Principles of Laue diffraction

Moffat, K.

doi:10.1107/97809553602060000670

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 8.2, pp. 167-168 | 1 | 2 |

Section 8.2.2. Principles of Laue diffraction

K. Moffat^a ^*

^aDepartment 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
Correspondence e-mail: moffat@cars.uchicago.edu

8.2.2. Principles of Laue diffraction

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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 $[d^{*}_{\max}]$ is illuminated by a polychromatic X-ray beam spanning the wavelength (energy) range from $[\lambda_{\min}\ (E_{\max})]$ to $[\lambda_{\max}\ (E_{\min})]$ . All reciprocal-lattice points that lie between the Ewald spheres of radii $[1/\lambda_{\min}]$ and $[1/\lambda_{\max}]$ , and within a radius $[d^{*}_{\max}]$ of the origin O where $[d^{*}_{\max} = 1/d_{\min}]$ , the resolution limit of the crystal, are in a diffracting position for a particular wavelength λ, where $[\lambda_{\min} \leq \lambda \leq \lambda_{\max}]$ 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 $[\lambda_{\min}]$ to $[\lambda_{\max}]$ .

Figure 8.2.2.1| top | pdf |

Laue diffraction geometry. The volume element dV stimulated in a Laue experiment lies between $[d^{*}]$ and $[d^{*} + \hbox{d}d^{*}]$ , between the Ewald spheres corresponding to λ and $[\lambda + \hbox{d}\lambda]$ , and between φ and $[\varphi + \hbox{d}\varphi]$ , where φ denotes rotation about the incident X-ray beam direction. The entire volume stimulated in a single Laue exposure lies between 0 and $[d_{\max}^{*}]$ , between the Ewald spheres corresponding to $[\lambda_{\min}]$ and $[\lambda_{\max}]$ , and between values of θ ranging from 0 to 2π.

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 reciprocal-lattice point onto a point on the detector. A ray may contain only a single reciprocal-lattice point hkl with spacing $[d^{*}]$ , in which case the corresponding Laue spot arises from a single wavelength (energy) and structure amplitude, or it may contain several reciprocal-lattice points, such as $[hkl, 2h 2k 2l \ldots nhnknl \ldots]$ , 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 energy-overlap 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 $[d^{*}_{\max}]$ of the crystal increased, more and more Laue spots would be multiple and the energy-overlap 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 energy-overlap problem is evidently of restricted extent. However, the magnitude of the energy-overlap problem varies with resolution: reciprocal-lattice 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 X-ray structure amplitudes from a single Laue spot requires the derivation and application of a wavelength-dependent correction factor known as the wavelength normalization curve or λ-curve. 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 co-prime.

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 co-prime 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, $[V_{v}]$ , is given by $[V_{v} = 0.24\ d^{*4}_{\max} (\lambda_{\max} - \lambda_{\min}),]$ and contains $[N_{v}]$ reciprocal-lattice points where $[N_{v} = V_{v} / V^{*}]$ and $[V^{*}]$ is the volume of the reciprocal unit cell (Moffat, 1997). $[N_{v}]$ can be large, particularly for crystals that diffract to high resolution and thus have larger values of $[d^{*}_{\max}]$ . Laue patterns may therefore contain numerous closely spaced spots and exhibit a spatial-overlap problem (Cruickshank et al., 1991). The value of $[N_{v}]$ 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 $[1/\lambda]$ and $[1/(\lambda + d\lambda )]$ ]. The details are modified slightly for mosaic crystals of finite dimensions subjected to an X-ray 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 $[d^{*}_{\min}]$ (where $[d^{*}_{\min} = 2\sin \theta / \lambda_{\max}]$ and θ is the Bragg angle for that Laue spot) through the Laue spot at $[d^{*}]$ to either $[d^{*}_{\max}]$ or $[2\sin \theta / \lambda_{\min}]$ , 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 $[(\lambda_{\max} - \lambda_{\min})]$ . Decreasing the wavelength range both generates fewer spots and increases the signal-to-noise 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 X-ray exposures of 100 to 150 ps, corresponding to the duration of a single X-ray 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 better-established and more familiar monochromatic techniques, are presented in Table 8.2.2.1.

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Advantages and disadvantages of the Laue technique

This table is adapted from Moffat (1997). See also Ren et al. (1999).

Advantages

Shortest possible exposure time, well suited to rapid time-resolved studies that require high time resolution.

Insensitive to all temporal fluctuations in the beam incident on the crystal, whether arising from the source itself, the optical components of the beamline or the shutter train. (Sensitive only to unusual fluctuations of the shape of the incident spectrum with time.)

All spots in a local region of the detector have an identical profile; none are (geometrically) partial.

Requires a stationary crystal and relatively simple optical components, therefore images are easy to acquire.

A large volume of reciprocal space is surveyed per image, hence fewer images are necessary to survey the entire unique volume.

High redundancy of measurements readily obtained, particularly at high resolution.

Disadvantages

Energy overlaps must be deconvoluted into their components if complete data are to be obtained, particularly at low resolution.

Spatial overlaps are numerous, particularly for mosaic crystals, and must be resolved.

Completeness at low resolution may be low, which would lead to significant series-termination errors in Fourier maps.

The rate of heating owing to X-ray absorption can be very high.

The wider the wavelength range, the higher the background under each spot; a trade-off is unavoidable between coverage of reciprocal space and accuracy of intensity measurements.

Spot shape is quite sensitive to crystal disorder.

More complicated wavelength-dependent corrections must be derived and applied to spot intensities to yield structure amplitudes.

References

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International Tables for Crystallography (2006). Vol. F. ch. 8.2, pp. 167-168