Module mermin 

The Mermin dielectric function of a free-electron gas, and the Mermin-ELF inelastic model built from a fit of oscillators to an optical ELF.

Sources and what was and was not read

Mermin’s relaxation-time dielectric function is N. D. Mermin, “Lindhard dielectric function in the relaxation-time approximation”, Phys. Rev. B 1, 2362 (1970), doi:10.1103/PhysRevB.1.2362. That paper is closed access and was not opened, so its own equation numbers are not known here. The formula is taken from the account in 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”, read 2026-10-07), section 2.1.1, whose eq. (4) cites Mermin (1970) as its reference 33:

                    (1 + iγ/ω) [ε_L(q, ω + iγ) - 1]
ε_M(q, ω) = 1 + ---------------------------------------------   (dV2022 eq. (4))
                 1 + (iγ/ω) [ε_L(q, ω + iγ) - 1] / [ε_L(q, 0) - 1]

(energies ħω, ħγ; ε_L the Lindhard function, super::lindhard_gas). The Mermin-ELF (MELF) of the MELF-GOS method is (dV2022 eq. (3), without the threshold step Θ(E - E_th))

Im[-1/ε(q, ω)] = Σ_j (A_j/(ħω_j)²) Im[-1/ε_M(q, ω; ω_j, γ_j)],

where ε_M(q, ω; ω_j, γ_j) uses the Lindhard function of a gas of plasma frequency ω_j, and A_j, ω_j, γ_j are fitted to the optical ELF by dV2022 eq. (5) (see super::mermin_fit); at q = 0 the Mermin-type ELF is the Drude-Lorentz one (super::drude).

Checks made here instead of a comparison with Mermin’s own paper (all in the unit tests):

• γ -> 0 gives the Lindhard function to 1e-8 (the analytic continuation of the Lindhard function to complex frequency is checked against the real-frequency form of super::lindhard_gas);
• q -> 0 gives the Drude function 1 - ω_p²/(ω² + iγω);
• ω -> 0 gives the static Lindhard function ε_L(q, 0) (the local-equilibrium property that motivates the formula);
• the f-sum rule ∫ ω Im[-1/ε_M] dω = (π/2) ω_p² at finite q. We did not find it stated in the sources read, so it is a numerical check of the transcription, not an input.

The independent algebraic form of Latyshev and Yushkanov, arXiv:1212.6260 (eq. (5.5″), read as described in super::lindhard_gas), is the same function with x + iy for ω + iγ.

Complex-frequency Lindhard function

ε_L(q, ω) = 1 + (k_TF²/q²) [1/2 + (A(z - u) + A(z + u))/(8z)] with A(s) = (1 - s²) Log((s + 1)/(s - 1)) (principal logarithm of the ratio, analytic off [-1, 1] and vanishing at infinity) and u = (ω + iγ)/(q k_F). For γ -> 0+ this gives the real-frequency Lindhard function with the electron-hole continuum as its imaginary part. When |z ± u| > 4 the bracket is evaluated from the same series as the real-frequency code (every term of which is O(1/|s|^m)), avoiding cancellation.

The model

MerminPenn uses the MELF loss function in the DIIMFP of the same nonrelativistic kinematics as the Penn models (super::penn, S2017 eqs. (2)-(3)):

p(T, ω) = (1/(π T')) ∫_{q-}^{q+} (dq/q) Im[-1/ε(q, ω)],

and integrates it over ω (in ln ω, from 1e-8 T to T) for the inverse IMFP and the stopping power. This is our own evaluation of a published construction; the omission of the threshold step is a choice documented in docs/data-provenance.md (a fitted Drude tail extends below any real band gap).

Comparison with the single-pole model (synthetic Drude fixture)

For the synthetic Drude plasmon of the tests (E_p = 20 eV, γ = 5 eV, 240 tabulated energies; the fit returns it to 1e-13), ratios Mermin / single-pole of the stopping power S and of the inverse IMFP 1/λ (default tolerances, release build; outputs of this code, not reference values):

   E / eV     S ratio   1/λ ratio
     30.0      1.9222     2.0818
    100.0      0.9832     1.0235
   1000.0      1.0002     1.0173
  10000.0      0.9991     1.0121
  50000.0      0.9988     1.0100

The stopping powers agree to 2e-3 from 1 keV (both obey the f-sum); below about 100 eV the models differ strongly, as the full Penn model does from the single pole (see super::full_penn). Nothing was tuned.

Structs

MerminGas

The Mermin dielectric function of a free-electron gas with a relaxation energy ħγ (see the module docs). Atomic units (Hartree, bohr).

MerminPenn

The Mermin-ELF inelastic model: a fit of Mermin oscillators to one optical ELF, extended to finite momentum by the Mermin function, and the DIIMFP, IMFP and stopping power built from it. See the module docs.

Constants

DEFAULT_MERMIN_TOLERANCE

Default relative tolerance of the integrals of the Mermin model.