Maximum current in a converging beam.
The following information is taken from O. Klemperer and M. E. Barnett, Electron Optics, Cambridge University Press, 1971, pages 272-5.
A homogeneous, homocentric beam that initially converges to a point will spread because of its space-charge.
The maximum current I that can be passed through a tube of radius r and length 2z is given by
sqrt(I/V^1.5) = (D*4/2.35)*sqrt(pi*epsilon0)*(e/2m)^0.25*tan(theta),
where eV is the particle energy, D is a defined function that has the value 1.271 here, and tan(theta) = r/z.
For electrons this gives
I/V^1.5 = 3.85*10^-5*tan(theta)^2.
For ions of charge Ze and mass M the constant is multiplied by sqrt(Z*m/M), where m is the electron mass.
The radius of the beam at the centre of the tube is r/2.35.