Understanding of atomic physics in dense plasmas is crucial in astrophysics, cosmology, planetary science, and inertial confinement fusion research,1–6 and is also fundamental for understanding the interplay of a quantum atom with a plasma under extreme conditions.7–14 In recent decades, important improvements have been made in the investigation of the radiative properties (emission, absorption, and scattering) of dense plasmas.

Among a variety of dense-plasma effects, ionization potential depression (IPD) is of fundamental interest and has attracted attention from the high-energy-density community for generations.7,15–19 In dense plasmas, the changes in the energy level structure (level shifts) and the corresponding IPD and line shifts have significant impact on the ionic charge state distribution and the equation of state, as well as on photon emission and absorption and on almost all radiative and thermodynamic properties. High-precision experiments that enable determination of the energy perturbations of atomic levels are rare, since it is rather difficult to isolate well-diagnosed dense hot samples. On the other hand, self-consistent quantum theory with spectroscopic precision is likewise not an easy task (this is a finite-temperature multiparticle Coulomb system) and the development of models to calculate the IPD is a task that continues to the present day.20–48

To create hot near-solid-density samples, most experiments have relied on the interaction of high-energy-density lasers with solids, while on the theoretical side, a wide variety of works have aimed at improving the screening potential.39,40

As high density is mandatory to study dense-plasma effects, atomic radiation suffers from photoabsorption due to high populations in absorbing states. This is a serious obstacle for high-precision studies, owing to opacity broadening, opacity-induced line shifts, and asymmetries.27 To minimize photoabsorption effects, X-ray transitions are the first choice, because the line center opacity is proportional to the wavelength: τ₀ ∝ λ. For this reason, most experiments to study dense-plasma effects on atoms and ions have been performed with high-energy-density lasers that allow effective ionization of atoms in dense plasmas and subsequent excitation of highly charged ions in the X-ray spectral range. Among a wide variety of possible X-ray transitions, those emitted from H- and He-like ions have attracted particular interest, because here the line identification is comparatively simple.13,14,46,49

Owing to the nature of the laser–plasma interaction and heating, the energy of the radiation emission and the ionization potential of the highly charged ions are of the order of the electron temperature kT_e. Therefore, X-ray transitions at low charge states (corresponding to comparatively low temperatures of the order of the Fermi energy) can hardly been studied, since their intensity is exponentially small. The gap to the study of X-ray transitions at low temperatures (X-ray transitions corresponding therefore essentially to transitions involving inner-shell holes) was closed with the development of powerful X-ray radiation sources. X-ray excitation in solids at low temperatures with synchrotrons works well, but the creation of an inner-shell hole is overall a rare phenomenon in the lattice, and consequently the number of inner-shell holes generated is small. Therefore, the solid is not effectively heated, and studies are essentially limited to temperatures below 1 eV. The situation is dramatically different for X-ray free-electron laser (XFEL) installations. These provide more than ten orders of magnitude higher intensities than synchrotrons. Therefore, generation of inner-shell holes effects almost every atom in the lattice, since the photon density in the light pencil is of the order of solid density. Therefore, the number density of subsequently emitted photo and Auger electrons is also near solid density. This allows effective heating to the order of the Fermi energy or even above. Consequently, near-solid-density matter can be studied at low temperatures (of the order of the Fermi energy) up to some 100 eV, while intense X-ray radiation is emitted from refilling of the inner-shell holes. Therefore, XFELs have also been employed to study IPD, and even isoelectronic sequences can be investigated.32–35

Controversy regarding IPD in dense plasmas is exacerbated by the lack of general models with spectroscopic precision. Among the wide variety of IPD models, the Stewart–Pyatt (SP) model17 and the Ecker–Kröll (EK) model18,19 are frequently employed in high-energy-density physics (see, e.g., Refs. 13, 14, and 21). However, all the energy levels of one ionization state are affected in almost the same manner, indicating that the energy shifts of the line transitions approximately vanish. Moreover, the previously claimed agreement34,35 of the EK model with XFEL data on IPD turned out to be questionable,36,41–45 because the polarization and lattice term functions of the EK model have been suppressed and simply replaced by a fitting constant.35

Spectroscopic precision of generalized atomic structure calculations in dense plasmas is therefore urgently required, but is not an easy task. For example, even in Hartree–Fock–Slater (HFS) methods, calibration shifts of more than 20 eV are applied38,50 to match the data. Note that the standard imprecision of multiconfiguration Hartree–Fock methods, including configuration interaction and intermediate coupling, is of the order of only a few 10⁻¹³ m in the X-ray spectral range: for example, an imprecision of 2 × 10⁻¹³ m for the He-like resonance line of aluminum (transition energy of about 1.6 keV) corresponds to an imprecision in energy of less than 0.5 eV.

The use of density functional theory (DFT) to interpret XFEL data on IPD34,37 is particular challenging, since the ionized electron is thermalized. This is a questionable assumption in highly nonequilibrium systems and in handling the concept of atomic/ionic ionization: although the mean thermal energy of the ionized electron is subtracted from the finally calculated IPD,37 the bound-electron energy is already affected by the thermalized ionized electron in the calculation process. Thus, the self-consistent total energy with a nonthermalized ionized electron should be considered in the whole system instead of a simple subtraction of the thermal energy. However, determining the state of the nonthermalized ionized electron is an unsolved problem in the calculation of IPDs. For similar reasons (related to the highly nonequilibrium nature of the 10 fs XFEL interaction with solids), electrons are strongly heated while the lattice structure is still present, implying that the standard characterizations of the sample by, for example, coupling parameters are not entirely meaningful.36 In addition, the use of pseudopotentials and approximations for the exchange potential of a few-electron system with core-hole states is rather challenging in the framework of DFT models.44,47,48,50 In fact, the final results for the energy levels have been artificially shifted by34,37 +35–40 eV to match the experimental data for the IPD. The method of calibration shifts becomes unclear when isoelectronic sequences are under investigation: usually, a calibration shift is only applied for the well-known cold K_α transition, and then the same shift is applied for all other ionic transitions of the isoelectronic sequence generated by the codes. In addition, it has been observed that for the 1s²2s² configuration, the discrepancies are much larger (despite the calibration shifts) than for other transitions. This might also indicate difficulties in the DFT methods applied to obtain a correct description of the subshell structure where the Pauli principle is important.

