874 THE BELL SYSTEM TECHNICAL JOTTRXAL, JULY 1957 



In a beam with large ripples, nearly all electrons have their maximum 

 radii and zero radial velocity at the same z-plane. Those with sufficiently 

 large maximum radius will have enough transverse kinetic energy to 

 surmount the space-charge forces at the beam waist, and pass through 

 or close to the axis. Others, with smaller maximum radii, will spiral 

 about that axis. Bolder and Klemperer^ have observed a similar di- 

 vision of electrons into "crossovers" and non-crossovers, in electron- 

 optical systems without magnetic fields. 



Positive ions tend to neutralize the electronic space charge at the 

 beam waists, broadening the region in which crossover occurs. The cross- 

 over trajectories thereupon overlap one another, resulting in multi- 

 valued transverse particle velocities in this region. In a first-order 

 (linearized) study of wave propagation along the beam, one must re- 

 place the actual multivelocity charge motions with a single "fluid" of 

 charge, whose velocity at any point is the average of the particle ve- 

 locities there. It is clear that the e-velocity of the stream is u, and the 

 radial velocity zero. The tangential velocity, ve = (^r)av , however, is 

 more complicated. 



Owing to the partial or total neutralization of electronic space charge 

 at the beam waists, and their large radii elsewhere, the crossover elec- 

 trons will encounter virtually no space-charge forces in their paths. Their 

 transverse paths will consequently be circles about fLxed centers, de- 

 scribed with angular velocity equal to the cyclotron frequency. Their 

 angular velocity about the beam axis is given by Busch's Theorem: 



*=2 



1 + 5 



r- 



where 



K = -^M-^j = ^max?*min (3l) 



is a positive quantity, as Bc/B is negative. Here, Tc is the radius at which 

 a particular electron left the cathode, and r is its radius in the drift 

 region. The angular velocity, 6, is greater than coc/2 at all times, and 

 exceeds coc in the waist region of the beam. The average value of ve at 

 any point here, therefore, is greater than UcT and presvunabl}^ varies from 

 point to point in some unknown way. 



If Ve is left unspecified, and the assumptions adopted of zero space- 

 charge forces and radial velocity over a finite length of beam: 



,^° = 0, r. =0, f=0, (32) 



the radial component of the force equation (7) in Euler coordinates can 



