Calculation of Stopping Power for protons in Biological Human parts View PDF

The loss of energy for particles in the matter studied for many years as result for collision phenomena, where the charged particle loss its kinetic energy in different ways of reaction dependent on the velocity and nuclear charge for incident charged particle and also on the nature of target particles [1].

A charged particle moving rapidly through matter loses energy primarily by ionizing and exciting atoms. An important goal of theoretical understanding of these processes is the prediction of the average rate of energy loss of the particle per unit distance traveled as a function of the particle’s energy. This fundamental quantity is called the stopping power of the material for that particle [2].

The electronic stopping term governs the energy losses caused by the electronic interactions, which can be further divided into several different contributions depending on the nature of the interaction. Hence, the stopping power can be written as [3].

It is often denoted by the symbol dE/dx and expressed in the units MeV/cm. Dividing the stopping power by the density ρ of the material gives the closely related mass stopping power,- dE/ρdx, which can be expressed in [2].

The stopping power depends on the type and energy of the particle and on the properties of the material it passes. Since the production of an ion pair (usually a positive ion and a negative ion (electron) requires a fixed amount of energy, the density of ionization along the path is proportional to the stopping power of the material. Stopping power refers to the property of the material while energy loss per unit path length describes what happens to the particle [4]. The stopping power of light and heavy particles depends mainly and directly on the projectile's charge and speed, and the increase in the particle's energy leads to a decrease in the linear stopping power. This means that it is inversely proportional to the projectile's energy [5].

In 1913, Niels Bohr derived nuclear formula for the Stopping power of heavy charged particles, based on classical physics. In 1934, Beth developed a formula similar to the Bohr formula to calculate the Stopping power by relying on quantum theory within the limits of high [6]. It has been shown that, depending on the value of the parameter η = / v , where is the atomic number of a bare ion moving with velocity v, the energy losses can be described classically if η>>1, and the quantum perturbation approach is applicable at the opposite limiting case of small η [7]. From the collision principle of Bohr  classic theory and the principle of momentum and transmitted energy to quantum

Beth’s Theory, the final Stopping power for Beth – Bloch is [8].

<I> is the rate of ionization potential of the electrons, Y( Z₁) relative stopping function in terms of the atomic number of the falling particle ,which is a small amount containing Bloch  correction. The rule says the mass stopping power for the substance containing several elements is taken to be equal to the weighted sum of the mass stopping power of the constituent atoms [9].

There has never been a detailed calculation of the stopping of low- energy light ions in solids which has been shown to be even modestly accurate for abroad range of targets. This may be because there is no accurate way to calculate the dynamic charge distribution of light ions inside solids [10]. Ziegler was able devise a formula for calculating the ability to Bloch active light ions (H, He, Li) at energies above 1MeV. Experimental proton stopping power data are summarized by Anderson and Ziegler. Reliable data are available for many elements over a wide range of energies [8]. However, in order to obtain values for all elements over a continuous range of proton energies, the authors fit curves through the available experimental data to generate coefficients for use in a semi-empirical parameterization of the stopping power as a function of proton energy E (keV) and the target atomic number Z₂. S is assumed to be proportional to E^0.45 for E <25 keV, except for Z₂ 6, where it is proportional to E^0.25; for 25 keV E 10000 MeV, Ziegler F, et al. (1977) [11]. 

and the coefficients Ai for each Z₂, available from TRIM, For 10 MeV E 2 GeV, TRIM2 includes four additional coefficients A₉ through A₁₂, for use in the parameterization [11].

Stopping power Was calculated by Ziegler and applied to the elements only in the three equations (4,5, and 6) through our research, for the first time the stopping power was calculated on six different types of parts of the human (Water, Bone, Muscle, Tissue, Trachea, Adipose tissue) and with an energy range (1-100000)KeV. The program has been implemented in a language Matlab 2012.

Ziegler equations

By program SRIM 2013 which was implemented in Matlab program we have used the major Ziegler equation in low and high energies where it is represented as follows:

S1 = S_LOWS_HIGH  / (S_LOW+S_HIGH)       (8)

Where: (S1) Represent stopping power total, (S_LOW) Low energy Stopping power, (S_HIGH) High energy stopping power

Zieglers equation of stopping power are divided into three regions [8].

• The first region is the low energy region with energy range of (1-10) KeV and is shown in the following formula

• The second region of the Ziegler equation for the stopping power with energy range of (10-1000) KeV divided into two regions is shown in the following formulas

1. Low energy equation in the second region as follows

1. High energy equation in the second region as follows

1. The third region of the Ziegler equation for the stopping power with energy range of ( 1000 – 100000) KeV is shown in the following formula

From A₁ to A₈ is constant as shown in table 1 of the water and oxygen.

Table 1: Values of constants for protons.

Table 2: Correlation coefficient values when comparing the protons stopping power resulting from Ziegler equation and SRIM2013.

Fitting equations

At low energies E < 400 keV, stopping power theory is mostly evaluated using the Thomas-Fermi statistical model of the atom. The energy loss processes are divided up into electronic energy losses and elastic energy losses to the screened nuclei. For hydrogen projectiles, the nuclear stopping power is very small for all energies of interest here. (~1-2% at 10 keV and increasing in relative importance with decreasing energy). Electronic stopping was found to increase in high energies and proportional to the speed projectile, one basic conclusion of Andreson and Ziegler study is that the entire experimental material considered does not give indication of any influence of projectile mass on stopping powers [11]. By program SRIM 2013 which was implemented in MATLAB program, we have used a curved fitting with Ziegler equations we concluded that results are very good as in the figures (Figure 1-Figure 6). Table 1 shows values of constant H and O and Table 2 shows values of constant.

Through the research results that we reached to calculate the total stopping power by using Ziegler equations and with an energy range (1-100000) KeV, it was found that the nuclear stopping power decreases at low energies less than 400KeV and decreases with the decrease in energy, while the electronic stopping power is greater than the nuclear stopping power at these energies and the more they increase energy increases the irritation of the atoms, and consequently, the electronic stopping power increases at high energies greater than 400KeV, where it was found through calculations that they depend on the speed projectile and undepend on the mass projectile. By calculating the Ziegler equations to calculate the total stopping power in the studies tissue, it was found that results of the curve match are close to the practical results.

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