Constant-tune cyclotrons

In this paper we demonstrate that cyclotrons can be made to have precisely constant betatron tunes over wide energy ranges. particular, show the horizontal tune and does not follow Lorentz factor gamma, while still perfectly satisfying isochronous condition. To make demonstration developed a technique based on calculation of entirely from geometry realistic non-hard-edge closed orbits. We present two particular cyclotron designs, one compact ring cyclotron. The design is backed up by 3-dimensional finite element magnet calculation, also here.