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Study of the electronic stopping power of proton in different materials according to the Bohr and Bethe theories

  • Shahla A. S. Alruhaimi , Widad Hamza Tarkhan EMAIL logo and Ahlam Habeeb Hussien
Published/Copyright: February 22, 2025
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Open Engineering
From the journal Open Engineering Volume 15 Issue 1

Abstract

In this research, a theoretical study was conducted to calculate the electronic stopping force using the Bohr and Bethe equations for charged particles (protons) located on the elements (13Al, 14Si, 29Cu, 32Ge, 47Ag, 78Pt, 79Au, and 82Pb) within the energy range of [0.01–1000] MeV. When using the Bethe equation to calculate the proton stopping power in the target elements, it was found that the greatest value was at the energy of 0.5 MeV, and a cutoff occurs at energy of 0.1 MeV in Al and 0.5 MeV in Si, Ag, Cu, Ge, Ag, Pt, Au, and Pb, but while using the Bohr equation, it is at an energy of 0.5 MeV in the elements Al, Si, Ag, Cu, Ge, Ag, Pt, Au, and Pb, and the cut occurs at an energy of 0.1 MeV in Al and 0.5 MeV in Si, Ag, Cu, Ge, Ag, Pt, Au, and Pb. The process varies in energy loss due to the nature and type of incident particles; these data can be used in many industrial applications. The stopping power of the Bohr and Bethe equations for the proton shell in these media was compared after programming them in MATLAB and using data from the SRIM 2012 program and data obtained from the PSTAR program. A semi-empirical equation was also extracted as a factor to multiply the Bohr equation so that its results were close to the PSTAR results for the electronic stopping force of the proton in the elements Al and Si within the energy range of 1–1,000 MeV, and the results were consistent with it. The experimental equations were extracted using curve fitting using the MATLAB language. For other elements within the same energy range, the results were consistent with the PSTAR results.

Keywords: electronic stopping power; Bohr theory; Bethe theory; SRIM; PSTAR; experimental equation; mean ionization potential; Bohr velocity; curve fitting for MATLAB language

1 Introduction

The energy of heavy charged particles, such as protons, is gradually transferred to the material until it stops, and the transition is primarily made through inelastic collision with the electrons of the material atoms that the protons pass through. These collisions result in the excitation of these atoms (i.e., the transition of one of the atom’s electrons from its orbit to another orbit of higher energy) or its ionization, that is, the production of the s electron in the atom [1].

Both elastic and inelastic interactions with electrons and nuclei can result in a deceleration of the movement of a heavy ion through materials. As in the case of charged particles being deflected by an electric field, elastic interactions retain the system’s overall momentum and kinetic energy, when it comes to inelastic interactions, the overall momentum is preserved, but the kinetic energy is not, as seen in the target atoms’ excitation, for instance. Atomic collisions that are both elastic and inelastic are known as nuclear stops, whereas collisions involving electrons are known as electronic stopping power [2].

The stopping power of a medium is defined as the average amount of energy lost by charged particles per unit path length [3,4]. Collisions and radiative energy combine to form the stopping power. The first is the most important and is caused by the collision interaction of incident particles and atomic electrons. To reduce reliance on medium density, mass collision stopping power is commonly employed (ρ) [4].

The SRIM-2012 program can be used to calculate total stopping power, which uses a quantum mechanical treatment of ion-atom collisions to calculate the stopping power and range of ions (10 eV to 2 GeV/amu) in the matter. Within the SRIM manual, the term “ion” is used to denote the moving atom, while all target atoms are referred to as such. Ziegler and Biersack described the calculation in detail. Stopping power is the average unit of energy loss experienced by charged particles per unit path length in the medium [3,5].

The energy loss can be expressed as S(E) = −dE/dx, expressed with the unit of MeV/m or sometimes expressed by unit MeV cm2/g because of the energy loss per area density dx = ρ ds, where ρ is the density in g/cm3 and ds is the length in cm. Stopping power depends on the charge of a projectile (particle) and the nature of the target (material) [6].

The method of calculating the speed, charge and mass of the falling particle as well as the properties of the target material depends on the speed, charge and mass of the falling particle (i.e. the target material) so that the process differs in energy loss due to the nature and type of the falling particle. Bethe (in 1930) used the first-born approximation to calculate the electronic stopping power using quantum mechanics (the electronic energy loss) [7,8].

2 Classical Bohr theory

The famous stopping formula of Bohr from 1913 provided the first theoretical model for energy loss for charged particles infiltrating matter [9]. This model investigated the progressive loss of particle kinetic energy to the medium’s particles and the many collisions that lead to energy loss, including inelastic collisions. This happens at high particle energies when falling heavy ions collide with atoms of the material medium’s electrons. Heavy ions interact with atomic nuclei in addition to the elastic collisions at low particle energies. Additionally, inelastic collisions play a significant role in the process of losing the electronic stopping power (electronic collision) [10], which can be written as follows [11]:

(1) S = 4 π Z 1 2 Z 2 e 4 m v 2 ln Cm v 3 Z 1 e 2 w ,

where:

(2) L Bohr = ln Cm v 3 Z 1 e 2 w ,

(3) S = 4 π Z 1 2 Z 2 Ne 4 m v 2 L Bohr .

3 Quantitative Bethe theory

The Quantum theory proposed by Bethe is a quantum mechanics approach to halting the motion of a charged particle. It shares a formula with Bohr’s theory, with the key distinction being the substitution of the logarithm parameter in Bohr’s equation with the parameter (2 mv 2/I), where (I) represents the decay rate. In instance 11 of resonance, the ionization potential is equal to 3.6 eV, the Bethe formula is true when the velocity of the incoming particle is more significant than Bohr velocity, and the Bethe formula is [12,13] given by

(4) S = 4 π Z 1 2 Z 2 Ne 4 m v 2 ln 2 mv I − ln 1 − v 2 c 2 − v 2 c 2 ,

(5) S = 4 π Z 1 2 Z 2 Ne 4 m v 2 L Bethe .

