Module drude 

An analytic Drude-Lorentz energy-loss function with closed-form sum rules.

This is a synthetic model ELF for tests, examples and benchmarks: its parameters are chosen by the caller, not fitted to any material, and its sum rules are known exactly, so a numerical sum-rule evaluation can be checked against them.

Model

Each oscillator i has a strength A_i (eV²), a resonance energy E_i (eV) and a width γ_i (eV). The inverse dielectric function is

1/ε(W) = 1 + Σ_i A_i / (W² - E_i² + i γ_i W),

so the energy-loss function is

ELF(W) = Im[-1/ε(W)] = Σ_i A_i γ_i W / ((W² - E_i²)² + γ_i² W²),   W > 0.

One term with A = E_p² and E_i = E_p is the Mermin-type optical ELF of Shinotsuka et al., Surf. Interface Anal. 49, 238 (2017), doi:10.1002/sia.6123, eq. (16) (a_i = 1); there it describes a free-electron plasmon of energy E_p and width γ.

Closed-form sum rules

The two sum rules of the same paper, eqs. (18) and (19), are, for the full range 0 < W < ∞ and in energy units,

∫_0^∞ W ELF(W) dW         = (π/2) Σ_i A_i           (f-sum)
(2/π) ∫_0^∞ ELF(W) / W dW = Σ_i A_i / E_i²          (P_eff, without the n(0)⁻² term)

Both follow from the two elementary integrals (by residues: the integrands are even in W, and (W² - E²)² + γ² W² = |W² - E² + iγW|² vanishes at W = ±sqrt(E² - γ²/4) ± iγ/2)

∫_0^∞ γ W² / ((W² - E²)² + γ² W²) dW = π/2,
∫_0^∞ γ    / ((W² - E²)² + γ² W²) dW = π / (2 E²).

The derivation is ours; no external source for these two integrals was consulted. The unit tests check both against an independent numerical quadrature. With A = E_p² and E_i = E_p (a free-electron plasmon), the f-sum gives the plasma energy E_p back and P_eff = 1, the values the paper states for a complete ELF (P_eff -> 1 as ΔE_max -> ∞, text below eq. (19)).

Structs

DrudeLorentz

A sum of Drude-Lorentz oscillators: an analytic, synthetic ELF with closed-form sum rules (see the module docs).

DrudeLorentzOscillator

One Drude-Lorentz oscillator (see the module docs).