M. Month

Particle trapping during passage through a high-order nonlinear resonance

Abstract A theory of particle trapping and transport during a single passage through a high-order nonlinear resonance is developed. The main result is an expression for the trapping efficiency as a function of two scaling variables which are related to the resonance excitation width, the nonlinear detuning and the speed of passage through the resonance. The question of what phase-space region trapped particles are drawn from and the question of adiabaticity are discussed. The theory is then verified with a computer simulation for crossing a fifth order resonance.

Analysis of the high frequency longitudinal instability of bunched beams using a computer model

Abstract The effects of high frequency longitudinal forces on bunched beams are investigated using a computer model. These forces are thought to arise from the transfer of energy between the beam and various structures in the vacuum chamber, this coupling being characterized by a longitudinal impedance function. The simulation is performed with a passive cavity-like element. It is found that the instability can be generated if three conditions are fulfilled: (1) the impedance must be sufficiently large, (2) the induced field must have a fast wake, and (3) the frequency of the induced field must be high enough. In particular, it is shown that the coasting beam threshold criterion for the longitudinal impedance accurately describes the onset of instability, if local values along the bunch of energy spread and current are used. It is also found that the very fast initial growth rate is in good agreement with linear theory and that the coasting beam overshoot expression may be used as a rough guide of the limiting growth for unstable bunches. Concerning the wake field, it is shown how the instability tends to disappear as the fields persist longer. It is furthermore demonstrated that as the wavelength of the unstable mode is increased, initially unstable conditions begin to weaken and vanish. This, it should be emphasized, is primarily a result of the strong correlation between the unstable mode frequency and the time rate of attenuation of the induced fields. ISR parameters are used throughout and a correspondence between the microwave instability observed in the ISR bunches and the simulated instability is suggested.

Theory for the fast blowup of particle bunches in accelerators

Abstract A theory for the fast longitudinal blowup of individual bunches (the microwave instability) is given. The predictions of this theory are consistent with experimental observations of the behavior of bunched proton beams. A numerical simulation is used to test the theory. Very good agreement is obtained. It is remarked that since the development of the instability is on a time scale where electrons and protons are equivalent, the theory should be applicable to the bunch lengthening phenomenon in electron storage rings.

Coupling impedance structure above the tube cutoff frequency

Abstract Previous studies of the equations for the beam induced electromagnetic fields, which occur when a smooth vacuum chamber is disturbed by a periodic sequence of double discontinuities, are extended. From these equations the coupling impedance of the beam to these cavity-like discontinuities is obtained directly. A method is proposed for handling the equations in the microwave region (above the cutoff frequency for chamber propagation). It is shown how field singularities develop as the cutoff frequency is crossed. In fact for a single cutoff mode, the lowest being the TM01 mode, many resonances arise with a close but regular spacing in frequency. Their high multiplicity and close spacing give these tube resonances a quality very different from the “normal” cavity resonances. Resistive loading leads to a decrease in the resonant impedance with the resulting resonance width being very narrow. The resonant coupling impedance is studied in detail as a function of frequency including the parametric dependence on the cavity dimensions, the ring and tube radii, and the resistive loading of the cavity. In addition to demonstrating clearly these tube resonances the mathematical procedure developed here can be used to study the frequency dependence of the induced fields and the corresponding coupling impedance in general.

Aperture limitation due to random errors in superconducting magnets

Abstract The presence of the current carrying coils within the iron shield of superconducting magnets has the effect of making the magnetic field within the magnet aperture sensitive to coil block positioning errors. In high current devices where a substantial fraction of the magnet aperture is to be used to stack particle beams, criteria based on a multipole analysis at the magnet center are inadequate. An analysis of the effect of coil block errors taking proper account of the multipole variation across the beam aperture is given. The implications for current accumulation within a given magnet aperture are considered in terms of both the distortions introduced into the linear beam orbits and the induced non-linear resonances.

