Space group
[1] The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged.
A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography Hahn (2002).
The space groups in three dimensions were first enumerated in 1891 by Fedorov[4] (whose list had two omissions (I43d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies[5] (whose list had four omissions (I43d, Pc, Cc, ?)
The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies.
[citation needed] Burckhardt (1967) describes the history of the discovery of the space groups in detail.
A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations.
The number of replicates of the asymmetric unit in a unit cell is thus the number of lattice points in the cell times the order of the point group.
The elements of the space group fixing a point of space are the identity element, reflections, rotations and improper rotations, including inversion points.
glide, which is a fourth of the way along either a face or space diagonal of the unit cell.
For (overall dimension, lattice dimension): The 65 "Sohncke" space groups, not containing any mirrors, inversion points, improper rotations or glide planes, yield chiral crystals, not identical to their mirror image; whereas space groups that do include at least one of those give achiral crystals.
An inversion and a mirror implies two-fold screw axes, and so on.
Arithmetic crystal classes may be interpreted as different orientations of the point groups in the lattice, with the group elements' matrix components being constrained to have integer coefficients in lattice space.
Conway, Delgado Friedrichs, and Huson et al. (2001) gave another classification of the space groups, called a fibrifold notation, according to the fibrifold structures on the corresponding orbifold.
In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain.
It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space.
This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z3.
Including time reversal there are 1651 magnetic space groups in 3D (Kim 1999, p.428).
Number of original and magnetic groups by (overall, lattice) dimension:(Palistrant 2012)(Souvignier 2006) Table of the wallpaper groups using the classification of the 2-dimensional space groups: For each geometric class, the possible arithmetic classes are Note: An e plane is a double glide plane, one having glides in two different directions.
They are found in seven orthorhombic, five tetragonal and five cubic space groups, all with centered lattice.
The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R. The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A-face centered or C-face centered lattices.
