Longitudinal and Quadrupolar Coupling Impedance of an Elliptical Vacuum Chamber With Finite Conductivity in Terms of Mathieu Functions

The resistive wall impedance of an elliptical vacuum chamber in the classical regime with infinite thickness is known analytically for ultra-relativistic beams by means of the Yokoya form factors. Starting from the longitudinal electric field of a point charge moving at arbitrary speed in an elliptical vacuum chamber, which we express in terms of Mathieu functions, in this paper we take into account the finite conductivity of the beam pipe walls and evaluate the longitudinal and quadrupolar impedance for any beam velocity. We also obtain that the quadrupolar impedance of a circular pipe is different from zero, approaching zero only for ultra-relativistic particles. Even if some of the results, in particular in the ultrarelativistic limit, are already known and expressed in terms of form factors, this approach is the first step towards the calculation of the general problem of a multi-layer vacuum chamber with different conductivities and of elliptic cross section

Measurements of the CERN PS longitudinal resistive coupling impedance

The longitudinal coupling impedance of the CERN PS has been studied in the past years in order to better understand collective effects which could produce beam intensity limitations for the LHC Injectors Upgrade project. By measuring the incoherent quadrupole synchrotron frequency vs beam intensity, the inductive impedance was evaluated and compared with the impedance model obtained by taking into account the contribution of the most important machine devices. In this paper, we present the results of the measurements performed during a dedicated campaign, of the real part of the longitudinal coupling impedance by means of the synchronous phase shift vs beam intensity. The phase shift has been measured by using two different techniques: in one case, we injected in the machine two bunches, one used as a reference with constant intensity, and the second one changing its intensity; in the second case, more conventional, we measured the bunch position with respect to the RF signal of the 40 MHz cavities. The obtained dependence of the synchrotron phase with intensity is then related to the loss factor and the resistive coupling impedance, which is compared to the real part of the PS impedance model

Chromaticity dependence of the transverse effective impedance in the CERN Proton Synchrotron

The current knowledge of the transverse beam coupling impedance of the CERN Proton Synchrotron (PS) has been established with beam-based measurements at different energies. The transverse coherent tune shift as a function of the beam intensity has been measured in order to evaluate the total effective imaginary part of the transverse impedance in the accelerator at the energies of 7, 13 and 25 GeV. Measurements have been performed changing the vertical chromaticity for each vertical tune scan with intensity. The data analysis revealed an increase of impedance with chromaticity for all the considered energies. The transverse impedance can be compared with the previously evaluated theoretical impedance budget taking into account the individual contribution of several machine devices

Impedance measurements and simulations on the TCT and TDI LHC collimators

The LHC collimation system is a critical element for the safe operation of the LHC machine and it is subject to continuous performance monitoring, hardware upgrade and optimization. In this work we will address the impact on impedance of the upgrades performed on the injection protection target dump (TDI), where the absorber material has been changed to mitigate the device heating observed in machine operation, and on selected secondary (TCS) and tertiary (TCT) collimators, where beam position monitors (BPM) have been embedded for faster jaw alignment. Con- cerning the TDI, we will present the RF measurements per- formed before and after the upgrade, comparing the result to heating and tune shift beam measurements. For the TCTs, we will study how the higher order modes (HOM) intro- duced by the BPM addition have been cured by means of ferrite placement in the device. The impedance mitigation campaign has been supported by RF measurements whose results are in good agreement with GdfidL and CST simula- tions. The presence of undamped low frequency modes is proved not to be detrimental to the safe LHC operation

Improved techniques of impedance calculation and localization in particle accelerators

