I want to know whether the amount of refraction of a given monochromatic light depends solely upon the density of the of the medium ( increase the density to increase the angle of refraction), or there are other parameters like the number of free electrons in the atoms of the material, atomic size etc.
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there are other parameters like the number of free electrons in the atoms of the material, atomic size etc.
Close. While density of particles does matter, it also depends on the material property. More precisely, it is closely related to how the electrons react when situated under electromagnetic oscillation.
Each bound electrons has its natural frequency of oscillation. Incident light is an oscillating electromagnetic field, and it shakes the bound electrons inside atoms. Difference of the frequencies between them decides the refractive index. Thus, the atomic/molecular properties of the medium is important.
If the medium has abundant free electrons, it is called metal. In that case, incident light creates electric current alternating on the surface of medium. this oscillation counterbalances the incident light's electric field, thus no light can be refracted. More precisely, the intensity of incident light decreases as it penetrates into metal. In this case, in a mathematical sense, we call the refractive index is imaginary number.
If you are familiar with university level physics and mathematics, I suggest you to read through some electromagnetic texts (such as Griffiths or Jacksons), or dedicated optics text (such as Hecht), as detailed calculations are made there.
Of course these are non-quantum theory, but it will provide enough predictions.
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I want to know whether the amount of refraction of a given monochromatic light depends solely upon the density of the of the medium ( increase the density to increase the angle of refraction), or there are other parameters like the number of free electrons in the atoms of the material, atomic size etc
There are a number of factors at play, the major one is density. Basically what you are asking is how to calculate the frequency-dependent dielectric function of a given material (the square of the index of refraction is called the dielectric function), which is a material property.
Equivalently you could ask how to calculate the frequency-dependent absorption, since it is related to the dielectric function by some algebra and a KK transform.
The absorption is proportional to the number density of atoms $n$ $$ \mu=n\sigma\;, $$ where $\sigma$ is the absorption cross-section $$ \sigma(\omega)=\frac{4\pi^2 \omega}{c}\sum_m |\langle \Psi_m|\epsilon\cdot \vec d|\Psi_0\rangle|^2 \delta(E_0+\omega-E_m)\;, $$ where $\vec d$ is the dipole operator for the system of electrons, $\epsilon$ is the polarization of the light, $\Psi_m$ is an eigenfunction of the Hamiltonian of the electronic system having eigenvalue $E_m$, and $\omega$ is the frequency.
Basically, you can think of that complicated part as roughly and integral over the electronic density of states of the material under consideration. This has to be calculated numerically, for example, using density functional theory, or some other numerical technique. Or it can be measured for the material of interest.
