Relation between absorption and refraction

[Gobil]

Hi All,

this is puzzling me; we have a real part of the refractive index that governs refraction, ie. scattering, and we have an imaginary part, which describes the absorption of the light. The two are related, as you would expect, and a mathematical relationship exists between the two. The Kramers-Kronig relation. When you see graphs of the real and imaginary part of the refractive index, there is usually an abrupt change in the absorption at some photon energy, i.e. an absorption edge, where the photon now has enough energy to ionise the atom at that level. this is fine. If you look at the REAL part, there is a gradual change around this edge, this I have a problem with...

Considering a photon with an energy just below the absorption edge, it will not be absorbed so strongly as it does not have enough energy to ionise this level. HOWEVER, it is being scattered very strongly as it is near a resonant frequency. Classically, I guess it kinda makes sense, its close to the frequency, but in terms of quantum mechanics, I don't get it, it does not have enough energy to be effected by that electron..

so why is the absorption spectrum so abrupt, and the scattering spectrum so smooth?

many thanks!

 

[Simon Bridge]

In QM you get a scattering contribution from everywhere in the material, but the resonant absorption comes only from specific energy levels.

 

[Gobil]

so what you´re saying is that in QM you don´t need to match the energy of the photon to the resonant electron energy to have scattering. but you need to match the photon energy to the electron energy to get absorption?

 

[Born2bwire]

In a material you need to consider the bulk behavior, not just the behavior of the constituent atoms. When we consider the bulk behavior, we generally find that there are numerous phonon modes and dispersion lines that arise. So you do not generally see a set of singular individual absorption peaks but instead there are a distribution of absorptions due to the density of states that arise from the mutual interaction of all the bulk's atoms.

As for the scattering, that arises due to the coupling of the photon's fields with the electron cloud of the bulk's atoms. The electromagnetic field from the photon disturbs the electron cloud which in turn creates its own electromagnetic wave in response. You can think of the total sum of these waves as giving rise to the scattering of the photon. In this manner, you do not need to be at an absorption level to achieve scattering. I believe that you can look into Griffith's introductory quantum mechanics books and find a few sections regarding scattering.

 

[Gobil]

In a material you need to consider the bulk behavior, not just the behavior of the constituent atoms. When we consider the bulk behavior, we generally find that there are numerous phonon modes and dispersion lines that arise. So you do not generally see a set of singular individual absorption peaks but instead there are a distribution of absorptions due to the density of states that arise from the mutual interaction of all the bulk's atoms.

hmm, it depends where you look in the spectrum, from the sof x rays onwards, in metals for example, we see sharp rises in absorption lines where a new shell of electrons can be ionised by the x rays. they are quite sharp and distinct. this I can understand as the photon can either ionise the electron and be absorbed or just go through the material. this also holds true for a single atom.

As for the scattering, that arises due to the coupling of the photon's fields with the electron cloud of the bulk's atoms. The electromagnetic field from the photon disturbs the electron cloud which in turn creates its own electromagnetic wave in response. You can think of the total sum of these waves as giving rise to the scattering of the photon. In this manner, you do not need to be at an absorption level to achieve scattering. I believe that you can look into Griffith's introductory quantum mechanics books and find a few sections regarding scattering.

considering the same atom and photon, the photon can be greatly affected by scattering in quite a large range of energies around the resonant absorption edge, for example, even if it has a much lower energy than the ionization energy. so I´m still a little hazy on this. why is absorption abrupt and scattering smooth, both are the same physical interaction...(?)

 

[Born2bwire]

hmm, it depends where you look in the spectrum, from the sof x rays onwards, in metals for example, we see sharp rises in absorption lines where a new shell of electrons can be ionised by the x rays. they are quite sharp and distinct. this I can understand as the photon can either ionise the electron and be absorbed or just go through the material. this also holds true for a single atom.



considering the same atom and photon, the photon can be greatly affected by scattering in quite a large range of energies around the resonant absorption edge, for example, even if it has a much lower energy than the ionization energy. so I´m still a little hazy on this. why is absorption abrupt and scattering smooth, both are the same physical interaction...(?)

