# What causes dispersion?

sophiecentaur
Gold Member
But the wave DOES interact with individual atoms. And yes, it causes scattering.

Look at gases. The molecules in atmospheric or lower pressure gases are too far from another to allow light to interact with more than one molecule at a time, except on the rare occasions when molecules are undergoing collision.

The phases of light scattered from different air molecules are indeed out of phase.

Now, a wave of light encounters many air molecules over one wave. With the result that while the retarding/refracting effects of air molecules add up over many molecules, the scattered waves being out of phase undergo destructive interference. This is not complete because the positions of air molecules are random. There is Rayleigh scattering in air. But it is still relatively weak - much of the light can pass through long column of air and is left over from scattering, yet appreciably retarded.

I was actually thinking of solids and liquids (which have high refractive indices but isn't Rayleigh scattering due to polarisation of the molecules in a gas and not the atoms? It's an elastic scattering phenomenon and I don't know how atomic energy levels would work apart from at frequencies corresponding to line spectra.

I was actually thinking of solids and liquids (which have high refractive indices but isn't Rayleigh scattering due to polarisation of the molecules in a gas and not the atoms? It's an elastic scattering phenomenon and I don't know how atomic energy levels would work apart from at frequencies corresponding to line spectra.

Monoatomic gases like noble gases or quicksilver vapour undergo Rayleigh scattering, refraction/retardation and far UV absorption qualitatively in the same manner as multiatomic gases like nitrogen or oxygen. The orbits of electrons are perturbed/distorted by external electric field, and if the frequency is sufficient then the electronic excited states of the molecules span the whole molecule.

sophiecentaur
Gold Member
Thanks for clearing that up.

Well I want to get down to the nub of what it is that causes the dispersion. So far the only explanation given involves absorption and emission of photons, if there is a better way of looking at it then I'm all ears.

Ignoring photons, look at field as continuous phenomenon.

The differences in light speed come because medium is usually "softer" than vacuum. In a medium, electric field causes movement of charges, and the field from the moved charges usually opposes and weakens the inducing field. Just like in a mechanical medium, a softer medium with less resistance to compression or shear will transmit sound at a lower speed, so in case of electromagnetic field a medium with higher permittivity (weaker electric field) transmits the wave at a lower speed.

Now, a mechanical analogue of dispersion would be elasticity dependent on time/frequency. For example, a substance which on rapid deformation exerts high elastic stress - but once the deformation ends, will plastically relax to a lower stress (but not to zero) over some timescale.

If such substance were subjected to waves with high frequency, it would transmit them at a high speed due to high stiffness.

If the waves are very low frequency, the substance would be relaxed all the time, so it would exert only the final stiffness and accordingly transmit sound at the lower speed.

If the substance were subjected to waves at the frequency in the timescale of relaxation then large amounts of energy would be turned to heat, since large force is exerted to compress the substance, but only small force would be restoring it.

Saw
Gold Member
The mechanical analogue of "polarization" is "strain". And the analogue of "polarizability" is "compressibility".

The differences in light speed come because medium is usually "softer" than vacuum. In a medium, electric field causes movement of charges, and the field from the moved charges usually opposes and weakens the inducing field. Just like in a mechanical medium, a softer medium with less resistance to compression or shear will transmit sound at a lower speed, so in case of electromagnetic field a medium with higher permittivity (weaker electric field) transmits the wave at a lower speed.

I find this interesting but conflicting with the understanding I had so far.

The speed of a mechanical wave in a medium is:

(a) Directly proportional to restoring (elastic) force of the material, i.e. how much the medium resists strain or deformation and hence with which force it recover the equilibrium position.
(b) Inversely proportional to inertia of the material.

For example:

- In a string, (a) is tension and (b) is the longitudinal density.
- In sound, (a) is compressibility and (b) is volume density.

In the case of EM waves, I had thought that permittivity (which is related to how easily the material polarizes) plays the role (b) of inertia, not (a). In fact, permittivity is placed in the denominator of the formula, like density, thus clearly denoting that the speed is inversely proportional to this factor. Of course, one could say that speed is proportional to how low permittivity is, but that is a convoluted approach. We also talk about optical density of the material. I also found somewhere the formula below, which I don’t fully understand and which someone might be able to explain, where the slowing down of light is related to mass density (M) together with other factors like how far the electron orbits from the nucleus (R) …

$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq % aH3oaAdaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaaabaGaeq4T % dG2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaaaacqGH9aqpda % qadaqaamaalaaabaGaeqyWdiNaamOtamaaBaaaleaacaWGbbaabeaa % aOqaaiaaiodacqaH1oqzdaWgaaWcbaGaaGimaaqabaGccaWGnbaaaa % GaayjkaiaawMcaamaabmaabaWaaSaaaeaacaaIYaGaamyzamaaCaaa % leqabaGaaGOmaaaakiaadkfadaahaaWcbeqaaiaaikdaaaaakeaaca % aIZaGaeuiLdqKaamyraaaaaiaawIcacaGLPaaaaaa!51C4! \frac{{{\eta ^2} - 1}}{{{\eta ^2} + 1}} = \left( {\frac{{\rho {N_A}}}{{3{\varepsilon _0}M}}} \right)\left( {\frac{{2{e^2}{R^2}}}{{3\Delta E}}} \right)$$

Ultimately, what all this would mean is that light slows down because, every time it interacts either with an atom or, as the answer to the FAQ says, with the bulk of a material, it makes a brief stop: more stops and longer ones mean more delay.

