Why is there a sum over $\mathbf{k}$ in Debye's calculation

Question

In Einstein's calculation for the specific heat of solids, the expression for the average energy in 1d is $$\langle E\rangle = \hbar\omega\left(n_B(\beta\hbar\omega)+\frac{1}{2}\right)$$ In the book I'm reading, the author says:

Debye decided that the oscillation modes of a solid were waves with frequencies ω(k) = v|k| with v the sound velocity—and for each k there should be three possible oscillation modes, one for each direction of motion. Thus he wrote an expression entirely analogous to Einstein’s expression $$\langle E\rangle = 3\sum_{\mathbf{k}}\hbar\omega (\mathbf{k})\left(n_B\left(\beta\hbar\omega(\mathbf{k})\right)+\frac{1}{2}\right)$$

And then he goes on to do some calculations and arrive at the specific heat expression. My problem is that I have no idea why there is that summation over k. Summing over the k's means summing over all the possible wavelengths, right? But why should that be? Shouldn't that equation be true for some sort of average wavelength to fit the average energy without the sum?

Answer 1

Debye's idea was to consider all the possible modes of the lattice, a more realistic hypothesis with respect to Einstein that considered just one instead. Then, each mode has its Planck distribution and so, your avearge is the sum of all the averages with respect to each mode. This is the reason why you get that sum. The factor 3 accounts for the possible oscillation modes that can be in three different directions, quite similar to the two polarization modes of photons.