Hollow atoms and hollow ions (HI) have been promoted as a concept since experiments on ion beams meeting a surface at shallow angles of incidence found that the observed spectra were incompatible with an earlier concept, namely, that neutralization of highly charged ions near a surface would proceed by tunneling of individual electrons from the solid to the traveling ion. Instead, it was conjectured that a wave of many electrons would move to the many empty high-lying levels of the ion faster than to the few low-lying vacancy states (hence the name “hollow atoms”), and the de-excitation of the hollow atoms/ions would proceed internally to the ion, with many opportunities for Auger processes that would restore a high ion charge state.1–5

An early classical beam–foil experiment6 in which the angle of incidence was 90° demonstrated hollow ion X-ray emission with many features that were also later recognized in experimental spectra of high-density plasmas, such as dense laser-produced plasmas,7–10 high-current dense Z-pinch plasmas,11 vacuum sparks,12 and X-ray free-electron laser (XFEL) interactions with matter.13

However, the origin of sometimes very high intensities (of the order of the usual resonance-line intensities) of the emission resulting from transitions of the type K⁰L → K¹L⁻¹ + ℏω_N in dense plasmas remains a matter of discussion to this day. For example, standard collisional–radiative modeling and analytical estimates demonstrate that HI emission should be very low, i.e., below bremsstrahlung,12 and would be incompatible with observations. The precision of the atomic structure calculations1,6,14–16 that are used to identify hollow ion X-ray emission is rather high (multiconfiguration Hartree–Fock or Dirac–Fock including configuration interaction and intermediate coupling), and inaccuracies in the atomic data (energy levels, spontaneous transition rates, and autoionization rates) can therefore be excluded as a source of this incompatibility.

The continued interest in hollow ion X-ray emission from transitions of the type K⁰L → K¹L⁻¹ + ℏω_N in dense plasmas is due to its advanced spectroscopic diagnostic capabilities.17 Of particular interest for high-energy density science is the very small opacity of hollow ion X-ray emission, i.e., the absorbing ground states are the excited states K¹L⁻¹, which are weakly populated even in very dense plasmas.12 This is important for the study of phenomena in the interior of very dense objects via X-ray spectroscopic methods.18–21

For dense plasmas, essentially two basic ideas have been put forward to explain the anomalous, high-intensity hollow ion X-ray emission:(i)charge exchange between interpenetrating plasmas when highly ionized ions propagate into the residual gas;^12,22–24(ii)short-pulse intense radiation fields inducing K-hole vacancies (e.g., X-ray laser-driven matter) via the impact of X-ray photons on atoms on time scales comparable to the autoionization time scale.²⁵

Atomic population kinetics including charge exchange processes are exceedingly difficult to describe, owing to the resonance properties of the cross section and the nonlinearity induced by the populations of the donor atom.21 Therefore, collisional–radiative treatments of charge exchange processes in dense laser-produced plasmas and comparison with experimental data are currently rather scarce.24

For XFEL radiation interacting with matter the situation is nowadays quite different. Early collisional–radiative simulations25,26 identified the essential elements responsible for the high-intensity hollow ion X-ray emission and the charge state dynamics some time before XFEL facilities became readily available to researchers. Double core hole vacancies and subsequent intense hollow ion X-ray emissions are induced by successive impact of photons on K² configurations with rates smaller than, or of the order of, the autoionization rates.25 “Beating the Auger clock” is therefore an essential element in the charge state dynamics and short-pulse hollow ion X-ray emission.25,26

Apart from direct photoionization from the K shell, which induces higher charge states, the ionization dynamics and charge state distributions in dense plasmas are additionally influenced by collisional ionization. This has been well identified in simulations:25 if the XFEL photon energies are above the ionization threshold for the H-like bound state K¹, heavy overionization of the material occurs. However, if the XFEL energy is tuned below the ionization threshold for the ionization of H-like magnesium, the population of the bare nucleus is not zero, but is instead determined by collisional ionization.

The situation where collisional L-shell ionization leads to important changes in the charge state distribution, while a secondary process is preferentially ionizing the core electrons (and is responsible for X-ray emission), is known from the atomic physics of suprathermal electrons:27 thermal collisional ionization from the L shell leads to qualitative distortions of the charge state distribution, while the suprathermal electrons strongly drive X-ray emission. We note that the influence on the ionization dynamics via collisional ionization of excited states is well known from the early days of atomic physics in plasmas.28,29

With the increasing availability of XFEL installations,30–32 hollow ion X-ray emission was finally observed in the interaction of intense short-pulse radiation fields with matter.13,33 Numerous subsequent analyses based on collisional–radiative simulations12,13,34 (which are very similar to those originally developed in Refs. 25 and 35) reconfirmed the essential elements of high-intensity hollow ion X-ray emission, namely, photoionization, resonance pumping, collisional ionization, and Auger time scale.

