4. Diffraction patterns

The menu items Crystal-Structure Factor (Alt-C + Ctrl-V), Drawing-Diffraction (Alt-D + Ctrl-D), Image-Blochwave (Alt-I + Ctrl-B) are used to display diffraction patterns such as spots, Kikuchi lines, High Order Laue Zone lines, powder or ring and dynamical Convergent Beam Electron Diffraction patterns. The symmetry of the CBED pattern is used to determine the point and space group of small crystals. Point group is determined by inspecting the symmetry of whole CBED pattern and the symmetry of the (0,0,0) disk.  Screw axes and glide planes can be identified by setting special diffraction conditions and looking at Gjonnes-Moodie black crosses and dark lines.

4.1 Spot pattern

A typical spot pattern is shown on figure 4.1. The enclosing window contains a tool box with tools to print the spot pattern, , to save it, , to move the Center of the Laue Circle down, , up, , left, , or right, , to put back the CLC at (0,0,0), , and to find the closest zone axis direction, . The drawing of the spot pattern is controlled using a tabbed panel with 7 different panels to change the diffraction conditions, i.e. accelerating voltage, camera length, deviation parameter, number of Laue zones, zone axis, etc.

Figure 4.1

The drawing High Order Laue Zones is controlled by the "crystal" panel, the plot of Kikuchi and HOLZ lines by the option panel and the distortion of the pattern due to a tilted film by the normal panel. Indices can be assigned to the spot and lines, the Laue circle can be plotted too. The lattice side and lattice angle panel control the distortions that result of small lattice parameters changes. The avalanche panel is used to plot all the non equivalent zone axis pattern of a given stereographic quadrant. Tilting of the pattern is made possible by selecting the green cross and moving it around using the mouse.

 

4.2 Kikuchi lines

Figure 4.2 shows a tilted Kikuchi line pattern with indexed spots and lines. The lines density is controlled by the Kikuchi threshold slider. The "Dots" check box enables one to see the reflections as dots which diameter and brightness are function of their structure factors' amplitude or as constant diameter spots with a brightness proportional to the structure factors' amplitude.

Figure 4.2

On figure 4.2 the center of the Laue circle has been moved on to reflection (0,2,6). 

 

4.3 HOLZ lines

The HOLZ check box enlarges the (0,0,0) spot and plots the HOLZ lines inside (figure 4.3):

Figure 4.3

The acceptance angle slider makes controls the angular spread of the generated reflections, a larger acceptance angle will allow more reflections to be plotted at the expense of a longer plotting time. Its value and the deviation parameter are adjusted for each crystal file. Each spot or line is indexed when the mouse is placed on it and clicked. Spot or line information is displayed at the top of the drawing area (figure 4.4) and the couple spot-line is highlighted.

Figure 4.4

It can be verified that the distance from the Laue circle - spot intersection to the associated HOLZ line is twice the Bragg angle. The HOLZ shift check box introduces a so called High Voltage shift that partly takes into account the dynamical diffraction effects (figure 4.5):

Figure 4.5  (Si [-2,-1,2] 100 kV)

 

The HOLZ shift for Si [1,2,2] is approximately -500 V, so that the HOLZ pattern is plotted using an accelerating voltage of 99.495 kV. The HOLZ shift can be as large as several kilovolts or more. It depends on several factors such as the atoms density projected on the plane perpendicular to the zone axis, the atomic number, etc. 

Figure 4.6 (Au 200 kV, [001])

 

Figure 4.7 shows the dynamical calculation by the Blochwave method of the HOLZ lines (Si [1,2,2], 100 kV). This figure should be compared with figure 4.5. 25 reflections from the ZOLZ, FOLZ and SOLZ were included. The specimen thickness of the calculation was 220 nm. Strong dynamical effects affect lines (3,-9,-1) and (1,-9,-3). These reflections from the FOLZ have a dynamical path of type (2,0,2).

