Module lindhard_gas 

The Lindhard dielectric function of a free-electron gas at zero temperature (random-phase approximation), in Hartree atomic units (m_e = e = ħ = 1), and the loss-function integrals built from it.

This is the building block of the full Penn algorithm (super::full_penn): Penn expands the optical energy-loss function over the loss functions of free-electron gases of all plasma frequencies (Shinotsuka et al., Surf. Interface Anal. 49, 238 (2017), doi:10.1002/sia.6123, eq. (4), with ε_L “the dielectric function from the Lindhard model of a free electron gas”, citing J. Lindhard, Mat. Fys. Medd. Dan. Vid. Selsk. 28, no. 8 (1954)).

Sources and what was and was not read

Lindhard (1954) was not opened (no open copy was found). The function is implemented in its standard closed form, written here in the variables z = q/(2 k_F) and u = ω/(q k_F):

Re ε(q, ω) = 1 + (k_TF²/q²) [ 1/2 + (A(z - u) + A(z + u)) / (8 z) ],
A(s) = (1 - s²) ln|(s + 1)/(s - 1)|,      k_TF² = 4 k_F/π,
Im ε(q, ω) = A_c(q, ω) / q³,
A_c = (k_F² - a²)₊ - (k_F² - (a + q)²)₊,    a = ω/q - q/2,    ω > 0.

These were compared with an opened source: A. V. Latyshev and A. A. Yushkanov, arXiv:1212.6260v1 (open-access preprint, PDF pages read 2026-10-07), section 5 “Degenerate plasma”: the integral B(k, z) on p. 14 (∝ ∫_{-1}^{1} (1 - P_x²) dP_x / ((P_x - z/k)² - (k/2)²), with k = q/k_F, z = (ω + iν)/(k_F v_F)), its closed form b(k, z) on pp. 14-15, and the dielectric function eq. (5.5″) on p. 15, ε = 1 - (3 x_p²/4k²)(x + iy) b(k,z) b(k,0) / (x b(k,0) + i y b(k,z)) with x_p = ω_p/(k_F v_F). Its abstract states that for collision frequency ν -> 0 (y -> 0) the result is Lindhard’s formula, which is then ε = 1 - (3 x_p²/4k²) b(k, x). Compared numerically (atomic units, v_F = k_F, ω_p² = 4 k_F³/(3π)) at six (k_F, q, ω) points:

• the real part above equals the closed form b(k, z) of p. 15 to 1e-10 (the -2 and the two ln terms; their ln arguments are the ones of A(z ∓ u) combined);
• the imaginary part equals the residue (+i0) contribution of the p. 14 integral at its two poles P_x = z/k ± k/2 when they lie in [-1, 1], i.e. our A_c/q³, to 1e-15 (the printed closed form b gives only the real part under principal-branch logarithms, so the imaginary part is compared through the integral, not the closed form).

This is a comparison with a secondary source that derives the formula, not with Lindhard’s paper; the original equation numbers are not known here. The further checks are:

• the imaginary part is derived here directly from the Fermi golden rule (Im ε = (4π/q²) π · 2 ∫ d³k/(2π)³ f_k (1 - f_{k+q}) δ(ω - ΔE), an elementary Fermi-sphere area), and the unit tests check the real part against it by a Kramers-Kronig integral;
• the static limit ε(q, 0) -> 1 + k_TF²/q² for q -> 0 (Thomas-Fermi screening) and the exact static form 1 + (k_TF²/q²)(1/2 + (1 - z²)/(4z) ln|(1+z)/(1-z)|);
• the long-wavelength limit ε(q -> 0, ω) -> 1 - ω_p²/ω²;
• the f-sum rule ∫ ω Im[-1/ε] dω = (π/2) ω_p² at every q (plasmon delta function included), the property Penn’s algorithm relies on (it is what makes the expanded ELF satisfy the optical f-sum rule).

The plasma frequency and Fermi wavenumber are related by ω_p² = 4π n, n = k_F³/(3π²), i.e. k_F = (3π/4)^(1/3) ω_p^(2/3) (S2017 below eq. (8)).

The loss function

For ω > 0, Im[-1/ε] = Im ε / |ε|² inside the electron-hole continuum (|a| < k_F) and, above the continuum, a delta function at the plasmon energy ω_pl(q) (the root of Re ε = 0 above the continuum edge ω_+ = q k_F + q²/2) of weight π / (∂Re ε/∂ω). That weight is the one of S2017 eq. (7) at q -> 0: with Re ε = 1 - ω_p²/ω², ∂Re ε/∂ω = 2/ω_p at ω = ω_p and the weight is π ω_p/2.

Structs

LindhardGas

A free-electron gas of Fermi wavenumber k_F (bohr⁻¹), zero temperature.

LindhardPlasmon

The undamped plasmon of a Lindhard gas at one momentum.

Functions

fermi_wavenumber_au

k_F(ω_p) in atomic units.