Bethe formula

In nuclear physics and theoretical physics, 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 describes[1] the energy loss per distance travelled of swift charged particles (protons, alpha particles, atomic ions, but not electrons[Footnote 1]) traversing matter (or, alternatively, the stopping power of the material). The non-relativistic version was found by Hans Bethe in 1930; the relativistic version (shown below) was found by him in 1932 (Sigmund 2006).

The Bethe formula is sometimes called "Bethe-Bloch formula", but this is misleading (see below).

The formula

The relativistic version of the formula reads:

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

where
\( \beta \) = 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
\( \varepsilon_0 \) vacuum permittivity
Stopping Power of Aluminum for Protons versus proton energy, and the Bethe formula

Here, the electron density of the material can be calculated by \( n=\frac{N_{A}\cdot Z\cdot\rho}{A \cdot M_{u}} \) , where \( \rho \) is the density of the material, Z, A its atomic number and mass number, respectively, \( N_A \) the Avogadro number and \( M_{u} \) the Molar mass constant.

In the figure to the right, the small circles are experimental results obtained from measurements of various authors (taken from http://www.exphys.uni-linz.ac.at/Stopping/); the red curve is Bethe's formula. Evidently, Bethe's theory agrees very well with experiment at high energy. Only below 0.3 MeV, the curve is too low; here, corrections are necessary (see below).

For low energies, i.e., for small velocities of the particle ( \( \beta \ll 1 \) ), the Bethe formula reduces to

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

At low energy, the energy loss according to the Bethe formula therefore decreases approximately as \( 1/v^2 with increasing energy. It reaches a minimum for approx. \( E = 3Mc^2 \) , where M is the mass of the particle (for protons, this would be about at 3000 MeV). For highly relativistic cases ( \beta \approx 1 \) ), the energy loss increases again, logarithmically due to the transversal component of the electric field.
The mean excitation potential
The mean excitation potential I of atoms, versus atomic number Z, in eV, divided by Z

In the Bethe theory, the material is completely described by a single number, the mean excitation potential I. Felix Bloch has shown in 1933 that the mean ionization potential of atoms is approximately given by

\( I = (10eV) \cdot Z \) (3)

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 accurate tables of I as a function of Z (see below),the use of such a table will yield better results than the use of formula (3).

The figure shows normalized values of I, taken from a table.[2] The peaks and valleys in this figure lead to corresponding valleys and peaks in the stopping power. These are called "Z2-oscillations" or "Z2-structure" (where Z2 means the atomic number of the target).
Corrections to the Bethe formula

The Bethe formula is only valid for energies high enough so that the charged atomic particle (the ion) does not carry any atomic electrons with it. At smaller energies, when the ion carries electrons, this reduces its charge effectively, and the stopping power is thus reduced. But even if the atom is fully ionized, corrections are necessary.

Bethe found his formula using quantum mechanical perturbation theory. Hence, his result is proportional to the square of the charge z of the particle. The description can be improved by considering corrections which correspond to higher powers of z. These are: the Barkas-Andersen-effect (proportional to \( z^3 \) , after Walter H. Barkas and Hans Henrik Andersen), and the Bloch-correction (proportional to \( z^4 \) ). In addition, one has to take into account that the atomic electrons are not stationary ("shell correction").

These corrections have been built into the programs PSTAR and ASTAR, for example, by which one can calculate the stopping power for protons and alpha particles.[3] The corrections are large at low energy and become smaller and smaller as energy is increased.

At very high energies, Fermi's density correction[2] has to be added also.
The problem of nomenclature

In describing programs PSTAR and ASTAR, the National Institute of Standards and Technology[3] calls formula (1) "Bethe's stopping power formula".

On the other hand, in the 2008 Review of Particle Physics[4] the formula was called "Bethe-Bloch equation", even though Bloch's expression (3) did not appear in the formula. As of the most recent edition, this seems to have disappeared with the formula being called only the "Bethe formula".[5]
See also

Stopping power (particle radiation)
Hans Bethe

Footnote

^ For electrons, the energy loss is slightly different due to their small mass and their indistinguishability, and since they suffer much larger losses by Bremsstrahlung

References

Sigmund, Peter (2006). Particle Radiation and Radiation Effects. Springer Series in Solid State Sciences, 151. Berlin Heidelberg: Springer-Verlag. ISBN 3-540-31713-9

^ H. Bethe und J. Ashkin in "Experimental Nuclear Physics, ed. E. Segré, J. Wiley, New York, 1953, p. 253
^ a b Report 49 of the International Commission on Radiation Units and Measurements, "Stopping Powers and Ranges for Protons and Alpha Particles", Bethesda, MD, USA (1993), http://www.icru.org/index.php?option=com_content&task=view&id=74
^ a b NISTIR 4999, Stopping Power and Range Tables, www.physics.nist.gov/PhysRefData/Star/Text/programs.html
^ 2008 Review of Particle Physics, C. Amsler et al., Physics Letters B 667 (2008) 1
^ K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010) and 2011 partial update for the 2012 edition.

External links

The Energy Loss of charged particles passing through matter
Original Publication: Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie in "Annalen der Physik", Vol. 397 (1930) 325 -400
Passage of charged particles through matter, with a graph
Stopping power for protons and alpha particles
Stopping Power graphs and data

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