Introduction

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The trigonal and hexagonal unit-cell information given in the tables below is reference material only, as are the vector dot product equations.

Introduction

These sections on indexing and unit-cell refinement are mainly concerned with the methods by which a unit cell is determined and its lattice parameters refined. In contrast to single-crystal diffraction where information is obtained for the position of the reflection in three dimensions (conventionally in terms of the three orientation angles ω,χ,φ), powder diffraction leads to the recording of one-dimensional information (conventionally 2θ from angle dispersive and d or 1/d from wavelength dispersive instruments). The first step in reconstructing the three dimensional information from powder diffraction data is the determination of a unit cell solely from the positions of the peaks. These are determined primarily by the Bragg equation λ = 2d sinθ, though other factors such as poor instrument alignement may apparently shift their positions away from their expected values. Given that d spacings can be measured, how does one obtain a unit cell?

Reciprocal-Space Metric Tensor

The d spacings in a material are related to the d* spacings of the reciprocal lattice according to the simple inverse relationship 1/d = d*. The latter is related to the reflection indices h,k,l according to the vector equation d* = ha* + kb* + lc* (which was introduced earlier in the section on reciprocal space). The magnitude of d* is obtained from the vector dot product ( • ) as follows:

The final equation shows how the measured d spacings are determined by the reflection indices and a maximum of six independent parameters, which may be labelled A, B, C, D, E, and F as in the equation below:

This equation is sometimes referred to as the reciprocal-space metric tensor equation. It is the key equation used in the indexing of powder diffraction data and for the refinement of unit-cell parameters.

Symmetry Constraints

The above equation can be considerably simplified by the application of symmetry. The earlier section on crystal systems listed the constraints on the real-space unit-cell parameters for the various crystal systems, while that on diffraction symmetry listed the constraints that apply to the reciprocal space parameters. The table from the latter page is repeated below (with the independent variables shown in red):

Note: The trigonal unit-cell parameters given here are for the case of rhombohedral cell axes with a rhombohedral lattice. For the primitive trigonal unit cells, the parameters and constraints are identical to those of the hexagonal crystal system.

Applying these constraints to the reciprocal-space metric tensor equation leads to the following simplified equations:

In terms of real-space unit-cell parameters, these may be written as

The expressions for the general triclinic case and the trigonal with rhomobohedral axes are more complicated, and are therefore not given here.

Thus given a set of unit-cell parameters it is relatively straightforward to calculate the d spacings of the peaks in the powder diffraction data for all possible values of hkl; for angle dispersive diffraction, the equivalent 2θ may also be calculated. Indexing involves the application of these equations in the opposite sense, i.e. one starts with d values and then one tries to obtain the hkl values together with a self-consistent set of lattice parameters. Note that in contrast to the calculation of d, one cannot simply invert the above equations in order to index the powder diffraction data.

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