Bethe formula

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The Bethe formula, found by Hans Bethe in 1933, 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).

1 The formula
2 Corrections to the Bethe formula
3 See also
4 Weblinks

[edit] The formula

The formula reads:

$- \frac{dE}{dx} = \frac{4 \pi}{m_e c^2} \cdot \frac{nz^2}{\beta^2} \cdot \left(\frac{e^2}{4\pi\varepsilon_0}\right)^2 \cdot \left[\ln \left(\frac{2m_e c^2 \beta^2}{I \cdot (1-\beta^2)}\right) - \beta^2\right]$

where

$β$	$= v / c$
$v$	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 $n=\frac{N_{A}\cdot Z\cdot\rho}{A}$ , where $ρ$ is the density of the material, $Z, A$ the atomic number and mass number, resp., and $N A$ the Avogadro number.

Stopping Power of Aluminum for protons versus proton energy, and the Bethe formula

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 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 $( \beta \ll 1)$ , the Bethe formula reduces to

$- \frac{dE}{dx} = \frac{4 \pi nz^2}{m_e v^2} \cdot \left(\frac{e^2}{4\pi\varepsilon_0}\right)^2 \cdot \left[\ln \left(\frac{2m_e v^2 }{I}\right)\right]$ .

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. $E = 3 M c 2$ , where $M$ is the mass of the particle (for protons, this would be about at 3000 MeV). For highly relativistic cases $( \beta \approx 1)$ , the energy loss increases again, logarithmically.

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:

[edit] 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, if the energy is increased.