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 cm^{2}, 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 cm^{3} deposited through atomic
displacements by neutrons, which are characterized by the spectral fluence ϕ(E)
in n/cm^{2} MeV, and is given by:

| (2) |

where N is the number of atoms per cm^{3} 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 cm^{2} 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:

_{A}the number of atoms per cm

^{3}; from the above equation one finds

| (4) |

and, conversely,

| (5) |

### References

[ASTM E722-09 (2009)] ASTM E722–09 (2009).

[ASTM E722-14 (2014)] ASTM-722–14 (2014).

[Consolandi et al. (2006)] C. Consolandi et al. (2006), Systematic Investigation of Monolithic Bipolar Transistors Irradiated with Neutrons, Heavy Ions and Electrons for Space Applications, Nucl. Instr. and Meth. in Phys. Res. B 252 (2006), 276, doi:10.1016/j.nimb.2006.08.018; http://www.sciencedirect.com/science/article/pii/S0168583X0600913X.

[Coulter and Parkin (1977)] C. A. Coulter and D. M. Parkin, Trans. Am. Nucl. Soc. 27 (1977), 300.

[Coulter and Parkin (1979)] C. A. Coulter and D. M. Parkin, Jou. Nucl. Mat. 85&86 (1979), 611.

[Coulter and Parkin (1980)] C. A. Coulter and D. M. Parkin, Jou. Nucl. Mat. 88 (1980), 249.

[Leroy and Rancoita (2007)] C. Leroy and P.G. Rancoita (2007), Particle Interaction and Displacement Damage in Silicon Devices operated in Radiation Environments Reports on Progress in Physics 70, 493-625, doi:10.1088/0034-4885/70/4/R0; http://iopscience.iop.org/0034-4885/70/4/R01/.

[Leroy and Rancoita (2011)] C. Leroy and P.G. Rancoita (2011), Principles of Radiation Interaction in Matter and Detection - 3rd Edition -, World Scientific, Singapore, ISBN-978-981-4360-51-7; http://www.worldscientific.com/worldscibooks/10.1142/8200.

[Leroy and Rancoita (2016)] C. Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, ISBN-978-981-4603-18-8 (printed); ISBN.978-981-4603-19-5 (ebook); http://www.worldscientific.com/worldscibooks/10.1142/9167; it is also partially accessible via google books.

[Lindhard, Nielsen, Scharff and Thomsen (1963)] J. Lindhard, V. Nielsen, M. Scharff and P. V. Thomsen, Danske Vidensk. Selsk. Mat.-Fys. Medd. 33 vol. 10 (1963), 10

[Namenson, Wolicki and Messenger (1982)] A. I. Namenson, E. A. Wolicki and G.C. Messenger, IEEE Trans on Nucl. Sci. 29 (1982) 1018.

[Ougouag et al. (1990)] A. M. Ougouag et al., IEEE Trans on Nucl. Sci. 37 (1990) 2219.