A Novel Approach to Calculate the Range of High-Energy Charged Particles Within a Medium

Abstract

The determination of energy loss of charged particles as they pass through a medium and consequently the calculation of their range within the medium are of tremendous importance in various areas of physics from both theoretical and practical perspectives. Previous works have derived approximate equations regarding the range of ions within a medium, focusing on providing simple solutions for practitioners in the radiotherapy field that do not require significant computational cost, unlike traditional Monte Carlo methods. These solutions focus on radiotherapy and are limited to specific ions’ initial speeds, which should be up to 0.65c (where c is the speed of light in vacuum). In this paper, solutions for significantly larger initial velocities are explored. A new analytical equation for determining the range of charged particles within a medium for initial velocities between 0.6c and 0.9c is presented. This equation provides excellent results when compared to the accurate numerical solution. Beyond its theoretical and mathematical interest, this solution is also reliable for radiotherapy applications. It provides excellent results for protons with initial energies between 200 MeV and 350 MeV and has the major advantage of being expressed in terms of elementary functions, making its use more straightforward compared to other approaches.

1. Introduction

The Bethe equation serves as a fundamental framework in particle and radiation physics, providing a quantitative description of the energy loss of charged particles as they pass through a material medium [1,2]. The average energy lost by ions per unit distance as they travel through matter is expressed as follows [1]:

$$-{\displaystyle \frac{dE}{dx}}={\displaystyle \frac{4\pi n{z}^2}{m_e}}{\left({\displaystyle \frac{e^2}{4\pi {\epsilon }_0}}\right)}^2\left[{\displaystyle \frac{1}{v^2\left(x\right)}}\right]\left\{\left.\mathrm{l}\mathrm{n}\left[{\displaystyle \frac{2m_e{v}^2\left(x\right)}{I\left(1-{\beta }^2\right)}}\right]-{\beta }^2\right\}\right.$$

(1)

In Equation (1), n represents the electron density of the material; e is the electron charge; $m_e$ is the electron mass; I is the mean excitation potential; z is the charge number (i.e., the multiple of the elementary charge) of the incident particle, and $\beta \equiv v\left(x\right)/c$, where v is the particle’s speed and c is the speed of light in vacuum. It is important to emphasize that a particle’s energy depends on its speed, which is, in turn, a function of distance. Equation (1) can also be rewritten as follows:

$$-{\displaystyle \frac{dE}{dx}}={\displaystyle \frac{A{m}_p}{v^2\left(x\right)}}\left[\ln {\displaystyle \frac{B{v}^2\left(x\right)}{1-{\beta }^2}}-{\beta }^2\right]$$

(2)

where $A={\displaystyle \frac{4\pi n{z}^2}{m_e{m}_p}}{\left({\displaystyle \frac{e^2}{4\pi {\epsilon }_0}}\right)}^2$, $m_p$ is the particle mass (i.e., the mass of the particle travelling through the medium), and $B={\displaystyle \frac{2m_e}{I}}$. An analytical table presenting the values of $A$ and $B$ for protons in various media is provided in reference [2].

It is important to highlight that differential Equation (1) lacks an analytical solution. To derive an approximate solution for the range of charged particles in the medium, Grimes et al. adopted the approach presented below based on the relativistic definition of particle kinetic energy [1]:

$$E={(\gamma -1)m}_p{c}^2, \gamma ={\left(1-{\beta }^2\right)}^{-{\displaystyle \frac{1}{2}}}$$

(3)

According to Grimes et al. [1],

$${\displaystyle \frac{dE}{dx}}={\displaystyle \frac{dE}{dv}}{\displaystyle \frac{dv}{dx}}=m_{p}v\left(x\right){\gamma }^3\left(v\right){\displaystyle \frac{dv}{dx}}$$

(4)

Subsequently, by substituting Equation (4) into Equation (2), it is easily concluded that

$${\displaystyle \frac{A}{v^3\left(x\right)}}\left\{\left.\mathrm{l}\mathrm{n}{\displaystyle \frac{\mathrm{B}{v}^2\left(x\right)}{1-{\beta }^2}}-{\beta }^2\right\}\right.=-{\gamma }^3\left(v\right){\displaystyle \frac{dv}{dx}}$$

(5)

The particle’s range, $R$, is defined as [1,3,4]

$$R=\underset{v_0}{\overset{0}{\int }}{\displaystyle \frac{dx}{dv}}dv=-\underset{0}{\overset{v_0}{\int }}{\displaystyle \frac{dx}{dv}}dv$$

(6)

where $v_0$ is the initial speed of the particle.

