Energy-loss straggling

Code: lindhard/src/ion/stopping/straggling.rs (StragglingModel, variance_per_atom).

Model

The energy loss of an ion along a path fluctuates. For a thick target in the free-electron picture, the variance of the energy loss per unit areal density is (Bohr 1948)

\[ \Omega_B^2 = 4\pi z^2 Z_2 (e^2)^2 \quad \text{(J}^2\,\text{m}^2 \text{ per target atom)}, \]

independent of energy in the non-relativistic regime, with \( z \) the projectile charge from an effective-charge model. The relativistic variant multiplies it by \( (1 - \beta^2/2)/(1 - \beta^2) \) (Bethe and Livingston 1937; Fano 1963). For a compound the variances add by atom fraction, \( \sum_j x_j \Omega_j^2 \).

Selecting it

Library only (variance_per_atom, variance_per_atom_material). The BCA engine does not use it at present and there is no TOML key: the electronic loss along each free flight is deterministic.

Assumptions

• Free, stationary target electrons; every electron of the target contributes, whatever its binding.

Validity

High energy, \( v \gg v_0 Z_1^{2/3} \). Below about 1 MeV/u Bohr’s value is an overestimate, because the bound electrons contribute less than free ones. The corrections that reduce it at intermediate energy (Chu; Yang-O’Connor-Wang) are declined, and the Lindhard-Scharff low-velocity correction is omitted (not verified); see Validity ranges and declined terms.

Verification status

Not verified against the original papers (docs/data-provenance.md).

References

• N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 18 (8) (1948).
• H. A. Bethe and M. S. Livingston, Rev. Mod. Phys. 9, 245 (1937).
• U. Fano, Ann. Rev. Nucl. Sci. 13, 1 (1963).
• W. K. Chu, Phys. Rev. A 13, 2057 (1976) (declined correction).
• Q. Yang, D. J. O’Connor, Z. Wang, Nucl. Instrum. Methods B 61, 149 (1991) (declined correction).