Compton scattering shows that light behaves both like a wave and a particle. At a large wavelength, light behaves like a wave, but at a small wavelength it behaves like a particle, called photons. When light hits an electron the photons scatter, changing wavelength. Compton scattering is determined by how large the wavelength the light being sent in is than the Compton wavelength of the electron which is 2.4 pm.
So let's talk about Compton scattering. Compton scattering is another one of those really important events that happened at the beginning of the 20th century that indicated that photons were real. They really, like really does behave like a particle, sometimes. So here's the idea.
When light has a very very very large wavelength, it behaves like a wave. So if you send the light into a metal where the electrons can go back and forth, the electrons are just going to go up and down and up and down and up and down with the electric field associated with that electromagnetic wave. However, when the light has a very small wavelength it behaves like a particle. So here we've got photons and when you take a particle and you throw it at another particle, this particle is not going to go like that. You are going to have a collision. And so we actually get a collision like this. Light comes in, hits the electron and then they go off at different angles. And that's what happens when we have light incident on an electron. As long as it's got a small enough wavelength to really behave like a particle. Alright. So let's see what else happens in this process.
So, we've got this collision and we're going to do the, exactly the same thing that we always do with every collision between two particles. We go through and we conserve energy and we conserve momentum. This is going to be an elastic collision because afterwards, what have we got? Well, we got an electron and we got a photon. So we're not having any inelastic, it's not like you can deform the electron. So when we go through and we want to conserve energy and momentum, well jeez. We've got to know what the energy of the photon is and what its momentum is. Well we know form the blackbody radiation spectrum and also from the photoelectric effect that the energy associated with the photon is h times the frequency.
We also know and this actually kind of came from Compton scattering, that the momentum of the photon is given by h over lambda. Planck's constant divided by the wavelength of the radiation. We can also write this as the energy divided by the speed of light in the vacuum. Alright.
Now when you go through this energy momentum process which I'm not going to do on the board for you right now because it is kind of complicated. In fact it's surprisingly complicated. But the answer's not that bad. The answer tells us that in order to conserve energy and momentum, I've got to have a shift in the wavelength. So that means that the outgoing photon has a different wavelength than the incoming one. this is impossible classically. You cannot have a wave do that. But in quantum, you can. He's behaving like a particle and he's just going to change his energy and whenever that requires of the wavelength where it's going to change. So we have a shift in the wavelength which is given by h over mc, m is the mass of the electron, c is the speed of light, h of course is Planck's constant times one minus the cosine of the angle that the photon went off at, alright.
So, what's interesting is that this shift in wavelength is associated with h over mc. This is dimensionless. So that means that h over mc must play the role of a wavelength. In fact it does, it's called the Compton wavelength of an electron. Now, if h over mc is really really really small compared to the wavelength of light, then the shift in wavelength is relatively really small. So no problem. If on the other hand h over mc is very very very large compared to the wavelength of the radiation that you're sending in to hit the electron, then this shift in wavelength is going to be huge. So the Compton scattering is determined by how big the wavelength of the light that you're sending in is, in comparison to the Compton wavelength. Alright.
We'll get a large effect if the wavelength is much smaller than the Compton wavelength and we'll get a small effect if the wavelength is much bigger. So that gives us a quantitative explanation of what is meant by what I said earlier, large wavelength it behaves like a wave, small wavelength behaves like a particle. Large compared to what? Small compared to what? Large or small compared to the Compton wavelength of the electron. That Compton wavelength is really really really small. 2.4 pecometers. So a pecometer is a trillionth of a meter. So this actually is about 100 times smaller than the atom. So you got to send in light with a really tiny wavelength, 100 times smaller than an atom in order to get an appreciable Compton scattering effect. But we've done it and this is the way it comes out.