By substituting Equation (5) into Equation (6),

$$R={\displaystyle \frac{1}{A}}\underset{0}{\overset{v_0}{\int }}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{v^3}{\ln \frac{B{v}^2\left(x\right)}{1-{\beta }^2}-{\beta }^2}}dv$$

(7)

To provide an approximate solution to Equation (7) in terms of the exponential integral function, Grimes et al. use the following approximation [1]:

$$\ln {\displaystyle \frac{B{v}^2\left(x\right)}{1-{\beta }^2}}-{\beta }^2\cong \ln \left[B{v}^2\left(x\right)\right]$$

(8)

This approach yields excellent results for proton energies up to 300 MeV, which corresponds to $v_0/c\cong 0.65$. For larger initial speeds, there is a non-negligible error in the range calculations, and Grimes et al.’s approach cannot be used. Another interesting approach was presented by Martinez et al. [2], who derived a series solution for range calculation. However, the applicability of this model is limited, as it is valid only for protons with initial energies up to 200 MeV ($v_0/c\cong 0.57$). In this paper, a new approach focused on high-energy charged particles ($v_0/c>0.6$) is proposed. The proposed approach enables range calculation for charged particles in a medium with large initial speeds (up to 0.9c). This paper is organized as follows: In Section 2, a new analytical approach for range calculation, with a focus on large initial speeds, is presented. In Section 3, the results obtained using the proposed analytical equation are compared to the exact numerical results from Equation (7) and the results derived by Grimes et al.’s approximation (based on Equation (8)). In Section 4, the results are discussed, and remarks regarding the applications of the proposed equation are provided.

2. An Approximate Analytical Approach for Range Calculation

Equation (7) can also be written as follows:

$$\begin{gathered} R={\displaystyle \frac{c^4}{A}}{\int }_0^{{\beta }_0}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{\ln \left(B{c}^2\right)+\ln \left({\beta }^2\right)-\ln \left(1-{\beta }^2\right)-{\beta }^2}}d\beta \iff \\ R={\displaystyle \frac{c^4}{A}}{\int }_0^{{\beta }_0}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{\Omega \left(\beta \right)}}d\beta \end{gathered}$$

(9)

where $\beta \equiv v/c$, ${\beta }_0\equiv v_0/c$, and

$$\Omega \left(\beta \right)\equiv \ln \left(B{c}^2\right)+\ln \left({\beta }^2\right)-\ln \left(1-{\beta }^2\right)-{\beta }^2$$

(10)

Considering $m_e=9.109\times {10}^{-31} \mathrm{k}\mathrm{g}$ and $I=75 \mathrm{e}\mathrm{V}$ (this is the mean ionization potential for ${\mathrm{H}}_2\mathrm{O}$), the coefficient B results in $B=1.5163\times {10}^{-13} {\mathrm{s}}^2·{\mathrm{m}}^{-2}$ [1,5]. The speed of light equals to $c=\mathrm{299,792,458} \mathrm{m}·{\mathrm{s}}^{-1}$. The function $\Omega$ can be approximated as a linear function in the domain $0.3\le \beta \le 0.9$, as presented in Figure 1. The linear behaviour does not depend on the B-factor (which depends on the medium through which the charged particles travel), as is evident from Equation (10) (since $\ln \left(B{c}^2\right)$ is a constant term regardless of the exact value of $B$). To simplify the integral given by Equation (9), a linear approximation around the central point of the linear region (which is ${\beta }_L=0.6$) will be considered using a Taylor series expansion:

$$\Omega \left(\beta \right)\cong \Omega \left({\beta }_L\right)+{\displaystyle \frac{1}{1!}}{\displaystyle \frac{d\Omega }{d\beta }}\left(\beta -{\beta }_L\right)=\ln \left(B{c}^2\right)-{\beta }_L^2+\ln \left({\beta }_L^2\right)-\ln \left(1-{\beta }_L^2\right)+2\left({\displaystyle \frac{1}{{\beta }_L}}-{\displaystyle \frac{{\beta }_L^3}{{\beta }_L^2-1}}\right)(\beta -{\beta }_L)$$

