International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 4.5, pp. 472-474   | 1 | 2 |

Section 4.5.2.5. Processing diffraction data

R. P. Millanea*

4.5.2.5. Processing diffraction data

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Since the diffraction pattern from a fibre is two-dimensional, it can be collected with a single exposure of a stationary specimen. Diffraction data are collected either on film, which is subsequently scanned by a two-dimensional microdensitometer to obtain a digitized representation of the diffracted intensity, or using an electronic area detector (imaging plate, CCD camera, wire detector etc.) (Fraser et al., 1976[link]; Namba, Yamashita & Vonderviszt, 1989[link]; Lorenz & Holmes, 1993[link]). We assume here that the diffraction pattern is recorded on a flat film (or detector) that is normal to the incident X-ray beam, although other film geometries are easily accommodated (Fraser et al., 1976[link]). The fibre specimen is usually oriented with its axis normal to the incident X-ray beam, although, as is described below, it is sometimes tilted by a small angle to the normal in order to better access reciprocal space close to the meridian. The diffraction and camera geometry are shown in Fig. 4.5.2.1[link]. Referring to this figure, P and S denote the intersections of the diffracted beam with the sphere of reflection and the film, respectively. The fibre, and therefore reciprocal space, is tilted by an angle β to the normal to the incident beam. The angles μ and χ define the direction of the diffracted beam and θ is the Bragg angle. Cartesian and polar coordinates on the film are denoted by (u, v) and [(r, \varphi)], respectively, and D denotes the film-to-specimen distance.

[Figure 4.5.2.1]

Figure 4.5.2.1 | top | pdf |

Fibre diffraction geometry (see text). O is the origin of reciprocal space and OA is normal to the incident X-ray beam. Reciprocal space is rotated about OY so that the Z axis is inclined at an angle β to OA. Q is the projection of P onto the plane containing the incident beam and OY.

Inspection of Fig. 4.5.2.1[link] shows that the cylindrical [(R, \psi, Z)] and spherical [(\rho, \psi, \sigma)] polar coordinates in reciprocal space are related to μ and χ by [\rho = (1/\lambda) [2(1 - \cos \mu - \cos \chi)]^{1/2}, \eqno(4.5.2.44)] [Z = (1/\lambda) [\sin \beta (1 - \cos \mu \cos \chi) + \cos \beta \sin \chi]^{1/2}, \eqno(4.5.2.45)] [R = (\rho^{2} - Z^{2})^{1/2}, \eqno(4.5.2.46)] [\sin \psi = {\sin \mu \cos \chi \over R \lambda} \eqno(4.5.2.47)] and [\tan \sigma = R/Z. \eqno(4.5.2.48)]

The coordinates on the film are related to μ and χ by [u = D \tan \mu \eqno(4.5.2.49)] and [v = D \cos \mu \tan \chi, \eqno(4.5.2.50)] and we also have that [r = D \tan 2 \theta. \eqno(4.5.2.51)]

Use of the above equations allows the reciprocal-space coordinates to be calculated from film-space coordinates, and vice versa. The film coordinates [(u, v)] represent a relatively undistorted map of reciprocal space [(R, Z)], except near the v (vertical) axis of the diffraction pattern. The meridian of reciprocal space does not map onto the film. Inspection of Fig. 4.5.2.1[link] shows that the only point on the meridian that does appear on the film is at [Z = \lambda^{-1} \sin \beta]. The region close to the meridian that appears on the film can therefore be manipulated by adjusting the fibre tilt.

The film-to-specimen distance can be determined by including with the specimen a crystalline power that gives a diffraction ring of known spacing and adjusting the film-to-specimen distance so that the calculated and observed rings coincide. A nonzero fibre tilt leads to differences between the upper and lower halves of the diffraction pattern, and these differences can be used to determine the tilt. This can be done by either calculating the ρ and χ values for several sets of the same reflection above and below the equator and using the relationship [\tan \beta = {\sin \chi_{U} + \sin \chi_{L} \over \lambda^{2} \rho^{2}}, \eqno(4.5.2.52)] where [\chi_{U}] and [\chi_{L}] refer to the upper and lower [(\chi  \lt  0)] reflections (Millane & Arnott, 1986[link]; Lorenz & Holmes, 1993[link]), or by finding the tilt that minimizes the differences between optical densities at the same reciprocal-space coordinates above and below the equator (Fraser et al., 1976[link]). The optical densities may also be corrected for the effects of film (or detector) nonlinearity (Fraser et al., 1976[link]) and the effects of variable absorption owing to the oblique passage of the beam through the film using expressions given by Fraser et al. (1976)[link] and Lorenz & Holmes (1993)[link].

