International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 2.2, pp. 2641
https://doi.org/10.1107/97809553602060000577 Chapter 2.2. Singlecrystal Xray techniques
J. R. Helliwell^{a}
^{a}Department of Chemistry, University of Manchester, Manchester M13 9PL, England The diffraction of beams of Xrays from single crystals involves very specific geometries that form the basis for the measurement of intensities used in crystal structure analysis. In terms of the incident beam itself, the two key approaches available involve either a monochromatic beam or a polychromatic `Laue' white beam. Starting from Bragg's law and the Ewald reciprocalspace construction, the methods for the collection of diffraction data, i.e. reflection intensities, using commonly available apparatus are then described. Monochromatic beam measuring methods such as rotating/oscillating crystal, stills, Weissenberg, precession and fourcircle diffractometry are covered. The mathematical relationships between the reciprocallattice point (relp) coordinates in reciprocal space and the corresponding diffraction spot positions at the detector (flat, cylindrical or Vshaped) are given in detail. These coordinate transformations represent an idealized situation of relps (i.e. as points) and of diffracted rays as lines. Deviations from ideality arise from practical considerations such as the incident beam spectral purity, its divergence or convergence to the sample from the source via the optics, as well as the crystal perfection (`mosaicity'), and of the pointspread factor of the detector. The reflection rocking curves and diffraction spot shapes and sizes are practical manifestations of these effects. 
In Chapter 2.1 Classification of experimental techniques there are given the various common approaches to the recording of Xray crystallographic data, in different geometries, for crystal structure analysis. These are:
The reasons for the choice of order are as follows. Laue geometry is dealt with first because it was historically the first to be used (Friedrich, Knipping & von Laue, 1912). In addition, the first step that should be carried out with a new crystal, at least of a small molecule, is to take a Laue photograph to make the first assessment of crystal quality. For macromolecules, the monochromatic still serves the same purpose. From consideration of the monochromatic still geometry, we can then describe the cases of singleaxis rotation (rotation/oscillation method), singleaxis rotation coupled with detector translation (Weissenberg method), crystal and detector precession (precession method), and finally threeaxis goniostat and rotatable detector or area detector (diffractometry).
Method (a) uses a polychromatic beam of broad wavelength bandpass (e.g. 0.2 2.5 Å); if the bandwidth is restricted (e.g. to δλ/λ = 0.2), then it is sometimes referred to as narrowbandpass Laue geometry. The remaining methods (b)–(f) use a monochromatic beam.
There are textbooks that concentrate on almost every geometry. References to these books are given in the respective sections in the following pages. However, in addition, there are several books that contain details of diffraction geometry. Blundell & Johnson (1976) describe the use of the various diffraction geometries with the examples taken from protein crystallography. There is an extensive discussion and many practical details to be found in the textbooks of Stout & Jensen (1968), Woolfson (1970, 1997), Glusker & Trueblood (1971, 1985), Vainshtein (1981), and McKie & McKie (1986), for example. A collection of early papers on the diffraction of Xrays by crystals involving, inter alia, experimental techniques and diffraction geometry, can be found in Bijvoet, Burgers & Hägg (1969, 1972). A collection of papers on, primarily, protein and virus crystal data collection via the rotationfilm method and diffractometry can be found in Wyckoff, Hirs & Timasheff (1985). Synchrotron instrumentation, methods, and applications are dealt with in the books of Helliwell (1992) and Coppens (1992).
Quantitative Xray crystal structure analysis usually involves methods (c), (d), and (f), although (e) has certainly been used. Electronic area detectors or image plates are extensively used now in all methods.
Traditionally, Laue photography has been used for crystal orientation, crystal symmetry, and mosaicity tests. Rapid recording of Laue patterns using synchrotron radiation, especially with protein crystals or with small crystals of small molecules, has led to an interest in the use of Laue geometry for quantitative structure analysis. Various fundamental objections made, especially by W. L. Bragg, to the use of Laue geometry have been shown not to be limiting.
The monochromatic still photograph is used for orientation setting and mosaicity tests, for protein or virus crystallography, and computer refinement of crystal orientation following initial crystal setting.
Precession photography allows the isolation of a specific zone or plane of reflections for which indexing can be performed by inspection, and systematic absences and symmetry are explored. From this, spacegroup assignment is made. The use of precession photography is usually avoided in smallmolecule crystallography where autoindexing methods are employed on a singlecrystal diffractometer. In such a situation, the burden of data collection is not huge and symmetry elements can be determined after data collection. This is also now carried out on electronic area detectors in conjunction with autoindexing principally at present for macromolecular crystallography but also for chemical crystallography.
In the following sections, the geometry of each method is dealt with in an idealized form. The practical realization of each geometry is then dealt with, including the geometric distortions introduced in the image by electronic area detectors. A separate section deals with the common means for beam conditioning, namely mirrors, monochromators, and filters. Sufficient detail is given to establish the magnitude of the wavelength range, spectral spreads, beam divergence and convergence angles, and detector effects. These values can then be utilized along with the formulae given for the calculation of spot bandwidth, spot size, and angular reflecting range.
The main book dealing with Laue geometry is Amorós, Buerger & Amorós (1975). This should be used in conjunction with Henry, Lipson & Wooster (1951), or McKie & McKie (1986); see also Helliwell (1992, chapter 7). There is a synergy between synchrotron and neutron Laue diffraction developments (see Helliwell & Wilkinson, 1994).
The single crystal is bathed in a polychromatic beam of Xrays 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 reciprocallattice 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 reciprocallattice 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.
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).
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 unitcell parameters are known from a monochromatic experiment, then a Laue spot at a given yields λ since d is then known. Conversely, precise unitcell lengths cannot be determined from a Laue pattern alone; methods are, however, being developed to determine these (see Carr, Cruickshank & Harding, 1992).