(11)

Equation (11) can be simply written as

$$\Omega \left(\beta \right)\cong c_1\beta +c_2$$

(12)

where

$$c_1=2\left({\displaystyle \frac{1}{{\beta }_L}}-{\displaystyle \frac{{\beta }_L^3}{{\beta }_L^2-1}}\right)$$

(13)

and

$$c_2=\ln \left(B{c}^2\right)-{\beta }_L^2+\ln \left({\beta }_L^2\right)-\ln \left(1-{\beta }_L^2\right)-2\left(1-{\displaystyle \frac{{\beta }_L^4}{{\beta }_L^2-1}}\right)$$

(14)

For $B=1.5163\times {10}^{-13} {\mathrm{s}}^2·{\mathrm{m}}^{-2}$ (this is the case for ions travelling in ${\mathrm{H}}_2\mathrm{O}$), we obtain $c_1=4.01$ and $c_2=6.18.$

It is also interesting to compare the functions within the integral shown in Equation (9). In particular, the exact function is

$$f_1\left(\beta \right)={\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{\ln \left(B{c}^2\right)+\ln \left({\beta }^2\right)-\ln \left(1-{\beta }^2\right)-{\beta }^2}}$$

(15)

whereas the function based on the linear approximation of Equation (12) is

$$f_2\left(\beta \right)={\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{c_1\beta +c_2 }}$$

(16)

The results ($f_1\left(\beta \right)$ and $f_2\left(\beta \right)$ with respect to $\beta$) are shown comparatively in Figure 2 (in Figure 2a in the domain $0.01\le \beta \le 0.4$ and in Figure 2b in the domain $0.4\le \beta \le 0.9$). Therefore, Equation (16) is a reliable approximation of Equation (15).

The comparison between the accurate $\Omega$ function and the linear approximation is presented in Figure 1 (considering ${\mathrm{H}}_2\mathrm{O}$ as the medium of the travelling ions). Using the linear approximation presented by Equation (12), Equation (9) simplifies to

$$R={\displaystyle \frac{c^4}{A}}{\int }_0^{{\beta }_0}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{c_1\beta +c_2}}d\beta$$

(17)

or

$$R={\displaystyle \frac{c^4}{c_{2}A}}{\int }_0^{{\beta }_0}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{\lambda \beta +1}}d\beta$$

(18)

where $\lambda =c_1/c_2$. The integral on the right-hand side of Equation (18) can be evaluated analytically, yielding

$$\begin{gathered} R={\displaystyle \frac{c^4}{\lambda c_{2}A}}\left[\mathit{tan}(\mathit{si}{n}^{-1}{\beta }_0)-\mathit{si}{n}^{-1}{\beta }_0+{\displaystyle \frac{2}{({\lambda }^2-1)\sqrt{1-{\lambda }^2}}}\left({\mathit{tan}}^{-1}{\displaystyle \frac{\lambda }{\sqrt{1-{\lambda }^2}}}-{\mathit{tan}}^{-1}{\displaystyle \frac{\lambda +tan\frac{{\mathit{sin}}^{-1}{\beta }_0}{2}}{\sqrt{1-{\lambda }^2}}}\right) \\ +{\displaystyle \frac{1}{\lambda +1}}\left(1-{\displaystyle \frac{1}{1-tan\frac{{\mathit{sin}}^{-1}{\beta }_0}{2}}}\right)+{\displaystyle \frac{1}{\lambda -1}}\left(1-{\displaystyle \frac{1}{tan\frac{{\mathit{sin}}^{-1}{\beta }_0}{2}+1}}\right)\right] \end{gathered}$$

(19)

The analytical evaluation of the integral

$$I={\int }_0^{{\beta }_0}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{{\beta }^3}{\lambda \beta +1}}d\beta$$

(20)

is presented in the Appendix A Section.

