# Finding plasmon dispersion

I need help with a problem in solid state physics:

Find the dispersion relation of long-wavelength plasmons in a simple tetragonal crystal in the case of almost empty band built of s-type orbitals. What happens in the case when $$m_{xx} << m_{zz}$$ ?

This is what I did for now (with some help from a friend):

I started from:

$$\omega^2 = \frac{c^2 k^2}{\epsilon(\omega)}$$

For wave vector k=0 (this is very long wavelength approximation) I have formula (from my lectures, but there is also in Ashcroft & Mermin, (26.19)):

$$\epsilon(\omega) = 1 - \frac{\Omega_p^2}{\omega^2}$$, where $$\Omega_p$$ is plasma frequency:

$$\Omega_p^2 = \frac{4\pi e^2 n}{m}$$, where m is (I suppose effective?) electron mass, which needs to be found. Effective mass tensor is defined by:

$$m_{ij} = \frac{\hbar^2}{\partial^2 E(k) / \partial k_i \partial k_j}$$

where E(k) for tetragonal lattice with s-type orbitals is:

$$E(k) = E_s - J \cos(k_x a) - J \cos(k_y a) - J \cos(k_z c)$$.

So what "m" shoud I insert in equation for $$\Omega_p$$, maybe $$\sqrt{2m_{xx}^2 + m_{zz}^2}$$ ? In that case last part would be easy.. . Is this making sense ? :uhh:

Thanks.