Transverse collective instability excited by a non-uniform v-shift in intense beams☆

Abstract A model which describes the effect of a non-uniform space charge v-shift on the resistive wall instability associated with transverse collective oscillations of a beam of particles is proposed. It successfully predicts the overall characteristics of the instability as observed at the CERN ISR: (1) the observation of a current limitation in spite of the proportionate increase in v-spread as charge is stacked; and (2) the fact that the onset of instability, beam losses occur at a particular edge of the beam. The model consists simply of the original model of the resistive wall instability, augmented with the effect on the zero-order betatron tune of the equilibrium space-charge forces. It is shown that, under certain circumstances, the space charge distortion of the equilibrium betatron tune distribution function can permit the instability. An important result is that the stable region in W-space (W is a complex quantity depending on the beam characteristics and its environment) is not diminished by the distortion, but rather shifted. A consequence of this result is that an optimization of design should allow stability, even as the beam intensifies and distortion increases.

Error analysis of ν value measurements and an application

Abstract By perturbing the closed orbit in an accelerator and comparing the shift in beam position with the theoretical expression for a dipole perturbation by a least squares procedure, an estimate of the average ν value of the beam is obtained. A statistical analysis of the experimental data is presented. This yields point estimates of the parameters (i.e., ν value, amplitudes and betatron phases) together with interval estimates and confidence levels. In this way, ν value measurements have been made at various times in the Alternating Gradient Synchrotron (AGS) cycle and at various average radii (i.e., momenta) in order to obtain the time dependent functional relationship between ν and radius, R. Assuming a linear relationship between ν and R, least squares fits and standard errors have been obtained for various times in the AGS cycle. Since the rate of change of ν with radius is very nearly proportional to the sextupole moment in each of the AGS magnets, estimates of the sextupole moment as a function of the field strength in the main ring magnets (i.e., time in the AGS cycle) have also been obtained.

Lectures in Accelerator Theory

Lecture I deals with the behavior of particles in the nonlinear field arising from the electromagnetic interaction of colliding beams. The case treated, that of counter-rotating proton beams crossing each other at a non-zero angle, has the simple feature that the force between the beam is one dimensional. In lecture II, an analysis of the development of traveling waves on particle beams is presented. The situation studied is that of a uniform beam current in a circular accelerator and the excitation for the coherent motion is induced by the resistivity of the vacuum chamber wall. Finally, in lecture III, a description of the current accumulation process used at the proton storage rings at CERN (The ISR) is given. Particle pulses of rather low average current are injected and stored along the length and width of the vacuum chamber. The efficiency is very high and large currents (over 40 amperes) have been achieved.

Anomalous electron bunch length due to fast instability

Abstract A theory of bunch lengthening in electron storage rings is proposed. The equilibrium bunch length is determined by a balancing of beam induced fields tending to cause coherent instability and bunch frequency spread tending to prevent the buildup of coherent modes (Landau damping). The mechanism is related to the existence of a single bunch, single mode “fast” longitudinal instability. For such an instability to develop, the growth rate must be sufficiently faster than the synchrotron frequency. It is suggested that the source of the anomalous electron bunch length is a broad band resistive impedance at high frequency which could be produced by the combination of many closely spaced high frequency resonators, such as vacuum flanges. It follows that bunch lengthening and the observed higher mode heating are directly correlated. A scaling law for the equilibrium bunch length is derived. The scaling parameter is g =( hV cos ϕ s )/ I , with h , V and ϕ s the usual rf parameters and I the average beam current. Data taken in SPEAR I and II, data in which g extends in value by more than three orders of magnitude, can be fit with an appropriate choice of coupling impedance. The impedance functions for SPEAR I and II are taken to be the same, a reflection of the fact that the high frequency sources are chamber discontinuities rather than structures connected with the rf systems. A parameter search leads to an impedance characterized by a central frequency ∼5 GHz, a width (fwhm) ∼1.8 GHz and a peak value ∼0.2 M Ω . In addition to the agreement obtained between predicted and experimentally observed bunch lengths, the measured higher mode resistance (i.e., heat dissipated) in SPEAR is also found to be in agreement with the theoretical predictions.

Dynamic beam cleaning by a nonlinear resonance

Abstract The general framework for the dynamic cleaning of a stored proton beam by passing the beam through a nonlinear resonance is developed. The limitations and advantages of this technique are discussed. The method is contrasted with physical beam scraping, which is currently in use at the CERN ISR.