In this section we will discuss

1. What is phonon and its representation?

2. Properties of phonon. Here also we discuss difference between photon and phonon.

3. U process or Umklapp process

4. Normal process or n process

The energy in a lattice vibration or an elastic wave is quantised. This is similar to the quantisation of electromagnetic energy.

**What is phonons ?**

Quantum of electromagnetic energy is called **phonon**.

Pictorial representation of o phonon

But **photon** is quantum of electromagnetic field including electromagnetic radiation.

A photon travels with the velocity of light where as phonon with the velocity of sound in a solid.

Remember, photons do not required any medium for their propagation where as phonons need a material to travel. Thus, phonons can never travel through vaccume.

Mathematically,

The energy of lattice vibration or phonons is given by

E= hw/2(pi)

Here, h is planck's constant

w is angular frequency of vibration

When space lattice is excited in nth state, then n phonons each of energy E are present.

One phonon is released when The lattice transists from n th mode to (n+1)th mode. Similarly during the transistion from n th mode to (n-1) th mode, a phonon is said to be destroyed. In this way, phonons are created and destroyed during the transition from one mode of vibration to another .

**Properties of phonon **

1. Phonon is produced as a result of the quantisation of lattice vibration.

2. Phonon contain both wave as well as particle nature. In other words, we conclude that wave particle duality is also applicable to phonon.

3. Phonon like photons have zero spin. Therefore phonons follows Bose-Einstein statistics and are indistinguishable particle.

4. The momentum of a phonon is called crystal momentum. It I'd represented as hw/2(pi) where k is wave vector.

In physical sense, a phonon on a lattice really has no momentum but when its particle nature is taken into account during the interaction with photon, electron and neutron, it behaves as if it possesses momentum.

5. Energy of a phonon is given by

E= hfk

Where f is frequency and k is phonon vector.

6. Due to Bose-Einstein's distribution law, the increase in temperature of crystal enhance the number of phonons and decrease in temperature destroys the phonons.

7. The phonon do nothing conserve in number as they can be created or destroyed.

8. The vibration spectrum of the phonon wave occupies a wide range of frequency from 10 ' 4 to 10 ' 12 Hz.

9. The low frequency part of the spectrum lies in the infrared region and is due to thermal vibrations in a crystal similar to the thermal excited photons in black body radiation.

10. Phonon travels with velocity of sound in solid.

11. The low frequency part of spectrum corresponds to acoustic region and comprises sound waves in a crystal similar lattice.

12. The number of phonons in a crystal is temperature dependent.

a. At finite temperature, the crystal are full of phonons.

b. At low temperature, acoustic modes are more populated than the optical modes.

c. At high temperatures, the atomic vibrations amplitude increases.

**Normal process**

In case of Bragg's diffraction that is in an elastic scattering of X ray photon by a crystal, we have

** K'= K+G**

Where K and K' are the incident and scattering wave vectors.

G is reciprocal lattice vector of crystal.

If we multiply the above equation by h/2(pi)

We get

Here the entire crystal recoil with momentum hG/2(pi) .

In the case of **elastic scattering** the frequency of scattered photon is same as that of the incident photon.

The process in which the frequency of incident photon is the same as that of scattering photon is known as *Normal process* or** N - process**.

In this case no phonon is created or destroy.

**Umklapp process or U-process **

In inelastic scattering of a photon by a crystal, the frequency of the incident photon changes and a new phonon is emitted along with an emitted or created during scattering, then due to the conservation of wave vector, we have

If a phonon of wave vector k is absorbed during the scattering, then according to the wave vector conservation law,

The emission and absorption of phonon in inelastic scattering is shown as

The process in which the frequency of the photon changes is known as Umklapp process or** U process**.

Umklapp is a German word which means flipping over. Apart from Bragg's reflection, a phonon is created or absorbed in this process. The momentum is transferred to the crystal as a whole.

This is all about phonons in elementary lattice vibration.

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