Multislice

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.

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. &=E\Psi(x,t) \end{align} 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: \begin{align} \Psi({\mathbf{r}}) &= \Psi_{0}({\mathbf{r}}) + \int{G({\mathbf{r,r'}})V({\mathbf{r'}})\Psi({\mathbf{r'}})d{\mathbf{r'}}} \end{align} where G(\mathbf{r,r'}) is the Green's function that represents the amplitude of the electron wave function at a point \mathbf{r} due to a source at point \mathbf{r'} . Hence for an incident plane wave of the form \Psi(r)=\exp(i\mathbf{k\cdot r}) the Schrödinger equation can be written as {{NumBlk|:| \begin{align} \Psi({\mathbf{r}}) = \exp(i{\mathbf{k\cdot r}}) - \frac{m}{2\pi\hbar^2}\int\frac{\exp(ik\cdot {\mathbf })} V({\mathbf{r'}})\Psi({\mathbf{r'}}) dr' . \end{align} }} We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the \hat{z} -direction, i.e., \mathbf{k} = (0, 0, k) . Now we consider a wave-function \Psi(r)=\phi(\mathbf{r}) \exp(i\mathbf{k\cdot r}) with a modulation function \phi({\mathbf{r}}) for the amplitude. Equation ( ) becomes then an equation for the modulation function, i.e., \begin{align} \phi({\mathbf{r}}) &= 1 - \frac{m}{2\pi\hbar^2}\int{\frac{\exp[ik|{\mathbf{r-r'}}|-i{\mathbf{k}}\cdot({\mathbf{r-r'}})]} V({\mathbf{r'})\phi({\mathbf{r'}})}dr'} \end{align} . Now we make substitutions with regards to the coordinate system we have adhered, i.e., \begin{align} {\mathbf{k}} \cdot ({\mathbf{r-r'}}) &= k(z-z') \\ |{\mathbf{r-r'}}| &\approx (z-z') + ({\mathbf{X-X'}})^2/{2(z-z')} \end{align} where \boldsymbol{X}=\begin{pmatrix}x\\y\end{pmatrix} . Thus \begin{align} \phi({\mathbf{r}}) = 1 -i\frac{\pi}{E\lambda} \int \int \limits_{z'=-\infty}^{z'=z} V({\mathbf{X'}},z') \phi({\mathbf{X'}},z') \frac{1} {i\lambda (z-z')} \exp\left(ik\frac \tilde{\phi}(\mathbf{u},z) = \tilde{\phi}(\mathbf{u},z=0)\exp(\pi i \lambda \mathbf{u}^2 z) Where \mathbf{u} is the reciprocal lattice coordinate, z is the depth in the sample, and \lambda is the wavelength of the electron wave (related to the wave vector by the relation k = 2\pi / \lambda ). In the case of the small-angle approximation ( \theta \sim 100 mRad) we can approximate the phase shift as \Delta z . For 100 mRad the error d - S is on the order of 0.5% since \cos(0.1) = 0.995 . For small angles this approximation holds regardless of how many slices there are, although choosing a \Delta z greater than the lattice parameter (or half the lattice parameter in the case of perovskites) for a multislice simulation would result in missing atoms that should be in the crystal potential. Additional practical concerns are how to effectively include effects such as inelastic and diffuse scattering, quantized excitations (e.g. plasmons, phonons, excitons), etc. There was one code that took these things into consideration through a coherence function approach called Yet Another Multislice (YAMS), but the code is no longer available either for download or purchase. The code is updated quite often with new features and a user mailing list is maintained. Freely available on their website.

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