The main use of Laue photography has in the past been for adjustment of the crystal to a desired orientation. With smallmolecule crystals, the number of diffraction spots on a monochromatic photograph from a stationary crystal is very small. With unfiltered, polychromatic radiation, many more spots are observed and so the Laue photograph serves to give a better idea of the crystal orientation and setting prior to precession photography. With protein crystals, the monochromatic still is used for this purpose before data collection via an area detector. This is because the number of diffraction spots is large on a monochromatic still and in a proteincrystal Laue photograph the stimulated spots from the Bremsstrahlung continuum are generally very weak. Synchrotronradiation Laue photographs of protein crystals can be recorded with short exposure times. These patterns consist of a large number of diffraction spots.
Crystal setting via Laue photography usually involves trying to direct the Xray beam along a zone axis. Angular missetting angles in the spindle and arc are easily calculated from the formula where Δ is the distance (resolved into vertical and horizontal) from the beam centre to the centre of a circle of spots defining a zone axis and D is the crystaltofilm distance.
After suitable angular correction to the sample orientation, the Laue photograph will show a pronounced blank region at the centre of the film (see Fig. 2.2.1.2). This radius of the blank region is determined by the minimum wavelength in the beam and the magnitude of the reciprocallattice spacing parallel to the Xray beam (see Jeffery, 1958). For the case, for example, of the Xray beam perpendicular to the a*b* plane, then where and R is the radius of the blank region (see Fig. 2.2.1.2), and D is the crystaltoflatfilm distance. If λ_{min} is known then an approximate value of c, for example, can be estimated. The principal zone axes will give the largest radii for the central blank region.
In Laue geometry, several relp's can occur in a Laue spot or ray. The number of relp's in a given spot is called the multiplicity of the spot. The number of spots of a given multiplicity can be plotted as a histogram. This is known as the multiplicity distribution. The form of this distribution is dependent on the ratio λ_{max}/λ_{min}. The multiplicity distribution in Laue diffraction is considered in detail by Cruickshank, Helliwell & Moffat (1987).
Any relp nh, nk, nl (n integer) will be stimulated by a wavelength λ/n since d_{nhnknl} = d_{hkl}/n, i.e. However, d_{nhnknl} must be d_{min} as otherwise the reflection is beyond the sample resolution limit.
If h, k, l have no common integer divisor and if 2h, 2k, 2l is beyond the resolution limit, then the spot on the Laue diffraction photograph is a singlewavelength spot. The probability that h, k, l have no common integer divisor is Hence, for a relp where d_{min} d_{hkl} 2d_{min} there is a very high probability (83.2%) that the Laue spot will be recorded as a singlewavelength spot. Since this region of reciprocal space corresponds to 87.5% (i.e. 7/8) of the volume of reciprocal space within the resolution sphere then 0.875 × 0.832 = 72.8% is the probability for a relp to be recorded in a singlewavelength spot. According to W. L. Bragg, all Laue spots should be multiple. He reasoned that for each h, k, l there will always be a 2h, 2k, 2l etc. lying within the same Laue spot. However, as the resolution limit is increased to accommodate this many more relp's are added, for which their hkl's have no common divisor.
The above discussion holds for infinite bandwidth. The effect of a more experimentally realistic bandwidth is to increase the proportion of singlewavelength spots.
The number of relp's within the resolution sphere is where = 1/d_{min} and V* is the reciprocal unitcell volume.
The number of relp's within the wavelength band λ_{max} to λ_{min}, for , is (Moffat, Schildkamp, Bilderback & Volz, 1986) Note that the number of relp's stimulated in a 0.1 Å wavelength interval, for example between 0.1 and 0.2 Å, is the same as that between 1.1 and 1.2 Å, for example. A large number of relp's are stimulated at one orientation of the crystal sample.
The proportion of relp's within a sphere of small d* (i.e. at low resolution) actually stimulated is small. In addition, the probability of them being single is zero in the infinitebandwidth case and small in the finitebandwidth case. However, Laue geometry is an efficient way of measuring a large number of relp's between and as singlewavelength spots.
The above is a brief description of the overall multiplicity distribution. For a given relp, even of simple hkl values, lying on a ray of several relp's (multiples of hkl), a suitable choice of crystal orientation can yield a singlewavelength spot. Consider, for example, a spot of multiplicity 5. The outermost relp can be recorded at long wavelength with the inner relp's on the ray excluded since they need λ's greater than λ_{max} (Fig. 2.2.1.3 ). Alternatively, by rotating the sample, the innermost relp can be measured uniquely at short wavelength with the outer relp's excluded (they require λ's shorter than λ_{min}). Hence, in Laue geometry several orientations are needed to recover virtually all relp's as singles. The multiplicity distribution is shown in Fig. 2.2.1.4 as a function of λ_{max}/λ_{min} (with the corresponding values of δλ/λ_{mean}).
There is an interesting variation in the angular separations of Laue reflections that shows up in the spatial distributions of spots on a detector plane (Cruickshank, Helliwell & Moffat, 1991). There are two main aspects to this distribution, which are general and local. The general aspects refer to the diffraction pattern as a whole and the local aspects to reflections in a particular zone of diffraction spots.
The general features include the following. The spatial density of spots is everywhere proportional to 1/D^{2}, where D is the crystaltodetector distance, and to 1/V*, where V* is the reciprocalcell volume. There is also though a substantial variation in spatial density with diffraction angle ; a prominent maximum occurs at
Local aspects of these patterns particularly include the prominent conics on which Laue reflections lie. That is, the local spatial distribution is inherently onedimensional in character. Between multiple reflections (nodals), there is always at least one single and therefore nodals have a larger angular separation from their nearest neighbours. The blank area around a nodal in a Laue pattern (Fig. 2.2.1.2) has been noted by Jeffery (1958). The smallest angular separations, and therefore spatially overlapped cases, are associated with single Laue reflections. Thus, the reflections involved in energy overlaps – the multiples – form a set largely distinct, except at short crystaltodetector distances, from those involved in spatial overlaps, which are mostly singles (Helliwell, 1985).
