Calculations of space-charge tune shifts in storage rings with extremely short bunches and small bunch spacing

Space-charge tune shifts can affect the beam quality in storage rings. Laslett's method, a classical approach for calculating space-charge tune shifts, may not hold valid assumptions for rings with extremely short bunches and small bunch spacing, such as in steady-state microbunching (SSMB) storage rings. To overcome this limitation, we have developed new formulas that are suitable for arbitrary bunch lengths and spacings for calculating space-charge tune shifts. Additionally, simplified formulas have been provided for the calculations of space-charge tune shifts specifically in the context of extremely short bunch lengths. Numerical computations were carried out to demonstrate the validity of these newly derived formulas. Moreover, the formulas were applied to compute the space-charge tune shifts in an SSMB storage ring characterized by typical parameters.