Calculation of beam envelopes in storage rings and transport systems in the presence of transverse space charge effects and coupling

Abstract

In the following report we describe a method for calculating the envelopes of coasting beams in linear coupled storage rings and transport systems in the presence of transverse space charge forces. This work is an extension of earlier calculations [2, 3] to include coupled beam optics and energy deviations. The extension is achieved by defining a 5-dimensional ellipsoid in thex−p x −z−p z −Δp/p space. The motion of this ellipsoid under the influence of the external fields and the instantaneous space charge forces can be described by five generating orbit vectors which can be combined into a 5-dimensional matrix $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} (s)$$ . This “bunch-shape matrix”, $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{B} (s)$$ , contains complete information about the configuration of the bunch. The solution of the equations of motion is carried through in the thin lens approximation.