We calculate the impedance and the wake functions for laminated structures with parallel-planes and circular geometries. We critically examine the approximations used in the literature for the coupling impedance in laminated chambers and find that most of them are not justified because the wall surface impedance is large. A comparison between the flat and the circular geometry impedance is presented. We use the wake fields calculated for the Fermilab Booster laminated magnets in realistic beam simulations using the Synergia code. We find good agreement between our calculation of the coherent tune shift at injection energy and the experimental measurements. In this paper we calculate the impedance and the wake functions for laminated structures with parallel-planes and circular geometries. First the coupling impedance is derived as a function of the wall surface impedance. Then the surface impedance is calculated by solving Maxwell's equations inside the lamination and the crack regions. We find that the commonly used resistive-wall approximations, good for metallic pipes with small surface impedance, are not valid in the laminated structures where the surface impedance is large. Realistic Synergia simulations of the Booster machine with wake fields predict transverse coherent tune shifts in good agreement with the experiment.

The stable region of the Fermilab Booster beam in the complex coherent-tune-shift plane appears to have been shifted far away from the origin by its intense space charge making Landau damping appear impossible. Simulations reveal a substantial buildup of electron cloud in the whole Booster ramping cycle, both inside the unshielded combined-function magnets and the beam pipes joining the magnets, whenever the secondary-emission yield (SEY) is larger than (almost equal to)1.6. The implication of the electron-cloud effects on the space charge and collective instabilities of the beam is investigated.

The Fermilab Booster is a nearly 40-year-old proton synchrotron, designed to accelerate injected protons from a kinetic energy of 400 MeV to 8 GeV for extraction into the Main Injector and ultimately the Tevatron. Currently the Booster is operated with a typical intensity of 4.5 x 10¹² particles per beam, roughly twice the value of its design, because of the requirement for high particle flux in various experiments. Its relatively low injection energy provides certain challenges in maintaining beam quality and stability under these increasing intensity demands. An understanding of the effects limiting this intensity could provide enhanced beam stability and reduced downtime due to particle losses and subsequent damage to the accelerator elements. Design of future accelerators can also benefit from a better understanding of intensity effects limiting injection dynamics. Chapter 1 provides a summary of accelerator research during the 20th century leading to the development of the modern synchrotron. Chapter 2 puts forth a working knowledge of the terminology and basic theory used in accelerator physics, and provides a brief description of the Fermilab Booster synchrotron. Synergia, a 3d space-charge modeling framework, is presented, along with some simulation benchmarks relevant to topics herein. Emittance, a commonly used quantity characterizing beam size and quality in a particular plane, is discussed in Chapter 3. Space-charge fields tend to couple the motion among the planes, leading to emittance exchange, and necessitating a simultaneous measurement to obtain a complete emittance description at higher intensities. A measurement is described and results are given. RMS beam emittances are shown to be in keeping with known Booster values at nominal intensities and emittance exchange is observed and accounted for. Unmeasurable correlation terms between the planes are quantified using Synergia, and shown to be at most an 8% effect. Results of studies on the coherent and incoherent shifts of transverse (betatron) frequencies with beam intensity at injection energies are presented. In Chapter 4 the coherent frequency shifts are shown to be due to dipole- and quadrupole-wakefield effects. The asymmetry of the Booster beam chamber through the magnets, as well as the presence of magnet laminations, are responsible for the magnitudes and for the opposing signs of the horizontal and vertical tune shifts caused by these wakefields. Chapter 5 details the procedures for obtaining a linear coherent-tune-shift intensity dependence, yielding -0.009/10¹² in the vertical plane and +0.001/10¹² in the horizontal plane. Data demonstrate a requirement of several hundred turns to accumulate to its maximal value. Two independent studies are compared, corroborating these results. In Chapter 6, a measure of the incoherent tune shift with intensity puts an upper limit on the magnitude of the direct space-charge effect in the Fermilab Booster. A prediction is made for the representative incoherent particle tune shift using a realistic Gaussian distribution, allowing for growth of the beam envelope with intensity, and found to be 0.004/10¹². The tune-spread dependence obtained by quantification of the resonant stopband width from beam-extinction measurements was measured at 0.005/10¹², similar to the predicted value. These will be shown to be one order of magnitude smaller than the space-charge term from the Laslett tune shift for a fixed-size, uniform beam.

