Seven distinct unit cells can be recognised for 3-D lattices. The seven unit cells are associated with seven systems of crystal axes called the seven crystal systems.

Only occasionally does a crystal have the same external shape as that of its unit cell; however, it must possess the symmetry of the unit cell for that system.

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The system to which a crystal belongs can be identified from its observed or external symmetry. Note...

The first five systems shown here possess decreasing symmetry. They can be obtained from distorting the cubic cell. Important...

All crystals fall into one of the seven crystal systems, according to the way in which their symmetry elements restrict the shape of the unit cell. Symmetry is primary, the relationships between unit cell edge lengths and angles are secondary and arise from the symmetry.

The crystal with the highest symmetry belongs to the cubic system. The three crystal axes are
orthogonal and equal. They are related by three secondary triad axes along the diagonals of the unit cell cube, which define the system. The unit cell shape is restricted to a cube.

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The trigonal, tetragonal, and hexagonal systems are characterised by a single triad, tetrad and hexad axis respectively. One crystal axis (the z axis given by c) is taken parallel to the axis of rotation (except for the primitive rhombohedral) and the other two are perpendicular to it and equal. The cell lengths given by a and b are equal to each other.

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An orthorhombic crystal has diad symmetry in three perpendicular directions, along which the crystal axes are taken. The unit cell has three right angles with the three cell lengths unspecified.

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A monoclinic crystal has diad symmetry in one direction only, with a mirror plane perpendicular. One axis is taken parallel to the diad axis, the others are perpendicular to it. The cell shape is thus restricted to a
parallelepiped with two right angles. One angle is unspecified, as are the three cell lengths.

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If a crystal has no symmetry, there are no restrictions on the shape of the unit cell. The unit cell is a general
parallelepiped with lengths and angles not specified, this is the triclinic system.