An outstanding property of hollow ion X-ray emission is related to the response time and duration after creation of the hollow ion configuration via various atomic physics processes. As the hollow ion configurations K⁰L are autoionizing states, the response time τ_HI of the X-ray emission21,36 is very short and of the order of the autoionizing time scale, i.e., τHI−1=A+C+Γ≈Γ, where A characterizes the spontaneous radiative decay rate, C the collisional depopulation rate of the hollow ion configuration, and Γ the autoionization rate. The relation τHI−1≈Γ holds provided that the collisional rate C does not considerably exceed the autoionizing rate Γ (which requires near-solid densities) and provided that the radiative decay rate A is not larger than the autoionizing rate (for high Z-elements, A might be larger than Γ). Therefore, the hollow ion emission induced by intense photon sources corresponds to emission times close to the length of the X-ray pulse itself. The hollow ion X-ray emission therefore acts like an X-ray emission switch25,26 at times when the plasma density is highest (i.e., when the pulse interacts with the matter) and contributions from the low-density recombination regime are negligible. This situation constitutes an important advantage in high-energy-density research compared with the usual resonance lines; i.e., in time-integrated spectra, the usual resonance lines typically accumulate high intensity from the recombination phase at low density, thereby making spectroscopic diagnostics via line intensity or line shape analyses rather challenging. Consequently, hollow ion X-ray emission was proposed for the analysis of dense plasma properties25 and was later applied to temperature diagnostics in FEL experiments.37

The fact that the relaxation time of a system is given by the inverse sum of collisional and noncollisional rates allows one to “clock” the collisional processes with respect to the noncollisional ones. This has been well explored in the framework of the concept of “collisional mixing of relaxation times” (CMRT) in dense plasmas.38 If collisions occur on time scales shorter than, or of the order of, those of the noncollisional ones (e.g., radiative decay or autoionization), collisional depopulation contributes considerably to the dynamics. Its efficiency is in general an interplay of the collisional rates with the radiative decay rates and the autoionizing rates. Both lead to a disintegration of the atomic level, but the higher the collisional rates are compared with autoionization rates and radiative decay rates, the more strongly is the distribution of level populations (e.g., charge state distribution) driven by collisions instead of autoionization or radiative decay. We note that the concept of CMRT and its potential to clock collisions with noncollisional ones has also been applied in the context of XFEL experiments.13,34

Experimentally obtained two-dimensional intensity maps (i.e., transition wavelengths, XFEL wavelengths, and intensity)13 can in principle be reconstructed via combined plasma and atomic physics simulations.39 However, the simulations are very complex, and it remains a great challenge to match their results with observations (owing to uncertainties in laser intensity, intensity distribution over the focal spot, intensity evolution over time, line of sight, and other lateral integrations). Spectroscopic diagnostics based on hollow ion X-ray emission is therefore an important element to advance the field.

The first step in the analysis of hollow ion emission concerns atomic structure calculations that include perturbations due to the dense hot plasma environment.40–54 Below, such calculations are presented for isoelectronic sequences involving single and double core hole states. With the help of a diagrammatic technique, we demonstrate in a second step that these data already allow some conclusions to be drawn about the evolution in time, independent of complex plasma simulations. Finally, the technique is employed to visualize the impact of ionization potential depression on ionization pathways.

Calculations of hollow ion atomic structure (energy levels, transition probabilities, wavelengths) have been performed with the multiconfiguration Hartree–Fock method including intermediate coupling, configuration interaction, and relativistic corrections using Cowan’s atomic structure codes.55

