International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 
International Tables for Crystallography (2012). Vol. F, ch. 8.2, pp. 205206
Section 8.2.2. Principles of Laue diffraction ^{a}Department of Biochemistry and Molecular Biology, The Center for Advanced Radiation Sources, and The Institute for Biophysical Dynamics, The University of Chicago, Chicago, Illinois 60637, USA 
The principles of Laue diffraction have been reviewed by Amorós et al. (1975), Cruickshank et al. (1987, 1991), Helliwell et al. (1989), Cassetta et al. (1993), Moffat (1997), and Ren et al. (1999).
Assume that a stationary, perfect single crystal that diffracts to a resolution limit of is illuminated by a polychromatic Xray beam spanning the wavelength (energy) range from to . All reciprocallattice points that lie between the Ewald spheres of radii and , and within a radius of the origin O where , the resolution limit of the crystal, are in a diffracting position for a particular wavelength λ, where and will contribute to a spot on the Laue diffraction pattern (Fig. 8.2.2.1). All such points diffract simultaneously and throughout the exposure, in contrast to a monochromatic diffraction pattern in which each point diffracts sequentially and briefly as it traverses the Ewald sphere. A Laue pattern may alternatively be thought of as the superposition of a series of monochromatic still patterns, each arising from a different wavelength in the range from to .
Each Laue spot arises from the mapping of a complete ray (a central line in reciprocal space, emanating from the origin) onto a point on the detector. In contrast, each spot in a monochromatic pattern arises from the mapping of a single reciprocallattice point onto a point on the detector. A ray may contain only a single reciprocallattice point hkl with spacing , in which case the corresponding Laue spot arises from a single wavelength (energy) and structure amplitude, or it may contain several reciprocallattice points, such as , in which case the Laue spot contains several wavelengths (energies) and structure amplitudes. In the former case, the Laue spot is said to be single, and in the latter, multiple. The existence of multiple Laue spots is known as the energyoverlap problem: one spot contains contributions from several energies. It seems to have been thought by Pauling, Bragg and others that, as the wavelength range and the resolution limit of the crystal increased, more and more Laue spots would be multiple and the energyoverlap problem would dominate. Cruickshank et al. (1987) showed that this was not so. Even in the extreme case of infinite wavelength range, no more than 12.5% of all Laue spots would be multiple. The energyoverlap problem is evidently of restricted extent. However, the magnitude of the energyoverlap problem varies with resolution: reciprocallattice points at low resolution are more likely to be associated with multiple Laue spots than to be single (Cruickshank et al., 1987).
The extraction of Xray structure amplitudes from a single Laue spot requires the derivation and application of a wavelengthdependent correction factor known as the wavelength normalization curve or λcurve. This curve and other known factors relate the experimentally measured raw intensities of each Laue spot to the square of the corresponding structure amplitude. The integrated intensity of a Laue spot is achieved automatically by integration over wavelength, rather than in a monochromatic spot by integration over angle as the crystal rotates. If, however, a Laue spot is multiple, its total intensity arises from the sum of the integrated intensities of each of its components, known also as harmonics or orders nhnknl of the inner point hkl where h, k and l are coprime.
Laue spots lie on conic sections, each corresponding to a central zone [uvw] in reciprocal space. Prominent spots known as nodal spots or nodals lie at the intersection of well populated zones and correspond to rays whose inner point hkl is of low coprime indices. All nodal spots are multiple and all are surrounded by clear areas devoid of spots.
The volume of reciprocal space stimulated in a Laue exposure, , is given by and contains reciprocallattice points where and is the volume of the reciprocal unit cell (Moffat, 1997). can be large, particularly for crystals that diffract to high resolution and thus have larger values of . Laue patterns may therefore contain numerous closely spaced spots and exhibit a spatialoverlap problem (Cruickshank et al., 1991). The value of is up to an order of magnitude greater than the typical number of spots on a monochromatic oscillation pattern from the same crystal. Since the overall goal of a diffraction experiment is to record all spots in the unique volume of reciprocal space with suitable accuracy and redundancy, a Laue data set may contain fewer images and more spots of higher redundancy than a monochromatic data set (Clifton et al., 1991). This is particularly evident if the crystal is of high symmetry.
Kalman (1979) provided derivations of the integrated intensity of a single spot in the Laue case and in the monochromatic case. Moffat (1997) used these to show that the duration of a typical Laue exposure was between three and four orders of magnitude less than the corresponding monochromatic exposure. The physical reason for this significant Laue advantage lies in the fact that all Laue spots are in a diffracting position and contribute to the integrated intensity throughout the exposure. In contrast, monochromatic spots diffract only briefly as each sweeps through the narrow Ewald sphere [more strictly, through the volume between the closely spaced Ewald spheres corresponding to and ]. The details are modified slightly for mosaic crystals of finite dimensions subjected to an Xray beam of finite cross section and angular crossfire (Ren et al., 1999; Z. Ren, unpublished results).
Exposure times are governed not merely by the requirement to generate sufficient diffracted intensity in a spot – the signal – but also to minimize the background under the spot – the noise. The background under a Laue spot tends to be higher than under a monochromatic spot, since it arises from a larger volume of reciprocal space in the Laue case. This volume extends from (where and θ is the Bragg angle for that Laue spot) through the Laue spot at to either or , whichever is the smaller (Moffat et al., 1989). Since both the signal and the noise in a Laue pattern are directly proportional to the exposure time, their ratio is independent of that parameter. The ratio does depend on the wavelength range . Decreasing the wavelength range both generates fewer spots and increases the signaltonoise ratio for each remaining spot by diminishing the background under it. This is analogous to decreasing the oscillation range in a monochromatic exposure.
The choice of appropriate exposure time in the Laue case is complicated, but the central fact remains: both in theory and in practice, Laue exposures are very short with respect to monochromatic exposures (Moffat et al., 1984; Helliwell, 1985; Moffat, 1997). Satisfactory Laue diffraction patterns have been routinely obtained with Xray exposures of 100 to 150 ps, corresponding to the duration of a single Xray pulse emitted by a single 15 mA bunch of electrons circulating in the European Synchrotron Radiation Facility (ESRF) (Bourgeois et al., 1996).
The advantages and disadvantages of the Laue technique, compared to the betterestablished and more familiar monochromatic techniques, are presented in Table 8.2.2.1.

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