Abstract
The microwave instability driven by the coherent synchrotron radiation (CSR) has been previously studied [S. Heifets and G. V. Stupakov, Phys. Rev. ST Accel. Beams 5, 054402 (2002)] neglecting effect of the shielding caused by the finite beam pipe aperture. In practice, the unstable mode can be close to the shielding threshold where the spectrum of the radiation in a toroidal beam pipe is discrete. In this paper, the CSR instability is studied in the case when it is driven by a single synchronous mode. A system of equations for the beam-wave interaction is derived and its similarity to the 1D free-electron laser theory is demonstrated. In the linear regime, the growth rate of the instability is obtained and a transition to the case of continuous spectrum is discussed. The nonlinear evolution of the single-mode instability, both with and without synchrotron damping and quantum diffusion, is also studied.
Highlights
A relativistic electron beam moving in a circular orbit in free space can radiate coherently if the wavelength of the synchrotron radiation exceeds the length of the bunch.In accelerators coherent radiation of the bunch is usually suppressed by the screening effect of the conducting walls of the vacuum chamber [1,2,3]
If the radiation reaction force is directed so that it drives the growth of the initial fluctuation, one can expect an instability that leads to microbunching of the beam and an increased coherent radiation at short wavelengths
In Ref. [4] the growth rate of the beam instability driven by the coherent synchrotron radiation (CSR) was found using the so-called ‘‘CSR impedance’’ [5,6] that neglects the shielding effect of the walls and assumes a continuous spectrum of radiation
Summary
A relativistic electron beam moving in a circular orbit in free space can radiate coherently if the wavelength of the synchrotron radiation exceeds the length of the bunch. [4] the growth rate of the beam instability driven by the coherent synchrotron radiation (CSR) was found using the so-called ‘‘CSR impedance’’ [5,6] that neglects the shielding effect of the walls and assumes a continuous spectrum of radiation. The instability develops at wavelengths where effects of the shielding [1] should be taken into account This issue was only qualitatively addressed in [4], because the ‘‘free space’’ CSR impedance [5,6] is not applicable near the shielding threshold where the spectrum of the synchrotron radiation is discrete, and one has to take into account that the instability may be driven by a single synchronous mode rather than a continuous spectrum. We employ a model of a perfectly conducting toroidal chamber (in general, of arbitrary cross section) as the beam environment We show that this model allows solutions for both linear and nonlinear regimes.
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