Change in wavelength, photon hits a free electron.

In summary, the problem asks for the change in wavelength of a photon after colliding with a free electron. Using conservation of energy and momentum, the Compton scattering equation is derived and simplified for the case of 180 degree recoil. The change in wavelength is found to be 2h/mec.
  • #1
AbigailM
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0

Homework Statement


A photon with initial momentum p collides with a free electron having
a mass m that is initially at rest. If the electron and photon recoil in opposite
directions, what will be the change in the photon’ wavelength? (Hint: use
relativistic forms for energy and momentum.)

Homework Equations


Conservation of Energy:
[itex]hf_{i}+m_{e}c^{2}=hf_{f}+\sqrt{p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}[/itex]

Conservation of Momentum:
[itex]\boldsymbol{p_{i}}=\boldsymbol{p_{f}}+\boldsymbol{p_{e}}[/itex]

The Attempt at a Solution


I won't go through the whole derivation as it's quite a bit of latex but:
If you square both equations above and introduce hf into the conservation of momentum equation, you can equate the two equations and rearrange. This will give you the compton scattering equation:

[itex]\lambda_{2}-\lambda_{1}=\frac{h}{m_{e}c}(1-cos\theta)[/itex]

If the recoiling electron and photon are to be in opposite directions this is an angle of 180°.
Plugging this into the compton scattering equation gives:

[itex]Δ\lambda=\frac{2h}{m_{e}c}[/itex]

,which is the change in wavelength.

Does this look ok? As always everyone, thanks for the help!
 
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  • #2
Looks ok. But it seems to me that the intent of the problem was to set up the conservation equations for the specific case of 180 degree recoil. That makes the messy algebra simpler than the general case.

However, if you can go through the algebra for the general case and then substitute the specific value of theta at the end, then that should certainly count as a solution!
 

1. How does a change in wavelength affect the energy of a photon?

A change in wavelength directly affects the energy of a photon. The shorter the wavelength, the higher the energy of the photon. This is because the energy of a photon is directly proportional to its frequency, and since wavelength and frequency are inversely related, a shorter wavelength corresponds to a higher frequency and therefore, a higher energy photon.

2. What happens when a photon hits a free electron?

When a photon hits a free electron, it transfers its energy to the electron. This can cause the electron to be excited and jump to a higher energy level, or it can completely free the electron from its atom, resulting in the emission of a new photon.

3. How does the energy of a photon affect the probability of it interacting with a free electron?

The energy of a photon directly affects the probability of it interacting with a free electron. Higher energy photons have a higher probability of interacting with free electrons compared to lower energy photons. This is because higher energy photons have shorter wavelengths, making them more likely to collide with electrons.

4. Can a change in wavelength cause a free electron to move?

Yes, a change in wavelength can cause a free electron to move. When a photon hits a free electron, it transfers its energy to the electron, causing it to gain kinetic energy and move. This can result in the electron being excited or being completely freed from its atom.

5. How does the speed of a free electron compare to the speed of light?

The speed of a free electron is much lower than the speed of light. In most cases, the speed of a free electron is around 1% of the speed of light. However, when a free electron is accelerated, such as in a particle accelerator, it can reach speeds close to the speed of light.

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