Conservation of momentum in relativistic disintegration

[Frank Einstein]

Homework Statement

Hi everybody, I was studyng a photon disintegrating in one electron and one positron in the presence of a heavy atomic core; I want to find the final momentum of the atomic core

Homework Equations

Conservation of energy: Eγ=2me+Sqrt(Mn²+Pn²)
Conservation of momentum Pγ=Pn
Relation between the energy and momentum of a maseless particle Pγ=Eγ

The Attempt at a Solution

From the conservation of energy: (Eγ-2me)²=Mn²+Pn²; and since Pn=Eγ, operating after developing (Eγ-2me)², I obtain Pn=4me-(Mn²/me). If I introduce the values in units of atomic mass, I obtain a negative value of Pn, the momentum of the atomic core.
Can anyone please tell me what am I doing wrong?
Thanks.[/B]

 

[haruspex]

I obtain Pn=4me-(Mn2/me).

I get a different equation. Please post your working.

 

[Frank Einstein]

I get a different equation. Please post your working.

This is what I have calculated:

I start from Eγ=Pn and Eγ=2me+Sqrt[Mn^2+Pn^2]

Eγ-2me=Sqrt[Mn^2+Pn^2]

(Eγ-2me)^2=Mn^2+Pn^2

(Eγ)^2+4(me^2)-4meEγ=Mn^2+Pn^2. Then, (Eγ)^2 on the left and Pn^2 on the right cancel each other.

4me^2-4meEγ=Mn^2

4me^2-Mn^2=4meEγ=4mePn

Pn=(4me^2-Mn^2)/me=4me-(Mn^2)/me

 

[haruspex]

4me^2-Mn^2=4meEγ=4mePn

Pn=(4me^2-Mn^2)/me=4me-(Mn^2)/me

Fine until the last step. However, correcting it will not fix your sign problem, so the error must be much earlier, in one of your initial equations. I am not a nuclear physicist, so cannot comment on those.

 

[jbriggs444]

Relation between the energy and momentum of a massless particle Pγ=Eγ

To be more correct, in units where c=1 and dealing with massless particles, |P| = E. If you got a negative momentum then you must have had a negative energy. If you have a negative energy then something does not add up.

I am having a horrible time trying to decrypt the ASCII math in the original post without a cheat sheet for the variable names.