Figure.1. The conventional unit cells (thick black outline) of the 14 Bravais lattices. [crystallographic symmetry] 1

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1 [crystallographic symmetry] The crystallographic space groups. Supplementary to { 9.6:324} In the 3-D space there are 7 crystal systems that satisfy the point (e.g., rotation, reflection and inversion) and translational invariance of lattices underlying all crystal structures. They are the systems which are defined by symmetry sets displaying one 1-fold axis (triclinic), one 2-fold axis parallel to the unique b! -axis (monoclinic), three mutually perpendicular 2-fold axes (orthorhombic), one 4-fold axis parallel to the unique c! -axis (tetragonal), four 3-fold axes along space diagonals (cubic), one 3-fold axis (trigonal), and one 6-fold axis (hexagonal). After considering the distinct unit cells embeddable within the 7 crystal systems with possible added lattice points called centering, a total of 14 Bravais lattices are obtained. They are: triclinic-p, monoclinic-p and -B (1 st setting), orthorhombic-p, -I, and -F, tetragonal-p and I, hexagonal-p, trigonal-r, and cubic-p, I, and F, where P, I, F, and A (equivalently B and C, not shown) stand for primitive, body-centering, face-centering, and one-face centering, respectively. R denotes the rhombohedral unit-cell representation of the trigonal system. The conventional unit cells of the 14 Bravais lattices are shown in Figure 1. Figure.1. The conventional unit cells (thick black outline) of the 14 Bravais lattices. Crystallographic point symmetry operations refer to rotations, reflections and inversion (and any combination of them) with respect to a point that often is called the origin, which leave the lattice invariant. This implies that only 1-, 2-, 3-, 4, and 6-fold rotations are possible. The point [crystallographic symmetry] 1

2 symmetry operations can be categorized into 32 sets that satisfy the mathematical definition of group, hence 32 crystallographic point groups there are. A crystallographic space group, on the other hand, must include additional symmetry operations as well as the primitive lattice translations that leave the 3-dimensional lattice invariant. Primitive translations, represented by Seitz operators, in general include fractional (smaller than the dimensions of a primitive lattice) translation coupled with a rotation or reflection. These are known as screw rotation and glide reflection operations. There are 230 crystallographic space groups in total: 73 of them completely definable by point symmetry operations and unit-cell translations are called symmorphic space groups; the rest of them, called nonsymmorphic space groups, necessarily include in addition screw rotation and/or glide reflection operations involving fractional unit-cell translations. All important details and definitions regarding crystallography space groups can be found in the authoritative Volume A of the International Tables for Crystallography, edited by Theo Hahn for the International Union of Crystallography (IUCr). Study materials on various crystallographic topics are available from the IUCr website, Figure 2 (a and b) shows a typical page of a space group in the International Tables. Brief annotations of the key entries in a table are given in the bottom panels. We do not comment on the subgroups and supergroups of crystallographic space groups. In the following subsections we remark on a few issues that are relevant to crystal structure determination by means of diffraction. [crystallographic symmetry] 2

3 Figure 2a. A typical starting page of a crystallographic space group in the International Table (top panel) and explanations of various entries (bottom panel). [crystallographic symmetry] 3

4 Figure 2b. The continuing page of the P 4 / n crystallographic space group in the International Table (top panel) and explanations of various entries (bottom panel). Further listing of the subgroups and supergroups as well as the pages for a second choice of origin are omitted. [crystallographic symmetry] 4

5 Systematic absences The presence of translational symmetry in a lattice may cause the diffracted intensity of reflections having certain combination of Miller indices to extinct, which crystallographers called systematic absences. One obvious example is the systematic absences due to centering. For example, in a face-centered cubic lattice every atom has a symmetrically equivalent position shifted by (0, 1/2, 1/2), (1/2, 0, 1/2), or (1/2, 1/2, 0). The unit cell has a volume 4 times of that of the primitive cell and renders many of the hkl reflections to be extinct. The systematic absences resulted from different centering are given in Table 1. Note that the International Tables list the conditions of allowed reflections. Systematic absences caused by glide planes and screw axes are less straightforward to obtain. Table 1. Systematic absences in different Bravais lattices caused by centering. Bravais lattice P Extinct reflections None I h + k + l = 2n +1 F h + k = 2n +1 or k +1= 2n +1 or h + l = 2n +1 A k + l = 2n +1 B h + l = 2n +1 C h + k = 2n +1 R!h + k + l = 3n +1 and 3n + 2 All the space groups that produce the same rules of systematic absences are collectively called a diffraction group, which usually contains only a few members. Therefore, careful analysis of the systematic absences in the diffraction patterns is an important step in determination of crystal structures because it significantly narrows down the possibilities of space group symmetry. Clearly, single-crystal data are superior in identifying systematic absences than those from powder diffraction because of the ability in removing ambiguity of multiplicity and in discerning reflections having nearly the same d-spacing. [crystallographic symmetry] 5