Damage Function for Neutrons, 1MeV Neutron Equivalent and Hardness Parameter

Fast neutrons, as well as other types of particles, produce defects mainly by displacing silicon atoms from their lattice positions to interstitial locations, i.e., generating vacancy-interstitial pair, the so-called Frenkel pairs. Neutrons with sufficient large energies (i.e. fast neutrons) can cause the displacement of a silicon or any other atom inside the lattice, referred to as primary displacement, from its location. The displaced arom can, in turn, generate secondary displacements, following a cascading effect. In literature (see for instance[Namenson, Wolicki and Messenger (1982), Ougouag et al. (1990), ASTM E722-09 (2009), ASTM E722-14 (2014)]), the damage effect due to displacements induced by neutrons is expressed by the damage function D(E) in units of MeV cm², also called the displacement kerma function, whose value, for instance, in Si at 1 MeV is the ASTM standard [ASTM E722-09 (2009), ASTM E722-14 (2014)] D(1 MeV)≈95 MeV mb. The damage function accounts for both the cross section for displacing silicon atoms and the energy released in creating displacements. It is given by:

(1)

where E is the incoming neutron energy, σ_k(E) is the cross section for the kth reaction in which f_k(E,E_R) is the probability that a recoil energy E_R is generated and, finally, P(E_R) is the partition energy, i.e., the part of the recoil energy deposited in displacements calculated, for instance, in the framework of the Lindhard screened potential scattering theory, based on the Thomas-Fermi model and further developments[Lindhard, Nielsen, Scharff and Thomsen (1963), Coulter and Parkin (1977), Coulter and Parkin (1979), Coulter and Parkin (1980)]. The damage function is given in literature (e.g., see [Ougouag et al. (1990)] and references therein). The damage functions were obtained in [ASTM E722-09 (2009), ASTM E722-14 (2014)] for E_d = 25 and 20.5 eV, respectively, in Si and 10 eV in GaAs, where E_d is the is the so-called displacement threshold energy. 

The energy density E_dis is the energy per cm³ deposited through atomic displacements by neutrons, which are characterized by the spectral fluence ϕ(E) in n/cm² MeV, and is given by:

(2)

where N is the number of atoms per cm³ in the bulk silicon and E _min is the lowest neutron kinetic energy for displacement. E_dis, as computed by means of Eq. (2), is the Non Ionizing Energy Loss (NIEL in units of MeV/cm) deposited by the neutrons per cm² of the absorber (e.g., treatment in Chapter 4 of [Leroy and Rancoita (2011], Chapter 7 of [Leroy and Rancoita (2016)]: as long as E_min is ≲ 10 keV, E_dis varies slightly. In the case of the Triga reactor RC:1 neutron spectrum, the E_dis value varies by no more than 0.5 % for E_min values in the energy range up to 10 keV (e.g., see [Consolandi et al. (2006), Leroy and Rancoita (2007)] and references therein).

In the current sr-niel calculator (e.g., see [sr-niel.org (2016)]) the NIEL and NIEL doses are determined by means of the damage functions from [ASTM E722-09 (2009), ASTM E722-14 (2014)].

The displacement damages induced by neutrons can be normalized to displacement damages induced by 1 MeV neutrons by means of the hardness parameter (κ) also termed hardness factor. For fission neutrons with spectral fluence ϕ(E), the 1 MeV equivalent neutron fluence is given by:

(3)

Φ_eq^1 MeV is the fluence of 1 MeV neutron needed for generating the same amount of displacement damage compared to the fission fluence Φ_n.

From Eqs. (2, 3), we can derive the hardness parameter κ as:

(4)

and, conversely,

(5)

References