4 Results and discussion

The electronic stopping power of charged protons striking the materials (13Al, 14Si, 29Cu, 32Ge, 47Ag, 78Pt, 79Au, and 82Pb) in the energy range of 0.01–1,000 MeV was theoretically calculated using the Bohr and Bethe equation. The results were compared with the information from the PSTAR websites and the SRIM 2012 program and results are represented in Tables 1–4, as shown in Figure 1.

Table 1

Electronic stopping power of a proton in Al and Si (MeV cm2/mg)

P in Al P in Si
E PSTAR S Bohr S Bethe SRIM E PSTAR S Bohr S Bethe SRIM
0.01 2.80 × 10−1 −2.56 × 101 −1.69 × 101 2.80 × 10−1 0.01 3.32 × 10−1 −2.72 × 101 −1.81 × 101 3.13 × 10−1
0.05 4.66 × 10−1 −1.07 × 100 −6.76 × 10−1 4.66 × 10−1 0.05 5.47 × 10−1 −1.23 × 100 −8.20 × 10−1 5.38 × 10−1
0.1 4.50 × 10−1 3.38 × 10−1 2.43 × 10−1 4.50 × 10−1 0.1 5.04 × 10−1 2.92 × 10−1 1.93 × 10−1 4.96 × 10−1
0.5 2.51 × 10−1 4.73 × 10−1 3.19 × 10−1 2.51 × 10−1 0.5 2.57 × 10−1 4.79 × 10−1 3.19 × 10−1 2.55 × 10−1
0.8 1.97 × 10−1 3.69 × 10−1 2.48 × 10−1 1.97 × 10−1 0.8 2.01 × 10−1 3.76 × 10−1 2.51 × 10−1 1.99 × 10−1
1 1.75 × 10−1 3.24 × 10−1 2.18 × 10−1 1.75 × 10−1 1 1.75 × 10−1 3.30 × 10−1 2.20 × 10−1 1.76 × 10−1
2 1.11 × 10−1 2.06 × 10−1 1.38 × 10−1 1.11 × 10−1 2 1.12 × 10−1 2.11 × 10−1 1.40 × 10−1 1.12 × 10−1
5 5.74 × 10−2 1.06 × 10−1 7.09 × 10−2 5.74 × 10−2 5 5.82 × 10−2 1.09 × 10−1 7.24 × 10−2 5.87 × 10−2
10 3.40 × 10−2 6.21 × 10−2 4.16 × 10−2 3.40 × 10−2 10 3.46 × 10−2 6.39 × 10−2 4.26 × 10−2 3.48 × 10−2
20 1.98 × 10−2 3.59 × 10−2 2.40 × 10−2 1.98 × 10−2 20 2.02 × 10−2 3.70 × 10−2 2.47 × 10−2 2.03 × 10−2
40 1.15 × 10−2 2.07 × 10−2 1.39 × 10−2 1.15 × 10−2 40 1.17 × 10−2 2.14 × 10−2 1.43 × 10−2 1.18 × 10−2
60 8.35 × 10−3 1.51 × 10−2 1.01 × 10−2 8.35 × 10−3 60 8.56 × 10−3 1.56 × 10−2 1.04 × 10−2 8.60 × 10−3
80 6.71 × 10−3 1.21 × 10−2 8.13 × 10−3 6.71 × 10−3 80 6.89 × 10−3 1.25 × 10−2 8.36 × 10−3 6.91 × 10−3
100 5.69 × 10−3 1.03 × 10−2 6.89 × 10−3 5.69 × 10−3 100 5.84 × 10−3 1.06 × 10−2 7.08 × 10−3 5.86 × 10−3
200 3.53 × 10−3 6.34 × 10−3 4.28 × 10−3 3.53 × 10−3 200 3.63 × 10−3 6.55 × 10−3 4.40 × 10−3 3.64 × 10−3
300 2.78 × 10−3 4.95 × 10−3 3.36 × 10−3 2.78 × 10−3 300 2.85 × 10−3 5.11 × 10−3 3.46 × 10−3 2.86 × 10−3
400 2.40 × 10−3 4.24 × 10−3 2.91 × 10−3 2.40 × 10−3 400 2.46 × 10−3 4.38 × 10−3 2.99 × 10−3 2.47 × 10−3
600 2.02 × 10−3 3.52 × 10−3 2.46 × 10−3 2.02 × 10−3 600 2.08 × 10−3 3.64 × 10−3 2.53 × 10−3 2.08 × 10−3
800 1.85 × 10−3 3.17 × 10−3 2.25 × 10−3 1.85 × 10−3 800 1.90 × 10−3 3.27 × 10−3 2.32 × 10−3 1.90 × 10−3
1,000 1.75 × 10−3 2.96 × 10−3 2.14 × 10−3 1.75 × 10−3 1,000 1.80 × 10−3 3.06 × 10−3 2.20 × 10−3 1.81 × 10−3
Table 2

Electronic stopping power of a proton in Cu and Ge (MeV cm2/mg)

