Quantitative Electron Crystallography

Nowadays, electron crystallography is ever-increasingly active because it has incomparable and unique advantages over X-ray crystallography. However, compared to X-ray crystallography, electron crystallography confronts with a critical problem, i.e. dynamical diffraction problem which makes structure analysis very complicated. To solve the problem, crystallographers have developed several numerical methods for many years. These methods cannot find a straightforward relationship between simulated results and crystal structures. They can only verify whether the proposed model is correct or not. To solve a structure by electron crystallography, it is necessary to develop an expression for electron dynamical diffraction. Up to date, there is no a close analytical expression to describe electron dynamical diffraction. The only way is to derive the approximate formulae with different methods. Based on Schr?dinger equation. Qibin Yang derived a analytical expression in 2001,as follows:In this paper, its modification is obtained by further considering the second-order differential termof Schr?dinger equation. It is pointed out that in which case the modification is necessary. Generally, the exit wave function highly dominated by zero order Laue zone (ZOLZ), but in some cases high order Laue zone (HOLZ) effects needs to be considered. Compared to the ZOLZ, the study on the HOLZ effect on exit wave function is fairly difficult, since ZOLZ effect can be separated from HOLZ one in reciprocal space while they are overlap with each other in high resolution image. Firstly, same as the study of ZOLZ, we take two routes to calculate HOLZ effect, i.e. numerical simulation method and analytical expression. The calculated results showed that the two methods almost are equivalent with each other. Theoretical analysis showed that the effects of HOLZ are influenced by 4 factors: i.e. thickness of the sample, wavelength of incident electron beam, the lattice planar spacing perpendicular to the incident beam and the atomic distribution along a column. The calculated results verified that the above conclusions are valid.The analytical expression can calculate diffraction not only for incident beam along Laue zone but also for incident beam deviated from Laue zone. This is quite important because electron beam always deviated a little bit from exact Laue zone in practical microscope operation.