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tokens: speed projected onto the beam direction. A.2. Plasma Instability A multipeaked electron distribution function , such as that shown in Figure 2 with the addition of the thermal electron peak at small velocity (not in the figure, that only shows the scattering source) is unstable to the electrostatic “bump-on-tail” instability. This instability is essentially inverse Landau damping, and occurs whenever the derivative of the distribution function with respect to some component of electron velocity is positive. Electron plasma waves with phase velocities ( ω and k are the plasma wave frequency and wave vector, related by the dispersion relation , where is the plasma frequency and the Debye length) for which grow at a rate of (Chen 1974 )where is the normalized one-dimensional electron distribution function of a density n beam of high velocity (Compton recoil) electrons and n e is the background electron density. If the fast electrons are only a small fraction of the total electron density and their speeds are much greater than the electron thermal velocity, conditions that are generally met, then for waves excited by the fast electrons and , where v is a velocity on the rising part of the electron distribution function. Then the growth rate becomeswhere is not normalized to n beam , but is the full electron distribution function. The rapid decrease of drag deceleration with increasing speed, shown by the factor in Equation ( 27 ), acts to create a minimum in (the large peak in from thermal electrons at very small velocity is not shown in Figure 2 ) even when S ( v ) is a monotonically decreasing function, as it can be for power-law photon spectra or a mixture of two- and three-photon annihilation (not shown). For example, if , making the non-relativistic approximation , then . For all but the most steeply decreasing S ( v ), and for v exceeding a few thermal velocities, and the plasma is unstable. In the opposite limit of scattering by very energetic gamma-rays , , and again the plasma is unstable (for quantitative growth rates Equations ( 28 ) and ( 29 ) need to be corrected for the relativistic increase in electron mass). This mechanism of producing unstable electron distributions may operate anywhere an intense flux of gamma-rays, not necessarily from positron annihilation, whose spectrum is not too steep, is incident upon a non-relativistic plasma. The required intensity is determined by the competition between the growth rate γ (Equations ( 28 ) and ( 29 )), proportional to the gamma-ray flux, and the collisional damping rate of the plasma wave that depends on the plasma density and temperature. For a plasma frequency comparable to the frequencies MHz of observed FRBs, is sufficient to grow the instability by ∼10 e -folds to saturation in the sub-millisecond timescale of FRBs. A.3. Energetic Failure This apparently attractive model fails on energetic grounds. The total Klein–Nishina cross-section for a 511 keV photon is cm 2 , so that