Why does an electron have minimum kinetic energy when its momentum is 2h/λ?

In summary: For θ<180º, the wavelength of the scattered photon is slightly greater than the wavelength of the incident photon. It should be clear that the difference is equal to ##2\frac h{m_ec}##. This quantity is large compared to λ. That’s why the question uses the word ‘near’.
  • #1
blueberryRhyme
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Homework Statement
An X-ray beam of wavelength λ is directed along the x-axis in the positive x direc- tion and collides with carbon atoms. Electrons with a range of energies and veloc- ities are produced via the Compton Effect. The electrons with an x-component of momentum of nearly 2h/λ are produced by photons that are scattered at an angle closest to:
A. 0 radians B. π/4 radians C. π/2 radians D. 3π/4 radians E. π radians
Relevant Equations
Lambda = h/p, Compton shift = h/mc (1-cos theta)
Solution given:
The minimum kinetic energy electrons will arise from a change in photon energy on scattering that is a minimum and this will arise from the smallest wavelength change of the photon. The Compton scattering formula is
∆λ = (h/mc)(1 − cos φ) which is minimised when 1 = cos φ. This occurs for φ = 0-why electron would have minimum Kinetic energy when it’s momentum is 2h/lambda ?.
 

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  • #2
Hi @blueberryRhyme. It looks like there is a mistake here.

The question asks what photon scattering angle (θ) results in the electron having an x-momentum of ‘nearly’ 2h/λ.

But the given solution appears to be the answer to a completely different question! (The question being “what value of θ results in the minimum kinetic energy of the electron?).

Are you sure you are looking at the solution to the correct problem?

Also, the original question seems poorly posed to me. However, note that if the electron’s x-momentum changes by ‘nearly’ 2h/λ, this means the photon’s x-momentum must have changed by ‘nearly’ -2h/ λ (conservation of momentum).

The photon’s initial x-momentum was h/λ, so you need to ask yourself what must have happened to the photon if its x-momentum has changed by (nearly) -2h/λ ?

You should be able to see how your answer relates to the Compton scattering equation ##Δλ= \frac h{m_ec}(1-cosθ )## though you don't need the equation to answer the question.
 
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  • #3
Steve4Physics said:
Hi @blueberryRhyme. It looks like there is a mistake here.

The question asks what photon scattering angle (θ) results in the electron having an x-momentum of ‘nearly’ 2h/λ.

But the given solution appears to be the answer to a completely different question! (The question being “what value of θ results in the minimum kinetic energy of the electron?).

Are you sure you are looking at the solution to the correct problem?

Also, the original question seems poorly posed to me. However, note that if the electron’s x-momentum changes by ‘nearly’ 2h/λ, this means the photon’s x-momentum must have changed by ‘nearly’ -2h/ λ (conservation of momentum).

The photon’s initial x-momentum was h/λ, so you need to ask yourself what must have happened to the photon if its x-momentum has changed by (nearly) -2h/λ ?

You should be able to see how your answer relates to the Compton scattering equation ##Δλ= \frac h{m_ec}(1-cosθ )## though you don't need the equation to answer the question.

The solution is posted by my lecturer so I’m sure that it’s the solution for this question. My attempted solution assumed that the final photon momentum would be -h/ λ ( the photon scattering angle is 180). Is there anything wrong with my attempted solution?
 
  • #4
The solution is posted by my lecturer so I’m sure that it’s the solution for this question. My attempted solution assumed that the final photon momentum would be -h/ λ ( the photon scattering angle is 180). Is there anything wrong with my attempted solution?
 
  • #5
blueberryRhyme said:
Is there anything wrong with my attempted solution?
Your solution, if I understand it correctly, ends with a question. Namely, why ##0 \ne 180^\circ##?
 
  • #6
PeroK said:
Your solution, if I understand it correctly, ends with a question. Namely, why ##0 \ne 180^\circ##?
my solution is 180 degree (pi radian) but the solution provided is 0 degree. That’s why I was asking why shouldnt it be 180 degree :)
 
  • #7
blueberryRhyme said:
my solution is 180 degree (pi radian) but the solution provided is 0 degree. That’s why I was asking why shouldnt it be 180 degree :)
I think you have may a problem with the assumption that your lecturer is infallible to the point of not even handing out the solution to a different problem by mistake!
 
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  • #8
PeroK said:
I think you have may a problem with the assumption that your lecturer is infallible to the point of not even handing out the solution to a different problem by mistake!
Ahhh alright. Just want to make sure that my solution is reasonable before I go to discuss with my lecturer about the wrong solution. Thanks for helping 😇
 
  • #9
Just to add a little to what @PeroK has said:
180º is correct.
0º is incorrect.

And you should be aware that the magnitudes of the initial and final photon momenta can’t both be exactly h/λ. That would mean the photon energy (hc/λ) wouldn’t change - the scattered electron would have gotten it’s kinetic energy from nowhere!

For θ=180º, the wavelength of the scattered photon is slightly less than the wavelength of the incident photon. It should be clear that the difference is equal to ##2\frac h{m_ec}##. This quantity is small compared to λ. That’s why the question uses the word ‘nearly’.
 
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Related to Why does an electron have minimum kinetic energy when its momentum is 2h/λ?

1. What is the Compton Shift in terms of momentum?

The Compton Shift is a phenomenon in which the wavelength of a photon changes after it interacts with a particle. In terms of momentum, this means that the photon's momentum increases or decreases depending on the direction of the shift.

2. How is the Compton Shift related to the Compton Effect?

The Compton Shift is a result of the Compton Effect, which is the scattering of photons by charged particles. The shift in wavelength is caused by the transfer of energy and momentum between the photon and the particle.

3. What factors influence the magnitude of the Compton Shift?

The magnitude of the Compton Shift is influenced by the energy and momentum of the incoming photon, as well as the mass and velocity of the scattering particle. Additionally, the angle at which the photon is scattered also affects the magnitude of the shift.

4. How is the Compton Shift used in scientific research?

The Compton Shift is used in various fields of science, including astrophysics, nuclear physics, and quantum mechanics. It is used to study the properties of particles and their interactions, as well as to understand the behavior of electromagnetic radiation.

5. Can the Compton Shift be observed in everyday life?

Yes, the Compton Shift can be observed in everyday life through various phenomena such as the blue color of the sky, the redshift of distant galaxies, and the absorption of X-rays by atoms. These observations provide evidence for the existence of the Compton Shift and its role in understanding the behavior of particles and radiation.

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