International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 2.2, pp. 36-37

Section 2.2.6. Diffractometry

J. R. Helliwella

aDepartment of Chemistry, University of Manchester, Manchester M13 9PL, England

2.2.6. Diffractometry

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The main book dealing with single-crystal diffractometry is that of Arndt & Willis (1966[link]). Hamilton (1974[link]) gives a detailed treatment of angle settings for four-circle diffractometers. For details of area-detector diffractometry, see Howard, Nielsen & Xuong (1985[link]) and Hamlin (1985[link]).

2.2.6.1. General

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In this section, we describe the following related diffractometer configurations:

  • (a) normal-beam equatorial geometry [ω, χ, [\varphi] option or ω, κ, [\varphi] (kappa) option];

  • (b) fixed χ = 45° geometry with area detector.

(a) is used with single-counter detectors. The kappa option is also used in the television area-detector system of Enraf–Nonius (the FAST). (b) is used with the multiwire proportional chamber, XENTRONICS, system. (FAST is a trade name of Enraf–Nonius; XENTRONICS is a trade name of Siemens.)

The purpose of the diffractometer goniostat is to bring a selected reflected beam into the detector aperture or a number of reflected beams onto an area detector of limited aperture (i.e. an aperture that does not intercept all the available diffraction spots at one setting of the area detector) [see Hamlin (1985[link], p. 431), for example].

Since the use of electronic area detectors is now increasingly common, the properties of these detectors that relate to the geometric prediction of spot position will be described later.

The single-counter diffractometer is primarily used for small-molecule crystallography. In macromolecular crystallography, many relp's are almost simultaneously in the diffraction condition. The single-counter diffractometer was extended to five separate counters [for a review, see Artymiuk & Phillips (1985[link])], then subsequently to a multi-element linear detector [for a review, see Wlodawer (1985[link])]. Area detectors offer an even larger aperture for simultaneous acquisition of reflections [Hamlin et al. (1981[link]), and references therein].

Large-area on-line image-plate systems are now available commercially to crystallographers, whereby the problem of the limited aperture of electronic area detectors is circumvented and the need for a goniostat is relaxed so that a single axis of rotation can be used. Systems like the R-AXISIIc (Rigaku Corporation) and the MAR (MAR Research Systems) fall into this category, utilizing IP technology and an on-line scanner. A next generation of device beckons, involving CCD area detectors. These offer a much faster duty cycle and greater sensitivity than IP's. By tiling CCD's together, a larger-area device can be realized. However, it is likely that these will be used in conjunction with a three-axis goniostat again, except in special cases where a complete area coverage can be realized.

2.2.6.2. Normal-beam equatorial geometry

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In normal-beam equatorial geometry (Fig. 2.2.6.1[link] ), the crystal is oriented specifically so as to bring the incident and reflected beams, for a given relp, into the equatorial plane. In this way, the detector is moved to intercept the reflected beam by a single rotation movement about a vertical axis (the [2\theta] axis). The value of [\theta] is given by Bragg's law as sin−1(d*/2). In order to bring d* into the equatorial plane (i.e. the Bragg plane into the meridional plane), suitable angular settings of a three-axis goniostat are necessary. The convention for the sign of the angles given in Fig. 2.2.6.1[link] is that of Hamilton (1974[link]); his choice of sign of [2\theta] is adhered to despite the fact that it is left-handed, but in any case the signs of ω, χ, [\varphi] are standard right-handed. The specific reciprocal-lattice point can be rotated from point P to point Q by the [\varphi] rotation, from Q to R via χ, and R to S via ω (see Fig. 2.2.6.2[link] ).

[Figure 2.2.6.1]

Figure 2.2.6.1 | top | pdf |

Normal-beam equatorial geometry: the angles ω, χ, ϕ, 2θ are drawn in the convention of Hamilton (1974[link]).

[Figure 2.2.6.2]

Figure 2.2.6.2 | top | pdf |

Diffractometry with normal-beam equatorial geometry and angular motions ω, χ and ϕ. The relp at P is moved to Q via ϕ, from Q to R via χ, and R to S via ω. From Arndt & Willis (1966[link]). In this specific example, the ϕ axis is parallel to the ω axis (i.e. χ = 0°).

