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, p. 210   | 1 | 2 |

Section 11.1.4. Exploring all possible directions to find a good set of basis vectors

M. G. Rossmanna*

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

11.1.4. Exploring all possible directions to find a good set of basis vectors

| top | pdf |

The polar coordinates ψ, ϕ will be used to define the direction t, where ψ defines the angle between the X-ray beam and the chosen direction t. The Fourier analysis is performed for each direction t in the range [0 \lt \psi \leq \pi/2, 0 \lt \varphi \leq 2\pi]. A suitable angular increment in ψ was determined empirically to be about 0.03 rad (1.7°). For each value of ψ, the increment in ϕ is taken to be the closest integral value to [(2\pi \sin \psi)/0.03]. This procedure results in ~7300 separate, roughly equally spaced, directions.

For each direction t, the distribution of the corresponding F(k) coefficients is surveyed to locate the largest local maximum at k = l. The ψ and ϕ values associated with the 30 largest maxima are selected for refinement by a local search procedure to obtain an accuracy of 10−4 rad (~0.006°). If the initial angular increment (0.03 rad) used for the hemisphere search was reduced, then it would not be necessary to refine quite as many local maxima. However, to increase the efficiency of the search procedure, the ratio of angular increments to the number of refined positions was chosen to minimize the total computing time. The F (l) values of the refined positions are then sorted by size. Directions are chosen from these vectors to give a linearly independent set of three basis vectors of a primitive real-space unit cell. These are then converted to the basis vectors of the reciprocal cell. The components of the three reciprocal-cell axes along the three camera axes are the nine components of the crystal orientation matrix [A] ([link]).

The final step in the selection of the best [A] matrix is to choose various nonlinear combinations of the refined vectors that have the biggest F (l) values. That set of three vectors which gives the best indexing results is then chosen to represent the crystal orientation matrix [A]. A useful criterion is to determine the nonintegral Miller indices h′ from ([link]) using the [A] matrix and the known reciprocal-lattice vectors x. Any reflection for which any component [|{\bf h} - {\bf h}'|] is bigger than, say, 0.2 is rejected. The best [A] matrix is chosen as the one with the least number of rejections. In most cases, the best combination corresponds to taking the three largest F (l) values.

The program goes on to determine a reduced cell from the cell obtained by the above indexing procedure (Kim, 1989[link]). The reduced cell is then analysed in terms of the 44 lattice characters (Burzlaff et al., 2005[link]; Kabsch, 1993[link]) in order to evaluate the most likely Bravais lattice and crystal system.


Burzlaff, H., Gruber, B., Zimmermann, H. & de Wolff, P. M. (2005). Crystal lattices. In International tables for crystallography, Vol. A. Space-group symmetry, edited by Th. Hahn, Part 9. Heidelberg: Springer.
Kabsch, W. (1993). Automatic processing of rotation diffraction data from crystals of initially unknown symmetry and cell constants. J. Appl. Cryst. 26, 795–800.
Kim, S. (1989). Auto-indexing oscillation photographs. J. Appl. Cryst. 22, 53–60.

to end of page
to top of page