Validity ranges and declined terms

This page is docs/stopping-models.md, included verbatim so there is one copy. It collects the validity ranges of the electronic stopping models side by side, and records the terms that are deliberately not implemented and why (the clean-room rules exclude coefficient sets that are only published as tables). The model pages are Lindhard-Scharff, Oen-Robinson, Bethe-Bloch, user tables, Bragg additivity and straggling; the references cited below are listed in full on those pages and at the end of this one.

Code: lindhard/src/ion/stopping/. All models return the stopping cross section per atom in J m² (to_ev_1e15_cm2 converts to eV·10⁻¹⁵ cm²) and expose the ranges below at run time through ElectronicStopping::validity. Ranges are advisory; the models do not refuse energies outside them (Bethe-Bloch returns NotApplicable where its bracket is not positive, and user tables return OutOfTableRange). v0 = α c is the Bohr velocity, Z1 the ion charge number.

What is deliberately not implemented

The clean-room rules (CONTRIBUTING.md) rule out tabulated coefficient sets from SRIM/ZBL, ICRU reports and similar. As a result:

• Barkas (L1) term: declined (see “Literature findings” below).
• Shell correction C/Z2: declined. Callers may set a constant from a cited source (BetheBloch::shell_over_z); default 0.
• Density effect δ: only the single-oscillator form is implemented (opt-in, BetheBloch::with_density_effect); the tabulated multi-oscillator parameter sets are declined. density_delta can still be set as a constant.
• Mean excitation energy I: defaults to the Bloch rule 10 eV · Z2 (rough); supply a cited measured value with with_mean_excitation_ev.
• Biersack-Varelas interpolation joining the low- and high-energy regimes: documented but not implemented (issue #44, closed; see “Biersack-Varelas interpolation” below). Fitted ZBL/Biersack-Varelas coefficients are not used.
• Straggling: Chu and Yang-O’Connor-Wang corrections declined (see below) and the Lindhard-Scharff low-velocity correction (not verified) is omitted; Bohr plus an optional relativistic factor only.
• Heavy-ion effective charge: the Barkas empirical form is available in bethe and is not applied by default.

Biersack-Varelas interpolation (issue #44, closed)

Status: documented, not implemented.

The joining form

The interpolation joins a low-energy and a high-energy stopping branch harmonically:

1/S = 1/S_low + 1/S_high     (equivalently S = S_low S_high / (S_low + S_high))

Two open-access sources were read (rendered pages) and show this form:

• M. V. Moro, PhD thesis, Universidade de Sao Paulo (2017), doi:10.11606/T.43.2017.tde-18092017-095345, eq. (3.7), printed p. 40. It gives s_low = A1 E^0.45 and s_high = (A2/E) ln(1 + A3/E + A4 E) (E in reduced units as defined there), attributes the form to Varelas and Biersack (1970), refined by Andersen and Ziegler (1977), and says A1 to A4 are fitted to experimental data.
• P. de Vera et al., arXiv:2608.15368 (2026), eq. (9), p. 3: the same s = s_low s_high / (s_low + s_high), citing Varelas and Biersack, and Andersen and Ziegler.

NISTIR 4999 (Berger 1992), section 3.5.2, p. 8, also mentions the “fitting formula of Varelas and Biersack (1970)” used with ICRU 49 coefficients (no equation given).

The original papers were not consulted (closed access; Unpaywall and Semantic Scholar report no open copy, ScienceDirect returned 403):

• C. Varelas and J. P. Biersack, Nucl. Instrum. Methods 79, 213 (1970), doi:10.1016/0029-554X(70)90141-2.
• J. P. Biersack and L. G. Haggmark, Nucl. Instrum. Methods 174, 257 (1980), doi:10.1016/0029-554X(80)90440-1.

Citation correction: “Biersack and Varelas, NIM 194, 93 (1982)” is not a Biersack-Varelas paper. Crossref resolves NIM 194, 93-100 (1982) to Biersack and Ziegler, “Refined universal potentials in atomic collisions” (doi:10.1016/0029-554X(82)90496-7). It is not a source for this join.

Why it is not implemented

The published s_high = (A2/E) ln(1 + A3/E + A4 E) is a fully fitted functional form (de Vera et al. eqs. (10)-(11), following ICRU 49; A1 to A4 are all fitting parameters) built so that it never crosses zero. For positive coefficients the 1 + keeps the logarithm’s argument above 1, so s_high > 0 at every E whatever A3 is. The harmonic join tends to s_low at low energy even with A3 = 0: s_high then tends to the finite value A2 A4 while s_low = A1 E^0.45 goes to 0. The fitted A3/E term only sets how fast s_high grows as E goes to 0. The coefficients come from the Andersen-Ziegler / ICRU 49 fits, which the clean-room policy (CONTRIBUTING.md) does not allow.

lindhard’s only sourced high-energy branch is BetheBloch, which has no such structure: its bracket crosses zero inside the crossover region. As S_high goes to 0+, the join S_low S_high / (S_low + S_high) goes to 0, not to S_low. Evaluated with LindhardScharff and BetheBloch defaults (Bloch term on, I = 10 eV Z2), in units of 1e-15 eV cm^2/atom (computed by a Python mirror of the formulas, not by the Rust models):

Bethe-Bloch is negative for H in Si below roughly 90 keV, for P in Si up to about 400 keV/u, and for He in Au up to about 450 keV/u. Neither workaround is acceptable. Falling back to S_low below the Bethe zero makes S jump from about 0 to the full LS value (about 27 for H in Si near 90 keV), which breaks continuity. Propagating NotApplicable leaves the model undefined over the whole low-energy regime, so ions slowing down through it cannot be transported. Restricting the join to where Bethe-Bloch is positive violates the low-energy limit and gives badly wrong stopping near the zero. A fix would need a high branch that stays positive at low E. The options are the fitted A1 to A4 (Tier C data), or a regularisation of our own, such as grafting the ln(1 + ...) structure onto Bethe by identifying A2 and A4 with Bethe quantities and setting A3 = 0. That identification is our own construction, which neither source makes. Neither option is admissible.

