2018_EJRNL_PP_ROBERT_WARNOCK_1.pdf
Terbatas  Ratnasari
» Gedung UPT Perpustakaan
Terbatas  Ratnasari
» Gedung UPT Perpustakaan
The longitudinal charge density of an electron beam in its equilibrium state is given by the solution of
the Haïssinski equation, which provides a stationary solution of the Vlasov-Fokker-Planck equation. The
physical input is the longitudinal wake potential. We formulate the Haïssinski equation as a nonlinear
integral equation with the normalization integral stated as a functional of the solution. This equation can be
solved in a simple way by the matrix version of Newtons’s iteration, beginning with the Gaussian as a first
guess. We illustrate for several quasirealistic wake potentials. Convergence is extremely robust, even at
currents much higher than nominal for the storage rings considered. The method overcomes limitations of
earlier procedures, and provides the convenience of automatic normalization of the solution.