Longitudinal plasmon oscillation

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In the discussion on longitudinal plasmon oscillation, participants explore the implications of zero dielectric permittivity as stated in Kittel's solid state physics book. The central question revolves around how longitudinal polarization waves can exist when permittivity is zero, suggesting a lack of response. It is noted that while D = εE implies no displacement field at ε = 0, Maxwell's equations indicate that free longitudinal solutions can arise at frequencies above plasma frequency. The conversation highlights that the longitudinal dielectric constant is dependent on both frequency and wavevector, leading to the conclusion that a longitudinal electric field can exist even when the dielectric constant is zero. This nuanced understanding of electromagnetic wave behavior under specific conditions is crucial for grasping the nature of longitudinal modes.
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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?
 
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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.
 
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?
 
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.
 

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