Multislice approach for calculate dynamical selected area electron diffraction patterns (SAED) and high resolution electron microscope images (HREM) is shown on Figure 1. The calculations are orginized in 8 panes to calculate:
Multislice approach for calculate dynamical selected area electron diffraction patterns (SAED) and high resolution electron microscope images (HREM) is shown on Figure 1. The calculations are orginized in 8 panes to calculate:
The toolbar contains tool buttons to:
The theory behind the multislice calculation is available here.
The Fresnel propagator propagates the wave function from one slice to the next. It appears as Fresnel fringes located at the origin of the unit cell (primitive Bravais lattice) or at positions that reflects the Bravais lattice. It is a complex function that is displayed as a Real and Imaginary parts. The image tiling controls allows to duplicate or tile the propagator in the x and y direction.
Figures 2, 3 shows the the Fresnel propagator of Si [001] duplicated 10 × 10 times.
Such propagators allow to filter out, during the convolution, the contributions of reflections forbidden by the Bravais lattice. Without this filtering effect, the forbidden reflections will acquire some (although very weak) intensity due to numerical rounding errors. It is always possible to describe a centered lattice as a primitive one in order to use a primitive Bravais lattice propagator (this involves more work to describe the unit cell).
Figures 4a and shows 4b show the Phase Object Function of of KAlSi3O8. Controls allows for:
Using Weickenmeier-Kohl atomic form factors makes the calculations slower for the absorption correction due to phonon and core scattering is calculated per atom and per scattering angle.
Position the mouse on a maximum and click to identify the atom (or one of the atoms of the column) (Figures 5a, 5b, 5c). The imaginary part of the phase object function would be the ideal amplitude image of a one unit cell thick crystal. This is very useful to identify contrasts on high resolution images and wave functions. A file of a super-cell with the 98 elements shows this possibility (Fig. 5a).
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Figure 5a Periodic table of the elements (3 × 3). |
Figure 5b POF periodic table (real). |
Figure 5c POF periodic table (imaginary). |
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Figure 6a Periodic table of the elements. |
Figure 6b Projected potential, periodic table (real). |
Figure 6c Projected potential, periodic table (imaginary). |
A popup menu attached to the image allows for identifiying the atomic columns, viewing the image in 3-D, ... (Fig. 7a, 7b).
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Figure 7a Popup menu attached to show atomic columns info. |
Figure 7b 3-D view of the projected potential. |
The columns position are calculated per element and for all the elements. The different elemnt can be displayed one by one or all at once (Figures. 8a, 8b).
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Figure 2a Table of super-cells. | Figure 2b Imaging options. |
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Figure 2c Multislice, HOLZ reflections. | Figure 2d Multislice, Frozen lattice controls. |
The tool buttons of the super-cells table allows to:
The Initialize calculates the phase object of the first super-cell and sets the probe shape (Fig. 4).
The Aberrations coeeficients can be changed either using sliders (on the left) or directly in the text fields on the right. When selected the text field background color is yellow. When the text field is selected (yellow color) using the:
The formula text field displays the mathematical description of the aberration in orthogonal coordinates.
The notation adopted for the optical Aberrations coefficent follows either the notation of Krivanek and Haider, as well as a notation describing the wavefront aberration (Wnm). The wavefront aberration is simpler to remember since n provides the power of the spatial frequency and m the rotational symmetry. For example W40 is the spherical aberration coefficient C30 or C3 that describes the wavefront aberration:
As another example three fold astigmatism W23 (Krivanek) or A2 (Haider) is labelled W33 with formula:
that clearly shows that this aberration scales as the third power of the spatial frequency (u3) and has a rotational symmetry 3.
Figure 3a Selected super-cell displayed in projection panel. |
Figure 3b 3-D view of the super-cells table. |
The buttons Start, Stop starts and stops the calculation. When finished the HAADF image is displayed (Fig. 5).
The HAADF wave-function can be saved (checkbox Save to HRSTEM imager) in order to introduce other probe Aberrations in real time (HRSTEM Imager).
