Bethe-Bloch and the density effect

Code: lindhard/src/ion/stopping/bethe.rs (BetheBloch, EffectiveCharge, density_effect_single_oscillator).

Model

At high velocity the electronic stopping cross section per target atom is (Bethe 1930, 1932; Fano 1963; the same equation as the Particle Data Group review of the passage of particles through matter):

\[ S = \frac{4\pi (e^2)^2 z^2 Z_2}{m c^2 \beta^2} \left[ \frac{1}{2}\ln\frac{2 m c^2 \beta^2\gamma^2 W_\mathrm{max}}{I^2}

• \beta^2 - \frac{\delta}{2} - \frac{C}{Z_2} + L_1 + L_2 \right], \]

\[ W_\mathrm{max} = \frac{2 m c^2 \beta^2\gamma^2}{1 + 2\gamma m/M + (m/M)^2}, \]

with \( m \) the electron mass, \( M \) the ion mass, \( z \) the projectile effective charge, \( I \) the mean excitation energy, \( \delta \) the density-effect correction and \( C \) the shell correction. \( \beta^2 \) is computed as \( \tau(\tau + 2)/(1 + \tau)^2 \) with \( \tau = E/Mc^2 \), which stays accurate at low energy.

• Bloch term (Bloch 1933), on by default: \( L_2 = -y^2 \sum_{n \ge 1} \frac{1}{n(n^2 + y^2)} \), \( y = z\alpha/\beta \).
• Mean excitation energy: by default the Bloch rule \( I = 10\ \mathrm{eV} \cdot Z_2 \) (Bloch 1933), a rough estimate (about 20 % in \( I \), about 2 % in \( S \)). The library takes a measured, cited value through BetheBloch::with_mean_excitation_ev.
• Shell correction \( C/Z_2 \) and density effect \( \delta \): 0 by default; the library accepts constants from a cited source (shell_over_z, density_delta).
• Barkas term \( L_1 \): not implemented (see Validity ranges and declined terms).

Effective charge

The CLI uses the bare charge; the Barkas form is available in the library and is not applied by default.

Single-oscillator density effect (opt-in)

The dielectric formulation of the density effect (Fermi 1940; Sternheimer 1952), in the form given by Fano (1963), is

\[ \delta = \sum_i f_i \ln\!\left(1 + \frac{L^2}{\omega_i^2}\right) - \frac{L^2 (1 - \beta^2)}{\beta^2\omega_p^2}, \qquad \sum_i \frac{f_i\, \omega_p^2}{\omega_i^2 + L^2} = \frac{1}{\beta^2} - 1. \]

Specialised to one oscillator (\( f = 1 \), \( \omega_0 = I/\hbar \)) the constraint solves in closed form, \( L^2 = \omega_p^2(\beta\gamma)^2 - \omega_0^2 \), and

\[ \delta = \ln\!\left[(\beta\gamma)^2 \left(\frac{\hbar\omega_p}{I}\right)^2\right] - 1

• \frac{(I/\hbar\omega_p)^2}{(\beta\gamma)^2} \quad \text{for } \beta\gamma > \frac{I}{\hbar\omega_p}, \]

and 0 below, with \( \hbar\omega_p = \hbar\sqrt{n e^2/\varepsilon_0 m} \) the free-electron plasma energy. The one-oscillator reduction is ours, not quoted from a paper. It is enabled with BetheBloch::with_density_effect (library only).

Selecting it

stopping = "bethe-bloch" in [physics], StoppingChoice::BetheBloch: bare charge, Bloch term on, Bloch-rule \( I \), no shell or density correction, all loss nonlocal.

Use it only for problems that stay above its range. Where the bracket is not positive the model returns NotApplicable, and the transport stops with that error rather than silently switching model. For any ion that slows down in the target (a semi-infinite substrate always) the run fails once the energy falls below the bracket’s zero: for example, 300 keV H into Si with stopping = "bethe-bloch" stops with “bethe-bloch is not applicable at 85306 eV”. No low-to-high energy interpolation is implemented, for the reasons given in Validity ranges and declined terms.

Assumptions

• First Born approximation plus the Bloch correction; the projectile charge is fixed by the effective-charge model.
• Without shell and Barkas corrections, expect errors of a few percent at 1 to 10 MeV/u.

Validity

\( v \ge 3 v_0 Z_1^{2/3} \) up to 1 GeV/u (the upper end because the density effect is off by default). The single-oscillator density effect switches on only at \( \beta\gamma > I/\hbar\omega_p \), about 5 to 6 for Si (a proton of about 5 GeV), so it is zero over the whole advisory range. Real materials switch the effect on much earlier (around \( \beta\gamma \approx 1.5 \) for Si), because their oscillator spectrum is spread: the one-oscillator form underestimates \( \delta \) through the transition and is only the correct asymptote for ultra-relativistic ions.

Verification status

Hand-calculated proton-in-Si values (1, 10 and 100 MeV, \( I = 140 \) eV, CODATA constants) are unit tests; they are our own arithmetic, not taken from any stopping table. The density-effect form is tested through its limits (\( \delta \to 0 \) continuously at threshold, and \( \delta \to 2\ln(\beta\gamma\, \hbar\omega_p / I) - 1 \) at large \( \beta\gamma \)). The Bloch rule, the Bloch series and the Barkas effective charge have not been checked against the original papers (docs/data-provenance.md).

References

• H. Bethe, Ann. Phys. 397, 325 (1930); H. Bethe, Z. Phys. 76, 293 (1932).
• F. Bloch, Ann. Phys. 408, 285 (1933).
• U. Fano, Ann. Rev. Nucl. Sci. 13, 1 (1963).
• Particle Data Group, “Passage of particles through matter”, in the Review of Particle Physics.
• W. H. Barkas, Nuclear Research Emulsions I (Academic Press, 1963).
• L. C. Northcliffe, Ann. Rev. Nucl. Sci. 13, 67 (1963).
• E. Fermi, Phys. Rev. 57, 485 (1940).
• R. M. Sternheimer, Phys. Rev. 88, 851 (1952).