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 lawwhere d is the interplanar spacing and θ 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. 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. 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 1 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. Note the restriction on the resolution limit occurring because of the flat plate, whereas with a cylindrical geometry one can record diffraction data for larger 2θ angles and increase the resolution limit. The cylindrical neutron sensitive image plate is actively used today for macromolecular neutron crystallography diffraction data collection at reactor neutron sources (see Blakeley, 2009) where a LAue DIffractometer (`LADI') instrument reaches 2θ measurements of 180°. Historically such a `wrap around' cylindrical geometry was the basis of both the original `Bernal rotation camera' and the original `Weissenberg camera' used in the past with photographic film for X-ray diffraction measurements from small-molecule organic and inorganic crystals.

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. 1). This region of reciprocal space is referred to as the accessible or stimulated region. Fig. 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 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 Section 1.2). The spot marked N is one example of a nodal spot (see Section 1.4).

For a Laue spot at a given θ, only the ratio λ/d is determined, and 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 θ yields λ since d is then known. Conversely, precise unit-cell lengths cannot be determined from a Laue pattern alone; methods are, however, available to help determine these (see Carr et al., 1992).

The maximum Bragg angle is given by the equation

The main use of Laue photography has in the past been for adjustment of the crystal to a desired orientation. With small-molecule 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 can give a clear indication of the crystal orientation and setting. Currently, with small-molecule or protein crystals for structure determination, the monochromatic still or small-angular-range rotation diffraction image is used for this purpose before data collection via an area detector. Synchrotron-radiation Laue photographs of protein crystals can be recorded with ultra-short exposure times such as into the sub-nanosecond range (Moffat, 2001). For a review see Ren et al. (1999). These patterns consist of a large number of diffraction spots. There has been a trend to narrower wavelength bandpasses to optimize intensity signal to noise and this leads to fewer diffraction spots. This can lead to auto-indexing difficulties in determining the crystal setting.

Crystal setting via Laue photography usually involves trying to direct the X-ray beam along a zone axis. Angular mis-setting angles ɛ in the spindle and arc are easily calculated from the formulawhere 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 crystal-to-film 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). This radius of the blank region is determined by the minimum wavelength in the beam and the magnitude of the reciprocal-lattice spacing parallel to the X-ray beam (see Jeffery, 1958). For the case, for example, of the X-ray beam perpendicular to the a*b* plane, thenwherewhere R is the radius of the blank region (see Fig. 2) and D is the crystal-to-flat-film 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.

The treatment above is for when the incident X-ray beam is able to pass through the sample, i.e. in what is referred to as transmission Laue diffraction. For the cases of huge crystals of inorganic/small-molecule materials transmission of the X-ray beam does not occur and crystal setting is then made in a `back-reflection' mode of measurement. This is used by crystal growers targetting industrial applications of such large crystals. See e.g. Amorós et al. (1975), pp. 103–105.

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 et al. (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 within 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 single-wavelength spot. The probability that h, k, l have no common integer divisor isHence, 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 single-wavelength 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 single-wavelength 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 integer divisor.

The above discussion holds for infinite bandwidth. The effect of a more experimentally realistic bandwidth is to increase the proportion of single-wavelength spots.

The number of relp's within the resolution sphere iswhere = 1/d_min and V* is the reciprocal unit-cell volume.

The number of relp's within the wavelength band λ_max to λ_min, for , is (Moffat et al., 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 infinite-bandwidth case and small in the finite-bandwidth case. However, Laue geometry is an efficient way of measuring a large number of relp's between and as single-wavelength spots. In any case multiplet spots are now readily deconvoluted to their single component intensities where there are enough observations e.g. from various crystal orientations (see Nieh et al., 1999, and Ren et al., 1999).

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 single-wavelength 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. 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). The multiplicity distribution is shown in Fig. 4 as a function of λ_max/λ_min (with the corresponding values of δλ/λ_mean).

Figure 3 A multiple component spot in Laue geometry. A ray of multiplicity 5 is shown as an example. The inner point A corresponds to d and a wavelength λ, the next point, B, is d/2 and wavelength λ/2. The outer point E corresponds to d/5 and λ/5. Rotation of the sample will either exclude inner points (at the λmax surface) or outer points (at the λmin surface) and so determine the recorded multiplicity.

Figure 4 The variation with M = λmax/λmin of the proportions of relp's lying on single, double and triple rays for the case . From Cruickshank et al. (1987).

