Module shell_elf 

Inner-shell optical ELFs built from the hydrogenic K-shell (1s) and L-subshell (2s, 2p) photoionization formulas, for ShellResolvedChannels.

Sources (opened 2026-10-08; Karzas and Latter opened 2026-10-09)

• The shell ELF. P. de Vera, S. Taioli, P. E. Trevisanutto, M. Dapor, I. Abril, S. Simonucci and R. Garcia-Molina, Int. J. Mol. Sci. 23, 6121 (2022), doi:10.3390/ijms23116121 (open access, PMC9181504; “dV2022”, as in super::inner_shell), eq. (2): the inner-shell part of the ELF of a target with N molecules per unit volume, ν_j atoms of element j per molecule, is

  Im[-1/ε(k, E)]_inner = (2π² ħ² e² N / E) Σ_j ν_j Σ_nℓ df_nℓ^j(k, E)/dE Θ(E - B_nℓ^j),

  written in atomic units (the electron mass is set to 1). The prefactor with the mass restored is fixed by the f-sum rule of H. Shinotsuka et al., Surf. Interface Anal. 49, 238 (2017), doi:10.1002/sia.6123 (open copy PMC5524379), eq. (18), N_eff = (2 / (π (ħΩ_a)²)) ∫ W ELF dW with (ħΩ_a)² = ħ² n_a e² / (ε₀ m_e) (SI, the form super::sum_rules uses): requiring N_eff = ∫ df for one shell of atoms at density n_a gives, in the optical limit k -> 0,

  ELF_nℓ(W) = (π/2) (ħΩ_a)² (1/W) df_nℓ/dW,   W >= B_nℓ,   zero below,

  which is dV2022 eq. (2) for one shell ((π/2) 4π = 2π² with the Gaussian Ω² = 4π n e²/m). The shell’s oscillator strength is then exactly the shell’s N_eff (tested).

• The oscillator-strength density. F. Salvat, “Inelastic collisions of fast charged particles with atoms. Relativistic plane-wave Born approximation”, arXiv:2605.22442 (2026), eq. (6.2): the optical oscillator strength is proportional to the dipole photoabsorption cross section, df/dW = (m_e c / (2π² e² ħ)) σ_ph(W) (Gaussian e²); and eq. (6.1a): a subshell with q_a of its 2|κ_a| places filled contributes q_a / 2|κ_a| times the closed-subshell value, i.e. the oscillator strength scales with the occupancy.

• The hydrogenic K-shell cross section. M. Stobbe, Ann. Phys. (Leipzig) 7, 661 (1930) (not read), as printed by A. I. Mikhailov, A. V. Nefiodov and G. Plunien, Phys. Lett. A 368, 391 (2007), doi:10.1016/j.physleta.2007.04.027 (open copy arXiv:0704.2180), eq. (1): for one electron in the 1s state of a hydrogen-like ion of charge Z,

  σ_K = α a₀² (2⁹ π² / (3 Z²)) exp(-4ξ cot⁻¹ ξ) / ((1 + ξ⁻²)⁴ [1 - exp(-2πξ)]),

  with ξ = 1/sqrt(ε_γ - 1), ε_γ = W/I the photon energy over the ionization energy I = Z² E_h / 2, valid for 1 <= ε_γ << 2 (αZ)⁻² (dipole approximation, nonrelativistic). Its eq. (3) is the threshold value (ξ -> ∞, exp(-4)), its eq. (2) the high-energy limit α a₀² (2⁸ π / (3 Z²)) ε_γ^(-7/2), and its eq. (6) says that for two independent K electrons the cross section is 2 σ_K.

• The hydrogenic L-subshell cross sections. W. J. Karzas and R. Latter, “Electron Radiative Transitions in a Coulomb Field”, Astrophys. J. Suppl. 6, 167 (1961) (“KL1961”; open scan at https://articles.adsabs.harvard.edu/pdf/1961ApJS....6..167K, read from the page images, the text layer of the scan being unreliable in the equations). For a dipole transition from the bound state (n, ℓ) of a hydrogen-like ion to the continuum, averaged over the magnetic substates (eq. (27), one electron), the cross section is the sum, eq. (38), of the ℓ -> ℓ-1 part, eq. (36),