Accurate subtraction of background diffraction is important in order to obtain accurate intensity measurements. One approach to estimating background diffraction is to fit a global background function, usually expanded as a polynomial (Lorenz & Holmes, 1993[link]) or a Fourier–Bessel (Millane & Arnott, 1985[link]) series, to optical densities at a set of points on the diffraction pattern that represent background alone. The background function may or may not be circularly symmetric. The background function is subtracted from the whole diffraction pattern. Another approach, suitable only for Bragg diffraction patterns, is to fit a plane under each reflection, either to the peripheral regions of the reflection or as part of a profile-fitting procedure (Fraser et al., 1976[link]). A different plane is required for each reflection. A third approach, more suitable for continuous diffraction patterns, is to fit a one-dimensional polynomial in angle, for each value of r on the film, possibly as part of a deconvolution procedure (Makowski, 1978[link]). Recently, Ivanova & Makowski (1998[link]) have described an iterative low-pass filtering technique for estimating the background on diffraction patterns from poorly oriented specimens in which there is little space between the layer lines for sampling the background.

A polarization correction is applied to the diffraction pattern, where for unpolarized X-rays (laboratory sources) the polarization factor p is given by (Fraser et al., 1976[link]) [p = (1 + \cos^{2} 2\theta)/2. \eqno(4.5.2.53)] The diffraction pattern is usually mapped into reciprocal space [(R, Z)] for subsequent analysis. The mapping is performed by assigning to the intensity [I (R, Z)] at position [(R, Z)] in reciprocal space the value given by (Fraser et al., 1976[link]) [I (R, Z) = {I(u (R, Z)\hbox{; } v (R, Z)) \over \cos^{3} \mu \cos^{3}\chi}, \eqno(4.5.2.54)] where [I (u\hbox{; } v)] denotes the intensity on the film. The functions [u (R, Z)] and [v (R, Z)] can be derived from the equations given above. Note that equation (4.5.2.54)[link] includes, implicitly, the Lorentz factor.

Subsequent processing depends on whether the diffraction pattern is continuous (i.e. from a noncrystalline specimen) or Bragg (i.e. from a polycrystalline specimen). Diffraction patterns from partially crystalline specimens that contain both components have been analysed using a combination of both approaches (Arnott et al., 1986[link]; Park et al., 1987[link]).

For a diffraction pattern containing continuous diffraction on layer lines, one usually extracts the cylindrically averaged transform [I_{l} (R)] from the intensity [I (R, Z)] on the diffraction pattern mapped into reciprocal space. This involves correcting for the effects of coherence length and disorientation expressed by equation (4.5.2.19[link]), and for the overlap of the smeared layer lines that results from their increasing width with increasing R. The diffracted intensity [I (\rho, \sigma)] in polar coordinates in reciprocal space is equal to the sum of the diffraction [I_{l} (\rho, \sigma)] due to each (overlapping) layer line so that [I (\rho, \sigma) = {\textstyle\sum\limits_{l}} I_{l} (\rho, \sigma). \eqno(4.5.2.55)] Referring to equations (4.5.2.55)[link], (4.5.2.19)[link] and (4.5.2.20)[link] shows that if the smearing due to disorientation dominates over that due to coherence length, then for fixed ρ, equation (4.5.2.55)[link] represents a convolution along σ of the layer-line intensities [I_{l}(\rho \sin \sigma_{l})] with the Gaussian angular profile in equation (4.5.2.19)[link]. By mapping the intensity [I (R, Z)] into polar coordinates as [I (\rho, \sigma)], or by simply sampling [I (R, Z)] for fixed ρ and equally spaced samples of σ, [I_{l} (R)] can be calculated from [I (\rho, \sigma)] by deconvolution, usually by some appropriate solution of the resulting system of linear equations (Makowski, 1978[link]). If the effects of coherence length are significant, as they often are, then equation (4.5.2.55)[link] does not represent a convolution since the width of the Gaussian smearing function depends on σ through equation (4.5.2.20)[link]. However, the problem can still be posed as the solution of a system of linear equations and becomes one of profile fitting rather than deconvolution (Millane & Arnott, 1986[link]). This allows the layer-line intensities to be extracted from the data beyond the resolution where they overlap, although there is a limiting resolution, owing to excessive overlap, beyond which reliable data cannot be obtained (Makowski, 1978[link]; Millane & Arnott, 1986[link]). This procedure requires that [\alpha_{0}] and [l_{c}] be known; these parameters can be estimated from the angular profiles at low resolution where there is no overlap, or they can be determined as part of the profile-fitting procedure.

