Calculation of the total mass stopping power for electrons in some human body tissues in the energy range 0.01-1000 MeV

The mass collision energy loss (dE/dX), the mass radiative energy loss (Srad/) and the total mass stopping power of electrons in the energy range of 0.01 MeV up to 1000 MeV has been calculated for Lung, Urea and Skin. The results of the present work for the mass collision stopping power of electrons in Lung, Urea and Skin are in excellent agreement with the standard results given by ESTAR program, where the maximum percentage error between the present calculated values and that of ESTAR program in Lung tissue, Urea and Skin tissue is 0.27%, 0.3% and 0.8% respectively. The mass radiative energy loss of electrons in the same energy range is also calculated using a modified equation, and the results are found to be in very good agreement with the standard published values. The employed modified equation used to calculate the mass radiative energy loss of electrons is valid in the energy range of electrons from 0.01 MeV up to 1000 MeV and gives accurate results. As the results of total stopping power calculation are concerned, they are found in excellent agreement with the published results, where the error is less than 1%.


Introduction
The stopping power and the range of electrons in a media are characteristics of the stopping media. The knowledge of such physical quantities, since such quantities are very important in the electron radiotherapy and in the calculation of radiation dose. When the energy of the electrons is more than the ionization energy of the molecule, the electrons shall interact with biological cells by ionization and excitation with atomic electrons. During the passage of electrons through the human tissues, electron can emit electromagnetic radiation which is known as bremsstrahlung. The energy loss due to ionization and excitation is known as collisional energy loss. The calculation of electron's range and the energy loss in biomedical materials was the subject of many researches [1][2][3][4][5]. The calculation of energy loss due to emission of bremsstrahlung is more complicated than the calculation of energy loss due to ionization and excitation, and there is no exact formula used in such calculations, but there are only an approximate equations [6][7][8], that will be given for the calculation of radiative loss for electrons only, since the energy loss by bremsstrahlung radiation is important, which is less importance for heavy charged particles. Adwan N. H. Al-Ajili et al., [9] had been used an approximate equation to calculate the energy loss due to emission of bremsstrahlung in the energy range 0.1-10 MeV, The result obtained by the approximate equations didn't give appropriate results, and it was found that a high error to calculate the results when it compared with the standard values due to Berger and Seltzer [1] and had been showed an error up to 18.5 % at 10 MeV with that standard values. The Error could be reached more higher at energies near 1000 MeV. The limitation of the relation is that it is not valid for such calculation and does not give any idea for high energy region.
Aman Allah et al. [10], had been used the same approximate formula as reference [9]. It can be recognized from their graphs that, the error is high in the energy range 0.1-10MeV, where the effect of the energy loss by emission of bremsstrahlung had no importance effect, since the energy loss by this process was considerable above 10 MeV.
The present study is undertaken to calculate the mass collisional energy loss with minimum possible error and to modify the approximate formula to calculate the energy loss by bremsstrahlung in intermediate and high energy region.

Mass collisional of electrons stopping powers
Electron loses its energy by ionization and excitation by the orbital electrons in the medium. Mass stopping Power (dE/dX) can be defined as the rate of energy loss per unit path length of an electron or positron by excitation and ionization which was known as "collisional energy loss". The mass collision stopping powers for electrons and positrons are given by [8]: is for shell correction accounting for non-participation of K-shell electrons at low energies and can be calculated by the formula [8];  is for the polarization or density effect correction in condensed media [11] 4.6052X+C X ˃X 1 4.6052X + a(X1-X) m + C X 0 < X < X l (X) = 0 for non -conducting materials X < X o (X o ) 10 2(x-xo) for conducting materials X < Xo.
The parameters X o , Xi, a, m, and C Parameters for elements and many compounds and mixtures were published [34,35]. : is the Fine structure constant = 1/137 : is the density of the medium (g/cm 3 )

The mass radiative stopping power
Electrons and positrons passing through matter undergo acceleration (deceleration) in the interaction with the strong electric field of the nucleus and the atomic electrons. This leads to emission of x-ray and gamma radiation, the process which is known as Bremsstrahlung. The total power radiated, in non relativistic limits, is given by Larmor formula [12]: ( ) in SI unite relativistic generalization is given by the Liénard-Wiechert potentials [13] ( ) in cgs unite where a: is the acceleration. e: is the electron charge, and c is the speed of light. The power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as: (6) where The acceleration of electron under the effect of electric field is given by: which means, ( ) The energy flux density (S) of the radiation field can be found by computing its Pointing vector: It can be seen from equation 8b that the intensity of radiation produced is proportional to (Z/M) 2 . So it be noticed that a relatively unimportant energy loss mechanism for electrons below about 10 MeV in low-Z materials and it is completely negligible for heavy charged particles.
In a similar way to that for collision losses a radiative stopping power (dE/dX) rad (or sometimes it is written as S rad ), and also a mass radiation stopping power (S rad /) can be define. The mass radiative stopping power is the rate of energy loss by electrons or positrons that results in the production of bremsstrahlung. The Bethe-Heitler theory leads to the following formula for the mass radiative stopping power [14]: where B is a slowly varying function of Z and E, varying between 5.33 and 15 for energies in the range from less than 0.5 MeV to 100 MeV. Reference [6]: give the following formula for the mass radiative stopping power at high energies (complete screening:  » 1/ Z 1/3 ): Leo [7] gives another approximate equations to calculate the energy loss by Bremsstrahlung at low and high energy region.
where E is the energy of the electron.
The fitting parameters a, b, c and d are given in following The total stopping power for light charged particles is equal to the sum of both collisional and bremsstrahlung stopping powers [1,8].

Calculations
The mass collision energy loss of electrons in the energy range of electrons from 0.01 MeV up to 1000 MeV has been calculated for Lung, Urea and Skin using Eq. (1), and the correction for k-shell accounting for non-participation of K-shell electrons at low energies and the polarization or density effect correction ( ) in condensed media (-parameter) have been considered. The results are presented in Table 1.
The mass radiative energy loss of electrons in the same energy range is calculated by using of Eq. (13b), and the results are compared with most new published results and are given in Table 2. The total mass stopping power is obtained as the sum of both collisional and bremsstrahlung stopping powers as illustrated in Eq. (14), where the results are compared with the standard published results and are presented in Table 3.

Results and discussion
The results of the present work for mass collision stopping power of electrons in Lung, Urea and Skin were in excellent agreement with the standard results of reference [15]. The maximum percentage error between the present work values and that of ESTAR program in Lung tissue, Urea and Skin tissue is 0.27%, 0.3% and 0.8% respectively.
The mass radiative energy loss of electrons in the same energy range is calculated using Eq.(13b). The Maximum percentage error between the present work values and that of ESTAR program in Lung tissue, Urea and Skin tissue is 6.2%, 6.6%, and 5.86% respectively, this error appear at low energy region and it was not important and could be neglected in that energy region. It can be noticed that the results are agreement with the energy range of electrons (0.06-1000 MeV), where the error was very low and less than 1%.

Conclusions
It can be concluded that, the employed equation used in the calculation of the mass radiative energy loss of electrons is valid in the energy range of electrons from 0.06 MeV up to 1000 MeV gives accurate results. When the total stopping power was concerned, one can found that the results are in good agreement with the published results, where the error is less than 1%.