Module phonon 

Insulator energy-loss channels for low-energy electrons: Fröhlich longitudinal-optical (LO) phonon scattering (emission and absorption) and polaron trapping.

Both are extra channels of the event loop in crate::electron::transport, each with its own inverse mean free path next to the elastic and inelastic tables. They are opt-in per layer (InsulatorChannels) and off by default: they describe an electron in a polar insulator, and a layer that is a metal (or any layer the caller does not opt in) has neither. The choice is recorded in the run metadata.

Sources

The model is Fröhlich’s electron–LO-phonon coupling (H. Fröhlich, “Electrons in lattice fields”, Adv. Phys. 3, 325 (1954), doi:10.1080/00018735400101213) as applied to low-energy electron transport by J. Llacer and E. L. Garwin, J. Appl. Phys. 40, 2766 and 2776 (1969) (doi:10.1063/1.1658076 for part II), and the polaron-trapping rate of J. P. Ganachaud and A. Mokrani, Surf. Sci. 334, 329 (1995), doi:10.1016/0039-6028(95)00474-2. None of these three papers was open to us (no open-access copy; the publisher pages answer a bot challenge, see docs/data-provenance.md). The equations implemented here are taken from two open-access papers that restate them, and the two agree:

• Ding et al. (2021): Z. J. Ding, C. Li, B. Da, J. Liu, “Charging effect induced by electron beam irradiation: a review”, Sci. Technol. Adv. Mater. 22, 932 (2021), doi:10.1080/14686996.2021.1976597 (CC BY 4.0), Section 2.3: Eq. (14) (LO-phonon creation frequency), Eq. (15) (annihilation frequency), the Bose occupation in the text after Eq. (14), and Eq. (16) (angular distribution). The review credits these to H.-J. Fitting and J.-U. Friemann, Phys. Status Solidi A 69, 349 (1982), building on Llacer and Garwin.
• Taioli & Dapor (2025): S. Taioli and M. Dapor, “Advancements in Secondary and Backscattered Electron Energy Spectra and Yields Analysis: from Theory to Applications”, arXiv:2404.07521v5 (2025): Eq. (33) (phonon-emission inverse mean free path), Eq. (34) (Bose occupation) and Eq. (37) (polaron inverse mean free path C exp(-γE)), both credited there to Ganachaud and Mokrani (1995).

Fröhlich LO-phonon scattering

With x = ħω/E, n the Bose occupation of the mode at temperature T (Ding et al. (2021) text after Eq. (14); Taioli & Dapor Eq. (34)),

n(T) = 1 / (exp(ħω / k_B T) - 1)

the inverse mean free paths for emitting (+, needs E > ħω) and absorbing (-) one phonon of energy ħω are

λ₊⁻¹(E) = (n + 1) / a₀ · (1/ε∞ - 1/ε₀) · x/2 · ln[(1 + √(1 - x)) / (1 - √(1 - x))]
λ₋⁻¹(E) =  n      / a₀ · (1/ε∞ - 1/ε₀) · x/2 · ln[(√(1 + x) + 1) / (√(1 + x) - 1)]

λ₊⁻¹ is Taioli & Dapor Eq. (33) (where the coupling is written (ε₀ - ε∞)/(ε₀ ε∞), the same quantity). Both are also Ding et al. (2021) Eqs. (14) and (15), which give the rates per unit time f±; dividing by the speed v = √(2E/m*) gives the forms above with a₀ = 4πε₀ħ²/(m e²) when the effective mass m* is the free-electron mass, which is the choice made here (as in Taioli & Dapor Eq. (33)). ε₀ and ε∞ are the static and high-frequency relative permittivities of the material, a₀ the Bohr radius.

Detailed balance. With E' = E - ħω the two logarithms coincide: both equal ln[(√E + √E')² / ħω]. Hence E λ₊⁻¹(E) = ((n + 1)/n) E' λ₋⁻¹(E'): the rate per unit time times the density of states (∝ √E) of a transition E → E' and of its reverse E' → E stand in the ratio (n + 1)/n = exp(ħω/k_B T). This follows from the two formulas above by algebra; the tests check it, analytically and by Monte Carlo.

Angle. The polar deflection θ in either process follows Ding et al. (2021) Eq. (16), with E and E' the energies before and after the event (E' = E ∓ ħω):

dw/dΩ = (EE')^(1/2) / [π (E + E' - 2(EE')^(1/2) cos θ) ln R],
R = (E + E' + 2(EE')^(1/2)) / (E + E' - 2(EE')^(1/2))

It is normalised over the sphere. Its cumulative distribution in μ = cos θ from -1 is F(μ) = ln[(a + b)/(a - bμ)] / ln R (a = E + E', b = 2(EE')^(1/2)), which inverts to μ = (a - (a + b) R^(-u)) / b for u uniform on [0, 1). That inversion is our own algebra on Eq. (16). The azimuth is uniform.

Polaron trapping

The trapping inverse mean free path is (Taioli & Dapor Eq. (37), from Ganachaud and Mokrani 1995)

λ_pol⁻¹(E) = C exp(-γ E)

with C (here in m⁻¹) and γ (eV⁻¹) material constants. A trapped electron deposits its whole remaining energy at the trapping point and its history ends.

Material parameters

Every parameter is a validated input with a provenance string, recorded in the run metadata. The only committed material values are the SiO₂ ones of FrohlichPhonon::sio2 (Ding et al. (2021) Section 2.3). No C, γ is committed for any material: the source that tabulates them (Ganachaud and Mokrani 1995) could not be opened, so PolaronTrapping::new takes them from the caller. The gaps are listed in docs/data-provenance.md.

Structs

FrohlichPhonon

One Fröhlich LO-phonon mode of a polar material at a temperature: the parameters of the emission and absorption channels. See the module docs for the equations and their sources.

InsulatorChannels

The insulator channels switched on in one layer. The default has none, which is what a metal (or any layer not opted in) gets.

PolaronTrapping

Polaron trapping with inverse mean free path C exp(-γE) (Taioli & Dapor Eq. (37), Taioli & Dapor, arXiv:2404.07521v5, from Ganachaud & Mokrani, Surf. Sci. 334, 329 (1995)).

Enums

PhononError

Errors from building an insulator channel.

Sio2LoMode

The two LO-phonon modes of SiO₂ quoted in Ding et al. (2021) Section 2.3 (Ding, Li, Da, Liu, Sci. Technol. Adv. Mater. 22, 932 (2021), doi:10.1080/14686996.2021.1976597), who cite J.-C. Kuhr and H.-J. Fitting, J. Electron Spectrosc. Relat. Phenom. 105, 257 (1999) for them.

Functions

cos_theta_cdf

The cumulative distribution F(μ) of μ = cos θ under Ding et al. (2021) Eq. (16) for a phonon event E -> E' (the inverse of sample_cos_theta).

sample_cos_theta

cos θ of a phonon event that takes the electron from e_ev to e_after_ev (both positive and different), at cumulative probability u in [0, 1): the inverse of the distribution of Ding et al. (2021) Eq. (16), see the module docs. ln R is evaluated as 2 ln[(√E + √E')² / |E - E'|] to avoid the cancellation in E + E' - 2√(EE').