Quantitative Electron Diffraction (QED)

Copyrights 2006, EMLab Software

 

 

 

I. Introduction. 2

II. GET STARTED.. 3

III. DISPLAY IMAGES. 3

V. HOLZ LINE FITTING FOR STRAIN MAPPING.. 11

V.1 Using xlines to process CBED patterns. 11

V.2 Using Hough to Detect Lines. 12

V.3 Using xindex to assign (hkl) indices to experimental HOLZ lines. 14

V.4 Define fitting parameters. 14

V.5 Fit 16

V.6 Examples. 18


I. Introduction

 

The QED program is designed for quantitative analysis and simulation of electron diffraction patterns. The program comes with a basic package for displaying and simulation of selected area and convergent beam electron diffraction patterns for crystals using kinematical approximation. The optional packages (installed separately) include 1) analysis of high-order Laue zone lines (HOLZ), 2) radial distribution function analysis of amorphous and nano-crystalline materials, and 3) simulation of electron diffraction patterns by dynamic diffraction.

 

The program uses a command window for calling/executing functions in the program and a graphics display for interactive processing and feedback (see Fig. 1). The command window uses the DOS interface. Some knowledge of DOS commands is assumed for using QED program (For users unfamiliar with DOS, google introduction to DOS for a number of helpful writeup's for DOS. The most commonly used commands for QED are, cd, copy, del, rename, and dir). Interaction with display is achieved through the use of a mouse pointer. A basic model with at least two buttons (right and left clicks) is required.

 

Figure 1 Screen snapshot of QED program showing a command window and a display window.

This manual describes the use of QED program. The description is organized based on tasks. A summary of functions and their usage are provided in the appendix. The conventions used in this manual are bold italics for commands and functions in Courier font.

 

II. GET STARTED

 

To install the program, insert the install CD and follow the instructions. At the end of the install process, you should see the following message, " ". To run QED, click on start on your desktop, choose accessories and choose command prompt. A DOS command window should appear on your desktop. Type cd /emaps/examples/ to change working directory to /emaps/examples/ . Type qed to start qed program. You should see the prompt of qed >. To enter a command, make sure that the command window is active, you can do this by left click anywhere in the window.

 

Hint: To simplify the starting process, create a short cut to command prompt by right click on command prompt in the start menu and choose create short cut from the list of popup menu items. Drag the short cut to your desktop. Use the short cut to start command prompt.

 

Hint: DOS programs can be run inside QED by adding x or / in front of the command, for example, xdir list the files in current working directory and xcd can be used to change directory.

 

III. DISPLAY IMAGES

 

 

To display a diffraction pattern or an image in general, use read command. For example,

 

read si133.img

 

followed by

 

show

 

in the directory of  /emaps/examples/  will read image file si133.img and display the image in the graphics window.

 

QED accepts three types of image format, the img format, the raw format and Gatan Mac Fixed Format (.grw file). To read a raw data file, use

 

read -r si133.raw

 

for example, will read the raw image in the file of si133.raw. A prompt will appear in the command window and ask you for 1) the number of bytes to skip at the beginning of the file, the data type, whether to swap bytes and number of row and number of columns in the image. The raw image format is supported by most scientific image processing programs. For example, both Gatan DigitalMicrograph and NIH ImageJ or NIH Image support raw data format. In ImageJ, use save as and choose raw format. In DigitalMicrograph use save, data only format (.dat). A byte swap is required for the raw format of ImageJ, but not for Gatan .dat file.

Since the raw data image format does not contain information about the image, information the data type and the size of the image must be entered by hand. The byte swap is related to how numbers are stored in computers, PC and Mac use different formats and byte swap is used for format conversion.

 

A typical image consists of an array of data arranged in rows and columns:

 

 

QED uses (row, column) coordinate to display and specify image coordinates. The top left corner is (0,0) for example. For each data points, the data can be a byte (256 maximum), a short integer (2 bytes), an unsigned short integer (2 bytes), a real number (4 bytes) or complex (8 bytes). QED used an integer called data type to represent different data format, specifically, 0 is for byte, 1 for short integer and 2 for real. Others are specific to data format, for example, 11 is used for unsigned integer and 3 is used for complex.

