DEFW

home    software section

Simulation of section topographs of dislocations

Y. Epelboin

Laboratoire de Minéralogie-Cristallographie

Universités P.M. Curie and D. Diderot,U.A. C.N.R.S. 009

Case 115, 75252 Paris Cedex 05, France

Tel: +33(1)4427 5211

Fax: +33(1)4427 3785

Yves.Epelboin@lmcp.jussieu.fr

Contents

• Summary
• How to run DEFW?
• How to prepare the data?
• Set of axes used in the program
• Network of integration
• Data file
• Output
• Image file
• Short bibliography
• Installing DEFW

Summary

DEFW is a simulation program devoted to the simulation of section topographs of dislocations. It must be used only when the incident wave along the entrance surface of the crystal can be considered as a spherical wave (Kato approximation [1]). In the case of plane wave topographs, i.e. when using a good monochromator please use DEF4.

DEFW runs on any modern workstation. It needs less than 1 Mbyte of memory and a moderate time of computation. The result is an image written as non formatted Fortran file. To display it as an image it is usually necessary to convert these data into raw floating point numbers acceptable by most visualisation programs such as Triton or any visualization program written in C language. A program named "convert" is available in the package to convert DEFW images to raw data i.e. to a series of floating point values without any header.

How to run DEFW?

All explanations are given for a Unix machine. However the program may be easily compiled on any PC. A 486 or better processor is recommended.

DEFW is designed to run in batch mode. In most cases the computation needs from a few minutes up to a few tens of minutes without any dialogue, thus it is unnecessary to use an interactive display. All data are edited in a file called topoX.tmp. The name of the file containing the computed image is given in the script file used to start the computation.

A basic script looks like:

      # use Korne shell
      #
      #!/bin/ksh
      #
      # 
      # If the image file exists, it must be deleted before starting
      # the computation, otherwise the program will stop.
      # 
      #
      rm -f image.img
      # 
      # Now we start the program. 
      # All lines before EOF are data for DEFW. It is limited to the 
      # name of the image file since all other data are given in a file
      # named topoX.tmp, which name cannot be changed
      #
      defw <<EOF
      image.img
      EOF      

These lines must be written in a file named defw.bat, for instance. Make it executable, using the command: "chmod 700 defw.bat" and submit it as a batch. How to submit a batch depends on your system. Ask your supervisor to know how to do this. You may also run interactively this job by typing the name of the file: "defw.bat" or starting the program and giving the name of the image file.

How to prepare the data?

What do you need?

In this part you will find a brief comment about the data described later in details.

• Information about the crystalline materials
• Diffraction data
• Geometry of the dislocation
• the dimensions of the simulated image

Information about the materials

You need:

• the parameters (in Angströms) as well as the angles (in degrees) of the unit cell.
• the elastic coefficients cij for this materials.
• the indices of the normale to the entrance surface, oriented towards the materials.

The elastic stiffnesses are 3 to up 21 independent values - their number depends on the crystallographic symetry of the crystal. You will have to write half the tensor, i.e. you must know which components are null. See for instance [4].

Diffraction data

All explanations refer to set of axes described below. You must know:

• the dielectric coefficients Kih for the -h and +h reflections. You may use the program Fhkl to compute these data.
• the Bragg angle. Once again you may use Fhkl to compute it.
• the asymetry angle, i.e. the angle between the transmitted beam and the normale to the entrance surface of the crystal. See the section set of axes below for more explanations.
• the indices of the diffraction vector oriented from the transmitted towards the reflected beam.

Note that the Kih are complex numbers and that you must calculate, for each reflection, both the real and imaginary parts. The ratio of the imaginary and real part depend on the choice of the origin in the unit cell. These data are often negative. The program computes the extinction distance and the width at half the maximum of the rocking curve (See the example below) The extinction distance and the width of the rocking curve are a good means to check the consistency of your data.

Geometry of the dislocation

You must know, from an experimental topograph:

• the orientation of the dislocation given either in the crystallographic axes or in the macroscopic axes. See the section below where the axes are explained.
• choose a reference point along the dislocation

Dimensions of the simulated image

The dimensions of the computed image are computed, as well as the number of pixels along a line or a column, from the knowledge of the thickness of the crystal, the Bragg and the asymetry angle. The width of the image is the width of the beam in the film, for an incidence plane. It corresponds to a line in the image. The height of the image is the vertical dimension. More explanations are given in the section set of axes.

