Multislice
[3] 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.
The mapping from a crystal structure to its image or electron diffraction pattern is relatively well understood and documented.
Hence, the use of numerical techniques in simulating results for different crystal structure is integral to the field of electron microscopy and crystallography.
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.
[1] In this work, the authors describe scattering of electrons using a physical optics approach without invoking quantum mechanical arguments.
In 1957, Cowley and Moodie showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beams.
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 Schrödinger equation can be written as We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the
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.
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 use of FFTs allows a significant computational advantage over the Bloch Wave method in particular, since the FFT algorithm involves
The most important step in performing a multislice calculation is setting up the unit cell and determining an appropriate slice thickness.
Other programs exist but unfortunately many have not been maintained (e.g. SHRLI81 by Mike O’Keefe of Lawrence Berkeley National Lab and Cerius2 of Accerlys).
The Java applet is ideal for a quick introduction and simulations under a basic incoherent linear imaging approximation.
The ACEM code accompanies an excellent text of the same name by Kirkland which describes the background theory and computational techniques for simulating electron micrographs (including multislice) in detail.
The atomic scattering factors in ACEM are accurately characterized by a 12-parameter fit of Gaussians and Lorentzians to relativistic Hartree–Fock calculations.
This program uses a mouse-drive graphical user interface and is written by Roar Kilaas and Mike O’Keefe of Lawrence Berkeley National Laboratory.
While the code is no longer developed, the program is available through the Electron Direct Methods (EDM) package written by Laurence D. Marks of Northwestern University.
A structure file must be provided as input in order to run use this code, which makes it ideal for advanced users.
This software is specifically developed to run in Mac OS X by Roar Kilaas of Lawrence Berkeley National Laboratory.
This is a software for multislice simulation was written in FORTRAN 77 by J. M. Zuo, while he was a postdoc research fellow at Arizona State University under the guidance of John C. H. Spence.
[14] The Quantitative TEM/STEM (QSTEM) simulations software package was written by Christopher Koch of Humboldt University of Berlin in Germany.
OpenCL accelerated multislice software written by Adam Dyson and Jonathan Peters from University of Warwick.
[15][16] This method is based on the Pauli equation, incorporating the effects of spin and orbital angular momentum of electrons in elastic scattering.
The multislice method has been extended to model inelastic scattering mechanisms involving phonons[17] and magnons.
[18] The frozen phonon multislice method (FPMS) was developed to simulate the impact of thermal vibrations on electron diffraction.
[17] First introduced in the early 1990s, FPMS approximates phonon-induced distortions by averaging over static snapshots of atomic displacements sampled from thermal distributions.
[18] FMMS follows an analogous approach to FPMS but is implemented within the Pauli multislice framework, allowing for the simulation of inelastic magnetic scattering processes.
