Simple expressions based on relativistic impulse approximation for obtaining accurate whole atom Compton Profiles from photon scattering doubly differential cross sections

There is continuing interest in the measurement of and the theoretical calculations of the values for Compton profiles (CPs) due to their many applications. Since CPs can only be obtained by measurement of physical observables such as photon or electron scattering doubly differential cross sections (DDCS), the accuracy of their values depends partly on the validity of the formulas used to transform DDCS into CP. Here such expressions, to be referred to as RKJ, based on relativistic impulse approximation (RIA) were derived for all K, L, M and N subshell nonrelativistic hydrogenlike wavefunctions. The derivation was done such that much of the relativistic and screening factors cancel, while the relativistic kinematic factors are retained making it possible to obtain CP from DDCS to within a few percent error over at least 99{\%} of the electron momentum distribution range at any energy ($\omega )$ in which RIA is valid and for atoms having a nuclear charge as high as 100. Good results over limited regimes of scattering angle and $\omega $ were obtained from two simpler RKJ expressions, one derived from the zero photon scattering angle limit the other from the high energy limit of full RKJ expression.