Successive approximations for charged particle motion

Single particle dynamics in electron microscopes, ion or electron lithographic instruments, particle accelerators, and particle spectrographs is described by weakly nonlinear ordinary differential equations. Therefore, the linear part of the equation of motion is usually solved and the nonlinear effects are then found in successive order by iteration methods. A Hamiltonian nature of these equations can lead to simplified computations of particle transport through an optical device when a suitable computational method is used. Many ingenious microscopic and lithographic devices were found by H. Rose and his group due to the simple structure of the eikonal method. In the area of accelerator physics the eikonal method has never become popular. Here I will therefore generalize the eikonal method and derive it from a Hamiltonian quite familiar to the accelerator physics community. With the event of high energy polarized electron beams and plans for high energy proton beams, nonlinear effects in spin motion have become important in high energy accelerators. I will introduce a successive approximation for the nonlinear effects in the coupled spin and orbit motion of charged particles which resembles some of the simplifications resulting from the eikonal method for the pure orbit motion

Orbit and Optics Improvement by Evaluating the Nonlinear BPM Response in CESR

We present an improved system for orbit and betatron phase measurement utilizing nonlinear models of BPM pickup response. We first describe the calculation of the BPM pickup signals as nonlinear functions of beam position using Green's reciprocity theorem with a two-dimensional formalism. We then describe the incorporation of these calculations in our beam position measurements by inverting the nonlinear functions, giving us beam position as a function of the pickup signals, and how this is also used to improve our measurement of the betatron phase advance. Measurements are presented comparing this system with the linearized pickup response used historically at CESR.Comment: 7 pages, 11 figure

Equilibrium ion distribution in the presence of clearing electrodes and its influence on electron dynamics

Here we compute the ion distribution produced by an electron beam when ion-clearing electrodes are installed. This ion density is established as an equilibrium between gas ionization and ion clearing. The transverse ion distributions are shown to strongly peak in the beam's center, producing very nonlinear forces on the electron beam. We will analyze perturbations to the beam properties by these nonlinear fields. To obtain reasonable simulation speeds, we develop fast algorithms that take advantage of adiabatic invariants and scaling properties of Maxwell's equations and the Lorentz force. Our results are very relevant for high current Energy Recovery Linacs, where ions are produced relatively quickly, and where clearing gaps in the electron beam cannot easily be used for ion elimination. The examples in this paper therefore use parameters of the Cornell Energy Recovery Linac project. For simplicity we only consider the case of a circular electron beam of changing diameter. However, we parameterize this model to approximate non-round beams well. We find suitable places for clearing electrodes and compute the equilibrium ion density and its effect on electron-emittance growth and halo development. We find that it is not sufficient to place clearing electrodes only at the minimum of the electron beam potential where ions are accumulated

Transverse emittance dilution due to coupler kicks in linear accelerators

One of the main concerns in the design of low emittance linear accelerators (linacs) is the preservation of beam emittance. Here we discuss one possible source of emittance dilution, the coupler kick, due to transverse electromagnetic fields in the accelerating cavities of the linac caused by the power coupler geometry. In addition to emittance growth, the coupler kick also produces orbit distortions. It is common wisdom that emittance growth from coupler kicks can be strongly reduced by using two couplers per cavity mounted opposite each other or by having the couplers of successive cavities alternation from above to below the beam pipe so as to cancel each individual kick. We therefore analyze consequences of alternate coupler placements. We show here that for sufficiently large Q values, alternating the coupler location from before to after the cavity leads to a cancellation of the orbit distortion but not of the emittance growth, whereas alternating the coupler location from before and above to behind and below the cavity cancels the emittance growth but not the orbit distortion. These compensations hold even when each cavity is individually detuned, e.g. by microphonics. Another effective method for reducing coupler kicks that is studied is the optimization of the phase of the coupler kick. This technique is independent of the coupler geometry but relies on operating on crest. A final technique studied is symmetrization of the cavity geometry in the coupler region with the addition of a stub opposite the coupler, which reduces the amplitude of the off axis fields and is thus effective for off crest acceleration as well. We show applications of these techniques to the energy recovery linac (ERL) planned at Cornell University

Synchro-Betatron Stop-Bands due to a Single Crab Cavity

We analyze the stop-band due to crab cavities for horizontal tunes that are either close to integers or close to half integers. The latter case is relevant for today's electron/positron colliders. We compare this stop-band to that created by dispersion in an accelerating cavity and show that a single typical crab cavity creates larger stop-bands than a typical dispersion at an accelerating cavity. We furthermore analyze whether it is beneficial to place the crab cavity at a position where the dispersion and its slope vanish. We find that this choice is worth while if the horizontal tune is close to a half integer, but not if it is close to an integer. Furthermore we find that stop-bands can be avoided when the horizontal tune is located at a favorable side of the integer or the half integer. While we are here concerned with the installation of a single crab cavity in a storage ring, we show that the stop-bands can be weakened, although not eliminated, significantly when two crab cavities per ring are chosen suitably.Comment: 7 pages, 9 figure

Compensation of wake-field-driven energy spread in Energy Recovery Linacs

Energy Recovery Linacs provide high-energy beams, but decelerate those beams before dumping them, so that their energy is available for the acceleration of new particles. During this deceleration, any relative energy spread that is created at high energy is amplified by the ratio between high energy and dump energy. Therefore, Energy Recovery Linacs are sensitive to energy spread acquired at high energy, e.g. from wake fields. One can compensate the time-correlated energy spread due to wakes via energy-dependent time-of-flight terms in appropriate sections of an Energy Recovery Linac, and via high-frequency cavities. We show that nonlinear time-of-flight terms can only eliminate odd orders in the correlation between time and energy, if these terms are created by a beam transport within the linac that is common for accelerating and decelerating beams. If these two beams are separated, so that different beam transport sections can be used to produce time-of-flight terms suitable for each, also even-order terms in the energy spread can be eliminated. As an example, we investigate the potential of using this method for the Cornell x-ray Energy Recovery Linac. Via quadratic time-of-flight terms, the energy spread can be reduced by 66%. Alternatively, since the energy spread from the dominantly resistive wake fields of the analysed accelerator is approximately harmonic in time, a high-frequency cavity could diminish the energy spread by 81%. This approach would require bunch-lengthening and recompression in separate sections for accelerating and decelerating beams. Such sections have therefore been included in Cornell's x-ray Energy Recovery Linac design