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.

Wyckoff positions of forms

The number of possible forms of a point group is infinite. They are easily classified in terms of Wyckoff positions of point groups.

• A Wyckoff position of a crystal form consists of all those crystal forms of a point group P for which the face poles are positioned on the same set of conjugate symmetry elements of P.
• A Wyckoff position of a point form consists of all those point forms of a point group P for which the points are positioned on the same set of conjugate symmetry elements of P.

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 symmetrically equivalent general faces is a general crystal form. The multiplicity (number 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-characteristic 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.

Basic vs. limiting forms

In a Wyckoff position, forms of different inherent symmetries may occur.

• Forms with the lowest inherent symmetry are called basic forms
• Forms of higher inherent symmetry are called limiting forms

Limiting forms always have the same multiplicity and oriented symmetry as the corresponding basic forms because they belong to the same Wyckoff position. The face poles (for face forms) or points (for point forms) 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.

Example

In the point group 4mm, the pyramid {h0l} has inherent symmetry C = 4mm and multiplicity 4; its face symmetry is .m. In the same group, the prism {100} has inherent symmetry C = 4/mmm and multiplicity 4; its face symmetry is again .m.

• For the pyramid, C = P and thus the form is characteristic. For the prism, C ⊃ P and thus the form is non-characteristic.
• Both forms lie on the mirrors perpendicular to the secondary symmetry directions; both forms are special.
• The prism can be seen as the limiting result of opening the pyramid at its vertex: the pyramid is the basic form, wheras the prism is a limiting form (the only one in this case). The face poles of the prism stay also on the mirror plane perpendicular to the fourfold axis which belong to C, supergroup of P, but not to P.

List of crystal (face) and point forms

The list of face forms includes 47 or 48 forms, depending on whether the dihedron is separated into sphenoid and dome or not.

Open face forms and their dual point forms

Face form	Point form	Inherent symmetry
Pedion	Single point	[math]\infty[/math]m
Pinacoid	Line segment through origin	[math]\infty[/math]m /m
Dihedron	Line segment	mm2
Rhombic prism	Rectangle through origin	mmm
Rhombic pyramid	Rectangle	mm2
Trigonal pyramid	Trigon	3m
Tetragonal pyramid	Square	4mm
Hexagonal pyramid	Hexagon	6mm
Ditrigonal pyramid	Truncated trigon	3m
Ditetragonal pyramid	Truncated square	4mm
Dihexagonal pyramid	Truncated hexagon	6mm
Trigonal prism	Trigon through origin	[math]\bar62m[/math]
Tetragonal prism	Square through origin	4/mmm
Hexagonal prism	Hexagon through origin	6/mmm
Ditrigonal prism	Truncated trgion through origin	[math]\bar62m[/math]
Ditetragonal prism	Truncated square through origin	4/mmm
Dihexagonal prism	Truncated hexagon	6/mmm

Closed face forms and their dual point forms

Face form	Point form	Inherent symmetry
Rhombic disphenoid	Rhombic disphenoid	222
Rhombic bipyramid	Quad	mmm
Trigonal bipyramid	Trigonal prism	[math]\bar62m[/math]
Tetragonal bipyramid	Tetragonal prism	4/mmm
Hexagonal bipyramid	Hexagonal prism	6/mmm
Ditrigonal bipyramid	Edge-truncated trigonal prism	[math]\bar62m[/math]
Ditetragonal bipyramid	Edge-truncated tetragonal prism	4/mmm
Dihexagonal bipyramid	Edge-truncated hexagonal prism	6/mmm
Tetragonal disphenoid	Tetragonal disphenoid	[math]\bar42m[/math]
Rhombohedron	Trigonal antiprism	[math]\bar3m[/math]
Tetragonal scalenohedron	Tetragonal disphenoid cut off by pinacoid	[math]\bar42m[/math]
Ditrigonal scalenohedron*	Trigonal antiprism sliced off by pinacoid	[math]\bar3m[/math]
Tetragonal trapezohedron	Twisted tetragonal antiprism	422
Trigonal trapezohedron	Twisted trigonal antiprism	32
Hexagonal trapezohedron	Twisted hexagonal antiprism	622
Tetartoid or pentagono-tritetrahedron	Snub tetrahedron	23
Pentagon-dodecahedron	Irregular icosahedron	[math]m\bar3[/math]
Diploid or Didodecahedron	Cube & octahedron & pentagon-dodecahedron	[math]m\bar3[/math]
Gyroid or Pentagon-trioctahedron	Cube & octahedron & pentagon-trioctahedron	432
Tetrahedron	Tetrahedron	[math]\bar43m[/math]
Tetragon-tritetrahedron	Cube & two tetrahedra	[math]\bar43m[/math]
Trigon-tritedrahedron	Tetrahedron truncated by tetrahedron	[math]\bar43m[/math]
Hexatetrahedron	Cube truncated by two tetrahedra	[math]\bar43m[/math]
Cube	Ochtaedron	[math]m\bar3m[/math]
Octahedron	Cube	[math]m\bar3m[/math]
Rhomb-dodecahedron	Cuboctahedron	[math]m\bar3m[/math]
Trigonotrioctahedron	Cube truncated by octahedron	[math]m\bar3m[/math]
Tetragonotrioctahedron	Cube & octahedron & rhomb-dodecahedron	[math]m\bar3m[/math]
Tetrahexahedron	Octahedron truncated by cube	[math]m\bar3m[/math]
Hexaoctahedron	Cube truncated by octahedron and by rhomb-dodecahedron	[math]m\bar3m[/math]

• The special case of ditrigonal scalenohedron where the dihedral angles are 60º is a hexagonal scalenohedron.

Note: the disphenoids are sometimes improperly called "tetrahedra"

See also

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