Crystal structure

In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a unit cell, a set of atoms arranged in a particular way; which is periodically repeated in three dimensions on a lattice. The spacing between unit cells in various directions are called its lattice parameters. The symmetry properties of the crystal are embodied in its space group. A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage, electronic band structure, and optical properties.

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A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal. The positions of the atoms inside the unit cell are described by the symmetric unit or basis, the set of atomic positions (xi,yi,zi) measured from a lattice point.
For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space. A Wigner-Seitz cell is a particular kind of primitive cell which has the same symmetry as the lattice

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The crystal system is the point group of the lattice (the set of rotation and reflection symmetries which leave a lattice point fixed), not including the positions of the atoms in the unit cell. There are seven unique crystal systems. The simplest and most symmetric, the cubic (or isometric) system, has the symmetry of a cube. The other six systems, in order of decreasing symmetry, are hexagonal, tetragonal, rhombohedral (also known as trigonal), orthorhombic, monoclinic and triclinic. Some crystallographers consider the hexagonal crystal system not to be its own crystal system, but instead a part of the trigonal crystal system.

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A Bravais lattice is a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In three dimensions, there are 14 unique Bravais lattices (distinct from one another in that they have different space groups) in three dimensions. All crystalline materials recognized till now (not including quasicrystals) fit in one of these arrangements. The fourteen three-dimensional lattices, classified by crystal system, are shown to the right.
The crystal structure is one of the lattices with a unit cell, which contains atoms at specific coordinates, at every lattice point. Because it includes the unit cell, the symmetry of the crystal can be more complicated than the symmetry of the lattice.
The crystallographic point group or crystal class is the set of non-translational symmetries that leave a point in the crystal fixed. There are 32 possible crystal classes.
The space group of the crystal structure is composed of the translational symmetries in addition to the symmetries of the point group. There are 230 distinct space groups.
Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials. In particular dislocations in the crystal lattice allow shear at much lower stress than that needed for a perfect crystal structure.

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Crystal structures can be divided into 32 classes, or point groups, according to the number of rotational axes and reflection planes they exhibit that leave the crystal structure unchanged. Twenty of the 32 crystal classes are piezoelectric. All 20 piezoelectric classes lack a center of symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance which has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.
There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnetic can be reversed. However, it is important to note that, although they are called ferroelectrics, the effect is due to the crystal structure, not the presence of a ferrous metal.
Incommensurate crystals have period-varying translational symmetry. The period between nodes of symmetry is constant in most crystals. The distance between nodes in an incommensurate crystal is dependent on the number of nodes between it and the base node.

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