To estimate the accuracy of the present studies, we compare in Table I relevant transition and ionization energies. For the Kα transitions as well as the K edges, we have considered a rather limited set of K-, L- and M-shell configurations (for clarity in the discussion of hollow ions, we also mention empty shells, e.g., 3s⁰) in a neutral atom (i.e., the total number of bound electrons equals the nuclear charge; for details see Ref. 56): 1s²2s²2p⁶3s², 1s²2s²2p⁶3s¹3p¹, 1s²2s²2p⁶3s⁰3p², 1s¹2s²2p⁶3s²3p¹, 1s¹2s²2p⁶3s¹3p², and 1s²2s²2p⁵3s²3p¹. For the K edge in a solid, we have performed a statistical average of the energy levels originating from the M-shell configurations 1s²2s²2p⁶3s² (g = 1), 1s²2s²2p⁶3s¹3p¹ (g = 4), and 1s²2s²2p⁶3s⁰3p² (g = 5) to the respective lowest energy levels (i.e., 1s¹2s²2p⁶3s²3p¹ and 1s¹2s²2p⁶3s¹3p²), because the edge is defined as the lowest energy for which a transition occurs. The physical motivation for the statistical average in a solid is given by the fact that in reality, the Stark effect leads to mixed states in the band structure where ionization can occur only to states that are not occupied and allowed by the Pauli principle. As can be seen from Table I, the present approach leads to remarkably good agreement with the experimental data. The last column of the table shows the results of rather sophisticated theoretical calculations that are likewise in good agreement with the experimental data except for the K edge of solid material.

For further demonstration of the high precision of the current method, let us consider the different spin transitions of He-like magnesium in vacuum, i.e., the He-like resonance and intercombination lines of the α and β transitions, i.e., W2 = 1s2p¹P₁–1s²¹S₀, Y2 = 1s2p³P₁–1s²¹S₀, W3 = 1s3p¹P₁–1s²¹S₀, and Y3 = 1s3p³P₁–1s²¹S₀. Taking into account only the simplest necessary configurations up to the M-shell as above (i.e., 1s², 1s2s, 1s3s, 1s3d, 1s2p, and 1s3p), the present calculations deliver ΔE(W2) = 1352.5 eV, ΔE(Y2) = 1343.0 eV, ΔE(W3) = 1579.9 eV, and ΔE(Y3) = 1577.3 eV. These data are in very good agreement (within one eV) with the reference data from the U.S. Institute for Standards and Technology (NIST):58 ΔE_NIST(W2) = 1352.2 eV, ΔE_NIST(Y2) = 1343.1 eV, ΔE_NIST(W3) = 1579.2 eV, and ΔE_NIST(Y3) = 1577.2 eV. Therefore, the present approach has spectroscopic precision and, consequently, predictive power.

Corrections for dense-plasma effects have been included in the Hamiltonian via a plasma potential V̂plasma:Ĥ=Ĥ0+V̂plasma,(1)where Ĥ0 is the unperturbed Hamiltonian. For V̂plasma, we employ the analytical b-potential59 that has proven to provide excellent agreement up to near-solid density plasmas at rather low temperatures (of the order of the Fermi energy).60 Even high-precision line-shift data obtained in dense laser-produced plasmas are well described.59 We note that the particular form of the b-potential even allows the derivation of high-precision analytic formulas for line shifts, either in terms of gamma functions or in terms of a fourth-order series development employing integer-order-k radial matrix elements rk.59

The analytical b-potential is now widely applied in atomic structure calculations of ions in dense plasmas,61–66 since it greatly simplifies atomic structure calculations while preserving high precision (in atomic units):V̂plasma=NfreeRi1+1x−11−rRix−1,(2)wherex=3−bπNfreeRiTe,(3)and N_free is the number of free electrons in the ion sphere with radius R_i and electron temperature T_e. To keep the ion sphere neutral,Zn−Nfree−Nbound=0(4)andZn−Nbound=4π3Ri3ne(5)are assumed, where N_bound is the number of bound electrons, Z_n the nuclear charge, and n_e the electron density. The parameter b characterizes the self-consistent electron distribution in the ion sphere; for example, for a Maxwellian plasma, b ≈ 2. In the limit of high temperature, the b-potential turns into the uniform electron gas model (x = 3).

We note that degeneracy effects and Pauli blocking are not very important if the electron temperature is a few times larger than the Fermi energy E_F (E_F = 7.08 eV for magnesium). Moreover, detailed simulations39,67 show that essentially for temperatures below about half the Fermi energy, important differences compared with Maxwell–Boltzmann statistics are expected.

The present atomic structure calculations are carried out in a framework of a self-consistent field Hartree–Fock approach taking into account the exact nonlocal exchange terms, configuration interaction, and intermediate coupling. Because no calibration shifts or scaling parameters are applied, the method has predictive power for transitions where no reference data exist (e.g., the transitions in hollow ions along the isoelectronic sequence that are of interest in the present study).