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 et al., 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/D², where D is the crystal-to-detector distance, and to 1/V*, where V* is the reciprocal-cell volume. There is also though a substantial variation in spatial density with diffraction angle θ; 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 one-dimensional 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) 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 crystal-to-detector 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. A development of this involving gnomonic projections can be even more effective (Cruickshank et al., 1992).

A useful means of transformation of the flat-detector 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. 5 shows the graphical relationships involved [taken from International Tables for X-ray Crystallography, Vol. II (Evans & Lonsdale, 1959)], for the case of a Laue pattern recorded on a plane film, between the incident-beam direction SN, which is perpendicular to the 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 crystal-to-film 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

Figure 5 Geometrical principles of the spherical, stereographic, gnomonic and Laue projections. From Evans & Lonsdale (1959).

In a monochromatic still exposure, the crystal is held stationary and a near-zero wavelength bandpass (e.g. δλ/λ = 0.001) beam impinges on it. For a small-molecule 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 reciprocal-cell 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 rotating-crystal monochromatic methods.

The diffraction spots lie on curved arcs where each curve corresponds to the intersection of a cone with a film (or detector). With a flat film the intersections are conic sections. The curved arcs are most obviously recognizable for the protein crystal case where there are a large number of spots.

Crystal setting follows the procedure given in Section 1.2 whereby angular mis-setting angles are given by equation (3). When viewed down a zone axis, the pattern on the 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 θ, by the formulae

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. a charge-coupled device (CCD) or pixel area detector or image plate. The use of a repeated rotation or oscillation, for a given exposure, is simply to average out any time-dependent 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 Section 3.5.

Fig. 6(a) to (d) are taken from International Tables for X-ray Crystallography Vol. 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.

Figure 6 (a) Elevation of the sphere of reflection. O is the origin of the reciprocal lattice. C is the centre of the Ewald sphere. The incident beam is shown in the plane. (b) Plan of the sphere of reflection. R is the projection of the rotation axis on the equatorial plane. (c) Perspective diagram. P is the relp in the reflection position with the cylindrical coordinates ζ, ξ, φ. The angular coordinates of the diffracted beam are ν, ϒ. (d) Stereogram to show the direction of the diffracted beam, ν, ϒ, with DD′, normal to the incident beam and in the equatorial plane, as the projection diameter. From Evans & Lonsdale (1959).

With the nomenclature of Table 1:

Table 1 Glossary of symbols used to specify quantities on diffraction patterns and in reciprocal space The equatorial plane is the plane normal to the rotation axis. θ Bragg angle 2θ Angle of deviation of the reflected beam with respect to the incident beam Unit vector lying along the direction of the incident beam Unit vector lying along the direction of the reflected beam s = The scattering vector of magnitude . s is perpendicular to the bis­ector of the angle between and . s is identical to the reciprocal-lattice vector d* of magnitude λ/d, where d is the interplanar spacing, when d* is in the diffraction condition. In this notation, the radius of the Ewald sphere is unity. This convention is adopted because it follows that in Volume II of International Tables for X-ray Crystallography (p. 175). Note that in Section 1 Laue geometry the alternative convention (|d*| = 1/d) is adopted whereby the radius of each Ewald sphere is 1/λ. This allows a nest of Ewald spheres between and to be drawn ζ Coordinate of a point P in reciprocal space parallel to a rotation axis as the axis of cylindrical coordinates relative to the origin of reciprocal space ξ Radial coordinate of a point P in reciprocal space; that is, the radius of a cylinder having the rotation axis as axis τ The angular coordinate of P, measured as the angle between ξ and [see Fig. 6(b)] φ The angle of rotation from a defined datum orientation to bring a relp onto the Ewald sphere in the rotation method (see Fig. 8) μ The angle of inclination of to the equatorial plane ϒ The angle between the projections of and onto the equatorial plane ν The angle of inclination of to the equatorial plane ω, χ, φ The crystal setting angles on the four-circle diffractometer (see Fig. 11). The φ used here is not the same as that in the rotation method (Fig. 8). This clash in using the same symbol twice is inevitable because of the widespread use of the rotation camera and four-circle diffractometer

Fig. 6(a) givesFig. 6(b) gives, by the cosine rule,andand Fig. 6(a) and (b) give

The following special cases commonly occur:

(a) μ = 0, normal-beam rotation method, thenand

(b) μ = −ν, equi-inclination (relevant to Weissenberg upper-layer photography), then

(c) μ = +ν, anti-equi-inclination

(d) ν = 0, flat cone

In this section, we will concentrate on case (a), the normal-beam 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. Z_F is parallel to the rotation axis and ζ. Y_F is perpendicular to the rotation axis and the beam. International Tables for X-ray Crystallography Vol. II (1959, p. 177) follows the convention of y being parallel and x perpendicular to the rotation-axis direction, i.e. (Y_F, Z_F) ≡ (x, y). The advantage of the (Y_F, Z_F) notation is that the x-axis direction is then the same as the X-ray beam direction.

