Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Indexing Electron Diffraction Patterns Indexing with the orientation of the electron beam unknown
Indexing Electron Diffraction Patterns AimsBefore you startIntroductionMathematics relating the real space to the electron diffraction patternLaue zonesKikuchi linesUsing polycrystalline materials in the TEMConvergent beam electron diffraction (CBED)Using other methods in conjunction with electron diffractionSummaryQuestionsGoing further
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Indexing with the orientation of the electron beam unknown

If we do not know the beam orientation, it is rather more difficult to find which reciprocal plane is the one projected down onto the film.

One approach is to consult tables of angles and distance ratios for the low index reflections for the structure of the crystal we are imaging. Again, we will use copper as an example.

Table of angles

The angles in this table are the angles between the reciprocal lattice vectors given at the sides of the table in the appropriate row and column. Such a table can be extended to include planes with negative indices.

  111 200 220 113 222 133
111 - 54.7° 35.3° 29.5° collinear 22.0°
200 - - 45.0° 72.5° 54.7° 76.7°
220 - - - 64.7° 35.3° 50.0°
113 - - - - 29.5° 26.0°
222 - - - - - 22.0°
133 - - - - - -

Table of distance ratios

The dimensionless numbers in the central portion are the ratios of the 1/dhkl values for the planes given at the sides of the table (column/row).

  1/dhkl (nm-1) 111 200 220 113 222 133
1/dhkl (nm-1)   4.8 5.54 7.83 9.19 9.60 12.07
111 4.8 1 1.15 1.63 1.91 2.00 2.52
200 5.54 - 1 1.41 1.66 1.73 2.18
220 7.83 - - 1 1.17 1.22 1.54
113 9.19 - - - 1 1.04 1.31
222 9.60 - - - - 1 1.26
133 12.07 - - - - - 1

When making such tables spots forbidden by the lattice type should be excluded. Thus for copper, which has an F lattice, the reflections are those with h,k,l all even or all odd.

Now we pick two spots on the diffraction pattern and measure the angle between them and the ratio of their distances from the 000 spot - and see if they correspond to any of the values in the tables.

View an example

Once we know for sure what two of the non-collinear dots are, we can index the rest of the pattern by vector addition.

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