# Beam of particles in momentum space

1. Oct 14, 2013

### Habeebe

I'm mostly concerned with whether or not I understand this problem intuitively in order to answer the final part of this problem.

1. The problem statement, all variables and given/known data
Discuss the implications of Liouville's theorem on the focusing of beams of charged particles by considering the following case. An electron beam of circular cross section (radius R0) is directed along the z-axis. The density of electrons across the beam is constant, but the momentum components transverse to the beam are disctributed uniformly over a circle of radius p0 in momentum space. If some focusing system reduces the beam radius from R0 to R1 find the resulting distribution of the transverse momentum components. What is the physical meaning of this result? (Consider the angular divergence of the beam.)

2. Relevant equations
Aellipse=πr1r2

3. The attempt at a solution

I answered that the circle in momentum space would become an ellipse of equal area, thereby satisfying the equation $R_0^2=R_1R_p$ where $R_p$ is the radius of the ellipse along the momentum axis. The next part is what I'm feeling sketchy on: the physical meaning. It seems like the focusing causes an increased tendency of the beam to want to converge/diverge, that is, the divergence of the beam is increased proportional to the change of radius from R0 to R1.

Does this sound right?