Static Green’s functions for a coaxial cavity including an innovative representation

This paper deals with the three-dimensional potential equation in cylindrical coordinates and its Green’s function for a geometry close to a typical Time Projection Chamber field-cage, namely a coaxial cavity. The derivations of three different representations of the Green’s function are described in detail. Each one is a sum over two of the three labels of the eigenfunctions of the homogeneous boundary value problem; the coefficients correspond to one-dimensional Green’s functions in the third variable, which are found by the method of particular integrals. The third representation is an innovative one using modified Bessel function of purely imaginary order, which can also be derived by a Sommerfeld-Watson transformation. In combination, these three representations allow fast converging calculations of the potential and the electric field for nearly any position of the point charge and therefore for any desired space charge configuration within the cavity.