Section 2.2.1.1. General

The single crystal is bathed in a polychromatic beam of X-rays containing wavelengths between λ_min and λ_max. A particular crystal plane will pick out a general wavelength λ for which constructive interference occurs and reflect according to Bragg's law where d is the interplanar spacing and is the angle of reflection. A sphere drawn with radius 1/λ and with the beam direction as diameter, passing through the origin of the reciprocal lattice (the point O in Fig. 2.2.1.1) , will yield a reflection in the direction drawn from the centre of the sphere and out through the reciprocal-lattice point (relp) provided the relp in question lies on the surface of the sphere. This sphere is known as the Ewald sphere. Fig. 2.2.1.1 shows the Laue geometry, in which there exists a nest of Ewald spheres of radii between 1/λ_max and 1/λ_min. An alternative convention is feasible whereby only a single Ewald sphere is drawn of radius 1 reciprocal-lattice unit (r.l.u.). Then each relp is no longer a point but a streak between λ_min/d and λ_max/d from the origin of reciprocal space (see McKie & McKie, 1986, p. 297). In the following discussions on the Laue approach, this notation is not followed. We use the nest of Ewald spheres of varying radii instead.

Figure 2.2.1.1| top | pdf | Laue geometry. A polychromatic beam containing wavelengths λmin to λmax impinges on the crystal sample. The resolution sphere of radius is drawn centred at O, the origin of reciprocal space. Any reciprocal-lattice point falling in the shaded region is stimulated. In this diagram, the radius of each Ewald sphere uses the convention 1/λ.

Any relp (hkl) lying in the region of reciprocal space between the 1/λ_max and 1/λ_min Ewald spheres and the resolution sphere 1/d_min will diffract (the shaded area in Fig. 2.2.1.1). This region of reciprocal space is referred to as the accessible or stimulated region. Fig. 2.2.1.2 shows a predicted Laue pattern from a well aligned protein crystal. For a description of the indexing of a Laue photograph, see Bragg (1928, pp. 28, 29).

Figure 2.2.1.2| top | pdf | A predicted Laue pattern of a protein crystal with a zone axis parallel to the incident, polychromatic X-ray beam. There is a pronounced blank region at the centre of the film (see Subsection 2.2.1.2). The spot marked N is one example of a nodal spot (see Subsection 2.2.1.4).

For a Laue spot at a given , only the ratio λ/d is determined, whether it is a single or a multiple relp component spot. If the unit-cell parameters are known from a monochromatic experiment, then a Laue spot at a given yields λ since d is then known. Conversely, precise unit-cell lengths cannot be determined from a Laue pattern alone; methods are, however, being developed to determine these (see Carr, Cruickshank & Harding, 1992).

The maximum Bragg angle is given by the equation