| Abstract: | The mathematical formulation of the problem of determining the electrodes for the formation of intense beams of charged particles reduces to the solution of the Cauchy problem for the Laplace equation. One can proceed either by separating the variables 1] or on the basis of the theory of analytic continuation 2–5], This approach can be used for plane or axisymmetric flows. An algorithm for the construction of the analytic solution, which can also be used in the threedimensional case, is given below. It is assumed that the beam boundary coincides with the coordinate surface x1=0 of an orthogonal system xi (i=1,2,3). The solution is put in the form of a series in x1 with coefficients dependent on x2 and x3, determined from recurrence relations. The case of emission limited by space charge and temperature generally gives rise to difficulties due to the divergence of the series which makes it impossible to calculate the zero equipotential by the indicated method.As an example, the formation of beams with an elliptic cross section is considered in the following cases: (1) periodic variation of the z- component of the velocity; (2) nonmonotonic variation of the potential in one-dimensional flow between planes z=const; (3) a beam accelerated in accordance with a 3/2 law.In the construction of the expansions the conditions on the boundary are satisfied exactly by the first two terms of the series. |
