= Maxwellian beam threshold for a positive velocity slope
{c}
{title2=$n_b/n_p>\sqrt e\,V\sigma_b^2\sigma_p^{-3}e^{-V^2/(2\sigma_p^2)}$}
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 $V\sigma_b/\sigma_p^2\ll1$; 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.
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