Introductory Chapter: Ion Beam Technology and Applications

1. Introduction

Ion beams are charged particle beams comprising ions accelerated in particle accelerators, and they have a wide variety of applications in science, medicine, and engineering since ion beam technology is a key tool that has found vital applications. In the present day, the applications of ion beams for scientific and commercial purposes are remarkable. Especially in nuclear medicine, the charged particles produced by particle accelerators are used to treat and diagnose cancer. For this aim, not only ion beam technology but also particle accelerator technology used in ion beam technology also stands out. It is obvious that ion beam technology has branched into many fields in science and technology. In addition to therapy, low- and high-energy ion beams effectively lead to new developments in space applications, materials science, and atomic and nuclear physics [1, 2, 3, 4, 5, 6, 7, 8].

In ion beam applications, there are modern technologies including the use of energetic ion beams for different science fields. Fundamental areas of application include microelectronics, space, energy, physics, materials sciences, etc. Frankly, the underlying science in the interactions between the ions beam and atoms in matter should be well explained. Hence, the interactions of charged particles with matter are experimentally investigated, and the obtained measurements are compared with the specialized simulation and theoretical data. Though fundamental physical processes are well understood, the credibility of the obtained data explanations is restricted by insufficient physical data, the examination of the unevaluated experimental data, and many discrepancies beyond the error limits in data reported by the authors. Notably, two major data are needed to understand the profile of the investigated material, namely, stopping power and cross-section data. The cross-section depends on processes such as direct, elastic and nonelastic scatterings, and equilibrium and pre-equilibrium reaction processes [9, 10]. On the other hand, the stopping power gives data defining the slowing of the ion in the material such as elements, compounds, alloys, etc. For this aim, the charged particle interactions between ions and materials should be well defined [8, 11].

2. Theoretical framework for ion beam interactions with matter

The charged particles interacting with matter lose their energy through ionization and excitation of the atoms. Therefore, the stopping power of matter may be explained by the mean energy loss per unit path length $\left(\mathit{dT}/\mathit{dx}\right)$. Then, mass stopping power for heavy particles is given by [11]:

$${\left(\frac{\mathit{dT}}{\mathit{\rho dx}}\right)}_c={\left(\frac{{\mathit{dT}}_s}{\mathit{\rho dx}}\right)}_c+{\left(\frac{{\mathit{dT}}_h}{\mathit{\rho dx}}\right)}_c$$E1

where c is collision interactions in which s and h represent soft and hard collisions. The soft collision term by Bethe can be written as:

$${\left(\frac{{\mathit{dT}}_s}{\mathit{\rho dx}}\right)}_c=\frac{2C{m}_0{c}^2{z}^2}{{\beta }^2}\left[\mathit{ln}\left(\frac{2m_0{c}^2{\beta }^{2}H}{I^2\left(1-{\beta }^2\right)}\right)-{\beta }^2\right]$$E2

Where $C\equiv \pi \left(\frac{N_{A}Z}{A}\right)r_0^2=0.150Z/A{\mathrm{cm}}^2/\mathrm{g}$ where $r_0=e^2/m_0{c}^2=2.818\times {10}^{-15}$ is classical electron radius, $m_0{c}^2$ represents the rest-mass energy of an electron, and $\beta =v/c$. H means arbitrary energy boundary between soft and hard collisions.

The equation can be simplified as follows:

$$k\equiv \frac{2C{m}_0{c}^2{z}^2}{{\beta }^2}=0.1535\frac{Z{z}^2}{A{\beta }^2}\frac{\mathit{MeV}}{g/{\mathit{cm}}^2}$$E3

On the other hand, the hard-collision statement can be given by:

$${\left(\frac{{\mathit{dT}}_h}{\mathit{\rho dx}}\right)}_c=k\left[\mathit{ln}\left(\frac{T_{\mathit{max}}^{\prime }}{H}\right)-{\beta }^2\right]$$E4

where $T_{\mathit{max}}^{\prime }$ is written as:

$$T_{\mathit{max}}^{\prime }\cong 2m_0{c}^2\left(\frac{{\beta }^2}{1-{\beta }^2}\right)=1.022\left(\frac{{\beta }^2}{1-{\beta }^2}\right)\mathit{MeV}$$E5

