Module damage 

Displacement-damage models: the Lindhard partition, NRT and Kinchin-Pease.

These are model estimates of the number of displacements made by a primary knock-on atom (PKA) of a given energy. They are kept apart from the event counts of a full-cascade simulation (vacancies, interstitials, replacements, crate::tally::CascadeDefects), which the BCA engine produces by following every recoil. The two answer different questions and are reported side by side, never added or substituted for each other.

Damage energy (Lindhard partition)

Of a PKA’s energy T, the part eventually given to atomic motion (the damage energy) is T_dam = T / (1 + k g(ε)), J. Lindhard, V. Nielsen, M. Scharff and P. V. Thomsen, “Integral equations governing radiation effects”, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (10) (1963), with the analytic fit g(ε) = 3.4008 ε^(1/6) + 0.40244 ε^(3/4) + ε of M. T. Robinson (in Nuclear Fusion Reactors, British Nuclear Energy Society, London, 1970, p. 364), as adopted by the NRT standard below (coefficients checked against J. F. Ziegler, J. P. Biersack, M. D. Ziegler, SRIM: The Stopping and Range of Ions in Matter (SRIM Co., 2015), Eq. 7-52, p. 7-28, a secondary source). Here ε is the Lindhard reduced energy and k the Lindhard-Scharff electronic stopping coefficient of the PKA (atomic number Z1, mass A1) in the target (Z2, A2), as in crate::ion::stopping::lindhard_scharff:

ε = T a A2 / (Z1 Z2 e^2 (A1 + A2)),   a = 0.8853 a0 (Z1^(2/3) + Z2^(2/3))^(-1/2),
k = 0.0793 Z1^(2/3) Z2^(1/2) (A1 + A2)^(3/2) / ((Z1^(2/3) + Z2^(2/3))^(3/4) A1^(3/2) A2^(1/2)).

For a self-ion (Z1 = Z2 = Z, A1 = A2 = A) these reduce to the forms written in the NRT standard, ε = T / (86.931 Z^(7/3)) (T in eV) and k = 0.1337 Z^(1/6) (Z/A)^(1/2) (checked in a test).

NRT and Kinchin-Pease

M. J. Norgett, M. T. Robinson and I. M. Torrens, “A proposed method of calculating displacement dose rates”, Nucl. Eng. Des. 33 (1975) 50:

N_NRT = 0                      for T_dam < E_d,
      = 1                      for E_d <= T_dam < 2 E_d / 0.8,
      = 0.8 T_dam / (2 E_d)    for T_dam >= 2 E_d / 0.8,

with E_d the displacement threshold and 0.8 the displacement efficiency the authors took from BCA simulations. The older G. H. Kinchin and R. S. Pease, “The displacement of atoms in solids by radiation”, Rep. Prog. Phys. 18 (1955) 1 count is 0, 1, or T / (2 E_d) with thresholds E_d and 2 E_d; kinchin_pease is evaluated on whatever energy it is given (the tallies pass the damage energy, the usual “modified” form).

NRT is not a cascade count. R. E. Stoller, M. B. Toloczko, G. S. Was, A. G. Certain, S. Dwaraknath and F. A. Garner, “On the use of SRIM for computing radiation damage exposure”, Nucl. Instrum. Methods B 310 (2013) 75, discuss how BCA full-cascade vacancy counts and NRT displacement estimates differ and recommend that, when a displacement dose comparable with the NRT standard is wanted, it be computed from the damage energy with the NRT formula rather than taken from the full-cascade vacancy count. NRT is also known to overestimate the number of defects that survive in-cascade recombination (as seen in molecular dynamics); it is a standard exposure unit, not a prediction of surviving defects. This crate therefore reports both and lets the user choose.

Compound and multi-element targets (an approximation)

NRT (Norgett, Robinson and Torrens 1975) and the Lindhard partition it uses are defined for a monatomic target. For a layer with more than one element, the tally (crate::tally::IonTally) evaluates the partition with:

• Z1, A1: the PKA’s own atomic number and mass;
• Z2, A2: the atom-fraction-weighted mean atomic number and mean mass of the layer the PKA starts in (non-integer in general);
• E_d: the PKA element’s own displacement threshold in that layer.

This is an approximation outside NRT’s monatomic definition. It is this crate’s own convention; no published source is claimed for this particular averaging. For a monatomic layer it reduces exactly to the standard NRT evaluation. Treat NRT numbers for compounds as an exposure index that is comparable within this crate, not as a value that follows any compound-target standard.

Constants

NRT_EFFICIENCY

NRT displacement efficiency (Norgett, Robinson and Torrens 1975).

Functions

damage_energy_ev

Damage energy T / (1 + k g(ε)), eV, of a PKA of energy t_ev. Returns 0 for non-positive t_ev.

kinchin_pease

Kinchin-Pease displacements for energy energy_ev and threshold e_d_ev.

lindhard_k

Lindhard-Scharff electronic stopping coefficient k of the PKA in the target (see the module docs).

lindhard_reduced_energy

Lindhard reduced energy ε of a PKA of energy t_ev (Z1, mass a1 u) in a target (Z2, a2 u). Z and masses may be non-integer (averaged) values.

nrt_displacements

NRT displacements for damage energy t_dam_ev and threshold e_d_ev.

robinson_g

Robinson’s fit g(ε) = 3.4008 ε^(1/6) + 0.40244 ε^(3/4) + ε.