more often takes place for simpler initial configurations with a smaller number of shells. The wider distribution of the ion yields corresponds to the cascades from the configurations with one excited electron.