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Electron

  Ratio Technique     3 of 8
 

Geometry

The geometry of an electron diffraction experiment is shown here.

The Bragg Law for small angles approximates to: l = 2dq  Electron diffraction
From the diagram: Equation image
Therefore: Equation image or Equation image
The distance,  r, of a diffraction spot from the direct beam spot on the diffraction pattern, varies inversely with the spacing of the planes, d, that generate that spot.

Note: no lenses have been shown. They merely alter the effective camera length, L. Often the value of lL is referred to as the camera constant of the microscope.

Indexing the pattern

Vectors imageAny 2-D section of a reciprocal lattice can be defined by two vectors so we only need to index 2 spots. All others can be deduced by vector addition.

If the crystal structure is known, the ratio procedure for indexing is:

  1. Choose one spot to be the origin. Note: it does not matter which spot you choose.
  2. Measure the spacing of one prominent spot, r1. Note: for greater accuracy measure across several spots in a line and average their spacings.
  3. Measure the spacing of a second spot, r2. Note: the second spot must not be collinear with the first spot and the origin.
  4. Measure the angle between the spots, f.
  5. Prepare a table giving the ratios of the spacings of permitted diffraction planes in the known structure.  Hint: start with the widest spaced plane (smallest r). You only ever need to do this once for each structure.  Blank tables are provided here...
  6. Take the measured ratio r1/r2 and locate a value close to this in the table.
  7. Assign the more widely-spaced plane (usually with lower indices) to the shorter r value.
  8. Calculate the angle between pair of planes of the type you have indexed.  Equation and example...
  9. If the experimental angle,f , agrees with one of the possible values - accept the indexing. If not, revisit the table and select another possible pair of planes.
  10. Finish indexing the pattern by vector addition. Example of indexing provided here...
 

 
 

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