Module lattice_doppler 

Doppler broadening by nuclei bound in a harmonic crystal.

Lamb’s incoherent harmonic model, the one SAMMY’s crystal-lattice model (DOPUSH, after NJOY’s LEAPR) expands in phonons: a neutron of energy E absorbed by a nucleus of mass ratio A leaves an energy ε to the lattice, drawn from

P(ε) = (1/2π) ∫ e^{iεt} Φ(t) dt,   Φ(t) = exp(R·[G(t) − G(0)]),
G(t) = ∫ ρ(ω)/ω · [coth(ω/2kT)·cos ωt − i·sin ωt] dω,   R = E/(A + 1),

ρ the phonon density of states, so the cross section is

σ_L(E) = e^{−2W}·σ(E + R) + ∫ P_c(ε)·σ(E + R − ε) dε,   2W = R·G(0),

the first term the zero-phonon line and P_c = P − e^{−2W}·δ. P has mean R and variance 2R·kT_eff with kT_eff = ½∫ρ·ω·coth(ω/2kT) dω, so a free gas at T_eff has the same two moments.

σ is sampled once on a uniform grid of step h, the integral is the trapezoid rule on that grid, and P_c comes on the grid’s nodes from a fast Fourier transform of Φ − e^{−2W}, sampled at t = j·dt up to 100/ω_max, which resolves the phonon spectrum’s structure, whose weight e^{−2W}·2W the one-phonon term carries. A resonance of total width Γ makes the trapezoid rule’s error ≈ e^{−πΓ/h}, so h ≤ πΓ/30 for every resonance the kernel reaches, h ≤ π·ω_max/100, and h is a tenth of every File-3 knot interval within reach.

Structs

PhononSpectrum

A crystal’s phonon density of states ρ(ω), ω in eV: the values ρ_i at ω = i·step, ρ_0 = 0, quadratic in ω up to the first point, linear between the others and zero beyond the last, scaled to unit area.

Enums

LatticeError

Errors from lattice broadening.

Functions

broaden_in_lattice

Value and temperature derivative of channel at every target energy, broadened by nuclei bound in a harmonic crystal with phonon spectrum spectrum at temperature_k. negative_values counts the negative values; nothing is roundoff-limited.