IS-1141 Chemistry (UC-4)
TID-4500, July 1, 1965
UNITED STATES ATOMIC ENERGY COMMISSION
Research and Development Report
THE REDUCED CELL AND ITS
CRYSTALLOGRAPHIC APPLICATIONS
by
Stephen L. Lawton and Robert A. Jacobson
April 1965
Ames Laboratory
at
Iowa State University of Science and Technology
F.H. Spedding, Director
Contract W-7405 eng-82
IS-1141
THE REDUCED CELL AND ITS CRYSTALLOGRAPHIC APPLICATIONS
Stephen L. Lawton and Robert A. Jacobson
ABSTRACT
This report describes the reduced cell and its applications to
structural crystallography. Typical applications which are discussed
are its use as the standard choice of the unit cell in a triclinic lattice
and the use of its scalars in identifying in a lattice the cell of highest
symmetry. The report also describes two FORTRAN computer pro-
grams which may be used to locate the reduced cell in a lattice, to
calculate its parameters and to derive the matrix for the transformation
of the original cell to the reduced cell.
I. INTRODUCTION
Preliminary investigations of a crystalline substance usually
begin with an identification of its crystal symmetry, such as its crystal
system, lattice type, space group and cell parameters. Such informa-
tion may be obtained either by powder or single crystal X-ray diffraction
techniques. Once the symmetry has been established, however, it is
sometimes desirable to locate and identify its REDUCED CELL as well.
The "true" reduced cell is defined as that cell whose axes are the
three shortest non-coplanar translations in the lattice; consequently
there is only one such cell in any one lattice. It is, by convention, the
standard choice for the triclinic cell. But more important, this cell,
and the method of finding it, provides a simple direct method for
identifying and locating in the lattice the cell of highest symmetry,
starting from any cell in any arbitrary orientation. This fact along im-
mediately suggests two useful applications: its use in powder work in
assigning the cell of highest symmetry to a pattern indexed in the tri-
clinic system by a method such as that due to Ito, and for the alignment
of single crystals on a single crystal stage in which the crystals are
mounted in a completely random orientation. It also serves as a
"fingerprint" and can thus be used not only for comparing two crystalline
forms of a compound for similarities in their lattice but can also be used
to verify whether or not two crystalline forms actually correspond to the
same compound. Furthermore, by the same reasoning, any two or
more cells in a lattice may be linked together via the reduced cell since
the same reduced cell can always be found regardless of the starting
point. The reduced cell is thus an important one; consequently the
method of finding it, as well as a discussion of its applications, is the
purpose of this report and is fully discussed with the aid of detailed
examples.
The concluding portion of this report describes two computer pro-
grams written in full Fortran for the IBM 7074 computer. The first of
the two programs, RCELL, is strictly a cell reduction program which
obtains the reduced cell by the method discussed in this report. The
second, TRACER, is an expanded version of RCELL and may be used
not only for obtaining the reduced cell, but also for general cell trans-
formations as well as matrix multiplication and matrix inversion of
3 x 3 transformation matrices.
E. APPLICATIONS
Once a unit cell has been identified from single crystal or powder
diffraction data, one question often asked is whether or not a unit cell
of symmetry higher than the observed one actually exists and if so what
is it, how is it oriented relative to the observed one, and what are its
dimensions? The reduced cell and the method of finding it provides the
answer.
1. THE 43 REDUCED CELLS
In 1928 P. Niggli showed that there are only 43 unique reduced cell
types. He showed that by considering all the possible combinations of axial
lengths and interaxial angles in the fourteen Bravais lattices, there re-
sult just these 43 cells whose axes correspond to the three shortest
non-coplanar translations in the lattice.
TRACER
1. GENERAL INFORMATION
Program TRACER, written in IBM 7074 Fortran language, is an
expanded version of RCELL. It is a computer program for general cell
transformations in direct space (using matrices supplied by the user),
for cell reductions only or for general cell transformations followed by
cell reduction. Typical examples of its uses are
1. Transformation of lattice axes in direct space and reciprocal
space (1/A) from an old cell to a new cell, e.g., Monoclinic P21/n to
P21/c, using a transformation matrix supplied by the user.
2. Transformation of a primitive triclinic or monoclinic cell to its
reduced cell.
3. Reduction of primitive monoclinic cells, using the cell reduction
technique incorporated in the program, to locate a better monoclinic cell
with shorter and more orthogonal axes.
4. Two or more transformations in sequence, using matrices sup-
plied by the user, to transform each cell consecutively to the next cell
and to calculate the cell parameters of each intermediate cell and the
final cell, e.g., F-triclinic to P-triclinic to I-orthorhombic.
5. Two or more transformations in sequence, transforming the
first N cells to new cells using matrices supplied by the user and then
letting the program transform the Nth cell to its reduced cell, e.g., F-
triclinic to P-triclinic to reduced cell.
6. Matrix multiplication of two or more transformation matrices.
The tabulated transformations of lattices for the triclinic and mono-
clinic systems appearing in Appendix VII cover transformations frequently
encountered in crystallography and may be used in routine work with the
program. Suppose, for instance, one has a face-centered triclinic cell
and it is desired to obtain the reduced cell. Before the reduced cell can
be found the face-centered cell must first be converted to a primitive
cell. Being unique, the program is always able to locate it starting with
any arbitrary primitive triclinic cell in the lattice. The same reduced
cell is always obtained, as will the matrix for the transformation of the
original centered cell to the reduced cell, regardless of the intermediate
primitive cell. The transformations of pages 174 - 175 will be found
particularly useful in this regard for obtaining such intermediate primi-
tive cells at this step. (Note that they may be used on any centered cell
belonging to any one of the seven crystal systems.)
The essential input consists of the six lattice parameters (real or
reciprocal), the matrices to be used for the consecutive transformations
of cells which will not be reduced by the program as well as any
alphanumeric information identifying each cell. the reduced cell, it
desired, does not have to be obtained directly from the original cell but
may be obtained from a cell previously obtained by other transformations
(see No. 5 above); it must, however, be the last cell in any sequence of
transformations, that is, after the reduced cell is obtained "by the pro-
gram" one may not transform the cell to a new cell without reloading the
program. A maximum of eight consecutive transformations may be ap-
plied to any one original cell. Provision has been made for allowing
more than one compound to be run without reloading the program.
The output consists of the matrices used and generated for the
lattice axes in direct space, the lattice parameters (real and reciprocal)
of the original cell, all intermediate cells and the final cell, and the
sine and cosine values of all angles in each cell. In the case of the
reduced cell the program prints out two cells, as in the case of program
RCELL, the second being just as rearrangement of the first and cor-
responds to the convention established for the triclinic reduced cell,
namely, that cell whose edges are the three shortest non-coplanar
translations in the lattice, labeled so as to have c