Answer to Question #12845 Submitted to "Ask the Experts"

I am confused about the fraction of the electron energy that is converted to bremsstrahlung. In a bibliography I found the following expressions: the fraction of electron energy that appears as bremsstrahlung is ZE/3000 (Cherry et al. 2012; Evans 1955) and ZE/700 (Attix and Roesch 1968).

Is there something I don't consider? Which expression is preferable, or which one do you use in your calculations?

The different expressions that you cite are all valid, but the difficulty comes in knowing when each expression is applicable. Your question reminds me that we often adopt and use various technical recipes and rules-of-thumb expressions that have a historical context and development that may have been neglected or forgotten over time, which leaves us in danger of misapplying the expression under conditions that aren’t relevant. The expressions you cite tend to fall into this category.

I have used some of them often and have seen them used by others, sometimes in an inappropriate fashion. You state that the expression, ZE/3000, as shown in Evans (1955). and elsewhere, represents "the fraction of electron energy that appears as bremsstrahlung." In actuality, however, if you have the opportunity to go back and reread the section in Evans that describes how this expression came about, you will find that it was intended to apply to the whole energy distribution of beta particles associated with decay of a particular radionuclide, characterized by a particular maximum beta energy. The assumption is that all of the beta radiation gets stopped in a thick target attenuator; the term, Z represents the atomic number of the target material in the expression, and E represents the maximum beta particle energy (note, though, that the energy used in the development to represent the energy associated with the typical beta particles was the average energy for the beta spectrum). The fraction of the beta ray energy that appears as bremsstrahlung is not the same as the fraction of a monoenergetic electron’s energy that appears as bremsstrahlung.

In fact, in the pages prior to Evans' discussion of bremsstrahlung from beta radiation, he develops the expression for the fraction I/E, of an initially monoenergetic electron's energy that appears as bremsstrahlung when the electron is stopped in a thick target. The expression he provides for the monoenergetic electron case (for energies up to about 2.5 MeV) is

I/E = 0.0007ZE, which is equivalent to ZE/1400   (1)

where I is the bremsstrahlung energy, and E is the electron's initial energy. Evans goes on to incorporate this expression into his development when considering the distribution of energies associated with beta decay.

If you are able to relook at the expression you cite from Attix and Roesch (1968), I believe you will find that the quantity ZE/700 does not represent the average fraction of an electron's energy lost by bremsstrahlung but rather represents the ratio of the radiative mass stopping power to the collision mass stopping power—i.e.,

(dE/ρdx)_rad/(dE/ ρdx)_coll = ZE/n,   (2)

where the recommended values of n are 700 for electron energies above 3 MeV and (700 + 200 log₁₀[E/3]) for energies between 0.01 MeV and 3 MeV. I do not have the Attix and Roesch text available, but this development is covered in Attix (1986). This stopping power ratio provides a snapshot in time of the relative energy loss by bremsstrahlung compared to collision losses, but it only applies to the electron at the specific energy, E. It does not apply to the cases considered above in which an electron (or beta particles) of some initial energy, E, loses all its energy in a thick target of atomic number Z.

For a thick target in which an electron will lose all of its energy, the ratio of radiative to collision mass stopping powers may be integrated over all energies from 0 to the initial electron energy and the result divided by the initial electron energy to obtain the radiation yield, which is the fraction of the electron energy represented by the bremsstrahlung radiation when the electron loses all its energy in a given material. These radiation yields are published in column 6 of Appendix E of the Attix (1986). As an example, consider a 1 MeV electron being stopped in tin (Z = 50); the text shows the radiation yield as 0.03666. If we apply equation 1 above we would get an estimate 0.036, quite close. At 2 MeV, equation 1 would yield 0.071, which compares to the Attix table value of 0.06284, not quite as good agreement, but still a reasonable estimation. Greater deviations may be found at other energies.

In summary, these convenience expressions can be quite useful for providing quick estimations of the expected bremsstrahlung radiation for electrons or beta particles losing their energies in thick target materials. None of them provide exact results for all energies, and their use must be accompanied by a recognition of their limitations. Thanks again for the question. It provided a good opportunity for some useful review.

George Chabot, PhD

References

Attix FH, Roesch WC. Radiation dosimetry. 2nd ed. New York: Academic Press; 175; 1968.

Attix FH. Introduction to radiological physics and radiation dosimetry. Hoboken, New Jersey: Wiley & Sons; 176; 1986.

Cherry SR, Sorenson JA, Phelps ME. Physics in nuclear medicine. 4th ed. Amsterdam: Elsevier; 65; 2012.

Evans R. The atomic nucleus. New York: McGraw Hill Book Company, Inc.; 619; 1955.