DEBYE THEORY OF HEAT CAPACITY OF SOLIDS

Debye avoided the most serious assumption for Einstein's theory namely that the vibration in a crystal lattice are independent. The concept of atomic structure has been introduced in this theory. A crystal is considered to be a continuous elastic medium. The atoms or ions or molecules in the crystal exhibit a simple harmonic motion and there will be interaction between the different oscillators. The normal mode which characterized the motion of the crystal as a whole have long wavelengths compared to crystal spacings. The various vibrational modes are distinguished by their unique frequency. Debye further assumed that the properties of the crystal could be determined from superposition of contributions made by individual atoms.

                              The number of oscillations in the frequency range υ and υ + dυ is given as dn. Using quantum mechanical principle dn is evaluated as 

dn =    4π²v  . ʋ² dʋ ....................(1)

             c³

V - volume of the crystal, c - velocity of light and υ - frequency of oscillator. 

There are two kinds of waves with in elastic materials. They are transverse waves with velocity c₁ with two directions of polarization for any given direction of propagation and longitudinal waves with velocity c₁. The total number of modes of vibration between ʋ and ʋ + dʋ will be

dn = (   2  +  1 ) 4π²V. ʋ² dʋ  ............(2)

           ct²     cl³

One mole of crystal contains N number of atoms or ions or molecules which will have (3N-6) vibrational degrees of freedom. As N is large ; it's approximated to 3N. Hence the upper limit for the allowed vibrational frequency ʋD is equal to 3N.

∫ dn = ∫₀ᵛᴰ 4π² V. v² (2 + 1 ) ʋ² dʋ = 3N

                                  ct³  cl³

     

           4π²V ( 2  +  1  )  ʋ³D  = 3N

                      ct³    cl³      3

( 2  + 1 ) =           9N              ..........(3)

 ct³   cl³       4π² V ʋ³D

Substitute equation (3) into equation (2)

dn =         9N         . 4π²V. ʋ² dʋ = 

          4π² V ʋ³D

       9Nʋ²dʋ       ...................(4)

         ʋ³D

The average energy of oscillator, vibrational modes in a crystal is assumed to be same as the average energy of oscillator in a black body as given by Planck. 

            E =       E.       

                   eE/kT-1

∴ Energy of dn vibrational modes = dn. E

    = (  9Nʋ²dʋ )  (      E          )

            ʋ³D            eE/kT− 1

  = (  9Nʋ²dʋ )   (       hʋ       )

         ʋ³D             ehʋ/kT − 1