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I am trying to calculate the Debye temperature, $\theta_D$, of copper using the following:

$$ \theta_D = \frac{\hbar v_s}{k_B} \left( \frac{6\pi^2N}{V} \right)^{1/3} $$

I have the following values: $\rho = 8900$ kgm$^{-3}$, $v_s = 3800$ms$^{-1}$, atomic mass $ M_a=63.5$gmol$^{-1}$. Now, the speed of sound, $v_s$ is not correct for copper according to online tables, and it seems to be closer to $4600$ms$^{-1}$.

However, I also know that the Debye temp for copper is about $343$K.

Using the fact that,

$$ {N\over V} = \frac{N_A\rho}{M_a} = 8.44\times10^{27} $$

where $N_A$ is Avogadro's number, I get,

$$ \begin{align} \theta_D &= \frac{(1.055\times10^{-34})(4600)}{(1.381\times10^{-23})}\cdot \left( 6\pi^2\cdot8.44\times10^{27} \right)^{1/3} \\\\ &=279K \end{align} $$

Which just isn't right. And using the value for $v_s$ provided in the question gives an even lower answer of $230$K...which isn't right either.

Am I missing something here?

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  • $\begingroup$ The number density should of the order 10^28. $\endgroup$ – nasu Oct 6 '18 at 11:43
  • $\begingroup$ @nasu doing that gives an answer of around 600K, which is even worse! $\endgroup$ – monkeyofscience Oct 6 '18 at 20:38
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There are few things going on here. The first is that you seem to be mixing units for density and the molar mass, using kg in one case, and g in the other. If you fix that, you will correctly get a number density on the order of $10^{28}$. However, you still won't find good agreement with the $~345K$ value you expect. Why is this?

Well, there's a second and subtler thing going on, which is that you are using a single speed of sound. In reality, the speed of sound is different in the (one) longitudinal and (two) transverse directions. If you instead use a mean speed calculated through $$\bar{v}_s = 3^\frac{1}{3}\left( \frac{1}{v^3_{\mathrm{transverse}}}+\frac{2}{v^3_{\mathrm{longitudinal}}}\right)^{-\frac{1}{3}} $$ you'll get a lot closer to the experimental value. You'll note that this isn't an ordinary average velocity, it's simply a constant defined in the derivation of the Debye temperature.

The third thing worth mentioning is that the speed of sound is going to depend on how your sample of copper was made. The value $v_s=3800$m/s is the longitudinal sound speed in thin copper rods, whereas $v_s=4600$m/s is a (rather low) value for the bulk material.

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