Find the ratio of the Compton shift of a photon scattered from a free electron at a scattering angle 30.00 degrees to that of a photon scattered at an angle of 45.00 degrees.
Before we solve for this ratio, let’s consider what a Compton shift of a photon is in the first place. Say we have an electron that’s sitting at rest. Imagine further that a photon approaches this electron, itself having a wavelength we’ll call 𝜆, and runs into this electron at rest. What happens as a result of this collision is that the photon and electron are scattered. The electron will go off at some angle we could call 𝜙. And the photon will scatter off at some probably different angle we can call 𝜃.
This scattered photon though is not the same as the photon that ran into the electron. The scattered photon has had a wavelength shift. And Compton was the one who discovered that that shift can be calculated in terms of the scattering angle of the photon 𝜃. What Compton found is that the shift in the wavelength of the photon, before and after its collision, is equal to Planck’s constant divided by the rest mass of the electron that the photon runs into times the speed of light 𝑐 all multiplied by the quantity one minus the cosine of 𝜃, where 𝜃 is the scattering angle of the photon.
Incidentally, the shift of the photon’s wavelength is to a longer wavelength after the collision. The collision causes it to lose energy. In our particular scenario, we want to solve for a ratio of these Compton shifts, for two different scattering angles given to us in the problem statement. We could label these two wavelength shifts Δ𝜆 sub 30 and Δ𝜆 sub 45. And by the Compton shift equation, we can write these shifts in terms of the scattering angle of each photon.
In solving for this ratio, we see that the prefactor ℎ over 𝑚 sub zero times 𝑐 appears in both numerator and denominator and therefore cancels out. Our fraction simplifies then to one minus the cosine of 30.00 degrees divided by one minus the cosine of 45.00 degrees.
To four significant figures, this ratio is 0.4574. This tells us that the wavelength shift of a photon scattered at 45 degrees is roughly twice the wavelength shift of a photon scattered at 30 degrees.