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INFN Screened Relativistic Non-Ionising Energy Loss

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Introduction

The non-ionizing energy-loss (NIEL) accounts for the amount of energy - lost by a particle passing through a medium -, which was imparted to create atomic displacements. It can be expressed in units of

where is the displacement stopping power (e.g., see discussion in Sect. 4.2.1 of [Leroy and Rancoita (2011)]) and the minus sign indicates that the energy is lost by the incoming particle to create atomic displacements inside the absorber. For instance, in case of (elastic) Coulomb scattering on nuclei, the displacement stopping power can be calculated as

where [see Eq. (1.71) at page 25 of [Leroy and Rancoita (2012)]

is the number of atoms per cm3 in the absorber, rA and A are the density and atomic weight of the medium, respectively; N is the Avogadro constant, E is the kinetic energy of the incoming particle; ER and ER max are the recoil kinetic energy and the maximum energy transferred to the recoil nucleus, respectively; Ed is the so-called displacement threshold energy, i.e., the minimum energy necessary to permanently displace an atom from its lattice position; the expression of L(ER) is the Lindhard partition function discussed, for instance, in Sects. 4.2.1 and 4.2.1.2 in [Leroy and Rancoita (2011)] (see also references therein and [Jun (2001)]); finally, ds(E,ER)/d ER is the differential cross section for elastic Coulomb scattering for electrons or protons on nuclei. Furthermore, the non-ionizing energy-loss (NIEL) can be expressed in units of MeV cm2/g as

with c =x rA, rA the absorber density in g/cm3 and dEde/dc the displacement mass-stopping power.

In the framework of the screened relativistic NIEL approach, the values of NIEL in units of MeV cm2/g for electrons (e.g., see Sect 2 of [C. Baur et al. (2014)] ) are obtained using the Mott differential cross section [Mott (1929)] of electrons on nuclei calculated from the practical expression discussed in [Boschini et al. (2013)] (see also [Linjian et al. (1995)]) and accounting for the effects due to both screened nuclear potentials and form factors discussed in [Boschini et al. (2012)] and Sects. 1.3.1-1.3.3 of [Leroy and Rancoita (2012)] (see also references therein). The Coulomb contribution to the overall NIEL for protons and nuclei above 50 keV/nucleon is obtained using the elastic cross section derived from treatment of the nucleus-nucleus screened Coulomb scattering discussed in [Boschini et al. (2011)] and Sect. 2.1.4.2 in [Leroy and Rancoita (2011)] (see also [Fernandez-Vera et al. (1993), Butkevick (2002), Boschini et al. (2010)]). That treatment allows one to deal with a scattering occurring up to relativistic energies.

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, for protons in Si and GaAs absorbers, the NIEL contribution resulting from hadronic interactions was estimated, for instance, by Jun et al. (2003) at Ed =10 eV. These authors found that for protons with energies above about 20 (30) MeV this is the dominant NIEL process - with respect to that due to elastic Coulomb scattering - in a Si (GaAs) absorber. A discussion about this topic can be found in Sect. 4.2.1.4 of [Leroy and Rancoita (2011)]).

NIEL for compounds can be determined by means of Bragg's rule, i.e., the overall NIEL in units of MeV cm2/g is obtained as a weighted sum in which each material contributes proportionally to the fraction of its atomic weight. For instance, in case of a GaAs medium ones obtains (e.g. see [Jun et al. (2009)] and Eq. (2.20) at page 15 in [ICRUM (1993)]):

where and AGa [AAs] are the NIEL (in units of MeV cm2/g) and the atomic weight of Gallium [Arsenic], respectively.

NIEL Dependence in GaAs Solar Cells

Baur et al. (2014) studied the reduction of solar-cells' maximum power resulting from irradiations with electrons and protons in both GaAs single junction and GaInP/GaAs/Ge triple junction solar cells. The results obtained by them indicate how:
  1. the dominant radiation damaging mechanism is due to atomic displacements,
  2. the relative maximum power degradation is almost independent of the type of incoming particle, i.e.,
  3. to a first approximation, the fitted semi-empirical function expressing the decrease of maximum power depends only on the absorbed NIEL dose, and
  4. the actual displacement threshold energy value (Ed =21 eV) accounts for annealing treatments, mostly due to self-annealing induced effects.
Thus, for a given type of solar cell, a unique maximum power degradation curve can be determined as a function of the absorbed NIEL dose. The latter expression allows one to predict the performance of those solar cells in space radiation environment. In [Baur et al. (2014)], the absorbed NIEL dose in units of MeV/g - imparted by particles with kinetic energy E - was obtained (e.g., see Eq. 4.150 at page 432 in [Leroy and Rancoita (2011)] as:

where F is the fluence in cm-2 of traversing particles (electrons or protons in the current investigations) and is the corresponding displacement mass-stopping power of the medium.

NIEL Tables for Electrons and Protons in GaAs

The values of NIEL in units of MeV cm2/g for electrons in GaAs obtained using the screened relativistic approach (discussed above) are available in Appendix 1 of [Baur et al. (2014)] (the table will be also available in Appendix of [Leroy and Rancoita (2015)]). These data are shown in Figure 1 for Ed =10, 21 and 25 eV.


Figure 1. NIEL in units of MeV cm2/g for electrons in GaAs as a function of the displacement threshold energy.

Baur et al. (2014) also determined NIEL for protons in GaAs: the table reporting these results is available in Appendix 1 therein (the table will be also available in Appendix of [Leroy and Rancoita (2015)]). These data are shown in Figure 2 for Ed =10, 21 and 25 eV. For proton energies lower than 200 ,keV, it was used the 4-terms analytical approximation of the ZBL cross section derived by Messenger et al. (2004)] and the Thomas--Fermi screening length (Eq. (2.73) at page 95 of [Leroy and Rancoita (2011)] as suggested by ICRUM (1993) for incident protons. Using such a cross section, the so obtained nuclear stopping powers typically agrees with those found in [ICRUM (1993)] within a few percent. For protons, the NIEL contribution resulting from hadronic interactions was estimated by Baur et al. (2014) using the one from Jun et al. (2003) at Ed = 10 eV. For displacement threshold energies of 21 and 25 eV, it was linearly reduced with respect to one found at 10 eV by the same amount found for the Coulomb contributions, i.e., by about 7.7 and 9.5%, respectively.


Figure 2. NIEL in units of MeV cm2/g for protons in GaAs as a function of the displacement threshold energy.

References

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NIEL dose dependence for solar cells irradiated with electrons and protons, in press, Proc. of the 14th ICATPP, September 23--27 2013, Villa Olmo, Como, Italy, S. Giani, C. Leroy, L. Price, P.G. Rancoita and R. Ruchti, Editors, World Scientific, Singapore, 698-713 (ebook version also available); http://arxiv.org/abs/1312.0402.
M.J. Boschini et al. (2010),
Geant4-based application development for NIEL calculation in the Space Radiation Environment, Proc. of the 11th ICATPP Conference, October 5-9 2009, Villa Olmo, Como, Italy, World Scientific, Singapore, 698-708, IBSN: 10-981-4307-51-3; http://www.worldscientific.com/worldscibooks/10.1142/7764 .
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Last update: Mon, 12 Mar 2018