The analytic potential [Eqs. (2)–(5)] enables the study of dense-plasma effects without the need for sophisticated and expensive atomic physics methods (e.g., self-consistent field iterations coupled to the Poisson equation, etc.44,46,48–50,52,54), while the level shifts depend on principal and orbital quantum numbers. Therefore, even shifts of line transitions can be obtained. These shifts are in excellent agreement with the data.59 The present method is thus in striking contrast to models that apply global shifts (implying vanishing shifts for line transitions) and to the methods of Refs. 51 and 68–71 that apply calibration shifts from about 10 to 100 eV to match the data.

Calibration shifts are usually obtained by comparing the results of simulations with the well-known experimental data for the Kα transition. It is then assumed that the same shift applies to all other transitions and processes [and even to hollow ion configurations (empty K shell) with different L-shell occupations]. This is a rather challenging procedure, even leaving aside those difficulties that arise in the construction of the wavefunctions and the application of local exchange approximations.21,56 Consequently, spectroscopic precision and predictive power (e.g., for unknown transitions in hollow ions) are not expected from the methods employed in Refs. 51 and 68–71.

The hollow ion X-ray emission resulting from transitions of the type K0LN→K1LN−1+ℏωN(HI) is given by variation of the L-shell occupation number N, i.e., N = 8 corresponds to O-like hollow ions, N = 7 to nitrogen-like ions, etc. K⁰ indicates that no electron is bound in the K shell. This shell is then called hollow if in higher-lying shells, two or more electrons are bound. The configurations K⁰L thus correspond to an isoelectronic sequence of hollow ion configurations depending on the number of bound electrons, N. The quantities ℏωN(HI) are the corresponding transition energies. As discussed above, hollow ion emission is strongly related to the photoionization processes of core hole configurations. We therefore study the ionization energies of free ions and ions in plasmas (i.e., ionization potential depression is taken into account) and present in Table II the respective ionization energies of the isoelectronic sequences.

Under real experimental conditions, different electron temperatures and densities have to be correlated with the various occupations of the L shell. The time-dependent average electron temperature can be estimated via a simple zero-dimensional model considering the evolution of the energy E in the atomic system. The system energy is driven by the power P_abs absorbed by the radiation field and the power P_rad emitted via radiative recombination and by spontaneous and stimulated emission. All elementary atomic physics processes are consistently calculated with Fermi–Dirac statistics and corresponding Pauli blocking factors. The link between the cold and hot solid is established with a probability formalism and generalized atomic physics processes,67 taking into account Fermi–Dirac statistics, while the fluorescence emission is calculated from the radiation transport equation.39 The evolution of the system energy is then given by39,67dEdt=Pabs−Prad.(6)

As has been demonstrated in one-dimensional hydrodynamic simulations, this zero-dimensional model provides a very efficient approximation to short-pulse (1–100 fs) XFEL radiation interacting with solid matter:72 owing to the femtosecond time scale, the density stays near solid density since ion motion is negligible. The evolution of the electron temperature is then strongly related to the atomic physics processes and the energy balance of the system. Comparing the electron temperatures obtained from the one-dimensional model with the temperatures obtained from the zero-dimensional model, very good agreement is found72 in the early stage of plasma evolution (which is of primary importance for hollow ion emission, as discussed above).

As has been demonstrated in Ref. 39, the maximum population of a certain charge state Z corresponds approximately to the mean ionization Z at this time if matter is essentially driven by short radiation pulses. We therefore attribute to the various L-shell occupations the following electron temperatures kT_e and electron densities n_e for an improved estimation of the reduction of the ionization potential. Because ion motion is negligible on these short time scales, the electron density values can be estimated from solid-density magnesium with the appropriate ionization: the number of bound electrons is given by the L-shell occupation number, while the number of free electrons N_free per ion is given by Eq. (4), i.e., N_free = Z_n − N_bound (e.g., four electrons for the L⁸ configuration). We therefore obtain the following parameter sets:39kT_e = 5 eV, n_e = 1.74 × 10²³ cm⁻³ for L⁸; kT_e = 25 eV, n_e = 2.18 × 10²³ cm⁻³ for L⁷; kT_e = 30 eV, n_e = 2.62 × 10²³ cm⁻³ for L⁶; kT_e = 40 eV, n_e = 3.05 × 10²³ cm⁻³ for L⁵; kT_e = 50 eV, n_e = 3.49 × 10²³ cm⁻³ for L⁴; kT_e = 60 eV, n_e = 3.93 × 10²³ cm⁻³ for L³; kT_e = 90 eV, n_e = 4.36 × 10²³ cm⁻³ for L²; kT_e = 115 eV, n_e = 4.80 × 10²³ cm⁻³ for L¹.

