The failure in Einsteins theory of specific heat at low temperature is due to assumption that the vibrations of all atoms are simple harmonic and have one and same frequency.

"Debye" improved the Einsteins theory by considering the atomic oscillators as a system of coupled oscillators having a continuous range of frequencies.

**Essential difference between Debye & Einstein model:**

Debye has considered the vibrational modes of a crystal as a whole, where as Einstein has considered the vibration of a single atom with the assumption that atomic vibrations are independent of each other.

Debye has made following assumptions:

- The solid is capable of vibrating elastically in many different modes.
- The frequency of vibration is different for different modes.
- The number of modes of vibration of waves solids are limited in number.
- The maximum frequency is the fundamental frequency of solid. The maximum frequency is frequency of shorter waves which the solid can transmit.

According to Debye, a solid can be treated as an elastic body in which vibrations of atoms generate "stationary waves" of both longitudinal & transverse types with velocities Vl & Vt respectively.

The velocities can be determined by elastic constants & densities of solids.

The frequencies range from zero to a definite upper limit.

The number of modes of longitudinal waves per unit volume with frequencies between '𝝂' & '𝝂+d𝝂 ' is represented by 4Π𝝂²d𝝂/(Vl)³.

The number of modes of transverse vibrations per unit volume with frequencies between '𝝂' & '𝝂+d𝝂 ' is represented by 8Π𝝂²d𝝂/(Vt)³. Here 8Π is taken in place of 4Π. Because transverse vibrations have two independent directions of vibration i.e. they are equivalent to two waves at right angles to each other.

So total number of modes of vibrations per unit volume with frequencies '𝝂' & '𝝂+d𝝂 ' is given by

4Π[(1/Vl³)+(2/Vt³)]𝝂²d𝝂

The total number of independent modes of vibrations is given by

4ΠV[(1/Vl³)+(2/Vt³)] ∫𝝂²d𝝂 ; V is volume of gm-mole of solid

## No comments:

## Post a Comment