Quantitative comparison between real space and Bloch wave methods in image simulation

Both Bloch wave (BW) and real space multislice (RS-MS) methods are based on the Schrödinger equation. The BW method is considered to be the most accurate and popular simulation method used in various simulations, and it has solved many crystallographic problems. In this paper, we verified that the aforementioned two methods are similar to each other with minor difference. First, the RS-MS method is implemented in the real space, while the BW method is carried out in the reciprocal space. Furthermore, by quantitatively calculating exit wavefunctions of four crystals, i.e., Cu, MgAl2O4, Mg44Rh7 and KNbO3, we found that the RS-MS method offers certain advantages over the BW method. Firstly, the RS-MS method takes all the structure factors into account to calculate crystal potentials, so accuracy loss caused from the crystal potential is effectively avoided. Secondly, the dimension of the scattering matrix for the RS-MS method is proportional to the area of the 2 dimensional (2D) unit cell which is perpendicular to the incident beam direction, whereas that for the BW method is proportional to the square of the area of the 2D unit cell. To reduce computation time, reduction of dimension of the scattering matrix for the BW method has to be performed, thus accuracy loss is inevitable even if compensated by the Bethe potential method. For crystals with a small 2D unit cell, both simulation methods work well. When the area of the 2D unit cell is larger than square of 2.0 nm, for example KNbO3, the size of the scattering matrix of the BW method is too large to be carried out by simulation and one should use the RS-MS method instead. This paper presents a tutorial comparison of the different methods.