The relationship between reduced cells in a general Bravais lattice

Gruber, B.

doi:10.1107/s0567739473001063

Cited by 70 publications

(79 citation statements)

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“…To attempt to maintain the relationship between the real and reciprocal axes would lead to instabilities similar to those found for angles. For lattices in which several Buerger reduced cells exist, the sets of lengths of the edges of these alternate cells are the same (Gruber, 1973). The same, of course, holds for the reciprocal lattice.…”

Section: Methodsmentioning

confidence: 59%

“…Gruber (1973). The negative-square-root solution is the same mirrored at 90 ° in either a or fl (however, the values of ), differ).…”

Section: Methodsmentioning

confidence: 95%

“…That is, a,b,c are the lengths of the three shortest non-coplanar vectors in the lattice. However, there may be as many as five alternate sets of angles, hence five Buerger reduced cells (Gruber, 1973). If there are alternate sets of angles, one such Buerger reduced cell may be selected as the reference cell by such criteria as the standardized presentation of the offdiagonal terms of the matrix tensor (the Niggli reduced cell, see Mighell, Santoro & Donnay, 1969) or minimum total surface area of the cell (Gruber, 1978).…”

Section: Introductionmentioning

confidence: 99%

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A perturbation stable cell comparison technique

Andrews¹,

Bernstein²,

Pelletier³

1980

Acta Cryst Sect A

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Section: Methodsmentioning

confidence: 59%

“…Gruber (1973). The negative-square-root solution is the same mirrored at 90 ° in either a or fl (however, the values of ), differ).…”

Section: Methodsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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A perturbation stable cell comparison technique

Andrews¹,

Bernstein²,

Pelletier³

1980

Acta Cryst Sect A

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“…Both cells are primitive. The Niggli cell is unique with regard to its shape (but can occur in different orientations in the lattice) whereas a Buerger cell may be ambiguous (Gruber, 1973). We say that [u,v,x,y,z] …”

Section: Description and Representation Of A Latticementioning

confidence: 99%

“…Thus we have first to ascertain all Buerger cells of the given lattice L. These cells may be found according to Gruber (1973Gruber ( , 1989. Secondly, having a particular Buerger cell B of L, we have to find all Buerger points that follow from B.…”

Section: Appendix B How To Determine All Buerger Points Of a Latticementioning

confidence: 99%

Classification of Lattices: a New Step

Gruber¹

1997

Acta Cryst Sect A

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From the classification of (three-dimensional) lattices into the 14 Bravais types, the finer classifications into the 44 Niggli characters and 24 Delaunay sorts are considered. The last two divisions are mutually incompatible and the Niggli characters show a disturbing 'asymmetry' with respect to the conventional parameters. The aim of the paper is to find a common subdivision of both the Niggli characters and Delaunay sorts that reveals no 'asymmetry'and is crystallographically meaningful. The first attempt based on separating the non-sharp inequalities (<__) into sharp inequalities (<) and equalities (=) in the systems defining the Niggli characters removed only the 'asymmetry', whereas the incompatibility with the Delaunay sorts remained. The second approach may be called the hyperfaces idea. To any lattice there are attached several points in E 5, its Buerger points. These Buerger points lie in two convex five-dimensional hyperpolyhedra J2 +, J2-. The division of lattices into classes is determined by the distribution of their Buerger points along the vertices, edges, faces, three-and fourdimensional hyperfaces and the interior of g2 + and ~Q-. The resulting classes are called genera. There are 127 of them. They form a subdivision of both the Delaunay sorts and the Niggli characters (and, consequently, also of the Bravais types) and their parameter ranges are open. Genera stand for a remarkably strong bond between lattices. The lattices belonging to the same genus agree in a series of important crystallographic properties. Genera are explicitly described by systems of linear inequalities. The five-dimensional geometrical objects obtained in this way are illustrated by their threedimensional cross sections. From these illustrations, a suitable notation of the genera was derived. Extensive tables enable the determination of the genus of a given lattice.

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Crystal Layers: Two‐Dimensional Lattices

Hermann

2011

Crystallography and Surface Structure

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The relationship between reduced cells in a general Bravais lattice

Cited by 70 publications

References 3 publications

A perturbation stable cell comparison technique

A perturbation stable cell comparison technique

Classification of Lattices: a New Step

Crystal Layers: Two‐Dimensional Lattices

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