The Evolution of the Phase Space Density of Particle Beams in External Fields*

In this paper the evolution of the phase space density of particle beams in external fields is presented proceeding from the conti ity equation in the six-dimensional (6D) phase space ( -space). Such a way the Robinson theorem, which includes the Liouville theorem as a special case, was proved in a more simple and consistent alternative way valid for arbitrary external fields, averaged fields of the beam (self-generated electromagnetic fields except intrabeam scattering) and arbitrary frictional forces (linear, nonlinear). It includes particle accelerators as a special case. The limits of the applicability of the Robinson theorem in case of cooling of excited ions ha resented. ving a finite living time are p INTRODUCTION In 1958 K.W.Robinson derived at once the sum of damping rates (decrements) of three particle oscillation modes in circular accelerators in the relativistic case [1]. He did an expansion of the power of frictional forces over the particle energy for the private case of radiative reaction force. That is why his final formulae do not include the term with the derivative of the power. Later, A.A.Kolomensky derived the formulae in the general form for the relativistic case and applied it to the ionization cooling [2]. He calculated separately damping rates for three directions in the curvilinear coordinate system and then took their sum. In order to derive damping increment, K.W.Robinson evaluated the determinant of the transfer matrix of the infinitesimal element of length of a particle orbit. The determinant determines the evolution of a 6D phase space volume of the beam or its density along the trajectory. P.Csonka for this purpose evaluated the infinitesimal 6D phase space volume as well and used some additional conditions to prove the theorem [3]. H.Wiedemann in the textbook [4] presented the proof of the theorem following Robinson’s idea but keeping the derivative of the power losses over the energy. Now the theorem in the particle accelerator community is named by Robinson theorem or Robinson’s damping criterion.