Bethe-Bloch formula

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The Bethe-Bloch formula (more precisely: Bethe formula, see below) describes the energy-loss by ionization of swift charged particles (protons, alpha particles, atomic ions, but not electrons) traversing matter.

Charged particles moving through matter interact with the electrons of atoms in the material. The interaction excites or ionizes the atoms. This leads to an energy loss of the traveling particle. The Bethe formula which was found by Hans Bethe in 1930, describes the energy loss per distance traveled (or the stopping power of the material traversed):

$- \frac{dE}{dx} = \frac{4 \pi}{m_e c^2} \cdot \frac{nz^2}{\beta^2} \cdot \left(\frac{e^2}{4\pi\varepsilon_0}\right)^2 \cdot \left[\ln \left(\frac{2m_e c^2 \beta^2}{I \cdot (1-\beta^2)}\right) - \beta^2\right]$ (1)

where

$β$	$= v / c$
$v$	velocity of the particle
$E$	energy of the particle
$x$	distance travelled by the particle
$c$	speed of light
$z\,e$	particle charge
$e$	charge of the electron
$m e$	rest mass of the electron
$n$	electron density of the target
$I$	mean excitation potential of the target
$\varepsilon_0$	permittivity of free space

Here, the electron density of the material can be calculated by $n=\frac{N_{A}\cdot Z\cdot\rho}{A}$ , where $ρ$ is the density of the material, $Z, A$ the atomic number and mass number, resp., and $N A$ the Avogadro number.

For low energies, i.e. for small (compared to c) velocities of the particle $( \beta \ll 1)$ , the energy loss according to formula (1) decreases approximately as $1 / v 2$ with increasing energy, and reaches a minimum for approx. $E = 3 M c 2$ , where $M$ is the mass of the particle. For highly relativistic cases $( \beta \approx 1)$ , the energy loss increases again, logarithmically; here, charged particles additionally experience energy loss due to the emission of bremsstrahlung.

Also, for higher projectile energies, the last term should be replaced by

$\frac{1}{2} \ln \left(\frac{2m_e c^2 \beta^2\gamma^2 T_{max}}{I^2}\right) - \beta^2$

where $T_{max} = \frac{2m_e c^2 \beta^2\gamma^2}{1+2\gamma m_e/M + (m_e/M)^2}$ , which at low energy would reduce to $2 m e c 2 β 2 γ 2$ , giving as expected the original low energy expression.^[1]

Felix Bloch has shown in 1933 that the mean ionization potential of atoms is approximately given by

$I = (10eV) \cdot Z$ (2)

where $Z$ is the atomic number of the atoms of the material. If this approximation is introduced into formula (1) above, one obtains an expression which is often called Bethe-Bloch formula. But since we have now more accurate tables of $I$ as a function of $Z$ , (for example, in ICRU Report 49 of the International Commission on Radiation Units and Measurements, 1993), the use of such a table will yield better results than the use of formula (2).

1 The problem of nomenclature
2 See also
3 Notes
4 External links

[edit] The problem of nomenclature

In describing programs PSTAR and ASTAR (for protons and alpha particles), the National Institute of Standards and Technology (USA) (www.physics.nist.gov/PhysRefData/Star/Text/programs.html) calls formula (1) "Bethe's stopping power formula", which is a reasonable designation.

On the other hand, in the newest Review of Particle Physics (W.-M. Yao et al., Journal of Physics G 33 (2006) 1), the formula is paradoxically called "Bethe-Bloch equation", even though Bloch's expression (2) does not appear in the formula.