Are there optical materials with larger mean deviation with little to zero angular dispersion?

Question

The following statement is from Concepts of Physics by Dr. H.C.Verma, from the chapter "Dispersion and Spectra", page 434, topic "Dispersive Power":

The mean deviation depends on the average refractive index $\mu$ and the angular dispersion depends on the difference $\mu_v-\mu_r$ [where $\mu_v$ and $\mu_r$ are the refractive indices of violet and red components of light]. It may be seen from figure (20.1) [something like the one below with only the blue curves] that if the average value of $\mu$ is small [Fluorite crown FK51A], $\mu_v-\mu_r$ is also small and if the average value of $\mu$ is large [Dense flint SF10], $\mu_v-\mu_r$ is also large. Thus, larger the mean deviation, larger will be the angular dispersion.

^{Text within "[" and "]" are my comments. Emphasis mine.}

^{Modified image from Wikipedia.}

I don't understand how the author concluded "larger the mean deviation, larger will be the angular dispersion" by looking at the optical properties of very few materials. Is it possible to have a material with a large mean refractive index but little to zero variation across the entire spectrum? Or in other words, is it possible to have $\color{red}{\text{Material X}}$ as shown in the above graph for which the refractive index is high but there is no huge variation like "Dense flint SF10"? To put in simpler terms, is the converse of the statement "larger the mean deviation, larger will be the angular dispersion" possible? If not, what prevents the existence of such a material?

Answer 1

Verma's statement is roughly correct i.e. in almost all cases it is correct. However while I don't know of any exceptions I can't guarantee no exceptions exist.

If you look at the refractive index as a function of frequency then it looks a bit like this:

( from Why does the refractive index depend on wavelength?)

The crossover in the middle happens when the frequency matches one of the absorption lines in the material, and I've called it a resonance because at the frequency the frequency of oscillation of the electrons in the solid matches the frequency of the electric field associated with the light. This shape of curve is characteristic of a resonant process, and you see curves like this in other resonant systems like mechanical resonances, so the curve is going to look similar for the refractive index of most materials.

The graph you show is at the left of my diagram i.e. the low frequency end. It's reversed compared to my diagram because your diagram has wavelength not frequency on the $x$ axis.

Anyhow, Verma's statement is true because if all curves have the same shape then for every material the ratio of $\mu_{av}$ to $\mu_v-\mu_r$ will be roughly similar. To get something like your material X would require a $\mu(\nu)$ curve that was very different. I'm not sure I would state no such material exists, but I don't know of one.