Maxwellian beam threshold for a positive velocity slope ($n_b/n_p>\sqrt{e}V{\sigma }_b^2{\sigma }_p^{-3}{e}^{-V^2/(2{\sigma }_p^2)}$)

A broad electron Maxwell-Boltzmann distribution of width ${\sigma }_p$ plus a narrow beam of width ${\sigma }_b$ centred at speed $V$ can have a positive resonant velocity slope. The beam slope is largest at $V-{\sigma }_b$. Treating the background slope as constant over the beam width gives the displayed density threshold. Replacing the Maxwellian tail at $V-{\sigma }_b$ by its value at $V$ also requires $V{\sigma }_b/{\sigma }_p^2\ll 1$; width separation alone does not ensure that exponentially sensitive approximation. A positive slope is the local resonant-drive criterion, while an exact unstable mode also requires the kinetic dispersion relation.