Theory | Lars Fredrik Fjaera

## Theory

#### The Bethe-Bloch Formula

Heavy charged particles going through matter lose energy mainly by collisions with bound electrons. In these collisions, two processes are dominant. There is excitation, which is when the electrons receive energy and thereby goes into a higher energy state, and there is ionization which is when the bound electrons get ejected from the atom. And so, the average energy loss per unit length is given by the Bethe-Bloch formula:

$\bigg&space;\langle&space;\frac{dE}{dx}&space;\bigg&space;\rangle&space;=K&space;z^2&space;\rho&space;\frac{Z}{A}\frac{1}{\beta^2}&space;\bigg[\frac{1}{2}\ln{\frac{2{m_e}c^2\beta^2\gamma^2T_{\text{max}}}{I^2}-\beta^2-\frac{\delta(\beta\gamma)}{2}-\frac{C}{Z}}&space;\bigg],$

where

$K&space;=&space;4\pi&space;N_0r_e^2m_ec^2&space;=&space;0.307&space;MeV&space;cm^2&space;g^{-1}$

and Tmax is the maximum energy that can be transferred to a free electron in a single collision given by

$T_{max}&space;=&space;\frac{2m_ec^2\beta^2\gamma^2}{1+2\frac{m_e}{m}\sqrt{1+\beta^2\gamma^2}+\frac{m_e^2}{m^2}}$

Often used is the mass stopping power due to its small variation over a wide range of particles. We obtain it by diving by the density of the absorbing material:

$\bigg&space;\langle&space;\frac{dE}{dx}&space;\bigg&space;\rangle&space;=&space;\frac{1}{\rho}\bigg&space;\langle&space;\frac{dE}{dx}&space;\bigg&space;\rangle$

The following table provides the variables used in the Bethe-Bloch equation:

 Symbol Definition Unit and/or value $\rho$ Density of material $g/cm^3$ $Z$ Atomic number of material $A$ Atomic mass of material $\beta$ $v/c$ of incident particle $m_e$ Electron mass $MeV/c^2$ $m$ Mass of incident particle $MeV/c^2$ $c$ Speed of light in vacuum $3\times&space;10^8&space;m/s$ $\gamma$ Lorentz factor $\sfrac{1}{\sqrt{1-\beta^2}}$ $I$ Mean excitation potential $eV$ $\delta(\beta&space;\gamma)$ Density effect correction $C$ Shell correction $N_0$ Avogadro's number $6.022&space;\times&space;10^{23}&space;mol^{-1}$ $r_e$ Classical electron radius $2.818&space;fm$ $v$ Speed of incident particle $m/s$

The mean excitation potential can be calculated empirically from the following formula:

$I&space;=&space;\left\{\begin{matrix}&space;I_0&space;\cdot&space;Z&space;+&space;I_c,&space;&&space;\text{if}&space;\&space;Z&space;<&space;13,&space;&&space;I_0&space;=&space;12&space;eV,&space;\&space;I_c&space;=&space;7&space;eV\\&space;I_0&space;\cdot&space;Z&space;+&space;I_c&space;\cdot&space;Z^{-0.19}&space;&&space;\text{if}&space;\&space;Z&space;\geq&space;13,&space;&&space;I_0&space;=&space;9.8eV,&space;I_c&space;=&space;58.8&space;eV&space;\end{matrix}\right.$

The δ and the C are the density- and shell corrections respectively. The density correction becomes important at high energies, while the shell correction is important at low energies. The shell correction arises when the velocity of the incoming particle approaches the orbital velocity of the bound electrons in the absorbing material. The Bethe-Bloch equation builds on the assumption that the electrons are stationary with respect to the incoming particle. This assumption breaks down for low velocities, and therefore the shell correction is needed, which is generally small. It is valid for η ≥ 0.1 and can be calculated by the empirical formula:

$C(I,\eta)&space;&&space;=&space;(0.422377\eta^{-2}+0.0304043&space;\eta^{-4}-0.00038106&space;\eta^{-6})\times&space;10^{-6}&space;I^2&space;\\&space;&&space;+(3.850190&space;\eta^{-2}-0.1667989&space;\eta^{-4}+0.00157955&space;\eta^{-6})&space;\times&space;10^{-9}&space;I^3,$

where I is the mean excitation potential in eV and η = βγ.

When the energy and thus the velocity of the incoming particle is high, its electric field will flatten and extend. This electric field tends to polarize atoms along its path. Due to this polarization, electrons far from the incoming particle will be shielded for the full electric field. Collisions with these shielded electrons will contribute less to the total energy loss than predicted by the Bethe-Bloch formula. The density effect increases with higher density since the polarization will be larger in condensed materials. The density correction is usually calculated using Sternheimer's parametrization given by:

$\delta(\beta&space;\gamma)&space;=&space;\left\{\begin{matrix}&space;2(\ln&space;10)x&space;-&space;\bar{C},&space;&&space;x&space;\geq&space;x_1\\&space;2(\ln&space;10)x&space;-&space;\bar{C}+a(x_1-x)^k&space;&&space;x_0&space;\leq&space;x&space;<&space;x_1\\&space;0&space;&&space;x&space;<&space;x_0&space;\&space;\text{(nonconductors)}\\&space;\delta_0&space;10^{2(x-x_0)}&space;&&space;x&space;<&space;x_0&space;\&space;\text{(conductors)}&space;\end{matrix}\right.$

The values x0, x1, C, and k depend upon the absorbing material. C in terms of plasma frequency is given by:

$\bar{C}&space;=&space;-\bigg&space;(&space;2&space;\ln\bigg(\frac{I}{\hbar&space;\omega_p}\bigg)+1&space;\bigg&space;)$

where the plasma frequency is

$\omega_p&space;=&space;\sqrt{\frac{N_e&space;e^2}{\pi&space;m_e}}$

#### Energy loss in composite materials

As the Bethe-Bloch formula depends both on the density, atomic- and mass number of the absorber, one need to use the effective values for composite materials. They are given by:

$Z_{\text{eff}}&space;=&space;\sum&space;a_i&space;Z_i$
$A_{\text{eff}}&space;=&space;\sum&space;a_i&space;A_i$

where ai is the amount (weight) of the different elements in the composite material.

#### Range of particles

The range of the particle depends on the type of material, the particle type and of course its energy. Obtaining a simple representation of the range of charged particles in matter is close to impossible due to different energy loss mechanisms. When a particle is traversing a material there can occur interactions with high energy transfers. One also has multiple Coulomb scattering in the material (particles will go in a zig-zag curve as opposed to a straight line) all of which lead to substantial range straggling. Ignoring these uncertainties and assuming that the energy loss is more or less continuous, one can use the continuous slowing down approximation (CSDA) to get an approximation of the range of the particle. In this approximation, the rate of energy loss at every point along the track is assumed to be equal to the total stopping power. The CSDA range is obtained by integrating the reciprocal of the total stopping power with respect to energy. CSDA is given by

$R(E_{\text{kin}})&space;=&space;\int_{0}^{E_{\text{kin}}}&space;\bigg&space;(\bigg&space;\langle&space;\frac{dE}{dx}&space;\bigg&space;\rangle&space;\bigg&space;)^{-1}&space;dE$

#### References

• Bichsel, H. & Klein, S.R. (Revisors) (2014), 32. Passage of Particles Through Matter, Chinese Physics C - Volume 38 Number 9
• Grupen, Claus & Shwartz, Boris (2008), Particle Detectors, 2nd Edition; Cambridge University Press
• Leo, William R. (1994), Techniques for Nuclear and Particle Physics Experiments, 2nd Edition; Springer-Verlag