International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, p. 158

Section 2.1.3.6.8. Diagrams of the general position (by K. Momma and M. I. Aroyo)

Th. Hahna and A. Looijenga-Vosb

2.1.3.6.8. Diagrams of the general position (by K. Momma and M. I. Aroyo)

| top | pdf |

Non-cubic space groups. In these diagrams, the `heights' of the points are z coordinates, except for monoclinic space groups with unique axis b where they are y coordinates. For rhombohedral space groups, the heights are always fractions of the hexagonal c axis. The symbols [+] and − stand for [+z] and [-z] (or [+y] and [-y]) in which z or y can assume any value. For points with symbols [+] or − preceded by a fraction, e.g. [{1 \over 2}+] or [{1 \over 3}-], the relative z or y coordinate is [{1 \over 2}] etc. higher than that of the point with symbol [+] or −.

Where a mirror plane exists parallel to the plane of projection, the two positions superimposed in projection are indicated by the use of a ring divided through the centre. The information given on each side refers to one of the two positions related by the mirror plane, as in [-\def\circhyp{\mathop{\bigcirc\hskip-.95em{{\raise0.15em\hbox{$\scriptscriptstyle{{\hbox to 1.5pt{}}\displaystyle{,}}$}}\hskip-0.05em \lower0.2em\hbox{\vrule height 0.9em}}}\;\;}\circhyp +].

Diagrams for cubic space groups (Fig. 2.1.3.10)[link]. Following the approach of IT (1935), for each cubic space group a diagram showing the points of the general position as the vertices of polyhedra is given. In these diagrams, the polyhedra are transparent, but the spheres at the vertices are opaque. For most of the space groups, `starting points' with the same coordinate values, x = 0.048, y = 0.12, z = 0.089, have been used. The origins of the polyhedra are chosen at special points of highest site symmetry, which for most space groups coincide with the origin (and its equivalent points in the unit cell). Polyhedra with origins at sites [({\textstyle{1\over 8}}, {\textstyle{1\over 8}}, {\textstyle{1\over 8}})] have been chosen for the space groups [P4_332] (212) and [I4_132] (214), and [({\textstyle{3\over 8}}, {\textstyle{3\over 8}}, {\textstyle{3\over 8}})] for [P4_132] (213). The two diagrams shown for the space groups [I\bar{4}3d] (220) and [Ia\bar{3}d] (230) correspond to polyhedra with origins chosen at two different special sites with site-symmetry groups of equal (32 versus [\bar{3}] in [Ia\bar{3}d]) or nearly equal order (3 versus [\bar{4}] in [I\bar{4}3d]). The height h of the centre of each polyhedron is given on the diagram, if different from zero. For space-group Nos. 198, 199 and 220, h refers to the special point to which the polyhedron (triangle) is connected. Polyhedra with height 1 are omitted in all the diagrams. A grid of four squares is drawn to represent the four quarters of the basal plane of the cell. For space groups [F\bar{4}3c] (219), [Fm\bar{3}c] (226) and [Fd\bar{3}c] (228), where the number of points is too large for one diagram, two diagrams are provided, one for the upper half and one for the lower half of the cell.

Notes:

  • (i) For space group [P4_132] (213), the coordinates [\bar{x}, \bar{y}, \bar{z}] have been chosen for the `starting point' to show the enantiomorphism with [P4_332] (212).

  • (ii) For the description of a space group with `origin choice 2', the coordinates x, y, z of all points have been shifted with the origin to retain the same polyhedra for both origin choices.

An additional general-position diagram is shown on the fourth page for each of the ten space groups of the [m\bar{3}m] crystal class. To provide a clearer three-dimensional-style overview of the arrangements of the polyhedra, these general-position diagrams are shown in tilted projection (in contrast to the orthogonal-projection diagrams described above).

The general-position diagrams of the cubic groups in both orthogonal and tilted projections were generated using the program VESTA (Momma & Izumi, 2011[link]).

References

Momma, K. & Izumi, F. (2011). VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Cryst. 44, 1272–1276.








































to end of page
to top of page