TRAJECTORIES OF LOW ENERGY ELECTRONS IN PARTICLE ACCELERATORMAGNETIC STRUCTURES

It has been observed recently that low energy electrons, or electron clouds formed within quadrupoles can remain trapped for time periods beyond a beam revolution period at Cesr [1]. This paper provides some results obtained from analysis and numerical tracking of low energy electrons in quadrupoles under conditions similar to those when the above mentioned data was collected. MOTION OF PARTICLES IN A NONUNIFORM MAGNETIC FIELD Electron clouds generated in positively charged beams are known to negatively interfere with the beam. Most of the simulations done for various accelerators for estimating the cloud buildup have been for the duration of the passage of a single train of bunches. However if an electron remains trapped for long enough in a magnetic structure such as a quadrupole after the passage of a train, this trapped electron will encounter the next passage of a bunch train. The electron may then impinge upon the walls and potentially produce secondary electrons. Such a mechanism could lead to a progressive enhancement in cloud densities with repeated train passages and can have adverse consequences on the properties of the beam. Trapping of electrons has been observed in the PSR [2]. In this experiment the proton beam was allowed to circulate and produce electrons and the beam was then deflected out of the storage ring, while the electron cloud was allowed to evolve. It was observed that electrons in quadrupoles persisted for a time period exceeding 50-100μs. More recently, experiments performed at Cesr observed electron trapping in quadrupoles in the presence of a beam [1]. The revolution period of the beam in Cesr is about 2.5 μs. In this paper, we report results obtained from single particle tracking and compare them with theoretical expectations. The parameters of our computations were similar to those of the operating conditions when the data reported in Ref. [1] was taken. A similar study was performed earlier for the KEKB [3]. This study clearly showed that trapping of electron in quadrupoles can be important and requires a careful study. Adiabatic Invariants In dynamical systems exhibiting periodic motion, the quantity ∮ p· dq, which is the phase space area encom∗Work supported by NSF (PHY-0734867, PHY-1002467, PHY1068662), US DoE (DE-FC02-08ER41538) † kgs52@cornell.edu passed by the trajectory, also known as the Poincare invariant is a conserved quantity [4]. If a system undergoes a very gradual change, this quantity remains conserved so long as the relative value of this change over a full period of motion is very small, (see for example Ref. [4]). A classic example is a simple pendulum where the length of the string is gradually altered at a rate such that over the course of the oscillation period, δl/l << 1. Then the quantity E/ω remains constant and is the adiabatic invariant. The same principal of adiabatic invariance can be applied to a particle executing cyclotron motion in a magnetic field (see for example Ref. [5]). For a particle in this situation, the magnetic moment μ = IA = mv ⊥ 2B (1) is conserved as long as dB/B << 1 over the cyclotron period Tc. This is equivalent to the following condition,