Longitudinal space charge effect in slowly converging or diverging relativistic beams

Beginning with the Green function for a rod beam in a round beam pipe we derive the space charge induced average energy change and rms spread for relativistic beams that are slowly converging or diverging in round beam pipes, a result that tends to be much larger than the 1/γ² dependence for parallel beams. Our results allow for beams with longitudinal-transverse correlation, and for slow variations in beam pipe radius. We calculate, in addition, the space charge component of energy change and spread in a chicane compressor. This component indicates source regions of coherent synchrotron radiation (CSR) energy change in systems with compression. We find that this component, at the end of example compressors, approximates the total induced voltage obtained by more detailed CSR calculations. Our results depend on beam pipe radius (although only weakly) whereas CSR calculations do not normally include this parameter, suggesting that results of such calculations, for systems with beam pipes, are not complete.