An important requirement for advanced studies is therefore the availability of an IPD model in which each level depends on the various quantum numbers. But this is not the only one. Also, what is actually observed experimentally should likewise be carefully analyzed. In fact, what is experimentally observed via a scan of the XFEL energy irradiating materials is the appearance or disappearance of K_α X-ray transitions depending on the tunable XFEL energy.32,35 Only in the absence of M-shell electrons is there a straightforward correlation of the appearance energies (e.g., the minimum XFEL energy that induces a certain type of K_α X-ray transition) with IPD. A more profound data analysis shows that M-shell electrons are present, and thus excitation and ionization phenomena are mixed. Therefore, an alternative way to extract IPD data in highly nonequilibrium matter has been proposed,41 namely, via simulation of the two-dimensional spectral properties (with the observed intensity depending on the spectral distribution and XFEL energy) and the best overall match of the three-dimensional map (line intensities vs transition energies and XFEL energies).

To advance the study of this topic, one of the first questions to ask is what type of model framework is needed. Analysis shows that spectroscopic precision of the absolute energy values requires not only the use of multielectron atomic wave functions that are fully antisymmetrized (Pauli principle) in a self-consistent field (e.g., Hartree–Fock), but also that the exact nonlocal exchange term is calculated in the self-consistent field iteration.

It is the purpose of the present work to shed light on the controversy regarding IPD of atoms in dense plasmas by developing a generalized parameter-free model with spectroscopic precision. The dual characteristics of bound and continuum states as well as excitation–ionization phenomena are realized in a generalized manner and are valid for any type of configuration, including core-hole states. A particular highlight of the present work is the analysis of the not yet fully understood data from compound materials on the same footing as the generalized model.

For elements like sodium, magnesium, or aluminum, the M-shell electrons are free in the near-solid state, while the K- and L-shell electrons are strongly bound. In the metallic solid state, the valence and conduction bands are generally partially overlapping. In dense plasmas, the particle interactions induce energy level splitting, broadening, and shift, resulting in a quasi-continuous distribution of the energy levels in the valence band, which partially extends into the conduction band (see the resonance-like magenta line in Fig. 1, which schematically illustrates the energy level distribution of the valence band). This implies the possibility for one electron excited to the valence band to become bound or free. We therefore generalize to the arbitrary case, assuming that the valence band electrons in dense plasmas are a mixed state of bound and free electrons and designate this mixed state as VB-BFC (valence band bound–free continuum).

The model for the valence-band-like features with bound and free states is best illustrated by a detailed example. Figure 1 shows the case for solid-like aluminum with the bound configurations 1s², 2s², and 2p⁶ (Ne-like core). In a very dense plasma, the average atomic size is limited, and all bound and free states are located within the average size. In the simplest geometrical representation, the size is represented by an atomic radius (the so-called ion-sphere radius R):$$R={\left(\frac{3}{4\pi n_i}\right)}^{\frac{1}{3}},$$(1)where n_i is the ionic density. Thus, the average free-electron density in the ion sphere is given by$$n_e=\frac{N_f}{\frac{4}{3}\pi R^3}=N_f{n}_i,$$(2)where N_f is the number of free electrons in the ion sphere. This averaged size limitation is coupled to the quantum mechanical wave equations to account for the fact that beyond the average ion sphere radius, the potential vanishes (because the number of free electrons and the number of bound electrons are equal in the average ion sphere to the charge of the nucleus). Representing the wave equation in spherical coordinates, the atomic radial coordinate and the radius of the ion sphere correspond to each other while the two other dimensions φ and θ are explicitly solved in the framework of the Racah algebra and the Wigner–Eckert theorem,51 (more details are given in the Appendix). Therefore, all bound and free states are essentially located within the ion sphere.

Figure 1(a) shows the case of neutral aluminum Al I with 13 bound electrons in shells 1s, 2s, 2p, 3s, and 3p while Fig. 1(b) depicts the case of Al II with 12 bound electrons and one free electron. If the shell occupation is designated according to$${\left(n_1{l}_1\right)}^{w_1}{\left(n_2{l}_2\right)}^{w_2}\cdot \cdot \cdot {(n_q{l}_q)}^{w_q}{\left(F\right)}^{N_f},$$(3)where w_i is the number of equivalent electrons in subshell (n_il_i). $\left(F\right)$ designates formally the “free-electron subshell.” The sum over all subshells including the free electrons provides$$\sum _1^q{w}_i+N_f=N_b+N_f=Z_n,$$(4)where N_b is the number of bound electrons and Z_n the nuclear charge. The case of Fig. 1(a), is therefore given by the shell occupation ${\left(1s\right)}^2{\left(2s\right)}^2{\left(2p\right)}^6{\left(3s\right)}^2{\left(3p\right)}^1{\left(F\right)}^0$, while the VB-BFC configuration is given by ${\left(VB-BFC\right)}^3={\left(3s\right)}^2{\left(3p\right)}^1{\left(F\right)}^0$. For the case of Fig. 1(b), one encounters the shell occupation ${\left(1s\right)}^2{\left(2s\right)}^2{\left(2p\right)}^6{\left(3s\right)}^2{\left(3p\right)}^0{\left(F\right)}^1$ and the VB-BFC configuration ${\left(VB-BFC\right)}^3={\left(3s\right)}^2{\left(3p\right)}^0{\left(F\right)}^1$.