From a knowledge of the form of the angular distribution, it is possible, e.g. from the gaps bordering conics, to estimate and λ_{min}. However, a development of this involving gnomonic projections can be even more effective (Cruickshank, Carr & Harding, 1992).
A useful means of transformation of the flatfilm Laue pattern is the gnomonic projection. This converts the pattern of spots lying on curved arcs to points lying on straight lines. The stereographic projection is also used. Fig. 2.2.1.5 shows the graphical relationships involved [taken from International Tables, Vol. II (Evans & Lonsdale, 1959)], for the case of a Laue pattern recorded on a plane film, between the incidentbeam direction SN, which is perpendicular to a film plane and the Laue spot L and its spherical, stereographic, and gnomonic points S_{p}, S_{t} and G and the stereographic projection S_{r} of the reflected beams. If the radius of the sphere of projection is taken equal to D, the crystaltofilm distance, then the planes of the gnomonic projection and of the film coincide. The lines producing the various projection poles for any given crystal plane are coplanar with the incident and reflected beams. The transformation equations are
In this section and those that follow, which deal with monochromatic methods, the convention is adopted that the Ewald sphere takes a radius of unity and the magnitude of the reciprocallattice vector is λ/d. This is not the convention used in the Laue section above.
Some historical remarks are useful first before progressing to discuss each monochromatic geometry in detail. The original rotation method (for example, see Bragg, 1949) involved a rotation of a perfectly aligned crystal of 360°. For reasons of relatively poor collimation of the Xray beam, leading to spottospot overlap, and background buildup, Bernal (1927) introduced the oscillation method whereby a repeated, limited, angular range was used to record one pattern and a whole series of contiguous ranges on different film exposures were collected to provide a large angular coverage overall. In a different solution to the same problem, Weissenberg (1924) utilized a layerline screen to record only one layer line but allowed a full rotation of the crystal but now coupled to translation of the detector, thus avoiding spottospot overlap. Again, several exposures were needed, involving one layer line collected on each exposure. The advent of synchrotron radiation with very high intensity allows small beam sizes at the sample to be practicable, thus simultaneously creating small diffraction spots and minimizing background scatter. The very fine collimation of the synchrotron beam keeps the diffractionspot sizes small as they traverse their path to the detector plane.
The terminology used today for different methods is essentially the same as originally used except that the rotation method now tends to mean limited angular ranges (instead of 360°) per diffraction photograph/image. The Weissenberg method in its modern form now employed at a synchrotron is a screenless technique with limited angular range but still with detector translation coupled to crystal rotation.
In a monochromatic still exposure, the crystal is held stationary and a nearzero wavelengthbandpass (e.g. δλ/λ = 0.001) beam impinges on it. For a smallmolecule crystal, there are few diffraction spots. For a protein crystal, there are many (several hundred), because of the much denser reciprocal lattice. The actual number of stimulated relp's depends on the reciprocalcell parameters, the size of the mosaic spread of the crystal, the angular beam divergence as well as the small, but finite, spectral spread, δλ/λ. Diffraction spots are only partially stimulated instead of fully integrated over wavelength, as in the Laue method, or over an angular rotation (the rocking width) in rotatingcrystal monochromatic methods.
The diffraction spots lie on curved arcs where each curve corresponds to the intersection with a film of a cone. With a flat film the intersections are conic sections. The curved arcs are obviously recognizable for the protein crystal case where there are a large number of spots.
Crystal setting follows the procedure given in Subsection 2.2.1.2 whereby angular missetting angles are given by equation (2.2.1.3). When viewed down a zone axis, the pattern on a flat film or electronic area detector has the appearance of a series of concentric circles. For example, with the beam parallel to , the first circle corresponds to l = 1, the second to l = 2, etc. The radius of the first circle R is related to the interplanar spacing between the (hk0) and (hk1) planes, i.e. λ/c (in this example), through , by the formulae
The main modern book dealing with the rotation method is that of Arndt & Wonacott (1977).
The purpose of the monochromatic rotation method is to stimulate a reflection fully over its rocking width via an angular rotation. Different relp's are rotated successively into the reflecting position. The method, therefore, involves rotation of the sample about a single axis, and is used in conjunction with an area detector of some sort, e.g. film, electronic area detector or image plate. The use of a repeated rotation or oscillation, for a given exposure, is simply to average out any timedependent changes in incident intensity or sample decay. The overall crystal rotation required to record the total accessible region of reciprocal space for a single crystal setting, and a detector picking up all the diffraction spots, is 180° + . If the crystal has additional symmetry, then a complete asymmetric unit of reciprocal space can be recorded within a smaller angle. There is a blind region close to the rotation axis; this is detailed in Subsection 2.2.3.5.
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.
With the nomenclature of Table 2.2.3.1:

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 normalbeam 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 rotationaxis direction, i.e. . The advantage of the notation is that the xaxis direction is then the same as the Xray 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 crystaltofilm distance.

Geometrical principles of recording the pattern on (a) a plane detector, (b) a Vshaped detector, (c) a cylindrical detector. 
For the case of a Vshaped 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, detectormisalignment errors have to be considered. These are three orthogonal angular errors, translation of the origin, and error in the crystaltofilm distance.
The coordinates and are related to filmscanner raster units via a scannerrotation 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).
The reciprocallattice coordinates, etc. used earlier, refer to an axial system fixed to the crystal, of Fig. 2.2.3.3 . Clearly, a given relp needs to be brought into the Ewald sphere by the rotation about the rotation axis. The treatment here follows Arndt & Wonacott (1977).
The rotation angle required, , is with respect to some reference `zeroangle' direction and is determined by the particular crystal parameters. It is necessary to define a standard orientation of the crystal (i.e. datum) when = 0°. If we define an axial system fixed to the crystal and a laboratory axis system XYZ with X parallel to the beam and Z coincident with the rotation axis then = 0° corresponds to these axial systems being coincident (Fig. 2.2.3.3).
The angle of the crystal at which a given relp diffracts is The two solutions correspond to the two rotation angles at which the relp P cuts the sphere of reflection. Note that , (Subsection 2.2.3.2) are independent of .
