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If a crystal can be divided by a plane into two mirror-image halves, then the plane is a plane of symmetry. A crystal may have zero, one, or multiple planes of symmetry. For example, a cube has nine planes of symmetry. A plane of symmetry is also known as reflection symmetry or mirror symmetry.
The crystallographic directions are geometric lines linking nodes (atoms, ions or molecules) of a crystal. Likewise, the crystallographic planes are geometric planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows: [1]
The direction of a symmetry element corresponds to its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction, then they are denoted as a fraction n / m or n /m. If two or more axes have the same direction, the axis with higher symmetry is shown.
In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of ...
Planes with different Miller indices in cubic crystals Examples of directions. Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers h, k, and ℓ, the Miller indices.
The isometric crystal system class names, point groups (in Schönflies notation, Hermann–Mauguin notation, orbifold, and Coxeter notation), type, examples, international tables for crystallography space group number, [2] and space groups are listed in the table below. There are a total 36 cubic space groups.
In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C 4 rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy) C 4 .
For example, reflecting across a mirror plane will switch all the points left and right of the mirror plane, but points exactly on the mirror plane itself will not move. We can test every symmetry operation in the crystal's point group and keep track of whether the specified point is invariant under the operation or not.