Two-Dimensional Axisymmetric Child-Langmuir Scaling Law

The classical one-dimensional Child-Langmuir law has been extended to two dimensions by numerical simulation in planar geometries [1]. By considering an axisymmetric cylindrical system with emission radius $r$, outer radius $R > r$, and gap length $L$, we further examine the space charge limit in two dimensions. The ratio of the observed current density limit J$_{CL2}$ to the theoretical one-dimensional value J$_{CL1}$ is found to be a monotonically decreasing function of the ratio of emission area ($r^2$) to gap separation ($L$). This result is in agreement with the planar results, where the emission area is proportional to the cathode width (r) [1]. The simulations were run in the particle in cell code, OOPIC [2]. [1] J. W. Luginsland, Y. Y. Lau, and R. M. Gilgenbach, Phys. Rev. Lett. 77, 4668 (1996). [2] J. P. Verboncoeur, A. B. Langdon, and N. T. Gladd, Comp. Phys. Comm. 87, 199 (1995).