The values of x_{0} and y_{0} are calculated from the particular crystal system parameters. The relationships between the coordinates x_{0}, y_{0}, z_{0} and ξ and ζ are X_{0} can be related to the crystal parameters by A is a crystalorientation matrix defining the standard datum orientation of the crystal.
For example, if, by convention, is chosen as parallel to the Xray beam at = 0° and c is chosen as the rotation axis, then, for the general case,
If the crystal is mounted on the goniometer head differently from this then A can be modified by another matrix, M, say, or the terms permuted. This exercise becomes clear if the reader takes an orthogonal case (α = β = γ = 90°). For the general case, see Higashi (1989).
The crystal will most likely be misaligned (slightly or grossly) from the ideal orientation. To correct for this, the misorientation matrices , , and are introduced, i.e. where , , and are angles around the X_{0}, Y_{0}, and Z_{0} axes, respectively.
For a given oscillation photograph, there is maximum value of the oscillation range, , that avoids overlapping of spots on a film. The overlap is most likely to occur in the region of the diffraction pattern perpendicular to the rotation axis and at the maximum Bragg angle. This is where relp's pass through the Ewald sphere with the greatest velocity. For such a separation between successive relp's of a*, then the maximum allowable rotation angle to avoid spatial overlap is given by where Δ is the sample reflecting range (see Section 2.2.7). is a function of , even in the case of identical cell parameters. This is because it is necessary to consider, for a given orientation, the relevant reciprocallattice vector perpendicular to . In the case where the cell dimensions are quite different in magnitude (excluding the axis parallel to the rotation axis), then is a marked function of the orientation.
In rotation photography, as large an angle as possible is used up to . This reduces the number of images that need to be processed and the number of partially stimulated reflections per image but at the expense of signaltonoise ratio for individual spots, which accumulate more background since . In the case of a CCD detector system, is chosen usually to be less than Δ so as to optimize the signaltonoise ratio of the measurement and to sample the rockingwidth profile.
The value of Δ, the crystal rocking width for a given hkl, depends on the reciprocallattice coordinates of the hkl relp (see Section 2.2.7). In the region close to the rotation axis, Δ is large.
In the introductory remarks to the monochromatic methods used, it has already been noted that originally the rotation method involved 360° rotations contributing to the diffraction image. Spot overlap led to loss of reflection data and encouraged Bernal and Weissenberg to devise improvements. With modern synchrotron techniques, the restriction on (equation 2.2.3.30) can be relaxed for special applications. For example, since the spot overlap that is to be avoided involves relp's from adjacent reciprocallattice planes, the different Miller indices hkl and h + l, k, l do lead in fact to a small difference in Bragg angle. With good enough collimation, a small spot size exists at the detector plane so that the two spots can be resolved. For a standardsized detector, this is practical for lowresolution data recording. This can be a useful complement to the Laue method where the lowresolution data are rather sparsely stimulated and also tend to occur in multiple Laue spots. Alternatively, a much larger detector can be contemplated and even mediumresolution data can be recorded without major overlap problems. These techniques are useful in some timeresolved applications. For a discussion see Weisgerber & Helliwell (1993). For regular data collection, however, narrow angular ranges are still generally preferred so as to reduce the background noise in the diffraction images and also to avoid loss of any data because of spot overlap.
In normalbeam geometry, any relp lying close to the rotation axis will not be stimulated at all. This situation is shown in Fig. 2.2.3.4 . The blind region has a radius of and is therefore strongly dependent on d_{min} but can be ameliorated by use of a short λ. Shorter λ makes the Ewald sphere have a larger radius, i.e. its surface moves closer to the rotation axis. At Cu Kα for 2 Å resolution, approximately 5% of the data lie in the blind region according to this simple geometrical model. However, taking account of the rocking width Δ, a greater percentage of the data than this is not fully sampled except over very large angular ranges. The actual increase in the blindregion volume due to this effect is minimized by use of a collimated beam and a narrow spectral spread (i.e. finely monochromatized, synchrotron radiation) if the crystal is not too mosaic.

The rotation method. The blind region associated with a single rotation axis. From Arndt & Wonacott (1977). 
These effects are directly related to the Lorentz factor, It is inadvisable to measure a reflection intensity when L is large because different parts of a spot would need a different Lorentz factor.
The blind region can be filled in by a rotation about another axis. The total angular range that is needed to sample the blind region is in the absence of any symmetry or in the case of mm symmetry (for example).
Weissenberg geometry (Weissenberg, 1924) is dealt with in the books by Buerger (1942) and Woolfson (1970), for example.
The conventional Weissenberg method uses a moving film in conjunction with the rotation of the crystal and a layerline screen. This allows:
The Weissenberg method is not widely used now. In smallmolecule crystallography, quantitative data collection is usually performed by means of a diffractometer.
Weissenberg geometry has been revived as a method for macromolecular data collection (Sakabe, 1983, 1991), exploiting monochromatized synchrotron radiation and the image plate as detector. Here the method is used without a layerline screen where the total rotation angle is limited to ; this is a significant increase over the rotation method with a stationary film. The use of this effectively avoids the presence of partial reflections and reduces the total number of exposures required. Provided the Weissenberg camera has a large radius, the Xray background accumulated over a single spot is actually not serious. This is because the Xray background decreases approximately according to the inverse square of the distance from the crystal to the detector.
The following Subsections 2.2.4.2 and 2.2.4.3 describe the standard situation where a layerline screen is used.
Normalbeam geometry (i.e. the Xray beam perpendicular to the rotation axis) is used to record zerolayer photographs. The film is held in a cylindrical cassette coaxial with the rotation axis. The centre of the gap in a screen is set to coincide with the zerolayer plane. The coordinate of a spot on the film measured parallel () and perpendicular () to the rotation axis is given by where is the rotation angle of the crystal from its initial setting, f is the coupling constant, which is the ratio of the crystal rotation angle divided by the film cassette translation distance, in ° min^{−1}, and D is the camera radius. Generally, the values of f and D are 2° min^{−1} and 28.65 mm, respectively.
