In the study of crystals, the position of a crystal face in space is given by the intercepts the face or plane makes on three (or four) imaginary lines, called ‘crystallographic axes’. The X-ray crystallographer can measure the ‘unit-cell’ dimensions in ångstrom units (Å), and the axial ratios express the relative, and not the absolute, lengths of the cell edges corresponding to the crystallographic axes. These ratios (or ‘parameters’) are often expressed reciprocally as ‘indices’, e.g. Miller indices.
| Crystallographic Axes |
|---|
| a (x) | b (y) | c (z) |
|---|
Intercepts of crystal face DEF in ångstroms, on a, b, and c axes, measured from origin. | OD | OE | OF |
20Å | 10Å | 40Å |
If b intercept is made equal to 1 the axial ratio is obtained for the crystal. | 20 | 10 | 40 |
10 | 10 | 10 |
2 | :1 | :4 |
Indices are obtained by dividing the intercepts of face DEF into those of the parametral plane, which is a face of the unit form with intercepts (111). | 1 | 1 | 1 |
2 | 1 | 4 |
Miller’s indices of face DEF are obtained by removing fractions. | 2 | 4 | 1 |
Note that if face DEF is selected as the parametral plane, then its indices would be 2/2, 1/1, 4/4 = 111.