**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 ð Þ *dT=dx* . Then, mass stopping power for heavy particles is given by [11]:

$$
\left(\frac{dT}{\rho d\mathbf{x}}\right)\_c = \left(\frac{dT\_s}{\rho d\mathbf{x}}\right)\_c + \left(\frac{dT\_h}{\rho d\mathbf{x}}\right)\_c \tag{1}
$$

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:

$$\left(\frac{dT\_s}{\rho dx}\right)\_c = \frac{2Cm\_0c^2x^2}{\rho^2}\left[\ln\left(\frac{2m\_0c^2\beta^2H}{I^2(1-\beta^2)}\right) - \rho^2\right] \tag{2}$$

Where *<sup>C</sup>* � *<sup>π</sup> NAZ A* � �*r*<sup>2</sup> <sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*150*Z=<sup>A</sup>* cm2*=*g where *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*=m*0*c*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>818</sup> � <sup>10</sup>�<sup>15</sup> is classical electron radius, *m*0*c*<sup>2</sup> represents the rest-mass energy of an electron, and *β* ¼ *v=c*. *H* means arbitrary energy boundary between soft and hard collisions.

The equation can be simplified as follows:

$$k \equiv \frac{2Cm\_0c^2z^2}{\beta^2} = 0.1535 \frac{Zz^2}{A\beta^2} \frac{\text{MeV}}{\text{g}/cm^2} \tag{3}$$

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

$$k \left(\frac{dT\_h}{\rho d\mathbf{x}}\right)\_c = k \left[ \ln \left(\frac{T\_{max}'}{H}\right) - \beta^2 \right] \tag{4}$$

where *T*<sup>0</sup> *max* is written as:

$$T'\_{\max} \cong 2m\_0 c^2 \left(\frac{\beta^2}{1-\beta^2}\right) = 1.022 \left(\frac{\beta^2}{1-\beta^2}\right) MeV \tag{5}$$

Eq. (4) can be written as:

$$\left(\frac{dT\_h}{\rho dx}\right)\_c = k \left[ \ln \left( \frac{2m\_0 c^2 \beta^2 T\_{\text{max}}'}{I^2 \left(1 - \beta^2\right)} \right) - 2\beta^2 \right] \tag{6}$$

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

$$\left(\frac{dT}{\rho dx}\right)\_c = 2k \left[ \ln \left( \frac{2m\_0 c^2 \beta^2}{(1-\beta^2)I} \right) - \beta^2 \right] \tag{7}$$

$$
\ln\left(\frac{dE}{\rho dx}\right) = 0.3071 \frac{Zx^2}{A\rho^2} \left[13.8373 + \ln\left(\frac{\rho^2}{1-\rho^2}\right) - \beta^2 - \ln\left(I\right) - \frac{\delta}{2}\right] \tag{8}
$$

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


*Introductory Chapter: Ion Beam Technology and Applications DOI: http://dx.doi.org/10.5772/intechopen.113103*

$$\delta(\text{iii}) \quad \delta(X) = \delta(X\_0) \times 10^{2(X - X\_0)} (X \le X\_0) \tag{9}$$

where X is [ log *β=* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> � <sup>p</sup> )] [11]. In Eq. (8), the term *<sup>I</sup>* 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:

ð Þ*i I* ≈19*:*0 *eV Z*ð Þ ¼ 1

$$(ii) \quad I \approx \text{11.2} + \text{11.7} \times Z \, eV \, (2 \le Z \le \text{13})$$

$$(iii) \quad I \approx \text{52.8} + \text{8.71} \times Z \, eV \, (\text{13} < Z) \tag{10}$$

The different correction terms can be added to Eq. (8) such as the shell correction terms ð Þ �C*=*Z . In addition to elements, Eq. (8) may be written for the compound and mixtures based on the assumption of Bragg's rule [11, 15]:

$$\mathcal{W}\_a = \frac{N\_a A\_a}{\sum\_b N\_b A\_b} \tag{11}$$

$$\left(\frac{dE}{\rho d\mathbf{x}}\right)\_{comp.} = \sum\_{a} W\_{a} \left(\frac{dE}{\rho d\mathbf{x}}\right)\_{a} \tag{12}$$

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

$$\left(\frac{dT}{\rho dx}\right)\_c = k \left[ \ln \left( \frac{\tau^2 (\tau + 2)}{2(I/m\_0 c^2)} \right) + F^\mp(\tau) - \delta - \frac{2C}{Z} \right] \tag{13}$$

where the term *<sup>τ</sup>* is *<sup>T</sup>=m*0*c*<sup>2</sup> and *<sup>C</sup>=<sup>Z</sup>* for elements is shell correction term. *<sup>F</sup>*<sup>∓</sup>ð Þ*<sup>τ</sup>* is written for the electrons and positrons as follows [11]:

$$F^{-}(\tau) \equiv \mathbf{1} - \boldsymbol{\beta}^{2} + \frac{\tau^{2}/\mathbf{8} - (2\tau + \mathbf{1})\ln 2}{\left(\tau + \mathbf{1}\right)^{2}}\tag{14}$$

$$F^{+}(\tau) \equiv 2\ln 2 - \frac{\beta^2}{12} \left\{ 2\Im + \frac{14}{\tau + 2} + \frac{10}{\left(\tau + 2\right)^2} + \frac{4}{\left(\tau + 2\right)^3} \right\} \tag{15}$$
