Form

Forme (Fr), Forma (It), 結晶形 (Ja)

Definition

For a point group P a form is a set of all symmetrically equivalent "elements", namely:

• in vector space, a crystal form or face form is a set of all symmetrically equivalent faces;
• in point space, a point form is a set of all symmetrically equivalent points.

The polyhedron or polygon of a point form is dual to the polyhedron of the corresponding face form, where "dual" means that they have the same number of edges but the number of faces and vertices is interchanged. The inherent symmetry of a form is a point group C which either coincides with the generating point group P or is a supergroup of it.

Forms in point groups correspond to crystallographic orbits in space groups.

Classification of forms

Forms are classified on the basis of their symmetry properties and of their orientation with respect to the symmetry elements of the point groups in which they occur.

General vs. special forms

A face is called general if only the identity operation transform the face onto itself. Each complete set of symmetrycally equivalent general faces is a general crystal form. The mulplicity (numer of faces of the form) of a general form is the order of the point group P. In the stereographic projection, the poles of general faces do not lie on any symmetry element of P.

A point is called general if its site symmetry is 1. A general point form is a complete set of symmetrically equivalent general points.

A face is called special if it is transformed into itself by at least one symmetry operation of P, in addition to the identity. Each complete set of symmetrically equivalent special faces is called a special crystal form. The face symmetry of a special face is the group of symmetry operations that transforms this face onto itself; it is a subgroup of P. The multiplicity of a special form is the multiplicity of the general form divided by the order of the face-symmetry group. In the stereographic projection, the poles of special faces lie on symmetry elements of P.

A point is called special if its site symmetry is higher than 1. A special point form is a complete set of symmetrically equivalent special points. The multiplicity of a special point form is the multiplicity of the general from divided by the order of the site symmetry group and is the same as that of the corresponding special crystal form.

Characteristic vs. non-characteristis forms

A form is called characteristic if its inherent symmetry coincides with the generating point group P.

A form is called non-characteristic if its inherent symmetry is a supergroup of the generating point group P.

See also

• Chapter 10 in the International Tables for Crystallography, Volume A