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For a given rf cavity voltage, there is a maximum attainable peak current in an ion storage ring. This occurs when the electric field from the beam space charge is balanced by the field from the rf cavity. In this limit, the linear charge density distribution is parabolic and incoherent synchrotron motion is suppressed. The beam energy spread cannot be determined from the bunch length which depends only on the beam current and rf voltage. This has been observed using the high energy electron-cooling system at the Indiana University Cyclotron Facility.PACS 29.20.Dh, 29.27.Bd, Beams in storage rings are "bunched" using radio frequency (rf) accelerating cavities operating on a harmonic, h, of the beam revolution frequency. The particle beam, however, has a finite momentum spread and consequently a finite frequency spread. All the particles contained within the bucket, or separatrix in longitudinal phase space formed by the rf potential, oscillate in frequency about the synchronous frequency. The frequency of modulation is called the synchrotron frequency. This is the principle of "phase focusing" in a synchrotron [1]. In low energy rings ( < 10 GeV), the synchrotron oscillation frequency is on the order of 1 kHz, and is as low as 1 Hz in high energy rings (l TeV).Since the cavity produces a conservative force field, it cannot change the beam longitudinal phase space density. Consequently, for a known rf voltage, the beam time spread normally provides a direct measurement of the beam momentum spread and the area occupied by the beam in phase space, or the beam longitudinal emittance which is proportional to the beam temperature. An electron-cooling system [2,3], however, can reduce the ion beam emittance to extremely small values. In the process of cooling, the electrostatic potential energy spread across the beam bunch becomes larger than the moving frame kinetic energy spread. In this new regime, the bunch shape is no longer determined by the longitudinal emittance and rf cavity potential, but by a balance between the space charge and rf cavity forces [4][5][6].Since in all cases of interest the bunch lengths are much greater than the radius of the surrounding vacuum chamber, the particle potential energy as a function of longitudinal position U(s) can be found by integrating the forces on the particle due to the magnetic and electric fields produced by the beam, from the vacuum chamber radius, r r , to the outer edge of the beam, r b \• r e pi(s)where y is the usual relativistic parameter, m the electron mass, c the speed of light, r € the classical electron radius, e the electron charge, and pi (s) the beam linear charge density. For the Indiana University Cyclotron Facility (IUCF) Cooler \n(r v /ri,) ~3.2. The space-charge force FSQ causing the beam bunch to lengthen is then dUis)/ ds. The force compressing the beam, F T u exerted by the rf cavity, having a voltage wave form given by V T f xsinihs/R) where R is the radius of the synchrotron storage ring ( ===== 13.8 m for the IUCF Cooler Ring), is given by h eV...

Synchrotron motion with rf phase modulation was studied experimentally. Poincarb maps in the resonant precessing frame were obtained from the experimental data and compared with the tori of the resonant Hamiltonian. Our experimental data revealed island structure in longitudinal phase space. Experimental results for synchrotron motion excited by phase modulation at the third harmonic of the synchrotron frequency are also reported. '

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