Linear Monoatomic vibration

Let us assume that 

1. Atoms are arranged linearly in a straight line. 

2. The interatomic spacing (a) is constant. 

3. The mass (m) of each atom in crystal is same. 

4. The atom supposed to be a rigid sphere. 

5. The crystal is under the influence of short range forces where the interactions between atoms are confined to only the nearest neighbour. 

6. Each atom in one dimensional monoatomic lattice is connected with a spring having spring constant a and the spring is massless. 

7. The force on the atom is directly proportional to the extension or contraction of its nearest neighbouring distance. 

8. All atom in linear array obey Hook's law. 

9. For simplicity we consider only the longitudinal mode of vibrations.

10. A linear chain of identical atoms extended along the X axis as shown in figure. 

Linear monoatomic vibrations


    At mean position, the distance of nth atom from the origin is (na). When the crystal lattice starts vibration, the atom get displaced by a small magnitude. 

    Here we discuss the result according to fig. 

Here w is angular frequency of atom and k is propagation constant .

 Here relation between them is 

        w = w(o) sin(ka/2)

This relation is called as dispersion  relation. For simplicity we consider only there positive root of above equation so as to have real frequency.

    Here w(o) is maximum angular frequency. 

Knowing the mass of the atom from standard values , we can obtain the expression for the frequency f, called the Einstein's frequency 

     f = (1/2pi) (a/m)-½

Here a is interatomic spacing constant. 

   If we plot a graph between propagation constant or a wave vector k along X axis and corresponding value of angular frequency along Y axis for a longitudinal wave of a linear one dimension lattice, then the dispersion curve will look like as 

Normal mode


   Normal mode 

 A correlated motion of the atoms which has a characteristic wave vector k and angular frequency w is known as normal mode of a lattice. 

   Once this motion get started, it will go on continuously, provided the dissipative forces such as friction are negligible. 

  The powerful bonds transmit the vibration of one atom to other atom so as to have a collective motion in the form of an elastic wave. 

   This collective motion of atoms is referred as the normal mode of a lattice. The number of normal modes coincides with the number of degree of freedom. The degree of freedom is equal to 3N where N is the number of atoms constituting the crystal. 

    This is all about linear monoatomic vibrations. 

    

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