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. 3741

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 .
References
Andrews, S. J., Hails, J. E., Harding, M. M. & Cruickshank, D. W. J. (1987). Acta Cryst. A43, 70–73.Google ScholarArndt, U. W. (1986). Xray positionsensitive detectors. J. Appl. Cryst. 19, 145–163. Google Scholar
Arndt, U. W. & Willis, B. T. M. (1966). Single crystal diffractometry. Cambridge University Press. Google Scholar
Arndt, U. W. & Wonacott, A. J. (1977). The rotation method in crystallography. Amsterdam: NorthHolland.Google Scholar
Bonse, U., Materlik, G. & Schröder, W. (1976). Perfectcrystal monochromators for synchrotron Xradiation. J. Appl. Cryst. 9, 223–230.Google Scholar
Brooks, I. & Moffat, K. (1991). Laue diffraction from protein crystals using a sealedtube Xray source. J. Appl. Cryst. 24, 146–148. Google Scholar
Cassetta, A., Deacon, A., Emmerich, C., Habash, J., Helliwell, J. R., McSweeney, S., Snell, E., Thompson, A. W. & Weisgerber, S. (1993). The emergence of the synchrotron Laue method for rapid data collection from protein crystals. Proc. R. Soc. London Ser. A, 442, 177–192.Google Scholar
Charpak, G., Demierre, C., Kahn, R., Santiard, J. C. & Sauli, F. (1977). Some properties of spherical drift chambers. Nucl. Instrum. Methods, 141, 449.Google Scholar
Greenhough, T. J. & Helliwell, J. R. (1982). Oscillation camera data processing: reflecting range and prediction of partiality. 2. Monochromatic synchrotron Xradiation from a singly bent triangular monochromator. J. Appl. Cryst. 15, 493–508.Google Scholar
Hamlin, R. (1985). Multiwire area Xray diffractometers. Methods Enzymol. 114A, 416–451.Google Scholar
Harrison, S. C., Winkler, F. K., Schutt, C. E. & Durbin, R. (1985). Oscillation method with large unit cells. Methods Enzymol. 114A, 211–236.Google Scholar
Hart, M. (1971). Bragg reflection Xray optics. Rep. Prog. Phys. 34, 435–490.Google Scholar
Hastings, J. B. (1977). Xray optics and monochromators for synchrotron radiation. J. Appl. Phys. 48, 1576–1584.Google Scholar
Hastings, J. B., Kincaid, B. M. & Eisenberger, P. (1978). A separated function focusing monochromator system for synchrotron radiation. Nucl. Instrum. Methods, 152, 167–171.Google Scholar
Helliwell, J. R. (1984). Synchrotron Xradiation protein crystallography: instrumentation, methods and applications. Rep. Prog. Phys. 47, 1403–1497. Google Scholar
Helliwell, J. R. (1992). Macromolecular crystallography with synchrotron radiation. Cambridge University Press.Google Scholar
Kohra, K., Ando, M., Matsushita, T. & Hashizume, H. (1978). Design of highresolution Xray optical system using dynamical diffraction for synchrotron radiation. Nucl. Instrum. Methods, 152, 161–166.Google Scholar
Lairson, B. M. & Bilderback, D. H. (1982). Transmission Xray mirror – a new optical element. Nucl. Instrum. Methods, 195, 79–83.Google Scholar
Lemonnier, M., Fourme, R., Rousseaux, F. & Kahn, R. (1978). Xray curvedcrystal monochromator system at the storage ring DCI. Nucl. Instrum. Methods, 152, 173–177. Google Scholar
Rabinovich, D. & Lourie, B. (1987). Use of the polychromatic Laue method for shortexposure Xray diffraction data acquisition. Acta Cryst. A43, 774–780.Google Scholar
Rossmann, M. G. (1985). Determining the intensity of Bragg reflections from oscillation photographs. Methods Enzymol. 114A, 237–280.Google Scholar
Witz, J. (1969). Focusing monochromators. Acta Cryst. A25, 30–42.Google Scholar