Threshold studies of the microwave instability in electron storage rings

K L F Bane,Y Cai,G Stupakov

doi:10.1103/physrevstab.13.104402

Abstract

We use a Vlasov-Fokker-Planck program and a linearized Vlasov solver to study the microwave instability threshold of impedance models: (1) a $Q=1$ resonator and (2) shielded coherent synchrotron radiation (CSR), and find the results of the two programs agree well. For shielded CSR we show that only two dimensionless parameters, the shielding parameter $\ensuremath{\Pi}$ and the strength parameter ${S}_{\mathrm{csr}}$, are needed to describe the system. We further show that there is a strong instability associated with CSR, and that the threshold, to good approximation, is given by $({S}_{\mathrm{csr}}{)}_{\mathrm{th}}=0.5+0.12\ensuremath{\Pi}$. In particular, this means that shielding has little effect in stabilizing the beam for $\ensuremath{\Pi}\ensuremath{\lesssim}2$; for larger $\ensuremath{\Pi}$ it is effective, with threshold current depending on shielding aperture as ${h}^{\ensuremath{-}3/2}$. We, in addition, find another instability in the vicinity of $\ensuremath{\Pi}=0.7$ with a lower threshold, $({S}_{\mathrm{csr}}{)}_{\mathrm{th}}\ensuremath{\approx}0.2$. We find that the threshold to this instability depends strongly on damping time, $({S}_{\mathrm{csr}}{)}_{\mathrm{th}}\ensuremath{\sim}{\ensuremath{\tau}}_{p}^{\ensuremath{-}1/2}$, and that the tune spread at threshold is small---both hallmarks of a weak instability.

Highlights

INTRODUCTIONIn the design of electron storage rings there is often a need to evaluate the threshold to the (longitudinal) microwave instability
In the design of electron storage rings there is often a need to evaluate the threshold to the microwave instability
For the task of finding the microwave threshold, we have shown that our Vlasov-Fokker-Planck and linearized Vlasov solvers agree quite well when applied to impedance models of (1) a Q 1⁄4 1 resonator and (2) the more difficult to compute coherent synchrotron radiation (CSR) wake between two parallel plates

Summary

INTRODUCTION

In the design of electron storage rings there is often a need to evaluate the threshold to the (longitudinal) microwave instability. We apply the programs to finding the instability threshold for two models of ring impedance: (1) the Q 1⁄4 1 resonator and (2) shielded coherent synchrotron radiation (CSR). The former model has often been used to represent the impedance of a storage ring Fifteen years ago Oide [5], and others [6], found that, under certain conditions, a different type, so-called weak instability, is possible These authors showed that a resistive impedance, one that generates an asymmetric bunch shape and results in negligible tune spread within the beam, can be the source of the weak instability.

VLASOV-FOKKER-PLANCK SOLVER

CALCULATIONS

CSR wake

Weak instability

CONCLUSION