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

[blueberryRhyme]

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 ?.

 

[Steve4Physics]

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.

 

[blueberryRhyme]

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?

 

[blueberryRhyme]

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?

 

[PeroK]

Is there anything wrong with my attempted solution?

Your solution, if I understand it correctly, ends with a question. Namely, why $0 \ne 180^\circ$?

 

[blueberryRhyme]

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 :)

 

[PeroK]

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!

 

[blueberryRhyme]

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

 

[Steve4Physics]

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’.