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