P in Cu P in Ge
E PSTAR S Bethe S Bohr SRIM E PSTAR S Bohr S Bethe SRIM
0.01 1.10 × 10−1 −2.95 × 101 −2.95 × 101 1.12 × 10−1 0.01 1.45 × 10−1 −2.87 × 101 −2.08 × 101 1.45 × 10−1
0.05 1.93 × 10−1 −2.02 × 100 −2.02 × 100 1.96 × 10−1 0.05 2.50 × 10−1 −2.05 × 100 −1.70 × 100 2.49 × 10−1
0.1 2.09 × 10−1 −1.79 × 10−1 −1.79 × 10−1 2.17 × 10−1 0.1 2.46 × 10−1 −2.32 × 10−1 −3.19 × 10−1 2.47 × 10−1
0.5 1.66 × 10−1 3.51 × 10−1 3.51 × 10−1 1.65 × 10−1 0.5 1.59 × 10−1 3.22 × 10−1 1.82 × 10−1 1.58 × 10−1
0.8 1.33 × 10−1 2.90 × 10−1 2.90 × 10−1 1.33 × 10−1 0.8 1.26 × 10−1 2.69 × 10−1 1.59 × 10−1 1.27 × 10−1
1 1.18 × 10−1 2.59 × 10−1 2.59 × 10−1 1.19 × 10−1 1 1.11 × 10−1 2.41 × 10−1 1.44 × 10−1 1.14 × 10−1
2 7.99 × 10−2 1.72 × 10−1 1.72 × 10−1 7.98 × 10−2 2 7.42 × 10−2 1.60 × 10−1 9.86 × 10−2 7.64 × 10−2
5 4.42 × 10−2 9.10 × 10−2 9.10 × 10−2 4.42 × 10−2 5 4.14 × 10−2 8.54 × 10−2 5.37 × 10−2 4.23 × 10−2
10 2.71 × 10−2 5.42 × 10−2 5.42 × 10−2 2.71 × 10−2 10 2.55 × 10−2 5.10 × 10−2 3.24 × 10−2 2.60 × 10−2
20 1.62 × 10−2 3.18 × 10−2 3.18 × 10−2 1.62 × 10−2 20 1.53 × 10−2 2.99 × 10−2 1.91 × 10−2 1.56 × 10−2
40 9.57 × 10−3 1.85 × 10−2 1.85 × 10−2 9.56 × 10−3 40 9.09 × 10−3 1.74 × 10−2 1.12 × 10−2 9.20 × 10−3
60 7.05 × 10−3 1.35 × 10−2 1.35 × 10−2 7.04 × 10−3 60 6.71 × 10−3 1.28 × 10−2 8.23 × 10−3 6.78 × 10−3
80 5.70 × 10−3 1.09 × 10−2 1.09 × 10−2 5.69 × 10−3 80 5.43 × 10−3 1.03 × 10−2 6.64 × 10−3 5.48 × 10−3
100 4.85 × 10−3 9.24 × 10−3 9.24 × 10−3 4.84 × 10−3 100 4.62 × 10−3 8.72 × 10−3 5.64 × 10−3 4.67 × 10−3
200 3.04 × 10−3 5.73 × 10−3 5.73 × 10−3 3.03 × 10−3 200 2.90 × 10−3 5.41 × 10−3 3.53 × 10−3 2.93 × 10−3
300 2.40 × 10−3 4.48 × 10−3 4.48 × 10−3 2.39 × 10−3 300 2.30 × 10−3 4.24 × 10−3 2.79 × 10−3 2.32 × 10−3
400 2.07 × 10−3 3.84 × 10−3 3.84 × 10−3 2.07 × 10−3 400 1.99 × 10−3 3.63 × 10−3 2.42 × 10−3 2.01 × 10−3
600 1.75 × 10−3 3.20 × 10−3 3.20 × 10−3 1.75 × 10−3 600 1.69 × 10−3 3.02 × 10−3 2.05 × 10−3 1.70 × 10−3
800 1.60 × 10−3 2.88 × 10−3 2.88 × 10−3 1.60 × 10−3 800 1.55 × 10−3 2.72 × 10−3 1.89 × 10−3 1.56 × 10−3
1,000 1.52 × 10−3 2.69 × 10−3 2.69 × 10−3 1.52 × 10−3 1,000 1.47 × 10−3 2.54 × 10−3 1.80 × 10−3 1.48 × 10−3
Table 3

Electronic stopping power of a proton in Ag and Pt (MeV cm2/mg)

P in Ag P in Pt
E PSTAR S Bohr S Bethe SRIM E PSTAR S Bohr S Bethe SRIM
0.01 9.40 × 10−2 9.28 × 10−2 −2.34 × 101 9.28 × 10−2 0.01 4.36 × 10−2 −3.17 × 101 −2.45 × 101 3.95 × 10−2
0.05 1.76 × 10−1 1.73 × 10−1 −2.21 × 100 1.73 × 10−1 0.05 8.68 × 10−2 −2.98 × 100 −2.65 × 100 7.74 × 10−2
0.1 1.95 × 10−1 1.94 × 10−1 −5.74 × 10−1 1.94 × 10−1 0.1 1.08 × 10−1 −7.64 × 10−1 −8.43 × 10−1 9.54 × 10−2
0.5 1.34 × 10−1 1.32 × 10−1 1.32 × 10−1 1.32 × 10−1 0.5 8.42 × 10−2 1.83 × 10−1 5.55 × 10−2 7.68 × 10−2
0.8 1.08 × 10−1 1.04 × 10−1 1.27 × 10−1 1.04 × 10−1 0.8 6.93 × 10−2 1.76 × 10−1 7.56 × 10−2 6.54 × 10−2
1 9.60 × 10−2 9.30 × 10−2 1.19 × 10−1 9.30 × 10−2 1 6.29 × 10−2 1.64 × 10−1 7.60 × 10−2 6.08 × 10−2
2 6.41 × 10−2 6.37 × 10−2 8.62 × 10−2 6.37 × 10−2 2 4.56 × 10−2 1.18 × 10−1 6.22 × 10−2 4.48 × 10−2
5 3.67 × 10−2 3.66 × 10−2 4.88 × 10−2 3.66 × 10−2 5 2.76 × 10−2 6.68 × 10−2 3.78 × 10−2 2.73 × 10−2
10 2.30 × 10−2 2.30 × 10−2 3.00 × 10−2 2.30 × 10−2 10 1.80 × 10−2 4.09 × 10−2 2.39 × 10−2 1.78 × 10−2
20 1.41 × 10−2 1.40 × 10−2 1.80 × 10−2 1.40 × 10−2 20 1.13 × 10−2 2.44 × 10−2 1.46 × 10−2 1.12 × 10−2
40 8.44 × 10−3 8.42 × 10−3 1.07 × 10−2 8.42 × 10−3 40 6.91 × 10−3 1.44 × 10−2 8.75 × 10−3 6.86 × 10−3
60 6.26 × 10−3 6.25 × 10−3 7.94 × 10−3 6.25 × 10−3 60 5.17 × 10−3 1.06 × 10−2 6.49 × 10−3 5.14 × 10−3
80 5.08 × 10−3 5.07 × 10−3 6.45 × 10−3 5.07 × 10−3 80 4.22 × 10−3 8.60 × 10−3 5.27 × 10−3 4.20 × 10−3
100 4.34 × 10−3 4.32 × 10−3 5.52 × 10−3 4.32 × 10−3 100 3.62 × 10−3 7.31 × 10−3 4.50 × 10−3 3.60 × 10−3
200 2.74 × 10−3 2.73 × 10−3 3.53 × 10−3 2.73 × 10−3 200 2.31 × 10−3 4.57 × 10−3 2.86 × 10−3 2.30 × 10−3
300 2.17 × 10−3 2.16 × 10−3 2.83 × 10−3 2.16 × 10−3 300 1.84 × 10−3 3.59 × 10−3 2.27 × 10−3 1.83 × 10−3
400 1.88 × 10−3 1.87 × 10−3 2.48 × 10−3 1.87 × 10−3 400 1.60 × 10−3 3.08 × 10−3 1.97 × 10−3 1.59 × 10−3
600 1.59 × 10−3 1.59 × 10−3 2.13 × 10−3 1.59 × 10−3 600 1.36 × 10−3 2.57 × 10−3 1.69 × 10−3 1.36 × 10−3
800 1.46 × 10−3 1.46 × 10−3 1.98 × 10−3 1.46 × 10−3 800 1.26 × 10−3 2.32 × 10−3 1.55 × 10−3 1.25 × 10−3
1,000 1.39 × 10−3 1.39 × 10−3 1.89 × 10−3 1.39 × 10−3 1,000 1.20 × 10−3 2.17 × 10−3 1.48 × 10−3 1.19 × 10−3
Table 4

