Fig. 1. Schematic illustration of valence-band-like features with bound and free states in the framework of an ion-sphere model (R is the radius of the ion sphere) coupled to atomic wave equations: (a) Case of Al I with 13 bound electrons; (b) case of Al II with 12 bound electrons and one free electron.

Fig. 2. Schematic illustration of core-hole vacancies production in the framework of the VB-BFC valence band model. Case (a) depicts Al I with 13 bound electrons corresponding to the excitation of a K-electron, while case (b) shows Al II with 12 bound electrons and one free electron corresponding to the ionization of a K-electron.

Fig. 3. He_β line-shift measurements of Cl¹⁵⁺ ions in dense laser-produced plasmas (open and solid circles) and comparison with the present generalized model (red crosses) for T_e = 600 eV. Very good agreement between theory and experiment is observed, demonstrating the spectroscopic precision of the present generalized model, even in dense plasmas.

Fig. 4. Averaged radius of the 2p and 3p orbitals for different aluminum ionic states in solid-density Al plasma. The black dashed line is the average atomic size (or the radius of the ion sphere) calculated from the crystal constant of Al (0.404 nm, corresponding to 7.64a₀).

Fig. 5. Distribution of photon absorption oscillator strengths with respect to excitation energy. The excitation energy is the XFEL photon energy to induce a 1s − 3p transition in channels 1–4 for aluminum with a Ne-like ion core. The labels 1–4 correspond to the production of nonresonance K_α radiation via the lowest-energy channels 1–4 listed in the text at the start of Sec. II D. The gray areas indicate the experimental uncertainty in measuring the XFEL photon energy corresponding to the appearance of the K_α lines. The label 5 corresponds to channel 5 of 1s ionization. The black and red vertical dashed lines show the 1s ionization energy of Ne-like Al in vacuum and in solid-density plasma, respectively. The plasma conditions used in the calculations with the present generalized model are T_e = 100 eV, n_i = 5.97 × 10²² cm⁻³, and n_e = N_fn_i.

Fig. 6. Spontaneous transition probabilities of K_α transitions with respect to transition energies, where the labels 1–5 are the same as in Fig. 4. The gray areas indicate the experimentally recorded transition intervals where the K_α radiation intensity increases from zero to a nearly constant value.

Fig. 7. Comparison of 1s − 3p excitation energies and corresponding photon absorption oscillator strengths in the framework of the free-atom model (hollow circles) and the present model (solid circles). Simulations are performed for the Ne-like core in pure aluminum plasma at T_e = 100 eV and n_i = 5.97 × 10²² cm⁻³ (corresponding to a mass density of ρ = 2.7 g/cm³).

Fig. 8. Theoretical 1s − 3p excitation energies for the B-like core of aluminum using Maxwell–Boltzmann and Fermi–Dirac statistics for the free electrons. The plasma density is n_i = 5.97 × 10²² cm⁻³, which corresponds to solid-density aluminum. The plasma temperatures are 10 and 100 eV. The excitation channel considered is (1s)2(2s)2(2p)13s23p3F3−(1s)1(2s)2(2p)13s23p4F3. The gray area is the experimental uncertainty interval. The red circles are the calculations in vacuum, the green circles show the calculations in the framework of the uniform electron gas model (UEGM), while the light blue circles/pink crosses and dark blue circles/red crosses are calculations from the present generalized model (GM) for 100 and 10 eV, respectively. The circles and crosses are calculated assuming Maxwell–Boltzmann (MB) and Fermi–Dirac (FD) statistics, respectively, for the free electrons.

Fig. 9. Comparison of experimental data (solid orange squares) with calculations (solid black circles) on IPD with the results of the present VB-BFC model for solid-density pure aluminum at n_i = 5.97 × 10²² cm⁻³. The solid black circles are calculated for the real experimental temperatures T_e related to different aluminum ions:⁴¹ 8 eV for Al³⁺, 30 eV for Al⁴⁺, 42.5 eV for Al⁵⁺, 58 eV for Al⁶⁺, 72 eV for Al⁷⁺, 104 eV for Al⁸⁺, and 132 eV for Al⁹⁺. The long-dashed red line “EK-fit C = 1” is a fit to the data from Ref. 34, while the short-dashed red line is the result from the original Ecker–Kröll model^18,19 for the respective plasma parameters. The short-dashed blue curve is the result from the Stewart–Pyatt model¹⁷ as given in Ref. 34.

Fig. 10. The same as in Fig. 5, but for the 1s − 3p excitation energy of Ne-like Al in Al₂O₃ plasma and for the excitation channels 1–4 (open circles). The plasma conditions for Al₂O₃ are set as n_i = 1.2 × 10²³ cm⁻³ and T_e = 100 eV. The 1s − 3p excitation energies in a pure solid-density Al plasma (n_i = 5.97 × 10²² cm⁻³ and T_e = 100 eV) are also included for comparison (solid circles). All the theoretical results are calculated with the present VB-BFC model.

Fig. 11. Comparison of IPD for different charge states (core states) of aluminum between experiment and the present theory for the solid-density compound Al₂O₃ (n_i = 1.2 × 10²³ cm⁻³). The plasma temperature used in the calculations is T_e = 100 eV (solid green circles). The red-bar vertical line for the F-like core shows the theoretical range of IPD between (n_i = 1.2 × 10²³ cm⁻³, T_e = 30 eV) and (n_i = 1.2 × 10²² cm⁻³, T_e = 100 eV). Also indicated in the figure are the results from the Ecker–Kröll model, where the lattice function Cne,Te,Z has been replaced by a simple fitting constant³⁴ (curves designated with “EK-fit C = 1”).

Fig. 12. The same as in Fig. 5, except that the 1s − 3p excitation channels with F-like core are considered in an Al₂O₃ plasma. Compound Al₂O₃: T_e = 100 eV, n_i = 1.2 × 10²³ cm⁻³.

Fig. 13. The same as in Fig. 5, except that the 1s − 3p excitation channels with F-like core are considered in an Al₂O₃ plasma. Compound Al₂O₃: T_e = 30 eV, n_i = 1.2 × 10²³ cm⁻³.

Fig. 14. The same as in Fig. 5 except that the 1s − 3p excitation channels with F-like core are considered in an Al₂O₃ plasma. Compound Al₂O₃: T_e = 100 eV, n_i = 1.2 × 10²² cm⁻³.

Fig. 15. The same as in Fig. 5, except that the 1s − 3p excitation channels with F-like core are considered in an Al₂O₃ plasma. Compound Al₂O₃: T_e = 30 eV, n_i = 1.2 × 10²² cm⁻³.