7 Crystal System
7 Crystal System
7 Crystal System
Triclinic (a ≠ b ≠
c and α ≠ β ≠ γ )
The triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It has
(itself) the minimum symmetry all lattices have: points of inversion at each lattice point and
at 7 more points for each lattice point: at the midpoints of the edges and the faces, and at the
center points. It is the only lattice type that itself has no mirror planes.
Crystal classes
The triclinic crystal system class names, examples, Schönflies notation, Hermann-Mauguin
notation, point groups, International Tables for Crystallography space group number, [1]
orbifold, type, and space groups are listed in the table below. There are a total 2 space groups.
In crystallography, the monoclinic crystal system is one of the seven lattice point groups. A
crystal system is described by three vectors. In the monoclinic system, the crystal is described
by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism
with a parallelogram as its base. Hence two vectors are perpendicular (meet at right angles),
while the third vector meets the other two at an angle other than 90°.
Two monoclinic Bravais lattices exist: the primitive monoclinic and the centered monoclinic
lattices, with layers with a rectangular and rhombic lattice, respectively.
Unit cell
Orthorhombic crystal system
In crystallography, the orthorhombic crystal system is one of the seven lattice point groups.
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 rectangular base (a by b) and
height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three
lattice vectors remain mutually orthogonal.
Bravais lattices
Unit cell
Tetragonal crystal system
There are two tetragonal crystal structure types. Bravais lattices: the simple tetragonal (from
stretching the simple-cubic lattice) and the centered tetragonal (from stretching either the
face-centered or the body-centered cubic lattice). One might suppose stretching face-centered
cubic would result in face-centered tetragonal, but face-centered tetragonal is equivalent to
body-centered tetragonal, BCT (with a smaller lattice spacing). BCT is considered more
fundamental, so that is the standard terminology.[1]
Unit cell
Trigonal crystal system
In crystallography, the trigonal crystal system is one of the seven crystal systems. A crystal
system is a set of point groups in which the point groups themselves and their corresponding
space groups are associated with a lattice system. The trigonal crystal system consists of
those five point groups that have a single
three-fold rotation axis (see table in
Crystal_system#Crystal_classes).
Sometimes the term rhombohedral
lattice system is used as an exact Example trigonal Example trigonal
synonym, whereas it is more akin to a crystals (quartz) rhombohedral crystals
subset. Crystals in the rhombohedral (dolomite)
lattice system are always in the trigonal
crystal system, but some crystals such as
alpha-quartz are in the trigonal crystal
system but not in the rhombohedral lattice
system (alpha-quartz is in the hexagonal
lattice system). There are 25 space groups
(143-167) whose corresponding point
groups are one of the five in the trigonal Hexagonal (R-centered)
Hexagonal lattice cell
crystal system, consisting of the seven unit cell
space groups associated with the rhombohedral lattice system together with 18 associated
with the hexagonal lattice system. The crystal structures of alpha-quartz in the previous
example are described by two of those 18 space groups (152 and 154) associated with the
hexagonal lattice system.[1] To distinguish: The rhombohedral lattice system consists of the
rhombohedral lattice, while the trigonal crystal system consists of the five point groups that
have seven corresponding space groups associated with the rhombohedral lattice system (and
18 corresponding space groups associated with the hexagonal lattice system). An additional
source of confusion is that all members of the trigonal crystal system with assigned
rhombohedral lattice system (space groups 146, 148, 155, 160, 161, 166, and 167), can be
represented with an equivalent hexagonal lattice with so called R-centering (rhombohedral-
centering); there is a choice of using a R-centered hexagonal or a primitive rhombohedral
setting for the lattice.[2][3]
Hexagonal crystal system
The hexagonal lattice system is one of the seven lattice systems, consisting of the hexagonal
Bravais lattice. It is associated with 45 space groups whose underlying lattice has point group
of order 24. It is often confused with the smaller hexagonal crystal system, which consists of
the 27 space groups such that all space groups with the same point group are in the hexagonal
lattice system, or with the larger hexagonal crystal family, consisting of the 52 space groups
in either the hexagonal or rhombohedral lattice systems.
Each is subdivided into other variants listed below. Note that although the unit cell in these
crystals is conventionally taken to be a cube, the primitive unit cell often is not. This is related
to the fact that in most cubic crystal systems, there is more than one atom per cubic unit cell.