Calculation of transverse and longitudinal space charge effects within the framework of the fully six-dimensional formalism

Abstract

In the following report we describe a method for calculating the envelope of a particle bunch in linear coupled storage rings and transport systems in the presence of transverse and longitudinal space charge forces using the (canonical) variablesx, p _x ,z, p _z , σ=s−v ₀·t,p _σ=ΔE/E ₀ of the fully six-dimensional formalism. This work is an extension of earlier calculations on transverse space charge forces [1] to include the synchrotron oscillations. The extension is achieved by defining a 6-dimensional ellipsoid in thex−p _x −z−p _z −σ−p _σ space. The motion of this ellipsoid under the influence of the external fields and the instantaneous space charge forces can be described by six generating orbit vectors which can be combined into a 6-dimensional matrixB(s). This “bunch-shape matrix”,B(s), contains complete information about the configuration of the bunch. The solution of the equations of motion is carried through in the thin lens approximation. The formalism can also encompass acceleration by cavity fields.