Threshold and bursting dynamics of the CSR instability in electron rings

For a Gaussian bunch in an electron ring with the shielded steady-state CSR impedance, Vlasov-Fokker-Planck calculations give a threshold in the dimensionless current $S_{\mathrm{csr}}$ of approximately $S_{\mathrm{csr}},\mathrm{th} = 0.5 + 0.12$ $\Pi$, with $\Pi$ the shielding parameter, plus a narrow low-threshold region near $\Pi \sim 0.7$ (2010). Determine whether the threshold and the period and amplitude of sawtooth bursting above it are predicted within 20 percent once the full ring impedance, potential-well distortion and finite chamber geometry are included, in low-momentum-compaction operation of several rings.

Bunch-by-bunch measurements of threshold current and burst spectra versus Pi in two or more rings, compared with Vlasov-Fokker-Planck solutions that use independently measured impedances.