Module sum_rules 

Sum rules of an optical energy-loss function.

Two checks of the internal consistency of an optical ELF, from Shinotsuka, Da, Tanuma, Yoshikawa, Powell, Penn, Surf. Interface Anal. 49, 238 (2017), doi:10.1002/sia.6123 (open copy: PMC5524379), in Hartree atomic units there and in eV here:

• f-sum (eq. (18)): the effective number of electrons per target unit (atom or molecule) taking part in losses up to W, N_eff(W) = (2 / (π (ħΩ_a)²)) ∫_0^W W' ELF(W') dW', with (ħΩ_a)² = ħ² n_a e² / (ε₀ m_e) for the target-unit density n_a. Without a target density, the report gives the effective electron density n_eff(W) = (ε₀ m_e / ħ²) (2/π) ∫_0^W W' ELF dW' instead (N_eff = n_eff / n_a).
• Perfect-screening (Kramers-Kronig) sum (eq. (19)): P_eff = (2/π) ∫ ELF(W)/W dW. Eq. (19) adds n(0)⁻², the inverse square of the static refractive index, which is zero for a metal and is not known from the ELF alone; the report gives the integral only, and the caller adds n(0)⁻² for an insulator. For a complete ELF P_eff -> 1 (text below eq. (19)).

The report also gives the mean excitation energy I from the same ELF, ln I = ∫ W ELF ln W dW / ∫ W ELF dW (the definition the Bethe limit uses; see super::penn).

Integration

The integrals run over the tabulated range only (nothing is extrapolated; OpticalElf interpolates linearly and the ELF is taken as zero outside the table). They are exact for the piecewise-linear interpolant: the f-sum by Simpson’s rule (exact for the quadratic W ELF(W) on each segment), P_eff in closed form (∫ (α + βW)/W dW = α ln(W₁/W₀) + β (W₁ - W₀)), and the ln W-weighted integral by an 8-point Gauss-Legendre rule per segment (the integrand is smooth on each segment; relative error below 1e-12).

Structs

SumRuleReport

The sum rules of one OpticalElf. See the module docs for the definitions. Built by SumRuleReport::new; printed by its Display implementation.