Section 8.1.7.2. Monochromatic SR beams: optical configurations and sample rocking width

In the case of the single-crystal monchromator, the actual curvature employed is very important in the diffraction geometry. For a point source and a flat monochromator crystal, there is a gradual change in the photon wavelength selected from the white beam as the length of the monochromator is traversed (Fig. 8.1.7.1a). For a point source and a curved monochromator crystal, one specific curvature can compensate for this variation in incidence angle (Fig. 8.1.7.1b). The reflected spectral bandwidth is then at a minimum; this setting is known as the `Guinier position'. If the curvature of the monochromator crystal is increased further, a range of photon wavelengths, , is selected along its length so that the rays converging towards the focus have a correlation of photon wavelength and direction (Fig. 8.1.7.1c). The effect of a finite source is to cause a change in incidence angle at the monochromator crystal, so that at the focus there is a photon-wavelength gradient across the width of the focus (for all curvatures) (Fig. 8.1.7.1d). The use of a slit in the focal plane is akin to placing a slit at the tangent point to limit the source size.

Figure 8.1.7.1| top | pdf | Single-crystal monochromator illuminated by SR. (a) Flat crystal. (b) Guinier setting. (c) Overbent crystal. (d) Effect of source size (shown at the Guinier setting for clarity). From Helliwell (1984). Reproduced with the permission of the Institute of Physics.

The double-crystal monochromator with two parallel or nearly parallel perfect crystals of germanium or silicon is a common configuration. The advantage of this is that the outgoing monochromatic beam is parallel to the incoming beam, although it is slightly displaced vertically by an amount , where d is the perpendicular distance between the crystals and θ is the monochromator Bragg angle (unless the second crystal is unconnected to the first, in which case it can be translated as well to compensate for that). The monochromator can be rapidly tuned, since the diffractometer or camera need not be re-aligned significantly in a scan across an absorption edge. Since the rocking width of the fundamental is broader than the harmonic reflections, the strict parallelism of the pair of the crystal planes can be relaxed or `detuned', so that the harmonic can be rejected with little loss of the fundamental intensity. The spectral spread in the reflected monochromatic beam is determined by the source divergence accepted by the monochromator, the angular size of the source and the monochromator rocking width (see Fig. 8.1.7.2). The double-crystal monochromator is often used with a toroidal focusing mirror; the functions of monochromatization are then separated from the focusing (Hastings et al., 1978).

Figure 8.1.7.2| top | pdf | Double-crystal monochromator illuminated by SR. The contributions of the source divergence, αV [less than or equal to γ−1, equation (8.1.2.4), depending on the monochromator vertical entrance slit aperture; see also Colapietro et al., 1992], and angular source size, , to the range of energies reflected by the monochromator are shown. From Helliwell (1984). Reproduced with the permission of the Institute of Physics.

The rocking width of a reflection depends on the horizontal and vertical beam divergence or convergence (after due account for collimation is taken), and , the spectral spreads and , and the mosaic spread, η. We assume that the mosaic spread η is , the angular broadening of a reciprocal-lattice point (relp) due to a finite sample. In the case of synchrotron radiation, and are usually widely asymmetric. On a conventional source, usually . Two types of spectral spread occur with synchrotron (and neutron) sources. The term is the spread that is passed down each incident ray in a divergent or convergent incident beam; the subscript refers to the conventional source type. This is because it is similar to the line widths and separation. At the synchrotron, this component also exists and arises from the monochromator rocking width and finite-source-size effects. The term is special to the synchrotron or neutron case. The subscript `corr' refers to the fact that the ray direction can be correlated with the photon or neutron wavelength. In this most general case, and for one example of a arising from the horizontal ray direction correlation with photon energy and the case of a horizontal rotation axis, the rocking width of an individual reflection is given by where and L is the Lorentz factor, .

The Guinier setting of an instrument (curved crystal monochromator case, Fig. 8.1.7.1b) gives . The equation for then reduces to (from Greenhough & Helliwell, 1982). For example, for , (0.01°), , and (0.05°), then . But increases as ζ increases [see Greenhough & Helliwell (1982), Table 5]. In the rotation/oscillation method as applied to protein and virus crystals, a small angular range is used per exposure. For example, the maximum rotation range per image, , may be 1.5° for a protein and 0.4° or so for a virus. Many reflections will be only partially stimulated over the exposure. It is important, especially in the virus case, to predict the degree of penetration of the relp through the Ewald sphere. This is done by analysing the interaction of a spherical volume for a given relp with the Ewald sphere. The radius of this volume is given by (Greenhough & Helliwell, 1982).

In Fig. 8.1.7.3, the relevant parameters are shown. The diagram shows in a plane, usually horizontal with a perpendicular (vertical) rotation axis, whereas the formula for above is for a horizontal axis. This is purely for didactic reasons since the interrelationship of the components is then much clearer.

Figure 8.1.7.3| top | pdf | The rocking width of an individual reflection for the case of Fig. 8.1.7.1(c) and a vertical rotation axis. From Greenhough & Helliwell (1982).