Multislice method for ordinary crystals

References:
Cowley, J. M. & Moodie, A. F. (1957). Acta Cryst., 10, 609-619.
Cowley, J. M. (1981). Diffraction Physics, 2nd ed., pp. 225-247. Amsterdam: North-Holland.

In the multislice approach of dynamical electron diffraction, the sample is divided into n thin slices perpendicular to the incident electron beam.

Suppose the incident electron beam is parallel to one of the crystallographic axis, say a3, of the object. Let Dzj , jj(x) and qj(x) represent respectively the thickness, the projected potential distribution function and the transmission function of the jth slice. Since Dzj is small, each slice can be treated as a phase object. We have

,     (1)

where

s = p / lU , U is the accelerating voltage,
,
F-1 denotes the operation of inverse Fourier transform,
Fj(h) is the 2-dimensional structure factor of the jth slice. Since the electron wavelength is much smaller than the linear dimension of the object and the distance from the object to the observation plane, the propagation of electrons through each slice becomes a Fresnel diffraction. Thus according to the Kirchhoff formula, the wave function exiting the bottom plane of the first slice is

,     (2)

where

* denotes the operation of convolution
is the propagation factor of the first slice. For simplicity the factor exp(-i2pDz j/l)  will be omitted in all wave functions. Thus, equation. (2) becomes

.     (3)

Accordingly, the wave function on the bottom plane of the second slice will be

.     (4)

Finally, the wave function on the bottom plane of the nth slice, i.e. the object wave function becomes

.     (5)

The diffraction wave function of the object is then the Fourier transform of equation (5). According to the convolution theorem it will be

,     (6)

where


,
F denotes the operation of Fourier transform.