The ISR Impedance between 40 KHz and 40 GHz

Different current dependent effects have been measured in the ISR over the years and used to obtain information on the impedance of the beam surroundings over a large frequency range. The transverse impedance at low frequencies f < 20 MHz was obtained from the response of the beam to an excitation (beam transfer function) and at medium frequencies 30 < f < 200 MHz from the growth rate of the head tail instability. Potential well bunch lengthening and synchrotron frequency shifts gave the longitudinal reactive impedance over low and medium frequencies while the observation of the microwave instability (turbulent bunch lengthening) gave its resistive part at high frequencies f ~ 1 GHz. The measured energy loss of an unbunched beam gave an integral over the resistive impedance up to frequencies of 7 to 70 GHz depending on the beam energy. To analyse all the measured data the approximate relation between transverse and longitudinal impedance was used and a model impedance was fitted which has the expected behavior at low, intermediate and very high frequencies where the impedance is dominated by the smooth resistive wall, the effect of higher frequency resonances and by diffraction respectively.