 

The img format is the default format for QED and other programs associated with QED. To convert other formats to img format, use

 

write filename.img

 

to output an img file.

 

The img format consists of a 8 byte header and followed by image data, which is similar to the grw of Gatan DigitalMicrograph except the byte order.

 

For users use  DigitalMicrograph program to convert image files, a more convenient format is the Mac Fixed Format, .grw, file, which can be read by QED using

 

Read -S filename.grw

 

Images are displayed in color in QED and in a default intensity range found by the program and at1x1 scale. To change these defaults, for example, to magnify image by 2, use

 

show -z 2

 

or

 

show -z -2

 

to demagnify by a factor of 2. To change the intensity ranges that are displayed, use

 

minmax value_1 value_2

 

value_1 should be smaller than value_2

 

To change the color of image display, use

 

color -image 1

 

For example, to switch to greyscale. Other acceptable colors are 2, 3, 4, and 5. Color 2 is the default color.

 

 

To move image around in the display window, use the display function:

 

display

 

left click within the image window (the first click activates the image window and is ignored). The image is divided into four quadrants, each corresponding to one of four possible directions. The display function is useful in zoom display mode for selecting the image area for display. The alternative to display function is to use -w option in show, for example

 

show -z 2 -w 250 250

 

shift image pixel (250, 250) to the top left corner.

 

To display the image intensity at a pixel, type

 

index

 

to index mode. Move the pointer to the desired pixel and left click. The pixel coordinate and image intensity should be displayed at the command window. Right click or type x to get out of index mode. Index mode has other functions, which will be introduced later.

 

The other way to check image intensity is by line profile. To do this, draw a line on the image using the line command using

 

line

 

and use left and right click to define the beginning and the end of the line. Make sure to save the line by by typing s. Exit the line mode by typing x. Use

 

profile

 

to plot the intensity along the line. To go back to the image, use

 

show

 

command. The line command has a default line width of 1. To average intensity over several pixels perpendicular to the line, use

 

line 20

 

for example to average over 20 pixels. To save line scan intensity, use

 

write -s filename

 

The file is in text format, which can be opened using a text editor such as notepad. The first column is the pixel number, the second column is intensity. The third column is reserved for other functions.
IV. Simulating Electron Diffraction Patterns

 

QED includes functions to simulate crystal electron diffraction pattern geometry and match with experimental patterns. To simulate, load a crystal first by

 

Load si.dat

 

for example, load silicon crystal defined in the file of si.dat. The file contains a set of commands, which define the atomic structure of silicon (see Fig. 3)

 

crystal silicon : dw = iso occ = 1

cell 5.4307 5.4307 5.4307 90.0 90.0 90.0

atom Si 0.125 0.125  0.125 0.4668

spg 227 2

 

Fig. 3 Crystal structure definition file

 

The first line defines the name of the crystal (silicon) and the style of structure file, which separated from the crystal name by :. In this case dw = iso tells the program that isotropic Debye-Waller factors are used and occ = 1 tells all atoms have occupancy of 1. The crystal structure is specified by the unit cell (a, b, c and a,b,g) using cell command and atomic positions in fractional coordinates using atom command. Each atom is defined its atomic symbol followed by 4 numbers (x, y, z and B). B is the isotropic Debye-Waller factor.

 

To simulate electron diffraction, type

 

Diffract

 

Fig. 4 Diffraction window and simulated diffraction pattern.

The diffraction pattern is plotted in a window separated from the image. Click on the sqaure symbol at the top right corner of the window to use the full display. A pattern similar to Fig. 4 should now show in the graphics display. The program plots electron diffraction using a set of default parameters, including high voltage (200 kv), zone axis ([001]), camera length (1000 mm) and diffraction mode (selected area).

