Materials are usually classified into gas, liquid, and solid from the viewpoint of mechanical properties. However, from the viewpoint of the arrangement of atoms, materials are classified as follows:
Most of materials with three-dimensional long-range order have periodicity.
This is the traditional definition of crystal.
A material with three-dimensional long-range order but no periodicity is called a quasicrystal.
First we focus on the periodicity of the crystal before considering the arrangement of atoms.
The repeating unit of atomic arrangement is called "unit cell". In other words, in a crystal, unit cells with the same shape, size and orientation are not overlapped and lined up with no gaps. The periodicity with which the unit cells are repeated can be represented by three independent vectors. These are called primitive lattice vectors for the lattice. Hereafter, we express the primitive vectors as a, b, and c.
The atomic arrangement of the crystal is invariant under the translation by a primitive vector. If one focuses on this periodicity, each unit cell may be represented by a certain point instead of considering the arrangement of all the atoms. The point is called lattice point. The lattice points are also regularly arranged with a periodicity determined by the primitive vectors. The geometric arrangement of the lattice points is called "crystal lattice".
The period can be represented by three independent vectors.
They are called primitive vectors.
Hereafter they are represented as a, b, c.
a, b, c should form a right-handed system.
The parallelepiped formed by the primitive vectors is regarded as the unit cell, though one can use other types of unit cells.
There are also infinite choices of primitive vectors, the highest-symmetry parallelepiped with the shortest primitive vectors is usually used.
The lengths of the primitive vectors are expressed as a, b, c. The angles between b and c, c and a, a and b are α, β, γ.
a, b, c, α, β, γ are called lattice constants or lattice parameters.
Lattice system, Crystal system, Bravais lattice, Crystallographic point group, and Spacegroup are frequently used to classify crystals.
The lattice system is the classification based on the shape of the unit-cell parallelepiped.
The shapes of parallelepipeds are classified as follows.
The symmetry of a crystal dominates its physical properties. Neumann pointed out that the symmetry of any property which the crystal shows cannot be lower than the symmetry of the crystal (Neumann's Principle)
In fact, if the arrangement of atoms is unchanged by the operation on the crystal, the physical response of the crystal should be also unchanged.
The symmetry of a system is decided by a set of symmetry operations. If a operation does not cause any change, the operation is called a symmetry operation.
Operations which should be considered for a crystal are translations, rotations, space inversion, and their combinations. For a crystal with magnetic order, time reversal operation should also be considered.
For considering macroscopic physical responses, the translation can be neglected in most cases, because the primitive vectors are quite small. The crystallographic point group is hence a set of symmetry operations except translation.
To represent the operations, Hermann-Mauguin notations are used in the international crystallographic association, while Schönflies notation is also often used in the field of molecules and clusters.
The definition and representation of each operation are summarized below.
Space inversion about a certain point.
Space inversion is an important issue in a wide range of research fields.
| Operation | Hermann-Mauguin | Schönflies |
| Inversion | 1 | i |
Rotation around a certain axis.
The rotation operations allowed in conventional crystals are only four type; Rotations by 180 (= 360/2) degrees, 120 (= 360/3) degrees, 90 (= 360/4) degrees, and 60 (= 360/6) degrees.
Note that quasicrystals have rotational symmetries that are forbidden in crystals.
The rotation by 360/n degrees is called nfold rotation.
| Operation | Hermann-Mauguin | Schönflies |
| Twofold rotation | 2 | C2 |
| Threefold rotation | 3 | C3 |
| Fourfold rotation | 4 | C4 |
| Sixfold rotation | 6 | C6 |
The absence of mirror reflection symmetry is an important issue in biological materials.
| Operation | Hermann-Mauguin | Schönflies |
| Mirror Reflection | m | σ |
Rotoinversion is the combination of a rotation about an axis and the inversion about a point on the axis.
The combination of a 360/n-degree rotation and inversion is called the nfold rotoinversion.
The rotation by 180 degrees about the z axis inverts the x and y coordinates.
Therefore, after the combination of the rotation about the z axis and the inversion about the origin,
the x and y coordinates are restored and the z coordinate is reversed.
The twofold rotoinversion is equivalent to the reflection about a mirror plane of z=0.
Rotoreflction is the combination of a rotation about an axis and the reflection about a mirror normal to the axis.
The combination of a 360/n-degree rotation and mirror reflection is called the nfold rotoreflection.
Because the mirror reflection is equivalent to twofold rotoinversion, a rotoreflection operation can be rewritten by a rotoinversion operation. The combination of 360/n-degree rotation and the reflection is the same as a rotation by 360/n + 180 degrees followed by a space inversion operation. Therefore,
Rotoinversion operations are used in Hermann-Mauguin notation, while rotoreflections in Schönflies notation.
| Operation | Hermann-Mauguin | Schönflies |
| Threefold rotoinversion | 3 | S6 |
| Fourfold rotoinversion | 4 | S4 |
| Sixfold rotoinversion | 6 | S3 |
There are several ways to classify crystals.
