3-D Green´s function modeling for moderately relativistic chargedparticle beams in cylindrical geometry

Summary form only given. Recently, utilization of integral (Green´s function) methods has gained a substantial attention in modeling intense (i.e. space-charge dominated) relativistic charged-particle beams. Such applications range from dynamics simulations of axially-symmetric charged-particle beams in grounded regular drift tubes to modeling space-charge effects in free-electron lasers. However, there are many situations, which require totally three-dimensional (3-D) treatment (see and references therein, where the corresponding approach has been published for the non-relativistic (υ/c ≤ 0.3) case). Although, the method under development sustains a full rela tivistic generalization, it seems to us that it is fairly suffi cient to employ the Darwin model to describe dynamics of intense moderately relativistic (vie <; 0.8 + 0.9) charged particle beams. This has another advantage of not dealing with the time-retardation issues as in the full relativistic treatment. Therefore, in the present contribution we extend treatment of space-charge field from non-relativistic (ιVc<;0.3) to moderately relativistic (v/c<; 0.8 + 0.9) char ged-particle beams. In the latter case one should not only account for irrotational (potential) part, but also include into the consideration rotational part of space-charge electric and magnetic fields. In cylindrical geometry, unlike the rectangular case, this give rises to a complicated problem because of interrelation of the transversal components of the vector-potential, electric and magnetic fields. We demonstrate the analytical solution to this problem and develop the totally 3-D first-post-Newtonian (quadratic in charged-particle velocities) approximation to the equations of motion of moderately relativistic charged particles moving in self space-charge and external electromagnetic fields.