A relevant process - which causes permanent damage to the silicon bulk structure - is the so-called displacement damage (e.g., see Chapter 7 of [Leroy and Rancoita (2016)], see also [Borchi et al. (1989), Consolandi et al. (2006), Leroy and Rancoita (2007)] and references therein). Displacement damage may be inflicted when a primary knocked-on atom (PKA) is generated. The interstitial atom and relative vacancy are termed Frenkel-pair (FP). In turn, the displaced atom may have sufficient energy to migrate inside the lattice and - by further collisions - can displace other atoms as in a collision cascade. This displacement process modifies the bulk characteristics of the device and causes its degradation. The total number of FPs can be estimated calculating the energy density deposited from displacement processes. In turn, this energy density is related to the Non-Ionizing Energy Loss (NIEL), i.e., the energy per unit path lost by the incident particle due to displacement processes.

Above 50 keV/nucleon (e.g., see here), in case of Coulomb scattering on nuclei (the reader can refer to Sections 2.2–2.2.2, 7.1–7.1.1.6 of [Leroy and Rancoita (2016)] for a comprehensive treatment) the non-ionizing energy-loss can be calculated using the Wentzel–Molière differential cross section:

| (1) |

discussed, for instance, in [Boschini et al. (2011)] and Section 2.2.2 of [Leroy and Rancoita (2016)], i.e.,

| (2) |

where E is the kinetic energy of the incoming particle, T is the kinetic energy
transferred to the target atom, L(T) - the so-called Lindhard partition
function - is the fraction of T deposited by means of
displacement processes. An analytical approximation of L(T) - the so-called Norgett-Robinson-Torrens
expression - can be found, for instance, in [Jun (2001), Messenger et al. (2003)]]
and in Equations (7.26, 7.27) of Sections 7.1.1.1 and 7.1.1.2 in [Leroy and Rancoita
(2016)] (see also references therein). T_{de} = T L(T) is the so-called damage
energy, i.e., the energy deposited by a recoil nucleus with kinetic energy T
via displacement damages inside the medium. The integral in Eq. (2) is
computed from the minimum energy T_{d} - the so-called threshold energy for
displacement, i.e., that energy necessary to displace permanently the atom
from its lattice position - up to the maximum energy T_{max} that can be
transferred during a single collision process. T_{d} is about 21 eV in silicon
(e.g., see Table 1 in [Jun, Xapsos, Messenger, Burke, Walters, Summers
and Jordan (2003)] and references therein) and 25 eV in lead (e.g., see
Table 22 at page 83 in [Was (2007)] and references therein). As already
remarked by Boschini and collaborators (2011), at high energy the Coulomb
NIEL - similarly to the nuclear stopping power - does not decrease with
energy as it is found by Ziegler, Biersack and Littmark (1985) or in other
calculations based on their universal screening potential derived in the
framework of a non-relativistic treatment of the screened Coulomb scattering.

For energies lower than 50-200 keV/nucleon, the scattering of protons and screened nuclei can be treated using the 4-terms analytical approximation of the ZBL cross section derived by Messenger et al. (2004) [see Eqs. (1--3, 15) and also references therein].

Furthermore, Jun and collaborators (2003) demonstrated that a relativistic treatment [Jun (2001)] of Coulomb scattering of protons - with kinetic energies above 50 MeV - upon silicon results into a non-ionizing energy loss which is larger than that expected from calculations using the Ziegler–Biersack–Littmark screened potential with a universal screening length (e.g., see [Ziegler, Biersack and Littmark (1985), Jun, Xapsos, Messenger, Burke, Walters, Summers and Jordan (2003), Messenger et al. (2003)]). The relativistic cross section used for treating the Coulomb scattering is the one derived by McKinley and Feshbach (1948) to describe the scattering of electrons on nuclei (e.g., see Sections 2.4.1, 7.1.1.7 of [Leroy and Rancoita (2016)] and references therein). Seitz and Koehler (1956) suggested that - when the mass of the projectile is much lower than the target rest-mass (e.g., see Section 13 of [Seitz and Koehler (1956)] and references therein) - this cross section can also describe - although screening effects are neglected - the scattering of protons and light nuclei, thus, providing - at high energy - a damage cross section which does not decrease with increasing energy. The data from Jun and collaborators (2003) - for protons with energies from 100 keV up to 1 GeV - are shown in Fig.1: the Ziegler–Biersack–Littmark (1985) screened potential was used to treat the Coulomb scattering of protons with energies lower than 50 MeV. In the same figure, the data obtained using Eq. (2) are also shown. There is an agreement to better than ≈ 6.5% - achieved at ≈ 1 GeV - between the results obtained by Jun and collaborators (2003) and the present screened relativistic (sr) calculations.

### References

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