Space charge impedances of a rectangular beam with longitudinal density modulations inside a rectangular chamber

Abstract

9This paper studies the space charge impedances of a rectangular beam inside a rectangular chamber, and the limiting 10 case, e.g., a rectangular beam between parallel plates, respectively. The charged beam has uniform density in vertical 11 direction and arbitrary distribution in horizontal direction. The method of separation of variables is used to calculate the 12 space charge potentials, fields, and impedances which are valid in the whole perturbation wavelength spectrum. 13Comparisons between the theoretical calculations and the numerical simulations are also provided and they match quite 14 well. It is shown that the rectangular beam shape may help to reduce the longitudinal space charge impedances. 19 chamber; line charge density modulations; separation of variables; parallel plates; finite element method.20 1. Introduction 21 In order to study the longitudinal beam instabilities due to the interactions between the perturbed beam 22 and the conducting vacuum chamber surrounding it, various space charge field models with different cross-23 sections of the beam and chamber have been investigated. Ref. [1] derived the longitudinal space charge 24 impedances of a round beam inside a rectangular chamber in the long-wavelength limits. Ref. [2] and Ref. 25 [3] studied the longitudinal resistive-wall instability and the space-charge driven microwave instability, 26 respectively, using a model consisting of a round beam inside a round vacuum chamber. In the research for 27 future linear colliders, some merits of flat (planar or rectangular) electromagnetic structures have been 28 found and aroused the interests of beam physicists, such as the reduced space charge forces [4] [5]. Ref. [6] 29 explored the properties of a planar beam between a pair of perfectly conducting plates. It concluded that, 30 comparing with the conventional axially symmetric configurations, the flat geometries of both the beam 31 and the chamber may help to reduce the longitudinal space charge fields. The two-dimensional (2D) 32 electrostatic space charge field of a rectangular beam inside a rectangular chamber was solved by Ref. [7] 33 using the method of separation of variables. While in this model, the field was induced only by the 34 unperturbed (constant) beam intensity without longitudinal modulations. The results are only valid when 35 the perturbation wavelengths of the longitudinal charge density are much larger than the transverse 36 dimensions of vacuum chamber, hence cannot be used directly in the study of short-wavelength instabilities, 37 such as microwave instability and micro-bunching instability. Another model of rectangular beam inside 38 rectangular chamber in Ref. [8] assumed the beam perturbation took place in the vertical direction, and 39 hence this model was devoted to study the transverse resistive wall instability instead of the longitudinal 40 one. 41 This paper introduced a three-dimensional (3D) space charge field model consisting of a rectangular 42 beam with sinusoidal longitudinal density modulations inside a r...

Approximate longitudinal space charge impedances of a round beam between parallel plates and inside a rectangular chamber

“…We choose the Cartesian coordinate xoy with o as the origin. Assume in the rest frame of the beam, the basic harmonic component of the space charge potential can be written in the form (17) where  > 0. Then the solutions to Eq.…”

“…Ref. [17] derived the potentials of an infinitely long beam with rectangular cross-section and uniform transverse charge density inside a rectangular chamber. The full beam width and height are 2a and 2b, respectively.…”

“…where ̅ for on-axis impedance [6,7] and ( ̅ ) for the average one [17], respectively. The identity of ( ̅ ) ( ̅ ) ̅ is used for the case of average impedance.…”

Analysis of the longitudinal space charge impedance of a round uniform beam inside parallel plates and rectangular chambers

“…Various LSC field models have been proposed and studied extensively by different methods in the existing literatures, such as Refs. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The LSC impedances in the entire wavelength spectrum of a beam in free space [8] and inside a round chamber [13][14][15][16] have been intensively studied.…”

Space charge and resistive wall impedance computation in the frequency domain using the finite element method

“…Meshing different material domains is possible, but yet involved to automate. An approach to obtain the longitudinal space charge impedance from the electrostatic potential calculated by finite element nodal shape functions is presented by Li and Wang [14].…”