Radiation length

In particle physics, the radiation length is a characteristic of a material, related to the energy loss of high energy particles electromagnetically interacting with it. It is defined as the mean length (in cm) into the material at which the energy of an electron is reduced by the factor 1/e.^[1]

Definition

In materials of high atomic number (e.g. tungsten, uranium, plutonium) the electrons of energies >~10 MeV predominantly lose energy by bremsstrahlung, and high-energy photons by e⁺e⁻ pair production. The characteristic amount of matter traversed for these related interactions is called the radiation length X₀, usually measured in g·cm⁻². It is both the mean distance over which a high-energy electron loses all but 1⁄e of its energy by bremsstrahlung,^[1] and 7⁄9 of the mean free path for pair production by a high-energy photon. It is also the appropriate length scale for describing high-energy electromagnetic cascades.

The radiation length for a given material consisting of a single type of nucleus can be approximated by the following expression:^[2]

$${\displaystyle X_{0}=716.4{\text{ g cm}}^{-2}{\frac {A}{Z(Z+1)\ln {\frac {287}{\sqrt {Z}}}}}=1433{\text{ g cm}}^{-2}{\frac {A}{Z(Z+1)(11.319-\ln {Z})}},}$$

where Z is the atomic number and A is mass number of the nucleus.

For Z > 4, a good approximation is^{[3][inconsistent]}. $${\displaystyle {\frac {1}{X_{0}}}=4\left({\frac {\hbar }{m_{\mathrm {e} }c}}\right)^{2}Z(Z+1)\alpha ^{3}n_{\mathrm {a} }\log \left({\frac {183}{Z^{1/3}}}\right),}$$

where

• n_a is the number density of the nucleus,
• ${\displaystyle \hbar }$ denotes the reduced Planck constant,
• m_e is the electron rest mass,
• c is the speed of light,
• α is the fine-structure constant.

For electrons at lower energies (below few tens of MeV), the energy loss by ionization is predominant.

While this definition may also be used for other electromagnetic interacting particles beyond leptons and photons, the presence of the stronger hadronic and nuclear interaction makes it a far less interesting characterisation of the material; the nuclear collision length and nuclear interaction length are more relevant.

Comprehensive tables for radiation lengths and other properties of materials are available from the Particle Data Group.^[2][4]

See also

• Mean free path
• Attenuation length
• Attenuation coefficient
• Attenuation
• Range (particle radiation)
• Stopping power (particle radiation)
• Electron energy loss spectroscopy

References

1. M. Gupta; et al. (2010). "Calculation of radiation length in materials". PH-EP-Tech-Note. 592 (1–4): 1. arXiv:astro-ph/0406663. Bibcode:2004PhLB..592....1P. doi:10.1016/j.physletb.2004.06.001.
2. S. Eidelman; et al. (2004). "Review of particle physics". Phys. Lett. B. 592 (1–4): 1–5. arXiv:astro-ph/0406663. Bibcode:2004PhLB..592....1P. doi:10.1016/j.physletb.2004.06.001. (http://pdg.lbl.gov/)
3. De Angelis, Alessandro; Pimenta, Mário (2018). Introduction to Particle and Astroparticle Physics (2 ed.). Springer. Bibcode:2018ipap.book.....D. doi:10.1007/978-3-319-78181-5. ISBN 978-3-319-78180-8.
4. "AtomicNuclearProperties on the Particle Data Group". Archived from the original on 2021-07-24. Retrieved 2008-01-26.