Dumb Question #215

[Bruno]

Thx Dinesh, very enlightening! Finally a description of light through solids that I can work with with my EE hat on.

[Dinesh]

You're welcome.

I now await dumb question #216 with breathless anticipation!

[holorefugee]

Dinesh,

One of your simplifications is either going to help me or lead me down the wrong path so I was hoping for a clarification. Is the index of refraction directly proportional to density in a specific material? Also, is index of refraction proportional to density across all transparent materials?

[dannybee]

I would think this would depend on the material,jello verses air,jello verses silver. How much is the photon is slowdown by the medium ..and speed up in the next medium . I think this is correct

My thinking is a bit foggy at12:30am

[Dinesh]

Dinesh,

One of your simplifications is either going to help me or lead me down the wrong path so I was hoping for a clarification. Is the index of refraction directly proportional to density in a specific material? Also, is index of refraction proportional to density across all transparent materials?

Colin
Sorry for late reply. I've been away and only just got back.

Anyway, short answer is "yes" with some exceptions for absorption bands and anomalous zones.

<-------------------For Math phobes, Math Ahead, return to your seats and fasten your seat belts!---------------------->

When light passes through a medium, we can use the "stretched spring" model to determine the behaviour of the atoms in response to the E field of the incoming light. This is the model wherein the electron/nucleus system is seen as a spring, with the "spring constant" being determined by the bond strength between electron and nucleus. This works for small oscillations around the mean position of the electron at rest.

When the primary wave passes through the medium, it generates secondary waves caused by the oscillations of the electrons. As the electrons oscillate, the distance between the centre of the electron cloud and that of the nucleus changes and so an electric dipole is formed. An electric dipole is two opposite charges separated by a distance, which may or may not be constant; clearly as the electrons oscillate, the dipole distance varies sinusoidally. This dipole causes a charge polarisation of the atomic system. The phase mismatch of the secondary wave is now a function of the polarisability of the atom, and so also is the index of the material. However, the polarisability of the atom is also a function of the "spring strength" of the electron-nucleus spring - the stronger the nucleus, the greater is the spring constant.

Consider an atomic system driven into oscillations by an incoming (primary) oscillating field. The primary wave causes a charge separation between electron cloud and nuclear center and so a new field is created between the centre of the electron cloud and the nuclear center . This gives rise to a dipole (two opposite charges separated by a small distance). The total field, called D (The Maxwell Displacement current caused by a displacement of the charge centers) now has two components, the field of the original primary wave which causes a polarisation P and the charge separation field eE, so

D = eE + P
where e is the permittivity of the material (it's behaviour between two capacitor plates) and E is the field of the incoming primary wave. The free space wave (the primary wave independant of the material) creates a polarisation of

P = e_0*E
where e_0 is the permittivity of free space. Thus the total polarisation of the atomic system,

P = D - P = (e - e_0)*E

However, the polarisabitlity of the atomic system can also be determined by the dipole strength/unit volume - the amount to which the spring "stretches", which is the charge multiplied by the separation, or

P = q*x*N
where x is the (instantaneous) charge separation and N is the number of atoms/unit volume. Putting this into a standard "spring equation", gives the refractive index as

n^2 = 1 + ((N*q^2)/(e_0*m))(1/(w_0^2 - w^2)

This (finally) gives the relationship between the material properties and the primary field. It's called a dispersion relation because it gives the variation of the refractive index with wavelength (OK, frequency!) as a function of the natural harmonic frequency of any material. It can now be shown that the phase lag between the primary wave and the secondary wave is dependent on the difference between the squares of the frequency of the primary wave (w^2) and the natural frequency of the material (w_0^2) in the denominator of the dispersion relation - The w_0 is dependent on the spring strength of the atom which in turn is dependent on the material density. From this, you can derive the phase lag for a given material and also show that for light, the index increases with the frequency of the primary wave, so the index for blue is higher than the index for red light.

[holorefugee]

Sorry to sound daft but this is important to my question. Does the "spring length" relate in a mathematical way to density of different solids. I have explored this question extensively with air and water but not different crystals or amorphic solids.

[Dinesh]

Does the "spring length" relate in a mathematical way to density of different solids. I have explored this question extensively with air and water but not different crystals or amorphic solids.

Actually, this sort of gets beyond the scope of anything pertinent on the forum and also bloody difficult to write down! Can I just give you vague hand-waving arguments and point to further studies?

The nub of the argument is that the dispersion relations connect the refractive index of a material to the permittivity of a material (epsilon, or e), or to the susceptibility of a material (chi) which can then be related to the atomic structure of the material ( The susceptibility and the permittivity are connected via the relative permittivity of the substance, so knowing one will lead to the other). So, an incoming primary wave causes polarisation and it's the degree of polarisability - the dipole moment - of the atom or molecule under the influence of the incoming primary wave that determines the index. But, the polarisability of the material is dependent on the permittivity e (or susceptibility, chi). So, in answer to the relationship of material density to the refractive index, we need to determine the connection of e or chi of the material to the density, or other physical parameters, of the material.

The most basic answer would be that the the number of atoms/unit volume (N) appears in the Cauchy dispersion relation:

n^2 = 1 + ((N*q^2)/(e_0*m))(1/(w_0^2 - w^2)

So, the larger the number of atoms/unit volume, the greater the density and so the greater the value of N, the greater the value of n since n^2 is proportional to N

A more detailed analysis can be carried out using the Clausius-Mossotti equation to connect e to density:

(e - e_0)/(e + 2e_0)*(M/d) = 4(pi)N_A*alpha/3
Here, M is the number of moles in the substance (the molar mass), d is the density, N_A is Avagadro's number and alpha is the molecular polarisability.
The Clausius-Mossotti equation applies only to isotropic, homogenous materials such as gases and some liquids. However, note that you now have a relationship connecting the epsilon to the density and polarisability of a material - the material parameters. The Lorentz-Lorenz relations also give you the index of a gas in terms of the gas constant. If you have a copy of Feynmann's "red books", I believe there's a discussion of the Clausius-Mossotti equation in book two, about half way through.

However, in a crystal or an anamorphic substance, the epsilon, or the related susceptibility (chi) is no longer a scalar. In fact, it's a tensor. Thus, the relationship connecting the epsilon (and through it, the dispersion relations and thus, on to the refractive index) is now a tensor equation. Also, you need to also take into consideration that the index of most materials is complex, with the real part being our familiar index and the complex part the absorption. In this case, the Kramers-Kronig relations may be more pertinent. I'm not even going to try and write down Kramers-Kronig, just point you to it!
http://www.rp-photonics.com/kramers_kro ... tions.html
http://academic.reed.edu/physics/course ... ture21.pdf

To look at the tensor nature of the index in crystals and anisotropic materials, you might also want to look up the Fresnel Ellipsoid and the Index Ellipsoid. These give the polarisability of an anisotropic substance in terms of the susceptibility tensor. The reason that crystals and anisotropic materials have a tensor epsilon is that the atomic (molecular) forces cause a polarisation that does not "line up" with the incoming primary wave. The "spring" does not stretch along the direction of the incoming force (the primary wave), but goes askew.

Of course, to truly answer this question I'd have to go into Quantum Field Theory. But, it's 8 in the morning and I'm sure Joy's wondering when her coffee is going to show up. So for the sake of marital harmony, I hope you accept that I can't go into the full QFT treatment here and now!

[holorefugee]

Thank you Dinesh. I have the Red books and will follow up on it. I don't need the whole QED description at all just some rules of thumb that can get me through some design phases. Thank you for taking the time to ponder my question. Please tell Joy that I still think she was named appropriately.