Electronic stopping power of a proton in Au and Pb (MeV cm2/mg)

P in Au P in Pb
E PSTAR S Bohr S Bethe SRIM E PSTAR S Bohr S Bethe SRIM
0.01 4.67 × 10−2 −3.19 × 101 −2.46 × 101 4.61 × 10−2 0.01 4.87 × 10−2 −3.17 × 101 −2.45 × 101 5.39 × 10−2
0.05 9.08 × 10−2 −3.00 × 100 −2.68 × 100 8.89 × 10−2 0.05 9.70 × 10−2 −3.01 × 100 −2.69 × 100 1.02 × 10−1
0.1 1.09 × 10−1 −7.75 × 10−1 −8.54 × 10−1 1.09 × 10−1 0.1 1.21 × 10−1 −7.88 × 10−1 −8.67 × 10−1 1.22 × 10−1
0.5 8.41 × 10−2 1.82 × 10−1 5.40 × 10−2 8.61 × 10−2 0.5 8.95 × 10−2 1.75 × 10−1 4.85 × 10−2 8.95 × 10−2
0.8 6.92 × 10−2 1.76 × 10−1 7.48 × 10−2 7.03 × 10−2 0.8 7.01 × 10−2 1.70 × 10−1 7.08 × 10−2 7.07 × 10−2
1 6.29 × 10−2 1.64 × 10−1 7.54 × 10−2 6.27 × 10−2 1 6.29 × 10−2 1.59 × 10−1 7.21 × 10−2 6.28 × 10−2
2 4.56 × 10−2 1.18 × 10−1 6.20 × 10−2 4.61 × 10−2 2 4.53 × 10−2 1.16 × 10−1 6.00 × 10−2 4.55 × 10−2
5 2.77 × 10−2 6.68 × 10−2 3.78 × 10−2 2.78 × 10−2 5 2.74 × 10−2 6.55 × 10−2 3.68 × 10−2 2.74 × 10−2
10 1.80 × 10−2 4.10 × 10−2 2.39 × 10−2 1.80 × 10−2 10 1.78 × 10−2 4.02 × 10−2 2.33 × 10−2 1.78 × 10−2
20 1.13 × 10−2 2.45 × 10−2 1.46 × 10−2 1.13 × 10−2 20 1.12 × 10−2 2.40 × 10−2 1.43 × 10−2 1.12 × 10−2
40 6.92 × 10−3 1.45 × 10−2 8.75 × 10−3 6.93 × 10−3 40 6.82 × 10−3 1.42 × 10−2 8.57 × 10−3 6.84 × 10−3
60 5.19 × 10−3 1.07 × 10−2 6.50 × 10−3 5.19 × 10−3 60 5.09 × 10−3 1.05 × 10−2 6.37 × 10−3 5.13 × 10−3
80 4.23 × 10−3 8.61 × 10−3 5.28 × 10−3 4.24 × 10−3 80 4.15 × 10−3 8.47 × 10−3 5.17 × 10−3 4.19 × 10−3
100 3.63 × 10−3 7.33 × 10−3 4.51 × 10−3 3.63 × 10−3 100 3.55 × 10−3 7.20 × 10−3 4.42 × 10−3 3.59 × 10−3
200 2.32 × 10−3 4.58 × 10−3 2.86 × 10−3 2.32 × 10−3 200 2.27 × 10−3 4.50 × 10−3 2.81 × 10−3 2.29 × 10−3
300 1.84 × 10−3 3.60 × 10−3 2.27 × 10−3 1.85 × 10−3 300 1.81 × 10−3 3.54 × 10−3 2.23 × 10−3 1.83 × 10−3
400 1.60 × 10−3 3.09 × 10−3 1.98 × 10−3 1.61 × 10−3 400 1.58 × 10−3 3.04 × 10−3 1.94 × 10−3 1.59 × 10−3
600 1.37 × 10−3 2.58 × 10−3 1.69 × 10−3 1.37 × 10−3 600 1.35 × 10−3 2.53 × 10−3 1.66 × 10−3 1.36 × 10−3
800 1.26 × 10−3 2.32 × 10−3 1.56 × 10−3 1.26 × 10−3 800 1.24 × 10−3 2.28 × 10−3 1.53 × 10−3 1.25 × 10−3
1,000 1.20 × 10−3 2.17 × 10−3 1.49 × 10−3 1.20 × 10−3 1,000 1.19 × 10−3 2.14 × 10−3 1.46 × 10−3 1.19 × 10−3
Figure 1 The electronic stopping power of a proton in Al.
Figure 1

The electronic stopping power of a proton in Al.