 

To change the program default for diffraction, use the following commands:

 

kv highvoltage, for example kv 100 change electron highvoltage to 100 kilovolts

 

zone u v w, for example zone 1 1 1 change to 111 zone axis

 

cl camera_length_in_mm, for example cl 2000 change to camera length to 2000 mm

 

xaxis h k l, for example xaxis 2 -2 0 change to the diffraction pattern orientation so that (2-20) is along x-axis (horizontal to the right)

 

kline no, to turn off Kikuchi lines (use kline yes   to turn it back on)

 

cbed disk_size_in_reciprocal_anstrom, for example cbed 0.25 change to CBED mode and disk radius 0.25 1/ansgtrom

 

zero_disk, to show (000) only

 

all_disk, to show all allowed diffraction disks

 

To tilt crystal and simulate electron diffraction pattern in off-zone axis orientations, use

 

Tilt x_tilt y_tilt, for tilt 1 0 tilt the crystal by 1 degree around the x-axis and 0 around the y-axis.

 

Tilt is only recommend for small angles around the zone axis. For large tilt, redefine the zone axis.

 

To take effect of changes made to diffraction conditions, issue diffract command to replot the pattern.

 

Hint: Use clear all command in view menu of the graphics window to clear any left over graphics in the display.
IV. Interactive Matching of Diffraction Patterns

 

Electron diffraction patterns can be plotted directly on experimental images in QED for interactive matching. To use this function follow instructions in section III to set up diffraction conditions and section II to display experimental images. Use map command to match simulation with experiment:

 

map filename.map

 

To run map, the crystal must be defined first, otherwise, an error message will be displayed about undefined crystal.

 

Fig. 5 Interactive matching of experiment CBED patterns

 

 

Once enter the map mode, the program offers five options to manipulate the diffraction pattern: d for change disk size, c for centering the disk, s for the scale of diffraction pattern (magnification), r for rotating the pattern and t for tilt, which keeps the diffraction disk fixed and moves Kikuchi/HOLZ lines. To switch between modes, type the character representing the mode while the graphics window is highlighted. You should see a confirmation in the DOS window indicating the mode of current operation. Once the matching is finished, type x to exit map mode. The mapping information will be stored in the file specified after the map command.

 

Once a map file is created, it can be re-plotted by using the command

 

Map filename.map -r  -n

 

To further adjust the matching, use

 

Map filename.map -r

 

Without the -n option, which switches off the interactive feature.

 

Map information is also given in six numbers after exiting map. These numbers can be used to recreate the mapping as well by using

 

Map  filename.map -p value_1 … value_6

 

To find out the index of Kikuchi/HOLZ lines and index of disks, use the

 

index

 

Command. To index a line, right click the graphics window to make it active and type l. You should see a confirmation of line: in command window. Position the point to the line that you want to index and left click. The index will be placed next to the mouse position. Use r option to index disks (reflections).

 

To change line colors or disk or text colors, use command

 

Color -options color_index

 

For example, color -line 2 changes the line color to red. Change of color only affects next plotting and does not affect lines already plotted. Other options are text, disk and box. The option for image has already been described in section II.

Fig. 6 shows the color index from 0 to 15 with 0 for white and 4 for green for example

 

 


V. HOLZ LINE FITTING FOR STRAIN MAPPING

 

 

One of the major applications of CBED is strain mapping or lattice parameter measurements. The principle of lattice parameter measurements is based on fitting high order Laue zone (HOLZ) lines. The intercept of HOLZ lines is very sensitive to small changes in crystal lattice, especially for high index reflections. Measuring lattice parameters involves several tasks, including measuring lines from the recorded CBED pattern, indexing of experimental lines and finally fitting. These tasks are achieved through the following automated functions:

 

1. Preprocessing of diffraction patterns for line detection using xlines

2. Detect lines using Hough transformation using hough

3. Index HOLZ lines using xindex

4. Define fitting parameters using xkfiti

5. Perform fit using xkfit

 

These functions are described in the following.

 

Hint: functions with name starting with x reside in DOS and can be run separated from QED program. To run DOS programs inside QED, add x or / in front of the command.

 

V.1 Using xlines to process CBED patterns

 

To run xlines requires an input image file and an output file to store the processed image. To run the program, type

 

xlines input_image output_image threshold -d

 

For example, xlines si133.img si133p.img 10 -d will detect deficiency lines in the diffraction pattern stored in si133.img for lines with intensity greater than 10. xlines works by applying a filter to enhance lines in an image. The intensity in the processed image, si133p.img,  represents the strength of line, which is proportional to the intensity difference at the line center and outside the line. A threshold is used to define the existence of a line. The threshold is defined as the line intensity or a factor times the estimated noise (sigma). The program estimates the noise by taking square root of the intensity.

