Fully nonlocal inelastic scattering computations for spectroscopical transmission electron microscopy methods

Abstract

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 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 non-locality 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. arXiv:1810.05031v1 [cond-mat.mes-hall]

Dynamical diffraction effects in STEM orbital angular momentum resolved electron energy-loss magnetic chiral dichroism

“…The experimental quantity that can be evaluated directly through the combined action of an OAM sorter [7] and an energy spectrometer in the TEM is the OAM-resolved loss function I β ( , E ), which can be formally defined as [6,9,10]…”

Orbital angular momentum resolved electron magnetic chiral dichroism

“…4 in Ref. [23] and is a parameter that depends only on the energy of the electron beam.  We consider a highly symmetrical cubic crystal, i.e., bcc iron.…”

“…where is an evolution operator defined in Ref. [23] and is the projection along the z axis of the wave vector of the inelastically scattered electron. The operator is invariant under symmetry operations of the crystal, so in our case (A3) By performing the substitution in Eq.…”

“…We make use of the approach described in Ref. [23] to simulate OAM-resolved core loss spectra. We adapt the procedure, which is reviewed here for completeness, from a momentum-defined to an angular-momentum-defined final state basis, based on the following approximations:…”

“…Proceeding as in Ref. [23], it is possible to rewrite as a superposition integral of the probe propagating in the sample and a counter-propagating wave from a single state of the detector for defined values of l, k, according to the expression (4) where the sum over a is performed over atomic positions at which the inelastic process of interest can occur. Given the infinitesimal distribution in k, is the three-dimensional Fourier transform of an elastically scattered Bessel beam with topological charge l, transverse wavevector k and energy propagating from the bottom to the top of the crystal.…”