Niggli reduction and Bravais lattice determination

Abstract

A new algorithm has been developed and coded in DigitalMicrograph (DM) to reduce a three-dimensional primitive cell to the Niggli cell and further convert to the Bravais-lattice unit cell. The core of this algorithm is the calculation of the three shortest non-coplanar vectors to compose the reduced cell. The reduced cell is converted into the real-space reduced cell and then to the Bravais-lattice unit cell. The symmetry-constrained unit cell is, in turn, converted back into the real-space reduced cell, the reciprocal reduced cell and the reciprocal primitive cell. The DM package demonstrates superior numerical stability and can tolerate large uncertainties in the experimentally measured input primitive cell, making it especially suitable for electron diffraction analysis. Additionally, the DM package can be used to calculate various crystallographic parameters including Bravais-lattice plane indices, zone-axis indices, tilt angles and the radius of the high-order Laue zone ring, thus facilitating the correct determination of the Niggli cell and the Bravais lattice.