 

One useful option of xlines is -mask row column radius, for example,

 

xlines si133.img si133p.img 10 -d -mask 529 478 340

 

will produce the image shown in Fig. 6.

 

Fig. 6 Image after processing using xlines

 

To use noise-based method, using xlines together with -n option, for example,

 

xlines si133.img si133p.img 2 -d -mask 529 478 340 -n

 

will process the image by define lines larger than 2*sigma.

 

Whether the intensity or noise based detection method produce the best result depends on image quality. For high quality images, the results from the two methods are similar.

 

V.2 Using Hough to Detect Lines

 

QED uses Hough transformation to measure the slope and intercept of HOLZ lines. A line in general is defined by

 

y=kx+c,

 

where k is the slope and c is the intercept. Hough transform works by plot the line intensity on a distance and angle diagram (Hough diagram) using

 

r=xcosq+ysinq

 

The points on a straight line defines a peak in the Hough diagram, which can be detected by a search for peaks.

 

To use Hough transform for line detection in QED, use

 

Hough -h output_image1 output_image2 filename.holz

 

The image1 contains the points on detected HOLZ lines, while image2 contains the calculated Hough diagram. The detected lines are stored in filename.holz.

 

QED Hough command uses a number of criteria to distinguish a HOLZ line from background noises or other features in CBED patterns. The two most criteria are 1) line intensity threshold and 3) Hough strength threshold. The program provides default values for these thresholds, which can be used as a starting point. For example, Fig. 7 shows lines detected from the processed image of Fig. 5 using

 

Hough -h image1.img image2.img si133p.holz -g 90 -s 1800

 

Fig. 7 Lines detected by Hough command. The lines are labeled using sequential numbers. Once indexed, hkl appears inside the bracket.

 

To deal with unwanted lines and to provide a manual indexing mechanism, Hough includes an interactive mode, in which lines can be indexed or deleted. To use this mode, type -i option, for example, command

 

Hough -h image1.img image2.img si133p.holz -g 90 -s 1800 -i

 

detects the line and stay in the interactive mode. Two options are offered, type I to enter indexing mode. Left click on the line and click on command window to activate it and enter h k l values in the window. Type - to delete lines, right lick on the line in this mode will "remove" it from the line list.

V.3 Using xindex to assign (hkl) indices to experimental HOLZ lines

 

QED uses an automated procedure to index experimental HOLZ lines. The procedure is based on matching of line slopes and intercepts with the simulated pattern (see section IV). The indexing works best when there is a reasonable match between the experimental and theoretical patterns.

 

To prepare for indexing, use procedures described in IV to prepare a map file.

 

To run xindex, type

 

xindex filename.holz filename.map

 

When the program finishes, it displays how many lines are indexed and how many lines with multiple indicies. The holz file will be updated with new indicies. The file can be inspected by using the more command in DOS. To run this in QED, use xmore followed by the filename, e.g. xmore si133p.holz.

 

To see which lines are indexed, use

 

hough -P filename.holz

 

The program will plot the indexed lines on the image display.

 

There are two options for xindex, one is the angular tolerance and the other is intercept error range. The program uses these two values to decide if the experimental line and theoretical line are in agreement. To change the default values (1 degree and 5% respectively), use

 

xindex filename.holz filename.map -t angle -i intercept_percentage

 

 

V.4 Define fitting parameters

 

Once the experimental HOLZ lines are measured and indexed following above procedures, they can be used for fitting. Before fitting, one important thing to consider is what to fit. For a pattern taken far away from interfaces and devices, fitting electron high voltage is a natural choice to get reference high voltage. For a strained crystal, which one or more of six cell parameters should be fitted depends on the diffraction geometry and the nature of strain field.

