Abstract
The limiting current density that can be achieved with thermal velocities in the source emission is calculated by phase-space ellipse transformations. The results check those of Pierce, and are applicable to a wider range of problems. Simple lens systems can realize the ellipse transformations which give the limiting density. Under certain conditions the maximum current density is obtained between the object plane and the image plane, a situation analogous to the ``waist'' formation in a beam transport system used with particle accelerators.
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