Symmetry of semi-reduced lattices

Abstract

The main result of this work is extension of the famous characterization of Bravais lattices according to their metrical, algebraic and geometric properties onto a wide class of primitive lattices (including Buerger-reduced, nearly Buerger-reduced and a substantial part of Delaunay-reduced) related to low-restricted semi-reduced descriptions (s.r.d.'s). While the `geometric' operations in Bravais lattices map the basis vectors into themselves, the `arithmetic' operators in s.r.d. transform the basis vectors into cell vectors (basis vectors, face or space diagonals) and are represented by matrices from the set {\bb V} of all 960 matrices with the determinant ±1 and elements {0, ±1} of the matrix powers. A lattice is in s.r.d. if the moduli of off-diagonal elements in both the metric tensors M and M(-1) are smaller than corresponding diagonal elements sharing the same column or row. Such lattices are split into 379 s.r.d. types relative to the arithmetic holohedries. Metrical criteria for each type do not need to be explicitly given but may be modelled as linear derivatives {\bb M}(p,q,r), where {\bb M} denotes the set of 39 highest-symmetry metric tensors, and p,q,r describe changes of appropriate interplanar distances. A sole filtering of {\bb V} according to an experimental s.r.d. metric and subsequent geometric interpretation of the filtered matrices lead to mathematically stable and rich information on the Bravais-lattice symmetry and deviations from the exact symmetry. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from analysis of a lattice metric to analysis of symmetry matrices [Himes & Mighell (1987). Acta Cryst. A43, 375-384], (ii) from the isometric approach and invariant subspaces to the orthogonality concept {some ideas in Le Page [J. Appl. Cryst. (1982), 15, 255-259]} and splitting indices [Stróż (2011). Acta Cryst. A67, 421-429] and (iii) from fixed cell transformations to transformations derivable via geometric information (Himes & Mighell, 1987; Le Page, 1982). It is illustrated that corresponding arithmetic and geometric holohedries share space distribution of symmetry elements. Moreover, completeness of the s.r.d. types reveals their combinatorial structure and simplifies the crystallographic description of structural phase transitions, especially those observed with the use of powder diffraction. The research proves that there are excellent theoretical and practical reasons for looking at crystal lattice symmetry from an entirely new and surprising point of view - the combinatorial set {\bb V} of matrices, their semi-reduced lattice context and their geometric properties.

Symmetry of hR and Pseudo-hR Lattices

“…The concept of a semi-reduced lattice description (s.r.d.) has been given elsewhere [9]. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from the analysis of a lattice metric to the analysis of symmetry matrices [6], (ii) from the geometric interpretation of isometric transformation based on invariant subspaces to the orthogonality concept [7] extended to splitting indices [8], (iii) and from predefined cell transformations to transformations derivable via geometric information [6,7].…”

“…Cell parameters were recalculated to the primitive form, which was not Niggli. The strict symmetry had geometric description 2 [1][2][3][4][5][6][7][8][9][10](1-10). Therefore, it was assumed that composite deformation ε 1• (1-10) + ε 2• (110) brings these monoclinic cells to the rhombohedral ones.…”

“…Classifications of unique lattice representatives obtained by the Niggli reduction or Delaunay reduction are commonly used techniques to assign the Bravais symmetry to a given lattice. Another approach, called the matrix method, directly derives isometric transformations from the lattices by B-matrices, which transform a lattice onto itself [1,5,6], or by the space distribution of orthogonalities [7], or by filtering predefined set V of 480 potential symmetry matrices [8,9]. The latter technique is applicable to a wide class of semi-reduced lattice descriptions, additionally forced by a geometric interpretation of symmetry operations.…”