Noise analysis of a single-velocity electron gun of finite cross section in an infinite magnetic field

The Rack-Llewellyn-Pierce noise analysis,
^6,10,12,13
which has been applied to the problem of microwave noise in electron beams of large transit angle, utilizes the sinusoidal solutions for a space-charge-limited, infinite parallel plane diode, as derived by Llewellyn
³
and Smullin,
¹¹
in the gun region and Ramo's solution for the lowest mode of a finite beam confined by an infinite longitudinal magnetic field in the drift space.
⁸
In this calculation the input boundary condition which must be specified is the velocity modulation at the cathode. This was calculated by Rack
⁷
as the mean square fluctuation of the instantaneous average velocity of all electrons crossing the plane of the potential minimum.