3. Results

Range calculations were performed using Equation (9) (with results derived numerically in MATLAB), the Grimes et al. approximation [1] (based on Equation (8)), and Equation (19), which is derived in this paper.

The results are presented in Figure 3 for H₂O ($A=1.4667\times {10}^{32} {\mathrm{m}}^3·{\mathrm{s}}^{-4}, B=1.5163\times {10}^{-13} {\mathrm{s}}^2·{\mathrm{m}}^{-2}$). Figure 3a shows the results for $0.3\le {\beta }_0\le 0.6$, while Figure 3b presents those for $0.6\le {\beta }_0\le 0.95$. Figure 3c illustrates the errors when using Grimes et al.’s approximation ($\epsilon$(%)) and when using Equation (19) proposed in this paper ((${\epsilon }^{\prime }$(%)). The error, in any case, is calculated using the following equation:

$$\epsilon \left(\%\right)={\displaystyle \frac{R_{num.}-R_{model}}{R_{num.}}}100\%$$

(21)

where $R_{num.}$ denotes the accurate numerical solution and $R_{model}$ is the approximate analytical model (Grimes et al.’s model or Equation (19)). As presented in Figure 3c, the proposed approach provides excellent results for $0.65\le v_0/c\le 0.9$. The results are also presented in

Table 1. The accurate numerical results for range calculations (Equation (9)), the results obtained using Grimes et al.’s approximation, and those obtained using the equation proposed in this paper (Equation (19)) for high-energy protons travelling in $H_{2}O$. ε (%) denotes the error when using Grimes et al.’s approximation, while ${\epsilon }^{\prime }$(%) represents the error when using the approach proposed in this paper.

$\mathbf{Range} \mathbf{Calculations}, \mathbf{PROTONS} \left({\mathit{H}}_2\mathit{O}\right)$
${\mathit{v}}_0/\mathit{c}$	$\mathit{R}$ (cm) (Equation (9))	$\mathit{R}$ (cm) (Grimes app.)	$\mathit{R} \left(\mathbf{c}\mathbf{m}\right)$ (Equation (19))	ε(%)	ε′(%)
0.50	14.89	14.93	14.62	−0.27	1.85
0.55	22.74	22.82	22.46	−0.35	1.25
0.60	34.07	34.25	33.78	−0.53	0.86
0.65	50.45	50.83	50.14	−0.75	0.62
0.70	74.34	75.15	74.03	−1.08	0.42
0.75	109.95	111.65	109.66	−1.52	0.26
0.80	165.02	168.70	164.95	−2.18	0.04
0.85	255.79	264.16	256.76	−3.17	−0.38
0.90	424.28	445.53	429.78	−4.77	−1.28
0.95	834.94	906.60	865.70	−7.90	−3.55

in the domain $0.50\le v_0/c\le 0.95$. For $v_0/c\ge 0.65$, Equation (19) is the most reliable solution.

In addition, it is important to emphasize once again that the coefficient $c_2=6.18$ in Equation (12) was derived considering $H_{2}O$ as the medium through which the charged particles travel. However, it is easy to provide general results that can be used for any charged particle and any medium. As already mentioned, the parameter $B$ depends on the mean ionization potential of the medium. Therefore, it is reasonable to obtain a linear function, as shown below:

$$\Omega \left(\beta \right)\cong c_1\beta +{c\prime }_2+\ln \left(B{c}^2\right)$$

(22)

where ${c\prime }_2=-3.34$ and $c_1=4.01$ (as calculated by Equation (13)). Equation (22) is the generalization of Equation (12) for any medium. It is also worth noting that even for $H_{2}O$, there are different values in the literature for the mean ionization potential, which affect the calculation of the B parameter. The 75 eV value used aligns with the guidelines established by the International Commission on Radiation Units and Measurements (ICRU) [6]. Alternatively, another report proposes a lower value of 67 eV [7]. In contrast, some authors have indicated a significantly higher value of 80.2 ± 2 eV [8], which was derived from experimental range measurements. However, these differences have a minor impact on range calculations [1].