The VB-BFC model divides the atom/ion in a dense plasma (inside the ion sphere) into core and valence-band-like parts. All electrons in the core are bound, and electrons in the valence-band-like structure are a bound–free mixture.

We note that it is very difficult to give the detailed valence band structure for an ion embedded in a dense plasma, and there are no general methods available to calculate such structures. But the continuous valence band structure for an ion in a plasma is formed owing to the broadening, shifting, and splitting of the valence levels because of the interactions from the plasma environments (similar to the formation of the valence band in the solid state). In fact, only two features of the valence band are important for the present calculations. One is the continuous structure of the valence band and the other is the bound–free mixture of the valence band. These two properties lead to two very important conclusions. The first is that the observation of continuous absorption in experiments does not necessarily imply the occurrence of ionization. The other is that the orbitals in the continuous valence band show distinct bound properties, and so we can treat them as a bound state in the VB-BFC model.

Moreover, the great advantage of this simple valence-band-like structure of bound and free states lies in its great potential for a unique and versatile description. In particular, it can be generalized straightforwardly to any core-hole configuration. This is of great importance when intense XFEL radiation interacts with matter and is schematically demonstrated in Fig. 2, where a K-electron is transferred to the VB-BFC structure. Figure 2(a) shows Al I with 13 bound electrons, i.e., the K-electron is transferred to a bound state in the VB-BFC structure. This transfer process corresponds to the elementary atomic physics process of excitation. Figure 2(b) depicts the transfer of a K-electron to the free state in the VB-BFC structure. This corresponds to ionization.

Stringent tests of the generalized model with its VB-BFC structure require X-ray spectroscopic data from near-solid-density materials with different types of core holes (holes in the K- and L-shells). These data have recently become available with the emergence of XFELs.32–35 Below, we will provide detailed comparisons of the present generalized model with the data and provide new insights into the excitation–ionization mechanisms in dense near-solid-density plasmas.

Despite the simplicity of the VB-BFC structured ion-sphere model, it has the great advantage that there are no free parameters (just the electron temperature and material density enter into the calculations). It therefore has predictive power and provides a unique footing for the calculation of dense-plasma effects on the structure of ions for any type of configuration. Moreover, this generalized model permits one to treat the exact nonlocal exchange terms related to all bound electrons (see also the Appendix), which is of primary importance for reaching spectroscopic precision.

To demonstrate the high precision of the generalized model, let us consider the well-established K_α and K_β transitions and the K_edge transsition.52Table I provides a comparison of the reference data52 and the present model.

In the framework of Eq. (3) of the generalized model, the configurations involved correspond to (VB − BFC)⁴ = (3s)²(3p)²(F)⁰ for the K-edge of Al I and (VB − BFC)⁴ = (3s)²(3p)¹(F)¹ for the K_α and K_β transitions in Al II. As can be seen from Table I, excellent agreement between the reference data and the present model is obtained.

To demonstrate the high precision of the generalized model for the isolated free-ion case, let us consider the well-established transitions of the He-like resonance and intercombination lines of aluminum for the α and β transitions of different spin states, nakely, $W2=1s2p\left({}^{1}p\right)-1s^2({}^{1}So)$, $Y2=1s2p\left({}^{3}p\right)-1s^2({}^{1}So)$, $W3=1s3p\left({}^{1}p\right)-1s^2({}^{1}So)$, $Y3=1s3p\left({}^{3}p\right)-1s^2({}^{1}So)$ (note that W2 is sometimes designated as He_α and W3 as He_β). In the framework of Eq. (3), these transitions correspond to VB − BFC = (3s)⁰(3p)⁰(F)⁰ for the α transitions and to VB − BFC = (3s)⁰(3p)¹(F)⁰ for the β transitions.

As can be seen from Table II, the generalized model provides a very good spectroscopic precision (better than 0.1%) for the absolute and relative line positions, indicating that angular coupling, including spin as well as exchange terms, is very well calculated.

Let us now demonstrate the high precision for highly charged ions in dense plasmas, which are currently of great interest.54–56 X-ray line-shift measurements are particular challenging, since they are based on the difference in the IPD between upper and lower states. As this difference depends critically on the n and l quantum numbers, it provides stringent tests for the accuracy and spectroscopic precision of a model. We note that the widely applied Stewart–Pyatt17 and Ecker–Kröll18,19 models are insufficiently precise: in fact, all energy levels of one ionization state are affected in almost the same manner by the dense plasma perturbation, thereby implying that the energy shift of a line transition of an ion with charge Z approximately vanishes, i.e., $\mathrm{\Delta }{E}_{\mathit{IPD}}^Z\left(j\to i\right)=E_{\mathit{IPD}}^Z\left(j\right)-E_{\mathit{IPD}}^Z(i)\approx 0$, where $E_{\mathit{IPD}}^Z\left(j\right)$ and $E_{\mathit{IPD}}^Z\left(i\right)$ are the IPD values of the upper i and lower j levels, respectively. Therefore, these types of models are (a) insufficiently detailed and thus are challenged by high-resolution spectroscopic data and (b) do not allow advanced investigations such as the present excitation–ionization studies of IPD.

