Model of fusion kinetics for electrostatic inertial confinement discharges in D$_2$

A model\footnote{A. V. Phelps, {\it Plasma Sources Sci Technol.} {\bf 20}, 043001 (2011).} of the collisional kinetics of low-pressure glow discharges in H$_2$ is extended to predict the energy spectrum of the protons from the D-D fusion reaction in the deuterium-filled, inertial electrostatic confinement device of Boris et. al.\footnote{D. R. Boris, et. al. {\it J. Appl. Phys.} {\bf 107}, 123305 (2010).} Deuterium and hydrogen cross sections are assumed equal. D$^+$, D$_2^+$, or D$_3^+$ ions injected into a potential minimum created by cathode grid wires produce positive and negative ions, fast neutrals, and electrons. D nuclei undergo D-D fusion reactions with the background D$_2$ and produce protons with Doppler shifted peaks above and below the reaction energy of 3.02 MeV. The model shows the highest fusion flux, a good fit to the proton spectrum, and at good fit of the calculated$^1$ anion energy distribution to experiment\footnote{D. R. Boris, et. al. {\it Phys. Rev. E} {\bf 80}, 036408 (2009).} with D$_3^+$ injection using an effective discharge voltage of 27 kV for an applied voltage of 70 kV. The calculated proton flux is $\sim 10^{-10}$ of the injected ion flux. Predicted deuterium ion energy distributions are very different from that unfolded$^2$ from the proton spectrum. With D$_2^+$ injection, the proton flux is reduced by about an order of magnitude.