Editing Debye model (section)

==== Two-dimensional crystal ====
The same derivation could be done for a two-dimensional crystal. The number of modes remains unchanged, because there are still three polarizations. The derivation is analogous to the previous two. We start with the same equation,
:<math>\sum_{\rm modes}3=3 N.</math>

And then the left hand side is rewritten and equated to <math>3N</math>
:<math> \sum_{\rm modes}3=\frac {3 A}{(2 \pi)^2} \iint d \mathbf k = \frac {3 A}{2 \pi v_{\rm s}^2} \int_{0}^{\omega_{\rm D}} \omega d \omega = \frac {3 A \omega_{\rm D}^2}{4 \pi v_{\rm s}^2} = 3 N ,</math>
where <math> A \equiv L^2</math> is the size of the system.

It can be rewritten as
:<math> \omega_{\rm D}^2 = \frac {4 \pi N}{A} v_{\rm s}^2 .</math>