The coordinates of a reflection on a flat film (Y_F, Z_F) are related to the cylindrical coordinates of a relp (ξ, ζ) [Fig. 7(a)] bywhich becomeswhere D is the crystal-to-film distance.

Figure 7 Geometrical principles of recording the pattern on (a) a plane detector, (b) a V-shaped detector, (c) a cylindrical detector.

For the case of a V-shaped cassette with the V axis parallel to the rotation axis and the film making an angle α to the beam direction [Fig. 7(b)], thenThis situation also corresponds to the case of a 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 International Tables for X-ray Crystallography Vol. II (1959). This avoids confusion with the ν of Table 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. 7(c)] then, for ϒ in degrees,which becomesHere, 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, detector-misalignment errors have to be considered. These are three orthogonal angular errors, translation of the origin, and error in the crystal-to-film distance.

The coordinates Y_F and Z_F are related to film-scanner raster units via a scanner-rotation 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 reciprocal-lattice coordinates ζ, ξ, ϒ, ν etc. used earlier, refer to an axial system fixed to the crystal, X₀Y₀Z₀ of Fig. 8. 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).

Figure 8 The rotation method. Definition of coordinate systems. [Cylindrical coordinates of a relp P (ξ, ζ, φ) are defined relative to the axial system X0Y0Z0 which rotates with the crystal.] The axial system XYZ is defined such that X is parallel to the incident beam and Z is coincident with the rotation axis. From Arndt & Wonacott (1977), reproduced with permission of Elsevier.

The rotation angle required, φ, is with respect to some reference `zero-angle' direction and is determined by the particular crystal parameters. It is necessary to define a standard orientation of the crystal (i.e. datum) when φ = 0°. If we define an axial system X₀Y₀Z₀ fixed to the crystal and a laboratory axis system XYZ with X parallel to the beam and Z coincident with the rotation axis, then φ = 0° corresponds to these axial systems being coincident (Fig. 8).

The angle of the crystal at which a given relp diffracts isThe two solutions correspond to the two rotation angles at which the relp P cuts the sphere of reflection. Note that Y_F, Z_F (Section 3.2) are independent of φ.

The values of x₀ and y₀ are calculated from the particular crystal system parameters. The relationships between the coordinates x₀, y₀, z₀ and ξ and ζ areX₀ can be related to the crystal parameters byA is a crystal-orientation matrix defining the standard datum orientation of the crystal.

For example, if, by convention, a* is chosen as parallel to the X-ray beam at φ = 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 , , are introduced, i.e.where Δφ_x, Δφ_y and Δφ_z are angles around the X₀, Y₀ and Z₀ axes, respectively.

Hence, the relationship between X₀ and h is

An interesting method of auto-indexing, without and with knowledge of crystal unit-cell parameters, from a single, small rotation, monochromatic diffraction pattern is that of Higashi (1990). This method may have utility with the new X-ray lasers and a single still diffraction pattern measured from a nano-crystal.

For a given oscillation photograph, there is a maximum value of the oscillation range, Δφ, 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 bywhere Δ is the sample reflecting range (see Section 7). is a function of φ, even in the case of identical cell parameters. This is because it is necessary to consider, for a given orientation, the relevant reciprocal-lattice 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 the rotation method several approaches are used. Firstly, 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 signal-to-noise ratio for individual spots, which accumulate more background since . Secondly, Δφ is chosen usually to be less than Δ so as to optimize the signal-to-noise ratio of the measurement and to sample the rocking-width profile. A diffraction image is read after the crystal rotation has finished. As disk storage has become cheaper and detector device readout times shorter the use of `very fine phi slicing' takes this approach of reflection profile measurement to the ideal limit. Thirdly, a continuous rotation scheme (`shutterless data collection') can be applied using pixel area detectors where each pixel can be read individually. Such schemes are also feasible with multiwire proportional chamber detectors (MWPCs).