Upperlayer photographs are usually recorded in equiinclination geometry [i.e. μ = −ν in equations (2.2.3.7) and (2.2.3.8)]. The Xraybeam direction is made coincident with the generator of the cone of the diffracted beam for the layer concerned, so that the incident and diffracted beams make equal angles (μ) with the equatorial plane, where The screen has to be moved by an amount where s is the screen radius. If the cassette is held in the same position as the zerolayer photograph, then reflections produced by the same orientation of the crystal will be displaced relative to the zerolayer photograph. This effect can be eliminated by initial translation of the cassette by .
The main book dealing with the precession method is that of Buerger (1964).
The precession method is used to record an undistorted representation of a single plane of relp's and their associated intensities. In order to achieve this, the crystal is carefully set so that the plane of relp's is perpendicular to the Xray beam. The normal to this plane, the zone axis, is then precessed about the Xraybeam axis. A layerline screen allows only relp's of the plane of interest to pass through to the film. The motion of the crystal, screen, and film are coupled together to maintain the coplanarity of the film, screen, and zone.
Setting of the crystal for one zone is carried out in two stages. First, a Laue photograph is used for small molecules or a monochromatic still for macromolecules to identify the required zone axis and place it parallel to the Xray beam. This is done by adjustment to the cameraspindle angle and the goniometerhead arc in the horizontal plane. This procedure is usually accurate to a degree or so. Note that the vertical arc will only rotate the pattern around the Xray beam. Second, a screenless precession photograph is taken using an angle of ∼7–10° for small molecules or 2–3° for macromolecules. It is better to use unfiltered radiation, as then the edge of the zerolayer circle is easily visible. Let the difference of the distances from the centre of the pattern to the opposite edges of the trace in the direction of displacement be called δ = DΔ so that for the horizontal goniometerhead arc and the dial: δ_{arc} = x_{Rt} − x_{Lt} and δ_{dial} = y_{Up} − y_{Dn} (Fig. 2.2.5.1 ). The corrections to the arc and camera spindle are given by where D is the crystaltofilm distance and is the precession angle.

The screenless precession setting photograph (schematic) and associated missetting angles for a typical orientation error when the crystal has been set previously by a monochromatic still or Laue. 
It is possible to measure δ to about 0.3 mm (δ = 1 mm corresponds to 14′ error for D = 60 mm and [Table 2.2.5.1, based on IT II (1959, p. 200)].
Alternatively, Δ = δ/D can be used if is small [from equation (2.2.5.1)]. Notes

Before the zerolayer photograph is taken, an Nb filter (for Mo Kα) or an Ni filter (for Cu Kα) is introduced into the Xray beam path and a screen is placed between the crystal and the film at a distance from the crystal of where is the screen radius. Typical values of would be 20° for a small molecule with Mo Kα and 12–15° for a protein with Cu Kα. The annulus width in the screen is chosen usually as 2–3 mm for a small molecule and 1–2 mm for a macromolecule. A clutch slip allows the camera motor to be disengaged and the precession motion can be executed under hand control to check for fouling of the goniometer head, crystal, screen or film cassette; s and need to be selected so as to avoid this happening. The zerolayer precession photograph produced has a radius of corresponding to a resolution limit . The distance between spots A is related to the reciprocalcell parameter a* by the formula
The recording of upperlayer photographs involves isolating the net of relp's at a distance from the zero layer of ζ_{n} = nλ/b, where b is the case of the b axis antiparallel to the Xray beam. In order to determine ζ_{n}, it is generally necessary to record a coneaxis photograph. If the cell parameters are known, then the camera settings for the upperlevel photograph can be calculated directly without the need for a coneaxis photograph.
In the upperlayer precession photograph, the film is advanced towards the crystal by a distance and the screen is placed at a distance The resulting upperlayer photograph has outer radius and an inner blind region of radius
A coneaxis photograph is recorded by placing a film enclosed in a lighttight envelope in the screen holder and using a small precession angle, e.g. 5° for a small molecule or 1° for a protein. The photograph has the appearance of concentric circles centred on the origin of reciprocal space, provided the crystal is perfectly aligned. The radius of each circle is where Hence, .
The main book dealing with singlecrystal diffractometry is that of Arndt & Willis (1966). Hamilton (1974) gives a detailed treatment of angle settings for fourcircle diffractometers. For details of areadetector diffractometry, see Howard, Nielsen & Xuong (1985) and Hamlin (1985).
In this section, we describe the following related diffractometer configurations:
(a) is used with singlecounter detectors. The kappa option is also used in the television areadetector 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, 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 singlecounter diffractometer is primarily used for smallmolecule crystallography. In macromolecular crystallography, many relp's are almost simultaneously in the diffraction condition. The singlecounter diffractometer was extended to five separate counters [for a review, see Artymiuk & Phillips (1985)], then subsequently to a multielement linear detector [for a review, see Wlodawer (1985)]. Area detectors offer an even larger aperture for simultaneous acquisition of reflections [Hamlin et al. (1981), and references therein].
Largearea online imageplate 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 RAXISIIc (Rigaku Corporation) and the MAR (MAR Research Systems) fall into this category, utilizing IP technology and an online 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 largerarea device can be realized. However, it is likely that these will be used in conjunction with a threeaxis goniostat again, except in special cases where a complete area coverage can be realized.
In normalbeam equatorial geometry (Fig. 2.2.6.1 ), 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 axis). The value of 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 threeaxis goniostat are necessary. The convention for the sign of the angles given in Fig. 2.2.6.1 is that of Hamilton (1974); his choice of sign of is adhered to despite the fact that it is lefthanded, but in any case the signs of ω, χ, are standard righthanded. The specific reciprocallattice point can be rotated from point P to point Q by the rotation, from Q to R via χ, and R to S via ω (see Fig. 2.2.6.2 ).