Table III compares experimental hollow ion transition energies (data extracted from a time-integrated X-ray image obtained from an experiment involving the interaction of an intense XFEL beam with solid-density magnesium69) with the results of the present simulations. The isoelectronic sequences are designated by the core hole configurations K⁰L. Owing to the numerous coupling possibilities of open shell configurations, the number of X-ray transitions is very large. We therefore define the interval of transition energies from the condition of gA-values larger than 10¹³ s⁻¹ (where g is the statistical weight and A the spontaneous transition rate).

It can be seen from Table III that for the configuration K⁰L¹, the plasma-free simulation seems to be in better agreement with experiment than the simulation including ionization potential depression. One reason might be that the related transitions K0LN→K1LN−1+ℏωN(HI) for N = 1 are the Lyman-alpha transitions, whose upper states are singly excited ones and are therefore non-autoionizing. In this case, the mechanism related to a short response time of the hollow ion X-ray emission due to temporal correlation with the X-ray pulse26 does not apply, and considerable intensity could originate from the long-lasting recombination phase at low density, as demonstrated by time-dependent atomic population kinetic simulations.26

Simulations of hollow ion X-ray emission are rather challenging owing to the dense-plasma effects on atomic structure and elementary atomic physics processes,20,44 as well as time-dependent phenomena. For example, the simulations presented in Ref. 13 indicate strong hollow ion X-ray emission in some cases, whereas almost no emission is observed in the experiment [see Fig. 2(b) of Ref. 13]. It is therefore of interest to base complementary analysis on X-ray spectroscopy and on atomic structure of ions in plasmas where dense-plasma effects are included with spectroscopic precision. The b-potential method [Eqs. (2)–(5)] has been shown to provide spectroscopic precision, while simpler methods, such as the well-known Ecker–Kröll73 and Stewart–Pyatt74 methods (which are frequently employed in simulations13), do not satisfactorily distinguish dense-plasma effects in terms of principal and orbital quantum numbers (and consequently provide almost the same energy shift for every level in a given charge state and therefore fail to provide spectroscopic precision).

We therefore propose below a diagrammatic technique in which the hollow ion X-ray emission is analyzed in terms of transition energies, XFEL energy, and isoelectronic sequence employing the present quantum-number-dependent perturbation of atomic structure [Eqs. (2)–(5)] taking into account the exact exchange terms.55

The analysis below of the various excitation channels for the isoelectronic sequence K⁰L is based on ionization energies, including ionization potential depression mapped together with XFEL energies.

The direct photoionization channel (I) starts from a single K hole and is given byK1LN+ℏωXFEL→K0LN+e,(7a)K0LN→K1LN−1+ℏωN(HI),(7b)K1LN−1+ℏωXFEL→K0LN.(7c)

The processes (7b) and (7c) are the finally required processes that characterize the resonance case for which hollow ion transition energies and XFEL photon energies are identical, i.e.,ℏωXFEL=ℏωN(HI).(7d)

As discussed in Ref. 12 with detailed simulations of the spectral distribution, the existence of the relation (7d) over the sequence is strongly related to ionization potential depression leading to a characteristic distribution of hollow ion emission over charge states.

Figure 1 shows the transition energies of the hollow ion isoelectronic sequence in relation to the XFEL resonance excitation (solid black squares connected by the solid black line). Channel I (red curves and symbols) indicates the necessary ionization energies to proceed from configuration K¹L to K⁰L [see (7a)], dashed red lines are calculations for the free-ion case, and solid red lines are calculations taking into account the dense plasma environment.

It can be seen from Fig. 1 that channel I, according to Eqs. (7a)–(7d), implies ionization energies that lie well above the curve related to resonance energies. Therefore, channel I cannot be related to the hollow ion X-ray emission driven by XFEL photons at resonance energies [Eq. (7d)]. Although dense-plasma effects reduce the relevant ionization energies (the solid red curve is located below the dashed one) the reduction is insufficient to explain the resonance features of the isoelectronic sequence. Of course, this conclusion can also be reached via collisional–radiative simulations,12,13,39 but it is interesting to see that atomic structure calculations alone provide more transparent insight into this phenomenon.

