# Permittivity real and imaginary parts with similar value possible?

Here's the context; I'm studying biological tissues that are supposed to behave like dielectrics. Using the modified cole-cole equation for theoretical predictions: $$\tilde{\varepsilon}_r (\omega )= \varepsilon_\infty + \sum\limits_n \frac{\Delta \varepsilon_n}{1 +(i \omega \tau_n)^{(1-\alpha_n)}} + \frac{\sigma_m }{i \omega \varepsilon_0 }$$ with $$\varepsilon_\infty$$ the one when $$\omega \tau >>1$$, $$\varepsilon_s$$ when $$\omega \tau <<1$$ and $$\Delta \varepsilon_n$$ the difference of the two previous. $$\tau$$ is the relaxation time (like the one in Debyes equation). $$\sigma_m/(i \omega \varepsilon_0)$$ is the ionic contribution with the ionic conductivity. The problem I'm having as of right now is that the second term, $$\sigma_m/(i\omega \varepsilon_0)$$ has such a great weight that the imaginary part of $$\tilde{\varepsilon}_r$$ (usually written either $$\varepsilon_r$$" or $$k$$) makes it as great as the real part (usually written $$\varepsilon_r$$') (y axis- permittivity, x axis- light frequenc Hz). The real part datas are in good agreement with previous studies. From my understanding, the real part is the polarizability of the material, and how much a field is transmitted, and the imaginary part represents the absorption/loss of energy in the matter.

In parallel, since it is known that the tissues I'm studying are lossy, I've derived the following equations $$n' = \sqrt{\frac{1}{2}\left(\sqrt{\varepsilon_r'^2 + \varepsilon_r"^2} +\varepsilon_r' \right)} \\ n'' = \sqrt{\frac{1}{2}\left(\sqrt{\varepsilon_r'^2 + \varepsilon_r"^2} -\varepsilon_r' \right)}$$ Using the results of the modified Cole-Cole equation in the expressions of refractive index above, the results for real and imaginary parts are almost identical (Y axis- diffraction axis, x axis, frequency of light) (orange is real and blue is imaginary here)

Here is the same graph, but with a log y-scale (good remark from a comment of
Jos Bergervoet, thank you)

I'm at a loss facing this. I thought that the sign change would've made a greater difference between the two parts and allowed me to better understand the matters behavior, but I'm more lost regarding it now than before. Any pointer/suggestion regarding this?

• It might be clearer if the second graph would also have a logarithmic vertical scale, like the first one, but as far as I can see the results may just be correct. If the imaginary part $\varepsilon''$ dominates, then the sign-flip of the real part $\varepsilon'$ will not make much difference. So the only region where it would make a difference is around $10^9$Hz, but there the second graph is unreadable... Commented Jul 8 at 17:29
• Thank you for the remark, I've added a third graph with a logarithmic y scale, we do indeed see the change at $10^9$ Hz Commented Jul 8 at 17:38

Since the right side of the second figure is not clear because of the inappropriate scale, I will only analyze the left half of the frequency range. Obviously, the imaginary part of the permittivity $$\varepsilon_\mathrm{r}^{\prime\prime}$$ is much larger than the real part $$\varepsilon_\mathrm {r}^{\prime}$$, which leads to $$\mathrm{Arg}(\tilde{\varepsilon}_\mathrm {r})\approx \pi/2$$. Meanwhile, $$n=\sqrt{\tilde{\varepsilon}_\mathrm {r}}$$, thus we can conclude that $$\mathrm{Arg}(n)\approx \pi/4$$. Therefore, $$n^{\prime\prime} \approx n^\prime$$ in this case.
So we can return to the question, when do the real part and imaginary part of the refractive index $$n$$ take on a similar vaule? It occurs when the $$\varepsilon_\mathrm{r}^{\prime\prime} \gg \varepsilon_\mathrm {r}^{\prime}$$. This implies the medium is more like conductor than dielectric! For electrolyte solutions (which are often described by the Debye model) this exactly happens in the low frequency region, where the alter of the external electric field is slower than the frequency of ionic oscillations. Thus a conduction current is formed and the solution looks like a conductor(the case that the question described). Conversely, the ionic motion cannot keep up with the direction change of the external electric field, and the medium will eventually behave as a dielectric in the real high frequency limit. And $$n^{\prime\prime} < n^\prime$$ in that case.
• @Laurier, that's right. You can see on the right side, $n^{\prime\prime}<n^{\prime}$ as predicted in the answer at a high frequency limit. In fact, this is a universal asymptotic behavior of the dispersion relation. Commented Jul 8 at 23:32