In Figure 1, we note that the most significant value of stopping power of the proton in Al when using Bohr and Bethe equations is at energy 0.5 MeV and a cutoff occurs at energy 0.1 MeV, and the most significant value of stopping power of the proton in Al when using SRIM and PSTAR data is at energy 0.05 MeV.

In Figure 2, the most significant value of stopping power of the proton in Si when using Bohr and Bethe equations is at energy 0.5 MeV and a cutoff occurs at energy 0.1 MeV, and the most significant value of stopping power of the proton in Si when using SRIM and PSTAR data is at energy 0.05 MeV.

Figure 2 The electronic stopping power of a proton in Si.
Figure 2

The electronic stopping power of a proton in Si.

In Figure 3, the most significant value of stopping power of the proton in Cu when using Bohr and Bethe equations is at energy 0.5 MeV and a cutoff occurs at energy 0.5 MeV, and the most significant value of stopping power of the proton in Cu when using SRIM and PSTAR data is at energy 0.1 MeV.

Figure 3 The electronic stopping power of a proton in Cu.
Figure 3

The electronic stopping power of a proton in Cu.

In Figure 4, the most significant value of stopping power of the proton in Ge when using Bohr and Bethe equations is at energy 0.5 MeV and a cutoff occurs at energy 0.5 MeV, and the most significant value of stopping power of the proton in Ge when using SRIM and PSTAR data is at energy 0.05 MeV.

Figure 4 The electronic stopping power of a proton in Ge.
Figure 4

The electronic stopping power of a proton in Ge.

In Figure 5, the most significant value of stopping power of the proton in Ag when using Bohr and Bethe equations is at energy 0.5 MeV and a cutoff occurs at energy 0.5 MeV, and the most significant value of stopping power of the proton in Ag when using SRIM and PSTAR data is at energy 0.1 MeV.

Figure 5 The electronic stopping power of a proton in Ag.
Figure 5

The electronic stopping power of a proton in Ag.

In Figure 6, the most significant value of stopping power of the proton in Pt when using Bohr equation is at energy 0.5 MeV and when using Bethe equation is at 0.8 MeV, and a cutoff occurs at energy 0.5 MeV, and the most significant value of stopping power of the proton in Pt when using SRIM and PSTAR data is at energy 0.1 MeV.

Figure 6 The electronic stopping power of a proton in Pt.
Figure 6

The electronic stopping power of a proton in Pt.

In Figure 7, the most significant value of stopping power of the proton in Au when using Bohr equation is at energy 0.5 MeV and when using Bethe equation is at 0.8 MeV, and a cutoff occurs at energy 0.5 MeV, and the most significant value of stopping power of the proton in Au when using SRIM and PSTAR data is at energy 0.1 MeV.

Figure 7 The electronic stopping power of a proton in Au.
Figure 7

The electronic stopping power of a proton in Au.

In Figure 8, the most significant value of stopping power of the proton in Pb when using Bohr equation is at energy 0.5 MeV and when using Bethe equation is at 1 MeV, and a cutoff occurs at energy 0.5 MeV, and the most significant value of stopping power of the proton in Pb when using SRIM and PSTAR data is at energy 0.1 MeV.

Figure 8 The electronic stopping power of a proton in Pb.
Figure 8

The electronic stopping power of a proton in Pb.

We note from the previous figures that the curves of the Bohr and Bethe equations diverge from the PSTAR and ASRIM curves in all energy ranges. We approximated the stopping power values extracted from the Bohr equation to PSTAR values within the power range of 1–1,000 MeV by finding an empirical relationship as a factor multiplied by the power values. The results obtained from the experimental Bohr equation showed good results in the Al and Si elements of equation (6), as shown in Table 5 and in Figures 9 and 10.

(6) S Bohr * = S Bohr × Fact ,

where fact = exp ( ( − 1 / I ) − 0.6 ) and I is mean ionization potential.

Table 5

Results of experimental equation (6) for the electronic stopping power of the proton (MeV cm2/mg)