Finally, we apply the present generalized model to a dense hot chlorine plasma, where high precision line-shift measurements of the He_β transition in Cl¹⁵⁺ ions have recently been performed.46 According to Eq. (3), the measured X-ray line transitions correspond in the generalized model to a transition from the upper state (1s)¹(2s)⁰(2p)⁰(3s)⁰(3p)¹(F)¹⁵, i.e., (VB − BFC)¹⁶ = (3s)⁰(3p)¹(F)¹⁵ to the lower state (1s)²(2s)⁰(2p)⁰(3s)⁰(3p)⁰(F)¹⁵, i.e., (VB − BFC)¹⁵ = (3s)⁰(3p)⁰(F)¹⁵. Figure 3 shows the experimental measurements (solid and open black circles with error bars46) and the simulations from the present model (red crosses connected with dashed line). Excellent agreement throughout the whole density interval is observed. Figure 3 thus demonstrates that spectroscopic precision of the present generalized model is even achieved for ions in near-solid-density plasmas.

Let us now discuss the possibility of extracting IPD data from spectroscopic XFEL experiments where a high-intensity (up to 10¹⁷ W/cm²), short-pulse (80 fs) XFEL beam was brought to interaction with a solid-density aluminum target. The recorded K_α spectra in the XFEL experiment32–35 mainly include two parts: the resonance K_α radiation in which the XFEL photon energy ℏω_XFEL equals the K_α transition energy $\hslash {\omega }_{K_{\alpha }}$, i.e.,$$\begin{array}{ll}\hfill & K^2{L}^x{(VB-BFC)}^y+\hslash {\omega }_{\mathit{XFEL}}\to K^1{L}^{x+1}{(VB-BFC)}^y\hfill \\ \hfill & \to K^2{L}^x{(VB-BFC)}^y+\hslash {\omega }_{K_{\alpha }},\hfill \end{array}$$(5)and the nonresonance K_α radiation in which the XFEL photon energy ℏω_XFEL exceeds the K_α transition energy $\hslash {\omega }_{K_{\alpha }}$, i.e.,$$\begin{array}{ll}\hfill & K^2{L}^x{(VB-BFC)}^y+\hslash {\omega }_{\mathit{XFEL}}\to K^1{L}^x{(VB-BFC)}^{y+1}\hfill \\ \hfill & \to K^2{L}^{x-1}{(VB-BFC)}^{y+1}+\hslash {\omega }_{K_{\alpha }}.\hfill \end{array}$$(6)

The formation of the resonance K_α spectra including hollow ions (i.e., double core-hole vacancies) has been well explored,41,57,58 whereas the nonresonance K_α radiation is still a matter of active discussion. In fact, the well-known and accepted expressions for IPD given by the Stewart–Pyatt17 and Ecker–Kröll19 models fail to describe the data.36 Their failure has been reconfirmed in very recent XFEL experiments.59 We note that sometimes the term “modified Ecker–Kröll model” has been employed to designate the replacement of the complex function C(n_e, T_e, Z) simply by 1. This is equivalent to a suppression of the Ecker–Kröll lattice and polarization effects by a fitted constant. Therefore, no agreement with the original Ecker–Kröll model can be observed36 (as will be discussed in more detail below).

In the XFEL experiments, the photon energy is scanned over a certain interval, and the question as to what is the lowest XFEL photon energy required to induce the nonresonance K_α transitions is closely related to the phenomenon of IPD. In the nonresonant regime, the lowest photon energy ℏω_XFEL to induce the K_α transition would correspond to the ionization potential E_i if the M-shell were absent, i.e., E_i = ℏω_XFEL in the process K²L + ℏω_XFEL → K¹L + e_photon(E = 0) followed by $K^1{L}^x\to K^2{L}^{x-1}+\hslash {\omega }_{K_{\alpha }}$. From this relation and the 1s ionization potential for the free atom/ion $E_i^{\mathit{free}}$, the IPD ΔE_IPD can be readily extracted via $\mathrm{\Delta }{E}_{\mathit{IPD}}=E_i^{\mathit{free}}-\hslash {\omega }_{\mathit{XFEL}}$.

However, the real situation is much more complex. First, the experimental data for the resonance transitions show that K_β lines exist.33,34 Therefore, bound orbital properties for the M-shell exist. Consequently, exciting one 1s electron to a 3p orbital is likewise a possible channel to create a 1s core-hole state. The 1s -to-3s and 1s -to- 3d transitions are so-called optical forbidden channels. However, in dense plasmas, these are both possible channels owing to Stark mixing. Second, the average radius of the M-shell electrons decreases considerably with increasing degree of ionization, as demonstrated in Fig. 4. Even for the neon-like aluminum ion (Al³⁺), the radius of the 3p orbital is much closer to the nucleus than that of the 3p orbital of neutral Al. Therefore, the bound features of the M-shell of charged atoms are more pronounced and very different compared with neutral Al.

A rigorous analysis via a two-dimensional spectral distribution (intensity vs photon energy of the target and vs the XFEL photon energy) likewise supports that K_β transitions overall exist but are severely broadened for all charge states.41 This implies that deducing the IPD from the threshold energy ℏω_XFEL for the appearance of the K_α transition is much more challenging. In fact, the real situation corresponds to the valence-band-like structure VB-BFC depicted in Fig. 1. As indicated in Fig. 2, this structure likewise implies in the generalized model that bound and free states in the VB-BFC structure are related to excitation and ionization, respectively.

