Calculation of Particle Trajectories in an Ion Linac with Self-Focusing Accelerating Gaps Having Quadrupolar Symmetry

In linear accelerators with drift tubes the bunching of particles around the synchronous one is obtained at the expense of focusing. Nevertheless, it is possible to ensure simultaneous radial and axial stabilities if, in the accelerating gaps, we abandon the cylindrical symmetry. If the potential function in a gap between two drift-tubes has quadrupolar symmetry, it is possible, with certain assumptionsabout the gap geometry, to expand this function as a double Fourier series. Thus we are able to calculate the transfer matrix from one drift tube to the next. Formulae giving axial and radial R. F. impulses, obtained in the thin-lens approximation for the quadrupole symmetry planes are easily applicable. The effect of the R.F. is due to "circular" and "quadrupolar" impulses; the former is independent of the meridian plane, whereas the sign of tne latter changes from one quadrupolar symmetry plane to the other. Tne radial and axial motions of the particles cease to be independent. The chosen potential functiorn enables us to calculate the second order transfer elements containing the coupling terms between the two degrees of freedom.