The value of Δ, the crystal rocking width for a given hkl, depends on the reciprocal-lattice coordinates of the hkl relp (see Section 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 (44)] can be relaxed for special applications. For example, since the spot overlap that is to be avoided involves relp's from adjacent reciprocal-lattice 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 standard-sized detector, this is practical for low-resolution data recording. Alternatively, a much larger detector can be contemplated and even medium-resolution data can be recorded without major overlap problems. These techniques could be useful in some time-resolved applications. For a discussion see Weisgerber & Helliwell (1993) and Helliwell (2005). For regular data collection, however, narrow angular ranges are 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 normal-beam geometry, any relp lying close to the rotation axis will not be stimulated at all. This situation is shown in Fig. 9. The blind region has a radius ofand 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 slightly greater percentage of the data than this is not fully sampled except over very large angular ranges. The actual increase in the blind-region volume due to this effect is minimized by use of a collimated beam and a narrow spectral spread (i.e. finely monochromated, synchrotron radiation) if the crystal is not too mosaic.

Figure 9 The rotation method. The blind region associated with a single rotation axis. From Arndt & Wonacott (1977), reproduced with permission of Elsevier.

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, which is unrealistic.

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). With crystals of high enough symmetry, i.e not triclinic, and use of a random orientation, complete sampling of the unique region of reciprocal space can be obtained from a single rotation axis setting. However, for precise anomalous scattering measurements careful setting of the crystal orientation is often preferred (see e.g. Einspahr et al., 1985; Nieh & Helliwell, 1995) and thus rotation about another axis to capture the blind-region missing data will be required. A reciprocal-lattice viewer of a processed diffraction data set, so as to scrutinize gaps in the measurements, is available at https://staraniso.globalphasing.org/cgi-bin/PDBpeep.cgi (Tickle et al., 2021).

The conventional Weissenberg method uses a moving film in conjunction with the rotation of the crystal and a layer-line screen. This allows:

(a) A larger rotation range of the crystal to be used (say 200°), avoiding the problem of overlap of reflections (referred to in Section 3.4 on oscillation photography).

(b) Indexing of reflections on the photograph to be made by inspection.

The Weissenberg method is not widely used now. In small-molecule crystallography, quantitative data collection is usually performed by means of a CCD area-detector diffractometer.

Weissenberg geometry has been revived as a method for macromolecular data collection at the Photon Factory (Sakabe, 1983, 1991), exploiting monochromated synchrotron radiation and the image plate as detector. Here the method is used without a layer-line 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 X-ray background accumulated over a single spot is actually not as serious as one might fear. This is because the X-ray background decreases approximately according to the inverse square of the distance from the crystal to the detector. For very weakly scattering crystals owing to very small sample size or large unit cell etc. this will of course become a real limitation. However, the use of such a camera has not been widely adoped at other synchrotron facilities.

The following Sections 4.2 and 4.3 describe the standard situation where a layer-line screen is used.

Normal-beam geometry (i.e. the X-ray beam perpendicular to the rotation axis) is used to record zero-layer 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 zero-layer plane. The coordinate of a spot on the film measured parallel (Z_F) and perpendicular (Y_F) to the rotation axis is given bywhere φ 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.

Upper-layer photographs are usually recorded in equi-incli­nation geometry [i.e. μ = −ν in equations (21) and (22)]. The X-ray-beam 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, whereThe screen has to be moved by an amountwhere s is the screen radius. If the cassette is held in the same position as the zero-layer photograph, then reflections produced by the same orientation of the crystal will be displacedrelative to the zero-layer photograph. This effect can be eliminated by initial translation of the cassette by .

The precession method with a layer-line screen 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 X-ray beam. The normal to this plane, the zone axis, is then precessed about the X-ray-beam axis. A layer-line 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. For X-rays this is now of mainly historical interest in that the precession camera is rarely used. However an undistorted view of a single plane of relp's and their associated intensities is still valuable and can be readily created from intensity measurements made with the rotation method and a diffractometer, as described above. With electrons there has been a renewed interest in the precession method (without a layer-line screen) since it improves the measured values of the electron intensity, which are basically free from dynamical effects (Vincent & Midgley, 1994). This is being applied to nano- and microcrystals of small molecules (see e.g. Nicolopoulos et al., 2007) and proteins (see e.g. Georgieva et al., 2008). This is complemented with rotation electron crystallography geometry (see Fig. 2 of Gemmi et al., 2019). Gemmi et al. (2019) also provide an up-to-date exposition and review of electron crystallography methods, analyses and results.

