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Difference between revisions of "Form"

From Online Dictionary of Crystallography

(List of crystal (face) forms)
(List of crystal (face) forms)
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==List of crystal (face) forms==
 
==List of crystal (face) forms==
47 or 48 forms exist, depending on whether the diedre is separated into sphenoid and dome or not. In parenthesis the [[eigensymmetry|inherent symmetry]] of the form.
+
47 or 48 forms exist, depending on whether the diedre is separated into sphenoid and dome or not.  
 
===Open forms===
 
===Open forms===
*Pedion (<math>\infty</math>''m'')
+
{|border="1"
*Pinacoid (<math>\infty</math>''m'' /''m'')
+
!Face form !!Inherent symmetry
*Diedre (''mm''2)
+
|-
*Rhombic prism (''mmm'')
+
|Pedion ||<math>\infty</math>''m''
*Rhombic pyramid (''mm''2)
+
|-
*Trigonal pyramid (3''m'')
+
|Pinacoid ||<math>\infty</math>''m'' /''m''
*Tetragonal pyramid (4''mm'')
+
|-
*Hexagonal pyramid (6''mm'')
+
|Diedre ||''mm''2
*Ditrigonal pyramid (3''m'')
+
|-
*Ditetragonal pyramid (4''mm'')
+
|Rhombic prism ||''mmm''
*Dihaxagonal pyramid (6''mm'')
+
|-
*Trigonal prism (<math>\bar62m</math>)
+
|Rhombic pyramid ||''mm''2
*Tetragonal prism (4/''mmm'')
+
|-
*Hexagonal prism (6/''mmm'')
+
|Trigonal pyramid ||3''m''
*Ditrigonal prism (<math>\bar62m</math>)
+
|-
*Ditetragonal prism (4/''mmm'')
+
|Tetragonal pyramid ||4''mm''
*Dihexagoanl prism (6/''mmm'')
+
|-
 +
|Hexagonal pyramid ||6''mm''
 +
|-
 +
|Ditrigonal pyramid ||3''m''
 +
|-
 +
|Ditetragonal pyramid ||4''mm''
 +
|-
 +
|Dihexagonal pyramid ||6''mm''
 +
|-
 +
|Trigonal prism ||<math>\bar62m</math>
 +
|-
 +
|Tetragonal prism ||4/''mmm''
 +
|-
 +
|Hexagonal prism ||6/''mmm''
 +
|-
 +
|Ditrigonal prism ||<math>\bar62m</math>
 +
|-
 +
|Ditetragonal prism ||4/''mmm''
 +
|-
 +
|Dihexagoanl prism ||6/''mmm''
 +
|}
 
===Closed forms===
 
===Closed forms===
*Rhombic disphenoid (222)
+
{|border="1"
*Rhombic bipyramid (''mmm'')
+
!Face form !!Inherent symmetry
*Trigonal bipyramid (<math>\bar62m</math>)
+
|-
*Tetragonal bipyramid (4/''mmm'')
+
|Rhombic disphenoid ||222
*Hexagonal bipyramid (6/''mmm'')
+
|-
*Ditrigonal bipyramid (<math>\bar62m</math>)
+
|Rhombic bipyramid ||''mmm''
*Ditetragonal bipyramid (4/''mmm'')
+
|-
*Dihexagonal bipyramid (6/''mmm'')
+
|Trigonal bipyramid ||<math>\bar62m</math>
*Tetragonal disphenoid (<math>\bar42m</math>)
+
|-
*Rhombohedron (<math>\bar3m</math>)
+
|Tetragonal bipyramid ||4/''mmm''
*Tetragonal scalenohedron (<math>\bar42m</math>)
+
|-
*Ditrigional scalenohedron (<math>\bar3m</math>)
+
|Hexagonal bipyramid ||6/''mmm''
*Tetragonal trapezohedron (422)
+
|-
*Trigonal trapezohedron (32)
+
|Ditrigonal bipyramid ||<math>\bar62m</math>
*Hexagonal trapezohedron (622)
+
|-
*Tetardohedron or pentagono-tritetrahedron (23)
+
|Ditetragonal bipyramid ||4/''mmm''
*Pentagonododecahedron (<math>m\bar3</math>)
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|-
*Diployd or Didodecahedron (<math>m\bar3</math>)
+
|Dihexagonal bipyramid ||6/''mmm''
*Gyroedron or pentagonotrioctahedron (432)
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|-
*Tetrahedron (<math>\bar43m</math>)
+
|Tetragonal disphenoid ||<math>\bar42m</math>
*Tetragonotritetrahedron (<math>\bar43m</math>)
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|-
*Trigonotritedrahedron (<math>\bar43m</math>)
+
|Rhombohedron ||<math>\bar3m</math>
*Hexatetrahedron (<math>\bar43m</math>)
+
|-
*Cube (<math>m\bar3m</math>)
+
|Tetragonal scalenohedron ||<math>\bar42m</math>
*Octahedron (<math>m\bar3m</math>)
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|-
*Rhombododecahedron (<math>m\bar3m</math>)
+
|Ditrigional scalenohedron ||<math>\bar3m</math>
*Trigonotrioctahedron (<math>m\bar3m</math>)
+
|-
*Tetragonotrioctahedron (<math>m\bar3m</math>)
+
|Tetragonal trapezohedron ||422
*Tetrahexahedron (<math>m\bar3m</math>)
+
|-
*Hexaoctahedron (<math>m\bar3m</math>)
+
|Trigonal trapezohedron ||32
 
