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?