Setting of the crystal for one zone is carried out in two stages. First, a monochromatic still identifies the required zone axis, which is placed parallel to the X-ray beam. This is done by adjustment to the camera-spindle angle and the goniometer-head 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 X-ray 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 zero-layer 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 goniometer-head arc and the dial: δ_arc = x_Rt − x_Lt and δ_dial = y_Up − y_Dn (Fig. 10). The corrections ɛ to the arc and camera spindle are given bywhere D is the crystal-to-film distance and is the precession angle.

Figure 10 The screenless precession setting photograph (schematic) and associated mis-setting angles for a typical orientation error when the crystal has been set previously by a monochromatic still.

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, based on International Tables for X-ray Crystallography Vol. II (1959, p. 200)].

Before the zero-layer photograph is taken, an Nb filter (for Mo Kα) or an Ni filter (for Cu Kα) is introduced into the X-ray beam path and a screen is placed between the crystal and the film at a distance from the crystal ofwhere r_s 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 r_s need to be selected so as to avoid this happening. The zero-layer precession photograph produced has a radius of corresponding to a resolution limit . The distance between spots A is related to the reciprocal-cell parameter a* by the formula

The recording of upper-layer 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 X-ray beam. In order to determine ζ_n, it is generally necessary to record a cone-axis photograph. If the cell parameters are known, then the camera settings for the upper-level photograph can be calculated directly without the need for a cone-axis photograph.

In the upper-layer precession photograph, the film is advanced towards the crystal by a distanceand the screen is placed at a distanceThe resulting upper-layer photograph has outer radiusand an inner blind region of radius

A cone-axis photograph (or screenless precession diffraction image) is recorded by using a small precession angle, e.g. 5° for a small molecule or 1° for a protein. The diffraction image has the appearance of concentric circles centred on the origin of reciprocal space, provided the crystal is perfectly aligned. The radius of each circle iswhereHence, .

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

(a) multiple axes [ω, χ, φ option or ω, κ, φ (kappa) option];

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

Configuration (a) is used with single-counter and area detectors. Configuration (b) is used with area detectors.

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] or for crystal alignment for the most precise anomalous scattering measurements. The use of electronic area detectors is now the most widespread.

The single-counter diffractometer used to be the primary device 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)], then subsequently to a multi-element linear detector [for a review, see Wlodawer (1985)]. Area detectors offered an even larger aperture for simultaneous acquisition of reflections [Hamlin et al. (1981), and references therein]. The CCDs or image plates have to a great extent been succeeded by the pixel area detectors (for an early reference see Broennimann et al., 2006).

With a multiple axes geometry (Fig. 11), a crystal can be 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 2θ axis). The value of θ is given by Bragg's law. 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. 11 is that of Hamilton (1974); his choice of sign of 2θ is adhered to despite the fact that it is left-handed, but in any case the signs of ω, χ, φ are standard right-handed. The specific reciprocal-lattice point can be rotated from point P to point Q by the φ rotation, from Q to R via χ, and R to S via ω (see Fig. 12).

Figure 11 Normal-beam equatorial geometry: the angles ω, χ, φ, 2θ are drawn in the convention of Hamilton (1974).

Figure 12 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). Reproduced with permission of Cambridge University Press through PLSclear. 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 ɛ 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, φ angles. When χ = ±90°, the ψ scan is simply a φ scan and ɛ is 0°.

The χ circle is a relatively bulky object whose thickness can inhibit the measurement of diffracted beams at high θ. 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, 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 φ axis is mounted on the κ axis. In this way, an unobstructed view of the sample is achieved.

The geometry with fixed χ consists of an ω axis, a φ axis and χ fixed. 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 (e.g. see Xuong et al., 1985) because of the generally random nature of any symmetry axis. For small-molecule crystallography fixed-χ instruments have χ = 54.74°, known as the `magic angle', as it provides a complete sphere of reciprocal space which can be collected by a set of three 180° ω scans at three different settings; the Bruker `SC-XRD' fixed-χ goniometer has this geometry. The angle of 54.74° chosen in such an apparatus is the angle between the body diagonal of a cube and the edges of the cube.

The tools required for making the necessary measurement of reflection intensities include beam-conditioning items, crystal goniostat and detectors.

In this section, we describe the common configurations for defining precise states of the X-ray beam. The topic of detectors is dealt with in Part 7 of the previous edition of International Tables for Crystallography, Vol. C (2006) (see especially Section 7.1.6 ). The impact of detector distortions on diffraction geometry is dealt with below in Section 7.4 of this present chapter.

