The effect of nonlinear synchrotron motion on the SOLEIL energy acceptance

The lengthening of the trajectory is determined by the momentum compaction factor α and the betatron oscillation contribution term ΔC/C₀. When the first order term of the momentum compaction factor α₁ is small, the second order term α₂ and ΔC/C₀ have to be included in the longitudinal equations of motion. In this case, the RF bucket form changes and the energy acceptance can be significantly reduced. This leads to a decrease in Touschek beam lifetime. In this paper we show that the value of α₂ in the standard low emittance lattice of SOLEIL is large and can reduce the energy acceptance of the machine. We discuss the magnitude of this reduction and the difficulties encountered in minimizing the value of α₂ in the case of a low emittance lattice. We also show that the lengthening of the trajectory induced by horizontal and vertical betatron oscillations vanishes when the chromaticities are corrected to zero