Lindhard-Scharff
Code: lindhard/src/ion/stopping/lindhard_scharff.rs (LindhardScharff).
Model
At ion velocities below \( v_0 Z_1^{2/3} \) the electronic stopping is proportional to the velocity (Lindhard and Scharff 1961). In the reduced units of Lindhard, Scharff and Schiøtt (1963):
\[ a = 0.8853\, a_0 \left(Z_1^{2/3} + Z_2^{2/3}\right)^{-1/2}, \qquad \varepsilon = \frac{E\, a\, M_2}{Z_1 Z_2 e^2 (M_1 + M_2)}, \qquad \rho = N x\, 4\pi a^2 \frac{M_1 M_2}{(M_1 + M_2)^2}, \]
the reduced electronic stopping is
\[ \left(\frac{d\varepsilon}{d\rho}\right)_e = k_L\, \varepsilon^{1/2}, \qquad k_L = \xi_e\, \frac{0.0793\, Z_1^{1/2} Z_2^{1/2} (A_1 + A_2)^{3/2}} {\left(Z_1^{2/3} + Z_2^{2/3}\right)^{3/4} A_1^{3/2} A_2^{1/2}}, \qquad \xi_e = Z_1^{1/6}, \]
with \( A_1, A_2 \) the masses in u. The equivalent dimensional form is
\[ S_e = \xi_e\, \frac{8\pi e^2 a_0\, Z_1 Z_2}{\left(Z_1^{2/3} + Z_2^{2/3}\right)^{3/2}}\, \frac{v}{v_0}. \]
The factor \( \xi_e \approx Z_1^{1/6} \) (order 1 to 2) is part of the Lindhard-Scharff result. A test checks the reduced form, with the rounded 0.0793, against the dimensional form to 1 % for all tested ion-target pairs.
A per-element multiplicative correction \( f(Z_2) \) can be attached in
the library (LindhardScharff::with_correction, default 1; it must be finite
and non-negative, and 0 switches the element’s stopping off). The CLI does
not expose it; use a user table for measured
stopping instead.
Selecting it
stopping = "lindhard-scharff" in [physics] (the default),
StoppingChoice::LindhardScharff. All of the loss is applied nonlocally,
along the free flights. It is also the nonlocal half of
stopping = "equipartition-ls-or" (Oen-Robinson).
Assumptions
- A free-electron-gas picture with Thomas-Fermi scaling; the stopping is smooth in \( Z_1 \) and \( Z_2 \) and has no shell structure (the measured \( Z_1 \) oscillations are not reproduced).
- No dependence on the chemical or physical state of the target.
Validity
\( v < v_0 Z_1^{2/3} \), that is \( E/A_1 \) below about \( 25\ \mathrm{keV} \cdot Z_1^{4/3} \). This is the regime of keV-range implantation and of cascade atoms. Above it the stopping peaks and turns over, which this model does not describe; see Bethe-Bloch for high velocities and Validity ranges and declined terms for why no interpolation joins the two.
The validation pages report a known offset: for B in Si the computed ranges
run long against the measurement, and at 10 to 20 keV part of the offset is
attributed to LS stopping being too small for that pair
(docs/validation.md,
level 3).
Verification status
The constants 0.8853, 0.0793 and \( \xi_e = Z_1^{1/6} \) are checked
against the dimensional form in a unit test (1 %); they have not yet been
checked digit by digit against the papers
(docs/data-provenance.md).
References
- J. Lindhard and M. Scharff, Phys. Rev. 124, 128 (1961).
- J. Lindhard, M. Scharff, H. E. Schiøtt, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (14) (1963).