These can be found with the following procedure. Letting Rational solutions of equations (11) can be found, if they exist, by substituting for the unknown elements B and B ik rational numbers generated systematically in some convenient way, e.g., by writing Bij = N../n ij' **ij (11) and by assigning to N and n Nij nij all possible integral values. In practical applications these integers can be limited to a small interval, for example (-5, 5), because the relationships of crystallographic interest usually involve simple rational numbers. If two lattices are not related exactly, equation (11) becomes The rational relationship between the two lattices will be acceptable or unacceptable depending on the magnitude of the values of Sij that can be tolerated in a given problem. 2. Applications a. Ambiguities in Determining Unit Cells From Powder Data The In the automatic indexing of powder patterns with the program of Visser (1969), the output consists of four candidate unit cells. algorithm described in the previous section has been used to relate these cells with one another and with the correct unit cell, where Not uncommonly, two or more of the four lattices were found to be in a derivative relationship with each other and/or with the true lattice of the crystal. known. An example has been encountered in the indexing of chromium phosphate hydrate. From a set of observed d-spacings, the best solution found by the indexing algorithm was a = 6.030, b = 11.471, c = 11.711 Å; A; α = a = 94.72, B = 97.72, Y = 99.49° °3 A. with a volume of the unit cell of 787.83 Å3. A single crystal analysis carried out with a Syntex automated diffractometer, on the other hand, gave a' = 6.003, b' = 6.007, c = 23.389 A; a' = 97.04, B' = 92.61, y' = 110.16° with a volume of 782.27 Å3. Both these cells are reduced, and one must A. therefore conclude that they define different lattices. The B-matrix algorithm, applied to the two cells, established that they are related by the transformation Since the determinant of the transformation matrix is unity and since there are fractional elements, the two lattices bear a composite relationship to each other. This example shows that the cell determined by the indexing program is closely related to the correct cell. This is a consequence of the fact that the two lattices, being in a composite relationship, have many d-spacings in common. In this particular case the observed diffraction lines, because of accidental absences, are consistent almost equally well with either cell a or with cell a¦. + i b. Relationship Between Cells Determined From Different Crystals of the Same Species There are lattices in which two or more unit cells are dimensionally similar. This may cause problems in single-crystal work where it is necessary to use two or more individuals of the same material to collect a complete set of data. An example of such a lattice has been reported by DeCamp (1976) and is summarized below. In a study of the structure of Condelphine Hydroiodide, two experimenters, using different crystals, described the crystal lattice by means of the two cells: and a = 9.34, b = 17.39, c = 9.10 A; α = A; 9 Å; a' = 94.85, B a' = 9.32, b' = 17.45, c' = 9.09 94.84, B' = 118.83, y' = 86.50° and the differences in lattice parameters were ascribed, at first, to experimental errors, especially absorption. Subsequent work revealed this not to be the case, because intensities of corresponding reflections differed far more than the experimental errors would justify. application of the B-matrix algorithm immediately established the relation between the two cells which are transformed into one another by means of the expression The In the original paper, reduction theory was used to show that the two cells describe the same lattice and to find the relation between them. The present method, however, has to be preferred in problems of this type, not only because it is more direct, but especially because there could be in the lattice more than two cells with similar parameters which would be revealed immediately by the B-matrix algorithm, but could not be detected by the reduction procedures. c. Relation Between Different Unit Cells of the Same Crystal Determined by an Automated 4-Circle Diffractometer In determining the unit cell of a crystal with a 4-circle diffractometer, there are cases in which accidental or systematic extinctions (as well as other causes) may lead to the choice in reciprocal space of a superlattice of the correct one. An example of this type of error was encountered in the study of the structure of 2,3-Dichloro-hydroxide-4nitro-diphenyl iodonium (C12H6C121 NO3). The solution of this structure was initially attempted by using the data collected on the basis of the unit cell a = 13.595, b = 4.638, c = 10.321 Å, a = y = 90°, B = 81.72° In the last stages of refinement, however, it was found that two different models refined equally well to an R factor of about 8%. This result could be ascribed to disorder in the structure or to an incorrect procedure in collecting the data. By mounting the crystal a second time, the following cell was determined: a' = 15.928, b' = 18.271, c' = 4.623, a' = B' = 90°, y' = 105.58 The B-matrix algorithm immediately revealed that the two cells are related by the transformation This matrix shows that by using the first cell, only half of the data were collected and used in the structure determination. With the second cell, the structure was successfully solved and refined. d. Studies of Related Structures In many problems of crystal chemistry it is important to establish the relationship between two or more structures. In this area the Bmatrix algorithm is particularly useful, as the following example illustrates. In a systematic study of compounds structurally related to palmierite, K2Pb(SO4)2, it was found that the powder pattern of the compound RbPb (Mn02) 2' although similar to that of palmierite, could not be indexed on the basis of a hexagonal cell theoretically derived from chemical considera tions and values of the ionic radii. The lattice of Rb2Pb(MnO2)2 was found to be monoclinic instead of reduced form This reduced form shows more specialization than the one required by a monoclinic C-centered cell, indicating that the lattice has derivative lattices of symmetry higher than monoclinic. The hexagonal lattice parameters of the Rb compound predicted on the basis of a preliminary analysis of the powder pattern and of the ionic radii involved, are [the parameters of palmierite reported in NBS Monograph 25 (Morris, McMurdie, Evans, Paretzkin, deGroot, Hubburd, and Carmel, 1976) are a = 5.4950(6), c = 20.849(4)] which gives the reduced form The B-matrix algorithm showed that the monoclinic lattice can be obtained from the hexagonal form with the transformations This indicates that the monoclinic lattice can be oriented in space in three different ways with respect to the hexagonal lattice, i.e., it is possible to mutually orient three monoclinic individuals so that a common hexagonal sublattice propagates from one to the other with little or no disturbance. These are the conditions required to have twinning (*) The transformation to obtain the conventional C-centered monoclinic cell from the reduced cell is (**) (I20/100/001) The transformation to obtain the conventional cell from the reduced cell is (100/110/113) |