To follow up, in our recently published paper “An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds by J. Donovan, A. Ducharme, J. J. Schwab, A. Moy, Z. Gainsforth, B. Wade, and B. McMorran" (
https://doi.org/10.1093/micmic/ozad069), it may be unclear to the reader how Eqs. 6 and 7 are derived from the theoretical calculations presented in the “The Differential Scattering Cross Section of the Yukawa Potential” section.
Here is a simple derivation of these equations.
Consider a compound with density ρ and molecular mass M. Its macroscopic cross section is the sum of the microscopic (differential) cross sections of each element j present in the compound
times the atomic density N
j, or:
The microscopic differential cross section and atomic density have units of cm
2/sr and atoms/cm
3, respectively. The atomic density of element j is given by:
,
with c
j as the mass fraction of element j and A
j as the atomic mass (in g/mol) of the element j. N
A is Avogadro’s number (in atoms/mol) and ρ is the density of the material (in g/cm
3).
The atomic density can also be expressed using atomic fractions a
j instead of the mass fraction:
with
So,
If we define the mean atomic number as the atomic number of each elemental constituent averaged over the macroscopic cross section above,
At small scattering angle, we have shown that dσ/dΩ is equal to
(p.7 of the paper), hence:
This expression is the mean atomic number we presented in Eqs. 6 and 7 in our paper “An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds” (Microscopy and Microanalysis (2023)).
I share credit for this explanation with Andrew Ducharme who improved the math and provided a clear interpretation of the probability of the expected atomic number.