Let us apply the model to the K_α-transition appearance data with the Ne-like core 1s²2s²2p⁶ in a pure dense Al plasma (other isoelectronic sequences are studied in the supplementary material). The three additional electrons in the VB-BFC structure, i.e., (VB − BFC)³ result in the following detailed configurations for the simplest successive ionization scheme: (VB − BFC)³ = (3s)²(3p)¹(F)⁰, (3s)²(3p)⁰(F)¹, (3s)¹(3p)⁰(F)², (3s)⁰(3p)⁰(F)³. The nonresonance K_α transitions related to the Ne-like core therefore correspond to five different channels, including the 1s ionization process (the terms on the left side of the “+” represent the cores and the terms on the right side of the “+” are the VB-BFC configurations):

1. 1.$1s^{2}2s^{2}2p^6+{(3s)}^2{(3p)}^1{(F)}^0\to 1s^{1}2s^{2}2p^6+{(3s)}^2{(3p)}^2{(F)}^0\to 1s^{2}2s^{2}2p^5+{(3s)}^2{(3p)}^2{(F)}^0$;
2. 2.$1s^{2}2s^{2}2p^6+{(3s)}^2{(3p)}^0{(F)}^1\to 1s^{1}2s^{2}2p^6+{(3s)}^2{(3p)}^1{(F)}^1\to 1s^{2}2s^{2}2p^5+{(3s)}^2{(3p)}^1{(F)}^1$;
3. 3.$1s^{2}2s^{2}2p^6+{(3s)}^1{(3p)}^0{(F)}^2\to 1s^{1}2s^{2}2p^6+{(3s)}^1{(3p)}^1{(F)}^2\to 1s^{2}2s^{2}2p^5+{(3s)}^1{(3p)}^1{(F)}^2$;
4. 4.$1s^{2}2s^{2}2p^6+{\left(3s\right)}^0{\left(3p\right)}^0{\left(F\right)}^3\to 1s^{1}2s^{2}2p^6+{\left(3s\right)}^0{\left(3p\right)}^1{\left(F\right)}^3\to 1s^{2}2s^{2}2p^5+{\left(3s\right)}^0{\left(3p\right)}^1{\left(F\right)}^3$;
5. 5.$1s^{2}2s^{2}2p^6+{\left(3s\right)}^0{\left(3p\right)}^0{\left(F\right)}^3\to 1s^{1}2s^{2}2p^6+{\left(3s\right)}^0{\left(3p\right)}^0{\left(F\right)}^4\to 1s^{2}2s^{2}2p^5+{\left(3s\right)}^0{\left(3p\right)}^0{\left(F\right)}^4$.

The first four channels represent excitation while channel 5 corresponds to ionization (see Fig. 2). Therefore, channels 1–4 correspond to the lowest-energy channels for the production of a core hole in the 1s shell. Each of channels 1–4 includes two steps. The first step is to excite one 1s electron to a 3p state creating a 1s hole state, and the second step is to decay one 2p electron to generate a K_α photon. The energy needed in these first steps is the lowest energy needed for the XFEL photon to induce the nonresonance K_α transition, while the energy required in the first step of channel 5 (1s ionization) is considerably larger. The energy change in the second step is the emitted K_α photon energy of the target element (aluminum) that is measured in the experiment.32–35

Taking channel 2 for the ions with Ne-like core as an example, 12 bound electrons and 1 free electron are located inside the ion sphere. Setting R, n_e, and T_e according to Eqs. (1) and (2), the self-consistent field theory is then used to calculate all the atomic parameters related to the configurations 1s²2s²2p⁶ + (3s)²(3p)⁰(F)¹, 1s¹2s²2p⁶ + (3s)²(3p)¹(F)¹, and 1s²2s²2p⁵ + (3s)²(3p)¹(F)¹. Finally, the excitation energy from 1s to 3p and the K_α transition energies are obtained for different plasma parameters (density and temperature). The same method of calculation is then applied for all the other channels.

By setting the plasma electronic temperature to T_e = 100 eV (the measured T_e is about 30–150 eV) and the plasma ion number density to n_i = 5.97 × 10²² cm⁻³ (corresponding to the solid density of aluminum ρ = 2.7 g/cm³, since ions are essentially immobile on a time scale of some 10 fs), the generalized model is used to calculate the energy change for each channel. The free-electron density n_e for each channel is calculated from the free-electron number N_f and plasma ion density n_i as given in Eq. (2). The energy change involving a 1s − 3p transition (the first step in each channel) is called the excitation energy and is the lowest energy needed for an XFEL photon to generate nonresonance K_α radiation. The energy change involving a 2p − 1s transition corresponds to a K_α photon (the second step in each channel). Although in an experiment, the energy of the target K_α emission is rather precisely measured, neither the energy corresponding to the 1s − 3p excitation nor the ionization energy of the 1s core electron is directly measured: only the threshold of the XFEL photon energy for the appearance of K_α radiation is measured. We therefore study the lowest-energy channels via coupling of the generalized model with the detailed multiconfiguration Dirac–Fock wave function equations, taking into account intermediate coupling and configuration interaction (see the Appendix). This creates all possible LSJ-split energy levels and the corresponding transition energies, as well as absorption oscillator strengths. Figures 5 and 6 present the results of simulations for channels 1–5 of the Ne-like ion core.

Figures 5 and 6 demonstrate that all energies for the 1s − 3p excitation from channels 1–4 fall into the experimentally measured region of the so-called K-edge energy35 and that all the corresponding K_α transition energies fall likewise into the measured interval. But for channel 5, which shows the 1s electron ionization, the ionization energy needed in the first step is about 102 eV (the difference between the red dashed line and the gray area) greater than the experimentally measured K-edge energy, which is even greater than the 1s ionization energy for the Ne-like Al in vacuum (dashed black line). The reason for this surprising result is that in the model, the ionized electron is included in the free-electron pool to exert shielding on the ionized F-like ion, and so four free electrons appear in the second step of channel 5. This results in greater plasma screening of the F-like core and increases the energy gap between the configurations 1s²2s²2p⁶ and 1s¹2s²2p⁶. In fact, we adopt two approaches to include the ionized 1s electron in the plasma screening inside the ion sphere after ionization (channel 5). The first is to consider the ionized 1s electron as being thermalized, and the second is to solve the wave function of the ionized 1s electron with zero energy in the Schrödinger equation. The difference between these two approaches is only 0.5 eV for the 1s ionization energy. All efforts to model the data using 1s ionization have failed. Instead, the excitation energies for all excitation channels from 1s to 3p agree well with the experimentally measured K-edge energy.