 

 QED uses xkfiti command to prepare fitting parameters. It takes values from the holz and map file to prepare a fitting parameter file (.fit). To run kfiti for refining high voltage, use

xkfiti filename.holz filename.map -fit V

 

To run kfiti for refining cell parameters, for example, for a in a cubic crystal, use

 

xkfiti filename.holz filename.map -fit a a a 0 0 0

 

In general, the format is -fit a b c A B G , where a b c A (for alpha), B (for Beta) and G (for gamma) stands for the six cell parameters. In some cases, two or three cell parameters are equal, for example, a=b for tetragonal crystals. In such cases, use the same symbol to make them equal. For example, -fit a a c 0 0 0 will fit a tetragonal crystal and  --fit a b c 0 B 0 will fit a monoclinic crystal. Here, 0 means constant, e.g. the program will use the default value in the .map file.

 

Other options for the program are used to control the fitting. These include:

 

-p DPar (parameter range in %, defualt 0.1)

-g DeltaG (dispersion correction, defualt 0.0)

-t DFTol (tolerance, defualt 0.00001);

-n MaxF (maximum iteration, defualt 100);

 

These options are used in case the default values do not yield satisfactory fitting results. The most important one is the -p option, which defines the approximate search range for refined parameters, for example, 0.1 means 10% . If the highvoltage is 200 kv, the search for best fit will be carried out in the range of 20 kV, which is sufficient in most cases.

 

A typical fit file looks like this:

 

#!FIT written by KFITI version 1

crystal silicon

5.4307 5.4307 5.4307 90.0 90.0 90.0

 

#hv

119.200000

#zone axis

1 3 3

#gx

0 -2 2

#kt

0.338351 -0.092898 -0.019885

#U0

0.0

#Dispersion Correction

0.000000

#number of pars

1

0 0 0 0 0 0 1

#fitting control parameters tol maxf chimn

0.000010

100

0.000005

#parameter range

0.100000

#experimental lines

23

4 -12 12 -1 -2276.8 6.14083 10.327 0.022722 1.0

-3 11 -9 -1 -2005.85 6.13695 8.09894 0.019694 1.0

1 7 -7 -1 -1039.39 1.93762 2.52946 0.003504 1.0

-3 -9 11 -1 -335.746 1.42495 0.66029 0.001165 1.0

4 12 -12 -1 -258.489 1.42141 0.61583 0.00115 1.0

-8 -8 12 -1 -50.4227 0.754357 0.603993 0.001036 1.0

9 7 -9 -1 388.414 0.557895 0.2708 0.000592 1.0

-5 -3 5 -1 52.5025 0.488179 0.37187 0.000643 1.0

-9 -5 9 -1 -19.2491 0.460337 0.463914 0.000783 1.0

7 3 -5 -1 290.607 0.286331 0.264783 0.000499 1.0

-14 -4 10 -1 573.162 0.198553 0.288772 0.000621 1.0

-14 -2 8 -1 284.702 0.068871 0.276118 0.000533 1.0

16 2 -6 -1 597.296 -0.050149 0.170963 0.000318 1.0

15 1 -5 -1 400.428 -0.105942 0.321129 0.000645 1.0

-15 3 3 -1 948.827 -0.405015 0.320811 0.000558 1.0

15 -5 1 -1 690.292 -0.781315 0.422496 0.000997 1.0

16 -6 2 -1 1025.5 -0.87553 0.304898 0.000566 1.0

7 -3 1 -1 1392.76 -0.962154 1.5522 0.002198 1.0

16 -8 4 -1 783.153 -1.14356 0.490069 0.001404 1.0

-14 10 -4 -1 1492.5 -1.43503 1.46828 0.00229 1.0

7 -5 3 -1 1237.39 -1.78093 0.840903 0.001866 1.0

-5 5 -3 -1 1303.75 -2.77028 1.27375 0.003975 1.0

10 -10 8 -1 2670.77 -3.41091 4.31137 0.006974 0

#!end_fit

 

The last part of the fit file list the experimental lines used for fit. The first three numbers are hkl index and -1 indicates a deficiency line. The four real numbers are measured intercept and slope and their estimated errors. The last number is the weight used for fitting. They are set to 1 as default. Lines that are mis-indexed or cannot be fitted are assigned with a weight of 0.