Figure 4.7 a, b  (Si 100kV [1,2,2])

The HOLZ lines position is affected by lattice deformation. The panels "Lattice side" and "Lattice angle" control the deformation. At present the 6 lattice parameters vary independently (this is indeed not observed in real crystals). Figure 4.8 shows the shift of the HOLZ lines as a result of a small change of a (a mere 0.6 % from 0.5450 to 0.5481):

Figure 4.8 (Si deformed lattice, b=0.5475)

One sees easily that the mirror symmetry of the pattern has been broken (this particularly evident below where FOLZ and SOLZ lines crosses. Figure 4.9 shows the corresponding dynamical calculation (Blochwave method):

Figure 4.9 a, b

The very same small HOLZ line shifts are observed where FOLZ and SOLZ lines crosses. The image size is 400 x 400 pixels. The calculation time (15 reflections) was 27 minutes for more than 90'000 iterations. This demonstrates that Java can be used for image simulation in electron microscopy.

See an animation of  Kossel-Moellenstedt fringes CBED patterns..

See an animation of a large angle Si [1,1,1] CBED patterns.

See an animation of Si [4,1,1] CBED patterns with kinematical HOLZ lines without kinematical HOLZ lines.

 

4.4 Powder pattern

Menu item Crystal-Structure factor (Alt-C + Ctrl-V) activates a dialogue that shows an ordered list of the reflections, their structure factors, distance, ..., up to some defined maximum (h,k,l) indices (figure 4.10):

Figure 4.10 (Si 100 kV)

The dialogue contains controls for changing the maximum (h,k,l) indices, tool buttons to save the reflections list as a text file, to print the window, to make the structure factor list, , to display a powder patter, , to reduce the list and show the multiplicity of the non-equivalent (h,k,l) reflections, , and to show a ring pattern, . Clicking on the button reduces the list from 169 reflections to 12. The atomic form factor type is also selectable in this window. It can at present be either from Doyle-Turner / Smith-Burge or Weickenmeier-Kohl sources. The Weickenmeier-Kohl form automatically calculates reflection dependent complex absorption structure factors due phonon absorption. Using DT/SB atomic form factors a constant absorption that only depend on the atom kind is used. It should be pointed out that the calculation time of the structure factor increases by a factor of about 20 (atom dependent). Thus it is always faster to study new structures using DT/SB atomic form factors.

The multiplicity of the lines replaces the sequence number (figure 4.11):

Figure 4.11 (Si 100 kV)

Clicking on the button opens the following powder pattern window (figure 4.12):

Figure 4.12 (Si 100 kV)

The powder pattern window contains a tool box with tools to print, save and change the background color of the drawing, . The line shape box allows to control the shape on the powder lines. When "Simple" is selected, it is possible to get the (h,k,l) indices of unlabeled lines (figure 4.13):

Figure 4.13 (Si 300 kV, line indexing)

The diffraction parameters are controlled using the tabbed panel "Controls". A calibration slider is available to match the calculated pattern lines position to some unknown phase keeping the camera length constant.

Clicking on the button opens the following ring pattern window (figure 4.14):

Figure 4.14 (ring pattern Si 200 kV)

The icon of the ring pattern window tool box is used to load an experimental ring pattern underneath the calculated one. The centered green cross allows to move the center of the calculated pattern over the experimental one. One can also use the arrow keys of the keyboard with either Shift, Alt, Ctrl or a combination of them to move the center. Then using the calibration slider one can match the patterns (figure 4.15):

Figure 4.15 (Matching an experimental and calculated ring pattern, no match with TiO2)

Finally, the "Line shape" box allows to overlay the powder pattern on the ring pattern (figure 4.16):

Figure 4.16 (Si ring and powder pattern)

 

4.5 Convergent Beam Electron Diffraction pattern

The CBED dialogue is found in menu Image-Blochwave (Alt-I_Ctrl-B). It is part of a larger object that performs HREM image simulation, CBED and rocking curves calculations. The toolbox at the upper left corner allows to save the CBED images, , to change the microscope settings, , to modify the specimen orientation, number of Laue zones, , and to activate the transfer function window,.

The "Blochwaver" dialogue is shown on figure 4.17:

Figure 4.17 (Blochwaver dialogue)

The CBED panel opens first. Its 2 panels control  the parameters of the CBED calculations parameters, i.e. incident beam convergence, illumination coherence and defocus when illumination is coherent. The 2 sliders on the sides of the diffraction pattern control the camera length and the deviation parameters. Selecting the green cross of this pattern allows to tilt the illumination. The "Half-convergence slider adjusts the diameter of the CBED disks.

