Some longitudinal dynamics of bunched beams

Abstract

We show the relation of the Rees G-function to the conventional complex impedance Z. Then we consider the Vlasov equation and beam instability, and present an example calculation of growth rate and coherent frequency shift. The working is presented in detail to show how the G-function enters the problem. A novel feature of the calculation is the use of purely real (i.e. non-complex) quantities and solely positive (i.e. physically measurable) frequencies throughout. As an aid to deeper understanding of the problem, we demonstrate that a single density wave, circulating clockwise in phase space, gives rise to two modulation sidebands; and further, by considering the driven response of the beam, we show that only clockwise waves are unstable because driven anti-clockwise waves cannot grow.