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Kossel-Moellenstedt

Analyzing Kossel-Moellenstedt fringes

The K-M fringes analysis window (Fig. 1) allows for the analyze CBED pattern with Kossel-Moellenstedt fringes pattern. It offers a tool to figure out the thickness of the crystal and allows for the measurement of structure factors.

The toolbar contains tool buttons to:

Figure 1 is plotted with parameters from the current crystal, accelerating voltage, etc. It is recommended to use the proper crystal. By default zone axis [001] is set and microscope parameters from the selected microscope are employed.

Figure1

Figure 1 The K-M fringes analysis window (current opended crystal file).

Follow these steps to analyze the K-M fringes pattern:


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Load the K-M pattern

The K-M fringes pattern is loaded as a background image (Fig. 2). It is still necessary the experimental zone axis [uvw], CLC (hkl) with the K-M pattern.

Figure2

Figure 2 Loading the K-M fringes.


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Setting the diffraction conditions

A low magnification SAED pattern taken under the diffraction conditions of the CBED K-M fringes is usually necessary to determine [uvw] zone axis indices and (hkl) CLC indices. Figuring out the [uvw] and (hkl) indices can be done using Kikuchi lines and/or HOLZ lines. An other possibility is to load the SAED into the diffraction pattern window and manually align the calculated pattern onto the experimental SAED. Adjusting precisely the beam half-convergence, camera length, deviation, accelerating voltage, and the acceptance angle is important for accurate fitting of the K-M fringes with calculated ones. diffraction conditions.

Figure2

Figure 2 Loading the K-M disks.

Introducing the measured [uvw] and (hkl) indices, camera length and beam convergence is shown in the good match shown in (Fig. 3). Note that the Keeper stores all these parameters (often not too easy to figure out) and allows to load them in the K-M analysis.

Figure3a Figure3b

Figure 3a Keeper [uvw] zone axis indices.

Figure 3b Keeper (hkl) CLC indices.

Figure3c Figure3d

Figure 3a Keeper accelerating voltage, camera length, beam-convergence, deviation.

Figure 3b Alignment of K-M pattern and calculated one.


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Getting the profile through the CBED disks

A mouse click identifies the reflections, (000) left and (200) right, at Bragg diffraction conditions.

To get a profile of the K-M fringes select the Profile mask (Figures 4a, 4d) and drag the profle on the K-M pattern using the yellow squares (Fig. 4b). The profile center must remain exactly between the reflections (green cross). When done the profile accross the (000) disk is blue and accross the (200) disk red.

Figure4a Figure4b

Figure 4a Activated profile.

Figure 4b Aligned profile.

Figure4c Figure4d

Figure 4c (000) disk blue, (200) disk red.

Figure 4d Alignment of the disks and profile.

The profile does not have to aligned with the disks nor does it have to cover the whole disks (Figures 5a, 5b). Every point of the profile contains the diffraction conditions to do a complete dynamical calculation. But the wider the profile the more accurate the thickness determination.

Figure5a Figurebb

Figure 5a Tilted profile.

Figure 5b Tilted shorter profile.


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2-Beams profile fitting

With the profile shown on Fig. 4c proceed with its fitting with a dynamically calculated profile (Bloch-wave) by clicking the make process tool button. In case it is not enabled, reselect the Profile mask.

Figure6

Figure 6 2-beams dynamical calculation profile (green) fitted to the profile disk (200) red.

A 2-beams dynamical fit is shown of Figure 6. The fit is using the 2-beams dynaminal formula and is only using (200) disk profile. A 157 nm crystal thickness is obtained. The fit can be further improved by checking the Deviation and (000) center. This is modifying the diffraction conditions using a Levenberg-Marquardt fitting. The final residual is 0.076 (Fig. 7).

Figure7

Figure 7 Levenberg-Marquardt fitting of deviation and (000) center.


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Many Beams profile fitting

The toolbar of the processing window contains tool buttons to:

The fitting controls are distributed in 2-panes:

Figure8a Figure8b

Figure 8a 2-Beams controls.

Figure 8b 2-Beams fitted.

Figure9a Figure9b

Figure 9a Many-Beams controls.

