Bethe Formula

The Bethe formula (also Bethe equation , Bethe-Bloch formula , Bethe-Bloch equation or braking formula ) indicates the energy loss per unit of path length, which fast charged heavy particles (e.g. protons , alpha particles , ions ) pass through when passing through Sustaining matter through inelastic collisions with the electrons; the transferred energy causes excitation or ionization in the material . This energy loss, also referred to as electronic deceleration or, inaccurately, ionization loss, depends on the speed and charge of the projectile particles and the target material.

The classic non- relativistic formula was drawn up by Niels Bohr as early as 1913 , the quantum-mechanical non-relativistic formula was drawn up in 1930, the quantum-mechanical- relativistic version shown below was drawn up by Hans Bethe in 1932 .

The Bethe-Bloch formula does not apply to incident electrons. On the one hand, the energy loss for these is different because they are indistinguishable from the shell electrons of the material. On the other hand, electrons have a significant energy loss due to bremsstrahlung because of their low mass . The energy loss of electrons can instead be described with the help of the Berger-Seltzer formula .

Other mechanisms that can contribute to the overall energy loss of faster charged heavy particles in matter are nuclear deceleration (elastic Coulomb collisions with atomic nuclei, see braking capacity ) and bremsstrahlung .

The formula

Braking power of aluminum for protons as a function of the energy of the proton, with Bethe formula without (red) or with corrections (blue)

When fast charged particles move through matter, they perform inelastic collisions with the shell electrons of the material. This leads to the excitation or ionization of the atoms. As a result, the traversing particle suffers an energy loss, which is approximated by the following formula. Its relativistic form is:

${\displaystyle -{\frac {\mathrm {d} E}{\mathrm {d} x}}={\frac {4\pi nz^{2}}{m_{\rm {e}}c^{2}\beta ^{2}}}\cdot \left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{2}\cdot \left[\ln \left({\frac {2m_{\rm {e}}c^{2}\beta ^{2}}{I\cdot (1-\beta ^{2})}}\right)-\beta ^{2}\right]\qquad \qquad }$ (1)

in which

${\displaystyle \beta }$	= ${\displaystyle v/c}$
${\displaystyle v}$	= instantaneous speed of the particle
${\displaystyle c}$	= Speed ​​of light
${\displaystyle E}$	= Energy of the particle
${\displaystyle x}$	= Path length
${\displaystyle z}$	= Number of charges of the particle ( = charge of the particle) ${\displaystyle z\cdot e}$
${\displaystyle \varepsilon _{0}}$	= Electric field constant
${\displaystyle e}$	= Elementary charge
${\displaystyle n}$	= Electron density of the material
${\displaystyle m_{\rm {e}}}$	= Mass of the electron
${\displaystyle I}$	= mean excitation potential of the material ( see below )

The electron density can also be calculated; is the density of the material, and the atomic or mass number of the material and the atomic mass unit . ${\displaystyle n}$ ${\displaystyle n={\frac {Z\cdot \rho }{A\cdot u}}}$ ${\displaystyle \rho }$ ${\displaystyle Z}$ ${\displaystyle A}$ ${\displaystyle u}$

In the picture on the right, the small circles mean measurement results from different work groups; the red curve represents the Bethe formula. Obviously, the agreement of Bethes theory with the experiments above 0.5 MeV is very good, especially if the corrections (see below) are added (blue curve).

For small energies, i.e. H. small particle velocities, the Bethe formula is reduced to ${\displaystyle (\beta \ll 1)}$

${\displaystyle -{\frac {\mathrm {d} E}{\mathrm {d} x}}={\frac {4\pi nz^{2}}{m_{e}v^{2}}}\cdot \left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{2}\cdot \ln \left({\frac {2m_{e}v^{2}}{I}}\right).}$

For small energies, the Bethe formula is only valid if they are still high enough that the traversing particle does not carry any shell electrons with it. Otherwise, its effective load will be reduced and the braking power will be less. For small energies there is a refined theory of electronic deceleration by Jens Lindhard , Morten Scharff and Hans E. Schiøtt ( LSS theory ). The Barkas formula is also used approximately for the effective number of charges:

${\displaystyle z_{\mathrm {eff} }=z\left(1-\exp \left(-125\beta z^{-{\frac {2}{3}}}\right)\right)}$

In general, the energy loss initially decreases with increasing energy and reaches a minimum at around , where the mass of the particle is (e.g. for protons around 3 GeV, which is no longer visible in the picture). Since for many in the particle , the energy loss in the vicinity of the minimum has the same value relevant radiation particles and absorbent materials about, particles (be combined often with an energy in the vicinity of the minimum and as MIPs minimum Ionizing Particles , dt. Minimum ionizing particles ) denotes . The rule of thumb for the specific energy loss of the MIPs is: ${\displaystyle 1/v^{2}}$ ${\displaystyle E=3m_{T}c^{2}}$ ${\displaystyle m_{T}}$

${\displaystyle -{\frac {1}{\rho }}{\frac {\mathrm {d} E}{\mathrm {d} x}}\approx 2{\frac {\mathrm {MeV} \mathrm {cm} ^{2}}{\mathrm {g} \;}}}$.

If the energy is even higher, the energy loss increases again. In the case of very high energies, particle reactions that lead to secondary particles must also be taken into account. The energy loss can therefore increase even more, depending on the material.

