Here’s an interesting insight into the mind of a physicist: they think “Oh boy, I’d sure love to know more about the nature of light!”
They’re solution? “Let’s barrage stuff with X-Rays and Gamma Rays!”
Well, it’s rather important to know how they actually interact with matter, I think, so I have no problems with that train of thought (physics student desu). Only potential downfall I can think of is your hair falls out, and so too do your teeth, and various other nasty things happen due to Radiation Sickness.
Let’s not go there quite yet.
So waaaay back whenever the hell this was (in the 1920’s), we were just getting a handle on X-rays and other junk, but physicists noted that whenever X-Rays scattered off of things (bounced off at some angle θ), the wavelength changed with the scattering angle.
They were like “Gee willikers, that’s wacky!”
Uhh, yeah, it is. Because classical Electrodynamics supposed doesn’t have any such predictions (in fact, it would predict the wavelength to remain absolutely unchanged). So they were like “obvi, we fucked up somewhere.” — It’s because (surprise!) light can be thought of as a particle due to the Wave-Particle Duality.
So this guy named Compton tried punching in the numbers (assuming one Photon interacted with one Electron, attributing particle-like characteristics to each) and derived the correct formula describing – what is now called – Compton Scattering. The formula is the following:
where that is the change in wavelength, as describe through an angle theta, with a factor of Planck’s Constant divided by the mass of the electron by the speed of light.
For a full derivation, you could just check out what they’ve written out on Wikipedia here (it ain’t bad, one might even call it decent), or you could hold your horses (Genghis) and wait up for my own derivation.
Either way, you’re learning!
Figure 2 shows a bit of the meat on those bones. The energy of the photon is given by Einstein‘s relation for more on that, check out this post on Photons! The electron’s recoil energy is given (again) by Einstein, this time from his Relativistic Energy equation