X-rays would not be a region of the spectrum where I feel it would make much sense to talk about permittivity and permeability which you are discussing with the Kramers-Kronig relation. At very high frequencies, the electromagnetic waves do not interact strongly with materials. In addition, the classical theory becomes supplanted by the quantum theory and the scattering and absorption processes there are different than in RF and microwave regions.

Scattering and absorption are not strictly the same physical interaction, unless you are talking about say inelastic scattering. And as I stated, normally the absorption pattern is smooth too. But I would also point out that in your example where the absorption lines are distinct that the scattering can also have similar peaks like in the form of Bragg diffraction. Absorption occurs when the energy of the photon is equivalent to the difference between the current state and some excited state of a system. This can occur smoothly over a bandwidth due to the density of states that arises from the bulk behavior in terms of phonons or it could be discrete peaks similar to single atom excitation. The scattering of the photon arises due to its interaction with the electron clouds of the bulk. If we look at inelastic scattering, this could occur by the absorption of the photon and then the emission of a photon of lower energy (Raman). It could also occur again by the interaction of the photon and the electron clouds (Compton).

So you haven't really stated what scattering and absorption processes you are talking about. If you are talking about Kramers Kronig relation, that implies to me you are talking about the classical theory where we can relate the real and imaginary parts of the permittivity and permeability via a Hilbert transform. But I would not blindly ascribe this to phenomenon above Terahertz where the classical theory needs to be augmented or replaced by the quantum. The physical explanation for the scattering depends on what scattering you are talking about. There's Thomson scattering (which I roughly described above) but there is also Compton, Bragg, Rayleigh, Mie, Raman, etc. many of which are just classical approximations of one of the other.

 

[Simon Bridge]

so what you´re saying is that in QM you don´t need to match the energy of the photon to the resonant electron energy to have scattering. but you need to match the photon energy to the electron energy to get absorption?

No - I am not being anywhere near so general. I'm saying your example involves this kind of process. What you described sounds like scattering from free electrons in a solid, where there is also a chance for absorption by individual atoms/states in the lattice.

In general you expect smoothness as a result of interactions involving many structures.
Resonances are overlaid on that, and you don't normally expect resonances for different processes to occur in the same sort of spectrum.

The others have made useful elaborations along these lines.

 

[Gobil]

X-rays would not be a region of the spectrum where I feel it would make much sense to talk about permittivity and permeability which you are discussing with the Kramers-Kronig relation. At very high frequencies, the electromagnetic waves do not interact strongly with materials. In addition, the classical theory becomes supplanted by the quantum theory and the scattering and absorption processes there are different than in RF and microwave regions.

well you also have a dielectric function in the x ray part of the spectrum. And I was under the impression this is exactly where the KK relation holds up and is used frequently.

So you haven't really stated what scattering and absorption processes you are talking about. If you are talking about Kramers Kronig relation, that implies to me you are talking about the classical theory where we can relate the real and imaginary parts of the permittivity and permeability via a Hilbert transform. But I would not blindly ascribe this to phenomenon above Terahertz where the classical theory needs to be augmented or replaced by the quantum.

So let's stick to xrays, near some bound absorption edge in a material. I don't know what type of scattering to expect here. let's take the L-edge of copper which is around 930eV. With two photon energies 900 eV and 1000 eV. The real part of the refractive index changes smoothly around the imaginary part (see graphic). So the phase velocity will slow down, even at energies below the absorption edge itself. Do you know what kind of scattering process this is? Anyway, classically I can kind of get this, the xrays are shaking the electron, even though its not in resonance, then when it hits resonance as the energy is increased, bam, its (more strongly) absorbed. From the quantum side of things I don´t understand it so well.