How Does Dimensionality Affect Phonon Polarizations?

[Kyuubi]

Homework Statement

Consider a crystal made up of two-dimensional layers of atoms, with rigid coupling between the layers. You may assume the motion of atoms is restricted to the plane of the layer. Calculate the phonon heat capacity in the Debye approximation, and show that the low temperature limit is proportional to $T^2$.

Relevant Equations

In 3D,
$\sum_n{(...)} = \frac{3}{8}\int4\pi n^2dn(...)$

In 2D,
$\sum_n{(...)} = \frac{a}{4}\int2\pi ndn(...)$

What is $a$?

When going from 3 to 2 dimensions, I am unsure about how the number of polarizations will be affected.

I know the following though:
The 1/8 factor becomes a 1/4 since we are now integrating over the positive quadrant in 2d rather than the positive octant in 3d.
The $4\pi n^2$ becomes a $2\pi n$ because we moved from spherical coordinates to just polar coordinates.

I am told that in 3D, an elastic wave has three polarizations: two transverse and one longitudinal. Do they remain unchanged in lower dimensions? My guess is that they should become 2. But if we lose a polarization every time we lower dimensions, then a photon gas in one dimension shouldn't exist, but it does since the book has a question about it :)

 

[Kyuubi]

Homework Statement: Consider a crystal made up of two-dimensional layers of atoms, with rigid coupling between the layers. You may assume the motion of atoms is restricted to the plane of the layer. Calculate the phonon heat capacity in the Debye approximation, and show that the low temperature limit is proportional to $T^2$.
Relevant Equations: In 3D,
$\sum_n{(...)} = \frac{3}{8}\int4\pi n^2dn(...)$

In 2D,
$\sum_n{(...)} = \frac{a}{4}\int2\pi ndn(...)$

What is $a$?

When going from 3 to 2 dimensions, I am unsure about how the number of polarizations will be affected.

I know the following though:
The 1/8 factor becomes a 1/4 since we are now integrating over the positive quadrant in 2d rather than the positive octant in 3d.
The $4\pi n^2$ becomes a $2\pi n$ because we moved from spherical coordinates to just polar coordinates.

I am told that in 3D, an elastic wave has three polarizations: two transverse and one longitudinal. Do they remain unchanged in lower dimensions? My guess is that they should become 2. But if we lose a polarization every time we lower dimensions, then a photon gas in one dimension shouldn't exist, but it does since the book has a question about it :)

It's been answered for me. Indeed the number of polarizations goes down. I suppose then that EM modes retain their 2 polarizations regardless of dimension (?)