The general equation for the coefficient λ is provided below:

$$\lambda ={\displaystyle \frac{{c\prime }_2+\mathit{ln}\left(B{c}^2\right)}{c_1}}$$

(23)

The accuracy of the linear approximation provided by Equation (12) for different mediums is easy to evaluate. For example, for $H_2$, $B_{H_2}=5.92\times {10}^{-13} {\mathrm{s}}^2·{\mathrm{m}}^{-2}$, and for $Xe$, $B_{Xe}=0.21\times {10}^{-13} {\mathrm{s}}^2·{\mathrm{m}}^{-2}$ [2]. Therefore, for $H_2$,

$${\Omega }_{H_2}\left(\beta \right)\cong 4.01\beta +7.54$$

(24)

For Xe,

$${\Omega }_{Xe}\left(\beta \right)\cong 4.01\beta +4.20$$

(25)

The graphs of Equation (10) and linear Equations (24) and (25) are presented in Figure 4.

Range calculations were also performed for H₂ and Xe using Equation (9) (with results derived numerically in MATLAB), the Grimes et al. approximation (based on Equation (8)), and Equation (19), which is derived in this paper. The conclusions are the same as for the case of H₂O. The results are presented in

Table 2. The accurate numerical results for range calculations (Equation (9)), the results obtained using Grimes et al.’s approximation, and those obtained using the equation proposed in this paper (Equation (19)) for high-energy protons travelling in $H_2$. $\epsilon$(%) denotes the error when using Grimes et al.’s approximation, while ${\epsilon }^{\prime }$(%) represents the error when using the approach proposed in this paper.

$\mathbf{Range} \mathbf{Calculations}, \mathbf{Protons} \left({\mathit{H}}_2\right)$$, {\mathit{A}}_{{\mathit{H}}_2} = 2.2\times {10}^{28} {\mathbf{m}}^3{\mathbf{s}}^{-4}, {\mathit{B}}_{{\mathit{H}}_2} = 5.92\times {10}^{-13} {\mathbf{s}}^2{\mathbf{m}}^{-2}$
${\mathit{v}}_0/\mathit{c}$	$\mathit{R}$$(\times {10}^4 \mathbf{m})$ (Equation (9))	$\mathit{R}$$(\times {10}^4 \mathbf{m})$ (Grimes app.)	$\mathit{R} (\times {10}^4 \mathbf{m})$ (Equation (19))	ε(%)	ε′(%)
0.50	0.0842	0.0844	0.0830	−0.2375	1.4252
0.55	0.1292	0.1296	0.1279	−0.3096	1.0062
0.60	0.1943	0.1952	0.1930	−0.4632	0.6691
0.65	0.2887	0.2905	0.2874	−0.6235	0.4503
0.70	0.4268	0.4308	0.4256	−0.9372	0.2812
0.75	0.6333	0.6418	0.6324	−1.3422	0.1421
0.80	0.9537	0.9722	0.9541	−1.9398	−0.0419
0.85	1.4837	1.5261	1.4899	−2.8577	−0.4179
0.90	2.4720	2.5802	2.5022	−4.3770	−1.2217
0.95	4.8960	5.2641	5.0593	−7.5184	−3.3354

and

Table 3. The accurate numerical results for range calculations (Equation (9)), the results obtained using Grimes et al.’s approximation, and those obtained using the equation proposed in this paper (Equation (19)) for high-energy protons travelling in $Xe$. $\epsilon$(%) denotes the error when using Grimes et al.’s approximation, while ${\epsilon }^{\prime }$(%) represents the error when using the approach proposed in this paper.

