Effective Atomic Numbers of Heterogeneous Materials

Abstract

FOR a single element, the three γ-ray processes—photoelectric, Compton and pair production, can be expressed as a function of photon energy hν and the atomic number Z of the element. At a given photon energy, the interaction is proportional to Zⁿ where n is between 4 and 5 for the photoelectric effect, 1 for the Compton effect, and 2 for pair production^1,2. For the purposes of γ-ray attenuation, a heterogeneous material, consisting of a number of elements in varying proportions, can be described as a fictitious element having an effective atomic number Z_eff. In the study of the X-ray energy absorption of certain biological specimens Spiers³ used the expression Z^2.94_eff = ɛα_iZ^2.94_i, where Z_i is the atomic number of the ith element and α_i its fractional electronic content. More recent measurements⁴ favour a value of 3.1 instead of 2.94 for the exponent for Z_eff. Glasser⁵, in radiological studies, used the expression Z³_eff = ɛp_iZ⁴_i/ɛp_iZ_i, where p_i is the fractional part by weight of the whole mixture occupied by the element the atomic number of which is Z_i. Extending this idea of the effective atomic number of a material for specific radiation, Hine⁴ pointed out that there is a different effective atomic number for each absorption process in a heterogeneous material, but he could obtain expressions only for the photoelectric and pair production processes as Z^3.1_eff = ɛp_iZ^3.1_i, and Z_eff = ɛp_iZ_i. It can easily be shown that p_i and α_i are very nearly equal⁶. The equations of Spiers and Hine may then be expressed as Zⁿ⁻¹ = ɛp_iZⁿ⁻¹_i, where n has values of 4 to 5, 1 and 2, respectively for the photoelectric, Compton, and pair production processes. Hence, an expression for the effective atomic number for the Compton effect could not be obtained. In this communication the validity of these differing expressions is examined and a single effective atomic number is suggested for a heterogeneous material.