Using these controls the setting of the diffraction geometry is easily set. Figure 4.18 shows a tilted incident beam where the center of the Laue circle is centered on reflection (1,-1,0). The indices of the reflections are given when one clicks the mouse on them.

Figure 4.18 (Control of the CBED geometry )

The iteration panel is shown on figure 4.19. This panel controls the thickness of the calculated CBED, the number of reflections, the atomic form factors. In this example, the first CBED image is calculated for a 20 nm thick TiO2 crystal oriented in the [0,0,1] direction, subsequent images are calculated each 10 nm. There will a total of 30 images. The number of reflections is selected with the deviation slider (right of the DP).

The slider "Number" sets the number of reflections that will be considered strong. Weak reflections will be introduced using Bethe potential. The "Boost" slider allows to boost the intensity of High Order Laue Zone reflections. The "Save" check box saves the CBED images as .txt files (not recommended). The LACBED check box to eliminate CBED spot overlaps when the incident beam convergence is very large.

Figure 4.19 (Control of the Blochwave calculation)

In the right panel, 2 types of drawings are produced, the diffraction pattern showing the geometry of the CBED calculation and the CBED pattern. The 2 sliders of the CBED panel control the thickness of the CBED that is shown and the gamma correction. Linear or logarithmic intensity scale is possible. Figures 4.20 and 4.21 show the progress of the CBED calculation for 2 thickness.

Figure 4.20 (CBED calculation in progress)

 

In this calculation among the 45 beams selected by the deviation slider, about 30 were considered strong, while the other were weak. Each iteration uses a set of strong reflections that depends on the incident beam direction. As a result the intensity inside the peripheral  CBED disks shows an  erratic behavior. When this can not acceptable, either increase the camera length (the spot will be moving outside the "photo film") or selected all reflections (this lengthens considerably the calculation time). It is always better to introduce at least a limit that considers as strong all the ZOLZ reflections.

Figure 4.21 (CBED calculation done)

   

A word of caution: the CBED calculations do not take into account systematic absences (forbidden reflections due a particular choice of the  Bravais lattice).  It naturally introduces other kinematical extinctions h (due to glide plane or screw axis) when they have a dynamical path i.e. when h = g1 + g2 where  g1 and g2 are not extinctions.  Figure 2.22 shows the Gjonnes-Moodie black cross and dark radial lines of BeO in the [100] (or [2-1-1 0] using Weber indices) zone axis direction. The spot (0,0,1) is at Bragg condition (figure 4.22):

Figure 4.22 (Gjonnes-Moodie black cross and dark radial lines, 22 to 37 nm thick BeO [1,0,0])

Finally figure 4.23 shows the CBED disk of Si [1,1,1] at 100 kV. Strong dynamical effects make HOLZ lines appear from place to place as "white" lines. In order to weaken this effect (observable on experimental patterns to a lesser extent) a much larger set of reflections would have to be included in the simulation.

Figure 4.23 a, b (Si 100 kV [1,1,1], 20 nm-1)

Figure 4.23 shows superimposed on the dynamical CBED pattern, the kinematical HOLZ lines (with holz shift correction). It is obvious that dynamical calculations are required when precise HOLZ line positions  are needed, simple HOLZ drawings are accurate enough only for beam convergence up to a few nm-1.

Figure 4.24    (Small angle CBED pattern, Si [4,1,1], 100 kV,  38 beams, 9.5 mrad)

4.6 Rocking curves

 

The rocking curves calculation is controlled by the panel shown on figure 4.24. The major difference with the CBED panel is the double arrow that sets the incident beam tilt direction. The Center of the Laue Circle is moved by selecting the green cross and moving it with the mouse or by using the specimen dialogue.

Figure 4.25 (Rocking beam control panel)

To change the direction of the rocking incident beam, click on the yellow head of the one of the arrows and drag it around. The curves are displayed on the plot panel where the sliders control the curve that is shown. When the "Save" box is checked the curves are saved in text form easily read by Mathematica.

Figure 4.26 Rocking curves (Si [1,2,2], (0,-2,2) at Bragg condition)