Eigenvalues of characteristic noise matrix of an electron beam with half-Maxwellian velocity distribution

A method is devised for determining the eigenvalue spectra of the characteristic noise matrix of electron beams with continuous velocity distributions. It is found that for an electron beam with a half-Maxwellian velocity distribution, all positive eigenvalues are equal to kTΔf while all negative eigenvalues approach zero. This finding is in agreement with the phenomenon of noise reduction in an electron beam drifting at a low average velocity (an average velocity comparable to the velocity spread).