Reduce form and cell mathmatical tools...

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To intelligently and effectively use crystallographic databases, mathematical and computer tools are required that can elucidate diverse types of intra- and interlattice relationships. Two such tools are the normalized reduced form and normalized reduced cell. Practical experience has revealed that the first tool--the normalized reduced form--is very helpful in establishing lattice metric symmetry as it enables one to readily deduce significant relationships between the elements of the reduced form.

Likewise research with crystallographic databases has demonstrated that the second tool--the normalized reduced cell--plays a vital role in determining metrically similar lattices. Knowledge of similar lattices has practical value in solving structures, in assignment of structure types, in materials design, and in nano-technology. In addition to using the reduced cell, it is recommended that lattice-matching strategies based on the normalized reduced cell be routinely carried out in database searching, in data evaluation, and in experimental work.

Key words: identification; lattice-matching strategies; lattice relationships; lattice similarity; metric lattice; normalized reduced cell and form; symmetry. **********

1. Introduction

The various crystallographic databases [1] now available constitute a large, comprehensive, and rapidly growing scientific resource, serving as an invaluable source of data for the intelligent design of materials, for crystal engineering, and for nanotechnology. To evaluate data entering these databases and to intelligently and effectively use this resource, diverse mathematical tools are required that can establish intralattice relationships or elucidate various types of interlattice relationships. Two such tools are the normalized reduced form and the normalized reduced cell--tools that are ideal for elucidating certain types of intra- and interlattice relationships. For example, with the normalized reduced form, one can determine lattice-metric symmetry and deduce other types of intralattice relationships.

With the normalized reduced cell, one can determine metrically similar lattices (1) via lattice matching techniques against the lattices in the crystallographic databases. Practical experience has revealed that these tools are very useful for routine and complex lattice analyses. Before proceeding with applications of these tools, it is necessary to define the normalized reduced cell and form.

1.1 Definitions

The reduced cell is a unique primitive cell of the lattice, which is based on the three shortest lattice translations. For the precise mathematical definition of the reduced cell and form and for procedures to calculate this cell, see [2] and NBS Technical Note 1290 [3].

The normalized reduced cell of a lattice is determined simply by dividing the cell edges of the reduced cell by the a-cell edge. The normalized reduced form is calculated from the normalized reduced cell and is defined by the vector dot products of the normalized reduced cell edge vectors:

a * a b * b c * c

b * c a * c a * b

As an example, consider the reduced cell for a typical triclinic crystal structure reported in the recent literature [4]:

[a.sub.t] = 9.6907[Angstrom] [b.sub.t] = 10.3119[Angstrom] [c.sub.t] = 11.2549[Angstrom]

[[alpha].sub.t] = 63.954[degrees] [[beta].sub.t] = 70.282[degrees] [[gamma].sub.t] = 87.414[degrees]

The corresponding normalized reduced cell and form are: Cell: a = 1.0000 b = 1.0641 c = 1.1614

[alpha] = 63.954[degrees] [beta] = 70.282[degrees] [gamma]...