EBSD Reflector Ranking
When using the standard Hough-based indexing algorithms provided by EBSD vendors, a list of possible reflectors must be selected. Often this list is suggested by the vendor, based on either x-ray kinematical structure factors, or on electron scattering factors (but still kinematical). For simple crystal structures, the kinematical ranking is usually similar to a ranking derived from dynamical scattering computations (see below), but for more complex structures the kinematical model can produce misleading results.
The EMsoft package provides the tools to compute a dynamical EBSD master pattern (MP). Once the MP is available, the integrated intensity of each Kikuchi band can be determined using the following relation:
where
is the normalized master pattern intensity on the Kikuchi sphere, with m the average intensity and sigma the standard deviation. The integration is carried out over a band of width 2theta_B with theta_B the Bragg angle. The quaternion q_hkl rotates the (hkl) Kikuchi band to the equatorial plane so that the integration can be carried out. This results in a list of dynamical integrated Kikuchi band intensities, which we refer to as beta parameters. A simple ranking of the beta parameters then results in a dynamical reflector ranking.
For a complete description of the algorithm as well as a series of example rankings for nickel, silicon, rutile, and forsterite, we refer to the following paper: S.I. Wright, S. Singh, and M. De Graef. "Reflector Selection for the Indexing of Electron Back-Scatter Diffraction Patterns". Microscopy and Microanalysis (2018, under review).
The EMEBSDreflectors program uses a short name list as input:
&EBSDreflectors
! smallest d-spacing to take into account [nm]
dmin = 0.05,
! name of master pattern file; relative to EMdatapathname
masterfile = 'undefined',
! name of the .csv, markdown or LaTeX output file (no extension) ; relative to EMdatapathname
listfile = 'undefined',
! sampling step size (degrees)
increment = 0.025,
! output type; available options 'latex', 'csv', 'markdown'
outputformat = 'csv',
! number of reflectors to be listed in final output
numlist = 20,
! number of parallel threads to be used for computation
nthreads = 1,
/The output can be generated in latex format, as a .csv file, or as a markdown (.md) file; the parameter listfile is the filename without extension, and the extension will be added automatically depending on the outputformat setting.
Using a master pattern file for fcc nickel at 20 kV, and numlist set to 10, the following program output is obtained (with outputformat set to 'markdown'):
| # | (hkl) | beta_hkl | Ikin+abs |
|---|---|---|---|
| 1 | ( 1 1 1) | 100.00 | 100.00 |
| 2 | ( 2 0 0) | 93.86 | 87.17 |
| 3 | ( 2 2 0) | 51.16 | 60.94 |
| 4 | ( 3 1 1) | 33.69 | 50.29 |
| 5 | ( 4 2 0) | 11.93 | 32.11 |
| 6 | ( 3 3 1) | 11.12 | 33.55 |
| 7 | ( 5 3 1) | 7.33 | 18.49 |
| 8 | ( 4 2 2) | 5.27 | 27.22 |
| 9 | ( 6 2 0) | 4.56 | 15.90 |
| 10 | ( 5 1 1) | 3.74 | 24.28 |
The column labeled beta_hkl contains the ranked integrated intensities, and the last column has kinematical intensities, including absorption (i.e., complex valued electron scattering factors with absorptive form factors). Note that the ranking order is nearly identical, with only a few switches.
A similar computation for the tetragonal rutile crystal structure (TiO_2) results in the following table (listing the top 15 reflectors):
| # | (hkl) | beta_hkl | Ikin+abs |
|---|---|---|---|
| 1 | ( 1 1 0) | 100.00 | 100.00 |
| 2 | ( 0 0 2) | 78.96 | 76.16 |
| 3 | ( 1 0 1) | 68.23 | 52.31 |
| 4 | ( 2 1 1) | 60.99 | 57.66 |
| 5 | ( 3 0 1) | 58.51 | 62.98 |
| 6 | ( 1 1 1) | 50.24 | 46.28 |
| 7 | ( 1 1 2) | 33.39 | 39.07 |
| 8 | ( 3 1 0) | 23.28 | 28.53 |
| 9 | ( 4 0 0) | 22.36 | 31.94 |
| 10 | ( 2 1 0) | 16.37 | 29.84 |
| 11 | ( 4 1 1) | 10.55 | 22.21 |
| 12 | ( 5 1 0) | 9.92 | 23.26 |
| 13 | ( 3 3 2) | 9.53 | 28.25 |
| 14 | ( 5 2 1) | 9.13 | 24.81 |
| 15 | ( 4 0 2) | 7.89 | 21.26 |
In this case, there are several differences between the kinematical and dynamical reflector ranking. As described in the paper cited above, the dynamical ranking using the beta parameters is in good agreement with a ranking based on averaged Hough peak intensities.