Within the topic of beam conditioning the following subtopics are dealt with: (a) collimation, (b) monochromators and (c) mirrors.

An exhaustive survey is not given, since a wide range of configurations are feasible. Instead, the commonest arrangements are covered. In addition, conventional X-ray sources are separated from synchrotron X-ray 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. With X-ray undulators the divergence angles, in both the horizontal and vertical directions, are very small e.g. ~20 µrad.

The angular reflecting range and diffraction-spot size are determined by the physical state of the beam and the sample. Hence, the idealized situation considered earlier of a point sample and zero-divergence beam will be relaxed. Moreover, the effects of the detector-imaging characteristics are considered, i.e. obliquity, parallax, point-spread factor and spatial distortions.

An extended discussion of instrumentation relating to conventional X-ray sources is given in Arndt & Willis (1966) and Arndt & Wonacott (1977). Witz (1969) has reviewed the use of monochromators for conventional X-ray sources. See also Aslanov et al. (1998).

It is generally the case that the Kα line has been used for single-crystal data collection via monochromatic methods. The continuum Bremsstrahlung radiation has been used for Laue photog­raphy usually at the stage of setting crystals but also used extensively for structure analysis from 1913 through to the 1920s and 1930s very notably by W. L. Bragg, R. G. Dickinson, S. Nishikawa, L. Pauling and R. W. G. Wyckoff (for a history of this period see Cruickshank, 1992).

The emission lines of interest consist of the Kα₁, Kα₂ doublet and the Kβ line. The intrinsic spectral width of the Kα₁ or Kα₂ line is ∼10⁻⁴, their separation (δλ/λ) is 2.5 × 10⁻³, 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 Kα₁, Kα₂ doublet in its entirety. The monochromator passes a certain, if small, component of a harmonic of the Kα₁, Kα₂ 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 short-wavelength 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 the previous edition of International Tables for Crystallography, Vol. C (2006), 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 θ is relatively small, the effect of the Kα₁, Kα₂ separation is generally 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 X-ray beam, where a is the crystal size, f the X-ray tube focus size (foreshortened), s the distance between the X-ray tube focus and the crystal, and η the crystal mosaic spread (Fig. 13).

Figure 13 Reflection rocking width for a conventional X-ray source. From Arndt & Wonacott (1977, p. 7), reproduced with permission of Elsevier. (a) Effect of sample mosaic spread. The relp is replaced by a spherical cap with a centre at the origin of reciprocal space where it subtends an angle η. (b) Effect of (δλ/λ)conv, the conventional source type spectral spread. (c) Effect of a beam divergence angle, γ. The overall reflection rocking width is a combination of these effects.

In the moving-crystal 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 X-ray tube as possible 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 entrance-slit size, a′ is the exit-slit size, and s′ the distance between them. Clearly, for a′ < a, the whole crystal is no longer bathed by the X-ray 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 single-counter diffractometer, tails on a diffraction spot can be eliminated by use of a detector collimator.

Spot-to-spot spatial resolution can be enhanced by use of focusing mirrors, which is especially important for large-protein 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, X-rays up to a certain critical energy are reflected. The critical angle θ_c is given bywhere 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 θ_c 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 X-ray 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 shorter-wavelength 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 signal-to-noise ratio is improved. Another reason for doing this, however, is to enhance spot-to-spot spatial resolution. The rocking width of the sample is not affected in the case of 1:1 focusing (object distance = image distance).

To summarize, the configurations are

(a) beam collimator only;

(b) filter + beam collimator;

(c) filter + beam collimator + detector collimator (single-counter case);

(d) graphite monochromator + beam collimator;

(e) pair of focusing mirrors + exit-slit assembly;

(f) focusing germanium monochromator + perpendicular focusing mirror + exit-slit assembly.

Configuration (a) is for Laue mode; (b)–(f) are for monochromatic mode; (f) can also be applied to conventional-source work.

In the utilization of synchrotron X-radiation (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 X-ray source. The properties of SR are dealt with in the previous edition of International Tables for Crystallography, Vol. C (2006), Section 4.2.1.5 and elsewhere; see also the Subject Index of that edition. 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 band-width and spot size. Laue geometry involves the use of the polychromatic SR spectrum as transmitted through the beryllium window that is often 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 by optical elements. 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 Laue-spot multiplicity. Second, the mirror can be used to focus the beam at the sample. The maximum-wavelength limit can be truncated by use of aluminium absorbers of varying thickness or by use of a transmission mirror (Lairson & Bilderback, 1982; Cassetta et al., 1993). The case of undulator or wiggler–undulator (`wundulator') Laue is described below.