$\mathbf{Range} \mathbf{Calculations}, \mathbf{Protons} \left(\mathit{X}\mathit{e}\right)$$, {\mathit{A}}_{\mathit{X}\mathit{e}} = 59.29\times {10}^{28} {\mathbf{m}}^3{\mathbf{s}}^{-4}, {\mathit{B}}_{\mathit{X}\mathit{e}} = 0.21\times {10}^{-13} {\mathbf{s}}^2{\mathbf{m}}^{-2}$
${\mathit{v}}_0/\mathit{c}$	$\mathit{R}$$(\times {10}^3 \mathbf{m})$ (Equation (9))	$\mathit{R}$$(\times {10}^3 \mathbf{m})$ (Grimes app.)	$\mathit{R} (\times {10}^3 \mathbf{m})$ (Equation (19))	ε(%)	ε′(%)
0.50	0.0494	0.0499	0.0485	−1.0121	1.8219
0.55	0.0748	0.0756	0.0740	−1.0695	1.0695
0.60	0.1111	0.1127	0.1106	−1.4401	0.4500
0.65	0.1632	0.1659	0.1630	−1.6544	0.1225
0.70	0.2386	0.2436	0.2390	−2.0956	−0.1676
0.75	0.3504	0.3598	0.3518	−2.6826	−0.3995
0.80	0.5221	0.5405	0.5257	−3.5242	−0.6895
0.85	0.8029	0.8418	0.8128	−4.8449	−1.2330
0.90	1.3193	1.4123	1.3509	−7.0492	−2.3952
0.95	2.5619	2.8577	2.6995	−11.5461	−5.3710

. The values of $A$ and $B$ shown in Table 2 and Table 3 were obtained from reference [2].

In addition, the accuracy of the provided results is not affected by the charged particles (e.g., protons, carbon ions, etc.) travelling through the medium. This is because the linear approximation leading to the integral in Equation (20) does not depend on the parameter $A$, as it is clearly shown in Equations (9)–(12).

4. Discussion

In this paper, we present a new approximate solution for the range of charged particles within a medium, with a focus on charged particles with high initial energy. The idea is to consider a linear approximation of the $\Omega$ function provided by Equation (10). Using this approach, it is possible to obtain an analytical solution to Equation (9). The solution is presented in Equation (19) and provides reliable results within the domain $0.6\le {\beta }_0\le 0.90$.

It is important to note that this is not the first attempt to provide an analytical equation for calculating the range of charged particles within a medium. As already mentioned, Grimes et al. used the approximation presented by Equation (8), resulting in

$$R={\displaystyle \frac{1}{A}}\underset{0}{\overset{v_0}{\int }}{\left[1-{\beta }^2\right]}^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{v^3}{\mathrm{l}\mathrm{n}\left[B{v}^2\left(x\right)\right]}}dv$$

(26)

The solution provided by Grimes et al. is given below.

$$R_G={\displaystyle \frac{1}{2A}}\sum _{n=0}^{\infty }{\displaystyle \frac{F_n}{B^{n+2}{c}^{2n}}}\mathrm{E}\mathrm{i}\left[\left(n+2\right)\mathrm{l}\mathrm{n}\left(B{v}_0^2\right)\right]$$

(27)

The first four values of $F_n$ (for $n=0$ to $n=3$) are 1, 3/2, 15/8, and 35/16, and $\mathrm{E}\mathrm{i}$ is the exponential integral function. This is an accurate solution for ${\displaystyle \frac{v_0}{c}}<0.65$. Since Solution (27) includes the special function ($\mathrm{E}\mathrm{i}$), alternative approaches have also been presented in the literature, providing solutions to the Bethe equation using only elementary functions. Martínez et al. presented a series solution in terms of elementary functions, focusing on the small initial velocities of the charged particles [2]. In particular, the aforementioned approach can only be used for protons with an initial energy of less than 200 MeV, which corresponds to ${\displaystyle \frac{v_0}{c}}<0.57$. On the contrary, the solution presented by Equation (19) is valid for high initial speeds and is the first of its kind reported in the literature.

