Continuous Slowing Down Approximation ( CSDA ) ranges of electrons for biomedical materials

Introduction Stopping media are characterized by their stopping power (SP), the inelastic mean free path (IMFP), the continuous slowing down approximation-range (CSDA-range) (R) and the energy straggling parameter. These physical quantities are important for application such as radiation biology, electron beam lithography, and chemical analyses of surface regions of a solid and in calculation of radiation dose in radiotherapy. The principal characteristic of ionizing radiation is that it has sufficient energy to break any chemical bond and to cause ionization in all materials. Whenever the energy of a particle exceeds the ionization potential of a molecule, a collision with the molecule might lead to ionization. The knowledge of the mean free path and CSDA-range of electrons is important, especially at low energies; in line with this, number of authors has made associated studies of biological compounds [1-5]. For electrons of low energies, the inelastic interaction characteristics, the stopping power, the mean free path and the CSDA-range cannot be obtained directly from experiments or from Bethe’s SP theory, the latter giving accurate SPs at energies larger than 10 keV. At lower energies, the theory is, in general, invalid. For low-energy electrons, a method has been used to estimate the mentioned characteristics, based on the use of the complex dielectric function εq, ωħqand ħω being the momentum and energy transfer, respectively. As mentioned by Akkerman and Akkerman [3] restrictions in these theories prevent their use for a wide range of non-organic and organic materials [3]. To calculate the mean free path and the CSDA-range, another method is to make use of the inelastic differential cross section (IDCS) suggested by [6] with the generalized oscillator strength (GOS). For this, the GOS has to be calculated from matrix elements that involve numerical integration of atomic wave functions. This calculation is too complicated. During the last few years, a number of optical data models have been proposed to compute the inelastic scattering of electrons, avoiding the calculation of the GOS from matrix elements. In recent years, [5] have calculated the IMFP and the CSDA-range in DNA (thymine–adenine or cytosine–guanine) for low and intermediate energy ranges. These calculations were also studied for liquid water, guanine and organic molecules in the energy range 20 eV–10 keV [1,5]. In this paper, we propose a method to obtain the CSDA ranges for electrons at intermediate energy (20–50000 keV) in terms of least square method. Results obtained by this procedure are compared with the available data, above 20 keV, derived from the Born–Bethe approximation. Previous empirical relations for CSDA Ranges The exact knowledge of range of electrons and positrons in several media is of practical interest for many applications in nuclear physics, radiation protection and semiconductor detector fabrication. The main effects produced by the passage of electrons through matter are: 1. Non radiative collision process and 2. Radiative collision process. Therefore the total energy loss during the passage of electron will be the sum of these two losses. In determining CSDA ranges fluctuations in energy losses are neglected and electrons are assumed to loss energy continuously along their track with a mean energy loss per unit path length given by the stopping power. Nelms [7] has calculated CSDA ranges using collisions loss expressions. Using collision loss expressions the following equation was solved numerically by Simpson’s 1/3 rd rule.