The collisions are casual, of course, but their number per macroscopic path length is generally large and this is the reason why average quantities are generally used. One of the most important quantity is the mean energy loss per units length, often called stopping power. Many theories have been developed during the first half of the twentieth century in order to characterize this quantity. The correct quantum‑mechanical calculation was first described, around 1932, by Hans Bethe, Bloch and other authors who gave the formula:

with:

2π N_{a }r_{e^}^{2 }m_{e}c^{^2}= 0.1535 MeVcm^{2}/g | β: v/c of incident particle |

r: electron radius (2.817 x 10_{e}^{-13} cm) | ρ: density of absorbing material |

M: electron mass_{e} | γ: |

N: Avogadro’s number_{a} | δ: density correction |

Z: atomic number of absorbing material | C: shell correction |

A: atomic weight of absorbing material | I: mean excitation potential |

z: charge of incident particle | W: maximum energy transferable in a single collision_{max} |

*W _{max}* can be calculated using the equation:

where: *s=m _{e}/M* and

*η=βγ.*

The mean excitation energy *I* depends by the orbital frequency of the absorbing material and there is not a precise formula to calculate that value. However values of *I* for several material have been deduced from measurements [**3**]. The last two terms in the parentheses of (2.3) are the density and the shell corrections and they have been inserted in the original formulation of Bethe‑Bloch in order to enhance the prediction of the formula at certain range compared to the experimental results. The density correction takes into account the effect of the electric field produced by incoming particles and is more evident at high velocity. Instead, the shell correction is noticeable when the velocity of incident particle is comparable to the orbital velocity of the bound electrons of the target material.

At this low energy some other complicated effects come into play and the Bethe-Bloch formula breaks down. When the velocity is comparable with the speed of orbital electrons of the target material the energy loss reach a maximum depending on the sign of the charge (Barkas effect) and for lower energy drops sharply. At higher energy (that means higher velocity) *dE/dx* is dominated by the *1/β ^{2}* factor and decreases until

*β*

*≅*

*0.96c*where a minimum is reached. Particle with this energy is usually indicated with the name of minimum ionizing particle (MIP).

Increasing the energy the losses do not increase so much due to the density effect (Fermi plateau) until the radiative components, such as the Cherenkov radiation and Bremsstrahlung, start to be relevant. The Cherenkov radiation arises when a charged particle in a medium moves faster than the speed of light in that same medium (*βc>c/n*, with *n*: index of refraction): in such case an electromagnetic shock wave is created, just as an aircraft that moves faster than sound.

Especially for light particles, such as electrons or positrons at very high energy, the Bremsstrahlung emission represents the main energy loss mechanism. The deflection and the deceleration of the particle due to the interaction with the nuclei of the target cause the emission of photons; this effect is much greater as lighter is the particle (in fact the emission probability by Bremsstrahlung varies as the inverse square of the particle mass) and higher is the atomic number of target material. While ionization loss rates rise logarithmically with energy , Bremsstrahlung losses rise linearly and dominate at high energy (just only above few tens of *MeV* in most material for electrons). In Figure 2.1 it is shown the mean energy loss (also known as stopping power) for muons that traverse a cupper target in the range of few hundreds of *keV* to tens of* TeV*.

Figure 2.1 Stopping power for positive muons in Copper [**4**]

### Correction to Bethe-Bloch for electrons and positrons

Electrons or positrons needs particular consideration. First, their small mass implies the possibility of a large deflection due to a single collision too; moreover the collisions are between identical particles, so that the calculation must take into account their indistinguishability. As result the maximum transferable energy in a single collision becomes:

with *T _{e}*: kinetic energy of the incident particle, and the Bethe-Bloch formula can be rearranged as:

with:

where suffix “*+*” means positrons and “*–*” means electrons.

**References**

*William R. Leo, Techniques for Nuclear and Particle Physics Experiments. Berlin and Heidelberg: Springer, 1987.**Particle Data Group PDG, Passage of particles through matter, Nuclear and Particle Physics, vol. 33, no. 27, pp. 258-270, July 2006.**P.V.Vavilov, Ionization losses of high energy heavy particles, Soviet Physics JETP, 5:749, 1957.**S. Meroli et al., Energy loss measurement for charged particles in very thin silicon layers , JINST,***6**P06013 doi: 10.1088/1748-0221/6/06/P06013*Claude Leroy, Pier Giorgio Rancoita, Principles of radiation interaction in matter end detection. Singapore: World Scientific Publishing, 2004.**International Commission on Radiation Units and Measurements. [Online]. http://www.icru.org/**H. Bichsel, Straggling of Heavy Charged Particles: Comparison of Born Hydrogenic-Wave-Function Approximation with Free-Electron Approximation, Phys. Rev. B1 (1970) 2854**S. M. Sze, Kwok Kwok Ng, Physics of Semiconductor Devices. John Wiley & Sons, 2007.*