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## 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 [tex]T^2[/tex].

You should note that

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

and

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

## Homework Equations

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

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

## The Attempt at a Solution

So I know that the specific heat is:

[tex] C_V = \left(\frac{\partial U}{\partial T}\right)_V [/tex]

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

[tex] g(\omega) = V \frac{1}{8\pi^3}4\pi \frac{\omega^2}{C_s^3} [/tex]

And I also have

[tex] g(\omega) = g(k) \frac{1}{\frac{d\omega}{dk}} [/tex]

where

[tex] \frac{d\omega}{dk} = C_s [/tex]