If the wave function entering the n-th slice is
then the wave function leaving the slice is given by:SIMULATEM: A PROGRAM FOR THE MULTISLICE SIMULATION OF IMAGES AND DIFFRACTION PATTERNS OF NON-CRYSTALLINE OBJECTS.
Alfredo Gómez Rodríguez & Luis Manuel Beltrán del Río Caballero
Departamento de Materia Condensada, Instituto de Física U.N.A.M. P.O. Box 20-364
México 01000 D.F.alfredo@fisica.unam.mx bluism@fisica.unam.mx
Summary
In this work the program SIMULATEM is presented. Simulatem can calculate images and diffraction patterns from arbitrary objects using the multislice approach. The basic algorithm is presented, the user interface is shown and various examples are given
Keywords: electron diffraction, simulation, multislice, high-resolution images.
Resumen
En este trabajo presentamos el programa SIMULATEM. Simulatem puede calcular imágenes y patrones de difracción de objetos arbitrarios usando el método de las multicapas. Presentamos el algoritmo básico, mostramos la interfase del usuario y damos varios ejemplos.
]]> Palabras clave: difracción de electrones, simulación, multicapas, imágenes de alta resolución.
1. Introduction
In materials science, electron microscope images are often used to elucidate the structure of samples but so far there is no direct way of achieving this; at this point comes into play electron microscope image simulation that allows the comparison of the actual images with synthetic ones generated from a known predicted structure. Electron microscope image synthesis is becoming an important tool in materials sciences.
2. Theoretical considerations
In this section we review briefly the full implementation of the multislice method of Cowley and Moody [1] that was used in Simulatem
2.1. The multislice method
then the wave function leaving the slice is given by:
(1)
where 
is the so-called interaction constant, 
is the projected potential from the n-th slice and * denotes the operation of convolution. The term 
represents propagation through a slice of thickness 
and is given by in terms of its Fourier transform as
(2)
where, 
2.2. The Gaussian fit to the potential.
In Simulatem the atomic scattering factors are expressed as
(3)
where the a1 and the bi are coefficients that are determined numerically to ensure that they produce an optimum fit to f(u). The coefficients were found by Herrera [2] using a non-linear least-squares fit [3], [4]
From equation (1) the projected potential at (x,y) in a slice due to an atom located at ( xj, yj, zj ) has been determined to be
(4)
where 
P is the complementary error function
(5)
and
(6)
(7)
The potential due to all the atoms can be obtained by adding the individual potentials.
2.4 Features of the present implementation.
In the present implementation no approximations were made concerning the size of the slice; that is, the integral of the potential along the slice is not approximated by an integral from minus infinity to plus infinity (as it is frequently assumed in the literature).
The propagator (eq. 2) is taken with no approximations concerning the square root appearing in its definition. No small angle approximation is made.No assumption is made concerning the crystallinity of the sample, the algorithm is devised from the onset in such a way that it can handle arbitrary objects; the only limitation is the tiling effect implicit in the use of the discrete Fourier transform algorithm.
3. The program.
In this section the main features of the program are presented.
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3.1. The samples
In the program the information concerning the sample is given in an ASCII file containing the coordinates of the atoms. Two formats are available: the Brookhaven PDB format (used widely by the protein data researchers) and a ( Xmol ) XYZ format in which the file simply contains the coordinates and atomic symbols of all the atoms in the sample.
At the moment this software is capable of handling structures with up to 30,000 atoms; in a way this defines the maximum bulk size of the structure ( as a cubic sample ) at around 70 angstroms wide, but the automatic default sampling mode will adjust the sample space in order to show all the structure within the image regardless of the sample geometry; of course SimulaTEM has manual settings that allow the user to set the sample size relative to the image size in accordance to his needs.
3.2. The microscopes
The program can simulate any TEM microscope. It contains data (spherical aberration, accelerating voltage, etc.) for a number of standard microscopes but the user can supply any parameters as needed.
3.3. Imaging and diffraction patterns
The primary outputs from the program are high-resolution images and diffraction patterns. The images can be calculated for any given defocus (or for Scherzer defocus if desired) and focal series can be easily produced. A small tilt in the illumination can be produced. The size and position of the objective aperture can be varied. The multislice parameters (width and number of slices) can be supplied by the user. A stigmator is included, it is intended as a pedagogical feature to illustrate the nature and correction of astigmatism.
