Bethe formula

The Bethe formula (also Bethe equation, Bethe -Bloch formula, Bethe- Bloch equation or braking formula ) is the energy loss per unit of path length of the fast charged heavy particles (eg, protons, alpha particles, ions) during passage through suffer matter by inelastic collisions with the electrons; the transmitted energy in the material causes excitation or ionization. This energy loss, also known as Electronic braking or inaccurate as Ionisationsverlust depends on the speed and load of the projectile particles and from the target material. The non- relativistic formula in 1930, the one shown below relativistic version in 1932 by Hans Bethe erected.

The Bethe -Bloch formula is not valid for incident electrons. Firstly, for this is the loss of energy due to their indistinguishability with the orbital electrons of the material differently. On the other hand comes with electrons due to their low mass, a significant energy loss by bremsstrahlung added. The energy loss of electrons can be described using the Berger Seltzer formula instead.

Other mechanisms that may contribute to the total matter - energy loss of fast charged heavy particles, the nuclear deceleration (elastic Coulomb collisions with the atomic nuclei, see stopping power ) and the bremsstrahlung.

The formula

Be fast charged particles moving through matter, they result from inelastic collisions with orbital electrons of the material. This leads to the excitation or ionization of the atoms. Characterized the particles traversing suffers an energy loss, which is represented by the following formula is approximately. Your relativistic form is:

In which

The electron density can be calculated with this; is the density of the material, and classification or mass number of the material and the atomic mass unit.

In the picture the small circles test results from various working groups; the red curve represents the Bethe formula dar. Apparently the conformity of Bethe's theory with the experiments above 0.5 MeV is very good, especially if the corrections (see below) are added (blue curve).

For small energies, ie small particle velocities, the Bethe formula reduces to

At low energies, the Bethe formula is only valid if they are still high enough that traversing particles leads no shell electrons with it. Otherwise, its effective charge is reduced by and the stopping power is smaller. There are small energies a more refined theory of electronic braking of Lindhart, Scharff and Schiott ( LSS theory).

In general, the energy loss with increasing energy first falls about with and reaches a minimum at about, the mass of the particle ( ie, for example, for protons at about 3 GeV, which is no longer visible in the picture). Since about for many relevant in particle radiation particles and absorber materials, the energy loss in the vicinity of the minimum has the same value, particles are combined often with an energy near the minimum and as MIPs (Minimum Ionizing Particles, dt minimum ionizing particles ) denotes. As a rule of thumb for the specific energy loss of the MIPs applies:

At still higher energies, the energy loss increases again. At very high energies and particle interactions need to be considered, leading to offspring. The energy loss can therefore rise even more in the material-dependent manner.

In radiation biology is called the energy output of ionizing particles according to the Bethe- Bloch equation the linear energy transfer ( ) and uses the unit keV / micron.

The mean excitation potential

In the scope of the Bethe formula (1) the penetrated material is described by a single constant, the mean excitation potential.

Felix Bloch 1933 showed that the average excitation potential of the atoms on the average about

Is, when the atomic number of the atoms of the material means. Substituting this value in formula (1 ) above, a, the results of an equation, often called the Bethe- Bloch equation. However, there are detailed tables of as a function of. With them, you will get better results than with formula (2).

Shown is the average excitation potential of the various elements is shown, which contains the information about the respective atom. The data comes from said ICRU Report. The peaks and valleys in the representation ( " Z2- oscillations ", where Z2 is the atomic number of the material) correspond to lower and higher values ​​of the stopping power; these oscillations is based on the shell structure of the atoms. As the picture shows, (2) applies only approximate formula.

Corrections to the Bethe formula

The Bethe formula was derived by Bethe using the quantum-mechanical perturbation theory, the result is therefore proportional to the square of the charge. A better description is obtained if one takes into account deviations corresponding to higher powers of, namely the Barkas -Andersen effect ( proportional, according to Walter H. Barkas and Hans Henrik Andersen ) and the Bloch correction ( proportional). The movement of the orbital electrons in an atom has to be taken into consideration of the material ( " shell correction ").

These corrections are, for example, in the programs PSTAR ASTAR and the National Institute of Standards and Technology ( NIST), which calculate the stopping power for protons and alpha particles incorporated. The corrections are large at low energies and are getting smaller, the greater the energy.

In addition comes at very high energies even Fermi's density correction added.