# Crystallographic Point Group and Physical Properties

We learn about crystal point group, which is closely related to the physical properties.

## Symmetry Operation

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.

#### Notations of 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.

### Inversion

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

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

### Mirror Reflection

The absence of mirror reflection symmetry is an important issue in biological materials.
 Operation Hermann-Mauguin Schönflies Mirror Reflection m σ

### Rotoinversion

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.

### Rotoreflection

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.

#### Equivalence between Rotoinversion and 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,

• Threefold rotoreflection is equivalent to the combination of a rotation by 120+180=300 degrees and an inversion. That is a sixfold rotoinversion.
• Fourfold rotoreflection is equivalent to the combination of a rotation by 90 + 180 = 270 degrees and an inversion. That is a fourfold rotoinversion.
• Sixfold rotoreflection is equivalent to the combination of a rotation by 60 + 180 = 240 degrees and an inversion. That is a threefold rotoinversion.

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 and physical properties

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
(−εxxE1εxyE2 +εxzE3, −εyxE1εyyE2 +εyzE3, −εzxE1εzyE2 +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.

## Lattice System and Crystal System

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.

• Cube: Parallelepiped surrounded by six squares
• Square Pillar: Parallelepiped surrounded by two squares and four rectangles
• Rectangular Parallelepiped: Parallelepiped surrounded by six rectangles
• Joined Two Regular Trianglar Pillars: Parallel piped surrounded by two rhombi with angles of 120 degrees and 60 degrees and four identical rectangles
• Rhombohedron: Parallelepiped surrounded by six identical rhombi
• Parallelepiped surrounded by four rectangles and two parallelograms
• Other parallelepiped

All the crystals are classified into these seven types of lattice systems; cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.

Note:
When the crystals are classified into point groups, as described later, not the shape of the unit cell but the symmetry operation of the atomic arrangement is essential. This classification is called crystal system (but not lattice system). There are seven types of crystal systems; cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic. Because the rhombohedral lattice has a trigonal rotation axis, it is classified into trigonal crystal system. If the crystals with hexagonal lattice do not have a sixfold rotation or rotoinversion axis, they are also classified into trigonal crystal system.

## 32 Crystallographic Point Groups

Crystals are classified into 32 point groups from the viewpoint of symmetric operations.

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 Point Groups

Cubic lattices have the following symmetry operations.

• Fourfold rotation (4) and rotoinversion about the x, y, zaxes and reflection mirrors (m) perpendicular to the axes.
• Threefold rotation and rotoinversions (3) about four diagonal axes of the cube.
• Twofold rotation axes (2) connecting the midpoints of two opposite sides and reflection mirrors (m) perpendicular to the axes.
The symmetry operations are described in the abovementioned order as 4/m 3 2/m.

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 Point Groups

Hexagonal lattices have the following symmetry operations.

• A sixfold rotation (6) axis, which should be set to z, and a reflection mirror (m) perpendicular to the axis.
• Twofold rotation (2) about three types of axes connecting neighboring lattice points on a plane perpendicular to the z axis and mirror reflection (m) perpendicular to the twofold axis.
• Three other twofold rotation (2) axes shifted by 30 degrees from the above-mentioned twofold axes and a mirror reflection (m) perpendicular to these twofold axis.
The symmetry operations are described in the abovementioned order as 6/m 2/m 2/m. Note that the inversion symmetry can be expressed as 2/m.

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

### Trigonal Point Groups

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.

• A threefold rotation and rotoinversion axis 3.
• Three twofold rotation axes perpendicular to the threefold axis and mirrors perpendicular to the twofold axes.
The symmetry operations are described in the abovementioned order as 3 2/m. Note that the inversion symmetry can be expressed as 2/m. One should note here again that in the crystals of a hexagonal lattice does not have neither sixfold rotation nor sixfold rotoinversion, the crystal is classified into a trigonal point group but not hexagonal.

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 Point Groups

Tetragonal lattices have the following symmetries.

• A fourfold rotation (4) and rotoinversion axis, which should be set to z, and reflection mirror perpendicular to the fourfold axis.
• Twofold rotation symmetries about the x and y axes, and mirror reflections perpendicular to the axes.
• Twofold rotation about the diagonal of the square, and reflection perpendicular to the diagonal axes.
The symmetry operations are described in the abovementioned order as 4/m 2/m 2/m Note that the inversion symmetry can be expressed as 2/m.

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 Point Groups

Orthorhombic lattices have the following symmetries.

• Twofold rotation about the x axis and a mirror reflection normal to the axis.
• Twofold rotation about the y axis and a mirror reflection normal to the axis.
• Twofold rotation about the z axis and a mirror reflection normal to the axis.
The symmetry operations are described in the abovementioned order as 2/m 2/m 2/m, Note that the inversion symmetry can be expressed as 2/m.

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 Point Groups

Monoclinic lattices have the following symmetries.

• Twofold rotation (2) about an axis, which should be defined as y-axis, and a mirror reflection normal to the axis.

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 Point Groups

Triclinic system has only one symmetry;

• Spatial inversion
The triclinic systems are classified as follows.
 Hermann-Mauguin (full) Hermann-Mauguin (short) Schönflies 1 1 Ci 1 1 C1

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Taka-hisa Arima