Measuring the Resonance Structure of Accelerator Impedance with Single Bunches

Abstract

We describe a new method used in the CERN SPS accelerator to measure the longitudinal impedance in the frequency range 100 MHz to 4 GHz. Single high intensity proton bunches were injected and their spectrum observed during slow debunching. The presence of different resonant impedances leads to line density modulation at the resonant frequencies. This instability reaches some maximum modulation amplitude which was recorded as a function of frequency for many bunches. Using sufficiently long bunches the SPS impedance structure was observed and previously unknown sources were identified. [S0031-9007(97) PACS numbers: 29.27.BdBeam intensity in particle accelerators is usually limited by collective instabilities which are the result of electromagnetic interaction of the beam with its environment. This interaction can be described by a coupling impedance Z͑v͒, the Fourier transform of the wake potential, proportional to the electric field integrated over one turn as seen by a probe particle following a source particle at time interval t. Knowledge of the impedance allows the behavior of intense beams to be explained or predicted [1,2].The main intensity limitation for the proton beam in the SPS, future injector for the Large Hadron Collider (LHC), comes from a longitudinal instability of a single bunch, known as the "microwave instability" since it is associated with the growth of high frequency signals. In this Letter we present results of measurements from which the longitudinal coupling impedance over a wide frequency range has been determined, and the source of the instability identified. Below we consider only longitudinal beam motion (and impedance), which in this case can be treated separately from transverse motion.Many of the vacuum chamber elements through which the beam travels have a cavitylike structure whose impedance can be represented as the sum of several individual resonances of the formcharacterized by a resonant frequency v r 2pf r , shunt impedance R sh , and quality factor Q. Measurements of impedance with a single bunch can give information only about the effective impedancethe actual impedance integrated over the spectrum of the bunch.For a stable bunch the effective impedance is defined by integration over the initial bunch spectrum centered at zero frequency. This effective impedance can be estimated from bunch lengthening (shortening) measurements based on the potential well distortion effect. Since the width of the bunch spectrum is inversely proportional to the bunch length, long bunches "see" only the low frequency part of the coupling impedance. Difficulties in resolving the "fine structure" of the machine impedance led to the creation of the widely used broadband impedance model, which replaces this structure by one resonator with very low Q (usually 1) and resonant frequency defined by the beam pipe cutoff frequency. The parameters of the SPS broadband model deduced from this type of measurement are Q 1, R sh 0.3 MV, and f r 1.3 GHz. This model has been used up to now also to describe...

Longitudinal coupled-bunch instability around 1 GHz at the CERN PS booster

“…Using Sacherer's theory of mode coupling [2], recently applied to the Fermilab Main Injector by Ng [3], one finds that for higher Q values R S /Q rather than R S only determines the threshold. Similar result were found also for Gaussian beams [4]. The case of the PSB manifolds with a resonance Q of 2400 is still different: Only one of the many lines of the (coupled-) bunch spectrum can interact with the cavity at one time.…”

Impedance issues in the CERN SPS

“…Attempts to determine a model by making an inventory of the different elements in the machine and computing the longitudinal impedance have been made [11] as well. We have developed a technique based on the measurement of the spectrum of unstable bunch modes to measure the fine structure of the machine impedance [12] and determine the dominant impedances for the model. Here we outline the procedure used and give the results.…”

Methods of observing the microwave instability above and below transition

“…Recent measurements of unstable bunch spectrum have allowed the sources of µw instability to be identified [17]. As a result in the SPS an extensive impedance reduction programme involving the work of a large team is close to completion.…”

“…Using the linearised Vlasov equation leads to the same matrix equation with RF on and off if the instability growth time ImΩ −1 is much less than synchrotron period 1/f s or debunching time t d = τ/(2|η| ∆p p ), see [17,18],…”

“…For a proton beam µw signals are very often observed during RF gymnastics involving adiabatic reduction of RF voltage or a debunching process. The range of observed frequencies can give important information about possible source of instability [16,17].…”