In addition to the mathematical and physical interest in solving the Bethe equation for high initial speeds, there is also practical interest in biomedical applications, particularly in radiotherapy. In particular, high-energy charged particles are used in cancer treatment through radiotherapy. It is interesting to note that the World Health Organization (WHO) reports cancer as the second leading cause of death worldwide, accounting for approximately 10 million deaths annually [9,10]. Over the past century, advancements in external beam energy have given rise to two primary treatment approaches. The first, radiotherapy, utilizes X-ray, gamma, and proton beams to induce ionizing damage [9,10,11,12,13,14,15,16]. The second approach, photodynamic therapy, employs photons to activate photosensitizers, generating reactive oxygen species with cytotoxic effects [17,18,19,20]. Traditionally, these two treatments have been applied separately, with photodynamic therapy primarily targeting surface tumours and radiotherapy addressing deeper tumours [9]. The latter case is the one of interest in this study. From this perspective, the methods presented in this paper can be practically applied by practitioners in radiotherapy. Protons with initial energies of 200 MeV to 350 MeV are used for radiotherapy purposes. These initial energies correspond to initial velocities in the range $0.56\le v_0/c\le 0.69$. As shown in Table 1, Equation (19) leads to accurate results within this domain. Additionally, it has the major advantage of being expressed in terms of elementary functions, in contrast to the solution by Grimes et al. [1]. As already noted, this has been the major advantage of Martínez et al.’s solution [2]; however, this solution is valid only for protons with initial energies up to 200 MeV. Therefore, this solution represents a step forward in finding simpler solutions for radiotherapy applications. Practitioners can apply Martínez’s solution for protons with initial energies up to 200 MeV and the one proposed in this paper (Equation (19)) for higher energies.

It is also interesting to discuss, from a practical perspective, the reasons for searching for approximate mathematical solutions to the Bethe equation and the range travelled by ions through matter. A key challenge in the medical procedures mentioned above is accurately modelling tissue–particle interactions, which also requires substantial computational resources. The precise simulation of radiation interactions is critical for various applications, particularly in optimizing the use of charged particles. To maximize their therapeutic potential, it is essential to understand energy deposition across different materials. Monte Carlo methods are commonly used to tackle these complexities [21]. Monte Carlo particle transport packages are advanced tools used to simulate the interactions of particle radiation with matter as it propagates through a medium [1,22,23,24]. These methods can accurately model system behaviour at all levels, including secondary and higher-order interactions, but they are computationally intensive and require significant processing time [1]. Therefore, many scientists in the radiotherapy field focus on finding the range of charged ions in a medium based on mathematical approaches using the Bethe equation [1,2].

It is important to emphasize that mathematical solutions, like the one presented here, are not meant to replace conventional Monte Carlo (MC) methods but rather serve as useful tools for rapid optimization. Since the analytical form provided can be quickly implemented with minimal computational cost, it is well suited for optimization scenarios where multiple iterations of dose calculation are required. Another significant advantage of the analytical solutions to the Bethe equation is that they enable the rapid simulation of complex cases involving heterogeneous tissue, relying on well-defined material constants such as electron density and mean ionization potential, rather than water-equivalent stopping power ratios (SPRs) [1]. A typical example demonstrating the importance of analytical methods is presented in [1], where a radiotherapy treatment planning scan for prostate cancer is discussed. Therefore, the approximate dynamics of particles in even complex media can be easily estimated.

5. Conclusions

In this paper, a new analytical equation for determining the range of charged particles within a medium for large initial speeds is presented. This is the first time that such an equation has been introduced for initial velocities up to 0.9c. The idea was to use a Taylor series expansion for the $\Omega$ function (Equation (10)) around $\beta =0.6$ and retain only the linear term. Using this approach, it is possible to obtain an analytical solution for Equation (9) (i.e., Equation (19)). The proposed equation was compared to the exact numerical results for different initial velocities and mediums (${\mathrm{H}}_2\mathrm{O}, {\mathrm{H}}_2,$ and Xe), as presented in Table 1, Table 2 and Table 3. The results were excellent for initial velocities in the range of 0.6c to 0.9c. In addition to its theoretical physics and mathematical importance, it was shown that Equation (19) also has practical significance in the field of radiotherapy. It was shown that Equation (19) provides excellent approximations for protons with initial energies in the range of 200 MeV to 350 MeV, which are used for medical purposes. It also has the advantage of being expressed in terms of elementary functions.