Section 5.3.3.2.2. Two-circle diffractometer

Lattice-parameter determination by the use of the two-circle (inclination) diffractometer, the so-called `Weissenberg diffractometer', is more troublesome than by means of the four-circle one, because only two rotations [ (or ) of the crystal, and (or γ) of the detector] are motor-driven under computer control, while two inclination angles (μ for the crystal and ν for the detector) must be set by hand.

The problem of application of the popular two-circle (Eulerian-cradle) diffractometer for measurements similar to those presented in §5.3.3.2.1 was discussed by Clegg & Sheldrick (1984). The main idea of their paper was to introduce equations combining setting angles, obtained for selected reflections, with reciprocal-cell parameters, for calculating the latter. The authors started with zero-layer reflections for which, for a crystal mounted about the c axis, where and is a zero-point correction.

The remaining parameter c had to be determined from the inclination angle μ, measured by hand. The use of zero-layer reflections was advantageous, apart from the simplicity of the formulae (5.3.3.4a,b,c,d), because they were less affected by crystal misalignment than were upper-layer reflections. However, a zero-point correction for had to be performed. For this purpose, the value was treated as an additional parameter in off-line least-squares refinement.

As the next step, the authors introduced equations for a general crystal orientation instead of an aligned crystal (cf. §5.3.3.2.1) and derived equations defining the setting angles for an arbitrary reflection useful for data collection from a randomly oriented crystal if preliminary lattice-parameter values had been assumed. This made possible measurements of reflections on a range of layers; only one crystal mounting was required. The matrix formulae suitable for Eulerian-geometry diffractometers are also given by Kheiker (1973, Chap. 3, Section 9) and Gabe (1980).

In order to perform precise refinement of all six cell parameters, Clegg & Sheldrick (1984) used least squares with empirical weights: where is the width of the hkl reflection. An additional (third) motor to control the μ circle was proposed.

The authors point out that the two-circle diffractometer, owing to its simpler construction in comparison with the four-circle one, is well suited to operations that require additional attachments; for example, for low-temperature operation.