Method for calculating the longitudinal space-charge tune shift suitable for arbitrary bunch lengths

The space-charge tune shift is a critical issue in circular proton, heavy-ion, and also low energy electron storage accelerators. Recently, some special electron rings with energies of several hundred MeV, nanometer-scale bunch lengths, and micrometer-scale bunch spacings have been proposed. In these novel rings, a significant longitudinal space-charge tune shift could be predicted by conventional methods. However, these methods may not be suitable when the bunches are extremely short. To address this issue, we derive a new set of formulas to calculate the longitudinal space-charge tune shift, which is applicable to arbitrary bunch lengths. These formulas incorporate the multibunch effects on the longitudinal space-charge tune shift, which have not been included in previous analyses of longitudinal space-charge tune shift calculations. The calculation results indicate that the tune shift predicted by our formulas is noticeably smaller than that obtained using the classical method. Moreover, our analysis demonstrates that the multibunch effect is not dominant in the longitudinal direction, even when the bunch spacing is extremely small.