Electron velocity after Compton Scattering

[Inferniac]

Homework Statement

In Compton scattering,how much energy must the photon have in order for
the scattered electron to achieve relativistic velocity?

Homework Equations

Compton scattering formula: $$λ'-λ=\frac{h}{mc}(1-cosθ)$$
$E=\frac{h}{λ}$,conservation of mass and momentum,possibly Lorentz transformations for velocity and kinetic energy?

The Attempt at a Solution

My train of thought goes like this:
Assume that θ=90°. That gives us $λ'-λ=\frac{h}{mc}$.
$λ'-λ$ can easily be turned into $E'-E$ using $E=\frac{h}{λ}$.
Using law of conservation of energy we solve for the kinetic energy of the scattered electron
Since we now know the electron's kinetic energy we can calculate its speed.

My problem lies with my first assumption.I don't know if it's correct. It was made after a hint that my professor made that we should use big angles.

Please note that I don't study physics in English so some things might require clarification.

Thank you for your time.

 

[szynkasz]

I think you should assume 180^o so that energy trasferred from photon to electron is maximal.

 

[BruceW]

Hi inferniac, welcome to physicsforums :)

The equation $E=\frac{h}{λ}$ is true in a natural system of units (where c=1), but in the rest of your post, it looks like you are keeping c not equal to 1. So maybe you forgot a c in the equation above?

Also, szynkasz has the right idea with the angle, although it is not usual in this forum to give the answer outright.

 

[Inferniac]

Thank you for the welcome.

You are absolutely correct,I misstyped $E=\frac{h}{λ}$ instead of $E=\frac{hc}{λ}$ the first time and I copy-pasted again.
Yes,assuming a 180° angle makes more sense.I'll see were it goes from here.

Thanks again for your time.

 

[paranormal]

I would like to point out that your approach is correct, but it is important to consider the relativistic effects on the electron's velocity. In Compton scattering, the energy of the photon is transferred to the electron, causing it to gain kinetic energy and increase its velocity. The amount of energy transferred depends on the angle of scattering, with larger angles resulting in more energy transfer.

In order for the scattered electron to achieve relativistic velocity, it must have enough energy to overcome its rest mass energy, which is given by the formula E=mc^2. This means that the photon must have enough energy to increase the electron's kinetic energy to at least its rest mass energy, and then some additional energy to account for the relativistic effects on its velocity.

Therefore, the minimum energy of the photon required for the electron to achieve relativistic velocity can be calculated using the conservation of energy equation: E_photon = E_electron + E_rest mass + E_relativistic.

Using the Compton scattering formula, you can calculate the energy transfer (E_electron) and then add the rest mass energy (E_rest mass) and the relativistic energy (E_relativistic) to determine the minimum energy of the photon. This will give you a more accurate answer for the electron's velocity after Compton scattering.

In addition, it is important to note that as the angle of scattering decreases, the energy transfer also decreases, and therefore the electron's velocity will not be as high. This is why your professor suggested using larger angles, as they will result in a higher velocity for the electron.

I hope this helps clarify your approach and the importance of considering relativistic effects in this scenario.