Normalbeam equatorial geometry: the angles ω, χ, ϕ, 2θ are drawn in the convention of Hamilton (1974). 
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 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, angles. When χ = ±90°, the ψ scan is simply a scan and is 0°.
The χ circle is a relatively bulky object whose thickness can inhibit the measurement of diffracted beams at high . Also, collision of the χ circle with the collimator or Xraytube housing has to be avoided. An alternative is the kappa goniostat geometry. In the kappa diffractometer [for a schematic picture, see Wyckoff (1985, 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 axis is mounted on the κ axis. In this way, an unobstructed view of the sample is achieved.
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 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) 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 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 axis by 180°. This renders the capillary axis horizontal and a different crystal axis vertical. Hence by rotation about this new crystal axis by , the blind region can be sampled.
The tools required for making the necessary measurement of reflection intensities include

In this section, we describe the common configurations for defining precise states of the Xray beam. The topic of detectors is dealt with in Part 7 (see especially Section 7.1.6 ). The impact of detector distortions on diffraction geometry is dealt with in Subsection 2.2.7.4.
Within the topic of beam conditioning the following subtopics are dealt with:
An exhaustive survey is not given, since a wide range of configurations is feasible. Instead, the commonest arrangements are covered. In addition, conventional Xray sources are separated from synchrotron Xray sources. The important difference in the treatment of the two types of source is that on the synchrotron the position and angle of the photon emission from the relativistic charged particles are correlated. One result of this, for example, is that after monochromatization of the synchrotron radiation (SR) the wavelength and angular direction of a photon are correlated.
The angular reflecting range and diffractionspot size are determined by the physical state of the beam and the sample. Hence, the idealized situation considered earlier of a point sample and zerodivergence beam will be relaxed. Moreover, the effects of the detectorimaging characteristics are considered, i.e. obliquity, parallax, pointspread factor, and spatial distortions.
An extended discussion of instrumentation relating to conventional Xray sources is given in Arndt & Willis (1966) and Arndt & Wonacott (1977). Witz (1969) has reviewed the use of monochromators for conventional Xray sources.
It is generally the case that the Kα line has been used for singlecrystal data collection via monochromatic methods. The continuum Bremsstrahlung radiation is used for Laue photography at the stage of setting crystals.
The emission lines of interest consist of the , doublet and the Kβ line. The intrinsic spectral width of the , or line is , their separation (δλ/λ) is , and they are of different relative intensity. The Kβ line is eliminated either by use of a suitable metal filter or by a monochromator. A mosaic monochromator such as graphite passes the , doublet in its entirety. The monochromator passes a certain, if small, component of a harmonic of the , line extracted from the Bremsstrahlung. This latter effect only becomes important in circumstances where the attenuated main beam is used for calibration; the process of attenuation enhances the shortwavelength harmonic component to a significant degree. In diffraction experiments, this component is of negligible intensity. The polarization correction is different with and without a monochromator (see Chapter 6.2 ).
The effect of the doublet components of the Kα emission is to cause a peak broadening at high angles. From Bragg's law, the following relationship holds for a given reflection: For proteins where is relatively small, the effect of the , separation is not significant. For small molecules, which diffract to higher resolution, this is a significant effect and has to be accounted for at high angles.
The width of the rocking curve of a crystal reflection is given by (Arndt & Willis, 1966) when the crystal is fully bathed by the Xray beam, where a is the crystal size, f the Xray tube focus size (foreshortened), s the distance between the Xray tube focus and the crystal, and η the crystal mosaic spread (Fig. 2.2.7.1 ).
In the movingcrystal method, Δ is the minimum angle through which the crystal must be rotated, for a given reflection, so that every mosaic block can diffract radiation covering a fixed wavelength band δλ from every point on the focal spot.
This angle Δ can be controlled to some extent, for the protein case, by collimation. For example, with a collimator entrance slit placed as close to the Xray tube source and a collimator exit slit placed as close to the sample as possible, the value of (a + f)/s can approximately be replaced by (a′ + f′)/s′, where f′ is the entranceslit size, a′ is the exitslit size, and s′ the distance between them. Clearly, for a′ a, the whole crystal is no longer bathed by the Xray beam. In fact, by simply inserting horizontal and vertical adjustable screws at the front and back of the collimator, adjustment to the horizontal and vertical divergence angles can be made. The spot size at the detector can be calculated approximately by multiplying the particular reflection rocking angle Δ by the distance from the sample to the spot on the detector. In the case of a singlecounter diffractometer, tails on a diffraction spot can be eliminated by use of a detector collimator.
Spottospot spatial resolution can be enhanced by use of focusing mirrors, which is especially important for largeprotein and virus crystallography, where long cell axes occur. The effect is achieved by focusing the beam on the detector, thereby changing a divergence from the source into a convergence to the detector.
In the absence of absorption, at grazing angles, Xrays up to a certain critical energy are reflected. The critical angle is given by where N is the number of free electrons per unit volume of the reflecting material. The higher the atomic number of a given material then the larger is for a given critical wavelength. The product of mirror aperture with reflectivity gives a figure of merit for the mirror as an efficient optical element.
The use of a pair of perpendicular curved mirrors set in the horizontal and vertical planes can focus the Xray tube source to a small spot at the detector. The angle of the mirror to the incident beam is set to reject the Kβ line (and shorterwavelength Bremsstrahlung). Hence, spectral purity at the sample and diffraction spot size at the detector are improved simultaneously. There is some loss of intensity (and lengthening of exposure time) but the overall signaltonoise ratio is improved. The primary reason for doing this, however, is to enhance spottospot spatial resolution even with the penalty of the exposure time being lengthened. The rocking width of the sample is not affected in the case of 1:1 focusing (object distance = image distance). Typical values are tube focalspot size, f = 0.1 mm, tubetomirror and mirrortosample distances ∼200 mm, convergence angle 2 mrad, and focalspot size at the detector ∼0.3 mm.