  σ_(nℓ -> E,ℓ-1) = (π e² / (m c ν)) (2^(4ℓ) / 3)
      ℓ² (n+ℓ)! {[1² + η²][2² + η²] ... [(ℓ-1)² + η²]} / ((2ℓ+1)! (2ℓ-1)! (n-ℓ-1)!)
      × (exp[-4η cot⁻¹ρ] / (1 - e^(-2πη))) (ρ^(2ℓ+2) / (1 + ρ²)^(2n-2))
      × [G_ℓ(ℓ+1-n; η; ρ) - (1 + ρ²)⁻² G_ℓ(ℓ-1-n; η; ρ)]²

  (the expression in braces is 1 for ℓ = 1), and the ℓ -> ℓ+1 part, eq. (37),

  σ_(nℓ -> E,ℓ+1) = (π e² / (m c ν)) (2^(4ℓ+6) / 3)
      (ℓ+1)² (n+ℓ)! (1² + η²)(2² + η²) ... [(ℓ+1)² + η²]
        / ((2ℓ+1) (2ℓ+1)! (2ℓ+2)! (n-ℓ-1)! [(ℓ+1)² + η²]²)
      × (exp[-4η cot⁻¹ρ] / (1 - e^(-2πη))) (ρ^(2ℓ+4) η² / (1 + ρ²)^(2n))
      × [(ℓ+1-n) G_(ℓ+1)(ℓ+1-n; η; ρ) + ((ℓ+1+n) / (1 + ρ²)) G_(ℓ+1)(ℓ-n; η; ρ)]²,

  with η = (Z² Ry / E)^(1/2) for a free electron of energy E (eq. (33)), ρ = η/n, and the real polynomials of its Appendix C, G_ℓ(-m; η; ρ) = Σ_(s=0)^(2m) b_s ρ^s (eq. (C.5)) with, eq. (C.8),

  b_0 = 1,   b_1 = 2mη/ℓ,
  b_s = -(1 / (s (s + 2ℓ - 1))) [4η (s-1-m) b_(s-1) + (2m+2-s)(2m+2ℓ+1-s) b_(s-2)].

  Kramers’ cross section is its eq. (39), σ^K = (2⁴ / (3√3)) (e² / (m c ν)) (1/n) (ρ² / (1 + ρ²))², and the Gaunt factor its eq. (40), g = σ / σ^K, tabulated in its Table 1 for n = 1 ... 6 and every ℓ against E / Z² Ry.

The formulas used

Inserting eq. (1) into eq. (6.2) with e² = α ħ c, a₀ = ħ / (m_e c α) and I = Z² E_h / 2, E_h = ħ² / (m_e a₀²) (our algebra; the test oscillator_strength_density_is_eq_6_2_of_the_stobbe_cross_section redoes it numerically with CODATA constants), the oscillator-strength density per K electron is

df_K/dW = (2⁷ / (3 I)) exp(-4 atan(t) / t) / ((1 + t²)⁴ [1 - exp(-2π/t)]),   t = 1/ξ = sqrt(W/I - 1),

equal to (2⁷ / (3 I)) e⁻⁴ at the edge. Integrated from the edge to infinity it gives about 0.435 per electron (computed here, by quadrature of the formula; the rest of the hydrogenic 1s oscillator strength goes to discrete excitations below the edge, which dV2022 eq. (2) leaves out with its step Θ(E - B)).

The L subshells. The photon energy is W = hν = E + I with I = Z² Ry / n², so W/I = 1 + 1/ρ² and ρ = 1/t with the same t = sqrt(W/I - 1). Salvat’s eq. (6.2) turns the prefactor π e² / (m c ν) of KL1961 eqs. (36) and (37) into 1/W: df/dW is the rest of the right-hand side over W. For n = 2 (η = 2ρ) the polynomials of eq. (C.8) are (our algebra)

G_1(0) = G_2(0) = 1,
G_1(-1; η; ρ) = 1 + 2ηρ - ρ²                              = 1 + 3ρ²,
G_1(-2; η; ρ) = 1 + 4ηρ + ((8η² - 10)/3) ρ² - 4ηρ³ + ρ⁴ = (1 + ρ²)(1 + (11/3) ρ²),
G_2(-1; η; ρ) = 1 + ηρ - ρ²                               = 1 + ρ²,

the bracket of eq. (37) is 2 (1 + 4ρ²) for 2s and 4 for 2p, and that of eq. (36) is -(8/3) ρ² / (1 + ρ²) for 2p. Per electron, with ε = W/I = 1 + t²,

df_2s/dW = (2¹⁰ / (3 I)) ((ε + 3) / ε⁵)    exp(-8 atan(t) / t) / [1 - exp(-4π/t)],
df_2p/dW = (2¹⁰ / (9 I)) ((3ε + 8) / ε⁶)   exp(-8 atan(t) / t) / [1 - exp(-4π/t)],

