Lattice symmetry

Otwinowski, Z.; Minor, W.

doi:10.1107/97809553602060000677

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

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International Tables for Crystallography (2006). Vol. F. ch. 11.4, p. 227 | 1 | 2 |

Section 11.4.3.1. Lattice symmetry

Z. Otwinowski^a ^* and W. Minor^b

^a UT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390-9038, USA, and ^bDepartment of Molecular Physiology and Biological Physics, University of Virginia, 1300 Jefferson Park Avenue, Charlottesville, VA 22908, USA
Correspondence e-mail: zbyszek@mix.swmed.edu

11.4.3.1. Lattice symmetry

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The relationship between a higher-symmetry cell and the reduced primitive triclinic cell can be described by $[[A] = [M][P], \eqno(11.4.3.5)]$ where [A] and [P] are matrices of the type $[\{a_{0}, b_{0}, c_{0}\}]$ , with [P] representing the reduced triclinic primitive cell, and [M] is one of the 44 matrices listed in IT A .³ If [A] is generated using equation (11.4.3.5) from an experimentally determined [P], owing to experimental errors it will not exactly satisfy the symmetry restraints. DENZO introduced a novel index that helps evaluate the significance of this violation of symmetry. This index is based on the observation that from [A] one can deduce the value of the unit cell, apply symmetry restraints to the unit cell and calculate any matrix [[A']] for the unit cell that satisfies these symmetry restraints. If [A] satisfies symmetry restraints, the matrix [U], where $[[U] = [A][A']^{-1}, \eqno(11.4.3.6)]$ will be unitary and $[[U]^{T} - [U]^{-1} = 0. \eqno(11.4.3.7)]$ The index of distortion printed by DENZO is $[\left\{{\textstyle\sum\limits_{i}} {\textstyle\sum\limits_{j}} \left([U]_{ij}^{T} - [U]_{ij}^{-1}\right)^{2}\right\}^{1/2}\Bigg/6, \eqno(11.4.3.8)]$ where i and j are indices of the $[3 \times 3]$ matrix [U].

The value of this index increases as additional symmetry restraints are imposed, starting from zero for a triclinic cell. Autoindexing in DENZO always finishes with a table of distortion indices for 14 possible Bravais lattices, but does not automatically make any lattice choice.

References

International Tables for Crystallography (2006). Vol. F. ch. 11.4, p. 227