More proper name for the dielectric constant is permittivity which is the generally a tensor describing the relationship between the electric displacement vector and the electric field vector at a certain frequency. For isotropic materials permittivity is a scalar but frequency dependent quantity. Frequency dependence of permittivity is called dispersion. For some materials permittivity vary but it is positive and can be described by a simply Debye model. For some materials at certain frequency ranges permittivity can be negative e.g. for plasma at RF frequencies or for metals at infrared frequencies. Negative permittivity means that the electric displacement vector and the electric field vector point in the opposite directions but it does not necessary mean that the electric energy stored is such medium is negative. For more details about for dispersive media have a look into a book on electrodynamics e.g. Jackson, J.D. Classical Electrodynamics. 3rd Ed. John Wiley & Sons, Inc., (1999).

In the simplest sense, a material having a positive dielectric constant implies it is acting as a capacitor. Now, there is no ideal dielectric in nature. Every dielectric material has certain amount of ions in it, which gives it certain conductive properties. Added with this, during dielectric measurements there is always a high chance of contact mismatch or impedance mismatch. At high frequencies, there may be resonance due to inductance of the cable, capacitance and resistance of the test object. Above the resonance frequency, inductive properties will dominate, which will give negative dielectric constants at those frequencies. In some cases, intentionally doping of ions is performed in the material, so that at certain frequencies resonance( due to the rotation of ions, electron spin) occur. Above that resonance frequency, the dielectric constant becomes, negative. This class of negative dielectric material are called metamaterials, which have applications in optical devices. So in short, negative dielectric constant is due to resonance conditions and phase reversal, which happens unintentionally(contact problems ) or intentionally (metamaterials).

In normal dispersion εr[1] is (pure, or nearly) real (εr,Real + j.0), a common case in (ideal) dielectrics. Symbolic examples are : ideal capacitors, ideal optical glasses, etc. However, we account some unusual cases, where intrude other effects, and the εr is not, any more, a pure real, that we are generally calling, as anomalous[4] dispersion case(s), where in limited sub-cases, we can find critical conditions for some unique negative εr[5-9].

1. εr is[2,3] the relative permittivity tensor of the dielectric

2. What is the electric displacement field https://www.physicsforums.com/threads/what-is-the-electric-displacement-field.763099/

3. Relative permittivity https://en.wikipedia.org/wiki/Relative_permittivity

4. Anomalous Dispersion, and Resonant Absorption http://webhome.phy.duke.edu/~rgb/Class/phy319/phy319/node50.html

5. Negative-index metamaterial https://en.wikipedia.org/wiki/Negative-index_metamaterial

6. CRYSTALS AT RESONANCE http://www.bliley.net/XTAL/docs/eng_bulls/E6/E-6_02.html

7. Basics of a Quartz Crystal Microbalance(see Fig 7) https://www.gamry.com/application-notes/physechem/basics-of-a-quartz-crystal-microbalance/

8. Dynamic Behavior in Fundamentals of Piezo Technology https://www.piceramic.com/en/piezo-technology/fundamentals/

9. Measurement of complex impedance of ultrasonic transducers http://www.ndt.net/article/ultragarsas/62-2007-vol.1_05-l.svilainis.pdf

I understand the idea of the displacement vector and electric field vector being pointed in opposite directions but the questions are: (a) can you use that value of permittivity without the negative sign to plot the data since the energy stored is not negative and (b) how reliable is the data before that negative value. I have done a airline sweep of high epsilon materials and have also used he 16453A; they both show negative values after a certain point in the frequency range? I could use some advice on what to do with the data? Thanks.