Figure 9b Many-Beams before fit.

The many-beams fitting can adjust supplementary parameters. Examples are:

Figure10a Figure10b

Figure 10a First fit without (000) center adjustment.

Figure 10b Second fit with (000) center adjustment.

Figure11a Figure11b

Figure 11a Checking the Debye-Waller fit.

Figure 11b Third fit with Debye-Waller adjustement.

Figure12a Figure12b

Figure 12a Adjusting (000) and (200) structure factors.

Figure 12b Residual is now 0.02.


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Many-Beams Parameters before the fits

     I N I T I A L   P A R A M E T E R S
 Acc. Volt.	100.00000
 Dilation  	0.98761
 Fog       	0.00000
 Scale     	1.00000
 Thickness 	157.21496
 clc_x      	0.00000
 clc_y      	0.00000
 CLC        	(1.0162, 4.0115, -12.0346)
 ooo_x      	0.00000
 ooo_y      	0.00000
 OOO        	(0.0000,0.0000,0.0000)
---------------------------------------------------------
 atom [0]	Al	 0.00000	 0.00000	 0.00000	 0.00500	 1.00000	 0.03400
 atom [1]	Al	 0.50000	 0.50000	 0.00000	 0.00500	 1.00000	 0.03400
 atom [2]	Al	 0.00000	 0.50000	 0.50000	 0.00500	 1.00000	 0.03400
 atom [3]	Al	 0.50000	 0.00000	 0.50000	 0.00500	 1.00000	 0.03400
 ---------------------------------------------------------
 (Vr, Vi)	(0, 0, 0)	0.00000	(20.26200, 0.86143)
 (Vr, Vi)	(2, 0, 0)	0.00074	(5.90540, 0.19101)
 (Vr, Vi)	(-7, 1, -3)	0.07954	(0.61792, 0.04835)
 (Vr, Vi)	(4, 0, 0)	-0.08941	(2.16122, 0.11282)
 (Vr, Vi)	(-2, 0, 0)	-0.09162	(5.90540, 0.19101)
 (Vr, Vi)	(6, 0, 0)	-0.27054	(1.07624, 0.07320)
 (Vr, Vi)	(-4, 0, 0)	-0.27423	(2.16122, 0.11282)

Many-Beams Parameters After the fits
     F I N A L   P A R A M E T E R S
 Acc. Volt.	100.00000
 Dilation  	1.00644
 Fog       	0.00000
 Scale     	14.28545
 Thickness 	149.84101
 clc_x      	0.00000
 clc_y      	0.00000
 CLC        	(0.9866, 3.9953, -11.9860)
 ooo_x      	0.00000
 ooo_y      	0.00000
 OOO        	(0.0000,0.0000,0.0000)
---------------------------------------------------------
 atom [0]	Al	 0.00000	 0.00000	 0.00000	 0.00548	 1.00000	 0.03400
 atom [1]	Al	 0.50000	 0.50000	 0.00000	 0.00548	 1.00000	 0.03400
 atom [2]	Al	 0.00000	 0.50000	 0.50000	 0.00548	 1.00000	 0.03400
 atom [3]	Al	 0.50000	 0.00000	 0.50000	 0.00548	 1.00000	 0.03400
 ---------------------------------------------------------
 (Vr, Vi)	(0, 0, 0)	0.00000	(19.98091, 1.21811)	(-0.28109, 0.34865)
 (Vr, Vi)	(2, 0, 0)	0.00074	(5.90540, 0.19101)	(0.01742, -0.00614)
 (Vr, Vi)	(-7, 1, -3)	0.07954	(0.61792, 0.04835)	(0.02634, 0.00083)
 (Vr, Vi)	(4, 0, 0)	-0.08941	(2.16122, 0.11282)	(0.02538, -0.00320)
 (Vr, Vi)	(-2, 0, 0)	-0.09162	(5.90540, 0.19101)	(0.01742, -0.00614)
 (Vr, Vi)	(6, 0, 0)	-0.27054	(1.07624, 0.07320)	(0.02823, -0.00061)
 (Vr, Vi)	(-4, 0, 0)	-0.27423	(2.16122, 0.11282)	(0.02538, -0.00320)