Screening lengths
Code: lindhard/src/ion/potential.rs (ScreeningLength).
Model
The screening length \( a(Z_1, Z_2) \) scales the separation in the screening function, \( x = r/a \). Three forms are implemented. Two of them use the Thomas-Fermi constant
\[ C_\mathrm{TF} = \left(\frac{9\pi^2}{128}\right)^{1/3} = 0.8853, \]
in the closed form of Firsov (1958, p. 535), printed as 0.8853 by Lindhard, Scharff and Schiøtt (1963, p. 8).
| Length | Formula | TOML ([physics]) | Rust |
|---|---|---|---|
| Universal | \( a_U = 0.8853\, a_0 / (Z_1^{0.23} + Z_2^{0.23}) \) | screening_length = "universal" | LengthChoice::Universal, ScreeningLength::Universal |
| Firsov | \( a_F = 0.8853\, a_0 / (Z_1^{1/2} + Z_2^{1/2})^{2/3} \) | screening_length = "firsov" | LengthChoice::Firsov, ScreeningLength::Firsov |
| Lindhard (Thomas-Fermi) | \( a_L = 0.8853\, a_0 / (Z_1^{2/3} + Z_2^{2/3})^{1/2} \) | screening_length = "lindhard" | LengthChoice::Lindhard, ScreeningLength::Lindhard |
Without screening_length the length paired with the potential is used:
universal for ZBL, Firsov for Kr-C and Molière, Lindhard for Lenz-Jensen.
The run echoes the length it used in summary.json
(input.physics.screening_length).
Assumptions
- The universal length is the one the ZBL screening function was fitted with; the other two come from Thomas-Fermi scaling arguments for the two-atom system.
- Mixing a screening function with a length other than its own is allowed (it is a model choice), but the fitted functions were fitted with their own length.
Validity
As for the potentials: screened Coulomb collisions from cascade energies upward.
Verification status
From docs/data-provenance.md:
the Firsov and Lindhard lengths have been verified against scans of the
primary papers (page references above). The universal length uses the same
constant: the SRIM book (Ziegler, Biersack and Ziegler, 2015, Ch. 2) confirms
the exponent 0.23 and prints the prefactor as 0.8853 in Eq. 2-73 (p. 2-30),
while the labels of its Figs. 2-16 and 2-17 (p. 2-32) print 0.8854, a 10⁻⁴
relative difference. The code follows the equation and uses the Thomas-Fermi
constant, 0.88534.
References
- O. B. Firsov, Sov. Phys. JETP 6, 534 (1958), pp. 535-536.
- J. Lindhard, M. Scharff, H. E. Schiøtt, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (14) (1963), p. 8.
- J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids (Pergamon, New York, 1985), ch. 2.
- J. F. Ziegler, J. P. Biersack, M. D. Ziegler, SRIM: The Stopping and Range of Ions in Matter (SRIM Co., 2015), ch. 2, pp. 2-29 to 2-32.