Solid Sate Physics

Fundamental types of crystal lattices and their symmetry operations

Fundamental types of crystal lattices and their symmetry operations.

Topics covered
a. Types and classes of crystals,

b. Symmetry operations in crystals

In this lecture we will follow through our basic knowledge gained in the last lecture. — lecture — I, II, and shed light on the most interesting properties of crystal lattices, viz. their symmetry properties. Based on their properties we will classify them into various types and classes.

ii. Lattices satisfy additional symmetry operations. But due to the constraint of translational symmetry the total number of symmetry operations that the lattices can satisfy is reduced to a minimum.

iii. This means in 2-dimensional lattice constructs we have only 5 types of lattices which satisfy additional symmetry operations. In 3-dimensional geometry there are a total of 14 classes of lattices.

iv. Thus in 3-dimensional lattices the 14 classes of Bravais lattices are categorized into 7 types or systems of fundamental lattices.

v. The extra symmetry operations are

various rotations,
inversion about a space point and
reflection about a plane passing through a lattice point or
their possible combinations.

Crystal structure: introduction to lattice properties.

Lattice and crystals. 
A lattice is a three-dimensional periodic array of identical building blocks. The building blocks are atoms or groups of atoms. The crystals usually come with imperfection of structure and impurities. 

The periodicity of crystals is well established by the experimental studies of X-ray, neutron and electron diffraction patterns. 

A solid is a crystal if the positions of the atoms in it are exactly periodic. Here is a diagram that represents this property ideally. 

i. Distance between two nearest neighbors is ‘a’ along x-axis and ‘b’ along y-axis, where x–, and y– axes are not necessarily orthogonal.  

ii. A perfect crystal maintains the periodicity for -infinity < x < infinity and -infinity < y < infinity . The points A, B, C are equivalents. That means for an observer at A, the environment at A is exactly the same as it is for an observer at B or C.
This is expressed by saying crystals have translational symmetry. e.g. if the crystal is translated by a vector R — joining two atoms, the appearance of the crystal remains unchanged.

The atoms have no restrictions as to which location they preside over, as long as that position can be occupied by any  atom, it can be taken over by any other given atom, and all others would relent.  

Imperfection in crystals. 
There are no perfect crystals though, defined the above way. All crystals have some degree of imperfections. There are 3 basic examples of imperfections. 

i. Atoms near the surface have a different environment than atoms deep inside the crystal. 

ii. Due to thermal vibrations, equilibrium position of atoms are distorted, which depends on temperature T.

iii. Atoms always contain foreign elements known as impurities. 

The effect of imperfections can be neglected in very ideal crystals. Imperfections lead to interesting physical properties of crystals. E.g. Resistivity of metals is a result of thermal vibrations of atoms. — We will discuss this at a later time, in this course. 

When atoms are replaced by geometrical points, geometrical patterns depicting the periodicity of the crystals are obtained. They do not have any physical contents. Such geometrical patterns are known as "Lattice" or "crystal Lattice". 

Bravais and non-Bravais lattices. 
There are two classes of lattices, Bravais and non-Bravais lattices.

Bravais Lattice: In a Bravais lattice all lattice points are equivalent, hence all atoms of the crystal are of the same kind. 

Non-Bravais Lattice: In a non-Bravais lattice some of the lattice points are not equivalent. 

This is easily understood by the following diagram.