A photon having energy E0=0.880 MeV is scattered by a free electron initially at rest such that

A photon having energy E0=0.880 MeV is scattered by a free...

Key Concepts

Compton Scattering

Compton scattering is the phenomenon where a photon collides with a free electron, resulting in a transfer of energy and momentum. This process leads to an increase in the photon's wavelength (a decrease in energy) and provides experimental evidence for the particle nature of light. It is governed by the Compton equation, which relates the change in the photon's wavelength to the scattering angle, reflecting the underlying conservation laws.

Conservation of Energy and Momentum

In any collision process, including photon-electron scattering, the total energy and momentum are conserved. This means that the sum of the energies and momenta before the collision equals the sum after the collision. In the context of photon scattering, these conservation laws are applied to derive relationships between the initial photon energy, the scattering angles, and the energies and momenta of both the photon and the recoiling electron.

Relativistic Kinematics

Photon-electron scattering often involves relativistic speeds, especially when the photon energy is high. Relativistic kinematics takes into account the effects of special relativity, incorporating factors such as the relativistic energy-momentum relation. This framework ensures that calculations of energy, momentum, and scattering angles remain accurate when dealing with particles moving at or near the speed of light.

Scattering Angle Relationships

The scattering angles of both the photon and the electron are crucial in determining the outcome of the collision in Compton scattering experiments. These angles are directly related to the energy transfer during the interaction. In some cases, symmetry in the scattering angles can simplify the application of conservation laws, enabling the derivation of explicit relationships between the scattering parameters.

Transcript

00:01 In this question, we are looking at the computer effect where we have a photon on the left coming in, hitting a stationary electron.

00:12 So this three electron are considered to be initially at rest, so it's stationary.

00:19 And subsequently, the electron gets scattered off in this direction, making an angle of deter with respect to the original motion of the original photon and the photon also get scattered off in the same angle, theta.

00:38 So both these theters are the same angle.

00:41 We want to find what is the angle, the scattering angle, and also to find the momentum and energy of the photons as well as the electron.

00:58 So first off, we want to apply some conservation of momentum.

01:07 So we know the momentum of the initial photon.

01:13 Here is related to the original energy of the photon divided by c.

01:19 So e0 divided by c is the momentum of this photon.

01:26 And by conservation of momentum, the final momentum in the x direction.

01:31 Alright, that's considered this the x direction, where the initial photon is traveling along must be conserved so the x direction of this momentum of the electron let me just call this pe as well as the final momentum of the photon call this p prime the x components must add up to give us the original momentum over here so that's pe cosytheidater plus p prime cosine theta must give us p .0.

02:15 On the other hand, if you consider the y direction, right, the y direction initially has no momentum in that direction, so the final momentum in this direction must also be zero.

02:29 Therefore, this y component momentum, the magnitude of them must be the same, right, so that they will add up to be zero in total.

02:42 So we have pe sine theta must be equals to p prime sine theta.

03:00 Therefore p prime pe must be equals to p prime.

03:05 This is for second equation.

03:15 Now we can simplify a little bit our first equation.

03:24 We can substitute actually pe as p prime, so this becomes this p prime.

03:39 So we have 2 times of p prime cosine tether equals to p .0.

03:48 Now of course we don't know what is p prime yet, but we know that it is related to its wavelength by h over lambda prime, when lambda prime is the wavelength.

04:01 Length.

04:05 So this is in terms of wavelength, this is actually equals to 2h cosine tether over lambda prime, is equals to h over lambda knot.

04:18 This is the original wavelength.

04:24 We arranging this a bit, we should get lambda prime equals to 2 times lambda not cosine theta.

04:34 Now this equation is quite important.

04:36 Because we can use the compton scattering equation.

04:42 So the compton scattering equation is lambda prime minus lambda not is equals to h over mc 1 minus cosine deter.

04:58 So this derived actually from conservation of energy plus momentum.

05:04 It will help us to solve this very quickly, because we already have our lambda...

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