The Space Group

The actual infinite lattices are obtained by parallel translations of the Bravais lattices as unit cells. Some Bravais cells are also primitive cells, others are not. For example, the body-centered cube is a unit cell but not a primitive cell. The primitive cell in this case is an oblique parallelepiped constructed by using as edges the three directed 

There are only two combinations possible for the triclinic system. They are named PI and PI. For the monoclinic system three point groups are to be considered and two lattice types. Combining P and I lattices, on one hand, and point group 2 and symmetry 2 on the other hand, the four possible combinations are P2, P21 72, and 72 The latter two, however, are equivalent only their origins differ. 

All three crystallographic axes are equivalent in the cubic system. The order of listing the symmetry elements is a, [111], [110], When the number 3 appears in the second position, it merely serves to distinguish the cubic system from the hexagonal one. 

The statistical analysis has also been applied separately to the data on inorganic and organic crystals. In both cases the extrapolated estimate for the total number of three-dimensional space groups was smaller than when all data had been considered together. The total 

The description of the symmetry elements of the space groups is similar to that of the point groups [9-19]. The main difference is that the order in which the symmetry elements of the space groups are listed may be of great importance, except for the triclinic system. The order of the symmetry elements expresses their relative orientation in space with respect to the three crystallographic axes. For the monoclinic system, the unique axis may be the c or the h axis. For the P2 space group, the complete symbol may be P112 or 

As we have reached in our discussion the system of the 230 three-dimensional space groups, it appears, as it indeed is, a perfect system. It was 

Ordered number Space group (Hermann-Mauguin) 

The problem of combining the point groups with Bravais lattices to provide a finite number of three-dimensional space groups was worked out independently by Federov and by Schoenflies in 1890. Since the centred cells contain elements of translational symmetry new symmetry elements, not of the point-group type are generated in the process. 

The three-dimensional space groups are characterized by numbers from 1 to 230 and symbols, standardized by the International Union of Crystallography. The Union also publishes International Tables that summarize in detail all of these groups and their interrelationships. 

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Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

The 230 space-group types are listed in full in International Tables for Crystallography, Volume A [48], Whenever crystal symmetry is to be considered, this fundamental tabular work should be consulted. It includes figures that show the relative positions of the symmetry elements as well as details concerning all possible sites in the unit cell (cf. next section). 

Taking into account these symmetry operations together with those corresponding to the translations characteristic of the different lattice types (see Fig. 3.4), it is possible to obtain 230 different combinations corresponding to the 230 space groups which describe the spatial symmetry of the structure on a microscopic... 

In the International Tables of Crystallography, for each of the 230 space groups the list of all the Wyckoff positions is reported. For each of the positions (the general and the special ones) the coordinate triplets of the equivalent points are also given. The different positions are coded by means of the Wyckoff letter, a, b, c, etc., starting with a for the position with the lowest multiplicity and continuing in alphabetical order up to the general position. 

Orthorhombic Space Groups. There are 59 of these space groups divided among three crystal classes 222(D2), mm2(C2l.), and mmm(DVt). Within each class there is at least one group associated with each of the four types of orthorhombic Bravais lattice, / C (or A), F, /. We shall make no attempt to derive these systematically, but a few examples and some useful observations are warranted. The complete list of the 230 space groups given in Appendix VIII should be consulted at this time. 

Lists of equivalent positions for the 230 space groups can be found in International Tables for X-ray Crystallography, a reference series that contains an... 

Space group A group of 230 known symmetry descriptions for the crystalline structures of solid substances. Bloss (1971) and other mineralogy textbooks list the 230 space groups and provide additional details. 

The English physicist William Barlow began as a London business man later he became interested in crystal structures and devoted his life to that study. In 1894, he published his findings of the 230 space groups. It is amazing that from consideration of symmetry three scientists in different countries arrived at the 230 space groups of crystals at about this time. Barlow then worked with ideas of close packing. He pictured the atoms in a crystal as spheres, which, under the influence... 

Obviously, much of the development of crystallography predates the discovery of diffraction of X-rays by crystals. Early studies of crystal structures were concerned with external features of crystals and the angles between faces. Descriptions and notations used were based on these external features of crystals. Crystallographers using X-ray diffraction are concerned with the unit cells and use the notation based on the symmetry of the 230 space groups established earlier. 

Symmetry is the fundamental basis for descriptions and classification of crystal structures. The use of symmetry made it possible for early investigators to derive the classification of crystals in the seven systems, 14 Bravais lattices, 32 crystal classes, and the 230 space groups before the discovery of X-ray crystallography. Here we examine symmetry elements needed for the point groups used for discrete molecules or objects. Then we examine additional operations needed for space groups used for crystal structures. 

Three investigators from different countries independently arrived at the 230 space groups around 1890 ... 

In the site symmetry compilation for the 230 space groups given in Appendix 4, the data are for a primitive cell and can be used directly. 

The 32 crystallographic point groups, first mentioned in Table 7.1, are now described in Table 7.8 (ordered by principal symmetry axes and also by the crystal system to which they belong). The 230 space groups of Schonflies and Fedorov were generated systematically by combining the 14 Bravais lattices with the intra-unit cell symmetry operations for the 32 crystallographic point... 

The monoclinic crystals now are listed with the b axis as the unique axis, but prior to 1940, another popular "setting" used c as the unique axis. Of the 230 space groups, 7 have two choices of unit cell, a primitive rhombohedral one (R) and, for convenience, a nonprimitive hexagonal one (H), with three times the volume of the rhombohedral cell. The 3x3 transformation matrices from rhombohedral (obverse, or positive, or direct) cipbj, Cr to hexagonal axes aH, bur Ch and vice versa are shown in the caption to Fig. 7.17. 

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