 

V.5 Fit

 

To fit, use the fit parameter file prepared by xkfiti. The fit file can also be edited by hand using a text editor, however, this is only recommended for experienced users. To run the program, type

 

xkfit filename.fit

 

After the program finishes, the results of fitting is displayed, which shows the best fit parameters and the difference between experimental and theoretical lines. A example  is shown below:

 

KFIT V2.0, COPYRIGHTED, EMLABSOFTWARE, ALL RIGHTS RESERVED

crystal silicon

 

                   NEAREST ZONE AXIS    1   3   3

 BASE G VECTOR ALONG X    0.00000  -2.00000   2.00000

 

 INCIDENT ELECTRON BEAM HIGH VOLTAGE   120.000000 KV

                          WAVELENGTH     0.033492  A

 

 THE UNIT CELL  a =   5.43070 Angstrom

                b =   5.43070 Angstrom

                c =   5.43070 Angstrom

            alpha =  90.00000 Degree

             beta =  90.00000 Degree

            gamma =  90.00000 Degree

 high voltage is a parameter

 

 TOTAL NUMBER OF LINES: 23

 

 H   K   L    m    sigma_m    b sigma_b   Weight

 4 -12 12  6.14083004  0.0227220003 -2276.80005  10.3269997  1.

 -3 11 -9  6.13695002  0.0196940005 -2005.84998  8.0989399  1.

 1 7 -7  1.93762004  0.00350399991 -1039.39001  2.52945995  1.

 -3 -9 11  1.42495  0.00116500002 -335.746002  0.660290003  1.

 4 12 -12  1.42140996  0.00115000003 -258.489014  0.615830004  1.

 -8 -8 12  0.75435698  0.00103599997 -50.422699  0.603992999  1.

 9 7 -9  0.557895005  0.000592000026  388.414001  0.270799994  1.

 -5 -3 5  0.488178998  0.000643000007  52.5024986  0.371870011  1.

 -9 -5 9  0.460337013  0.000783000025 -19.2490997  0.463914007  1.

 7 3 -5  0.286330998  0.000499000016  290.606995  0.264782995  1.

 -14 -4 10  0.198552996  0.000621000014  573.161987  0.288771987  1.

 -14 -2 8  0.0688709989  0.000532999984  284.701996  0.27611801  1.

 16 2 -6 -0.0501490012  0.000318000006  597.296021  0.170963004  1.

 15 1 -5 -0.105942003  0.000645000022  400.428009  0.321128994  1.

 -15 3 3 -0.405014992  0.000558  948.827026  0.320811003  1.

 15 -5 1 -0.781315029  0.000997000025  690.291992  0.422495991  1.

 16 -6 2 -0.875530005  0.000566000002  1025.5  0.304897994  1.

 7 -3 1 -0.962153971  0.00219799997  1392.76001  1.55219996  1.

 16 -8 4 -1.14356005  0.00140399998  783.153015  0.490069002  1.

 -14 10 -4 -1.43502998  0.00228999997  1492.5  1.46827996  1.

 7 -5 3 -1.78093004  0.00186600001  1237.39001  0.840902984  1.

 -5 5 -3 -2.77027988  0.00397500023  1303.75  1.27374995  1.

 10 -10 8 -3.41090989  0.00697400002  2670.77002  4.3113699  0.

 

 

 GOF = 0.16036E-01 AT THE LOWEST POINT OF SIMPLEX:

  0.12000E+03

 

 GOF = 0.16036E-01 AT THE LOWEST POINT OF SIMPLEX:

  0.12000E+03

 

 GOF = 0.16036E-01 AT THE LOWEST POINT OF SIMPLEX:

  0.12000E+03

 

 GOF = 0.16036E-01 AT THE LOWEST POINT OF SIMPLEX:

  0.12000E+03

 

 GOF = 0.92021E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11925E+03

 

 GOF = 0.92021E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11925E+03

 

 GOF = 0.92021E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11925E+03

 

 GOF = 0.92021E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11925E+03

 

 GOF = 0.92021E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11925E+03

 

 GOF = 0.92021E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11925E+03

 