In the most commonly used setting, the χ plane bisects the incident and diffracted beams at the measuring position. Hence, the vector d* lies in the χ plane at the measuring position. However, since it is possible for reflection to take place for any orientation of the reflecting plane rotated about d*, it is feasible therefore that d* can make any arbitrary angle [\varepsilon] with the χ plane. It is conventional to refer to the azimuthal angle ψ of the reflecting plane as the angle of rotation about d*. It is possible with a ψ scan to keep the hkl reflection in the diffraction condition and so to measure the sample absorption surface by monitoring the variation in intensity of this reflection. This ψ scan is achieved by adjustment of the ω, χ, [\varphi] angles. When χ = ±90°, the ψ scan is simply a [\varphi] scan and [\varepsilon] is 0°.

The χ circle is a relatively bulky object whose thickness can inhibit the measurement of diffracted beams at high [\theta]. Also, collision of the χ circle with the collimator or X-ray-tube housing has to be avoided. An alternative is the kappa goniostat geometry. In the kappa diffractometer [for a schematic picture, see Wyckoff (1985[link], p. 334)], the κ axis is inclined at 50° to the ω axis and can be rotated about the ω axis; the κ axis is an alternative to χ therefore. The [\varphi] axis is mounted on the κ axis. In this way, an unobstructed view of the sample is achieved.

2.2.6.3. Fixed χ = 45° geometry with area detector

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The geometry with fixed χ = 45° was introduced by Nicolet and is now fairly common in the field. It consists of an ω axis, a [\varphi] axis, and χ fixed at 45°. The rotation axis is the ω axis. In this configuration, it is possible to sample a greater number of independent reflections per degree of rotation (Xuong, Nielsen, Hamlin & Anderson, 1985[link]) because of the generally random nature of any symmetry axis.

An alternative method is to mount the crystal in a precise orientation and to use the [\varphi] axis to explore the blind region of the single rotation axis. It is feasible to place the capillary containing the sample in a vertically upright position via a 135° bracket mounted on the goniometer head. The bulk of the data is collected with the ω axis coincident with the capillary axis. This setting is beneficial to make the effect of capillary absorption symmetrical. At the end of this run, the blind region whose axis is coincident with the ω axis can be filled in by rotating around the [\varphi] axis by 180°. This renders the capillary axis horizontal and a different crystal axis vertical. Hence by rotation about this new crystal axis by [\pm\theta_{\max}], the blind region can be sampled.

References

Arndt, U. W. & Willis, B. T. M. (1966). Single crystal diffractometry. Cambridge University Press.
Artymiuk, P. & Phillips, D. C. (1985). On the design of diffractometers to measure a number of reflections simultaneously. Methods Enzymol. 114A, 397–415.
Hamilton, W. C. (1974). Angle settings for four-circle diffractometers. International tables for X-ray crystallography, Vol. IV, pp. 273–284. Birmingham: Kynoch Press.
Hamlin, R. (1985). Multi-wire area X-ray diffractometers. Methods Enzymol. 114A, 416–451.
Hamlin, R., Cork, C., Howard, A., Nielsen, C., Vernon, W., Matthews, D., Xuong, Ng. H. & Perez-Mendez, V. (1981). Characteristics of a flat multiwire area detector for protein crystallography. J. Appl. Cryst. 14, 85–93.
Howard, A., Nielsen, C. & Xuong, Ng. H. (1985). Software for a diffractometer with multi-wire area detector. Methods Enzymol. 114A, 452–472.
Wlodawer, A. (1985). Methods Enzymol. 114A, 551–564.
Wyckoff, H. W. (1985). Diffractometry. Methods Enzymol. 114A, 330–385.
Xuong, Ng. H., Nielsen, C., Hamlin, R. & Anderson, D. (1985). Strategy for data collection from protein crystals using a multiwire counter area detector diffractometer. J. Appl. Cryst. 18, 342–350.








































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