Other data are:

• the height of the image to be simulated (in microns).
• Options:
  • Resolution factor.
    The distance between calculated points, along the exit surface of the crystal, is too small for representation since it is choosen according to the network of integration (see the section Network of integration). An option factor, called kperfo in the program, allows to reduce their number. The representation step, i.e. the distance between pixels, along the surface, is kperfo x Pmean, where Pmean is the mean step of integration as explained below. In the following ouput Pmean = 0.8 microns as it may be understood, when looking at the description of the network of integration. kperfo = 2, as it may be seen in the data, line 19. Thus the step is 1.6 microns. In the output listing one finds an horizontal step of 1.57 microns only, since it takes into account the projection of the image from the exit surface to the plane of the film, assumed to be perpendicular to the diffracted beam.
  • an interpolation factor.
    When one computes all lines in the image its value is 1, when computing each second line its value is 2and 3 when computing each third line. Missing lines are interpollated which saves computing time but damages the resolution.
  • an integration option which will be explained in the section Network of integration
  • an option to choose the steps of integration, which is also explained below.

At first look it seems difficult to choose all these data. A simple means to check their consistency is the following: run first the program with the characters: "test" or "TEST" as first characters for the title. The program will write all the output as shown below but it will not run the integration. Thus you will be able to check your data, to verify if the size of the image is reasonnable (most image browsers will not accept more than 512x512pixels), the validity of the diffraction parameters (look at the extinction distance and at the width at half maximum of the rocking curve) and all other data.

Set of axes used in the program

The program uses a set of axes called experimental axes or macroscopic axes. The macroscopic axes are oriented as follow:

- axis x lies along the entrance surface, oriented from right to left. It is the projection of the transmitted direction s0 along the surface.
- axis z is the normale to the entrance surface, towards the inside of the crystal.
- axis y is perpendicular to the plane of incidence, in the plane of the surface. The origin is in the first computed line of the image which means that all lines in the image correspond to negative y values.

The dislocation orientation may be given either in this set of axes or in the crystallographic axes.

As explained before it is assumed that the film is perpendicular to the difracted beam. Steps are choosen to represent the image in the film, not along the exit surface of the crystal.

Network of integration

The diffraction equations [3] are a set of partial derivative hyperbolic equations which are integrated using a finite elements method [2]. The integration network, in a plane of incidence, is determined as follow.

The program computes the position of the extinction fringes in a perfect crystal and uses this information to create a network where the local steps vary according to the distance between the fringes [5]. It is minimum along the edges of the Borrmann fan (see fig.2), maximum in the center. A network option, called iopt, allows to choose the minimum and maximum values of the steps. The following table gives their values together with the value of the mean step called Pmean. Pmean is also the reference to choose the resolution in the computed image which is equal to Pmean x kperfo.

Steps are measured along the exit surface.

Option 1 is usually correct for most computations. However the only method to check the accuracy of the integration of TT equations is to run twice the program using option 1, 2 or 3 and to check if the results are the same. Since the computation time increases tremdously when using the values 2 or 3 this should be done in case of doubt only. Up to now option 1 has been satifying in all cases.

Another option, called, idef, allows to increase (or decrease) the accuracy of the computation. Where the dislocation crosses the transmitted beam (point D in fig.2 b) the phase of the wavefields vary rapidly and it may be useful to increase the precision of the integration by decreasing the integration step in these areas. This is done automatically if idef = 0. The program computes the position of the direct image and switches to the network as drawn in fig.2b in this area. For more explanations see [5]. If idef = 1, the computer ignores the existence of a direct image. Options -1 and -2 remain for compatibility with older versions and should not be used.

Data file

This is an example of data file. The numbers on the left, starting at 01, do not belong to the file. They are written to clarify the explanations.