 GOF = 0.91978E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 GOF = 0.91978E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 GOF = 0.91978E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 GOF = 0.91972E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 GOF = 0.91972E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 GOF = 0.91972E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 GOF = 0.91972E-02 AT THE LOWEST POINT OF SIMPLEX:

  0.11926E+03

 

 Results from fitting :

 H K L Experiment Fit Difference (theta Dist)

   4 -12  12     ( -0.1614   365.94)     ( -0.1597   366.95)     ( -0.0017    -1.01)

  -3  11  -9     ( -0.1615   322.59)     ( -0.1597   323.19)     ( -0.0018    -0.60)

   1   7  -7     ( -0.4764   476.69)     ( -0.4833   494.56)     (  0.0068   -17.87)

  -3  -9  11     ( -0.6119   192.87)     ( -0.6108   192.02)     ( -0.0011     0.84)

   4  12 -12     ( -0.6131   148.73)     ( -0.6108   148.26)     ( -0.0023     0.47)

  -8  -8  12     ( -0.9245    40.25)     ( -0.9231    38.92)     ( -0.0014     1.33)

   9   7  -9     ( -1.0619  -339.20)     ( -1.0618  -339.46)     ( -0.0001     0.26)

  -5  -3   5     ( -1.1167   -47.18)     ( -1.1277   -53.20)     (  0.0111     6.02)

  -9  -5   9     ( -1.1394    17.49)     ( -1.1452     9.60)     (  0.0059     7.89)

   7   3  -5     ( -1.2919  -279.38)     ( -1.2859  -282.04)     ( -0.0060     2.66)

 -14  -4  10     ( -1.3748  -562.19)     ( -1.3602  -552.56)     ( -0.0146    -9.62)

 -14  -2   8     ( -1.5020  -284.03)     ( -1.5050  -289.41)     (  0.0030     5.38)

  16   2  -6     (  1.5207   596.55)     (  1.5211   596.19)     ( -0.0004     0.36)

  15   1  -5     (  1.4652   398.20)     (  1.4569   404.01)     (  0.0083    -5.81)

 -15   3   3     (  1.1860   879.43)     (  1.1855   878.76)     (  0.0004     0.67)

  15  -5   1     (  0.9076   543.95)     (  0.9142   545.70)     ( -0.0066    -1.75)

  16  -6   2     (  0.8517   771.56)     (  0.8500   770.24)     (  0.0017     1.33)

   7  -3   1     (  0.8047  1003.64)     (  0.8083  1004.36)     ( -0.0036    -0.72)

  16  -8   4     (  0.7185   515.53)     (  0.7036   499.65)     (  0.0149    15.88)

 -14  10  -4     (  0.6086   853.30)     (  0.5897   849.22)     (  0.0189     4.09)

   7  -5   3     (  0.5116   605.83)     (  0.5154   610.36)     ( -0.0037    -4.53)

  -5   5  -3     (  0.3464   442.66)     (  0.3572   442.66)     ( -0.0108     0.00)

  10 -10   8     (  0.2852   751.38)     (  0.2835   895.97)     (  0.0017  -144.59)

 

The best fit in this case gives an effective voltage of 119.26 kV. The last part of the output file compares the line distance of the fit with experiment. In some cases, some of lines can not be fitted. The reason can be several, one is the wrong index by xindex program. In this case, the line can be simply removed by assigning a 0 weight in the .fit file for that line. This is the case for (10,-10,8) in si133p.fit file. In other cases, dynamic effect can play a role. Diffraction pattern distortion due to projector lens can also contribute to the poor fit.

 

V.6 Examples

 

The examples of fit are given in the folder cases. The si230s.img is  a diffraction pattern recorded near a device. The si230sp.img is processed image and si230sp.fit is the fit parameter file. The Si133 case was discussed in this manual, which was taken from silicon substrate along [133] zone axis. The si230.img was taken near [230] zone axis from si substrate. For practice, use si230 to calibrate high voltage and si230s to refine lattice. Follow the commands in si230.log file for the calibration. The file can be open by xmore si230.log command in qed or more si230.log in DOS. For lattice parameter measurement use commands in si230s.log file.

 

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