To summarize, the configurations are
(a) is for Laue mode; (b)–(f) are for monochromatic mode; (f) is a fairly recent development for conventionalsource work.
In the utilization of synchrotron Xradiation (SR), both Laue and monochromatic modes are important for data collection. The unique geometric and spectral properties of SR renders the treatment of diffraction geometry different from that for a conventional Xray source. The properties of SR are dealt with in Subsection 4.2.1.5 and elsewhere; see Subject Index. Reviews of instrumentation, methods, and applications of synchrotron radiation in protein crystallography are given by Helliwell (1984, 1992).
(a) Laue geometry: sources, optics, sample reflection bandwidth, and spot size. Laue geometry involves the use of the fully polychromatic SR spectrum as transmitted through the beryllium window that is used to separate the apparatus from the machine vacuum. There is useful intensity down to a wavelength minimum of ∼λ_{c}/5, where λ_{c} is the critical wavelength of the magnet source. The maximum wavelength is typically 3 Å; however, if the crystal is mounted in a capillary then the glass absorbs the wavelengths beyond ∼2.6 Å.
The bandwidth can be limited somewhat under special circumstances. A reflecting mirror at grazing incidence can be used for two reasons. First, the minimum wavelength in the beam can be sharply defined to aid the accurate definition of the Lauespot multiplicity. Second, the mirror can be used to focus the beam at the sample. The maximumwavelength limit can be truncated by use of aluminium absorbers of varying thickness or by use of a transmission mirror (Lairson & Bilderback, 1992; Cassetta et al., 1993).
The measured intensity of individual Laue diffraction spots depends on the wavelength at which they are stimulated. The problem of wavelength normalization is treated by a variety of methods. These include:

Each of these methods produces a `λcurve' describing the relative strength of spots measured at various wavelengths. The methods rely on the incident spectrum being smooth and stable with time. There are discontinuities in the `λcurve' at the bromine and silver Kabsorption edges owing to the silver bromide in the photographic emulsion case. The λcurve is therefore usually split up into wavelength regions, i.e. λ_{min} to 0.49 Å, 0.49 to 0.92 Å, and 0.92 Å to λ_{max}. Other detector types have different discontinuities, depending on the material making up the Xray absorbing medium. [The quantification of conventionalsource Lauediffraction data (Rabinovich & Lourie, 1987; Brooks & Moffat, 1991) requires the elimination of spots recorded near the emissionline wavelengths.]
The production and use of narrowbandpass beams may be of interest, e.g. δλ/λ 0.2. Such bandwidths can be produced by a combination of a reflection mirror used in tandem with a transmission mirror. Alternatively, an Xray undulator of 10–100 periods ideally should yield a bandwidth behind a pinhole of δλ/λ 0.1–0.01. In these cases, wavelength normalization is more difficult because the actual spectrum over which a reflection is integrated is rapidly varying in intensity. The spot bandwidth is determined by the mosaic spread and horizontal beam divergence (since γ_{H} γ_{V}) as where η = sample mosaic spread, assumed to be isotropic, γ_{H} = horizontal crossfire angle, which in the absence of focusing is (x_{H} + σ_{H})/P, where x_{H} is the horizontal sample size and σ_{H} the horizontal source size, and P is the sample to the tangentpoint distance; and similarly for γ_{V} in the vertical direction. Generally, at SR sources, σ_{H} is greater than σ_{V}. When a focusingmirror element is used, γ_{H} and/or γ_{V} are convergence angles determined by the focusing distances and the mirror aperture.
The size and shape of the diffraction spots vary across the film. The radial spot length is given by convolution of Gaussians as and tangentially by where L_{c} is the size of the Xray beam (assumed circular) at the sample, and and where ψ is the angle between the vertical direction and the radius vector to the spot (see Andrews, Hails, Harding & Cruickshank, 1987). For a crystal that is not too mosaic, the spot size is dominated by L_{c}. For a mosaic or radiationdamaged crystal, the main effect is a radial streaking arising from η, the sample mosaic spread.
(b) Monochromatic SR beams: optical configurations and sample rocking width. A wide variety of perfectcrystal monochromator configurations are possible and have been reviewed by various authors (Hart, 1971; Bonse, Materlik & Schröder, 1976; Hastings, 1977; Kohra, Ando, Matsushita & Hashizume, 1978). Since the reflectivity of perfect silicon and germanium is effectively 100%, multiplereflection monochromators are feasible and permit the tailoring of the shape of the monochromator resolution function, harmonic rejection, and manipulation of the polarization state of the beam. Two basic designs are in common use. These are (a) the bent singlecrystal monochromator of triangular shape (Lemonnier, Fourme, Rousseaux & Kahn, 1978) and (b) the doublecrystal monochromator.
In the case of the singlecrystal monochromator, 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. 2.2.7.2(a)]. For a point source and a curved monochromator crystal, one specific curvature can compensate for this variation in incidence angle [Fig. 2.2.7.2(b)]. 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, (δλ/λ)_{corr}, is selected along its length so that the rays converging towards the focus have a correlation of photon wavelength and direction [Fig. 2.2.7.2(c)]. 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 photonwavelength gradient across the width of the focus (for all curvatures) [Fig. 2.2.7.2(d)]. The use of a slit in the focal plane is akin to placing a slit at the tangent point to limit the source size.
The doublecrystal 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 the monochromator Bragg angle. The monochromator can be rapidly tuned, since the diffractometer or camera need not be realigned significantly in a scan across an absorption edge. Between absorption edges, some vertical adjustment of the diffractometer is required. Since the rocking width of the fundamental is broader than the harmonic reflections, the strict parallelism of the pair of crystal planes can be relaxed, i.e. 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. 2.2.7.3 ).
The doublecrystal monochromator is often used with a toroid focusing mirror; the functions of monochromatization are then separated from the focusing (Hastings, Kincaid & Eisenberger, 1978).
