- #1

Aleolomorfo

- 73

- 4

## Homework Statement

Finding the maximum mass ##M_x## which can be made from a collision of identical particles with mass ##m##, in the laboratory frame, in which one particle is at rest and the other one has energy ##E##. The reaction is the following: ##a+b \rightarrow a+b+x##.

## The Attempt at a Solution

I assume that the maximum mass is produced when the three resultant particles are at rest in the CM frame. So I use the invariant of the total momentum squared from the lab frame before the collision (##p##) and the CM frame after the collision (##k##):

$$p_1=(E,0,0,\sqrt{E^2-m^2}) \hspace{1cm} p_2=(m,0,0,0)$$

$$(p_1+p_2)^2 = 2m(E+m)$$

$$k_1=k_2=(m,0,0,0) \hspace{1cm} k_x=(M_x,0,0,0)$$

$$(k_1+k_2+k_x)^2=4m^2+4mM_x+m^2_x$$

I equal the invariants:

$$2m(E+m)=4m^2+4mM_x+M^2_x$$

After calculation I find a second order equation in ##M_x## whose solutions are:

$$-2m\pm\sqrt{2m^2+2mE}$$

I reject the solution with - because it is completely negative but also the solution with + is negative under a certain value of E. I do not know if this is because a mistake or it is the threshold energy of the reaction.