For a diffraction pattern from a polycrystalline specimen containing Bragg reflections, the intensities [I_{l} (R_{hk})] given by equation (4.5.2.24)[link] need to be extracted from the intensity [I (R, Z)] on the diffraction pattern mapped into reciprocal space. Each composite reflection [I_{l} (R_{hk})] is smeared into a spot whose intensity profile is given by equation (4.5.2.27)[link], and adjacent reflections may overlap. The intensity [I_{l} (R_{hk})] is equal to the intensity [I (R, Z)] integrated over the region of the spot, and the intensity at the centre of a spot is reduced, relative to [I_{l} (R_{hk})], by a factor that increases with the degree of smearing.

The c repeat can be obtained immediately from the layer-line spacing. Initial estimates of the remaining cell constants can be made from inspection of the [(R, Z)] coordinates of low-order reflections. These values are refined by minimizing the difference between the calculated and measured [(R, Z)] coordinates of all the sharp reflections on the pattern.

One approach to measuring the intensities of Bragg reflections is to estimate the boundary of each spot (or a fixed proportion of the region occupied by each spot) and integrate the intensity over that region (Millane & Arnott, 1986[link]; Hall et al., 1987[link]). For spots that overlap, an integration region that is the union of the region occupied by each contributing spot can be used, allowing the intensities for composite spots to be calculated (Millane & Arnott, 1986[link]). This is more accurate than methods based on the measurement of the peak intensity followed by a correction for smearing. Integration methods suffer from problems associated with determining accurate spot boundaries and they are not capable of separating weakly overlapping spots. A more effective approach is one based on profile fitting. The intensity distribution on the diffraction pattern can be written as [I (R, Z) = {\textstyle\sum\limits_{l}} {\textstyle\sum\limits_{h, \,  k}} I_{l} (R_{hk}, R, Z), \eqno(4.5.2.56)] where [I_{l} (R_{hk}, R, Z)] denotes the intensity distribution of the spot [I_{l} (R_{hk})], and the sums are over all spots on the diffraction pattern. Using equation (4.5.2.27)[link] shows that equation (4.5.2.56)[link] can be written as [I (R, Z) = {\textstyle\sum\limits_{l}} {\textstyle\sum\limits_{h, \,  k}} I_{l} (R_{hk}) S (R_{hk}\hbox{; } l/c\hbox{; } R\hbox{; } Z), \eqno(4.5.2.57)] where [S (R_{hk}\hbox{; } l/c\hbox{; } R\hbox{; } Z)] denotes the profile of the spot centred at [(R_{hk}, l/c)] [which can be derived from equation (4.5.2.27)[link]]. Given estimates of the parameters [l_{\rm lat}], [l_{\rm axial}] and [\alpha_{0}], equation (4.5.2.57)[link] can be written as a system of linear equations that can be solved for the intensities [I_{l} (R_{hk})] from the data [I (R, Z)] on the diffraction pattern. The parameters [l_{\rm lat}], [l_{\rm axial}] and [\alpha_{0}], as well as the cell constants and possibly other parameters, can also be refined as part of the profile-fitting procedure using nonlinear optimization.

A suite of programs for processing fibre diffraction data is distributed (and often developed) by the Collaborative Computational Project for Fibre and Polymer Diffraction (CCP13) in the UK (http://www.ccp13.ac.uk/ ) (Shotton et al., 1998[link]).

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