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Multislicer

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:

  1. Fresnel propagator image.
  2. Phase object function image.
  3. Projected potential image.
  4. Absorption potential image.
  5. Atom columns position.
  6. High Angle Annular Dark Field image.
  7. HREM images map.
  8. Super-cell HREM images map.
  9. Nano-diffraction patterns.

The toolbar contains tool buttons to:

The theory behind the multislice calculation is available here.

Figure1

Figure 1 Multislicer with phase object function (imaginary) duplicated 10 × 10 times.

Fresnel propagator

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.

Figure2

Figure 2 Fresnel propagator (Fourier) Si [001] duplicated 10 × 10 times.

Figure3

Figure 3 Fresnel propagator (complex) 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).


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Phase object function

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.

Figure4a Figure4b

Figure 4a KAlSi3O8 POF.

Figure 4b KAlSi3O8 POF.

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).

Figure5a Figure5b Figure5c

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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Projected Potential
Figure6a Figure6b Figure6c

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).

Figure7a Figure7b

Figure 7a Popup menu attached to show atomic columns info.

Figure 7b 3-D view of the projected potential.


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Atomic columns position

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).

Figure8a Figure8b

Figure 8a Popup menu attached to put atomic columns.

Figure 8b 3-D view of the projected potential.


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High Angle Annular Dark Field images

Figure1

Figure 1 The HAADF imager pane.

Figure2a Figure2b
Figure 2a Table of super-cells. Figure 2b Imaging options.
Figure2c Figure2d
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:

w40.gif (no angular dependence).

As another example three fold astigmatism W23 (Krivanek) or A2 (Haider) is labelled W33 with formula:

w33

that clearly shows that this aberration scales as the third power of the spatial frequency (u3) and has a rotational symmetry 3.

Figure3a

Figure3b

Figure 3a Selected super-cell displayed in projection panel.

Figure 3b 3-D view of the super-cells table.

Figure4

Figure 4 Probe with 3-Fold astigmatism.

The buttons Start, Stop starts and stops the calculation. When finished the HAADF image is displayed (Fig. 5).

Figure5

Figure 5 HAADF image simulated with 3-Fold astigmatism aberrated probe.

The HAADF wave-function can be saved (checkbox Save to HRSTEM imager) in order to introduce other probe Aberrations in real time (HRSTEM Imager).


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HREM images map

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.

Figure9

Figure 9 HRTEM image simulation.

The parameters to specify for HREM image simulations are:

Figure10a Figure10b Figure10c

Figure 10a Illumination controls.

Figure 10b Imaging controls.

Figure 10c Iteration controls.

Figure11a Figure11b Figure11c

Figure 11a Aberrations W22, W31, W33.

Figure 11b Aberrations W40, W60.

Figure 11c Aberrations W11 and phase plate shift.

Figure12a Figure12b Figure12c

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).

Figure13

Figure 13 YTi2O7 [111] defocus and thickness map.

Figure14

Figure 14 YTi2O7 [111] Montage, defocus and thickness map.


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Super-cell images

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.

Figure15a Figure15b

Figure 15a Table of super-cells.

Figure 15b HREM map.

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.

Figure16a Figure16b

Figure 16a Au Σ5310 grain boundary, 0 nm 3-Fold astigmatism.

Figure 16b Au Σ5310 grain boundary, 200 nm 3-Fold astigmatism.


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Nano-diffraction

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.

Figure17a Figure17b

Figure 17a Al [111] (ABCA sequence).

Figure 17a Al [111], first sub-slice (A).

The first super-cell must be loaded before selecting the multislice calculation method (Fig. 18).

Figure18

Figure 18 Loading the A sub-slice of Al [111] model.

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.

Figure19

Figure 19 Nano-diffraction panel with the A sub-slice.

The Nano-diffraction calculations are controlled by 3 panes:

Figure20a Figure20b

Figure 20a Cell pane.

Figure 20b Imaging pane.

Figure21a Figure21b

Figure 21a Multislice pane, HOLZ reflections.

Figure 21b Multislice pane, Frozen lattice.

The inicident probe (or wave-function) is initialized with the Initialize button (Fig. 22).

Figure22

Figure 22 Probe initializing.

The probe coherence can be coherent or uniform coherent:

Depending on the illumination type different controls are activated.


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Uniform Coherent
Figure23a Figure23b

Figure 23a Probe position.

Figure 23b Probe convergence.


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Uniform illumination

Figures 24a, 24b show the controls of a uniform probe.

Figure24a Figure24b

Figure 24a Beam Half-convergence setting.

Figure 24b Thermal magnetic noise (damping).


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Coherent illumination

Figures 25 shows the controls of a coherent probe.

Figure25a Figure25b

Figure 25a Aperture size and W00 (Cc) settings.

Figure 25b Coherence settings.

Figure25c Figure25d

Figure 25c Microscope settings.

Figure 25d Probe shape image with popup menu.

Nano-diffraction patterns calculated after increasing thicknesses.

Figure26a Figure26b

Figure 26a Nano diffration after sub-slice A.

Figure 26b Nano diffration after sequence ABC.

Figure26c Figure26d

Figure 26c Nano diffration after 6 sequences ABC.

Figure 26d Nano-diffraction with kinematical plot.