Comment on the validity of first-order paraxial theory for electron and ion sources with needle-type or pointed geometry

An essential feature of first-order paraxial theory is the assumption that if the potential along the axis of an axially symmetric system is known, then the potential (fields) near the axis can be obtained by a power-series expansion about points lying on axis. However, in traditional first-order theory, which commonly assumes systems with cylindrical symmetry, only terms up to second-order in the coordinate transverse to the beam axis are retained. In this letter we argue that a consequence of this restriction is that traditional first-order paraxial theory should not be applicable to electron and ion sources with pointed or needle-type geometries. In order to treat non-parallel trajectories which occur in the pointed geometries present, say, in field emission liquid metal ion sources, a modified paraxial theory is suggested which describes two-dimensional particle dynamics.This work has been supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-81008829     

Gaudin Model,KZ Equation and an Isomonodromic Problem on the Tours

This Letter presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe within the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural r-matrix structure. These Hamiltonians are used to build a nonautonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.