All the lattices in the table on the previous page are, by nature, centrosymmetric: each lattice point “sees” identical lattice points in opposite directions. If there are centers of symmetry at the lattice points, there are also centers midway between the lattice points. This is a common occurence that symmetry operations located at lattice points produce symmetry operations of the same type halfway between these primary symmetry operations. The only symmetry element present in the triclinic lattice is a center of symmetry. In the java frame at the right, the lattice points in the triclinic lattice are related, in pairs, through the action of the center of symmetry at ½, ½, ½.
The label 2/m (pronounced “2 upon m”) for the monoclinic lattice indicates a 2-fold rotation axis perpendicular to a mirror plane. In the monoclinic system, the b-axis is, by convention, the unique axis (b is perpendicular to the other two axes). Picturing an extended monoclinic lattice shows that rotation of 180° about the b-axis brings the lattice points into coincidence. The 010 set of planes act as mirror planes, reflecting lattice points on one side into those on the other side. The plane halfway between these unit cell faces is also a mirror plane (the shaded plane in the java frame to the right). In addition, a 2-fold axis in the center of the ac-plane relates the lattice points in a single cell.This is another example of a secondary symmetry element appearing midway between primary symmetry elements.
The symmetry of the orthorhombic lattice is mmm, indicating the presence of three mutually perpendicular mirror planes. Due to the orthogonal nature of all three axes, there are also three 2-fold rotation axes parallel to a, b, and c. Thus, the full symmetry of the orthorhombic axis is 2/m 2/m 2/m, but the mmm designation is most often used.
The first two symmetry labels of the 4/mmm designation for the tetragonal lattice refer to the 4-fold axis parallel to c and the mirror perpendicular to this axis. The mm labels refer to the mirror perpendicular to a and the mirror that makes an angle of 45° with this first mirror.
A rhombohedron can be considered to be a cube that has been stretched (or compressed) along the body-diagonal. There is then a 3-fold rotary inversion axis coincident with the body-diagonal which also lies in a mirror plane that is parallel to the body diagonal.
Similar to the tetragonal case, the 6/m part of the hexagonal label refers to a 6-fold axis parallel to the c-axis with a mirror plane perpendicular to it; the mm in the second and third positions refer to mutually perpendicular diagonal mirror planes parallel to c. That the c-axis is indeed a 6-fold rotation axis can be seen most easily by looking down the axis and “seeing” only lattice points.
The m3m designation for the cubic lattice indicates a mirror plane parallel to the unit cell face. There is a 3-fold rotation axis along the body diagonal and a mirror plane in which this axis lies.
