G.H." Miley

H. "Momota

National Institute for Fusion Science (NIFS-Repository)

§19. Virtual Cathode Formation in a Spherical Inertial Electrostatic Confinement Momota, H*, Miley, G.H. In a spherical inertial electrostatic confinement, plasma is produced between the spherical meshed cathode and a grounded spherical anode chamber. Ions are accelerated towards and passing through the meshed cathode, and focus near the spherical center. With this focused high-density ion core, one expects numerous fusion reactions with a simplest device. High-density profile of ions, however, yields a high potential hill structure and the kinetic energy of ions are subsequently decreases at the potential hill, which results in decreasing fusion reactions. Once electrons penetrate inside the potential hill, on the other hamd, they decrease the potential and occasionally fonn a potential well (a virtual cathode) inside the potential hill. The problem of decreasing ion energy due to the focusing ion beam will be resolved by performing the virtual cathode. Thus the virtual cathode seems one of the key issues in the development of a SIEC. The purpose of the present research is to analyze physical mechanism of virtual cathode formation and obtain quantitative condition to provide an optimum operation of a spherical inertial electrostatic confinement. The electrostatic potential at the birth region of ions is nearly zero. After acceleration by the cathode, the kinetic energy of ions attains the same as applied potential energy such as several tens keY. Their direction has small divergent around towards the spherical center. Electrons are, on the other hand, produced at the cathode mainly due to the thennal emission from its metallic structure. Here, the kinetic energy of electrons is much lower than that of ions and consequently the angular momentum of electrons is comparably small. According to the law of motion, ions of small angular momentum make certain ion cloud. Nevertheless, they are disable to access so close to the spherical center. A self-electrostatic potential also prevents ions from focusing close to the center. Majority of electrons is, in a certain case, produced at the cathode region and accelerated towards the center. In this case, electrons have a small angular momentum, and therefore, easily to penetrate inside the potentials hill to decrease the potential at the ion cloud. In another case where electrons are suffered from inelastic collisions and consequently in a thermal equilibrium state, the electron density is represented by the Boltzman's distribution. Formulae describing the number density profile of ions and electrons have been obtained theoretically in terms of respective number density and beam divergence at the cathode region. These expressions have been utilized to obtain the self-consistent electrostatic potential. Since Poisson's equation has nonlinear terms and seems hard to obtain an analytical solution, we carried out numerical calculations. One of the boundary conditions for the solution is the number density of respective species of particles at the cathode region. Another boundary condition is the vanishing radial electric field at the center. We allowed non-neutrality of local charge density either the total charge inside the cathode. One of our result concerns divergence of electrons in a low density (The density parameter: ea2n; (a) Ie oVg = 0.1.) plasma. In low-density plasma, an electrostatic potential hardly affects the density profile of ions. The size of high ion density core is a~; and the core density is 2n;(a)/~/, where a is the cathode radius and ~; is the divergence of ion beam at the cathode region. Electrons, on the other hand, suffer from effects of electrostatic potential significantly. Since the angular momentum of cathode origin electrons is small and they are accelerated towards the center, these electrons approach close to the center inside the potential hill. Consequently, these electrons create a certain virtual cathode at the center. The next result concerns high- number density at the cathode region. A higher number density at the cathode region (the density parameter: ea2n;(a) Ie oVg = 0.8) produces appreciable hill of the electrostatic potential. When the density parameter closes to unity, a bump in the number density profile of ions appears apart from the center. Ion cloud near the center changes to a bell shaped density profile. No bump on the potential profile or virtual cathode is, however, observed. It should be noted that no steady state solution has been obtained with a density parameter larger than unity. An infinitesimal change of the boundary value, for this case, brings appreciable change of the complex solution near the center. The last result deals with increased electron density at the cathode region. The bump in density profile of ions is suppressed by applying a high electron density at the cathode region. A large amount of electrons creates an electron core near the center, and consequently, an appreciable potential well (virtual cathode) appears. Due to the virtual cathode, ions with a small angular momentum crowd around the center to fonn a density peak of ions. The results obtained are in good agreement with the experimental results. Those include a bump on the density profile of ions in a high-density operation and virtual cathode fonnation. Further theoretical studies as well as experimental investigations are needed, however, to compare the results quantitatively. We found out no steady state solution for a case of the density parameter larger than unity. This suggests us an existence of a density limit for steady state equilibrium. If it is true, the power level of a steady state SIEC will be limited by the relation: Ni (aJ a2<2.5xl012Im. A certain pulsed operation has to be studied to avoid the problem. Studies on collisions such as ionization, recombination, and relaxation are inevitable to understand overall phenomena in an SIEC. 333 

Virtual Cathode Formation in a Spherical Inertial Electrostatic Confinement

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