There are 230
space groups
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
in three dimensions, given by a number index, and a full name in
Hermann–Mauguin notation
In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogis ...
, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a
point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
of the unit cell.
Symbols
In
Hermann–Mauguin notation
In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogis ...
, space groups are named by a symbol combining the
point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
identifier with the uppercase letters describing the
lattice type. Translations within the lattice in the form of
screw axes and
glide planes are also noted, giving a complete crystallographic space group.
These are the
Bravais lattices in three dimensions:
*P primitive
*I body centered (from the German ''Innenzentriert'')
*F face centered (from the German ''Flächenzentriert'')
*A centered on A faces only
*B centered on B faces only
*C centered on C faces only
*R rhombohedral
A reflection plane m within the point groups can be replaced by a
glide plane In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged.
Glide planes are noted by ''a'' ...
, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the
diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
structure.
*
,
, or
: glide translation along half the lattice vector of this face
*
: glide translation along half the diagonal of this face
*
: glide planes with translation along a quarter of a face diagonal
*
: two glides with the same glide plane and translation along two (different) half-lattice vectors.
A gyration point can be replaced by a
screw axis
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw ...
denoted by a number, ''n'', where the angle of rotation is
. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2
1 is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 3
1 is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector. The possible screw axes are: 2
1, 3
1, 3
2, 4
1, 4
2, 4
3, 6
1, 6
2, 6
3, 6
4, and 6
5.
Wherever there is both a rotation or screw axis ''n'' and a mirror or glide plane ''m'' along the same crystallographic direction, they are represented as a fraction
or ''n/m''. For example, 4
1/a means that the crystallographic axis in question contains both a 4
1 screw axis as well as a glide plane along a.
In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form
which specifies the Bravais lattice. Here
is the lattice system, and
is the centering type.
In Fedorov symbol, the type of space group is denoted as ''s'' (''symmorphic'' ), ''h'' (''hemisymmorphic''), or ''a'' (''asymmorphic''). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups.
Symmorphic
The 73 symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups, for example, the space groups P4/mmm (
, ''36s'') and I4/mmm (
, ''37s'').
Hemisymmorphic
The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Hemisymmorphic space groups contain the axial combination 422, which are P4/mcc (
, ''35h''), P4/nbm (
, ''36h''), P4/nnc (
, ''37h''), and I4/mcm (
, ''38h'').
Asymmorphic
The remaining 103 space groups are asymmorphic, for example, those derived from the point group 4/mmm (
).
List of triclinic
List of monoclinic
List of orthorhombic
{, class=wikitable
, +
Orthorhombic crystal system
In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a r ...
!Number
!
Point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
!
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
!Short name
!Full name
!
Schoenflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.
Schoenflies ...
!
Fedorov
!Shubnikov
!
Fibrifold In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by , who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine space gro ...
(primary)
!
Fibrifold In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by , who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine space gro ...
(secondary)
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12
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12
12
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1 2
1 2
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, 24, , I2
12
12
1, , I 2
1 2
1 2
1, ,
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