Eq. (4) can be written as:

$${\left(\frac{{\mathit{dT}}_h}{\mathit{\rho dx}}\right)}_c=k\left[\mathit{ln}\left(\frac{2m_0{c}^2{\beta }^2{T}_{\mathit{max}}^{\prime }}{I^2\left(1-{\beta }^2\right)}\right)-2{\beta }^2\right]$$E6

Then, the mass stopping power can be given as follows [11]:

$${\left(\frac{\mathit{dT}}{\mathit{\rho dx}}\right)}_c=2k\left[\mathit{ln}\left(\frac{2m_0{c}^2{\beta }^2}{\left(1-{\beta }^2\right)I}\right)-{\beta }^2\right]$$E7

$$\left(\frac{\mathit{dE}}{\mathit{\rho dx}}\right)=0.3071\frac{Z{z}^2}{A{\beta }^2}\left[13.8373+\mathit{ln}\left(\frac{{\beta }^2}{1-{\beta }^2}\right)-{\beta }^2-\mathit{ln}\left(I\right)-\frac{\delta }{2}\right]$$E8

where δ is density effect correction based on constants of the medium $a$, X₁, X₀, and C. δ includes three situations as follows [12]:

$$\left(\mathrm{i}\right)\delta \left(X\right)=4.6052X+a{\left(X_1-X\right)}^m+C\left(X_0<X<X_1\right)$$

$$\left(\mathrm{ii}\right)\delta \left(X\right)=4.6052X+C\left(X>X_1\right)$$

$$\left(\mathrm{iii}\right)\delta \left(X\right)=\delta \left(X_0\right)\times {10}^{2\left(X-X_0\right)}\left(X\le X_0\right)$$E9

where X is [$log(\beta /\sqrt{1-{\beta }^2}$)] [11]. In Eq. (8), the term I represents the mean excitation potential of the medium for materials [13]. This term was obtained by the quantum mechanical approaches and the experimental data given by Paul and Schinner [14] as follows:

$$\left(i\right)I\approx 19.0\mathit{eV}\left(Z=1\right)$$

$$\left(\mathit{ii}\right)I\approx 11.2+11.7\times Z\mathit{eV}\left(2\le Z\le 13\right)$$

$$\left(\mathit{iii}\right)I\approx 52.8+8.71\times Z\mathit{eV}\left(13<Z\right)$$E10

The different correction terms can be added to Eq. (8) such as the shell correction terms $\left(-\mathrm{C}/\mathrm{Z}\right)$. In addition to elements, Eq. (8) may be written for the compound and mixtures based on the assumption of Bragg’s rule [11, 15]:

$$W_a=\frac{N_a{A}_a}{\sum _b{N}_b{A}_b}$$E11

$${\left(\frac{\mathit{dE}}{\mathit{\rho dx}}\right)}_{\mathit{\text{comp}}.}=\sum _a{W}_a{\left(\frac{\mathit{dE}}{\mathit{\rho dx}}\right)}_a$$E12

where W_a is given the weight fraction of element (including N_a atoms), and A means the atomic weight. Furthermore, the mass stopping power for the electrons and positrons can be written as:

$${\left(\frac{\mathit{dT}}{\mathit{\rho dx}}\right)}_c=k\left[\mathit{ln}\left(\frac{{\tau }^2\left(\tau +2\right)}{2{\left(I/m_0{c}^2\right)}^2}\right)+F^{\mp }\left(\tau \right)-\delta -\frac{2C}{Z}\right]$$E13

where the term $\tau$ is $T/m_0{c}^2$ and $C/Z$ for elements is shell correction term. $F^{\mp }\left(\tau \right)$ is written for the electrons and positrons as follows [11]:

$$F^{-}\left(\tau \right)\equiv 1-{\beta }^2+\frac{{\tau }^2/8-\left(2\tau +1\right)\mathit{ln}2}{{\left(\tau +1\right)}^2}$$E14

$$F^{+}\left(\tau \right)\equiv 2\mathit{ln}2-\frac{{\beta }^2}{12}\left\{23+\frac{14}{\tau +2}+\frac{10}{{\left(\tau +2\right)}^2}+\frac{4}{{\left(\tau +2\right)}^3}\right\}$$E15

3. Conclusion

Considering both theoretical and experimental research and applications, this book will provide a broad perspective on developments in ion beam technology and the future of ion technology. Therefore, the book focuses on expanding and improving the application areas of ion beam technology.