Here we will discuss about acoustical and optical phonons in details. 

Today we will  cover 

1. Vibration of diatomic chain 

2. Dispersion relations for one dimensional diatomic lattice 

3. What is acoustical branch? (in details) 

4. What is acoustic phonons?  (in details) 

5. Graph of acoustical and optical phonons 

6. What is optical branch? (in details) 

7. What is optical phonons?  (in details) 

Vibration of diatomic chain 

     Let us take one dimensional diatomic lattice which has two atoms per primitive cell as in case of sodium chloride or the diamond structure.

Vibration of diatomic lattice
     Let m1 and m2 be the mass of these two atoms and suppose m1 < m2. The distance between two nearest neighbours be a. 

    Suppose the atoms are arranged along X axis and are placed at lattice sites 2n-3, 2n-2, 2n-1, 2n ,......... etc. The atom having small mass are placed at even numbered sites and atom with larger mass are placed at odd numbered sites.

       The assumption are supposed to be same as that for monoatomic lattice except that let x2n represent displacement of atom at 2n th site. 

  The equation of motion of two atoms can be written as 

Equation of motion of two atom


     Where alpha is the interatomic force constant. 

 The solution of both equation can be written as this respectively. 

Solution of equation


    In diatomic lattice, the two masses are not identical, so their amplitude of vibrations are taken to be different. Since both the both atoms are vibrating in identical manner, the frequency of both the atoms are taken same. 

 Now differentiate the solution of equation, we get 

Differentiate equation

After simplify these two equation we can write it in determinant form and it determinant is given by equation : 

Determinant

From this equation we can calculate two value of w² as 

Solution of angular momentum

    This above two equations are called dispersion relations for one dimensional diatomic lattice. 

Acoustical and optical phonons 

    The graph between the w and gives us two branches of the dispersion relations curve. 

One curve corresponding to w_ is called acoustical branch and other corresponding to w+ is called optical branch. 

Acoustical and optical phonons


Acoustical branch 

    The lower branch of the dispersion curve in the fig is known as acoustical curve or acoustical branch

For  k=0 ,sin(ka) = 0

   Hence  w_ = 0

For k<<1 , sin(ka) = ka 

    w_ = alpha [ 1/m1 + 1/m2 ]

In acoustical branch  the value of propagation constant k is restricted to the range between -(pi) /2a and +(pi) /2a as shown in fig. 

   For k = +-(pi) /2a, the frequency 

Acoustical branch frequency

  The amplitude of vibration for the acoustical branch are equal. 

   The vibration in the acoustical branch can be excited by force which compels the atoms in the crystal to move in the same direction. 

  As an example we may force a beam of longitudinal wave such as sound wave to be directed at the surface of a crystal to produce the desired effect. Because of this, the vibrations are termed as acoustical vibration and so on dispersion relations ,the acoustical branch is constituted. 

Discuss the case ,when acoustical beach vanishes 

      The acoustical branch vanishes when heavy particle of mass m2 tends to infinity. Then the mid point between each atom is tied down, thus isolating the atoms from one another. Physically each atom is independent of each other. 

   The acoustical branch remains unchanged when light particle of mass m1 tends to zero and the result of monoatomic lattice is obtained. 

What is acoustical phonon? 

When the atoms of lattice vibrate with the same phase and amplitudes about their mean position, the phonons are said acoustical phonons

    Acoustic phonons exhibit a linear relationship between frequency and phonon wave vector for long wavelength. The frequency of acoustic phonons tends to zero with longer wavelengths.

  Optical branch 

The upper beach of the dispersion curve is called optical curve. 

For k tends to zero, sin(ka) also tends to 0. 

that is neglegeble. 

   Now we can calculate value of w+ by putting sin(ka) =0 

  For ka tends to pi/2a and sin (ka)  tends to 1, we get 

Optical branch frequency


The optical branch has small vibrations over the entire range of k. Since the mode of the optical branch can be excited with visible light in solid which are partly ionic, we name the branch as optical branch. 

   The ratio of the amplitude is negative over the entire range of frequencies by the optical branch. 

What is optical phonons and why this is celled so? 

   Optical phonons are out of the phase movements of the atoms in the lattice. During vibrations, if one atom move towards the left then other atom move towards right. These phonons are called optical phonons because in most of ionic crystals such as NaCl, phonons are excited by infrared radiation. 

The electric field component of electromagnetic radiation compels positive sodium ion along the direction of the field and every negative chlorine ion moves in opposite direction to make the crystal vibrate.

       At Brillioun zone ,the optical phonons have a non zero frequency. Near the long wavelength limit, optical phonons show no dispersion. 

Optical phonons are responsible for the intersection of a solid with light because they have higher frequency. 

Forbidden band 

     There is a band of frequencies between the two branches acoustical and optical, which can not propagate. This band is called forbidden band. 


  This is all about acoustical and optical branch and its explanation. If you have any queries you can ask . Thank you. 

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