Lattice symmetry determination

Accurate determination of lattice parameters based on Niggli reduced cell theory by using digitized electron diffraction micrograph

In this paper, we used Niggli reduced cell theory to determine lattice constants of a micro/nano crystal by using electron diffraction patterns. The Niggli reduced cell method enhanced the accuracy of lattice constant measurement obviously, because the lengths and the angles of lattice vectors of a primitive cell can be measured directly on the electron micrographs instead of a double tilt holder. With the aid of digitized algorithm and least square optimization by using three digitized micrographs, a valid reciprocal Niggli reduced cell number can be obtained. Thus a reciprocal and real Bravais lattices are acquired. The results of three examples, i.e., Mg₄Zn₇, an unknown phase (Precipitate phase in nickel-base superalloy) and Ba₄Ti₁₃O₃₀ showed that the maximum errors are 1.6% for lengths and are 0.3% for angles.

Coincidence sites for rational lattices

The problem of identifying the coincidence site lattices produced by the disorientations of any rational lattice is investigated. A very simple new procedure for computing the CSLs is presented, based on Grimmer's reciprocity theorem and application of the reduction algorithm introduced by Buerger for finding reduced cells. The point symmetries of a lattice imply that any orientation relationship that produces a CSL can be alternatively expressed by a variety of different rotations. Two simple numerical methods (one based on integral matrix operations and one based on quaternions) are demonstrated for finding all the equivalent rotations that produce the same CSL and for obtaining a single “canonical” rotation as a representative of an equivalence class.

Are crystal polymorphs predictable? The case of sexithiophene

Using sexithiophene as a benchmark compound, we present a very effective strategy for searching the potential energy minima of a crystalline material, described in terms of rigid molecules with Coulombic and atom-atom interactions. The strategy involves uniform sampling of the many-body energy hypersurface, mechanical identification of all constraints deriving from the crystallographic symmetry, and a "sight-resight" method, originally introduced in wildlife ecology, for assessing the completeness of the search. Thousands of distinct potential energy minima, with a surprising variety of structural arrangements, are identified for sexithiophene. Despite the large number of competing minima, the system presents a small number of deep minima, with very different structures and not particularly congested in energy or density. The two deepest minima correspond to the structures of the two known experimental polymorphs, which are satisfactorily described.

Inherent structures of crystalline tetracene

We have systematically sampled the potential energy surface of crystalline tetracene to identify its local minima. These minima represent all possible stable configurations and constitute the "inherent structures" of the system. The crystal is described in terms of rigid molecules with Coulombic and atom-atom interactions. Hundreds of distinct minima are identified, mostly belonging to the space groups P (triclinic) and P2(1)/c (monoclinic), with a variety of structural arrangements. The deepest minimum corresponds to the high temperature-low pressure polymorph. This is the only polymorph with a completely described X-ray structure, which is satisfactorily described by the calculations. The next deep minimum is likely to correspond to the low temperature-high pressure polymorph, which has been experimentally identified but not yet fully described.

Computer programs for unit-cell determination in electron diffraction experiments

A set of computer programs for unit-cell determination from an electron diffraction tilt series and pattern indexing has been developed on the basis of several well-established algorithms. In this approach, a reduced direct primitive cell is first determined from experimental data, in the means time, the measurement errors of the tilt angles are checked and minimized. The derived primitive cell is then checked for possible higher lattice symmetry and transformed into a proper conventional cell. Finally a least-squares refinement procedure is adopted to generate optimum lattice parameters on the basis of the lengths of basic reflections in each diffraction pattern and the indices of these reflections. Examples are given to show the usage of the programs.