Magnetic focusing of electron beams in the presence of transverse velocity components

The behavior of nonlaminar electron beams having an originally Gaussian current density distribution in a uniform magnetic field is studied by analytical and numerical methods. The effects of transverse velocity components v
T
in the beam are described by a fictitious, equivalent perveance which is proportional to the square of
\bar{v}_{T}
, the rms value of v
T
. Simple expressions are given for the magnetic field required to achieve a certain beam transmission through a tunnel of given diameter, and for the optimum beam filling factor. In the Appendix a simple experimental method for the determination of
\bar{v}_{T}
is descried.