Input data

01    Image of dislocation in Silicon
02    5.43 5.43 5.43       lengths of the unit vector of the crystal cell
03    90 90 90             angles for the unit cell (cubic in this example)
04    1 -1 -1              normale to the entrance surface
05    -2 -2 0              diffraction vector
06    -0.3 1 2.11 1        orientation of the dislocation. 1 means macroaxes.
07     0.5 0.5 0           Burgers vector
08    16.6 7.96 7.96 0 0 0
      16.6 7.96 0 0 0
      16.6 0 0 0
       6.4 0 0
       6.4 0
       6.4            Elastic constants cij (cubic in this example)
09     0.70926        wave length
10    10.667          Bragg angle (positive value)
11   -10.667          Asymetry angle (- theta for symetric reflection)
12   -1.993e-06 -1.57e-08  KIhm  dielectric components -h values
13   -1.993e-06 -1.57e-08  KIhp  idem h values (real and imaginary parts)
14     300            Height of the image
15     0              Not used. For compatibility with other programs only
16    -32.17 0. 832.  Position of core of dislocation in the set of axes
17     0.00142        absorbtion per micron
18    800             Thickness of crystal
19    2 3 0 1         Options: kperfo, imp, idef, iopt
20    0.8             Step of integration for constant step algorithm
                      (used only if option idef=-2)

Explanations

Each section refers to the line number in the preceeding part. All input are to be written in free format.

1. title: up to 80 characters. It is registered in the first record of the simulated image. If "test" or "TEST" appears as first characters, the initialisation part only is performed. This is useful to check the data and to evaluate the computing time.
2. unit cell parameters in Angströms.
3. Angles of the unit cell, in degrees.
4. Indices of the normale to the entrance surface, oriented towards the inside of the crystal.
5. Indices of the diffraction vector oriented from the origin towards the diffraction node.
6. Orientation of the dislocation. Followed by 0 means that the data are given in the crystallographic axes, by 1 means that they are given in the experimental axes x,y,z as explained.
7. Indices of the Bürgers vector of the dislocation.
8. The 21 values of half the 6x6 matrix representing the elastic constants cij. The unit has no interest.
9. Wave length in Angströms.
10. Bragg angle in degrees. This is a positive value.
11. Asymetry angle in degree. This is the angle between the transmitted direction s0 and the normale to the entrance surface, as shown in the drawing there. It is negative when s0 lies to the left of the normale. Thus in a symetric reflection this angle is the opposite of the Bragg angle.
12. Dielectric constants for -h reflection. Real and imaginary parts are given in SI.
13. Same for the +h reflection. Program Fhkl may be used to compute these data.
14. Height of the image along the y direction in microns.
15. Real part of the dielectric constant for 000.
16. Coordinates of a point located along the dislocation line in the choosen set of axes. Data are in microns.
17. Absorbtion per micron. Also computed with Fhkl.
18. thickness of crystal in microns
19. Options kperfo, imp, idef, iopt
    • kperfo:
      reduction factor for distance between pixels along the exit surface: distance = Pmean x kperfo
    • imp:
      0 : each third line only is computed. Other are interpollated
      1 : all lines are computed
      2 : each second line is computed
      3 : same as 0
    • idef
      0 : varying step algorithm with change of network for the direct image area.
      1 : varying step algorithm without change near the direct image
    • iopt: see table in the section Network of integration.
20. Any value. Meaningless if idef >= 0 (but must be present).

Ouput

Example

The following example is the contents of to the file defw.log, automatically created when running DEFW, and corresponds to the data shown before.

         Program DEFW1. Version June 1993             
         --------------------------------
         Section topograph for dislocation
         ---------------------------------
    Image of dislocation in silicon



Diffraction data
----------------
wave length       .709260 Angstroms
Theta                .186 Deg.
Psi zero          -10.667 Deg.
Mu                .001420 per micron
Khi  H minus R  -1.9930 10(-6), Khi  H minus I   -.0157 10(-6)
Khi  H plus  R  -1.9930 10(-6), Khi  H plus  I   -.0157 10(-6)

rocking curve half width   2.260 seconds
extinction distance       34.972 microns




Defect Data
-----------

Parameters of the crystal
A =           5.430 angstroms
B =           5.430 angstroms
C =           5.430 angstroms
angles       90.000         90.000         90.000
Burgers vector                           .500           .500
.000
Dislocation (macro axes)                -.300          1.000
2.110
Normale to the surface                  1.000         -1.000
-1.000
Diffraction vector                     -2.000         -2.000
.000
Position of the core (microns)        -32.170           .000
832.000

Elastic stiffnesses
     16.600
      7.960    16.600
      7.960     7.960    16.600

       .000      .000      .000     6.400
       .000      .000      .000      .000     6.400
       .000      .000      .000      .000      .000     6.400

