Stationary longitudinal phase space distributions with space charge

An iterative technique has been employed to find longitudinal phase space distributions which are stationary under the influence of a constant, imaginary Z/n. In the positive mass regime (where Im(Z/n) is positive below transition or negative above transition), the main result is that, irrespective of the starting distribution, as the intensity is raised, the distribution tends towards one which has a parabolic line density. Since the locally elliptic distribution already has a parabolic projection, it is always stationary. In the negative mass regime, where the impedance augments the longitudinal focusing, centrally peaked stationary distributions cease to exist beyond a certain threshold intensity.<>