Another photoionization channel (II) starts from a fully occupied K shell but a different L-shell electron population, namely, an underpopulation of the L shell (N − 1 instead of N):K2LN−1+ℏωXFEL→K1LN−1+e,(8a)K1LN−1+ℏωXFEL→K0LN.(8b)

The dashed green line in Fig. 1 indicates the ionization energies for channel II [according to Eq. (8)]. It can be seen that only for configuration K⁰L⁸ does the resonance energy permit photoionization of the K shell according to channel II [Eq. (8a)]. For other configurations, the necessary ionization energies are larger than the corresponding resonance energies. For the dense-plasma case, however, channel II also permits resonance excitation for configurations K⁰L⁷ and K⁰L⁶. In the experiment, however, resonance excitation up to the configuration K⁰L¹ is observed, indicating that other channels are required to explain the emission.

Channel III indicates a coupling of different processes, namely, K-shell photoionization followed by autoionization (9b) or collisional ionization (9e) for an overpopulated L shell (N + 1 instead of N):K1LN+1+ℏωXFEL→K0LN+1+e,(9a)K0LN+1→K1LN−1+eAuger,(9b)K1LN−1+ℏωXFEL→K0LN.(9c)

Instead of autoionization (9b), electron collisional ionization of the L shell can also be considered, i.e.,K1LN+1+ℏωXFEL→K0LN+1+e,(9d)K0LN+1+ecoll→K0LN+2e,(9e)K0LN→K1LN−1+ℏωN(HI),(9f)K1LN−1+ℏωXFEL→K0LN.(9g)In general, all ionization processes from the L shell have to be considered, not only electron collisional ionization. Therefore, in principle, L-shell photoionization also contributes, i.e., if the photon energy exceeds the ionization threshold for K-shell electrons, it consequently exceeds those for the L-shell ionization; see the more detailed discussion below. Because the first K-shell photoionization (9a) and (9d) is related to an overpopulated L shell, ionization energies are reduced compared with (7a). However, as Fig. 2 demonstrates, even in dense plasmas, this channel does not provide ionization energies below the respective resonance energies.

Let us now consider K-shell photoionization followed by L-shell ionization [collisional ionization, indicated by e_coll in (10b) and photoionization indicated by ℏω_XFEL in (10c)], channel IV, i.e.,K2LN+ℏωXFEL→K1LN+e,(10a)K1LN+ecoll→K1LN−1+2e,(10b)K1LN+ℏωXFEL→K1LN−1+e,(10c)K1LN−1+ℏωXFEL→K0LN.(10d)As Fig. 2 indicates (green curves and symbols), this channel has ionization energies that are in the isolated ion case lower than the resonance energy for configurations up to about K⁰L⁷, while in the dense plasma case the ionization energies are lower than the resonance energy for configurations up to about K⁰L⁴.

For channel III, Fig. 2 demonstrates that L-shell ionization allows the movement of hollow ion configurations K⁰L to higher charge states (for a detailed discussion of phenomena induced by collisional ionization of L-shell electrons, see Refs. 27–29). Under real experimental conditions, L-shell collisional ionization can be quite effective: peak electron temperatures are of the order of 100 eV39,72 (see also the plasma parameters provided in Sec. III), while ionization energies are of the order of 300 eV. The scaling parameter for collisional processes (the so-called β parameter, which is the ratio of excitation energy to electron temperature) is therefore on the order of three, and collisional rate coefficients (e.g., for the ionization process K²L² + e → K²L¹ + 2e, with ionization energy 300 eV) are of the order of σexV≈2.7×10−11cm3s−1.21 At an electron density n_e = 10²³ cm⁻³, the collisional ionization rate at near-solid density is then about neσexV≈3×1012s−1.

