Bethe formula

The Bethe formula or Bethe–Bloch 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).

[3] 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 energy I (see below), the relativistic version of the formula reads, in SI units:[2] where c is the speed of light and ε0 the vacuum permittivity,

In the figure to the right, the small circles are experimental results obtained from measurements of various authors, while the red curve is Bethe's formula.

For low energies, i.e., for small velocities of the particle β << 1, the Bethe formula reduces to This can be seen by first replacing βc by v in eq.

For highly relativistic cases β ≈ 1, the energy loss increases again, logarithmically due to the transversal component of the electric field.

In the Bethe theory, the material is completely described by a single number, the mean excitation energy I.

The corrections mentioned have been built into the programs PSTAR and ASTAR, for example, by which one can calculate the stopping power for protons and alpha particles.

Stopping Power of Aluminum for Protons versus proton energy, and the Bethe formula without (red) and with corrections (blue)
Stopping Power of Aluminum for Protons versus proton energy, and the Bethe formula without (red) and with corrections (blue)
The mean excitation energy I of atoms, versus atomic number Z , in eV , divided by Z