Band Theory of Solids | Bloch theorem


Discuss the periodicily character of potential in a crystal and Bloch theorem in this reference

 Periodicity character of potential in a crystal

The potential energy of an electron at a distance r from an atomic nucleus of charge +Ze is given by,


When a crystal is formed, a number of such atomic nuclei are brought close to each other. The potential of an atom in such a case is sum of the potential energies due to individual nuclei. The variation of potential energy with distance in one dimensinal case is shown in figure –

The atomic nuclei have a positive charge and the electrostatic potential energy of an electron in the field of positive charge is attractive i.e negative.

To a reasonable approximation the nuclei have been considered at rest and the potential experienced by an electron in the crystal is assumed to be periodic with the period equal to a lattice constant a.

Bloch theorem : Wave equation in periodic potential

In order to consider the motion of an electron in a crystalline solid, we apply Schroedinger equation for electrons and find its solution under periodic boundary conditions.

The solution of Schroedinger equation was modified by Scientist Bloch by considering the symmetry properties of the potential in which the electron in a crystalline solid moves.

    1. Wave equation in periodic potential 

In free electron theory, the electron is supposed to move in a constant potential Vo and Schroedinger wave equation for an electron in one dimensimal, case is given by,

The solution of above equation,

i.e. Kinetic energy, (Ek) = p2/2m

Physical meaning of K is p/ℏ  i.e momentum of electron devided by ℏ.

Now consider the Schroedinger equation for an electron moving in one dimensional periodic potential. Thus, the potential energy of the electrion satisfy the equation as,

V(x) = V(x+a)   ………………3

Where a is the period equal to lattice constant.

Then the Schroedender equation can be written as,


    2. Bloch theorem

If an electron in an one dimensimal lattice of lattice constant a characterised by potential function, V(x) = V(x+a) then Schroedinger equation,

Then the wave functions ψ(x) of electron are in the form

So the solutions are plane waves modulated By Uk(x) which has the same periodicily as the lattice.

This theorem is known as Bloch theorem.


Read more –

CRYSTAL STRUCTURE | Classification of solids| Lattice and lattice points, Crystal structure| Bravais space lattice| Unit cell, Primitive cell| Miller indices.

CRYSTAL STRUCTURE | Packing fraction| sc , bcc, fcc| CRYSTAL DIFFRACTION| Bragg’s law |Von Laue equation of scattering vector

Crystal Structure | Reciprocal Lattice

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