While it is true that dipoles align against the E field and hence weaken it, it does not seem that such is the reason for an EM wave’s retardation, since the latter is an oscillation, so the dipoles align in one direction and then in the opposite direction, thus acting like an antenna that, instead of attenuating the E field, simply re-radiates the wave, thus leaving the E field unaffected.

Is this understanding correct or should I change it?

The simple wave equation is

2η/∂x2 = (1/c2)(∂2η/∂t2

The solution of this has velocity c independent of frequency. This is called the phase velocity.

However more complicated wave equations exist where the solution does not include a constant c but have velocity dependent upon frequency.

eg

4η/∂x4 = -(ρS/YI2)(∂2η/∂t2

These commonly arise from non linearities in the restoring force.

Since different frequency waves then travel at different velocities, a wave shape distorting effect called dispersion occurs.
The power in a wave depends upon its shape so if the shape changes the power changes, hence the term dispersion.

Sorry I don't have acces to my latex editor at the moment.

Please note SophieCentaur's comment that polarisation only occurs in trnasverse waves, not longitudinal ones and also that power loss only occurs at the instant of polarisation and not thereafter.

With waves with dispersion as every frequency has a different phase velocity we sometimes group velocity which refers to a (small) renge of frequencies that have a small range of velocities and therfore a signal containing only this range of frequencies will pass a substantial distance undistorted. They will, of course disperse eventually.

I find this interesting but conflicting with the understanding I had so far.

The speed of a mechanical wave in a medium is:

(a) Directly proportional to restoring (elastic) force of the material, i.e. how much the medium resists strain or deformation and hence with which force it recover the equilibrium position.
(b) Inversely proportional to inertia of the material.

For example:

- In a string, (a) is tension and (b) is the longitudinal density.
- In sound, (a) is compressibility and (b) is volume density.

In the case of EM waves, I had thought that permittivity (which is related to how easily the material polarizes) plays the role (b) of inertia, not (a). In fact, permittivity is placed in the denominator of the formula, like density, thus clearly denoting that the speed is inversely proportional to this factor. Of course, one could say that speed is proportional to how low permittivity is, but that is a convoluted approach.

While it is true that dipoles align against the E field and hence weaken it, it does not seem that such is the reason for an EM wave’s retardation, since the latter is an oscillation, so the dipoles align in one direction and then in the opposite direction, thus acting like an antenna that, instead of attenuating the E field, simply re-radiates the wave, thus leaving the E field unaffected.

Is this understanding correct or should I change it?

Compare an oscillator circuit including a capacitor.

A dielectric with dielectric permittivity will decrease the voltage of a capacitor, while leaving the charges unchanged. This also means that the increased capacity of the capacitor decreases the frequency of the oscillator circuit. That is the restoring force side - not the inertia side of the equation.

Likewise consider that the Hertz antenna is a limiting case of oscillator circuit - the plates are converted to ends of the antenna in surrounding medium. The frequency would respond to the permittivity of the surrounding medium in the same way.

@snorkack

I'm sorry, I don't follow where any of your examples are dealing with dispersion?

Saw
Gold Member
Compare an oscillator circuit including a capacitor.

A dielectric with dielectric permittivity will decrease the voltage of a capacitor, while leaving the charges unchanged. This also means that the increased capacity of the capacitor decreases the frequency of the oscillator circuit. That is the restoring force side - not the inertia side of the equation.

You bring in a good comparison, the role of the same dielectric when faced not with an electromagnetic wave but with a similar phenomenon, an oscillating current.

I fully agree that in this second example capacitance (which is a function of the permittivity of the dielectric material and of the geometry of the capacitor) plays against capacitative reactance, which is usually defined as a kind of inertia that opposes change in tension. To sum up, permittivity plays against inertia. Somehow we could say that here higher permittivity means better “conductivity” of the oscillation that the AC is. And the same applies to frequency: the higher the frequency, the lower the reactance and the higher the “conduction” of the oscillation.

However, the fact is that in the situation that is the object of our discussion (the dielectric is faced with an EM wave) permittivity and frequency play the opposite roles. The higher the permittivity and the higher the frequency, the lower the velocity, which looks more like inertia (since it plays against velocity) than like restoring force (which would play in favor of velocity). Someone could argue: “but you are comparing apples with pears: in one case you talk about the preservation of changes in tension and frequency; in the other, of the preservation of velocity". And I would agree. That is why I tended to believe that the two situations admit different solutions: in one case (AC - tension), permittivity acts against inertia, in the other (EM wave – velocity) it would be on inertia side.