An algorithm is proposed which enables one, starting from an arbitrary primitive cell of a three-dimensional Bravais lattice, to reach the Niggli form requisite for the lattice type determination.
To answer the question whether all derived (hklO) origins are in fact F faces, a detailed study of the morphology resulting from the bond strength in the calaverite crystal structure is required. This is not a simple task, even if at present one knows the atomic structure of calaverite, because the microscopic structure of the macroscopically fiat faces of an incommensurate crystal is still obscure. Concluding remarksWe still have only a partial understanding of the morphology of calaverite. Nevertheless, the complete indexing of the 92 independent forms of calaverite observed in nature shows the power of the application of the (incommensurate) modulation wave vector as a fourth base vector. The reason for the stability of the satellite faces (see Fig. 3) and the role which the so-called (hkl0) origins plays remain unclear, though the extended classical geometrical laws of crystal morphology seem to hold within a reasonable approximation.Deep thanks are expressed to J. D. H. Donnay with whom this investigation was started. Stimulating discussions with P. Bennema about the role of connected bonds in calaverite are gratefully acknowledged. Thanks are also due to the Stichting ZWO/SON and to the Stichting FOM for partial support of the present investigation. Cryst. 17, 614. DAM, B. (1985). Phys. Rev. Lett. 55, 2806-2809. DAM, B. (1986. PhD thcsis, Univ. of Nijmegen, The Netherlands. DAM, B. & JANNER, A. (1983). Z. Kristallogr. 165, 247-254. DAM, B. & JANNER, A. (1986 AbstractIt is known that the Buerger cell, a + b + c = abs min, is ambiguous.
Two formulae for the number of sublattices of a given index k of an n-dimensional lattice are presented. They are based on the decomposition of the index k into a product of prime numbers and have the form of a rational function of these primes. Compared with other known methods, they can give the result in a much quicker and more comfortable way.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.