Crystal Symmetry in Solids | Materials | Engineering

The definite ordered arrangement of the faces and edges of a crystal is known as crystal symmetry. A sense of symmetry is a powerful tool for the study of internal structure of crystals. It is a simplifying key to the endlessly various arrays of atoms which make-up crystalline solids, enabling us to think of them in terms of a familiar pattern.

Crystals possess different symmetries or symmetry elements. They are described by certain operations. A symmetry operation is one that leaves the crystal and its environment invariant. That is, after performing an operation on the body, if the body becomes indistinguishable from its initial configuration, the body is said to possess a symmetry element corresponding to that particular operation.

Symmetry operations performed about a point or a line are called point group symmetry operations.

Different point group symmetry elements exhibited by crystals-centre of symmetry or inversion centre, reflection symmetry, rotation symmetry.

Symmetry Elements in a Cubic Crystal:

One of the noticeable features of many crystals, is a certain regularity of arrangement of faces. The next regular feature we must notice is the frequent occurrence of similar faces (of the same size and shape) in parallel pairs on opposite sides of the crystal.

A cube possesses three such pairs of parallel and opposite faces. Hence a cube is said to show a centre of symmetry, i.e., the body centre of the cube is a centre of symmetry. This centre lies at equal distances from various symmetrical positions. Centre of symmetry is also known as inversion centre.

A crystal will possess an inversion centre if for every lattice point given by the position vector r⃗ there will be a corresponding lattice point at the positron -r⃗. Thus inversion is a symmetry operation in a crystal equivalent to reflection through a point. This is shown in Fig. 2.13.

The second kind of symmetry element in a crystal is a plane of symmetry or reflection symmetry. A crystal is said to possess reflection symmetry about a plane if it is left unchanged in every way after being reflected by the plane. The two symmetry elements, namely, centre of inversion and plane of reflection, may easily be understood by saying that inversion is a symmetry operation similar to reflection, with the only difference that reflection occurs in a plane through the lattice point, while inversion is equivalent to reflection through a point. The three straight planes of symmetry in a cube are shown in Fig. 2.14 and the six diagonal planes of symmetry in a cube are shown in Fig. 2.15.

The next kind of symmetry element is the symmetry about a line, known as axis of symmetry or rotation axis. A body is said to possess rotational symmetry about an axis if after rotation of the body about this axis by some angle ɸ_n, body appears as it did prior to rotation; i.e., the body is left invariant as a result of rotation.

The axis of symmetry may thus be defined as a line such that the crystal assumes a congruent position for every rotation of [360/n]°. The value of n decides the fold of the axis. Only one, two, four and six-fold rotation axes of symmetry alone are possible in a single crystal lattice.

If a cube is rotated about a line perpendicular to one of its faces at its mid-point, it will turn into a congruent position every 90° (in distinguishable from the position it occupied originally), four times during a complete revolution; the normal is thus an axis of four-fold symmetry, a tetrad axis, and a cube clearly possesses three such axes, one normal to each of three pairs of parallel faces. The three tetrad axes of a cube are shown in Fig. 2.16.

Let the cube be now rotated about a solid diagonal (body diagonal) through 120° to get congruence, and such a line is therefore a triad axis (Fig. 2.17).

Finally, a line joining the mid-points of a pair of opposite parallel edges proves to be a diad axis, and there are six such axes present in a cube. These are presented in Fig. 2.18.

The total number of crystallographic symmetry elements of the cubic system is thus given by:

Centre of symmetry – 1

Straight planes – 3

Diagonal planes – 6 (9 planes)

Tetrad axes – 3

Triad axes – 4 (13 axes)

Diad axes – 6

Five-Fold Rotation Axis Not Compatible with a Lattice:

Consider a lattice row XPQY as shown in Fig. 2.19.

Let the lattice translation be ‘a’ and the lattice have n-fold rotation axis of symmetry. Let us rotate the vectors PX and QY through angle ɸ_n = (360/n)° in the clockwise and anti-clockwise directions respectively. The tips of the vectors P₁ and Q₁ in the new positions must be lattice points if the lattice were to possess n-fold rotation axis of symmetry. As per the definition, rotation operation must leave the lattice invariant. Clearly, P₁ Q₁ must be parallel to PQ and must be equal to an integral multiple of the lattice translation ‘a’; i.e., P₁ Q₁ = ma, where m is an integer.

Thus cos ɸ_n = P₁R/PP₁

P₁R = PP₁ cos ɸ_n = a cos ɸ_n

SQ₁ = a cos ɸ_n

Therefore, P₁Q₁ = P₁R + SQ₁ + RS

Now P₁Q₁ = 2a cos ɸ_n + a = ma

Thus, 1 + 2 cos ɸ_n = m

2 cos ɸ_n = [m – 1] = N, where N is an integer

cos ɸ_n = (m-1/2) = N/2

The possible values of N can be obtained such that cos ɸ_n lies between +1 and -1. This gives the value of ɸ_n which decides the permitted values of n, the fold of the rotation axis.

Combination of Symmetry Elements:

The different symmetry elements (centre of inversion, reflection and rotation) can also be combined if they are compatible. The different combinations give rise to different symmetry points in the crystal. It must be noted that all the crystals do not possess all the symmetries discussed above.

The different crystal systems exhibit different symmetries. It is found that there are 32 compatible combinations of the above three point group-symmetry elements, called simply as point groups. Crystals belonging to different crystal systems can be classified on the basis of point groups.

Rotation-Inversion Axis:

A crystal structure is said to possess a rotation inversion axis if it is brought into self-coincidence by rotation followed by an inversion about a lattice point through which the rotation axis passes. See Fig. 2.20.

Let us now consider an axis normal to the paper passes through the centre, operating on a pole 1 to rotate it through 90° to the position 4, followed by inversion to the position 2; this compound operation is then repeated until the original position is again reached. Thus from position 2 the pole is rotated a further 90° and inverted to position 3; rotated a further 90° and inverted to position 4; rotated a further 90° and inverted to resume position 1. Crystals can possess only 1-, 2-, 3-, 4- and 6-fold rotation-inversion axes.

For example, the 5-, 7-, 8- and higher fold symmetries does not occur in crystals. The answer is that it is impossible to make a compact repeating pattern with these shapes. If we try to pack together some pentagons we will soon find that some space is always left vacant. The only Figs that can be packed together are the parallelogram (2-fold), the equilateral triangle (3-fold), the square (4-fold) and the hexagons (6-fold). These are the only axial symmetries that are found in crystals. The one-fold axial symmetry corresponds to absence of symmetry.

Space Groups:

In a crystal, point group symmetry operations can also be combined with translation symmetry elements, provided they are compatible. Such combinations are called space groups. There are 230 space groups exhibited by crystals. The study of symmetry elements of the different crystals enables one to classify the crystals and their properties based on different symmetries.