Section 2.2.3.2. Diffraction coordinates

Figs. 2.2.3.1(a) to (d) are taken from IT II (1959, p. 176). They neatly summarize the geometrical principles of reflection, of a monochromatic beam, in the reciprocal lattice for the general case of an incident beam inclined at an angle (μ) to the equatorial plane. The diagrams are based on an Ewald sphere of unit radius.

Figure 2.2.3.1| top | pdf | (a) Elevation of the sphere of reflection. O is the origin of the reciprocal lattice. C is the centre of the Ewald sphere. The incident beam is shown in the plane. (b) Plan of the sphere of reflection. R is the projection of the rotation axis on the equatorial plane. (c) Perspective diagram. P is the relp in the reflection position with the cylindrical coordinates . The angular coordinates of the diffracted beam are . (d) Stereogram to show the direction of the diffracted beam, , with DD′, normal to the incident beam and in the equatorial plane, as the projection diameter. From Evans & Lonsdale (1959).

With the nomenclature of Table 2.2.3.1:

Table 2.2.3.1| top | pdf | Glossary of symbols used to specify quantities on diffraction patterns and in reciprocal space Bragg angle Angle of deviation of the reflected beam with respect to the incident beam Unit vector lying along the direction of the incident beam Unit vector lying along the direction of the reflected beam s = The scattering vector of magnitude . s is perpendicular to the bisector of the angle between and . s is identical to the reciprocal-lattice vector d* of magnitude λ/d, where d is the interplanar spacing, when d* is in the diffraction condition. In this notation, the radius of the Ewald sphere is unity. This convention is adopted because it follows that in Volume II of International Tables (p. 175). Note that in Section 2.2.1 Laue geometry the alternative convention (|d*| = 1/d) is adopted whereby the radius of each Ewald sphere is 1/λ. This allows a nest of Ewald spheres between and to be drawn Coordinate of a point P in reciprocal space parallel to a rotation axis as the axis of cylindrical coordinates relative to the origin of reciprocal space Radial coordinate of a point P in reciprocal space; that is, the radius of a cylinder having the rotation axis as axis The angular coordinate of P, measured as the angle between and [see Fig. 2.2.3.1(b)] The angle of rotation from a defined datum orientation to bring a relp onto the Ewald sphere in the rotation method (see Fig. 2.2.3.3) The angle of inclination of to the equatorial plane The angle between the projections of and onto the equatorial plane The angle of inclination of to the equatorial plane The crystal setting angles on the four-circle diffractometer (see Fig. 2.2.6.1). The used here is not the same as that in the rotation method (Fig. 2.2.3.3). This clash in using the same symbol twice is inevitable because of the widespread use of the rotation camera and four-circle diffractometer. The equatorial plane is the plane normal to the rotation axis.

Fig. 2.2.3.1(a) gives Fig. 2.2.3.1(b) gives, by the cosine rule, and and Figs. 2.2.3.1(a) and (b) give

The following special cases commonly occur:

In this section, we will concentrate on case (a), the normal-beam rotation method (μ = 0). First, the case of a plane film or detector is considered.

The notation now follows that of Arndt & Wonacott (1977) for the coordinates of a spot on the film or detector. is parallel to the rotation axis and ζ. is perpendicular to the rotation axis and the beam. IT II (1959, p. 177) follows the convention of y being parallel and x perpendicular to the rotation-axis direction, i.e. . The advantage of the notation is that the x-axis direction is then the same as the X-ray beam direction.

The coordinates of a reflection on a flat film are related to the cylindrical coordinates of a relp (ξ, ζ) [Fig. 2.2.3.2(a)] by which becomes where D is the crystal-to-film distance.

Figure 2.2.3.2| top | pdf | Geometrical principles of recording the pattern on (a) a plane detector, (b) a V-shaped detector, (c) a cylindrical detector.

For the case of a V-shaped cassette with the V axis parallel to the rotation axis and the film making an angle α to the beam direction [Fig. 2.2.3.2(b)], then This situation also corresponds to the case of flat electronic area detector inclined to the incident beam in a similar way.

Note that Arndt & Wonacott (1977) use ν instead of α here. We use α and so follow IT II (1959). This avoids confusion with the ν of Table 2.2.3.1. D is the crystal to V distance. In the case of the V cassettes of Enraf–Nonius, α is 60°.

For the case of a cylindrical film or image plate where the axis of the cylinder is coincident with the rotation axis [Fig. 2.2.3.2(c)] then, for in degrees, which becomes Here, D is the radius of curvature of the cylinder assuming that the crystal is at the centre of curvature.

In the three geometries mentioned here, detector-misalignment errors have to be considered. These are three orthogonal angular errors, translation of the origin, and error in the crystal-to-film distance.

The coordinates and are related to film-scanner raster units via a scanner-rotation matrix and translation vector. This is necessary because the film is placed arbitrarily on the scanner drum. Details can be found in Rossmann (1985) or Arndt & Wonacott (1977).