On the design of systems for production of intense electron beams in the space-charge-limited emission regime

Numerical modeling of electron-optical systems in the space-charge-limited regime ({rho}-regime) includes finding the current distribution on the emitter in addition to calculation of the electron trajectories. The conventional scheme for solution of this problem is based on an iteration procedure in which the current is determined in each iteration by the three-halves law using values of the potential in the vicinity of the emitter. It is known that this scheme is in general incorrect, and convergence of the process is achieved only by the use of special relaxation procedures. Moreover, this method is applicable in principle only to systems with single-component, purely electrostatic, single-stream charged-particle fluxes, i.e., the three-halves law is satisfied only in such systems. A more-universal approach to study of the {rho}-regime consists of direct satisfaction of the natural physical condition (for a zero initial particle velocity) E{vert_bar}{sub {Gamma}{sub e}} = 0 (1) on the electric-field component that is normal to the emitter surface. A method has been proposed that implements (1) and is based on solution of the system of nonlinear equations f(E{sub i}) = {var_phi}{sub i}(I{sub 1}, I{sub 2}, ..., I{sub N}) = 0, i = 1, 2, ..., N (2) for the currents I{sub k}more » in the current pipes by Newton`s method. Here E{sub i} is the electric-field component that is normal to the emitter surface at the i-th starting point, N is the number of current pipes, and f(E) is a specially selected function that vanishes when E = 0. The form of the function f(E) greatly affects the convergence of the process. The function f(E) = E{sup 2} has been recommended for trajectory analysis. An important aspect of the procedure is the use of a diagonal approximation of a Jacobian matrix with elements U{sub ik} = {delta}f(E{sub i})/{delta}I{sub k} and calculation of that matrix only in the first two iterations.« less