Section 11.3.6.1. Determination of the Bravais lattice

The determination of possible Bravais lattices is based upon the concept of the reduced cell whose metric parameters characterize 44 lattice types as described in Part 9 of IT A. A primitive basis of a given lattice is defined there as a reduced cell if it is right-handed and if the components of its metric tensor satisfy a number of conditions (inequalities). The main conditions state that the basis vectors are the shortest three linear independent lattice vectors with either all acute or all non-acute angles between them. As specified in IT A, each of the 44 lattice types is characterized by additional equality relations among the six components of the reduced-cell metric tensor. As an example, for lattice character 13 (Bravais type oC) the components of the metric tensor of the reduced cell must satisfy

Any primitive triclinic cell describing a given lattice can be converted into a reduced cell. It is well known, however, that the reduced cell thus derived is sensitive to experimental error. Hence, the direct approach of first deriving the correct reduced cell and then finding the lattice type is unstable and may in certain cases even prevent the identification of the correct Bravais lattice.

A suitable solution of the problem has been found that avoids any decision about what the `true' reduced cell is. The essential requirements of this procedure are: (a) a database of possible reduced cells and (b) a backward search strategy that finds the best-fitting cell in the database for each lattice type.

The database is derived from a seed cell which strictly satisfies the definitions for a reduced cell. All cells of the same volume as the seed cell whose basis vectors can be linearly expressed in terms of the seed vectors by indices or +1 are included in the database. Each unit cell in the database is considered as a potential reduced cell even though some of the defining conditions as given in Part 9 of IT A may be violated. These violations are treated as being due to experimental error.

The backward search strategy starts with the hypothesis that the lattice type is already known and identifies the best-fitting cell in the database of possible reduced cells. Contrary to a forward directed search, it is now always possible to decide which conditions have to be satisfied by the components of the metric tensor of the reduced cell. The total amount by which all these equality and inequality conditions are violated is used as a quality index. This measure is defined below for lattice type 13 oC testing a potential reduced cell from the database for agreement. Positive values of the quality index indicate that some conditions are not satisfied. All potential reduced cells in the database are tested and the smallest value for is assigned to lattice type 13. This test is carried out for all 44 possible lattice types using quality indices derived in a similar way from the defining conditions as listed in Part 9 of IT A. For each of the 44 lattice types thus tested, the procedure described here returns the quality index, the conventional cell parameters and a transformation matrix relating original indices with respect to the seed cell to the new indices with respect to the conventional cell. These index-transformation matrices are derived from those given in Table 9.2.5.1 in IT A.

The results obtained by this method are shown in Table 11.3.6.1 for the example of a 1.5° oscillation data film containing 1313 strong diffraction spots which were located automatically. The space group of the crystal is and the cell constants are . The entry for the correct Bravais lattice oC with derived cell constants close to the true ones has a low value for its quality index and thus appears as a possible explanation of the observed diffraction pattern.

Table 11.3.6.1| top | pdf | Rating of lattice types implied by a given reduced cell Lattice type Quality index Conventional cell constants (Å, °) Reindexing transformation a b c α β γ 1 cF 999.0 119.3 137.3 119.1 121.1 77.5 122.6 2 hR 770.1 74.6 111.7 137.4 103.6 89.1 108.8 3 cP 769.7 62.1 63.5 92.9 90.0 90.1 107.2 5 cI 936.0 111.7 74.6 112.6 70.1 53.6 71.2 4 hR 769.5 101.1 111.9 119.1 116.5 89.1 116.4 6 tI 999.0 112.6 111.7 74.6 71.2 70.1 53.6 7 tI 999.0 111.7 74.6 112.6 70.1 53.6 71.2 8 oI 999.0 74.6 111.7 112.6 53.6 70.1 71.2 9 hR 772.7 62.1 74.6 296.6 90.5 105.8 125.5 10 mC 24.0 101.1 74.6 92.9 90.1 90.0 91.3 11 tP 174.8 62.1 63.5 92.9 90.0 90.1 107.2 12 hP 122.8 62.1 63.5 92.9 90.0 90.1 107.2 13 oC 23.8 74.6 101.1 92.9 90.0 90.1 88.7 15 tI 672.7 62.1 63.5 200.2 77.0 77.6 107.2 16 oF 999.0 74.6 101.1 200.2 90.5 111.8 88.7 14 mC 23.4 74.6 101.1 92.9 90.0 90.1 88.7 17 mC 999.0 101.1 74.6 111.7 71.2 116.4 88.7 18 tI 999.0 112.6 119.1 62.1 68.7 99.5 115.4 19 oI 999.0 62.1 112.6 119.1 64.6 68.7 80.5 20 mC 746.3 112.6 112.6 62.1 99.5 99.6 111.3 21 tP 748.0 63.5 92.9 62.1 90.1 107.2 90.0 22 hP 999.0 63.5 92.9 62.1 90.1 107.2 90.0 23 oC 747.8 112.6 112.6 62.1 80.5 99.6 68.7 24 hR 999.0 154.8 112.6 62.1 80.5 80.9 84.3 25 mC 746.1 112.6 112.6 62.1 80.5 99.6 68.7 26 oF 624.9 62.1 123.9 195.9 86.4 108.4 101.5 27 mC 499.7 123.9 62.1 112.6 80.5 119.7 78.5 28 mC 325.0 62.1 195.9 63.5 95.4 107.2 71.6 29 mC 99.8 62.1 123.9 92.9 90.0 90.1 78.5 30 mC 336.4 63.5 196.5 62.1 95.4 107.2 71.1 31 aP 0.2 62.1 63.5 92.9 90.0 89.9 72.8 32 oP 152.0 62.1 63.5 92.9 90.0 90.1 107.2 40 oC 413.0 63.5 196.4 62.1 84.5 107.2 108.9 35 mP 151.8 63.5 62.1 92.9 90.1 90.0 107.2 36 oC 400.3 62.1 195.9 63.5 84.6 107.2 108.4 33 mP 151.2 62.1 63.5 92.9 90.0 90.1 107.2 38 oC 100.1 62.1 123.9 92.9 90.0 90.1 101.5 34 mP 1.0 62.1 92.9 63.5 90.0 107.2 90.1 42 oI 661.3 62.1 63.5 200.2 103.0 102.4 107.2 41 mC 412.2 196.4 63.5 62.1 107.2 95.5 71.1 37 mC 400.1 195.9 62.1 63.5 107.2 95.4 71.6 39 mC 99.9 123.9 62.1 92.9 90.1 90.0 78.5 43 mI 999.0 74.6 200.2 63.5 103.0 127.3 68.2 44 aP 0.0 62.1 63.5 92.9 90.0 90.1 107.2