Classical Predictions of Scattered Radiation

[CWK]

I am currently reading the third edition of Modern Physics by Serway/Moses/Moyer and this quote, in reference to the Compton Effect, seems to conflict with various other online sources I have checked. I understand that the classical description does not give accurate predictions of what actually happens, but I am nonetheless interested in understanding its predictions.

... classical theory predicted that incident radiation of frequency $f_0$ should accelerate an electron in the direction of propagation of the incident radiation, and that it should cause forced oscillations of the electron and reradiation at frequency $f'$, where $f' \leq f_0$. Also, according to classical theory, the frequency or wavelength of the scattered radiation should depend on the length of time the electron was exposed to the incident radiation as well on the intensity of the incident radiation.

There is also a footnote that says:

This decrease in frequency of the reradiated wave is caused by a double Doppler shift, first because the electron is receding from the incident radiation, and second because the electron is a moving radiator as viewed from the fixed lab frame. See D. Bohm, Quantum Theory, Upper Saddle River, NJ, Prentice-Hall, 1961, p. 35.

A few conflicting sources:

By classical theory, when an electromagnetic wave is scattered off atoms, the wavelength of the scattered radiation is expected to be the same as the wavelength of the incident radiation.

- https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_(OpenStax)/Map:_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/06:_Photons_and_Matter_Waves/6.04:_The_Compton_Effect

Although classical electromagnetism predicted that the wavelength of scattered rays should be equal to the initial wavelength,[5] multiple experiments had found that the wavelength of the scattered rays was longer (corresponding to lower energy) than the initial wavelength.

- https://en.wikipedia.org/wiki/Compton_scattering

Using the classical wave theory, I would predict that the x-ray would cause the electron to oscillate at the same frequency as the incident radiation, as is stated by the latter two sources. The way I see it, the electric field of the incident radiation would cause the electron to oscillate along the same direction and at the same frequency as the electric field of the incident radiation, and this would in turn allow the magnetic field of the incident radiation to cause the electron to oscillate along the direction of propagation, still at the same frequency as the incident radiation.

I would appreciate it if someone more knowledgeable could inform me on what the classical model predicts and how it predicts that.

 

[hutchphd]

First I will point you to the Wikipedia article on Compton scattering

Most of the frequency shifts are caused by the recoil of the electron. This is largely determined by the quantum nature of the interaction. Also the degree to which the electron is bound to an atom or in a crystalline solid will mitigate this. Does that help?

 

[CWK]

First I will point you to the Wikipedia article on Compton scattering

Most of the frequency shifts are caused by the recoil of the electron. This is largely determined by the quantum nature of the interaction. Also the degree to which the electron is bound to an atom or in a crystalline solid will mitigate this. Does that help?

That is one of the articles I linked to in my post that I used as an example of a source that appears to conflict with my textbook

I understand that there are frequency shifts in reality, but what I am concerned with is understanding what the classical wave theory predicts. Does the classical wave theory predict that the frequency of the reradiated wave should be lower than the frequency of the incident radiation?

 

[kuruman]

Maybe this old article by no lesser person than C. V. Raman might answer your question. Disclaimer: I just scanned it without reading it. It's more of a classical explanation of the experimental results than a prediction.

 

[hutchphd]

I believe the classical theory in the limit of very low intensity will not predict the shift.

But that low intensity limit is not consistent with reality particularly for higher frequency light. ooops

I just noted @kuruman post and checked the paper. It is very interesting .
Prof Raman has to do a pretty extensive tap dance to make the classical theory yield the innelastic part but it is a lovely exercise in my opinion. You should look.

 

[CWK]

Maybe this old article by no lesser person than C. V. Raman might answer your question. Disclaimer: I just scanned it without reading it. It's more of a classical explanation of the experimental results than a prediction.

First of all, fantastic reference. I am impressed you were able to come up with it so fast.

It looks like the explanation of equation (9) answers my question.

$$\lambda ' = \lambda + \lambda \frac{v}{c} \left( 1 - \cos \chi \right) $$

where $v$ is the speed of the electron and $\chi$ is the angle from the direction of propagation of the incident ray from which the secondary radiation is observed.

So if the electron is bound to the atom, then $v/c \approx 0$. Classically, it would be expected that a higher intensity light would cause electrons to be emitted from the material, as @hutchphd mentioned. I am not yet entirely familiar with the details of Compton's experiment, but I expect that if he chose a material such that no free electrons were emitted when the x-rays strike the material and still measured a change in wavelength in the reradiated light, then that would be where the classical model fails. If electrons were emitted and their speed was known, then the above equation could be used to see if the experiment supports the classical model, which it would not and there would again be a failure to explain experiment.

In summary: the book accounts for the possibility of the electron moving and the Doppler shift that would result. If the electron were not moving, there would be no change in wavelength predicted.