We note that in Ref. 34, the K–M excitation energy in isolated neutral atoms (which is the first channel in the present VB-BFC model: see channel 1 for the case of an Ne-like core) was considered. This neutral-atom model assumes that all the excited electrons (both bound and free) from a relevant core (a K–L core in this case) are placed in the lowest-energy state in the M-shell. For example, in a pure aluminum plasma, the excitation energy for the F-like core (1s²2s²2p⁵)(3s²3p²) → (1s¹2s²2p⁵)(3s²3p³) is close to the so-called experimental K-edge energy for F-like Al, and the same holds true for other core charge states.34 However, the charge state with the free-electron configuration (plasma screening) cannot be included in this neutral-atom model. In fact, in Ref. 34, the excitation energy of K²LM⁰ − K¹LM¹ for different x values in vacuum were calculated. It was shown that the K–M excitation energies were all much greater than the experimental K-edge energies, and the differences increased with increasing degree of ionization (decreasing x).

By contrast, for the present VB-BFC model, all 1s − 3p (K–M) excitation energies for any core state with different VB-BFC structure are close to the experimental K-edge energy, as shown in Fig. 5. Taking the Ne-like core in a pure aluminum plasma as an example,34 the excitation energy for $\left(1s^{2}2s^{2}2p^6\right)-\left(1s^{1}2s^{2}2p^6\right)\left(3s^{0}3p^1\right)$ (optical dipole excitation) in vacuum is about 1573 eV, which is about 13 eV greater than the experimental K-edge energy. However, for the present VB-BFC model, the excitation energy for ${(1s)}^2{(2s)}^2{(2p)}^6{\left(F\right)}^3-{(1s)}^1{(2s)}^2{(2p)}^6{(3s)}^0{(3p)}^1{(F)}^3$ is about 1560 eV, which is in excellent agreement with the experimental K-edge energy of Ne-like Al. The results of calculations in the framework of the VB-BFC model clearly indicate that the experimentally measured K-edge energy is actually the excitation energy from 1s to 3p. Figure 7 compares the 1s − 3p excitation energies the channels 1–4 in solid-density plasma and in vacuum.

As demonstrated in Figs. 5–7, the VB-BFC structure has a significant impact on the 1s − 3p excitation energy, especially as the number of free electrons increases. Calculations in the framework of the free-atom model (which means that the quantum mechanical wave equations are solved for an atom/ion in vacuum) provide transition energies that are well outside the experimental interval (gray area), while the simulations in the framework of the present VB-BFC model are all located inside the experimental interval. Within the framework of the VB-BFC model, not only the neutral atom (channel 1) but also the ionic states (channels 2–4) of the 1s − 3p excitation energy fall into the experimental K-edge area. This implies that the experimentally measured lowest (threshold) XFEL photon energy to induce the K_α transition is just one possibility from many 1s − 3p excitation channels. Therefore, the VB-BFC structure provides a model of the interaction between core electrons and valence-band electrons for atoms/ions in dense plasmas.

Similar calculations have been performed for F-like, O-like, N-like, C-like, B-like, Be-like, and Li-like cores in a solid-density pure aluminum plasma (see the supplementary material). All the results strongly support the VB-BFC model and indicate that the lowest-energy channel to form the nonresonance K_α radiations is 1s − 3p excitation, and not 1s ionization.

Let us now investigate the differences between classical and quantum statistics, as well as different types of plasma potentials. For illustration, we consider the core excitation for near-solid-density heated material and take the B-like core of aluminum as an explicit example.

The top plot in Fig. 8 shows the 1s − 3p excitation energy and corresponding absorption oscillator strengths (red circles) for the transition channel ${(1s)}^2{(2s)}^2{(2p)}^1{\left(3s\right)}^2{\left(3p\right)}^3{\left(F\right)}^3-{(1s)}^1{(2s)}^2{(2p)}^1{(3s)}^2{(3p)}^4{(F)}^3$ for isolated ions where the 3 free electrons (F)₃ are not included (note that this transition channel corresponds to channel 4 of the B-like core; further details are presented in the supplementary material). As can be seen from the figure, the data calculated with the present generalized model lie essentially within the experimental uncertainty interval (albeit somewhat high-energy shifted), demonstrating good agreement between theory and experiment. The far-lying transitions of the theoretical calculations on the left- and right-hand sides of the gray area have very small probability and are not observed for the current signal/noise ratio of the experiment.

The second plot from the top in Fig. 8 shows the results (green circles) obtained for the uniform electron gas model (UEGM; see the Appendix for more details). The comparison with the top plot in Fig. 8 shows only slight improvements over the isolated case. The third and fourth plots in Fig. 8 present the results from the present generalized model for different electron temperatures kT_e = 100 and 10 eV, respectively. It can be seen that the results are greatly improved and that the theoretical data are centered around the gray area for temperatures of the order of some 10 eV. The visible reasonable agreement obtained already for the UEGM indicates that the results do not depend strongly on the particular type of the plasma potential and reinforce the conclusion that the 1s − 3p excitation channel is the primary element in the analysis.

