Space , Space groups I. Space symmetry In 3D a space filling symmetry is required, thus only the following symmetry axis are possible: 1, 2, 3, 4, 6. There are two additional symmetry operations (compared with 2D), which are so called combined symmetry operations including translations: Table 1: Additional Symmetry operations in space symmetry: Symmetry element, characterization, Hermann-Mauguin symbol symmetry operation symmetry element characterization H-M symbol screw screw axis 1) rotation by 2) translation along the axis glide 1) across the plane a, b, c, n, d 2) translation parallel to glide plane

II. Space groups According to the distinctive features of space symmetry described above, the 230 space groups are derived. Space = Bravais type + translational symmetry

III. Space group symbol The designation of the space group symbol is conducted according to the following sequence: 1.) Capital letter - represents the type - P: primitive; A, B, C: base centered; I: body centered; R: rhombohedral; F: face centered 2.) Symmetry elements - in a short form, ordered according to the viewing directions in a respective system (Table 2) - if there is more than one kind of symmetry element in one direction, only one of them is mentioned (mirror plane > glide plane; rotation axis > screw axis) Vice versa, the point groups symbol can be deduced out of the space group symbol by replacing screw axis through rotation axis and glide planes through mirror planes.

Example: Space group symbol of (TiO2): P42/mnm

P indicates a primitive Bravais lattice, 42 a screw axis, m indicate mirror planes and n indicates a glide plane. By replacing the screw axis through a rotation axis and the glide plane through a mirror plane the according symbol 4/mmm can be deduced.

Table 2: Viewing directions according to the 7 crystal systems position in symbol triclinic monoclinic orthorhombic tetragonal trigonal hexagonal cubic 1 - b a c c c 2 - b <111> 3 - c <110> - <210> <110> <...>: indicates that the viewing direction also can be a symmetry equivalent direction to the denoted one

IV. Relation between point- and space group On a crystal, there can only appear such planes, which belong to the corresponding point group. But there is no general correlation between the and the symmetry of the crystal. Questions 1.) To have a closer look on how the space groups are deduced: What space groups are possible in a monoclinic crystal system with the corresponding point group 2/m? 2.) Why are the Hermann-Mauguin symbols and not the Schönflies symbols used to denote space groups?

Literature W. Kleber, H.J. Bautsch, J. Bohm, Einführung in die Kristallographie, 18th ed., Oldenbourg, Berlin, 1998 W. Borchardt-Ott, Kristallographie, 7th ed. Springer, Berlin, Heidelberg, 2009