Rotating crystal pattern

Fig. 3-10 Rotating-crystal pattern of a quartz crystal (hexagonal) rotated about its c axis. Filtered copper radiation. (The stress are due to the white radiation not removed by the filter.) (Courtesy of B. E. Warren.)...

The maximum equi-inclination angle is normally 30° (Cmax = so that Cniiix IS the same as in the normal rotating crystal pattern, except that there is no blind region near the rotation axis (Figs. 5 and 22). 

Fig. 24) the reflections are easily indexed and the pattern is easily interpreted [9, pp. 15-68]. The cone pattern of Figure 25, which shows the layer lines measurable with a given crystal setting, corresponds to the rotating crystal pattern. 

Spinning a crystal during measurement of WAXS patterns is an old method that turns any scattering pattern into a fiber pattern. The rotational axis becomes the principal axis. Thereafter isotropization of the scattering data is simplified because the mathematical treatment can resort to fiber symmetry of the measured data. In the literature the method is addressed as the rotating-crystal method or oscillating-crystal method. 

A crystal pattern may possess rotational symmetry as well as translational symmetry, although the existence of translational symmetry imposes restrictions on the order of the axes. The fundamental translations (a in eq. (1) are the basis vectors of a linear vector... 

The space group G of a crystal is the set of all symmetry operators that leave the appearance of the crystal pattern unchanged from what it was before the operation. The most general kind of space-group operator (called a Seitz operator) consists of a point operator R (that is, a proper or improper rotation that leaves at least one point invariant) followed by a translation v. For historical reasons the Seitz operator is usually written R v. However, we shall write it as (R ) to simplify the notation for sets of space-group operators. When a space-group operator acts on a position vector r, the vector is transformed into... 

Optical analogues of one-dimensional parallel and rotational moire patterns in two perfect crystals are shown in Figures 6.6(a, b). 

The conclusion is that, for a rotating crystal, most sets of crystal planes give rise to four spots lying at particular points on the imaginary circle where the corresponding powder-pattern circle would have been. These points are symmetrically placed with respect to the plane that contains the incident X-ray beam and the rotation axis and to the plane that contains the incident X-ray beam and is normal to the rotation axis, as shown in fig. 4.9. A similar conclusion can be drawn for a stationary, highly oriented polymer fibre. Planes parallel to the rotation or fibre axis give rise to only two diffraction spots. 

FIGURE S.17 The method of rotating crystals diffraction the diffraction principle (left-up) and diffraction pattern (left-down) along the Ewald s rationalization on the reflection sphere of diffraction (right) after (Matter Dif action, 2003 X-Rays, 2003 Putz Lacrama, 2005 HyperPhysics, 2010). 

Axes of symmetry. An axis about which rotation of the body through an angle of 2njn (where n is an integer) gives an identical pattern 2-fold, 3-fold, 4-fold and 6-fold axes are known in crystals 5-fold axes are known in molecules. In a lattice the rotation may be accompanied by a lateral movement parallel to the axis (screw axis). 

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

For a two-dimensional array of equally spaced holes the difftaction pattern is a two-dimensional array of spots. The intensity between the spots is very small. The crystal is a three-dimensional lattice of unit cells. The third dimension of the x-ray diffraction pattern is obtained by rotating the crystal about some direction different from the incident beam. For each small angle of rotation, a two-dimensional difftaction pattern is obtained. 

A very narrow window produces monochromatic radiation that is still several orders of magnitude more intense than the beam from conventional rotating anode x-ray sources. Sucb beams allow crystallographers to record diffraction patterns from very small crystals of the order of 50 micrometers or smaller. In addition, the diffraction pattern extends to higher resolution and consequently more accurate structural details are obtained as described later in this chapter. The availability and use of such beams have increased enormously in recent years and have greatly facilitated the x-ray determination of protein structures. 

Figure 18.5 Schematic view of a diffraction experiment, (a) A narrow beam of x-rays (red) is taken out from the x-ray source through a collimating device. When the primary beam hits the crystal, most of it passes straight through, but some is diffracted by the crystal. These diffracted beams, which leave the crystal in many different directions, are recorded on a detector, either a piece of x-ray film or an area detector, (b) A diffraction pattern from a crystal of the enzyme RuBisCo using monochromatic radiation (compare with Figure 18.2b, the pattern using polychromatic radiation). The crystal was rotated one degree while this pattern was recorded.

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