An operation that does not change the atomic arrangement of a certain crystal is called a symmetry operation.
The atomic arrangement is exactly unchanged by some operations, and uniformly translated by some other operations.
Here we neglect the difference between them, and regard the both types of operations as symmetry operations.
Then the crystals are classified into crystal point groups, by considering the symmetry 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.
Symmetric operations which are necessary to describe the point group are listed below.
| 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 |
Symmetrical operation is closely related to physical properties.
Consider the electric polarization as a typical physical quantity at zero external field.
If the electric polarization appears along the z axis, the spatial inversion operation should reverse the electric polarization.
The atomic arrangement cannot overlap with the original one.
After all, the inversion operation cannot be a symmetric operation in crystals in which electric polarization exists.
Likewise, if the rotation about the x-axis or the y-axis is a symmetrical operation, there is no electric polarization in the z direction.
Next let us consider the external response such as the dielectric response and the piezoelectric response.
Dielectric constant is represented by using two subscripts (rank-2 tensor); one represents the component of the electric flux vector, and the other the component of the electric field.
The distortion is also expressed by using two subscripts; one represents the separation of two points, and the other the difference in the displacement between the two points.
Piezoelectric effect expresses the phenomenon that electric polarization is induced by applying distortion.
Hence the piezoelectric coefficient is represented by three suffixes (rank-3 tensor).
Consider symmetry operation on crystal measuring dielectric constant.
The applied electric field and the electric flux are changed their orientations by a symmetry operation of rotation, inversion, or mirror reflection.
However, the response should be unchanged because the symmetric operation does not affect the atomic arrangement (except for the translation operation).
In other words, the tensor representing the response must be the same.
The components of the tensor should have some relations.
Let's consider the dielectric constant of a crystal with sixfold symmetry around the z-axis.
The electric flux vector in the external electric field
(E1,E2,E3)
is expressed as
(εxxE1
+εxyE2
+εxzE3,
εyxE1
+εyyE2
+εyzE3,
εzxE1
+εzyE2
+εzzE3).
Any repitation of 60-degree rotations does not change the atomic arrangement.
The dielectric constant tensor hence does not change, either.
On the other hand, the applied electric field is
(−E1,−E2,E3).
Thus the electric flux vector is calculated as
(−εxxE1
−εxyE2
+εxzE3,
−εyxE1
−εyyE2
+εyzE3,
−εzxE1
−εzyE2
+εzzE3).
This must agree with the previous electric flux vector rotated 180 (=60 x 3) degrees about the z axis.
After all,
εxz
=εyz
=εzx
=εzy
=0.
A 60-degree rotation does not change the atomic arrangement.
The dielectric constant tensor does not change, either.
Because the electric field is changed to
(E1/2-sqrt(3)E2/2,
sqrt(3)E1/2+E2/2,
E3),
the electric flux vector is calculated as
(
εxx(E1/2-sqrt(3)E2/2)
+εxy(sqrt(3)E1/2+E2/2),
εyx(E1/2-sqrt(3)E2/2)
+εyy(sqrt(3)E1/2+E2/2),
εzzE3)
Comparing this vector with the one prior to the original electric flux, one can conclude
εxx=εyy,
and
εxy=εyx=0.
As mentioned above, the symmetry of a crystal is essential in determining the physical properties. Here it should be noted that one should also consider the time reversal operation for discussing the physical properties related to the magnetism. Though this is beyond the scope of this lecture, the similar discussion can be applied.
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 |
Taka-hisa Arima