The second interesting observation is that the results from the Maxwell–Boltzmann and Fermi–Dirac calculations differ only slightly from each other for the present experimental parameters of density and temperature. The reason is that for these temperatures, the impact of the Fermi–Dirac statistics (Pauli principle) is not large enough to induce considerable changes in the final transition energies. This will be different for temperatures much below 10 eV (e.g., temperatures much below the Fermi energy). It is therefore justified to employ the simpler Maxwell–Boltzmann statistics for the current analysis of XFEL data.

It is clear from basic atomic physics that a core hole can be produced either by ionization, by excitation, or by dielectronic capture. Core-hole production via dielectronic capture and inner-shell excitation phenomena are well known from dielectronic satellites (which are widely employed for plasma diagnostics): the upper states are autoionizing core-hole states populated by either dielectronic capture or inner-shell excitation. The upper state decays via spontaneous radiative decay (satellite emission) or by autoionization.60 Consequently, these channels of core-hole production have also been mentioned in the framework of XFEL interaction with matter.29,38

We emphasize an important point for the analysis: do dense plasma atomic structure models predicts bound M-shell states under real experimental parameter conditions? Only a bound M-shell will allow creation of core-hole states via resonance excitation; otherwise, only the ionization channel exists. Or, the other way around: if the model does not predict M-shell bound states for given experimental temperatures and densities, the resonance excitation channels K–M do not exist. For the experimental conditions of temperature and density (as indicated in Fig. 9), the present model predicts bound M-shell 3s and 3p states for the whole series of L-shell occupations (see Fig. 9), whereas other models fail to predict resonance excitation, since the M-shell states lie in the continuum. For example, the two-step HFS model38 predicts M-shell bound states only for temperatures greater than about 80 eV. Therefore, below this temperature (i.e., in the initial phase of XFEL interaction with matter) no 1s − 3p resonance excitation can be concluded. We note that the two-step HFS model has recently been refined;50 however, the principal points for the present discussion are essentially not changed. In addition, it is questionable to apply a periodic crystal structure to dense hot plasmas, since the charge states in each ion sphere are different. Likewise, in Ref. 61, bound 3s states were predicted at, for example, 2, 5, and 10 eV, but no 3p states were designated as bound states, and in Ref. 62, no bound states at all were predicted for 5 eV.

For all experimental cases in Fig. 9 (solid orange squares), the present generalized model predicts M-shell states that are bound (solid black circles). Therefore, effective IPD can be deduced also from resonance processes and compared with the experimental data (solid orange squares) as visualized in Fig. 9. The long-dashed red line “EK-fit C = 1” is a fit to the data as proposed in Ref. 34: the original function C(Z, n_e, T_e) related to lattice and polarization terms was replaced by a simple constant C = 1 (thereby removing the essence of the Ecker–Kröll model as outlined in Refs. 36 and 63).

The short-dashed red line is the result from the original Ecker–Kröll model (designated as “original EK” in Fig. 9)18,19 for the respective plasma parameters, while the short-dashed blue curve is from the Stewart–Pyatt model17 (designated as “SP” in Fig. 9) as proposed in Ref. 34. As can be seen from Fig. 9, neither the EK model nor the SP model fit the data. Moreover, these models also fail to explain the scaling along the isoelectronic sequences (charge state parameter in Fig. 9). On the other hand, the present generalized model (solid black circles) fits all experimental data (solid orange squares) and also explains well the scaling along Z. To study the temperature sensitivity of our model, we have also inserted in Fig. 9 our present calculations for fixed temperature (open blue circles). It can be seen that the differences are not very strong.

Surprisingly, the curve “EK-fit C = 1” not only matches the data, but also provides the correct scaling along charge states. However, this is an accidental coincidence related to the replacement of lattice- and polarization-dependent functions by a constant $C\left(n_e,T_e,Z\right)=1$: the fitting constant C = 1 provides an absolute shift to fit the data, while the suppression of the functional parameter dependences of $C\left(n_e,T_e,Z\right)$ provides a scaling roughly proportional to Z^4/3. Within the framework of Hartree–Fock calculations, it can be shown that the Z-scaling of the K-edge energies is roughly $\sim {\left(1+Z\right)}^{4/3}$ and therefore accidentally coincides with the EK model, where functional dependences of lattice and polarization effects are suppressed and replaced by $C\left(n_e,T_e,Z\right)=1$. As will be shown below, the curves “EK-fit C = 1” might be able to fit very specific data for solid aluminum and magnesium,34 but fail, for example, to describe compound materials. By contrast, the present generalized model is also in good agreement with the data without any additional assumptions.

To check the VB-BFC model for compound materials (or mixture), the K-edge energy for different charge states of aluminum are calculated for Al₂O₃. The results for the Ne-like core are given in Fig. 10.

Figure 10 demonstrates that the experimental K-edge energies34 for the aluminum in compound materials also agree well with the 1s − 3p excitation energy in the VB-BFC model. Smaller redshifts are found in the Al₂O₃ plasma, because the plasma density is higher. To illustrate the results of the VB-BFC model more comprehensively, we present in Fig. 11 the theoretical IPD from Li-like to Ne-like Al for an Al₂O₃ plasma, together with experimental data.

As there are quite a large number of 1s − 3p transitions falling into the experimental K-edge area, the 1s − 3p excitation energy selected to calculate the IPD is obtained by taking the energy of the excitation with the highest photon absorption oscillator strength in each channel and then averaging over the various channels.

The results for the IPD for all charge states are in excellent agreement with the experimental measurements. These results strongly support the VB-BFC model and also confirm that the K-edge energy measured in the experiment is only the energy of the 1s − 3p transition, rather than the energy of 1s ionization.

