Multislice method for large beam tilt with application to HOLZ effects in triclinic and monoclinic crystals

Three existing multislice formulations (MS) that are claimed to be applicable for large beam tilt are introduced in a unified form and the validity of these formulations for large beam tilt is evaluated. It is shown that one formula, which is called MSSBT (the MS for small beam tilt), is valid only for tilt angles less than 6 degrees, but the second one, which is called MSLBT (the MS for large beam tilt), can be used for tilt angles as large as 20 degrees, while the third one yields results very close to the MSLBT results for tilt angles up to 12 degrees. Simulations also show that the accuracy of the MSLBT is independent of the scattering power of the atoms. The reasons for the inaccuracy of these multislice formulations for beam tilt (MSBT) and the differences between the three MSBT formulations are discussed based on the complete Schrodinger equation, which includes backscattering effects. For calculating the higher-order Laue-zone (HOLZ) effects from triclinic and monoclinic crystals, it is pointed out that the slices have to be cut parallel to the ab plane of the crystal and MSBT formulations should be used. The computational formulations for potentials of the crystal slices and the propagator are derived based on the transformations between the crystallographic coordinate system and the working coordinate system. Calculations of the HOLZ reflections for the monoclinic crystal Na2Ti3O7 [001] are carried out. It is shown that for triclinic and monoclinic (when the beam direction is along the c axis) crystals HOLZ effects dynamically influence not only the intensity but also the symmetry of zero-order Lauezone (ZOLZ) diffraction patterns. Hence, in that case, the projection approximation for simulating the HRTEM images may not be used.