Laue geometry: sources, optics, sample reflection bandwidth and spot size

Helliwell, J. R.

doi:10.1107/97809553602060000669

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

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International Tables for Crystallography (2006). Vol. F. ch. 8.1, p. 162 | 1 | 2 |

Section 8.1.7.1. Laue geometry: sources, optics, sample reflection bandwidth and spot size

J. R. Helliwell^a ^*

^aDepartment of Chemistry, University of Manchester, M13 9PL, England
Correspondence e-mail: john.helliwell@man.ac.uk

8.1.7.1. Laue geometry: sources, optics, sample reflection bandwidth and spot size

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Laue geometry involves the use of the 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 $[\sim \!\!\lambda_{c}/5]$ , where $[\lambda_{c}]$ 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 ~2.5 Å.

The bandwidth can be limited somewhat under special circumstances. A reflecting mirror at grazing incidence can be used for two purposes. 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 a transmission mirror (Lairson & Bilderback, 1982; 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:

(i) use of a monochromatic reference data set;
(ii) use of symmetry equivalents in the Laue data set recorded at different wavelengths;
(iii) 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. The bromine and silver K absorption edges, in AgBr photographic film, lead to discontinuities in the λ-curve. Hence, the λ-curve is usually split up into wavelength regions, for example $[\lambda_{\min}]$ to 0.49 Å, 0.49 to 0.92 Å, and 0.92 Å to $[\lambda_{\max}]$ . Other detector types have different discontinuities, depending on the material making up the X-ray absorbing medium. Most Laue diffraction data now recorded on CCDs or IPs. The greater sensitivity of these detectors (expressed as the DQE), especially for weak signals, has greatly increased the number of Laue exposures recordable per crystal. Thus, multiplet deconvolution procedures, based on the recording of reflections stimulated at different wavelengths and with different relative intensities, have become possible (Campbell & Hao, 1993; Ren & Moffat, 1995b). Data quality and completeness have improved considerably.

The production and use of narrow-bandpass beams, e.g. $[\delta \lambda/\lambda \leq 0.2]$ , may be of interest for enhancing the signal-to-noise ratio. Such bandwidths can be produced by a combination of a reflection mirror used in tandem with a transmission mirror. Alternatively, an X-ray undulator of 10–100 periods should ideally yield a bandwidth behind a pinhole of $[\delta \lambda/\lambda \simeq]$ 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; nevertheless, high-order Chebychev polynomials are successful (Ren & Moffat, 1995a).

The spot bandwidth is determined by the mosaic spread and horizontal beam divergence (since $[\gamma_{H} \gt \gamma_{V}]$ ) as $[(\delta \lambda/\lambda) = \left(\eta + \gamma_{H}\right)\hbox{cot } \theta, \eqno(8.1.7.1) ]$ where η is the sample mosaic spread, assumed to be isotropic, $[\gamma_{H}]$ is the horizontal cross-fire angle, which in the absence of focusing is $[\left(x_{H} + \sigma_{H} \right)/P]$ , where $[x_{H}]$ is the horizontal sample size, $[\sigma_{H}]$ is the horizontal source size and P is the sample to the tangent-point distance. This is similar for $[\gamma_{V}]$ in the vertical direction. Generally, at SR sources, $[\sigma_{H}]$ is greater than $[\sigma_{V}]$ . When a focusing-mirror element is used, $[\gamma_{H}]$ and/or $[\gamma_{V}]$ are convergence angles determined by the focusing distances and the mirror aperture.

The size and shape of the diffraction spots vary across the detector image plane. The radial spot length is given by convolution of Gaussians as $[\left(L_{R}^{2} + L_{c}^{2} \sec^{2} 2\theta + L_{\rm PSF}^{2} \right)^{1/2} \eqno(8.1.7.2)]$ and tangentially by $[\left(L_{T}^{2} + L_{c}^{2} + L_{\rm PSF}^{2}\right)^{1/2}, \eqno(8.1.7.3) ]$ where $[L_{c}]$ is the size of the X-ray beam (assumed to be circular) at the sample, $[L_{\rm PSF}]$ is the detector point spread factor, $[\eqalignno{L_{R} &= D\sin \left(2\eta + \gamma_{R} \right)\sec^{2}2\theta, &(8.1.7.4)\cr L_{T} &= D\left(2\eta + \gamma_{T} \right)\sin \theta \sec 2\theta, &(8.1.7.5)}%(8.1.7.5)]$ and $[\eqalignno{\gamma_{R} &= \gamma_{V} \cos \psi + \gamma_{H} \sin \psi, &(8.1.7.6)\cr \gamma_{T} &= \gamma_{V} \sin \psi + \gamma_{H} \cos \psi, &(8.1.7.7)}%(8.1.7.7)]$ where 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 $[L_{c}]$ and $[L_{\rm PSF}]$ . For a mosaic or radiation-damaged crystal, the main effect is a radial streaking arising from η, the sample mosaic spread.

References

Andrews, S. J., Hails, J. E., Harding, M. M. & Cruickshank, D. W. J. (1987). The mosaic spread of very small crystals deduced from Laue diffraction patterns. Acta Cryst. A43, 70–73.Google Scholar

Campbell, J. W. & Hao, Q. (1993). Evaluation of reflection intensities for the components of multiple Laue diffraction spots. II. Using the wavelength-normalization curve. Acta Cryst. A49, 889–893.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

Lairson, B. M. & Bilderback, D. H. (1982). Transmission X-ray mirror – a new optical element. Nucl. Instrum. Methods, 195, 79–83.Google Scholar

Ren, Z. & Moffat, K. (1995a). Quantitative analysis of synchrotron Laue diffraction patterns in macromolecular crystallography. J. Appl. Cryst. 28, 461–481.Google Scholar

Ren, Z. & Moffat, K. (1995b). Deconvolution of energy overlaps in Laue diffraction. J. Appl. Cryst. 28, 482–493.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 8.1, p. 162