+
|-
 +
|Hexagonal trapezohedron ||622
 +
|-
 +
|Tetardohedron or pentagono-tritetrahedron ||23
 +
|-
 +
|Pentagonododecahedron ||<math>m\bar3</math>
 +
|-
 +
|Diployd or Didodecahedron ||<math>m\bar3</math>
 +
|-
 +
|Gyroedron or pentagonotrioctahedron ||432
 +
|-
 +
|Tetrahedron ||<math>\bar43m</math>
 +
|-
 +
|Tetragonotritetrahedron ||<math>\bar43m</math>
 +
|-
 +
|Trigonotritedrahedron ||<math>\bar43m</math>
 +
|-
 +
|Hexatetrahedron ||<math>\bar43m</math>
 +
|-
 +
|Cube ||<math>m\bar3m</math>
 +
|-
 +
|Octahedron ||<math>m\bar3m</math>
 +
|-
 +
|Rhombododecahedron ||<math>m\bar3m</math>
 +
|-
 +
|Trigonotrioctahedron ||<math>m\bar3m</math>
 +
|-
 +
|Tetragonotrioctahedron ||<math>m\bar3m</math>
 +
|-
 +
|Tetrahexahedron ||<math>m\bar3m</math>
 +
|-
 +
|Hexaoctahedron ||<math>m\bar3m</math>
 +
|}
 
<small>'''Note''': the disphenoids are sometimes improperly called "tetrahedra"</small>
 
<small>'''Note''': the disphenoids are sometimes improperly called "tetrahedra"</small>
  

Revision as of 14:24, 10 April 2007

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 consist of all those crystal forms of a point group P for which the face poles are positioned on the same set of conhjugate symmetry elements of P
  • A Wyckoff position of a point form consist of all those point forms of a point group P for which the points are positioned on the same set of conhjugate 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 symmetrycally equivalent general faces is a general crystal form. The mulplicity (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-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.

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) forms

47 or 48 forms exist, depending on whether the diedre is separated into sphenoid and dome or not.

Open forms

Face form Inherent symmetry
Pedion [math]\infty[/math]m
Pinacoid [math]\infty[/math]m /m
Diedre mm2
Rhombic prism mmm
Rhombic pyramid mm2
Trigonal pyramid 3m
Tetragonal pyramid 4mm
Hexagonal pyramid 6mm
Ditrigonal pyramid 3m
Ditetragonal pyramid 4mm
Dihexagonal pyramid 6mm
Trigonal prism [math]\bar62m[/math]
Tetragonal prism 4/mmm
Hexagonal prism 6/mmm
Ditrigonal prism [math]\bar62m[/math]
Ditetragonal prism 4/mmm
Dihexagoanl prism 6/mmm

Closed forms

Face form Inherent symmetry
Rhombic disphenoid 222
Rhombic bipyramid mmm
Trigonal bipyramid [math]\bar62m[/math]
Tetragonal bipyramid 4/mmm
Hexagonal bipyramid 6/mmm
Ditrigonal bipyramid [math]\bar62m[/math]
Ditetragonal bipyramid 4/mmm
Dihexagonal bipyramid 6/mmm
Tetragonal disphenoid [math]\bar42m[/math]
Rhombohedron [math]\bar3m[/math]
Tetragonal scalenohedron [math]\bar42m[/math]
Ditrigional scalenohedron [math]\bar3m[/math]
Tetragonal trapezohedron 422
Trigonal trapezohedron 32
Hexagonal trapezohedron 622
Tetardohedron or pentagono-tritetrahedron 23
Pentagonododecahedron [math]m\bar3[/math]
Diployd or Didodecahedron [math]m\bar3[/math]
Gyroedron or pentagonotrioctahedron 432
Tetrahedron [math]\bar43m[/math]
Tetragonotritetrahedron [math]\bar43m[/math]
Trigonotritedrahedron [math]\bar43m[/math]
Hexatetrahedron [math]\bar43m[/math]
Cube [math]m\bar3m[/math]
Octahedron [math]m\bar3m[/math]
Rhombododecahedron [math]m\bar3m[/math]
Trigonotrioctahedron [math]m\bar3m[/math]
Tetragonotrioctahedron [math]m\bar3m[/math]
Tetrahexahedron [math]m\bar3m[/math]
Hexaoctahedron [math]m\bar3m[/math]

Note: the disphenoids are sometimes improperly called "tetrahedra"

See also

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