A method to reduce N-dimensional lattices and solve N-dimensional Bézout's identities.

20^th May 2022, 14:00
Cyril Cayron
EPFL, Switzerland

Abstract

In order to predict deformation twins in metals and minerals, simple shears on different integral (reticular) planes should be determined. Calculating the shear vectors associated with a given shear plane requires to find a reduced unit cell attached to this plane. Different connected mathematical problems thus emerge: Bézout's identities, integer relations, and cell/lattice reduction. Quite ironically, instead of applying a pre-established lattice reduction method such as LLL, it appeared that simple shearing itself (hyperplanar shearing in dimension N > 3) can be used to reduce the unit cell, and solve the associated N-dimensional Bézout's identity. When applied recursively on different integral planes, hyperplane shearing also permits to reduce N-dimensional lattices by minimizing their metric "rhombicity". The talk will explain the different steps of the method and their geometrical meanings. Some examples with real crystals (N = 3) will be given to explain how the reduction method permits determine the Bravais lattice from diffraction measurements.