### 3. Interpolation and Combination

For the purpose of interpolation with respect to photon energy, the coherent
and incoherent scattering cross sections and the total attenuation coefficients
are approximated by log-log cubic-spline fits as functions of energy. For the
pair-production cross sections, the fitted quantity is the logarithm of the
quantity (1-E/E′)^{3}
*σ*_{PAIR}(E), where E is the
photon energy, E′ the threshold energy
for pair production, and *σ*_{PAIR}(E) is the cross section.
The fitting is done separately for pair production in the field of the atomic
nucleus (E′ = 1.022 MeV)
and in the field of the atomic electrons (E′ = 2.044 MeV).
The combined photoelectric absorption cross section for all shells is similarly
interpolated with use of log-log cubic-spline fits, but only at energies above
the K-shell absorption edge. Below this energy, interpolation is applied to the
logarithm of the photoelectric absorption cross section for each separate shell,
fitted as a linear function of the logarithm of the photon energy. The separate
fitting for each shell is necessary to avoid the error that would be incurred
by interpolating across absorption edges. Linear log-log fitting is equivalent
to assuming that the photoelectric cross section is proportional to a power of
the photon energy, and was found to provide more satisfactory fits than a
log-log cubic-spline fit near the absorption edges.

The interaction coefficients and total attenuation coefficients for
compounds are obtained as weighted sums over the corresponding coefficients
for elements. XCOM automatically calculates the weight factors, i.e., the
fractions by weights of the atomic constituents, from the chemical formula
for the compound entered by the user. For mixtures, the user must enter the
fractions by weight of the components.