InternationalCrystallography of biological macromoleculesTables for Crystallography Volume F Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 11.1, pp. 209-210
## Section 11.1.3. Fourier analysis of the reciprocal-lattice vector distribution when projected onto a chosen direction |

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

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. 11.1.3.1). Unlike the procedure of Kim (1989), 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.

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 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 and . If the maximum real cell dimension is assumed to be , then the maximum number of reciprocal-lattice planes between the observed limits of *p* is . Hence, the number of useful grid points along the direction **t** should be 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 can be given as , where *j* is the closest integer to and . Thus, the discrete Fourier transform of this frequency distribution will be given by the summation The transform is then calculated using a fast Fourier algorithm for all integer values between 0 and *m*/2 (Fig. 11.1.3.2). 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. 11.1.3.2). 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*/(*na*_{max}).

### References

Kim, S. (1989).*Auto-indexing oscillation photographs. J. Appl. Cryst.*

**22**, 53–60.