In radiation biology , the energy released by ionizing particles is called linear energy transfer ( ) according to the Bethe-Bloch equation, and the unit used is kiloelectron volts per micrometer (keV / µm). ${\displaystyle LET_{\infty }}$

The mean excitation potential

In the area of ​​validity of the Bethe formula (1), the penetrated material is only described by a single constant, the mean excitation potential , in addition to the particle density . ${\displaystyle n}$ ${\displaystyle I}$

Felix Bloch showed in 1933 that the mean excitation potential of the atoms is about

${\displaystyle I=(10\,\mathrm {eV} )\cdot Z\qquad \qquad }$ (2)

where means the atomic number of the atoms of the material. Inserting this quantity into formula (1) above leads to an equation that is often referred to as the Bethe-Bloch equation . But there are more precise tables of as a function of . With them you get better results than with formula (2). ${\displaystyle Z}$ ${\displaystyle I}$ ${\displaystyle Z}$

The mean excitation potential of elements divided by the ordinal number plotted against the ordinal number ${\displaystyle I}$ ${\displaystyle Z}$

The picture shows the mean excitation potential of the various elements, which contains the information about the respective atom. The data come from the aforementioned ICRU report. The peaks and valleys in the illustration (" oscillations", where the atomic number of the material means) correspond to lower or higher values ​​of the braking capacity; these oscillations are based on the shell structure of the atoms. As the picture shows, formula (2) only applies approximately. ${\displaystyle Z_{2}}$ ${\displaystyle Z_{2}}$

Corrections to the Bethe formula

The Bethe formula was derived by Bethe using quantum mechanical perturbation theory , the result is therefore proportional to the square of the charge . A better description is obtained if one also takes into account deviations that correspond 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 shell electrons in the atom of the material must also be taken into account (“shell correction”). ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle z^{3}}$ ${\displaystyle z^{4}}$

These corrections are built into the programs PSTAR and ASTAR of the National Institute of Standards and Technology (NIST), for example , which calculate the braking capacity for protons and alpha particles. The corrections are great at low energies and become smaller and smaller as the energy increases.

In addition, Fermi's density correction is added for very high energies .

literature

• P. Sigmund: Particle Penetration and Radiation Effects, General Aspects and Stopping of Swift Point Charges (= Springer Series in Solid State Sciences. Vol. 151). Springer, Berlin / Heidelberg 2006, ISBN 978-3-540-72622-7 .
• H. Bethe: On the theory of the passage of fast corpuscular rays through matter . In: Annals of Physics . tape 397 , no. 3 , 1930, p. 325-400 , doi : 10.1002 / andp.19303970303 (Original publication by Bethe).
• F. Bloch: To brake rapidly moving particles when passing through matter . In: Annals of Physics . tape 408 , no. 3 , 1933, pp. 285-320 , doi : 10.1002 / andp.19334080303 (Original publication by Bloch).
• N. Bohr: On the theory of the decrease of velocity of moving electrified particles on passing through matter . In: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . tape 25 , no. 145 , 1913, pp. 10–31 , doi : 10.1080 / 14786440108634305 (preliminary work by Bohr).
• N. Bohr: On the decrease of velocity of swiftly moving electrified particles in passing through matter . In: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . tape 30 , no. 178 , October 1915, p. 581–612 , doi : 10.1080 / 14786441008635432 (preliminary work by Bohr).

Web links

• Passage of charged particles through matter, including plot (English; PDF file; 512 kB)
• Stopping power for protons and alpha particles (Engl.)
• Stopping Power data and curves (Engl.)

Individual evidence

1. ↑ N. Bohr: On the theory of the decrease of velocity of moving electrified particles on passing through matter . In: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . tape 25 , no. 145 , 1913, pp. 10-31 , doi : 10.1080 / 14786440108634305 .
2. ↑ P. Sigmund: Particle Penetration and Radiation Effects, General Aspects and Stopping of Swift Point Charges (= Springer Series in Solid State Sciences. Vol. 151). Springer, Berlin / Heidelberg 2006, ISBN 978-3-540-72622-7 .
3. ↑ HA Bethe, J. Ashkin: Passage of radiation through matter . In: E. Segré (Ed.): Experimental Nuclear Physics . Vol. 1, Part II. New York 1953, pp. 253 .
4. ↑ cern.ch: Ionization ( Memento of the original from December 14, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
5. ^ Image source ( Memento from February 6, 2012 in the Internet Archive )
6. ↑ K. Bethge, G. Walter, B. Wiedemann: Kernphysik . 3rd edition, Springer, 2007, pp. 118, 121.
7. ↑ Jürgen Kiefer : Biological radiation effects. An introduction to the basics of radiation protection and the application of radiation. Heidelberg (Springer) 1981, ISBN 978-3-642-67947-6 , p. 47
8. ↑ Claude Amsler: Nuclear and Particle Physics . vdf Hochschulverlag AG, 2007, ISBN 978-3-8252-2885-9 , p. 116 ( limited preview in Google Book search).
9. ↑ F. Bloch: To slow down rapidly moving particles when passing through matter . In: Annals of Physics . tape 408 , no. 3 , 1933, pp. 285-320 , doi : 10.1002 / andp.19334080303 .
10. ↑ ^{a b c} ICRU Report 49, Stopping Powers and Ranges for Protons and Alpha Particles . International Commission on Radiation Units and Measurements, Bethesda, MD, USA (1993)
11. ↑ PSTAR and ASTAR Databases for Protons and Helium Ions