The multislice algorithm is a method for the simulation of the elastic scattering of an electron beam with matter, including all multiple scattering effects. The method is reviewed in the book by John M. Cowley, and also the work by Ishizuka. The algorithm is used in the simulation of high resolution transmission electron microscopy (HREM) micrographs, and serves as a useful tool for analyzing experimental images. This article describes some relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Some of the advantages and limitations of the technique and important considerations that need to be taken into account are described.
Background The multislice method has found wide application in electron microscopy and crystallography. The mapping from a crystal structure to its image or electron diffraction pattern is relatively well understood and documented. However, the reverse mapping from electron micrograph images to the crystal structure is generally more complicated. The fact that the images are two-dimensional projections of three-dimensional crystal structure makes it tedious to compare these projections to all plausible crystal structures. Hence, the use of numerical techniques in simulating results for different crystal structure is integral to the field of electron microscopy and crystallography. Several software packages exist to simulate electron micrographs. There are two widely used simulation techniques that exist in literature: the Bloch wave method, derived from Hans Bethe's original theoretical treatment, and the multislice method. This article focuses on the multislice method for simulation of dynamical diffraction, including multiple elastic scattering effects. Most of the packages that exist implement the multislice algorithm along with Fourier analysis to incorporate electron lens aberration effects to determine electron microscope image and address aspects such as phase contrast and diffraction contrast. For electron microscope samples in the form of a thin crystalline slab in the transmission geometry, the aim of these software packages is to provide a map of the crystal potential, however this inversion process is greatly complicated by the presence of multiple elastic scattering. The first description of what is now known as the multislice theory was given in the classic paper by Cowley and Moodie. In this work, the authors describe scattering of electrons using a physical optics approach without invoking quantum mechanical arguments. Many other derivations of these iterative equations have since been given using alternative methods, such as Greens functions, differential equations, scattering matrices or path integral methods, see for instance the book by Lianmao Peng, Sergei Dudarev and Michael Whelan. A summary of the development of a computer algorithm from the multislice theory of Cowley and Moodie for numerical computation was reported by Goodman and Moodie. They also discussed in detail the relationship of the multislice to the other formulations. Specifically, using Zassenhaus's theorem, this paper gives the mathematical path from multislice to 1. Schrödinger equation, 2. Darwin's differential equations, widely used for diffraction contrast Transmission electron microscopy (TEM) image simulations - the Howie-Whelan equations, 3. Sturkey's scattering matrix method. 4. the free-space propagation case, 5. The phase grating approximation, 6. A new "thick-phase grating" approximation, which has never been used, 7. Moodie's polynomial expression for multiple scattering, 8. The Feynman path-integral formulation, and 9. relationship of multislice to the Born series. The relationship between algorithms is summarized in Section 5.11 of Spence (2013), (see Figure 5.9).
Theory The form of multislice algorithm presented here has been adapted from Peng, Dudarev and Whelan 2003. The multislice algorithm is an approach to solving the Schrödinger equation:
− ℏ 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2 + V ( x , t ) Ψ ( x , t ) = E Ψ ( x , t ) {\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi (x,t)}{\partial x^{2}}}+V(x,t)\Psi (x,t)&=E\Psi (x,t)\end{aligned}}}
In 1957, Cowley and Moodie showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beams. Subsequently, the effects of dynamical diffraction can be calculated and the resulting simulated image will exhibit good similarities with the actual image taken from a microscope under dynamical conditions. Furthermore, the multislice algorithm does not make any assumption about the periodicity of the structure and can thus be used to simulate HREM images of aperiodic systems as well. The following section will include a mathematical formulation of the multislice algorithm. The Schrödinger equation can also be represented in the form of incident and scattered wave as:
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