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

International Tables for Crystallography (2006). Vol. F, ch. 11.1, pp. 209-210   | 1 | 2 |

Section 11.1.3. Fourier analysis of the reciprocal-lattice vector distribution when projected onto a chosen direction

M. G. Rossmanna*

aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA
Correspondence e-mail:

11.1.3. Fourier analysis of the reciprocal-lattice vector distribution when projected onto a chosen direction

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If the members of a set of reciprocal-lattice planes perpendicular to a chosen direction are well separated, then the projections of the reciprocal-lattice vectors onto this direction will have an easily recognizable periodic distribution (Fig.[link]). Unlike the procedure of Kim (1989[link]), which requires the input of a likely zone-axis direction, the present procedure tests all possible directions and analyses the frequency distribution of the projected reciprocal-lattice vectors in each case. Also, unlike the procedure of Kim, the periodicity is determined using an FFT.


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Frequency distribution of the projected reciprocal-lattice vectors for a suitably chosen direction of a diffraction pattern from a fibritin crystal (Tao et al., 1997[link]). Reproduced with permission from Steller et al. (1997[link]).

Let t represent a dimensionless unit vector of a chosen direction. Then, the projection p of a reciprocal-lattice point x onto the chosen vector t is given by [p = {\bf x} \cdot {\bf t}. \eqno(] To apply a discrete FFT algorithm, all such projections of the reciprocal-lattice points onto the chosen direction t are sampled in small increments of p. For the given direction, the values of the projections are in a range between the endpoints [p_{\min}] and [p_{\max}]. If the maximum real cell dimension is assumed to be [a_{\max}], then the maximum number of reciprocal-lattice planes between the observed limits of p is [(\>p_{\max} - p_{\min})/(1/a_{\max})]. Hence, the number of useful grid points along the direction t should be [m = (\>p_{\max} - p_{\min})na_{\max}, \eqno(] where n represents the number of grid points between successive reciprocal-lattice planes and is normally set to 5. Then, the frequency f(p) in the range [p \lt {\bf x} \cdot {\bf t} \lt p + \Delta p] can be given as [f(\>p)\Delta p = f(\>j)], where j is the closest integer to [(\>p - p_{\min})/\Delta p] and [\Delta p = na_{\max}]. Thus, the discrete Fourier transform of this frequency distribution will be given by the summation [F(k) = \sum\limits_{j= 0}^{m}\> f(\>j) \exp (2\pi ikj). \eqno(] The transform is then calculated using a fast Fourier algorithm for all integer values between 0 and m/2 (Fig.[link]). The Fourier coefficients that best represent the periodicity of the frequency distribution will be large. The largest coefficient will occur at k = 0 and correspond to the number of vectors used in establishing the frequency distribution. The next set of large coefficients will correspond to the periodicity that represents every reciprocal-lattice plane. The ratio of this maximum to F(0) will be a measure of the tightness of the frequency distribution around each lattice plane. Subsequent maxima will be due to periodicities spanning every second, third etc. frequency maximum and will thus be progressively smaller (Fig.[link]). The largest F(k) (when k = l), other than F(0), will, therefore, correspond to an interval of d * between reciprocal-lattice planes in the direction of t where d * = l/(namax).


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Fourier analysis of the distribution shown in Fig.[link]. The first maximum, other than F(0), is at k = 27, corresponding to [(1/d^{*})] = 41.9 Å and a value of F (27) = 97.0. Reproduced with permission from Steller et al. (1997[link]).


Kim, S. (1989). Auto-indexing oscillation photographs. J. Appl. Cryst. 22, 53–60.

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