Longitudinal plasmon oscillation

[scivet]

Kittel solid state physics book ( chapter 14)says when dielectric permittivity is zero, then longitudinal polarization wave possibly exists. It is hard to imagine how this is possible. Can anybody explain this?
If the permittivity is zero, then there shouldn'n be any response, right? How come the longitudinal mode-are generated?

 

[DrDu]

Given that D=\epsilon E, \epsilon=0 means that you can have an electric field E without an associated displacement field. If you take the full Maxwell equations you can see that for slightly higher frequencies there exist free longitudinal solutions of these equations which are not bound to external sources.

 

[scivet]

The epsilon is positive at the higher frequency than plasma, which means the transverse propagating wave. Right? So, epsilon is zero, then the electromagnetic wave equation says del^2 E is zero. How does this say the solution is longitudinal? It only says K, the wave vector must be also zero?

 

[DrDu]

The longitudinal dielectric constant is a function of both omega and wavevector k.
If D_L(\omega,k)=0 due to \epsilon_L(\omega,k)=0then clearly E_L(\omega, k) is a longitudinal solution of the free wave equation.