Beam of particles in momentum space

  • Thread starter Habeebe
  • Start date
  • #1
38
1
I'm mostly concerned with whether or not I understand this problem intuitively in order to answer the final part of this problem.

Homework Statement


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.)


Homework Equations


Aellipse=πr1r2

The Attempt at a Solution



I answered that the circle in momentum space would become an ellipse of equal area, thereby satisfying the equation [itex]R_0^2=R_1R_p[/itex] where [itex]R_p[/itex] 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?
 

Answers and Replies

Related Threads on Beam of particles in momentum space

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
3K
Replies
4
Views
594
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
3
Views
2K
Replies
1
Views
2K
Replies
1
Views
2K
Top