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# Bethe formula

The Bethe formula describes the energy loss per distance travelled of swift charged particles (protons, alpha particles, atomic ions, but not electrons) traversing matter (or, alternatively, the stopping power of the material). The non-relativistic version was found by Hans Bethe in 1930; the relativistic version (shown below) was found by him in 1932 (Sigmund 2006).

### The formula

$- frac\left\{dE\right\}\left\{dx\right\} = frac\left\{4 pi\right\}\left\{m_e c^2\right\} cdot frac\left\{nz^2\right\}\left\{beta^2\right\} cdot left\left(frac\left\{e^2\right\}\left\{4pivarepsilon_0\right\}right\right)^2 cdot left\left[ln left\left(frac\left\{2m_e c^2 beta^2\right\}\left\{I cdot \left(1-beta^2\right)\right\}right\right) - beta^2right\right]$

where

 $beta$ $= v/c$ $v$ velocity of the particle $E$ energy of the particle $x$ distance travelled by the particle $c$ speed of light $z,e$ particle charge $e$ charge of the electron $m_e$ rest mass of the electron $n$ electron density of the target $I$ mean excitation potential of the target

Here, the electron density of the material can be calculated by $n=frac\left\{N_\left\{A\right\}cdot Zcdotrho\right\}\left\{A\right\}$, where $rho$ is the density of the material, $Z, A$ its atomic number and mass number, respectively, and $N_A$ the Avogadro number.

In the figure to the right, the small circles are experimental results obtained from measurements of various authors (taken from http://www.exphys.uni-linz.ac.at/Stopping/); the red curve is Bethe's formula. Evidently, Bethe's theory agrees very well with experiment at high energy. Only below 0.3 MeV, the curve is too low; here, corrections are necessary (see below).

Sometimes, the Bethe formula is also called Bethe-Bloch formula, but this is misleading.

For low energies, i.e., for small velocities of the particle $\left(beta ll 1\right)$, the Bethe formula reduces to

$- frac\left\{dE\right\}\left\{dx\right\} = frac\left\{4 pi nz^2\right\}\left\{m_e v^2\right\}$
cdot left(frac{e^2}{4pivarepsilon_0}right)^2 cdot left[ln left(frac{2m_e v^2 }{I}right)right]. At low energy, the energy loss according to the Bethe formula therefore decreases approximately as $1/v^2$ with increasing energy. It reaches a minimum for approx. $E = 3Mc^2$, where $M$ is the mass of the particle (for protons, this would be about at 3000 MeV). For highly relativistic cases $\left(beta approx 1\right)$, the energy loss increases again, logarithmically.

The normalized mean excitation potential I which enters the Bethe formula is shown to the left. The numbers are taken from Report 49 of the International Commission on Radiation Units and Measurements, "Stopping Powers and Ranges for Protons and Alpha Particles" (1993). The peaks and valleys in this figure lead to corresponding valleys and peaks in the stopping power. These are called "Z2-oscillations" or "Z2-structure" (where Z2 means the atomic number of the target).

The Bethe formula is only valid for energies high enough so that the charged atomic particle (the ion) does not carry any atomic electrons with it. At smaller energies, when the ion carries electrons, this reduces its charge effectively, and the stopping power is thus reduced. But even if the atom is fully ionized, corrections are necessary:

### Corrections to the Bethe formula

Bethe found his formula using quantum mechanical perturbation theory. Hence, his result is proportional to the square of the charge $z$ of the particle. The description can be improved by considering corrections which correspond to higher powers of $z$. These are: the Barkas-Andersen-effect (proportional to $z^3$, after Walter H. Barkas and Hans Henrik Andersen), and the Bloch-correction (proportional $z^4$). In addition, one has to take into account that the atomic electrons are not stationary ("shell correction").

These corrections have been built into the programs PSTAR and ASTAR, for example, by which one can calculate the stopping power for protons and alpha particles (www.physics.nist.gov/PhysRefData/Star/Text/programs.html). The corrections are large at low energy and become smaller and smaller as energy is increased.