Question

I need to know the steps of solution for part (b,c,d) I have been trying for almost 2 days.

A 195 -keV X-ray photon coherently scatters off one of the valence electrons of a nitrogen atom. Assume that the scattering angle of the photon is ? = 15

Answer 1

Let's put this in context: Since the electron is a valence electron, this means it interacts weakly with the nucleus and we have to neglect the influence of it. We basically have an ELASTIC collision between the photon and the electron: TWO things are conserved: Energy of the system and Momentum of the system.

let's begin by remembering Compton Scattering formula:

where, for a photon:, the momentum of the photon:   and

where is the photon scattering angle. Meanwhile, for an electron:  Therefore:

for the speed of the electron we have: . Finally, a picture:

here is the electron recoil angle.

(a) For this, you probably used :   and used the electron rest energy that was given as 511 keV. You do get 192.497 keV which you rounded to 192.5 keV.

(b) Let's write down the conservation of momentum in the horizontal axis:

  

and for the vertical axis:

  

We have two unknowns: and . Solving for :

We then plug this into the second equation: .

Now plug this into the equation for the speed of the electron above:

in the recoil direction given by .

by the way, we had to convert the energy E' from keV to J by taking into account: so . therefore: Also: Since which we use in that formula for the speed of the electron.

(c) The photon is still traveling at the speed of light in the scattering direction

(d)We go back to the Compton scattering formula (simplifying the hc's):

this time we have therefore, solving for :