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:

(a) use of a monochromatic reference data set;

(b) use of symmetry equivalents and/or reflections recorded more than once in the Laue data set and measured at different wavelengths;

(c) calibration with a standard sample such as a silicon crystal.

Each of these methods produces a `λ-curve' 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 `λ-curve' at the bromine and silver K-absorption edges owing to the silver bromide in the photographic emulsion case. The λ-curve is then 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 X-ray absorbing medium. [The quantification of conventional-source Laue-diffraction data (Rabinovich & Lourie, 1987; Brooks & Moffat, 1991) requires the elimination of spots recorded near the emission-line wavelengths.]

The production and use of narrow-bandpass beams has increased in interest, e.g. δλ/λ ≤ 0.2 so as to maximize intensity signal to noise. An X-ray undulator of 10–100 periods ideally should yield a bandwidth behind a pinhole of δλ/λ ≃ 0.1–0.01. By using a slightly stronger magnetic field a hybrid device wiggler–undulator (`wundulator') can be used to produce broader `narrow-bandpass'. In these cases, wavelength normalization is more difficult because the actual spectrum over which a reflection is integrated is more rapidly varying in intensity, but modern Laue software caters for that (see Ren et al., 1999; Arzt et al., 1999; and Šrajer et al., 2000). The spot bandwidth is determined by the mosaic spread and horizontal beam divergence (since γ_H > γ_V) aswhere η = sample mosaic spread, assumed to be isotropic, γ_H = horizontal cross-fire 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 tangent-point distance; and similarly for γ_V in the vertical direction. Generally, at SR sources, σ_H is greater than σ_V. When a focusing-mirror 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 detector plane. The radial spot length is given by convolution of Gaussians asand tangentially bywhere L_c is the size of the X-ray beam (assumed circular for simplicity) at the sample, andandwhere ψ is the angle between the vertical direction and the radius vector to the spot (see Andrews et al., 1987). For a crystal that is not too mosaic, the spot size is dominated by L_c. For a mosaic or radiation-damaged 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 perfect-crystal monochromator configurations are possible and have been reviewed by various authors (Hart, 1971; Bonse et al., 1976; Hastings, 1977; Kohra et al., 1978). Since the reflectivity of perfect silicon and germanium is effectively 100%, multiple-reflection 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 single-crystal monochromator of triangular shape (Lemonnier et al., 1978) and (b) the double-crystal monochromator, either linked in a `channel cut' device or as separate crystals.

In the case of the single-crystal 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. 14(a)]. For a point source and a curved monochromator crystal, one specific curvature can compensate for this variation in incidence angle [Fig. 14(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. 14(c)]. This is used in dispersive measurements, e.g. see Arndt et al. (1982) and Renevier et al. (2003). 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. 14(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.

Figure 14 Single-crystal monochromator illuminated by synchrotron radiation: (a) flat crystal, (b) Guinier setting, (c) overbent crystal, (d) effect of source size (shown at the Guinier setting for clarity). From Helliwell (1984). Copyright IOP Publishing. Reproduced with permission. All rights reserved.

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 the monochromator Bragg angle. The monochromator can be rapidly tuned, since the diffractometer or camera need not be re-aligned 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. 15).

Figure 15 Double-crystal monochromator illuminated by synchrotron radiation. The contributions of the source divergence αV and angular source size Δθsource to the range of energies reflected by the monochromator are shown.

The double-crystal monochromator is often used with a toroid focusing mirror; the functions of monochromatization are then separated from the focusing (Hastings et al., 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 γ_H ≃ γ_V.

Two types of spectral spread occur with synchrotron and neutron sources (Greenhough & Helliwell, 1982; Schoenborn, 1983). 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 Kα₁, Kα₂ 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 (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 φ_R of an individual reflection is given bywhereand L is the Lorentz factor .

The Guinier setting of the instrument gives (δλ/λ)_corr = 0. The equation for φ_R then reduces to(from Greenhough & Helliwell, 1982). For example, for ζ = 0, γ_V = 0.2 mrad (0.01°), θ = 15°, (δλ/λ)_conv = 1 × 10⁻³ and η = 0.8 mrad (0.05°), then φ_R = 0.08°. But φ_R 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 (Section 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 et al. (1985) and Rossmann (1985).