When it could be revisited

• A published, coefficient-free high-energy branch that stays positive below the Bethe zero (for example a Bethe form with a closed-form, non-tabulated shell correction; the review of the omitted correction terms under #41, now closed, is recorded below) is found and can be cited to an equation.
• The A1 to A4 coefficients become available from a source whose licence permits use, or are derived from our own fits to data that may be used.
• The primary papers are read and show a high branch that stays positive without fitted coefficients, or that ties the ln(1 + ...) form to Bethe quantities in a way that can be cited.

Literature findings (issue #41, closed)

Each of the four omitted items was reviewed for a closed form whose coefficients come from a paper itself. The review was from the contributor’s knowledge of the literature; no paper text was available to check digits against, so every statement below about a paper’s content is “as recalled” and the declines are on that basis. Nothing was taken from ICRU/SRIM/NIST tables.

• Barkas L1: declined. Ashley, Ritchie & Brandt (Phys. Rev. B 5, 2393 (1972)) give L1 = F(b / x^(1/2)) / (Z2^(1/2) x^(3/2)) with F a function that is tabulated (and later fitted to it) rather than closed-form. Lindhard’s 1976 treatment of the Barkas effect (NIM 132, 1) gives asymptotic forms from a harmonic-oscillator model, whose numerical coefficients we could not reproduce with confidence from memory. Jackson & McCarthy (Phys. Rev. B 6, 4131 (1972)) and later fits carry fitted coefficients that cannot be verified here. No form with verifiable coefficients exists to us; an invented one is not acceptable.
• Shell correction: declined. Walske and Bichsel shell corrections come from hydrogenic-shell calculations published as tables in η = βγ and I; the compact empirical forms (e.g. the ICRU 49 expression) are fits with coefficients tuned to data and tables, which Tier C of CONTRIBUTING.md excludes. A first-principles computation (hydrogenic shell sums) is a research task of its own, not a closed form.
• Density effect: partly implemented. The Sternheimer parameter sets (x0, x1, a, m, C̄) are tables, which we do not ingest. What is admissible is the dielectric formulation itself, computed from a model; we implement its one-oscillator specialisation (see density_effect_single_oscillator). Its high-energy limit is exact, but it switches on too late to matter in range. A multi-oscillator version needs per-material oscillator strengths, which are tabulated material data. Callers wanting a realistic δ in the range 1 to 100 GeV/u should supply one from a cited source via density_delta.
• Chu / Yang-O’Connor-Wang straggling: declined. Chu’s correction (Phys. Rev. A 13, 2057 (1976)) is built from Hartree-Fock-Slater charge densities and is published as tables; Yang, O’Connor & Wang (NIM B 61, 149 (1991)) is a fit whose coefficients are fitted to data tables. Neither has a closed form with paper-sourced coefficients that we could verify.

Deviations from the issue text

• The Lindhard-Scharff k_L quoted in the issue omits the factor ξ_e = Z1^(1/6). It is included here, because with it the reduced form k_L ε^(1/2) agrees with the dimensional LS expression to better than 1 % for all tested ion/target pairs (and without it Z1 > 1 disagrees by Z1^(1/6)).
• The Oen-Robinson local-loss constants are flagged unverified in data-provenance.md.

References

• J. Lindhard and M. Scharff, Phys. Rev. 124, 128 (1961).
• O. S. Oen and M. T. Robinson, Nucl. Instrum. Methods 132, 647 (1976).
• 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).
• W. H. Bragg and R. Kleeman, Phil. Mag. 10, 318 (1905).
• E. Fermi, Phys. Rev. 57, 485 (1940).
• R. M. Sternheimer, Phys. Rev. 88, 851 (1952).
• N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 18 (8) (1948).
• M. V. Moro, PhD thesis, Universidade de São Paulo (2017), doi:10.11606/T.43.2017.tde-18092017-095345.
• P. de Vera et al., arXiv:2608.15368 (2026).
• M. J. Berger, NISTIR 4999 (1992).
• C. Varelas and J. P. Biersack, Nucl. Instrum. Methods 79, 213 (1970), doi:10.1016/0029-554X(70)90141-2 (not consulted).
• J. P. Biersack and L. G. Haggmark, Nucl. Instrum. Methods 174, 257 (1980), doi:10.1016/0029-554X(80)90440-1 (not consulted for this join).
• J. P. Biersack and J. F. Ziegler, Nucl. Instrum. Methods 194, 93 (1982), doi:10.1016/0029-554X(82)90496-7.
• J. C. Ashley, R. H. Ritchie, W. Brandt, Phys. Rev. B 5, 2393 (1972).
• J. Lindhard, Nucl. Instrum. Methods 132, 1 (1976).
• J. D. Jackson and R. L. McCarthy, Phys. Rev. B 6, 4131 (1972).
• W. K. Chu, Phys. Rev. A 13, 2057 (1976).
• Q. Yang, D. J. O’Connor, Z. Wang, Nucl. Instrum. Methods B 61, 149 (1991).