Answer to Question #4080 Submitted to "Ask the Experts"

Would you explain the energy dependence of the linear attenuation coefficient? What is the relation between the linear attenuation coefficient and energy?

We shall restrict the discussion here to ionizing photons, x rays and gamma rays. The linear attenuation coefficient provides an indication of how effective a given material is, per unit thickness, in promoting photon interactions. The larger the value of the attenuation coefficient, the more likely it is that photons of a given energy will interact in a given thickness of material. The magnitude of the coefficient varies with material and, as you imply, with photon energy. While specific values of the attenuation coefficient will vary among materials for photons of a specified energy, the generalized shapes of plots (neglecting fine details) of attenuation coefficient versus photon energy are similar among different materials. In general, such shapes show high values of the attenuation coefficient at low-photon energies that decrease as photon energy increases, go through a rather broad minimum value, and then increase as energy continues to increase. The reason for this overall shape is that the linear attenuation coefficient is made up of three major components, each of which depends on a different type of photon interaction. At lower energies, a process called the photoelectric effect is the dominant interaction mode that has a strong energy dependence, decreasing approximately as the inverse cube of the energy. At intermediate energies the dominant interaction is Compton scattering which shows a generalized decrease with increasing energy. Finally, at higher energies the dominant process becomes pair production, and this shows an increase as energy increases (the absolute minimum energy at which this process can occur is 1.022 MeV). Thus, at low energies it is the sharply decreasing photoelectric contribution that causes the decrease in the attenuation coefficient as energy increases; the Compton process becomes more dominant and continues to decline (but at a slower rate than did the photoelectric process); when the energy gets sufficiently high the pair production contribution will exceed the Compton and the attenuation coefficient will eventually begin to increase.

For comparison, we might compare two materials, water and lead, which have very different effective atomic numbers (a major determining factor of the specific values of the attenuation coefficients), about 7 for water and 82 for lead. In the water the attenuation coefficient value at 10 keV is about 5.0 per cm, the Compton contribution to the attenuation coefficient becomes equal to the photoelectric contribution at about 25 keV, and the pair production contribution equals that of the Compton process at about 30 MeV; the overall value for the attenuation coefficient decreases until an energy of about 100 MeV is reached, at which point the pair production contribution becomes large enough to offset the continuously decreasing Compton contribution. In lead, the photoelectric and Compton contributions are equal at about 600 keV, the Compton and pair production contributions are equal at about 2 MeV, and the attenuation coefficient begins to increase in value again at about 4 MeV.

In addition to the generalized shape discussed above, there are also some small but important discontinuities that occur at lower energies. These discontinuities demonstrate sharp changes in the photoelectric contribution to the attenuation coefficient when the photon energy just exceeds the binding energy of the electron in a shell of an atom of the given material. These sharp changes are most noticeable in higher atomic number materials because the binding energies of electrons in the inner electron shells of these materials are sufficiently high that they are comparable to some ionizing photon energies of interest. For example, lead shows a sharp increase in the attenuation coefficient at about 88 keV, which is about the binding energy of K-shell electrons in lead. In water such discontinuities, often referred to as absorption edges, would not be evident until the energy decreased below 1 keV.

I hope this answer is appropriate for your needs.

George Chabot, PhD, CHP