What are the dielectric constants of metals and i have heard that some materials have dielectric constants that are complex numbers. Please tell me which materials have complex Dielectric constants?
They all do. The dielectric constants of metals are all complex and are frequency dependent. For the most part, their imaginary parts of the permittivity are very large and are the most dominant for frequencies below the terahertz region. Above that it is difficult to generalize because then you start to near the various resonant frequencies of the permittivities.
The permittivity at DC is largely immaterial as a metal is a good conductor. At DC, they will all (though I'm sure someone knows an exception that will interject) behave like a perfect electrical conductor. That is, they will have infinite conductivity (or imaginary part).
The imaginary part of the permittivity is the loss factor. You can also relate it to the conductivity of the material. The very high conductivities of metals means that most electromagnetic waves, with the exception of those in very very low frequencies, will not penetrate any appreciable distance into a metal before being completely attenuated. The real part of the permittivity, that regulates wave phenomenon like refratction, will not be much of a factor since the waves cannot penetrate far into the material. Of course, this is still dependent on the frequency, the type of material that you are working with and the physical dimensions of the material.
Strictly speaking all materials have complex dielectric functions. In the case of metals, the dielectric function has not only a pronounced dependence on frequency but also on wavenumber. A model for the dielectric function of metals is the Lindhard or random phase dielectric constant.
The imaginary part of the permittivity is related to the ratio between the optical conductivity and the frequency omega. So it would appear that in the limit omega --> 0
the imaginary part of the permittivity should diverge, for any metal with finite conductivity, right ?
That is correct, although there is a limit to the range of frequencies over which the conductivity is valid. At DC, any good conductor will basically behave like a perfect electrical conductor for any static problems. Even for slowly varying problems, like with quasi-statics I feel this should probably hold true as well. This is of course reflected in what you stated in that the imaginary part of the permittivity will diverge. A perfect electrical conductor has an infinite imaginary part.
I assume by "perfect conductor" you mean a material with zero resistance, or, which is the
same, infinite conductivity sigma.
I believe the imaginary part of the permittivity should diverge at zero frequency even for a
"non perfect" electrical conductor, i.e. a conductor with a finite value of
sigma (finite DC electrical conductivity), as it is the omega in the denominator to make the
sigma/omega ratio diverge for omega --> 0, isn't it ?