It can be seen from Fig. 11 that the theoretical IPD for ionic Al in Al₂O₃ is greater than that in pure Al (Fig. 9). This is because the density of solid Al₂O₃ is greater than that of solid pure Al (which correspondingly results in a larger redshift). Also depicted in Fig. 11 are the results of the “EK-fit C = 1” curves as used in Ref. 34. It can clearly be seen that this fit fails to describe data from compound materials and therefore reveals that the “EK-fit C = 1” approach has no general meaning. On the other hand, the present generalized model provides on the same footing and without any additional assumptions a good match to the data of the compound materials too.

We note that different models to calculate the ion-sphere radii for compound plasmas have been discussed in the literature (see, e.g., Refs. 64 and 65). The corresponding variations lead to small corrections to the present analysis. We do not invoke these corrections in the discussion, because the purpose of the application of the generalized model proposed here is to demonstrate that even without any additional assumptions, it can reasonably be used to describe compound materials, even when previously developed models34 fail.

Both density and temperature have an effect on the 1s − 3p excitation energy, but the density turns out to be dominant. As shown in Figs. 12–15, the 1s − 3p excitation energy decreases as the plasma density increases and as the plasma temperature decreases. In the plasma parameter ranges in Figs. 12–15, the average 1s − 3p excitation energy changes from 1567 to 1597 eV, which corresponds to a change in the IPD from 88 to 58 eV (see the red bar-line in Fig. 11). The experimentally measured IPD for an F-like core is about 72 ± 5 eV, which falls into the theoretical range for the plasma parameters of Figs. 12–15.

We note that in the present work, we have considered in great detail the cases of Al and Al₂O₃ plasmas with respect to IPD and X-ray spectroscopic signatures using the VB-BFC model. This has allowed us to examine the cases of pure and compound materials and to present a clear analysis of the available spectroscopic observations. However, we have actually also carried out IPD calculations for different ionization states in Fe, Mg, Si, and SiO₂ plasmas. The preliminary results not only show good agreement with the experimental IPD values, but also confirm the good generalizability of the proposed model.

Finally, we draw attention to the fact that studies of low-Z materials at very high densities are important for astrophysics and inertial confinement fusion. Recently, very significant experimental results have been obtained at the National Ignition Facility,66 subsequent to earlier DFT calculations.67 The study of low-Z elements where the ground states of neutral isolated atoms fall within the L shell (e.g., carbon) provides a challenging situation to test the VB-BFC model with an L-shell valance band in very dense plasmas.

Plasma pressure ionization leads to a transition of valence electrons from a bound to a free state, while in the transition region the states must be a mixture of bound and free states. When the plasma density is low, the features of the bound state dominate. As the plasma density increases, orbital splitting, level broadening, and shifts become important, and the valence band energy distribution strongly overlaps with the conduction band. In this case, the features of the free state dominate.

The prediction of K-edge energies and center positions of X-ray transitions in near-solid-density finite-temperature plasmas with high precision through general and parameter-free models is of great importance both for fundamental studies and for the application of atomic physics to matter under extreme conditions. A generalized ion-sphere model has been developed and tested against a number of high-precision measurements. In the model, a valence-band-like structure that is either bound, free, or mixed (the so-called VB-BFC = valence band bound–free continuum) is introduced and coupled self-consistently with the Dirac-wave equations of multielectron bound states. This makes it possible to fully respect the Pauli principle, to take into account the exact nonlocal exchange terms, and to achieve a high precision. The high spectroscopic precision has been demonstrated with numerous data that include different spin states as well as X-ray line-shift measurements in dense plasmas.

Recent XFEL measurements on the IPD and the observed scaling of isoelectronic sequences have been reanalyzed. Overall excellent agreement between the proposed model and the data has been demonstrated where standard methods based on the 1s − core ionization have failed.

The analysis has also provided new insights in the interpretation of the experimental data. According to the VB-BFC model, the 1s hole state is formed by one 1s electron excited to the 3p orbital rather than by a 1s-electron ionization. The calculated excitation energy for the different channels of a given charge state are in good agreement with the experimentally measured lowest XFEL photon energy needed to generate K_α-line emission in the nonresonance case. Simultaneously, the proposed generalized model predicts theoretical K_α transitions that are also in good agreement with the experimental spectra. It has been demonstrated that the present model is much closer to the measurements than pure ionization models. In particular, the present model predicts bound M-shell states to allow resonance excitation, while other models fail to provide resonance excitation because bound M-shell electrons are not predicted (in particular, in the initial phase of near-solid-density plasma evolution). For the aluminum plasma environment studied in the XFEL experiment, the bound state characteristics in the M shell appear to be distinct and can be confirmed by the experimentally recorded resonance K_β lines (although seriously broadened).

All the results along the isoelectronic sequence show that the key point in the IPD calculations is the choice of the 1s − 3p transition channel requiring corresponding M-shell bound states. When the 1s − 3p transition channel is chosen, the IPD (within the current level of measurement errors) is not very sensitive to the particular type of screening potential or the calculational method.

A comparison between Maxwell–Boltzmann and Fermi–Dirac statistics demonstrates that under the real experimental parameter conditions, the differences are rather small and do not change the main conclusions.

Finally, it has been demonstrated that on the same footing as for pure materials, the generalized model leads to a sound interpretation of IPD data of compound materials without additional assumptions, while other methods and models fail or lack consistency.

The supplementary material encompasses the generalized channel analysis for Li-like until F-like ions.

Acknowledgment. This work was supported by the NSFC under Grant Nos. 11374315 and 12074395 and the Invited Scientist Program of CNRS at Ecole Polytechnique, Palaiseau, France.