Al Si
E (MeV) PSTAR S Bohr S Bohr* E (MeV) PSTAR S Bohr S Bohr*
1 1.72 × 10−1 3.24 × 10−1 1.79 × 10−1 1 1.75 × 10−1 3.30 × 10−1 1.82 × 10−1
2 1.09 × 10−1 2.06 × 10−1 1.13 × 10−1 2 1.12 × 10−1 2.11 × 10−1 1.16 × 10−1
5 5.69 × 10−2 1.06 × 10−1 5.81 × 10−2 5 5.82 × 10−2 1.09 × 10−1 5.97 × 10−2
10 3.38 × 10−2 6.21 × 10−2 3.41 × 10−2 10 3.46 × 10−2 6.39 × 10−2 3.51 × 10−2
20 1.97 × 10−2 3.59 × 10−2 1.97 × 10−2 20 2.02 × 10−2 3.70 × 10−2 2.03 × 10−2
40 1.14 × 10−2 2.07 × 10−2 1.14 × 10−2 40 1.17 × 10−2 2.14 × 10−2 1.17 × 10−2
60 8.33 × 10−3 1.51 × 10−2 8.29 × 10−3 60 8.56 × 10−3 1.56 × 10−2 8.55 × 10−3
80 6.70 × 10−3 1.21 × 10−2 6.66 × 10−3 80 6.89 × 10−3 1.25 × 10−2 6.87 × 10−3
100 5.68 × 10−3 1.03 × 10−2 5.64 × 10−3 100 5.84 × 10−3 1.06 × 10−2 5.82 × 10−3
200 3.53 × 10−3 6.34 × 10−3 3.48 × 10−3 200 3.63 × 10−3 6.55 × 10−3 3.59 × 10−3
300 2.77 × 10−3 4.95 × 10−3 2.72 × 10−3 300 2.85 × 10−3 5.11 × 10−3 2.81 × 10−3
400 2.39 × 10−3 4.24 × 10−3 2.33 × 10−3 400 2.46 × 10−3 4.38 × 10−3 2.40 × 10−3
600 2.02 × 10−3 3.52 × 10−3 1.93 × 10−3 600 2.08 × 10−3 3.64 × 10−3 2.00 × 10−3
800 1.85 × 10−3 3.17 × 10−3 1.74 × 10−3 800 1.90 × 10−3 3.27 × 10−3 1.80 × 10−3
1,000 1.75 × 10−3 2.96 × 10−3 1.62 × 10−3 1,000 1.80 × 10−3 3.06 × 10−3 1.68 × 10−3
Figure 9 The electronic stopping power of a proton in Al by experimental equation (6).
Figure 9

The electronic stopping power of a proton in Al by experimental equation (6).

Figure 10 The electronic stopping power of a proton in Si by experimental equation (6).
Figure 10

The electronic stopping power of a proton in Si by experimental equation (6).

In Figure 9, we note the convergence of the results obtained from the experimental equation (6) with the standard results obtained from the PSTAR data for Al in the energy range 0–1,000 MeV.

In Figure 10, we note the convergence of the results obtained from the experimental equation (6) with the standard results obtained from the PSTAR data for Si in the energy range 0–1,000 MeV.

As for the rest of the elements, the results of the Bohr equation were approximated from the results of PSTAR by applying curve fitting to them using the MATLAB language, when dividing the energy range 1–1,000 into three regions: 1–10, 10–200, and 200–1,000 MeV and extracting three equations as shown in Table 6 showed good results as shown in Tables 7–9 and Figures 11–16.

Table 6

Empirical equations using curve fitting for MATLAB language

Cu 1–10 MeV 10–200 MeV 200–1,000 MeV
F1 = a × E1 b+c F2 = (p1 × E2 + p2)/(E2 + q1) F3 = (p1 × E3 + p2)/(E3 + q1)
a = 0.1366 a = 0.1507 p1 = 0.001144
b = −0.4791 b = −0.7453 p2 = 0.3543
c = −0.01847 c = −5.301 × 10−6 q1 = −8.333
p1 = 0.001415
p2 = 0.3603
q1 = 3.832
Ge F1 = a × E1 b+c F2 = a × E2 b+c F3 = (p1 × E3 + p2)/(E3 + q1)
a = 0.1237 a = 0.14 p1 = 0.001117
b = −0.5118 b = −0.7396 p2 = 0.3357
c = −0.01268 c = −4.535 × 10−6 q1 = −7.396
Ag F1 = a × E1 b+c F2 = a × E2 b F3 = (p1 × E3 + p2)/(E3 + q1)
a = 0.1035 a = 0.1215 p1 = 0.001055
b = −0.5296 b = −0.7219 p2 = 0.3148
c = 0.007532 q1 = −8.29
Pt F1 = a × E1 b+c F2 = a × E2 b+c F3 = (p1 × E3 + p2)/(E3 + q1)
a = 0.08183 a = 0.08889 p1 = 0.0008994
b = −0.3478 b = −0.6905 p2 = 0.2933
c = −0.01888 c = −6.119 × 10−5 q1 = 5.762
Au F1 = a × E1 b+c F2 = a × E2 b F3 = (p1 × E3 + p2)/(E3 + q1)
a = 0.06849 a = a = 0.09409 p1 = 0.0009347
b = −0.4637 b = −0.7075 p2 = 0.2429
c = −0.005055 q1 = −15.02
Pb F1 = a × E1 b+c F2 = a × E2 b+c F3 = (p1 × E32 + p2 × E3 + p3)/(E32 + q1. × E3 + q2)
a = 0.0799 a = 0.088 p1 = 0.0009067
b = 0 b = −0.6912 p2 = 0.2705
c = −0.01697 c = −5.331 × 10−5 p3 = 0.6869
q1 = 0.8435
q2 = 0.4511
Table 7

Results of empirical equations using curve fitting for MATLAB language in Table 6

Cu Ge
E (MeV) PSTAR S Bohr S Bohr* E (MeV) PSTAR S Bohr S Bohr*
1 1.18 × 10−1 2.59 × 10−1 1.18 × 10−1 1 1.11 × 10−1 2.41 × 10−1 1.11 × 10−1
2 7.99 × 10−2 1.72 × 10−1 7.95 × 10−2 2 7.42 × 10−2 1.60 × 10−1 7.41 × 10−2
5 4.42 × 10−2 9.10 × 10−2 4.47 × 10−2 5 4.14 × 10−2 8.54 × 10−2 4.16 × 10−2
10 2.71 × 10−2 5.42 × 10−2 2.69 × 10−2 10 2.55 × 10−2 5.10 × 10−2 2.54 × 10−2
20 1.62 × 10−2 3.18 × 10−2 1.63 × 10−2 20 1.53 × 10−2 2.99 × 10−2 1.53 × 10−2
40 9.57 × 10−3 1.85 × 10−2 9.51 × 10−3 40 9.09 × 10−3 1.74 × 10−2 9.14 × 10−3
60 7.05 × 10−3 1.35 × 10−2 6.97 × 10−3 60 6.71 × 10−3 1.28 × 10−2 6.77 × 10−3
80 5.70 × 10−3 1.09 × 10−2 5.65 × 10−3 80 5.43 × 10−3 1.03 × 10−2 5.47 × 10−3
100 4.85 × 10−3 9.24 × 10−3 4.83 × 10−3 100 4.62 × 10−3 8.72 × 10−3 4.64 × 10−3
200 3.04 × 10−3 5.73 × 10−3 3.16 × 10−3 200 2.90 × 10−3 5.41 × 10−3 2.78 × 10−3
300 2.40 × 10−3 4.48 × 10−3 2.39 × 10−3 300 2.30 × 10−3 4.24 × 10−3 2.29 × 10−3
400 2.07 × 10−3 3.84 × 10−3 2.07 × 10−3 400 1.99 × 10−3 3.63 × 10−3 1.99 × 10−3
600 1.75 × 10−3 3.20 × 10−3 1.76 × 10−3 600 1.69 × 10−3 3.02 × 10−3 1.70 × 10−3
800 1.60 × 10−3 2.88 × 10−3 1.60 × 10−3 800 1.55 × 10−3 2.72 × 10−3 1.55 × 10−3
1,000 1.52 × 10−3 2.69 × 10−3 1.51 × 10−3 1,000 1.47 × 10−3 2.54 × 10−3 1.46 × 10−3
Table 8