In Fig. 16, 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 φ_R 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).

Figure 16 The rocking width of an individual reflection for the case of Fig. 14(c) and a vertical rotation axis. φR is determined by the passage of a spherical volume of radius ɛs (determined by sample mosaicity and a conventional-source-type spectral spread) through a nest of Ewald spheres of radii set by δ = ½[δλ/λ]corr and the horizontal convergence angle γH. From Greenhough & Helliwell (1982).

(c) The use of the X-ray laser to record `still' diffraction images from micro- and nano-sized crystal samples. Accurate modelling of the diffraction underpins the huge success of protein crystallography, using sophisticated models developed over many years. The properties of a synchrotron X-ray source monochromated beam were studied in great detail through numerous commissioning experiments with test samples (Helliwell et al., 1982). A similar strategy was undertaken for the characterization of a `white' X-ray beam with and without beam-conditioning elements (reflecting mirrors and filters). However, the X-ray laser as a new source poses new challenges for detailed characterization owing to its stochastic nature. The ultra-intense nature of the X-ray beam, whilst being formed of ultra-short time pulses, allows `only' a single diffraction pattern to be collected as there is no time to rotate the crystal sample. Finally, the smallest crystal samples now studied (in the nanometre size range) for structures with quite large unit cells leads to relatively few unit cells being irradiated and thereby the `grating interference function' is broadened. The use of many individual single-crystal X-ray diffraction patterns representing many orientations can make a merged single X-ray diffraction data set of good data completeness. In many studies of this kind a Monte Carlo approach to merging the still diffraction patterns (`stills') was used; typically many hundreds of thousands of these stills made up one data set. Attempts to refine the partiality of each measured diffraction spot have gradually been made, to good effect in terms of greatly reducing the number of nanocrystals used. For a description of the progress the reader is referred to the overview of Sauter (2015).

Whilst this section has focused on micrometre- and nanometre-sized crystals studied one by one by serial crystallography, a method initiated at XFELs, the use of a large single crystal is also possible. This allows many X-ray diffraction shots across the crystal, which is displaced and rotated between the XFEL shots, to be measured (Ago, 2014).

Electronic area detectors are real-time image-digitizing devices under computer control. The mechanism by which an X-ray photon is captured is different in the various devices available (i.e. gas chambers, television detectors, charge-coupled devices, pixel detectors) and is different specifically from film or image plates. Arndt (1986) has reviewed the various devices available, their properties and performances [see also Sections 7.1.6 and 7.1.8 of the previous edition of International Tables for Crystallography, Vol. C (2006); Section 7.1.8 deals with storage phosphors, image plates and pixel detectors]. Fourme (1997) provides an update on position-sensitive gas detectors.

(a) Obliquity. In terms of the geometric reproduction of a diffraction-spot 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 double-emulsion film, the thickness, g, is only ∼0.2 mm. Hence, even for a diffracted ray inclined at 2θ = 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 non-normal incidence. The obliquity effect causes a beam, of width w, to be recorded as a spot of widthFor example, if w = 0.5 mm and 2θ = 45°, then w′ is 0.7 mm. With an electronic area detector, obliquity effects are also present. In addition, the effects of parallax, point-spread factor and spatial distortions have to be considered.

(b) Parallax. In the case of a one-atmosphere xenon-gas 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 photon-capture 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′′, whereAs 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 drift-chamber design eliminated the effects of parallax (Charpak et al., 1977). In the case of a phosphor-based television system, the X-rays are converted into visible light in a thin phosphor layer so that parallax is negligible.

(c) Point-spread factor. Even at normal incidence, there will be some spreading of the beam size. This is referred to as the point-spread factor, i.e. a single pencil ray of light results in a finite-sized spot. In the TV-detector, CCD-with-phosphor and image-plate cases, the graininess of the phosphor and the system imaging the visible light contribute to the point-spread factor. In the case of a CCD used in direct-detection mode, i.e. X-rays impinging directly on the silicon chip, the point-spread factor is negligible for a spot of typical size.

(d) Spatial distortions. The spot position is affected by spatial distortions. These non-linear 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, non-linear, pixel-to-mm and mm-to-pixel correction curve or look-up table is thus established.

These are the special geometric effects associated with detectors. We have not discussed non-uniformity of response of detectors since this does not affect the geometry. Calibration for non-uniformity of response is discussed in Section 7.1.6 of the previous edition of International Tables for Crystallography, Vol. C (2006).