Displacement damage

Code: lindhard/src/ion/damage.rs (models), lindhard/src/tally/ (counts).

lindhard reports two different quantities side by side and never adds one to the other or substitutes one for the other:

• Model estimates of the number of displacements made by a primary knock-on atom (PKA) of a given energy: NRT and Kinchin-Pease, from the Lindhard partition of the PKA energy (results.damage.nrt).
• Event counts from the full-cascade simulation: vacancies, interstitials and replacements counted collision by collision as the BCA follows every recoil (results.damage.cascade).

Damage energy (Lindhard partition)

Of a PKA’s energy \( T \), the part eventually given to atomic motion is (Lindhard, Nielsen, Scharff and Thomsen 1963)

\[ T_\mathrm{dam} = \frac{T}{1 + k\, g(\varepsilon)}, \qquad g(\varepsilon) = 3.4008\, \varepsilon^{1/6} + 0.40244\, \varepsilon^{3/4} + \varepsilon, \]

with the analytic fit \( g \) of Robinson (1970) as adopted by the NRT standard, and

\[ \varepsilon = \frac{T a A_2}{Z_1 Z_2 e^2 (A_1 + A_2)}, \qquad a = 0.8853\, a_0 \left(Z_1^{2/3} + Z_2^{2/3}\right)^{-1/2}, \]

\[ k = \frac{0.0793\, Z_1^{2/3} Z_2^{1/2} (A_1 + A_2)^{3/2}} {\left(Z_1^{2/3} + Z_2^{2/3}\right)^{3/4} A_1^{3/2} A_2^{1/2}}, \]

the Lindhard reduced energy and the Lindhard-Scharff electronic stopping coefficient of the PKA (\( Z_1, A_1 \)) in the target (\( Z_2, A_2 \)). For a self-ion these reduce to the forms written in the NRT standard, \( \varepsilon = T / (86.931\, Z^{7/3}) \) (\( T \) in eV) and \( k = 0.1337\, Z^{1/6} (Z/A)^{1/2} \); a test checks this.

NRT and Kinchin-Pease

Norgett, Robinson and Torrens (1975):

\[ N_\mathrm{NRT} = \begin{cases} 0 & T_\mathrm{dam} < E_d, \\ 1 & E_d \le T_\mathrm{dam} < 2E_d/0.8, \\ 0.8\, T_\mathrm{dam} / (2 E_d) & T_\mathrm{dam} \ge 2E_d/0.8, \end{cases} \]

with \( E_d \) the displacement threshold and 0.8 the displacement efficiency the authors took from BCA simulations. The older Kinchin and Pease (1955) count is 0, 1, or \( T/(2E_d) \) with thresholds \( E_d \) and \( 2E_d \); the tallies evaluate it on the damage energy (the usual “modified” form).

NRT is not a cascade count. Stoller et al. (2013) discuss how BCA full-cascade vacancy counts and NRT estimates differ, and recommend that a displacement dose comparable with the NRT standard be computed from the damage energy with the NRT formula, not taken from the full-cascade vacancy count. NRT also overestimates the number of defects that survive in-cascade recombination; it is a standard exposure unit, not a prediction of surviving defects.

Cascade counts

During transport a target atom is displaced when it receives \( T > E_d \) (BCA transport, step 4). The counts are defined in lindhard::tally::CascadeDefects:

• Replacement: a moving atom comes to rest, immediately after the collision in which it displaced an atom of its own element, at that atom’s site, so it fills the site. A particle stops when its energy falls below its cutoff, so this count depends on the cutoffs (a recoil cutoff near \( E_d \) gives the most replacements); with follow_recoils = false it is always 0.
• Vacancies = displacements minus replacements.
• Interstitials: recoils that came to rest in the target and did not fill a site. Implanted beam particles are not counted here (they are in the range tally).

These are kept in total, per layer and as a depth profile (damage_profile.csv).

Selecting it

Always on: both are computed in every run. The inputs are the per-element energies: \( E_d \) (e_d_ev), \( E_b \) (e_b_ev) and \( E_s \) (e_s_ev), set per material or for an element in every layer with [physics.energies.<symbol>]. There are no model variants to choose.

Assumptions

• Compounds (this crate’s own convention). NRT and the Lindhard partition are defined for a monatomic target. For a layer with several elements the tally uses the PKA’s own \( Z_1, A_1 \), the atom-fraction-weighted mean \( Z_2, A_2 \) of the layer the PKA starts in (non-integer in general), and the PKA element’s own \( E_d \) in that layer. No published source is claimed for this averaging; it reduces exactly to standard NRT for a monatomic layer. Treat NRT numbers for compounds as an exposure index comparable within lindhard, not as a value following any compound-target standard.
• The cascade counts depend directly on \( E_d \), \( E_b \) and the recoil cutoff, and on the amorphous-target BCA itself (no recombination, no crystal structure).

Validity

NRT is a convention for damage accounting, defined for monatomic targets. The cascade counts are BCA counts of displacement events, not of surviving defects.

Verification status

The self-ion reductions of \( \varepsilon \) and \( k \) are checked in a test. The default \( E_d \) values of the element table follow the ASTM E521 convention and are not yet verified against the standard (docs/data-provenance.md); Si has no default, so an input must choose one.

References

• J. Lindhard, V. Nielsen, M. Scharff, P. V. Thomsen, “Integral equations governing radiation effects”, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (10) (1963).
• M. T. Robinson, in Nuclear Fusion Reactors (British Nuclear Energy Society, London, 1970), p. 364.
• M. J. Norgett, M. T. Robinson, I. M. Torrens, Nucl. Eng. Des. 33, 50 (1975).
• G. H. Kinchin and R. S. Pease, Rep. Prog. Phys. 18, 1 (1955).
• R. E. Stoller, M. B. Toloczko, G. S. Was, A. G. Certain, S. Dwaraknath, F. A. Garner, Nucl. Instrum. Methods B 310, 75 (2013).