The rocking width of a reflection depends on the horizontal and vertical beam divergences/convergences (after due account for collimation is taken) γ_{H} and γ_{V}, the spectral spreads (δλ/λ)_{conv} and (δλ/λ)_{corr}, and the mosaic spread η. We assume that η ω, where ω is the angular broadening of a relp due to a finite sample. In the case of synchrotron radiation, γ_{H} and γ_{V} are usually widely asymmetric. On a conventional source, usually .
Two types of spectral spread occur with synchrotron and neutron sources. The term (δλ/λ)_{conv} is the spread that is passed down each incident ray in a divergent or convergent incident beam; the subscript refers to 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 finitesourcesize effects. The term (δλ/λ)_{corr} 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. Usually, an instrument is set to have (δλ/λ)_{corr} = 0. In the most general case, for a (δλ/λ)_{corr} arising from the horizontal ray direction correlation with photon energy, and the case of a horizontal rotation axis, then the rocking width of an individual reflection is given by where and L is the Lorentz factor .
The Guinier setting of the instrument gives (δλ/λ)_{corr} = 0. The equation for then reduces to (from Greenhough & Helliwell, 1982). For example, for ζ = 0, γ_{V} = 0.2 mrad (0.01°), = 15°, (δλ/λ)_{conv} = 1 × 10^{−3} and η = 0.8 mrad (0.05°), then = 0.08°. 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 (Subsection 2.2.3.4). For example, 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). For discussions, see Harrison, Winkler, Schutt & Durbin (1985) and Rossmann (1985).
In Fig. 2.2.7.4 , the relevant parameters are shown. The diagram shows (δλ/λ)_{corr} = 2δ 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. For full details, see Greenhough & Helliwell (1982).
Electronic area detectors are realtime imagedigitizing devices under computer control. The mechanism by which an Xray photon is captured is different in the various devices available (i.e. gas chambers, television detectors, chargecoupled devices) and is different specifically from film or image plates. Arndt (1986 and Section 7.1.6 ) has reviewed the various devices available, their properties and performances. Section 7.1.8 deals with storage phosphors/image plates.
(a) Obliquity. In terms of the geometric reproduction of a diffractionspot position, size, and shape, photographic film gives a virtually true image of the actual diffraction spot. This is because the emulsion is very thin and, even in the case of doubleemulsion film, the thickness, g, is only ∼0.2 mm. Hence, even for a diffracted ray inclined at = 45° to the normal to the film plane, the `parallax effect', , is very small (see below for details of when this is serious). With film, the spot size is increased owing to oblique or nonnormal incidence. The obliquity effect causes a beam, of width w, to be recorded as a spot of width For example, if w = 0.5 mm and = 45°, then w′ is 0.7 mm. With an electronic area detector, obliquity effects are also present. In addition, the effects of parallax, pointspread factor, and spatial distortions have to be considered.
(b) Parallax. In the case of a oneatmosphere xenongas chamber of thickness g = 10 mm, the parallax effect is dramatic [see Hamlin (1985, p. 435)]. The wavelength of the beam has to be considered. If a λ of ∼1 Å is used with such a chamber, the photons have a significant probability of fully traversing such a gap and parallax will be at its worst; the spot is elongated and the spot centre will be different from that predicted from the geometric centre of the diffracted beam. If a λ of 1.54 Å is used then the penetration depth is reduced and an effective g, i.e. g_{eff}, of ∼3 mm would be appropriate. The use of higher pressure in a chamber increases the photoncapture probability, thus reducing g_{eff} pro rata; at four atmospheres and λ =1.54 Å, parallax is very small.
In general, we can take account of obliquity and parallax effects whereby the measured spot width, in the radial direction, is w′′, where As well as changing the spot size, the spot position, i.e. its centre, is also changed by both obliquity and parallax effects by . The spherical driftchamber design eliminated the effects of parallax (Charpak, Demierre, Kahn, Santiard & Sauli, 1977). In the case of a phosphorbased television system, the Xrays are converted into visible light in a thin phosphor layer so that parallax is negligible.
(c) Pointspread factor. Even at normal incidence, there will be some spreading of the beam size. This is referred to as the pointspread factor, i.e. a single pencil ray of light results in a finitesized spot. In the TVdetector and imageplate cases, the graininess of the phosphor and the system imaging the visible light contribute to the pointspread factor. In the case of a chargecoupled device (CCD) used in directdetection mode, i.e. Xrays impinging directly on the silicon chip, the pointspread factor is negligible for a spot of typical size. For example, in Laue mode with a CCD used in this way, a 200 µm diameter spot normally incident on the device is not measurably broadened. The pixel size is ∼25 µm. The size of such a device is small and it is used in this mode for looking at portions of a pattern.
(d) Spatial distortions. The spot position is affected by spatial distortions. These nonlinear distortions of the predicted diffraction spot positions have to be calibrated for independently; in the worst situations, misindexing would occur if no account were taken of these effects. Calibration involves placing a geometric plate, containing a perfect array of holes, over the detector. The plate is illuminated, for example, with radiation from a radioactive source or scattered from an amorphous material at the sample position. The measured positions of each of the resulting `spots' in detector space (units of pixels) can be related directly to the expected position (in mm). A 2D, nonlinear, pixeltomm and mmtopixel correction curve or lookup table is thus established.
These are the special geometric effects associated with the use of electronic area detectors compared with photographic film or the image plate. We have not discussed nonuniformity of response of detectors since this does not affect the geometry. Calibration for nonuniformity of response is discussed in Section 7.1.6 .
Acknowledgements
I am very grateful to various colleagues at the Universities of York and Manchester for their comments on the text of the first edition. However, special thanks go to Dr T. Higashi who commented extensively on the manuscript and found several errors. Any remaining errors are, of course, my own responsibility. Dr F. C. Korber is thanked for his comments on the diffractometry section. Dr W. Parrish and Mrs E. J. Dodson are also thanked for discussions. Mrs Y. C. Cook is thanked for typing several versions of the manuscript and Mr A. B. Gebbie is thanked for drawing the diagrams. I am grateful to Miss Julie Holt for secretarial help in the production of the second edition.
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