Theory of group applied to calculate field of multielectrode systems with symmetrical geometry of electrodes

摘要

Abstract: If electrodes of ion optical system possess definitegeometrical symmetry then the potential p of the systemfield can be written in the form of the sum offunctions p$-$alpha$/. These functions fall into one ofthe irresolvable representations of the group forsymmetry of field- forming surfaces of the system. Eachof the functions p$-$alpha$/ possesses a definitesymmetry in relation to the transform of the group andis the solution for a respective boundary valueproblem. Due to symmetry these new boundary valueproblems for p$- $alpha$/ are usually simpler than theoriginal problem for the potential p. In more detail isdealt the case, when the ion optical system'sfield-forming surfaces have the group symmetry C$-nv$/.It is group symmetry of the body with n planes ofsymmetry penetrating through the axis of symmetry inn-order. For n $EQ 2,3,4,... it furns quadrapole,hexapole, octapole,... systems respectively. Proposedmethod allows to receive confined analytical formulasfor electrostatic potential of different multipolesystems with reference to the width of the clearancebetween the lamellar electrodes. In so doing, Dirihle'sboundary value problem for multi-bound space may bereduced to the solution of Neyman's, Dirihle-Neyman'sor Dirihle's boundary value problem for the functionp$-$alpha$/, but in a single-bound space what is boundare the two symmetry planes and the part of surface inthe original electrode system. We have defined theconditions when such a solution is possible.!3