How Does Dimensionality Affect Specific Heat in the Debye Model?

[TFM]

Homework Statement

Consider a linear array of N similar atoms, the separation between nearest neighbours being a. Discuss the specific heat of the system on the basis of the Debye approximation and show that at low temperatures, the specific heat would be expected to be proportional to T.

Do the same thing for a 2-D square array and show that the expected low temperature dependence is now $$T^2$$.

You should note that

$$\int^{\theta_D/T}_0 \frac{x}{e^x - 1}dx \rightarrow$$ constant as $$\frac{\theta_D}{T} \rightarrow \infty$$

and

$$\int^{\theta_D/T}_0 \frac{x^2}{e^x - 1}dx \rightarrow$$ constant as $$\frac{\theta_D}{T} \rightarrow \infty$$

Homework Equations

$$U = \Sigma_{\omega} E_{\omega} -> \int^{\omega}_{0} \overline{E}(\omega)g(\omega)d\omega$$

$$\overline{E}(\omega) = 1/2 \hbar \omega + \frac{1}{exp\left(\left(\frac{\hbar\omega}{kT}\right) - 1\right)}$$

The Attempt at a Solution

So I know that the specific heat is:

$$C_V = \left(\frac{\partial U}{\partial T}\right)_V$$

And U is given by the above relevant equations.

However I am not sure how to do this for the 1D and 2D.

My notes have the 3D

$$g(\omega) = V \frac{1}{8\pi^3}4\pi \frac{\omega^2}{C_s^3}$$

And I also have

$$g(\omega) = g(k) \frac{1}{\frac{d\omega}{dk}}$$

where

$$\frac{d\omega}{dk} = C_s$$

 

[Gambla]

In 2D all that changes is your g($$\omega$$).

Then you can do the calculations for the specific heat.

The g you have is assuming that you have a sphere in k space. For 2D assume that you have a circle in k space.

Chapter 2 (around page 44) from Zieman is a great source for this type of stuff. Good luck!