The Fermilab Booster is a nearly 40-year-old proton synchrotron, designed to accelerate injected protons from a kinetic energy of 400 MeV to 8 GeV for extraction into the Main Injector and ultimately the Tevatron. Currently the Booster is operated with a typical intensity of 4.5 x 1012 particles per beam, roughly twice the value of its design, because of the requirement for high particle flux in various experiments. Its relatively low injection energy provides certain challenges in maintaining beam quality and stability under these increasing intensity demands. An understanding of the effects limiting this intensity could provide enhanced beam stability and reduced downtime due to particle losses and subsequent damage to the accelerator elements. Design of future accelerators can also benefit from a better understanding of intensity effects limiting injection dynamics. Chapter 1 provides a summary of accelerator research during the 20th century leading to the development of the modern synchrotron. Chapter 2 puts forth a working knowledge of the terminology and basic theory used in accelerator physics, and provides a brief description of the Fermilab Booster synchrotron. Synergia, a 3d space-charge modeling framework, is presented, along with some simulation benchmarks relevant to topics herein. Emittance, a commonly used quantity characterizing beam size and quality in a particular plane, is discussed in Chapter 3. Space-charge fields tend to couple the motion among the planes, leading to emittance exchange, and necessitating a simultaneous measurement to obtain a complete emittance description at higher intensities. A measurement is described and results are given. RMS beam emittances are shown to be in keeping with known Booster values at nominal intensities and emittance exchange is observed and accounted for. Unmeasurable correlation terms between the planes are quantified using Synergia, and shown to be at most an 8% effect. Results of studies on the coherent and incoherent shifts of transverse (betatron) frequencies with beam intensity at injection energies are presented. In Chapter 4 the coherent frequency shifts are shown to be due to dipole- and quadrupole-wakefield effects. The asymmetry of the Booster beam chamber through the magnets, as well as the presence of magnet laminations, are responsible for the magnitudes and for the opposing signs of the horizontal and vertical tune shifts caused by these wakefields. Chapter 5 details the procedures for obtaining a linear coherent-tune-shift intensity dependence, yielding -0.009/1012 in the vertical plane and +0.001/1012 in the horizontal plane. Data demonstrate a requirement of several hundred turns to accumulate to its maximal value. Two independent studies are compared, corroborating these results. In Chapter 6, a measure of the incoherent tune shift with intensity puts an upper limit on the magnitude of the direct space-charge effect in the Fermilab Booster. A prediction is made for the representative incoherent particle tune shift using a realistic Gaussian distribution, allowing for growth of the beam envelope with intensity, and found to be 0.004/1012. The tune-spread dependence obtained by quantification of the resonant stopband width from beam-extinction measurements was measured at 0.005/1012, similar to the predicted value. These will be shown to be one order of magnitude smaller than the space-charge term from the Laslett tune shift for a fixed-size, uniform beam.

The longitudinal phase space program ESME, modified for space charge and wall impedance effects, has been used to simulate transition crossing in the Fermilab Booster. The simulations yield results in reasonable quantitative agreement with measured parameters. They further indicate that a transition jump scheme currently under construction will significantly reduce emittance growth, while attempts to alter machine impedance are less obviously beneficial. In addition to presenting results, this paper points out a serious difficulty, related to statistical fluctuations, in the space charge calculation. False indications of emittance growth can appear if care is not taken to minimize this problem.

Space-charge effects on beam stabilities are studied for the proposed two-ring high-intensity Fermilab booster destined for the muon collider. This includes microwave instabilities and rf potential-well distortions. For the first ring, ferrite insertion is suggested to cancel the space-charge distortion of the rf wave form. To control the inductance of the ferrite during ramping and to minimize resistive loss, perpendicular biasing to saturation is proposed.