3.4. Measurements
In the image and diffraction pattern windows distances and angles can be measured.
3.5. Rotations
4. The user interface.
The main window is shown in figure 1. There one can see a large thumbnail of the structure, a thumbnail of the image and a thumbnail of the diffraction pattern. A schematic representation of the sample, the slices and the electron beam is shown in one of the windows. The contrast transfer function is shown at the bottom together with an indication of the aperture used. Both the phase and amplitude parts of the transfer are shown. In the image window one can see the actual high resolution image (bottom left). Optionally one can see a plot of where the atoms lie. There is a µ-mark. Distances and angles between directions can be measured, the results are shown in the measurements window. In the diffraction window (bottom right) the diffraction pattern is presented. The contrast in the image can be adjusted at will and measurement of distances and angles are also possible. In the illumination window the convergence of the beam can be specified. One can also apply a small tilt to the illumination. The radius and position of the objective aperture can be selected and there is an option to select the optimum aperture (that covering up to the first zero of the transfer function under Scherzer defocus conditions).In the multislice window the user can choose the image size (256X256, 512X512 or 1024X1024). The extent sampled in direct or reciprocal space can also be selected here. The number and size of the slices in the multislice algorithm can be selected at will (or one can use a default of one slice or 2 A slices).The focal series window is used to input the focal series parameters (defocus around which series is made, number of steps required). The focal series itself can be seen in the focal series composite image ( in the background of figure 1 ).
Figure 1. Illustrating the various window available in simulatem
5. Other features.
The program has a help facility that can be used also as a guide to the beginner. The various images (bright field, dark field and focal series) can be saved as bit map (bmp) or as tiff images and a text can be saved along the images (a text that contains the microscope parameters and comments provided by the user).
6. Examples.
In figure 2-a we show the image from an ordinary fcc gold particle along a [100] direction. The image corresponds to Scherzer defocus. In figure 2-b the corresponding diffraction pattern can be seen.
Figure 2. In part a the image at Scherzer defocus of a square [100] gold particle can be seen, In part b the corresponding pattern is presented.
Figure 3. Image from a DNA molecule. The imaging conditions correspond to Scherzer defocus at 100 KV.
Figure4. Diffraction pattern from the molecule shown in figure 3. The characteristic X-shaped pattern can be seen.
Figure 5a. image from a quasicrystalline particle oriented along the five-fold axis.
In figure 5 the image and pattern from a quasicrystal (icosahedral phase) can be seen. The ten-fold shape of the diffraction pattern clearly shows the quasicrystalline nature of the structure; the PDB file includes 2870 atoms.
Figure 6a. Synthetic image of three fcc nanoparticles with different orientations on a amorphous carbon substrate.
Figure 6b. Diffraction pattern of the set of particles with the cloudy pattern generated by the amorphous carbon.
As a final example of the versatility of the program we present in figure 6 a simulation involving several fcc particles in different orientations set atop an amorphous carbon substrate, the sample has several hundred atoms belonging to the nanoparticles and several thousand carbon atoms on the substrate.
7. Conclusion
Simulatem is a versatile program to calculate images and diffraction patterns from arbitrary objects. It is an auxiliary tool for the microscopist who has to compare his actual images with those theoretically predicted or expected.
At this moment this software is subject to momentary release limitations, for more information please contact us via E-mail ( see first page header ).
Acknowledgements
The authors are indebted to Mr. Alfredo Sánchez A. Cristina Zorrilla and Samuel Tehuacanero for the technical assistance.
References.
1 J.M. Cowley, A.F. Moodie. Acta Crystallogr. 12 (1959) 360 [ Links ]
2 R. Herrera. Un algoritmo para la simulación de imágenes y patrones de difracción de objetos arbitrarios en microscopía electrónica de alta resolución. Tesis doctoral. Centro de Investigación científica y de Educación superior de Ensenada. Baja California (1989). [ Links ]
3 P.R. Bevington. Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill. New York (1969)
4 R. Herrera. Diseño de un sistema de multicapas por el método de síntesis. Estudio comparativo de tres algoritmos. Tesis de Maestría. Centro de Investigación científica y de Educación superior de Ensenada. Baja California (1985). [ Links ] ]]>