Betatron Modes Research Articles

Electromagnetic dispersion characteristics of a high energy electron beam guided with an ion channel

Taking self-fields into consideration, dispersion properties of two types of electromagnetic modes for a high energy electron beam guided with an ion channel are investigated by using the linear perturbation theory. The dependences of the dispersion frequencies of electromagnetic waves on the electron beam radius, betatron frequency and boundary current are revealed. It is found that the electron beam radius and betatron frequency have different influences on the electromagnetic waves dispersion behavior by compared with the previous works. As the boundary current is taken into account, the TM modes will have two branches and a low-frequency branch emerged as the new branch in strong ion channel case. This new branch has similar dispersion behavior to the betatron modes. For TE modes, there are two branches and they have different dispersion behaviors in strong ion channel case. However, in weak ion channel case, the dispersion behaviors for both of the low frequency and high frequency branches are similar.

Electromagnetic Waves Dispersion and Interaction of an Annular Beam-Ion Channel System in Plasma Waveguide

A linear theory for the electromagnetic properties and interactions of an annular beam-ion channel system in plasma waveguide is presented. The dispersion relations for two families of propagating modes, including the electrostatic and transverse magnetic modes, are derived. The dependencies of the dispersion behavior and interaction for different wave modes on the thickness of the annular beam and betatron oscillation frequency are studied in detail by numerical calculations. The results show that the inner and outer radii of the beam have different influences on propagation properties of the electrostatic and electromagnetic modes with different betatron oscillation parameters. In the weak ion channel situation, the two types of electrostatic waves, that is, space charge and betatron modes, have no interaction with the transverse magnetic modes. However, in the strong ion channel situation, the transverse magnetic modes will have two branches and a low frequency mode emerged as the new branch. In this case, compared with the solid beam case, the betatron modes not only can interact with the high frequency branch at small wavenumber but also can interact with the low frequency branch at large wavenumber.

Effects of Boundary Current on Electromagnetic Dispersion Characteristics for a Relativistic Electron Beam**supported by the National ITER Project Foundation of China (Nos. 2013GB106001 and 2013GB106003)

With betatron oscillation characteristics of the electron beam and ion channel effect taken into account, dispersion characteristics of electrostatic modes and TM modes for a relativistic electron beam guided by ion channel are studied. Dispersion relations are derived and solved numerically to investigate the dependence of the dispersion characteristics for electrostatic modes and TM modes on the betatron oscillation frequency and the ratio of the relativistic electron beam radius to the waveguide radius. The effects of the boundary current on the dispersion characteristic of the TM modes and the interaction between the betatron modes and TM modes are analyzed. When considering the boundary current, for a strong ion channel, a new low-frequency branch of the TM modes arises and the interaction frequency between the betatron modes and the TM01 modes is increased with the same parameters.

Wave growth rate in a cylindrical metal waveguide with ion-channel guiding of a relativistic electron beam

This paper addresses the formulae and numerical issues related to the possibility that fast wave may be grown when a relativistic electron beam through an ion channel in a cylindrical metal waveguide. To derive the dispersion equations of the beam-wave interaction, it solves relativistic Lorentz equation and Maxwell's equations for appropriate boundary conditions. It has been found in this waveguide structure that the TM0m modes are the rational operating modes of coupling between the electromagnetic modes and the betatron modes. The interaction of the dispersion curves of the electromagnetic TM0m modes and the upper betatron modes is studied. The growth rates of the wave are obtained, and the effects of the beam radius, the beam energy, the plasma frequency, and the beam plasma frequency on the wave growth rate are numerically calculated and discussed.

Dispersion and interaction of waveguide modes in a relativistic electron beam with ion-channel guiding

A theory for the high-frequency eigenmodes of a cylindrical metallic waveguide partially filled with a relativistic electron beam and guided by an ion channel is presented. Equations that permit calculation of dispersion curves for five families of wave modes are derived. The dependence of the frequencies and the dispersion curves of azimuthally asymmetric modes on the ratio of the beam radius a to the waveguide radius R are studied in detail. The coupling of the dispersion curves of different modes is studied and illustrated for the partially filled waveguide. It was found that, in their course of interaction, HE modes cannot displace the betatron modes in a completely filled waveguide, which is different from the case of a magnetized electron beam. Physical and analytical explanations are presented for this difference.

Wave-mode dispersions in a relativistic electron beam with ion-channel guiding

An analysis of the eigenmodes of a cylindrical metallic waveguide partially filled with a relativistic electron beam, and guided with an ion channel, is presented. The analysis is performed in the rest frame of the beam with self-fields taken into account. Dispersion relations for five families of electrostatic and EM modes are derived and solved numerically to study the characteristics of azimuthally symmetric TM and TE waveguide modes, betatron modes, and space-charge modes. A strong dependence of the frequencies of these electromagnetic-electrostatic waves on the ratio of the radius of the waveguide to the beam radius is revealed. The physical insight given to the electrostatic modes, at large and small wavelengths, equally applies to the Trivelpiece-Gould modes in an electron beam with axial magnetic field guiding. Interactions between the dispersion curves of different modes are used to find the infinite-space dispersion relation from the finite-radius dispersion relation and to explain the transition from a weak to strong ion channel.

Untangling mixed modes in model-independent analysis of beam dynamics in circular accelerators

Model-independent analysis of beam dynamics in circular accelerators reveals spatial-temporal modes due to beam oscillations. Although such modes have been shown to be informative, their usefulness is limited by the lack of quantitative understanding in general. Here we present a simple approach to studying the modes analytically, which results in a quantitative understanding of the modes and an algorithm to extract the physical modes by untangling the mixed singular value decomposition modes. Particularly, we focus on the coupled betatron modes that are of great interest for high-energy colliders. A simple relationship between the coupling modes and the lattice functions is established, which not only provides a quantitative understanding but also lays the foundation for using these modes to measure machine properties and beam motion in phase space.