Results of empirical equations using curve fitting for MATLAB language in Table 6

Ag Pt
E (MeV) PSTAR S Bohr S Bohr* E (MeV) PSTAR S Bohr S Bohr*
1 9.60 × 10−2 2.16 × 10−1 9.60 × 10−2 1 6.29 × 10−2 1.64 × 10−1 6.30 × 10−2
2 6.41 × 10−2 1.48 × 10−1 6.42 × 10−2 2 4.56 × 10−2 1.18 × 10−1 4.54 × 10−2
5 3.67 × 10−2 8.05 × 10−2 3.66 × 10−2 5 2.76 × 10−2 6.68 × 10−2 2.79 × 10−2
10 2.30 × 10−2 4.85 × 10−2 2.30 × 10−2 10 1.80 × 10−2 4.09 × 10−2 1.79 × 10−2
20 1.41 × 10−2 2.87 × 10−2 1.40 × 10−2 20 1.13 × 10−2 2.44 × 10−2 1.12 × 10−2
40 8.44 × 10−3 1.68 × 10−2 8.47 × 10−3 40 6.91 × 10−3 1.44 × 10−2 6.90 × 10−3
60 6.26 × 10−3 1.23 × 10−2 6.32 × 10−3 60 5.17 × 10−3 1.06 × 10−2 5.20 × 10−3
80 5.08 × 10−3 9.94 × 10−3 5.14 × 10−3 80 4.22 × 10−3 8.60 × 10−3 4.25 × 10−3
100 4.34 × 10−3 8.45 × 10−3 4.37 × 10−3 100 3.62 × 10−3 7.31 × 10−3 3.64 × 10−3
200 2.74 × 10−3 5.26 × 10−3 2.65 × 10−3 200 2.31 × 10−3 4.57 × 10−3 2.30 × 10−3
300 2.17 × 10−3 4.12 × 10−3 2.16 × 10−3 300 1.84 × 10−3 3.59 × 10−3 1.84 × 10−3
400 1.88 × 10−3 3.53 × 10−3 1.88 × 10−3 400 1.60 × 10−3 3.08 × 10−3 1.61 × 10−3
600 1.59 × 10−3 2.94 × 10−3 1.60 × 10−3 600 1.36 × 10−3 2.57 × 10−3 1.38 × 10−3
800 1.46 × 10−3 2.65 × 10−3 1.46 × 10−3 800 1.26 × 10−3 2.32 × 10−3 1.26 × 10−3
1,000 1.39 × 10−3 2.48 × 10−3 1.38 × 10−3 1,000 1.20 × 10−3 2.17 × 10−3 1.19 × 10−3
Table 9

Results of empirical equations using curve fitting for MATLAB language in Table 6

Au Pb
E (MeV) PSTAR S Bohr S Bohr* E (MeV) PSTAR S Bohr S Bohr*
1 6.29 × 10−2 1.64 × 10−1 6.34 × 10−2 1 6.29 × 10−2 1.59 × 10−1 6.29 × 10−2
2 4.56 × 10−2 1.18 × 10−1 4.46 × 10−2 2 4.53 × 10−2 1.16 × 10−1 4.52 × 10−2
5 2.77 × 10−2 6.68 × 10−2 2.74 × 10−2 5 2.74 × 10−2 6.55 × 10−2 2.76 × 10−2
10 1.80 × 10−2 4.10 × 10−2 1.85 × 10−2 10 1.78 × 10−2 4.02 × 10−2 1.77 × 10−2
20 1.13 × 10−2 2.45 × 10−2 1.13 × 10−2 20 1.12 × 10−2 2.40 × 10−2 1.10 × 10−2
40 6.92 × 10−3 1.45 × 10−2 6.92 × 10−3 40 6.82 × 10−3 1.42 × 10−2 6.82 × 10−3
60 5.19 × 10−3 1.07 × 10−2 5.19 × 10−3 60 5.09 × 10−3 1.05 × 10−2 5.14 × 10−3
80 4.23 × 10−3 8.61 × 10−3 4.24 × 10−3 80 4.15 × 10−3 8.47 × 10−3 4.20 × 10−3
100 3.63 × 10−3 7.33 × 10−3 3.62 × 10−3 100 3.55 × 10−3 7.20 × 10−3 3.59 × 10−3
200 2.32 × 10−3 4.58 × 10−3 2.22 × 10−3 200 2.27 × 10−3 4.50 × 10−3 2.21 × 10−3
300 1.84 × 10−3 3.60 × 10−3 1.84 × 10−3 300 1.81 × 10−3 3.54 × 10−3 1.81 × 10−3
400 1.60 × 10−3 3.09 × 10−3 1.60 × 10−3 400 1.58 × 10−3 3.04 × 10−3 1.58 × 10−3
600 1.37 × 10−3 2.58 × 10−3 1.37 × 10−3 600 1.35 × 10−3 2.53 × 10−3 1.36 × 10−3
800 1.26 × 10−3 2.32 × 10−3 1.26 × 10−3 800 1.24 × 10−3 2.28 × 10−3 1.24 × 10−3
1,000 1.20 × 10−3 2.17 × 10−3 1.20 × 10−3 1,000 1.19 × 10−3 2.14 × 10−3 1.18 × 10−3
Figure 11 The electronic stopping power of a proton in Cu by curve fitting for MATLAB language from Table 6.
Figure 11

The electronic stopping power of a proton in Cu by curve fitting for MATLAB language from Table 6.

