In a Compton scattering experiment, a photon is scattered through an angle of 90.0^∘ and the ele

step 1 In this problem, we're going to talk about the Compton effect. So I'm going to briefly review wh

In a Compton scattering experiment, a photon is scattered...

Key Concepts

Relativistic Energy and Momentum Conservation

In any scattering process, especially in Compton scattering, both energy and momentum must be conserved. The conservation laws are used to relate the energies and momenta (and therefore the wavelengths) of the incident and scattered photon together with the recoil electron’s kinetic energy and direction, leading to a system of equations whose solution uniquely determines the unknown quantities if sufficient information is provided.

Compton Wavelength Shift Formula

A central equation in Compton scattering is ?? = (h/(m?c))(1 - cos ?), which gives the change in the photon’s wavelength as a function of its scattering angle. This formula encapsulates how the wavelength change is independent of the incident photon’s energy and depends solely on the scattering geometry, provided that other kinematic constraints are properly applied.

Compton Scattering

This is the phenomenon where an incident photon scatters off a target electron and suffers a change in wavelength (or energy) as a result of transferring a portion of its energy to the electron. It provided one of the first firm evidences for the particle?like behavior of light, and its description relies on quantum theory and special relativity.

Scattering Geometry and Kinematic Constraints

The geometry of the scattering process, namely the angles at which the photon and electron emerge, provides additional constraints that must be satisfied along with conservation laws. The angular information allows one to break the momentum conservation into components, ensuring that the process is fully determined if the given data provide enough independent equations to solve for the unknown quantities such as the scattered photon’s wavelength.

Transcript

00:01 In this problem, we're going to talk about the compton effect.

00:04 So i'm going to briefly review what this effect tells us.

00:08 So consider that we have an electron, actually a photon, that collides with an electron that's initially at rest.

00:17 And as a result, the photon is scattered with a scattering angle that's equal to theta.

00:27 What the compton effect tells us is that lambda prime, that is the one, the one that is the wavelength of the scattered photon is equal to the wavelength of the incoming photon plus h over the mass of the electron types c times 1 minus the cosine of the scattering angle favor and h over mc this is a constant that's useful to remember is equal to 2 .43 times 10 to the minus 12 meters okay and in our case we have a situation where we have an incoming photon that collides with an electron and as a result the photon is scattered at an angle of 90 degrees while the electron is scattered at an angle of 20 degrees and our goal is to find whether or not we can find the wavelength of the scatter photon, lambda prime with this information, and if we can, what is the value of lambda prime? so what i'm going to use here is both the conservation of momentum and the compton effect formula.

01:57 So from our formula, if we apply that theta is equal to 90 degrees, then lambda prime is equal to lambda plus h over mc.

02:09 Since the cosine of 90 is zero.

02:15 Also, let's write the conservation of momentum here.

02:19 The momentum, the initial momentum of the incoming photon is h over lambda.

02:25 The electron has no momentum.

02:28 And this is equal to, notice that you're working with the horizontal direction here.

02:35 So the outgoing photon has no momentum in the horizontal direction, only the electron, and the momentum of the electron is m times the speed of the electron, that i'm going to call it u, times the cosine of 20.

02:54 And in the vertical direction, initially there's no momentum, and in the end, we have the momentum of the photon, that's h over lambda prime, hoverlonda prime, minus the momentum of the electron, in the vertical direction, that's mu times the sine of 20.

03:22 So we obtain that h over lambda prime is mu times sine of 20...

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