Synchrotron oscillations with very small η

Synchrotron oscillations with very small η (Y_T^-2 - Y₀^-2) cannot be regarded as ordinary pendulum oscillations because the nonlinearity resulting from higher-order kinematic terms becomes sufficiently significant so as to lead to modification of the standard bucket. These dynamics are important in momentum debunching with rf rotation For instance, this dynamics has significant meanings in momentum debunching with rf rotation in the Debuncher Ring´ for the Fermilab TeV-I project.² RF rotation³ is used for quick reduction of large momentum spread to a small one. This large momentum spread demands that the Debuncher should have a very small n ≈ 0.002, because of engineering limitations of possible rf voltage. The above kinematic nonlinearity may eventually lead to undesirable diffusion in a debunched beam. In this paper, we shall discuss how the nonlinearity is suppressed by adjusting the path length of an off momentum particle with the hlep of sextupole correction magnets.