Figure 12 The electronic stopping power of a proton in Ge by curve fitting for MATLAB language from Table 6.
Figure 12

The electronic stopping power of a proton in Ge by curve fitting for MATLAB language from Table 6.

Figure 13 The electronic stopping power of a proton in Ag by curve fitting for MATLAB language from Table 6.
Figure 13

The electronic stopping power of a proton in Ag by curve fitting for MATLAB language from Table 6.

Figure 14 The electronic stopping power of a proton in Pt by curve fitting for MATLAB language from Table 6.
Figure 14

The electronic stopping power of a proton in Pt by curve fitting for MATLAB language from Table 6.

Figure 15 The electronic stopping power of a proton in Au by curve fitting for MATLAB language from Table 6.
Figure 15

The electronic stopping power of a proton in Au by curve fitting for MATLAB language from Table 6.

Figure 16 The electronic stopping power of a proton in Pb by curve fitting for MATLAB language from Table 6.
Figure 16

The electronic stopping power of a proton in Pb by curve fitting for MATLAB language from Table 6.

In Figure 11, we note the convergence of the results obtained from the experimental equations listed in Table 6 by applying a curve fitting to them using the MATLAB language at the three energy ranges 1–10, 10–200, and 200–1,000 MeV with the results obtained from PSTAR data for element Cu.

In Figure 12, we note the convergence of the results obtained from the experimental equations listed in Table 6 by applying a curve fitting to them using the MATLAB language at the three energy ranges 1–10, 10–200, and 200–1,000 MeV with the results obtained from PSTAR data for element Ge.

In Figure 13, we note the convergence of the results obtained from the experimental equations listed in Table 6 by applying a curve fitting to them using the MATLAB language at the three energy ranges 1–10, 10–200, and 200–1,000 MeV with the results obtained from PSTAR data for element Ag.

In Figure 14, we note the convergence of the results obtained from the experimental equations listed in Table 6 by applying a curve fitting to them using the MATLAB language at the three energy ranges 1–10, 10–200, and 200–1,000 MeV with the results obtained from PSTAR data for element Pt.

In Figure 15, we note the convergence of the results obtained from the experimental equations listed in Table 6 by applying a curve fitting to them using the MATLAB language at the three energy ranges 1–10, 10–200, and 200–1,000 MeV with the results obtained from PSTAR data for element Au.

In Figure 16, we note the convergence of the results obtained from the experimental equations listed in Table 6 by applying a curve fitting to them using the MATLAB language at the three energy ranges 1–10, 10–200, and 200–1,000 MeV with the results obtained from PSTAR data for element Pb.

5 Conclusion

Bethe’s mathematical approach for determining the charge, mass, and fundamental properties of the target material of an incident particle. We note that the maximam value of stopping power is at the energy 0.5 MeV and a cutoff occurs at energy 0.1 MeV in Al and 0.5 MeV in Si, Ag, Cu, Ge, Ag, Pt, Au, and Pb, while the that the maximam value of stopping power of proton in the elements Pt, Au at 0.8 MeV, Pb is at 1 MeV. But while using Bohr’s equation, that the maximam value of stopping power is at 0.5 MeV in the elements A l, Si, Ag, Cu, Ge, Ag, Pt, Au and Pb. and acutoff occurs at energy 0.1 MeV in Al and 0.5 MeV in Si, Ag, Cu, Ge, Ag, Pt, Au, and Pb. The process differs in energy loss due to the nature and type of incident particle. The stopping power of a proton in the matter is obtained because it has valuable applications in studying biological effects, radiation damage dosage rates, and energy dissipation at various absorber depths. It is also helpful in the design of detection systems, radiation technology, semiconductor detectors, shielding, and determining the proper target thickness.

Observing the results of the Bohr equation, we find that the stopping power values differ significantly from the PSTAR values. Therefore, semi-empirical equations were proposed, as well as the use of a curve fitting application using the MATLAB language to obtain results that match and are close to the PSTAR data values. Their analysis revealed that the values of the modified Bohr equation correspond to the PSTAR values. Enclosed within the energy spectrum of 1 to 1,000 MeV.

In future research, there will be a study on the Bethe equation, comparing its data with PSTAR and SRIM, and proposing a semi-experimental equation so that its results are compatible with them.

Acknowledgements

We acknowledge the support of the Department of Physics, College of Science, University of Kufa, Iraq.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: Shahla A. S. Alruhaimi, University of Kufa, College of Education for Girls. We, the authors, have accepted responsibility for the full content of this manuscript and agreed to submit it to the journal, reviewed all results, and approved the final version of the manuscript. The experiments were designed by Shahla Abdulsada Kadhim and Widad Hamza Tarkhan and implemented by Ahlam Habeeb Hussein. Simulations were conducted. Shahla Abdulsada Kadhim prepared the manuscript with contributions from all co-authors.

  3. Conflicts of interest: The authors state no conflict of interest.

  4. Data availability statement: Most datasets generated and analyzed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

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Received: 2024-02-16
Revised: 2024-05-23
Accepted: 2024-06-07
Published Online: 2025-02-22

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Keywords for this article
electronic stopping power; Bohr theory; Bethe theory; SRIM; PSTAR; experimental equation; mean ionization potential; Bohr velocity; curve fitting for MATLAB language
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