There is a close relationship between the shape of the unit cell and the symmetric operations.
Therefore, we first classify the crystal by the shape of the unit cell.
Because any crystal has the translational symmetry, one can define a parallelepiped-shaped unit cell.
Generally we set the parallelpiped as symmetric as possible.
Then the shapes of parallelepipeds are classified as follows.
All the crystals are classified into these seven types of lattice systems; cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.
Note:
Crystals are classified into seven crystal systems from the viewpoint of the shape of unit cell; the arrangement of lattice points.
Symmetrical operation must keep not only the arrangement of lattice points but also the atomic arrangement.
Because symmetry control dominates the properties and responses of crystals, it is useful to subdivide the crystal system.
This is the crystal point group.
Cubic lattices have the following symmetry operations.
The atomic arrangement of a cubic system does not neccesarily keep all the operations as a symmetry operation. With only four threefold symmetry axes, the cubic symmetry is preserved. The cubic systems are classified as follows depending on the presence or absence of other symmetric operations.
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 4/m 3 2/m | m 3 m | Oh |
| 4 3 m | 4 3 m | Td |
| 4 3 2 | 4 3 2 | O |
| 2/m 3 | m 3 | Th |
| 2 3 | 2 3 | T |
Hexagonal lattices have the following symmetry operations.
The atomic arrangement of a hexagonal system does not neccesarily keep these operations as a symmetry operation. With only a threefold rotation symmetry, the lattice could belong to a hexagonal system. However, ignoring the translation operation, the presence of the threefold rotation cannot distinguish the hexagonal system from the rhombohedral system. Therefore, in the crystal point group, crystals having sixfold rotation or rotoinversion symmetry is classified into hexagonal point groups, and the others are classified into trigonal point groups. Here one should note that some trigonal crystals have a rhombohedral Bravais lattice and others have a hexagonal Bravais lattice. The hexagonal point groups are classified as follows depending on the presence or absence of other symmetric operations.
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 6/m 2/m 2/m | 6/m m m | D6h |
| 6/m 2 m
6/m m 2 |
6 2 m
6 m 2 |
D3h |
| 6 m m | 6 m m | C6v |
| 6 2 2 | 6 2 2 | D6 |
| 6/m | 6/m | C6h |
| 6 | 6 | C3h |
| 6 | 6 | C6 |
If the crystal has only one threefold rotation axis and no sixfold rotation or rotoinversion symmetry, the lattice is classified into trigonal point groups. All the crystals with rhombohedral lattice belong to the point groups. The rhombohedral lattice has the following symmetries.
The atomic arrangement of a trigonal system does not neccesarily keep all the operations as a symmetry operation. With only a threefold symmetry axis, the trigonal symmetry is preserved. The crystals are classified as follows depending on the presence or absence of other symmetry operations.
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 3 2/m | 3 m | D3d |
| 3 m | 3 m | C3v |
| 3 2 | 3 2 | D3 |
| 3 | 3 | C3i or S6 |
| 3 | 3 | C3 |
Tetragonal lattices have the following symmetries.
The atomic arrangement of a tetragonal crystal does not neccesarily keep all the operations as a symmetry operation. With only a fourfold rotation or rotoinversion symmetry, the tetragonal symmetry is preserved. The tetragonal systems are classified as follows depending on the presence or absence of other symmetric operations.
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 4/m 2/m 2/m | 4/m m m | D4h |
| 4 2 m
4 m 2 |
4 2 m
4 m 2 |
D2d |
| 4 m m | 4 m m | C4v |
| 4 2 2 | 4 2 2 | D4 |
| 4/m | 4/m | C4h |
| 4 | 4 | S4 |
| 4 | 4 | C4 |
Orthorhombic lattices have the following symmetries.
The atomic arrangement of an orthorhombic crystal does not neccesarily keep all the operations as a symmetry operation. Three out of the six symmetry operations preserve the orthorhombic symmetry. The orthorhombic systems are classified as follows depending on the symmetric operations.
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 2/m 2/m 2/m | m m m | D2h |
| m m 2 | m m 2 | C2v |
| 2 2 2 | 2 2 2 | D2 |
Monoclinic lattices have the following symmetries.
Either operation can keep the monoclinic symmetry. The monoclinic systems can be classified as follows.
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 2/m | 2/m | C2h |
| m | m | Cs |
| 2 | 2 | C2 |
Triclinic system has only one symmetry;
| Hermann-Mauguin (full) | Hermann-Mauguin (short) | Schönflies |
| 1 | 1 | Ci |
| 1 | 1 | C1 |
ARIMA Taka-hisa