Collisional series resistance in electron beams

The standard reference books on electron optics, Glaser (1952), Sturrock (1955) and Grivet (1972), do not consider the possibility that an electron beam can exhibit a series resistance, even at zero frequency. The letter demonstrates that a series-resistance term must exist. Numerical estimates of the series resistance are given, from which it is shown that the resistance is not negligible for the high-density electron beams used in modern millimetre-wave oscillators and amplifiers. The consequences for noise theory and for the theory of crossed-field devices, in which the beam resistivity is increased by a large factor, are briefly discussed.