2.4. prismatique.hrtem

For specifying HRTEM systems, HRTEM simulation parameters, and running HRTEM simulations.

The algorithm used to perform HRTEM simulations is similar to the PRISM algorithm used to perform STEM simulations. See the documentation for the subpackage prismatique.stem for a discussion on the PRISM algorithm.

The PRISM algorithm can be used to perform STEM simulations as an alternative to the multislice algorithm. Here we discuss very briefly aspects of both the PRISM and multislice algorithms, although it is recommended that the user see Ref. [Ophus1] for more details on the former and Ref. [Kirkland1] for an exposition on the latter.

As discussed in the documentation for the class prismatique.thermal.Params, the intensity pattern for a given beam in a HRTEM experiment depends on the state operator \(\hat{\rho}_{t}\) of a transmitted beam electron, which is assumed to be a weighted sum of pure states

(2.4.1)\[\begin{split}&\hat{\rho}_{t}\left(\delta_{f};\mathbf{u}_{1},\ldots,\mathbf{u}_{N}; \boldsymbol{\delta}_{\beta}\right)\\&\quad=\left|\psi_{t}\left(\delta_{f}; \mathbf{u}_{1},\ldots,\mathbf{u}_{N}; \boldsymbol{\delta}_{\beta}\right)\right\rangle \left\langle \psi_{t}\left(\delta_{f};\mathbf{u}_{1},\ldots,\mathbf{u}_{N}; \boldsymbol{\delta}_{\beta}\right)\right|,\end{split}\]

where \(\left|\psi_{t}\left(\delta_{f};\mathbf{u}_{1},\ldots,\mathbf{u}_{N}; \boldsymbol{\delta}_{\beta}\right)\right\rangle\) being the state vector of a transmitted beam electron for a perfectly coherent beam and a sample in a frozen atomic configuration \(\left\{ \mathbf{u}_{j}\right\} _{j=1}^{N}\), with the \(\delta_{f}\) and \(\boldsymbol{\delta}_{\beta}\) implicitly specifying the defocus and beam tilt respectively. Note that the weighted sum is over the defocus, beam tilt, and \(\left\{ \mathbf{u}_{j}\right\} _{j=1}^{N}\).

Each \(\left|\psi_{t}\left(\delta_{f};\mathbf{u}_{1},\ldots,\mathbf{u}_{N}; \boldsymbol{\delta}_{\beta}\right)\right\rangle\) is calculated by

(2.4.2)\[\left|\psi_{t}\left(\delta_{f};\mathbf{u}_{1},\ldots,\mathbf{u}_{N}; \boldsymbol{\delta}_{\beta}\right)\right\rangle = \hat{Q}\left\{ S_{m_{x},m_{y}}\right\} \left(x,y\right),\]

(2.4.3)\[\hat{Q}\left\{ g\right\} \left(x,y\right)= \left(\hat{A}\circ\hat{D}\right)\left\{ g\right\} \left(x,y\right),\]

where the notation \(\hat{O}\left\{ g\right\} \left(q_{x},q_{y}\right)\) denotes an operator \(\hat{O}\) transforming a function \(g\) into a space with coordinates \(q_{x}\) and \(q_{y}\) [e.g. \(\hat{O}\) could be a two-dimensional Fourier transform]; the notation \(\left(\hat{O}_{1}\circ\hat{O}_{2}\right)\left\{ g\right\} \left(q_{x},q_{y}\right)\) denotes the composition of two operators \(\hat{O}_{1}\) and \(\hat{O}_{2}\) transforming a function \(g\) into a space with coordinates \(q_{1}\) and \(q_{2}\) with \(\hat{O}_{2}\) being applied first; \(\hat{A}\) is given by Eq. (2.8.14);

(2.4.4)\[\hat{D}\left\{ g\right\} \left(k_{x},k_{y}\right)\equiv \hat{\mathcal{F}}_{\text{2D}}\left\{ g\right\} \left(k_{x},k_{y}\right) e^{-i\chi\left(k_{x},k_{y}\right)}\xi\left(k_{x},k_{y}\right),\]

with \(\hat{\mathcal{F}}_{\text{2D}}\) being given by Eq. (2.8.17), \(\xi\left(k_{x},k_{y}\right)\) being an aperture function modelling the objective aperture, and \(\chi\left(k_{x},k_{y}\right)\) being the phase deviation due to coherent lens aberrations of the objective lens; and \(S_{m_{x},m_{y}}\) is the element of the \(S\)-matrix corresponding to a perfectly coherent plane wave incident at the sample with the beam tilt implicitly specified by \(\boldsymbol{\delta}_{\beta}\). See the documentation for the subpackage prismatique.stem for a discussion on \(S\)-matrices and the documentation for the class embeam.coherent.Aberration for a definition of \(\chi\left(k_{x},k_{y}\right)\). Note that \(\xi\left(k_{x},k_{y}\right)=0\) for all scattering directions \(\left(k_{x},k_{y}\right)\) blocked by the objective aperture, and \(\xi\left(k_{x},k_{y}\right)=1\) otherwise.

Modules

image	For specifying simulation parameters related to HRTEM image wavefunctions and intensities.
output	For specifying the output parameters for HRTEM simulations.
sim	For running HRTEM simulations.
system	For specifying simulation parameters related to the modelling of HRTEM systems.