Location of direct image               49.078       -270.825
260.559


network of integration
----------------------

Repartition of calculated points along the exit surface
direct image computed
from Y =   -257.921 to   -283.730 microns
planes number  55 to  76


horizontal step near the defect
  till point   17       .20 microns  till point   31       .40 microns
  till point   51       .80 microns  till point  115      1.60 microns
  till point  863       .20 microns  till point  892       .80 microns
  till point  897       .20 microns


Horizontal step far from defect core
  till point   17       .20 microns  till point   31       .40 microns
  till point   51       .80 microns  till point  201      1.60 microns
  till point  245       .80 microns  till point  250       .20 microns


Thickness of simulated crystal          798.49 microns
width of simulated image                295.60 microns
height of image                         303.46 microns
vertical calculated step                  4.72 microns
horizontal representation step            1.57 microns
vertical representation step              1.57 microns

number of calculated planes       79
number of lines in image         193
Number of points per plane       189

Explanations

Please note the indication of a version number. It is important to note it in case of error before reporting the problem. The results are presented in different sections as follow:

• Diffraction data
  The program shows the data. It computes the width at half maximum of the rocking curve and the extinction distance. These values are a good means to check the consistency of the data.
• Defect data
  The program writes the data about the crystal and the dislocation.
• Network of integration
  This section looks differently depending upon the value of option idef. It gives the sequencxe of steps along the exit surface of the crystal (See fig. 2). When idef = 1 the same network of integration is used in all the simulation, thus one sequence only is written. When idef = 0 a different network is used near the dislocation intersection with the transmitted beam [5]. The program indicates the y values of the area where it switches for this new network.

  It is the case in the example where planes 17 up to 31 use this second network.

• Characteristics of the image
  The thickness of the crystal is not always the given value since it must fit the network of integration. Thus it is recalculated and shown together with the width and height of the image. Then the program writes the values of the calculation step along y (called vertical in the listing) ( See fig. 1). It may be equal to the representation step in the same direction (iopt = 1), twice its value (iopt = 2) or three times its value (iopt = 3). It computes also the distances between pixels and the number of computed and represented lines in the image. The number of computed lines is smaller when some lines are interpollated.

Computed image

The name of the file containing the image is not in the data file topoX.tmp. It is part of the script used to run the program or may be input from the keyboard. See the section How to run DEFW.

The image file is written without format:

1. The first record contains the title of the simulation with a maximum of 80 characters.
2. The second record contains the number of pixels along a line and a column written as two integers.
3. The next records are the values of the pixels written line after line as floating point numbers.

This file contains some additional informations added by the Fortran subroutines for non formatted write. Thus it cannot be read directly by a visualisation routine written in C language. The program CONVERT is a utility to translate these data in asuitable form.

Utility program convert

convert is a utility program to convert the various image formats used in simulation programs. It is self explanatory.

CONVERT allows to:

1. option 1 converts an image written with format DEFxx to raw floating point numbers. The Fortran header is removed as well as the title and the image size. The resulting file may be read by C programs such as xtrn. The result consists of floating point numbers corresponding to the successive lines of the image.
   
2. options 2 and 3 are used forimages created by TRANSxx, simulation programs for white beam synchrotron topographs.
   
3. option 4 is used for programs written before 1979.

Bibliography

• [1] Kato N. (1963) J. Phys. Soc. Japan, 18,1785
• [2] Epelboin Y. (1985) Mater. Science and Engineer., 73, 1
• [3] Takagi S. (1969) J. Phys. Soc. Japan, 26, 1239
• [4] Nye J.F. (1985) "Physical Properties of Crystals", Clarendon Press
• [5] Epelboin Y. (1983) Acta Cryst. A39, 761

Installation

DEFW and CONVERT sources are available in the directory pub/topo, when logging as anonymous at ftp.lmcp.jussieu.fr/pub/sincris.

Read first 00readme which contains some additional explanations.

To install the program:

1. Uncompress the file using the command: uncompress defw.tar.Z
2. Untar using the command tar -xvf defw.tar or tar -xovf defw.tar for certain systems.
3. Check the contents of the directory defw and read the README.
4. Edit the makefile to choose the right options for your compiler. Examples are given (line starting with #) for some machines. Ask your supervisor in case of trouble.
5. Compile the program.
6. Rename the file topoX.test to topoX.tmp
7. Run defw as shown in the first part.
8. Check that the file named defw.log is identical to the output in the present document.

Please send your comments and your suggestions to Yves Epelboin, Yves.Epelboin@lmcp.jussieu.fr .

Last update 28/06/98 Y.E.

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