We have studied so far that the crystal can not propagate all the frequencies, they allow only those frequencies which fall in allowed band. These allowed bands are separated from each other by forbidden band.

If there are N number of atoms per Premitive cell the allowed frequencies split into N bands.

The field of elastic wave treated like a gas made up of quanta of the normal modes of the lattice or of phonons having the energy E = hw/2(pi)

And momentum of phonons is given by

p = hw/v2(pi)

Where v = velocity of sound

Whenever a crystal is heated it behaves as a box with phonon gas. Phonons are described by Bose-Einstein distribution function as photons.

The energy density E(f) of electromagnetic radiation consisting of photons lying between frequency f and f+df is given by

Where h is plank's constant, f is frequency, kB is Boltzmann's constant, c is the velocity of electromagnetic radiation and T is the temperature.

Every oscillating atom represents the normal modes of the lattice in which all the atoms of the crystal take part vibrating with the same frequency w.

The energy of quantum oscillator is expressed as

Where n = 0,1,2,.......... is the quantum number.

It consist of set of discrete level spaced at the interval hw/2(pi) .

The minimum portion of energy that can be absorbed or emitted by the lattice in the process of thermal vibrations corresponding to the transition of the normal mode being excited from the given level to the adjacent level and equal to **E=hw/2(pi)**.

Depending on the immensity of excitation of the normal mode, the crystal can emit a definite number of phonons. Hence if some mode is excited to the third level, in the above fig, the energy becomes 7E/2 .v

It means that the normal mode has produced three identical phonon each with an energy of E.

The graph between energy density and energy is this

This is all about qualitative description of phonon spectrum in solids.

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