Compton scattering event (Proof question)

In summary: You are trying to prove that the wavelength of the scattered photon is fixed at a constant value.You can use the relativistic energy-momentum relations for both the photon and the electron.You should be able to do something with those two equations.
  • #1
jisbon
476
30
Homework Statement
Prove that the scattered photon is fixed at a constant value with only the condition that both angles from the photon and electron sum up to 90 degrees after the collision.
Relevant Equations
-
Hi all,
Currently given a problem to prove that the scattered photon is fixed at a constant value with only the condition that both angles from the photon and electron sum up to 90 degrees after the collision. I can't seem to prove it and listed my steps below, was wondering if anyone can guide/correct my mistakes. Cheers.

For x-direction:
$$
P_{photon}=P'_{photon}cos\theta +P_{e}cos\phi
$$
$$
\frac{h}{\lambda_{0}}= \dfrac {h}{\lambda ^{'}}\cos \theta +P_{e}\cos \left( 90-\theta \right)
$$
$$\frac{h}{\lambda_{0}}= \dfrac {h}{\lambda ^{'}}\cos \theta +P_{e}\sin \left( \theta \right)$$
$$\begin{aligned}\dfrac {\lambda ^{0}}{h}=\dfrac {\lambda ^{'}}{h\cos \theta }+\dfrac {1}{P_{e}\sin \theta }\\ \lambda ^{0}=\dfrac {\lambda ^{'}}{\cos \theta }+\dfrac {h}{P_{e}\sin \theta }\end{aligned}$$

For y-direction:
$$
p_{e}\sin \phi =P'_{photon}\sin \theta
$$
$$
p_{e}\sin \left( 90-\theta \right) =P'_{photon}\sin \theta
$$
$$
p_{e}\cos \left(\theta \right) = \dfrac {h}{\lambda '}\sin \theta
$$
$$p_{e}\cot \left(\theta \right) = \dfrac {h}{\lambda '}$$
$$\lambda '=\dfrac {h}{P_{e}\cot \theta } $$

From the Compton equation,
$$\lambda '-\lambda _{0}=\dfrac {h}{m_{e}c}\left( 1-\cos \theta \right)$$I was wondering if this is correct because I can't seem to solve it moving on from these steps. Any advice help will be appreciated. Cheers
 
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  • #2
I guess what you are trying to prove is that there is only one solution where the sum of the angles is 90 degrees?

One approach is to show that the energy of the scattered photon is some function of the fixed quantities: namely the energy of the incident photon and the mass of the electron.

I would use Pythagoras theorem and avoid using ##\theta##.
 
  • #3
You know, I think, from classical mechanics that when two equal masses collide elastically in a 2-d collision, the sum of the scattering angles is 90o. Can you use that?
 
Last edited:
  • Informative
Likes etotheipi
  • #4
kuruman said:
You know, I think, from classical mechanics that when two equal masses collide elastically in a 2-d collision, the sum of the scattering angles is 90o. Can you use that?
But the masses are different if I'm not wrong? Phonon and electrons.
 
  • #5
PeroK said:
I guess what you are trying to prove is that there is only one solution where the sum of the angles is 90 degrees?

One approach is to show that the energy of the scattered photon is some function of the fixed quantities: namely the energy of the incident photon and the mass of the electron.

I would use Pythagoras theorem and avoid using ##\theta##.
Prove that the wavelength of the scattered photon is fixed at a constant value.
 
  • #6
jisbon said:
Prove that the wavelength of the scattered photon is fixed at a constant value.
Did you try using Pythagoras theorem for the momenta?
 
  • #7
kuruman said:
You know, I think, from classical mechanics that when two equal masses collide elastically in a 2-d collision, the sum of the scattering angles is 90o. Can you use that?
We have to use the relativistic energy-momentum relations for both the photon and the electron.
 
  • #8
I can't seem to get about using Pythagoras theorem, what do I put in as the two sides? What is the hypotenuse in this case? Thank you
 
  • #9
jisbon said:
I can't seem to get about using Pythagoras theorem, what do I put in as the two sides? What is the hypotenuse in this case? Thank you
By conservation of momentum we have ##\vec p = \vec p' + \vec p_e##. In general, the before and after momenta form a triangle. In this case the angle between ##\vec p'## and ##\vec p_e## is a right-angle, hence:
$$p^2 = p'^2 + p_e^2$$
That can be converted to an equation involving the energies. And you also have another equation from conservation of energy. You should be able to do something with those two equations.
 
  • #10
Bit late I suppose but all the same ...

Angle of Incidence: $$\tan\theta_i=\frac{\lambda_c+\lambda_i}{\lambda_i}\sqrt{\frac{\Delta\lambda}{2\lambda_c-\Delta\lambda}}$$Angle of Refraction: $$\tan\theta_r=\frac{\lambda_c-\lambda_r}{\lambda_r}\sqrt{\frac{\Delta\lambda}{2\lambda_c-\Delta\lambda}}$$Latter will be zero when ##\lambda_r=\lambda_c## corresponding to the given condition.
 

Related to Compton scattering event (Proof question)

1. What is a Compton scattering event?

A Compton scattering event refers to the phenomenon in which a photon interacts with an electron, causing the photon to lose energy and change direction. This process was first described by physicist Arthur Compton in 1923 and is an important concept in understanding the behavior of light and matter.

2. How does a Compton scattering event occur?

A Compton scattering event occurs when a photon collides with an electron. The photon transfers some of its energy to the electron, causing it to recoil and change direction. The scattered photon will have a longer wavelength and lower energy than the original photon, while the electron will have a higher energy and momentum.

3. What is the significance of Compton scattering?

Compton scattering is significant because it provides evidence for the particle-like behavior of light. It also helps to explain the behavior of X-rays and gamma rays, and is used in medical imaging and other applications.

4. How is a Compton scattering event measured or detected?

A Compton scattering event can be measured or detected using various techniques, such as X-ray diffraction, X-ray fluorescence, or Compton scattering spectroscopy. These methods involve analyzing the scattered photons to determine their energy and direction, which can provide information about the original photon and the electron it interacted with.

5. What is the mathematical formula for calculating the energy of a scattered photon in a Compton scattering event?

The energy of a scattered photon in a Compton scattering event can be calculated using the Compton scattering formula: Es = Ei / [1 + (Ei/mec2)(1 - cosθ)], where Es is the energy of the scattered photon, Ei is the energy of the incident photon, me is the mass of the electron, c is the speed of light, and θ is the scattering angle.

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