Solution to the Equations of Space-Charge Flow by the Method of the Separation of Variables

Abstract

The equations for irrotational, electrostatic laminar space-charge flow (no thermal velocities or normal magnetic field at the cathode) are set up in terms of the action function. The resulting nonlinear partial differential equation is first reduced, by the method of the separation of variables in cylindrical polar coordinates, to the solution of a set of first-order, nonlinear, ordinary differential equations in one coordinate. It is shown that it is possible to predict axially symmetric, electrostatic, hollow beams from the inside of hollow cylindrical cathodes with space-charge-limited emission which asymptote, theoretically, to other extremely thin cylinders—though thermal velocities would, in practice, limit such a convergence. The characteristics of a particular beam, and the electrode system to produce it are shown. The beams which result from the separation of variables in other coordinate systems are described. In spherical polar coordinates one obtains hollow axially symmetric beams from a conical cathode, which may asymptote to other cones; in a less familiar coordinate system, spiral coordinates, sheet beams from equiangular spiral cathodes result. Finally, it is shown how the method may be extended to include magnetic fields transverse to the cathode. Some new solutions to space-charge flow in crossed electric and magnetic fields from a space-charge-limited cathode are mentioned.

Moderately converging ion and electron flows in two-dimensional diodes

“…(18), which can be obtained to a very high degree of accuracy, while p = p 1 (s) be the solution of (27) (containing p 0 (s) plus a 2 1 perturbation); let p refer to both, depending on the context. Let also p = p 0 (m) be the solution of Eq.…”

“…(18). Finding other coefficient is straightforward, as well known: substituting the p(s) series into Eq.…”

“…Finding other coefficient is straightforward, as well known: substituting the p(s) series into Eq. (18) and equating the coefficient of each s power to zero, we find the so-called recurrence formulas; a first equation giving a 1 in term of a 0 , a second one giving a 2 in terms of a 0 and a 1 and so on. For example, the first equation is…”

“…This fact justifies the use of the m(s) transform and is simply explained by the singular points of the underlying equations, in the practically important case α 0 ≥ π /4. Equation (18) has singular points where g s1 is zero, that is, s = s k and s = s * k with s k = α 0 + i 1 + (k + 1 2 )π and k integer. The radius of convergence R s is given by the distance of s = 0 to the nearest singular point, that is, s −1 ; so R s = |s −1 | ∼ = 1 2 π − α 0 ; Eq.…”

“…More complicated and different beam flow equilibria were found elsewhere by variable separation. 18,19 In Sec. II we will calculate the beam compression, the space charge density and write the Poisson equation both in the z and in w coordinate; we also write the motion equation for as function of s without any paraxial approximation.…”