A Measurement of the Longitudinal Coupling Impedance in the Brookhaven Ags

The imaginary part of the longitudinal coupling impedance has been measured as a function of energy from 5 to ~ 28 GeV. This impedance is proportional to Δf = (fq - 2fd) where fd is the coherent dipole frequency and fq the coherent quadrupole frequency. These frequencies are obtained by stimulating coupled bunch oscillations. If the dominant impedance is due to inductive wall plus space charge effects, then one has (Z/n) = j[ΩoL - goZo/2ßγ2]1 where L is the inductance per turn and Ωo = 2γfo the particle rotation frequency. The expression (Z/n) = 4jΔfπ2hVocosφsB3/ 3Iofd can be used to find the impedance if the synchrotrop phase space distribution is proportional to (1 - r2)½. Io is the current per bunch, B = fo × Tl the bunch length and Vo is the external voltage. For a distribution given by (1 - r2) the right hand side should be multiplied by 27/4π2. If the latter is assumed, an inductive impedance of 20.4 Ω is obtained with a null at ≈ GeV (γtr = 8.5) for a transverse emittance of 22 πμ rad-m. At 5 GeV the reactance is negative but larger than the simple relation assumed for (Z/n) would predict. If the bunches are parabolic, then the inductive impedance would be 29.7 Ω with a null again at 6.6 GeV but only for an emittance of 2.5 μrad-m. Again the .5 GeV reactance is much too large. The significance of these results is discussed.