Criteria for using impulse approximation to obtain Compton scattering doubly differential cross sections

We find that the criterion often used for predicting when impulse approximation (IA) theory yields accurate doubly differential cross sections (DDCS), namely $/q \leq 1$, where $$ is the expectation value of the momentum distribution of the bound electron and q is the magnitude of the photon momentum transfer, which is much less restrictive than the assumptions on which IA theory is based ($/q << 1$), is not generally dependable. We examine the IA error $\Delta $, where $\Delta = (DDCS_{SM}-DDCS_{RIA})/DDCS_{SM}$ ($DDCS_{SM}$ and $DDCS_{RIA}$ are the peak magnitudes for S-matrix and relativistic IA derived DDCS respectively). One striking feature is that, for a given incident photon energy $\omega_i$ and nuclear charge Z, $\Delta $ goes from negative to positive as the scattering angle $\theta $ increases. Further, when $/q$ is held constant at a value less than unity, $\Delta $ changes sign at nearly the same $\theta $ for all Z. Therefore, when $\theta $ is large or small, $/q<<1$ is generally required in order for IA derived DDCS to be valid, while at intermediate $\theta $, $/q \approx 1$ is typically sufficient, since $\Delta $ is small. The $\theta $ at which $\Delta $ changes sign increases as $/q$ increases.