Kinetic energy of recoil atom in the photoelectric effect

[thatguy145]

Homework Statement

In the photoelectric effect it is generalley assumed that all the energy is given as kinetic energy to the electron while the atom is neglected. Do not neglect the recoil kinetic energy of the atom and calculate the kinetic energy of the photoelectron and the ionized atom and determine their ratio

Note: Etr (Energy transferred) = hf - E_B = E_K^e- + E_K^M+

Homework Equations

The above equation is relevant. I am not sure which other ones are relevant otherwise. most likely a few relativistic equations Ek = E-E0

The Attempt at a Solution

I already know the solution as it is given in the textbook except the author skipped quite a few steps.

The author gives Eke- = m(ionized atom)/M(total) * Etr and Ek(atom ionized) = me/M(total) * Etr
where m is the mass and Etr is defined above

I don't know how he gets this. I started off doing conservation of momentum:

xdir: h*nu/c = pecos(theta) + pacos(phi)
ydir: 0 = pesin(theta) + pasin(phi)

and it was suggested to me to use the relation p^2/2m and solve for the kinetic energy. I don't think that's correct. I also think solving the above equation won't yield the correct answer since it doesn't incorporate the binding energy and will require conservation of energy. At this point though I am wondering if I am going down the wrong rabbit hole since the author of the textbook just sort of wrote the equation down (when usually there is very strict derivation).

Any help would be appreciated. If clarifications are needed I will do that

 

[RedDelicious]

Ok, so you have the right idea. You need to apply conservation of momentum and energy to the system before and after the absorption and emission. You don't need to concern yourself with any angles for the momentum.

What do you think the initial energy of the atom and photon system is?