Single-mode coherent synchrotron radiation instability

In this paper we study a linear regime of single-mode coherent synchrotron radiation instability. We assume that the bunch is much longer than the wavelength of the modulation and consider a coasting beam model. The model developed in this paper considers a ring as a perfect toroid with a constant bending radius. We derived equations for the beam-mode interaction, found the growth rate for the instability, and estimated it for several machines. We have shown that in this case, due to the persistent interaction with a resonant mode, the beam becomes unstable even at low currents