Bethe-Block stopping power¶

Understanding stopping power is crucial to understanding of ion transport through matter. This material is reasonably common and spread out, but I’m referencing [Knoll2000] and ({Nastasi, et. al. 1996)

The general understanding of total stopping power, which includes nuclear and atomic stopping is often given through the following equation:

\[\frac{DE}{Dx}=c\left(\frac{Z}{v}\right)^{2}\]

This is particularly useful for deuterons, protons, and alphas. Notice that the stopping power is proportional to $Z^{2}$ and $v^{2}\therefore E$. This comes from the total Bethe Equation

\[-\frac{dE}{dx}=\frac{4\pi Z^{2}e^{4}}{m_{0}v^{2}}NZ\left[\ln\frac{2m_{0}v^{2}}{I}-\ln\left(1-\frac{v^{2}}{c^{2}}\right)-\frac{v^{2}}{c^{2}}\right]\]

A more involved Bethe-Bloch formulation uses the nuclear and the atomic stopping power, given respectively as

\[-\left.\frac{dE}{dx}\right|_{e}=\frac{4\pi Z_{1}^{2}e^{4}n_{e}}{m_{e}v^{2}}\ln\left[\frac{b_{max}}{b_{min}}\right]\]

\[ \begin{align}\begin{aligned}-\left.\frac{dE}{dx}\right|_{n}=\frac{4\pi Z^{2}e^{4}}{m_{0}v^{2}}\frac{NZ_{1}Z_{2}}{\left(\sqrt{Z_{1}}+\sqrt{Z_{2}}\right)^{\nicefrac{2}{3}}}\frac{M_{1}}{M_{1}+M_{2}}\\where :math:`b` is the collision parameter, and all other nomenclature\end{aligned}\end{align} \]

is as usual.

Figure 1: Electronic and nuclear stopping power as a function of $\varepsilon$

Bethe-Block stopping power¶

Bibliography