constant. So far we have set up the mathematical formulation of wave mechanics without addressing the scattering in the material. Further we need to address the transverse spread, which is done in terms of the Fresnel propagation function . The thickness of each slice over which the iteration is performed is usually small and as a result within a slice the potential field can be approximated to be constant . Subsequently, the modulation function can be represented as: We can therefore represent the modulation function in the next slice where, * represents convolution, and defines the transmission function of the slice. Hence, the iterative application of the aforementioned procedure will provide a full interpretation of the sample in context. Further, it should be reiterated that no assumptions have been made on the periodicity of the sample apart from assuming that the potential is uniform within the slice. As a result, it is evident that this method in principle will work for any system. However, for aperiodic systems in which the potential will vary rapidly along the beam direction, the slice thickness has to be significantly small and hence will result in higher computational expense. Practical considerations The basic premise is to calculate diffraction from each layer of atoms using Fast Fourier Transforms (FFT) and multiplying each by a phase grating term. The wave is then multiplied by a propagator, inverse Fourier Transformed, multiplied by a phase grating term yet again, and the process is repeated. The use of FFTs allows a significant computational advantage over the Bloch Wave method in particular, since the FFT algorithm involves steps compared to the diagonalization problem of the Bloch wave solution which scales as where is the number of atoms in the system. (See Table 1 for comparison of computational time). The most important step in performing a multislice calculation is setting up the unit cell and determining an appropriate slice thickness. In general, the unit cell used for simulating images will be different from the unit cell that defines the crystal structure of a particular material. The primary reason for this due to aliasing effects which occur due wraparound errors in FFT calculations. The requirement to add additional “padding” to the unit cell has earned the nomenclature “super cell” and the requirement to add these additional pixels to the basic unit cell comes at a computational price. To illustrate the effect of choosing