Study of inertial electrostatic confinement fusion using a finite-volume scheme for the one-dimensional Vlasov equation

While the majority of fusion energy research is focused on magnetic confinement, there have been several alternative confinement methods aimed at the development of smaller and less expensive reactors. A number of these alternative reactors are based on a spherically convergent beam of recirculating ions and include designs such as inertial electrostatic confinement (IEC), multigrid IEC, and the periodically oscillating plasma sphere concept. Here, a fully time-dependent GPU-based Vlasov solver was developed in order to study these spherically convergent devices. This code solves the Vlasov equation for a spherically symmetric system using a finite-volume method with a modified flux to account for electrode transparency. The solver accounts for secondary electron emission, interactions between the charged particles, and collisional effects such as ionization and charge exchange. This code was used to investigate a system similar to the ion-injected device described by Hirsch (see [R. L. Hirsch, J. Appl. Phys. 38, 4522 (1967)]), who had reported a neutron production rate for deuterium-deuterium reactions in the range of ${10}^6\mathrm{to}{10}^7$ neutrons per second, which was attributed to the formation of a virtual electrode structure near the center of the chamber. Attempts to reproduce this experiment [B. J. Egle, Ph.D. thesis, 2010] yielded similar fusion rates, though the majority of the reactions were found not to occur near the center of the chamber. The results of this Vlasov solver, considering only beam-beam and beam-background fusion reactions, show that beam-background reactions would be dominant in such an ion-injected device. This result is consistent with work by Baxter and Stuart, who proposed a simplified steady-state Boltzmann model. However, the result of both models are inconsistent with the experimental results, which indicate a higher neutron production rate, and an inverse pressure scaling trend. It is shown that the higher experimental rates may be explained by beam-target fusion between the ion beam and deuterium embedded on the inner surface of the cathode.

Figure 1

A system of $N$ spherically symmetric electrode grids. The simulation code takes the number of grids, as well as the radius and time-dependent potential of each grid, as user-defined parameters.

Figure 2

Convergence testing for the neutron production rate at $n_0$ = 1.0 mTorr, $V_C$ = $-110$ kV, with an ion injection current of 10 mA.

Figure 3

Schematic representation of the geometry used in simulating the ion-injected IEC device.

Figure 4

Representative snapshots of the phase-space density for ions as determined using the time-dependent Vlasov model. (Here, $n_0$ = 1.0 mTorr, $V_C$ = $-110$ kV, $I_{\mathrm{ion}}$ = 10 mA.)

Figure 5

Simulated neutron production rate over time (0.1 mTorr, $V_C$ = $-110$ kV). The vertical dashed line represents the time it takes an ion starting from rest at the outer edge of the chamber to reach the center, as determined semi-analytically.

Figure 6

The simulated NPR pressure scaling for a Hirsch-like device with pressure, for several cathode voltages.

Figure 7

The simulated NPR compared to Hirsch's results. Note that the pressure scaling trend is inverted in the experimental pressure range.

Figure 8

The effects of secondary electron emission and backscattered particles on the fusion rate (0.1 mTorr, $V_C$ = $-110$ kV).

Figure 9

The radial distribution of the particle number density (0.1 mTorr, $V_C$ = $-110$ kV). The spike of electron density at the electrode grid is due to secondary electron emission.

Figure 10

Neutron production rate density, radial distribution (0.1 mTorr, $V_C$ = $-110$ kV). The radius of the cathode grid is indicated by the vertical dashed line.

Figure 11

Comparison of the time-dependent FVM Vlasov code to the steady-state model of Baxter and Stuart.

Figure 12

The spatially distributed ion source function $S\left(x\right)$, as determined by Eq. (A7).