What Is the Wavelength of the Incident X-ray Photon in Compton Scattering?

[amani96]

Homework Statement

An X-ray photon is scattered at an angle of θ = 180.0° from an electron that is initially at rest. After scattering, the electron has a speed of 4.67 x 106m/s. Find the wavelength of the incident X-ray photon.

Homework Equations

hc/λ = hc/λ′ + KE

λ′ = λ + h/mc (1-cosθ)

The Attempt at a Solution

Ok, I solved the first equation for λ′and set both equal to each other:


hc/λ = hc/λ′ + KE
λ′= λ + (λ/hc)KE (hopefully my algebra isn't off...could somebody check?)

λ + (λ/hc)KE = λ + h/mc (1-cosθ) (subtract λ from both sides)

(λ/hc)KE = h/mc (1-cosθ) (multiply both sides 1/KE)

λ/hc = [h/mc (1-cosθ)]/KE (multiply both sides by hc/1)

λ = [h∧2/m(1-cosθ)]/KE

then plug in numbers...
But, do I have it right so far?

 

[mgb_phys]

Momentum is conserved, so all the new momentum of the electron came from the photon.
momentum of the electron = m v
momentum of the photon = h f = h c / wavelength.

 

[m2003]

I would like to provide some feedback on your solution attempt. Firstly, your algebra looks correct so far. However, it is important to note that the equation you used is only applicable for Compton scattering, where the photon loses energy and the electron gains energy. In this case, the electron's speed increasing indicates a loss of energy for the photon, which does not follow the Compton effect.

To solve this problem, we can use the conservation of energy equation:

Einitial = Efinal

Where Einitial is the energy of the incident photon (hc/λ) and Efinal is the sum of the energies of the scattered photon (hc/λ') and the kinetic energy of the electron (1/2mev^2).

Therefore, we can set up the equation:

hc/λ = hc/λ' + 1/2mev^2

Using the given values, we can rearrange this equation to solve for λ:

λ = h/(mc)(1 - cosθ)

Plugging in the values, we get:

λ = (6.63 x 10^-34 J s)/[(9.11 x 10^-31 kg)(3 x 10^8 m/s)](1 - cos 180°)

λ = 0.0012 nm

Therefore, the wavelength of the incident X-ray photon is 0.0012 nm.

It is important to also note that there are other factors that can affect the wavelength of the scattered photon, such as the atomic structure of the material it is scattered from. This solution assumes a single electron and does not take into account any other interactions. Further research and calculations may be needed for a more accurate answer.