Figure 9 shows the pane for the HRTEM image simulation. This pane makes available most of the controls necessary to calculate HREM images. The calculated images are displayed on the Map panel or on the Plot panel when the Plot> checkbox and the Image> radio button are selected.
The parameters to specify for HREM image simulations are:
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Figure 11a Aberrations W22, W31, W33. |
Figure 11b Aberrations W40, W60. |
Figure 11c Aberrations W11 and phase plate shift. |
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Figure 12a Drift, TM noise and vibrations. |
Figure 12b Multislice controls. |
Figure 12c Holography controls. |
Orthogonal unit cells can be sliced into very thin slices using the Multislice controls. First the 3-dimensional crystal potential is calculated either as a whole or using a patch technique that generates the potential of each atom species and then patches it at the atom position (Multislice Manager).
The Super-cell image pane allows for performing basically the same calculations done by the HREM map pane, except that a sequence of super-cell files is used. The sequence is shown as table on the Iteration (Fig. 15a). The incident wave-function is propadated through the stack of super-cells. Using the YTi2O7 [111] slices HREM images are just identical to what is calculated by the HREM map. More flexibility is provided by the Super-cell image pane (see precipitate calculations). Moreover the time consuming calculations (Phase Object Function) can be saved and reloaded for further simulations.
The table toolbar contains tool buttons allowing for:
The sequence of super-cells must follow the syntax anyname_0000, anyname_0001, ..., anyname_9999 with anyname NOT containing the _ character.
Several iterations can be performed using the same cell setting iteration number.
As for the HRTEM maps, the effect of imaging parameters are visible in real-time. In particular the effect of 3-fold astigmatism on the HREM images of grain boundaries is very educative (Fig. 16). On the HRTEM images the contrast does not any more correspond to the atomic columns position.
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Figure 16a Au Σ5310 grain boundary, 0 nm 3-Fold astigmatism. |
Figure 16b Au Σ5310 grain boundary, 200 nm 3-Fold astigmatism. |
The Nano-diffraction makes a multislice calculation of the diffraction pattern produced by a nano-sized coherent probe that can be positioned at any particular (x, y) point of a super-cell.
It uses a sequence of super-cells (i.e. sc_0000, sc_0001, ...) that have to be prepared. As an example Si [111] super-cells will be used. The whole Al [1,1,1] structure being too thick (Fig. 17a) for multislice calculations ( Al1113x5x1.txt) it is sliced into 3 sub-slices. The sequence of the 3 super-cells is A (Al1113x5x1_0000, B Al1113x5x1_0001, C Al1113x5x1_0002) is the stacking of FCC crystals (ABC). It is import to create as tetragonal as possible models in order to optimize the calculations. Here the Al1113x5x1 has lattice parameters (a, b, c) = (2.97613, 2.8637824, 0.70148057). It contains 360 atoms and is triclinic. The Wyckoff position of all atoms is a (inherited from the Si structure) and each atom has an occupancy of 1.0. In case it is necessary to set the occupancy of one atom to a lower value (vacancy) the Wyckoff symbol must be changed to any character other character (b, c, ...). Indeed atoms with same occupancy should have the same Wyckoff symbol.
The first super-cell must be loaded before selecting the multislice calculation method (Fig. 18).
Figure 18 shows the nano-diffraction pane. The super-cell of this figure is made of 3 x 5 orthogonal Al [111] cells. The viewing direction (zone axis) is [0, 0, 1]. Such large cells are required to sample reciprocal space with a sufficient accuracy.
The Nano-diffraction calculations are controlled by 3 panes:
The inicident probe (or wave-function) is initialized with the Initialize button (Fig. 22).
The probe coherence can be coherent or uniform coherent:
Depending on the illumination type different controls are activated.
Figures 25 shows the controls of a coherent probe.
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Figure 25a Aperture size and W00 (Cc) settings. |
Figure 25b Coherence settings. |
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Figure 25c Microscope settings. |
Figure 25d Probe shape image with popup menu. |
Nano-diffraction patterns calculated after increasing thicknesses.