  The complex interplay of elastic and inelastic scattering amenable to
different levels of approximation constitutes the major challenge for the
computation and hence interpretation of TEM-based spectroscopical methods. The
two major approaches to calculate inelastic scattering cross sections of fast
electrons on crystals-Yoshioka-equations-based forward propagation and the
reciprocal wave method-are founded in two conceptually differing schemes-a
numerical forward integration of each inelastically scattered wave function,
yielding the exit density matrix, and a computation of inelastic scattering
matrix elements using elastically scattered initial and final states (double
channeling). Here, we compare both approaches and show that the latter is
computationally competitive to the former by exploiting analytical integration
schemes over multiple excited states. Moreover, we show how to include full
nonlocality of the inelastic scattering event, neglected in the forward
propagation approaches, at no additional computing costs in the reciprocal wave
method. Detailed simulations show in some cases significant errors due to the
z-locality approximation and hence pitfalls in the interpretation of
spectroscopical TEM results.
