Answer to Question #106819 in Molecular Physics | Thermodynamics for MS

Question

Give the assumptions of Debye model. Hence calculate (gammaD)^3 where gamma is Debye cut -off frequency.

Answer 1

Assumptions of Debye model: It is a a model developed by the Peter Debay in 1912 for estimating the phenon contribution to the specific heat in a solid.

There should be low-frequency modes of oscillation in which large groups of atoms are moving together, and also high-frequency modes in which atoms are moving opposite to their neighbors.

The units of energy come in different sizes, proportional to the frequency of the modes of the vibration.

We know that the dispersion relation is

$\omega= vq$

v is the velocity of sound.

The density of the state is $D(\omega)=\dfrac{V\omega ^2}{2\pi^2v^3}---------(i)$

we know that,

$\Sigma_p\int_0^\omega D(\omega)d\omega=3N-----(ii)$

where $\omega$ is the Debay frequency

from the equation (i) and (ii)

$\Rightarrow \Sigma_p\int_0^\omega \dfrac{V\omega ^2}{2\pi^2v^3}d\omega=3N$

$\Rightarrow \dfrac{V\omega ^3}{6\pi^2v^3}=3N$

$\omega_D=(\dfrac{6\pi^2v^3 N}{V})^{1/3}$

$q_D=\dfrac{\omega_D}{v}=(\dfrac{6\pi N}{V})^{(1/3)}$