Last Update 06/ 02/ 2004
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This application presents a randomly selected Bravais (after Auguste Bravais, 1811-1863) lattice in comparison with a gray cube. A click on any other selected space group notation (according to Hermann-Mauguin) will display the corresponding figure. The space group classified as low symmetry is presented on the top of the list with the symbol P1. The letter P means primitive, for one point in the lattice cell. This space group belongs to the triclinic crystal system. Table 1 shows the selected space group symbols (in blue) and the crystal system to which they belong with the number of points in each unit cell.
Table 1. Space group notations, crystal systems and associated points.
| 1 | 2 | 2 | 4 | |
| Triclinic | P1 | |||
| Monoclinic | P2/m | C2/m | ||
| Orthorhombic | Pmmm | Cmmm | Immm | Fmmm |
| Rhombohedral | R3m | |||
| Tetragonal | P4/mmm | I4/mmm | ||
| Hexagonal | P6/mmm | |||
| Cubic | Pm3m | Im3m | Fm3m |
Character R is used for the notation of a primitive cell belonging
to the rhombohedral crystal system. If the base centered lattice is named
C,
its
coordinates of the base centered points are 0.5 a in the direction
of x and
0.5 b in the direction of y. A body centered
cell is represented by I, and this point has coordinates
0.5
a, 0.5 b and 0.5 c. Character F is used for face
centered cells, with a point centered on each of the 6 faces. Number 1
(the trivial one fold axis) is used to indicate absence of symmetry center.
Numbers 2,
3,
4
and 6 indicate the respective
rotation symmetry axis. In space group P2/m the two fold rotation
axis perpendicular to the mirror
plane is represented by m. When the mirror plane contains the
3
fold rotation axis the symbol is like R3m.
There are 230 space groups to classify the crystals according to the
respectively observed symmetry, distributed in seven crystal systems and
in 14 Bravais lattices.
A mouse push and drag action on the green bar for six of the crystal systems in this application will show either the edge variation either the angle variation or a combination of both in order to get a cube.
Exercises
1) How many mirror planes are expected to exist in space group R3m?
2) Explain why is it impossible to have a base centered cubic lattice, for example Cm3m?
Bibliography
International Tables for Crystallography, Volume A: Space-group symmetry, Kluwer Academic Publishers, Dordrecht, Co-publication with International Union of Crystallography, Edited by Theo Hahn, Fifth edition, April 2002.
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