In this thesis we mainly focus on particle accelerators applied to high energy physics research where a fundamental parameter, the luminosity, is maximized in order to increase the rate of particle collisions useful to particle physicists. One way to increase this parameter is to increase the intensity of the circulating beams which is limited by the onset of collective effects that may drive the beam unstable and eventually provoke beam losses or reduce the beam quality required by the particle physics experiments. One major cause of collective effects is the beam coupling impedance, a quantity that quantifies the effect of the fields scattered by a beam passing through any accelerator device. The development of an impedance budget is required in those machines that are planning substantial upgrades as shown in this thesis for the CERN PS case. The main source of impedance in the CERN LHC are the collimators. Within an impedance reduction perspective, in order to reach the goals of the planned upgrades, it was proposed to reduce the collimator impedance by means of their segmentation in the longitudinal direction. This motivated the study of electromagnetic techniques able to take into account the finite length of the device, such as the Mode Matching technique. This technique enabled us to study the impedance dependence on the device length and assess that no evident impedance reduction can be achieved by means of a collimator segmentation. The developed model allowed also for an accurate study of the impedance resonant-like behaviour below the beam pipe cut-off frequency in beam pipe flanges. These insertions are very common in particle accelerator and the resonant effect could drive harmful instabilities within circulating bunches. The possibility of detecting the high impedance sources by means of beam-based measurements represents another powerful investigation tool. In this thesis we improved the impedance localization technique based on the impedance-induced phase advance beating with intensity. We improved the theoretical background by means of macro particles simulations showing the effect of distributed and localized impedances, we quantified the impact of the noise over signal ratio in the measurement accuracy and we performed impedance localization measurements in the CERN PS, SPS, LHC and the Brookhaven RHIC

The mode matching method applied to beam coupling impedance calculations of finitelength devices

The infinite length approximation is often used to simplify the calculation of thebeam coupling impedance of accelerator elements. This is expected to be a reasonableassumption for devices whose length is greater than the transverse dimension but may beless accurate approximation for segmented devices. In this contribution we present thestudy of the beam coupling impedance of a finite length device: a cylindrical cavity loadedwith a toroidal slab of lossy dielectric. In order to take into account the finite length,we will decompose the fields in the cavity and in the beam pipe into a set of orthonormalmodes and apply the mode matching method to obtain the impedance. To validate our method,we will present comparisons between analytical formulas and 3D electromagnetic CSTsimulations as well as applications to the evaluation of the impedance of short beam pipeinserts, where the longitudinal and transverse dimensions are difficult to model innumerical simulations. Copyright © 2012 by IEEE

Impedance studies of 2D azimuthally symmetric devices of finite length

In particle accelerators, the beam quality can be strongly affected by the interaction with self-induced electromagnetic fields excited by the beam in the passage through the elements of the accelerator. The beam coupling impedance quantifies this interaction and allows predicting the stability of the dynamics of high intensity, high brilliance beams. The coupling impedance can be evaluated with finite element methods or using analytical approaches, such as field matching or mode matching. In this paper we present an application of the mode matching technique for an azimuthally uniform structure of finite length: a cylindrical cavity loaded with a toroidal slab of lossy dielectric, connected with cylindrical beam pipes. In order to take into account the finite length of the structure, with respect to the infinite length approximation, we decompose the fields in the cavity into a set of orthonormal modes. We obtain a complete set of equations using the magnetic field matching and the nonuniform convergence of the electric field on the cavity boundaries. We present benchmarks done with CST Particle Studio simulations and existing analytical formulas and codes, pointing out the effect of different material conductivities, finite length, and nonultrarelativistic particle beam velocity

RF coupling impedance measurements for particle accelerator devices

Bench measurements nowadays represent an important tool to estimate the coupling impedance of any particle accelerator device. The well-known technique based on the coaxial wire method allows to excite in the device under test a field similar to the one generated by an ultra-relativistic point charge. We discuss the basics of the coaxial wire method and review the formulae widely used to convert measured scattering parameters to longitudinal and transverse impedance data. We review, as well, bead-pull technique used in the design, construction and tuning of multi-cell accelerating structures. We discuss typical measurement examples of interest for the CERN Large Hadron Collider as well as other state of the art particle accelerator