equal to (2¹² / (3 I)) e⁻⁸ and (11 · 2¹⁰ / (9 I)) e⁻⁸ at the edge. Of the 2p value, (2¹³ / (27 I)) (ε + 3) / ε⁶ times the same last factor is the transition to the d continuum, eq. (37), and (2¹⁰ / (27 I)) / ε⁵ times it the transition to the s continuum, eq. (36). The same reduction for n = 1 gives back the K-shell formula above, an independent check of the transcription. The unit tests of this module evaluate eqs. (36), (37) and (C.8) as printed and compare them with these closed forms; lindhard/tests/penn_shell_elf.rs compares the closed forms with the 2s and 2p Gaunt factors of KL1961 Table 1. Integrated from the edge to infinity they give about 0.351 (2s) and 0.191 (2p) per electron (computed here, by quadrature of the formulas).

Our choices, not from the sources:

• The ionization energy I of the formula is set to the shell’s binding energy B from the SubshellBindingTable (for example EADL2017), so the edge of the ELF is the tabulated edge. For a hydrogen-like ion this is the formula itself; for a many-electron atom it amounts to a hydrogenic shell with the effective charge Z_eff = sqrt(2B/E_h) that reproduces B. No screening rule is applied beyond that.
• The oscillator strength is the per-electron value times the subshell occupancy from the same table (Salvat eq. (6.1a), Mikhailov et al. eq. (6)). L1 is the 2s formula. L2 (2p1/2) and L3 (2p3/2) are both the nonrelativistic 2p formula, each with its own tabulated binding energy and occupancy, so the 2p oscillator strength is shared between them in the ratio of their occupancies.
• The ELF is sampled on a log-spaced grid from B to B times ShellElfGrid::max_energy_factor and interpolated linearly between the samples, like every OpticalElf. The formula is nonrelativistic; the caller keeps the top of the grid well below 2 B (αZ)⁻².

Scope

Only the K, L1, L2 and L3 subshells are built; any other subshell (M and above) is an error. KL1961 eqs. (36) and (37) hold for every (n, ℓ), but their reduction for n >= 3 has not been done or checked here. The valence remainder (total ELF minus the inner shells) is not built here.

Nothing here is random: the tables are bit-identical whatever the number of threads.

Structs

ShellElfGrid

The sampling grid of a built shell ELF.

Functions

hydrogenic_2p_oscillator_strength_density_per_ev

The hydrogenic 2p (L2, L3) optical oscillator-strength density per electron, df/dW in eV⁻¹, at energy loss energy_ev for a subshell of binding energy binding_energy_ev: Karzas and Latter, Astrophys. J. Suppl. 6, 167 (1961), eqs. (36), (37) and (38) for n = 2, ℓ = 1 (the s and d continua together), converted by Salvat, arXiv:2605.22442, eq. (6.2) (module docs, “The formulas used”). Zero below the edge; at the edge, its limit (11 · 2¹⁰/(9B)) e⁻⁸.

hydrogenic_2s_oscillator_strength_density_per_ev

The hydrogenic 2s (L1) optical oscillator-strength density per electron, df/dW in eV⁻¹, at energy loss energy_ev for a subshell of binding energy binding_energy_ev: Karzas and Latter, Astrophys. J. Suppl. 6, 167 (1961), eq. (37) for n = 2, ℓ = 0, converted by Salvat, arXiv:2605.22442, eq. (6.2) (module docs, “The formulas used”). Zero below the edge; at the edge, its limit (2¹²/(3B)) e⁻⁸.

hydrogenic_k_oscillator_strength_density_per_ev

The hydrogenic K-shell optical oscillator-strength density per electron, df/dW in eV⁻¹, at energy loss energy_ev for a shell of binding energy binding_energy_ev (module docs, “The formula used”). Zero below the edge; at the edge, its limit (2⁷/(3B)) e⁻⁴.

hydrogenic_shell_elf

The hydrogenic optical ELF of one inner shell (module docs): the shell subshell of element z, with binding energy and occupancy from table, for atoms_per_m3 atoms of that element per m³ in the target material. Returns the InnerShell and its OpticalElf, ready for super::ShellResolvedChannels::new: the ELF starts at the binding energy and is zero below it.

hydrogenic_shell_elfs

hydrogenic_shell_elf for several shells of a target material, each given as (Z, subshell, atoms of that element per m³) and kept in the given order. For a compound with N molecules per m³ and ν_j atoms of element j per molecule, the density of element j is ν_j N (dV2022 eq. (2)).