International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A. ch. 10.1, pp. 764-766
Section 10.1.2.2. Crystal and point forms
a
Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany |
For a point group a crystal form is a set of all symmetrically equivalent faces; a point form is a set of all symmetrically equivalent points. Crystal and point forms in point groups correspond to `crystallographic orbits' in space groups; cf. Section 8.3.2 .
Two kinds of crystal and point forms with respect to can be distinguished. They are defined as follows:
General and special crystal and point forms can be represented by their sets of equivalent Miller indices and point coordinates x, y, z. Each set of these `triplets' stands for infinitely many crystal forms or point forms which are obtained by independent variation of the values and signs of the Miller indices h, k, l or the point coordinates x, y, z.
It should be noted that for crystal forms, owing to the well known `law of rational indices', the indices h, k, l must be integers; no such restrictions apply to the coordinates x, y, z, which can be rational or irrational numbers.
Example
In point group 4, the general crystal form stands for the set of all possible tetragonal pyramids, pointing either upwards or downwards, depending on the sign of l; similarly, the general point form x, y, z includes all possible squares, lying either above or below the origin, depending on the sign of z. For the limiting cases or , see below.
In order to survey the infinite number of possible forms of a point group, they are classified into Wyckoff positions of crystal and point forms, for short Wyckoff positions. This name has been chosen in analogy to the Wyckoff positions of space groups; cf. Sections 2.2.11 and 8.3.2 . In point groups, the term `position' can be visualized as the position of the face poles and points in the stereographic projection. Each `Wyckoff position' is labelled by a Wyckoff letter.
Definition A `Wyckoff position of crystal and point forms' consists of all those crystal forms (point forms) of a point group for which the face poles (points) are positioned on the same set of conjugate symmetry elements of ; i.e. for each face (point) of one form there is one face (point) of every other form of the same `Wyckoff position' that has exactly the same face (site) symmetry.
Each point group contains one `general Wyckoff position' comprising all general crystal and point forms. In addition, up to two `special Wyckoff positions' may occur in two dimensions and up to six in three dimensions. They are characterized by the different sets of conjugate face and site symmetries and correspond to the seven positions of a pole in the interior, on the three edges, and at the three vertices of the so-called `characteristic triangle' of the stereographic projection.
Examples
It is instructive to subdivide the crystal forms (point forms) of one Wyckoff position further, into characteristic and noncharacteristic forms. For this, one has to consider two symmetries that are connected with each crystal (point) form:
Examples
The eigensymmetries and the generating symmetries of the 47 crystal forms (point forms) are listed in Table 10.1.2.3. With the help of this table, one can find the various point groups in which a given crystal form (point form) occurs, as well as the face (site) symmetries that it exhibits in these point groups; for experimental methods see Sections 10.2.2 and 10.2.3 .
†These limiting forms occur in three or two non-equivalent orientations (different types of limiting forms); cf. Table 10.1.2.2.
‡In point groups and , the tetragonal prism and the hexagonal prism occur twice, as a `basic special form' and as a `limiting special form'. In these cases, the point groups are listed twice, as and as . |
With the help of the two groups and , each crystal or point form occurring in a particular point group can be assigned to one of the following two categories:
The importance of this classification will be apparent from the following examples.
Examples
The general forms of the 13 point groups with no, or only one, symmetry direction (`monoaxial groups') , are always noncharacteristic, i.e. their eigensymmetries are enhanced in comparison with the generating point groups. The general positions of the other 19 point groups always contain characteristic crystal forms that may be used to determine the point group of a crystal uniquely (cf. Section 10.2.2 ).4
So far, we have considered the occurrence of one crystal or point form in different point groups and different Wyckoff positions. We now turn to the occurrence of different kinds of crystal or point forms in one and the same Wyckoff position of a particular point group.
In a Wyckoff position, crystal forms (point forms) of different eigensymmetries may occur; the crystal forms (point forms) with the lowest eigensymmetry (which is always well defined) are called basic forms (German: Grundformen) of that Wyckoff position. The crystal and point forms of higher eigensymmetry are called limiting forms (German: Grenzformen) (cf. Table 10.1.2.3). These forms are always noncharacteristic.
Limiting forms5 occur for certain restricted values of the Miller indices or point coordinates. They always have the same multiplicity and oriented face (site) symmetry as the corresponding basic forms because they belong to the same Wyckoff position. The enhanced eigensymmetry of a limiting form may or may not be accompanied by a change in the topology6 of its polyhedra, compared with that of a basic form. In every case, however, the name of a limiting form is different from that of a basic form.
The face poles (or points) of a limiting form lie on symmetry elements of a supergroup of the point group that are not symmetry elements of the point group itself. There may be several such supergroups.
Examples
Whereas basic and limiting forms belonging to one `Wyckoff position' are always clearly distinguished, closer inspection shows that a Wyckoff position may contain different `types' of limiting forms. We need, therefore, a further criterion to classify the limiting forms of one Wyckoff position into types: A type of limiting form of a Wyckoff position consists of all those limiting forms for which the face poles (points) are located on the same set of additional conjugate symmetry elements of the holohedral point group (for the trigonal point groups, the hexagonal holohedry has to be taken). Different types of limiting forms may have the same eigensymmetry and the same topology, as shown by the examples below. The occurrence of two topologically different polyhedra as two `realizationsFace form' of one type of limiting form in point groups 23, and 432 is explained below in Section 10.1.2.4, Notes on crystal and point forms, item (viii).
Examples
|
Not considered in this volume are limiting forms of another kind, namely those that require either special metrical conditions for the axial ratios or irrational indices or coordinates (which always can be closely approximated by rational values). For instance, a rhombic disphenoid can, for special axial ratios, appear as a tetragonal or even as a cubic tetrahedron; similarly, a rhombohedron can degenerate to a cube. For special irrational indices, a ditetragonal prism changes to a (noncrystallographic) octagonal prism, a dihexagonal pyramid to a dodecagonal pyramid or a crystallographic pentagon-dodecahedron to a regular pentagon-dodecahedron. These kinds of limiting forms are listed by A. Niggli (1963).
In conclusion, each general or special Wyckoff position always contains one set of basic crystal (point) forms. In addition, it may contain one or more sets of limiting forms of different types. As a rule,7 each type comprises polyhedra of the same eigensymmetry and topology and, hence, of the same name, for instance `ditetragonal pyramid'. The name of the basic general forms is often used to designate the corresponding crystal class, for instance `ditetragonal-pyramidal class'; some of these names are listed in Table 10.1.2.4.
|
References
Niggli, A. (1963). Zur Topologie, Metrik und Symmetrie der einfachen Kristallformen. Schweiz. Mineral. Petrogr. Mitt. 43, 49–58.Google Scholar