absolutely...there is the a well known relationship first found in 1864 by Gladstone and Dale.basically this all has to do with the electron density, the dielectric constant and the index of refraction and how density is tied in.
look up the lorentz-lorenz law or the Gladstone-Dale relation. if you can get it, the little book by JAFFE called" Crystal Chemistry and refractivity" explains it all. any good book on OPTICAL MINERALOGY has it. i teach OPTICAL MINERALOGY classes at UCSB and i can give plenty of refs if need be.

Without some "stuff" to respond to the electromagnetic radiation (light, microwaves,
infrared, x-rays, UV) the index of refraction will be 1, the index for the vacuum.So there will be a weak correlation with density.

Much more important is how the electrons in the atoms that make up the solid respond to the electromagnetic field. Or, if the frequency of the radiation is low as in microwaves, how the ions or charged atoms respond to the electromagnetic field.

In metals the electrons are not bound and can move freely and are easy to move with an electromagnetic field. The index of refraction for a metal tends to be very large especially at low frequencies. At low frequencies the electrons move great distances before they must turn around as the electromagnetic field changes sign. The large excursions of the electrons in a metal combined with the large numbers produce a huge low frequency index of refraction.

On the other extreme, solid hydrogen (frozen at low temperatures) has few electrons and they are very tightly bound. They can hardly be moved by the electromagnetic field and the index of refraction for frozen hydrogen is low.

In between we have all sorts of materials including insulators and semiconductors. The index of refraction will depend very much on the density of electrons and how tightly they are bound to the atomic nucleus in the solid.

The details are rich, varied and important for much of the modern electronic and lightwave technology that we enjoy today.

Paul, this is a tough question -- The (mass) density of a material is related to both its electronic structure (i.e. the physical-chemical pattern of atoms) and to the atomic mass of the atoms themselves. For example, metallic iridium has a density of about 21g/cc, denser than gold at 19g/cc. However, individual gold atoms are heavier than Iridium atoms, in the metallic structure, the gold atoms are farther apart. (I seem to recall that both are denser than Uranium!) However, on the average, the spaces between atoms in differing materials with common physical characteristics (i.e. glasses or metals) tend to become more dense as heavier elements are chosen. i.e. Tungsten Nitride is a much denser coating material than Boron Nitride.

The index of refraction is a purely electronic property of the material. When a material interacts with light its properties are determined by the relative wavelength of the light to the characteristic distances and energies in the material. (For small wavelengths, one gets scattering such as the Compton effect when a gamma ray scatters off an electron). Visible light has a much longer wavelength than atomic spacing, so the effect is much more gentle. As the light passes a molecule, the molecule is subjected to an electric field due to the light itself. This field alters the electronic structure of the
molecule which usually produces its own field (called electric polarization) canceling some of the light wave. The effect is like balls separated by a spring which is pulled on by a magnet. Not only does the ball move, imparting potential energy to the spring, but the field is also changed. When the magnet is removed, the ball moves
back into position. The relative size of the polarization to the field is called the dielectric constant. For non-magnetic materials, the velocity of light is proportional to the square root of this constant. Finally, the index of refraction is just the ratio of the velocity of light in a vacuum to that in the material. (Light is slowed while in the material by the reaction of the polarization on its electric field).

So -- there is a relation, but not a direct one. Materials from the bottom of the periodic table often have high densities, but that also have lots of electrons in their shells, which make for interesting electronic properties: Neodymium makes good magnets, as does Samarium; prasedomium and europium make good light amplifiers. Finally, Lanthanium glasses are noted for having high indicies of refraction -- and low or negative chromatic abberation. These materials are chosen for their unique electronic structure properties -- but being from the bottom of the periodic chart -- typically have higher density.

Finally, please note that the dielectric constant is a strong function of frequency-- water has a dielectric constant of 81 at low frequencies and only about 2 at visible light. Further, resonances in the materials which cause adsorbtions also cause big changes in the index of refraction... Typically most materials with large indexes have a strong absorbtion close by--

It is fun to devise a way of measuring the index of refraction, mixtures of liquids often have indexes which vary with the concentration. For example, sugar dissolved in water is measured by checking its index. Can you imagine what industry needs to do this?

I will try to answer your question as best I can, but I am not a physicist, and I am (very) slightly afraid that my answer is not completely correct. Therefore, if you get an answer which contradicts my answer in any way, I would assume that the other answer is correct.

The short answer is: Yes, there is definitely some correlation between density and index of refraction. I have found a formula in an electricity and magnetism text which gives a relation between the index of refraction and the electron density as:

n^2 = 1 + K N , where n is the index of refraction, K is a (frequency dependent) factor, and N is a density of electrons (I'm cheating a little, but for the purposes of simplicity, N can be considered to be a density). So, you can see that the index is related to density since density is related to electron density.

Unfortunately, I must note that this formula apparently is derived using several approximations, and the book says: "Although the derivation as it stands is dubious, the equation does seem to provide a reasonable quasi-empirical formula that fits the properties of many materials." In other words, you were right on the ball when you guessed that there is a relationship between density and index of refraction, but there are certainly other things that come into play.

I might note that the index of refraction for ethanol is higher than for water, even though ethanol is considerably less dense than water. You might consider looking up as many materials as possible and making a plot of index versus density to better get a sense of the relation.