Bethe formula

The Bethe formula describes the mean energy loss per distance travelled of swift charged particles (protons, alpha particles, atomic ions) traversing matter (or alternatively the stopping power of the material). For electrons the energy loss is slightly different due to their small mass (requiring relativistic corrections) and their indistinguishability, and since they suffer much larger losses by Bremsstrahlung, terms must be added to account for this. Fast charged particles moving through matter interact with the electrons of atoms in the material. The interaction excites or ionizes the atoms, leading to an energy loss of the traveling particle.

The non-relativistic version was found by Hans Bethe in 1930; the relativistic version (shown below) was found by him in 1932. The most probable energy loss differs from the mean energy loss and is described by the Landau-Vavilov distribution.

The Bethe formula is sometimes called "Bethe-Bloch formula", but this is misleading (see below).

For a particle with speed v, charge z (in multiples of the electron charge), and energy E, traveling a distance x into a target of electron number density n and mean excitation potential I, the relativistic version of the formula reads, in SI units:

${\displaystyle -\left\langle {\frac {dE}{dx}}\right\rangle ={\frac {4\pi }{m_{e}c^{2}}}\cdot {\frac {nz^{2}}{\beta ^{2}}}\cdot \left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{2}\cdot \left[\ln \left({\frac {2m_{e}c^{2}\beta ^{2}}{I\cdot (1-\beta ^{2})}}\right)-\beta ^{2}\right]}$

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