Let us now discuss also photoionization rates. In fact, any K-shell photoionization is necessarily accompanied by L-shell photoionization: for example, taking a photon density of Ñ0=1023cm−3 (corresponding to an XFEL intensity of 4.5 × 10¹⁷ W/cm² at 1630 eV photon energy and 8.4 × 10¹⁶ W/cm² at 300 eV), L-shell photoionization from the K²L² configuration of magnesium (the ionization energy for the 1s²2p² configurations is about 300 eV) has a cross section σ^phi of about σ⁽⁾(L) ≈ 6.1 × 10⁻¹⁹ cm² at threshold,75 and the corresponding photoionization rates are σphic≈σ(L)(L)cÑ0≈1.8×1015s−1 (c ≈ 3 × 10¹⁰ cm/s is the speed of light). For comparison, let us consider K-shell photoionization from the same K²L² configuration (the ionization energy for the 1s²2p² configurations is about 1629 eV): σ⁽⁾(K) ≈ 1.1 × 10⁻¹⁹ cm² at threshold,75 with corresponding photoionization rate at the same XFEL intensity σphic≈σ(K)(K)cÑ0≈3.3×1014s−1. Therefore, at the respective threshold values and fixed photon density, K- and L-shell ionization rates differ by a factor of about six. If, however, L-shell photoionization is considered at the threshold value of the K shell [i.e., σ^phi = σ(K)], the situation is different: σ⁽⁾(K) ≈ 3.0 × 10⁻²¹ cm² and σphi⋅c≈σ(L)(K)cÑ0≈9.0×1012s−1. Although this rate is considerably lower than the respective value for K-shell photoionization, it is possibly of the same order as collisional ionization rates from the L shell (which is strongly dependent on electron temperature, owing to the exponential dependence of the rate coefficient on the β parameter). Therefore, L-shell photoionization might also be an efficient process to increase the charge state of hollow ions.

The importance of L-shell ionization processes manifests itself in an experimentally observed broad feature around the exact resonance position. This concept is depicted in Fig. 2 via the blue rectangles around the exact resonance position (red diffuse circles). It can be seen that for higher charge states, the vertical extension of the blue rectangles is rather large (of the order of 50 eV): consider, for example, the resonance position of the K⁰L² configuration near 1454 eV. Evidently, the resonance condition for the next ionization stage of the K⁰L¹ configuration near 1472 eV is not met. Nevertheless, important emission from the K⁰L¹ configuration is observed, indicated by the solid vertical arrow in Fig. 2, starting from the horizontal dashed black line that indicates the XFEL resonance energy at 1454 eV of the K⁰L² configuration.

We are therefore interested in considering cases that correspond to multiple L-shell ionization after K-shell photoionization, i.e., channel V according toK2LN+1+ℏωXFEL→K1LN+1+e,(11a)K1LN+1+ecoll/ℏωXFEL→K1LN+e,(11b)K1LN+ecoll/ℏωXFEL→K1LN−1+e,(11c)K1LN−1+ℏωXFEL→K0LN(11d)or channel VI according toK2LN+2+ℏωXFEL→K1LN+2+e,(12a)K1LN+2+ecoll/ℏωXFEL→K1LN+1+e,(12b)K1LN+1+ecoll/ℏωXFEL→K1LN+e,(12c)K1LN+ecoll/ℏωXFEL→K1LN−1+e,(12d)K1LN−1+ℏωXFEL→K0LN.(12e)

Figure 3 shows the ionization energies related to channels V and VI. It can be seen that for the dense-plasma cases, channel V has ionization energies that proceed toward the resonance of the K⁰L³ configuration (see the solid red line near 1437 eV), and channel VI proceeds toward the last possible configuration K⁰L¹ (the solid green line near 1472 eV).

Hollow ion X-ray emission of the type K0LN→K1LN−1+ℏωN(HI) induced by intense XFEL interactions with solid matter has been analyzed via atomic structure calculations that account for dense-plasma effects. Ionization potential depressions for isoelectronic sequences have been obtained for single and double core hole states. It has been demonstrated that a diagrammatic representation in terms of the isoelectronic sequences of the hollow ion X-ray transitions and XFEL energy enables a plasma-simulation-independent analysis of important pathways in the ionization dynamics. The crossing point of ionization potential energies for the various production channels with the curve of resonance excitation indicates whether the process is allowed or forbidden. The inclusion of a dense-plasma environment provides a visualization of the impact of the ionization potential depression on the possibility of opening forbidden channels via the shifts in potential energy. The diagrammatic method demonstrates how the observation of emission produced by the entire series of hollow ions leads to an understanding of the importance of L-shell ionization. More generally, this type of diagram indicates the interplay between photoionization, electron collisional ionization, and autoionization from the L shell, as well as ionization potential depression.

Acknowledgment. The work of C.J.F. was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (Contract No. 89233218CNA000001).

This research received no external funding.