Analytically Approximated Partition Functions
In the framework of the screened relativistic treatment the nuclear scattering occurs elastically, thus, the recoil nucleus does not modify its mass and atomic number. Furthermore, the partition of the deposited energy by the recoil nucleus with kinetic energy ER (e.g., see [Lindhard, Nielsen, Scharff and Tomsen (1963)]) is treated following Robinson’s analytical approximation (e.g., see [Robinson (1972)], [Norgett, Robinson and Torrens (1975)], [Jun (2001)], Sects. 4.2.1-188.8.131.52 in [C. Leroy and P.G. Rancoita (2011)], Chapter 7 in [C. Leroy and P.G. Rancoita (2016)]).
By means of the previously mentioned expression, one finds the energy Ede deposited by the recoil nucleus for creating atomic displacements (e.g., see Eq. (4.123) at page 405 in [C. Leroy and P.G. Rancoita (2011)]) and the energy Eion (= ER - Ede) deposited by ionizing processes.
Akkerman and Barak (2006) suggested a modified expression of the Robinson’s analytical approximation. It has to be remarked that Jun et al. (2009) determined how - for electrons in a silicom medium - the usage of the Norgett--Robinson--Torrens expression (for a recoil silicon with kinetic energy larger than about 30 keV) in [Summers et al. (1993)] or, alternatively, the one modified by Akkerman and Barak (2006) yields similar NIEL values.
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 .
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.
J. Lindhard, V. Nielsen, M. Scharff and P.V. Thomsen (1963), Kgl. Danske Vidensk. Selsk. Mat.-Fys. Medd. 33 (no. 10), 1.
M.J. Norgett, M. Robinson and I.M. Torrens (1975), Nucl. Engin. and Des. 33, 50.
M. Robinson, (1972), The dependence of radiation effects on the primary recoil energy, Proc. Int. Conf. Radiation-Induced Voids in Metal, Albany, NY, 397–429.