Coasting beam longitudinal coherent instabilities

Abstract

Nowadays, the performance of most machine are limited by coherent instabilities. It is one of the most important collective effects which prevents the current from being increased above a certain threshold without the beam quality being spoiled. An intense cool beam (high intensity contained in a small phase space volume) is always unstable. A small density perturbation in the particle distribution, which can be due to either previous beam manipulations or even statistical noise (related to the point-like aspect of the particles), can grow exponentially by driving the entire beam into an unstable process. In phase space, the beam blows up, and becomes hot. With regards to the scope of the lecture, the physical mechanisms which will be considered throughout can be applied to any type of machine: linear accelerators, circular accelerators, storage rings and beam: bunched beams continuous beams. 1 . INTRODUCTION In the first part of the lecture, we will limit ourselves to storage rings with a coasting beam which implies: Constant magnetic field no radio frequency applied. These conditions are met in proton or heavy-ion cooler rings. They are also met in pulsed machines during injection or extraction of a debunched beam. This regime does not exist on electron machines in which the RF is always on. With bunched beams, because of synchrotron motion, coherent instabilities manifest differently. However, the theoretical approach is essentially similar and easier to follow once the coasting beam case has been fully understood. The bunched beam case will be dealt with in the advanced course lecture. The source of the instability can only be an electromagnetic field. With a very weak (very low intensity) beam, individual particles behave essentially like single particles. The external guiding field imposes the trajectories and has been designed so that these trajectories are stable. With an intense beam, the large number of moving charges is responsible for the generation of an extra: space-charge field, or electromagnetic field as shown in Fig. 1. Fig. 1 If the intensity is large enough, this self field becomes sizeable in the sense that it can no longer be neglected when compared to the external guiding field. It can strongly influence the collective particle motion. We will also limit ourselves to situations in which the external guiding field magnitude largely dominates. In other words, particle behaviour is still almost single particle behaviour. The self field acts as a perturbation. As far as units are concerned, all formulae will be written in the International System units. Our formulae will make extensive use of the following standard quantities: e0 = 8.854 10-12 A s μ0 = 4 π 10-7 V s A-1m-1 c = 2.998 108 m s-1 e0μ0 c2 = 1 e = 1.602 10-19 A s (1) The self field follows Maxwell's equations. We will do our best to avoid detailed calculations of electromagnetic fields and therefore to make explicit reference to their form. However, since these equations govern the fields and therefore the source of the instabilities, it is advisable to keep them in mind. Let E and B be the electric field and magnetic induction respectively. Then, E and B can be drawn from the potentials V and A solutions of : ∆V 1 c2 ∂ 2V ∂t2 = ρ e0 where ρ is the particle charge density ∆A 1 c2 ∂ 2A ∂t2 = μ0j where j is the particle current density by means of E = grad V ∂ ∂t A and B = rot A (2) Obviously, boundary conditions which depend on the geometry and electromagnetic properties of the environment (vacuum chamber, surrounding magnets, etc.) strongly influence the solution and therefore the perturbed motion of particles. 2 . SINGLE-PARTICLE MOTION We will first ignore the self field and consider the unperturbed single particle motion. For a coasting beam, in the first order, the longitudinal motion is very simple. dp dt = e [ E + v × B ] (3) For the longitudinal component dp// dt = e [ E// + (v × B)// ] (4) there is no longitudinal field, no RF E// is nul and (v × B)// is a second order term (5) Therefore, dp// dt = 0 p// is a constant of the motion (6) The revolution period T around the orbit length L is written: T = 2 π ω = L βc (7) It depends on the particle momentum and can be expanded in terms of momentum deviation. Let us choose a reference (machine axis for instance) p//0 = m0γ0β0c (8) Let us then expand: β = β0 ( 1 + 1 γ0 2 p// p//0 p//0 ) and L = L0 ( 1 + α p// p//0 p//0 + ..) (9) L0 = 2 πR is the perimeter of the machine α = p// L ∂L ∂p// momentum compaction (10) in smooth machines α ≈ 1 Qx 2 where Qx 2 is the horizontal betatron tune Secondand higher-order contributions to orbit lengthening (transverse peak amplitude x and z dependence for instance) are neglected. It is usual to write: dT T = η dp// p// = dω ω with η = α 1 γ0 2 = 1 γt 2 1 γ0 2 (11) γt = 1 α1/2 defines the transition energy Et = m0γt c2 (12) Below transition: E0 Et and η > 0 particles with a positive momentum deviation circulate slower than the reference. When dealing with longitudinal (//) or transverse (⊥) instabilities the sign of η is essential. It indicates whether the slow particles at the tail, moving in the wake field, have a higher or lower energy than the fast particles at the front which create this wake field. In order to describe trajectories two coordinates are necessary. Let p/ / 0 and ω0 in rad s -1 be the momentum and angular revolution frequency respectively for the reference. We define two coordinates attached to the reference particle frame. τ and τ = dτ dt τ is expressed in s (seconds) and represents the time delay between the passing of the reference particle and the test particle at the same point around the circumference. τ is the time derivation of τ. The couple τ,τ defines the coordinates of the test particle in the longitudinal bidimensional phase space. With the definition of η as given above: τ = dT T = η dp// p// is a constant of the motion. Accordingly, the time delay: τ = τ0 + τ t is a linear function of time. The differential equation of motion is: τ = η p//0 dp// dt = ηe p//0 [E + v × B]// (t,θ) (13) Until now the right hand side has been null. Later on it will take into account the self field as a perturbation. Fig. 2 3 . LONGITUDINAL SIGNAL OF A SINGLE PARTICLE With a view to expressing the self field, one has to solve Maxwell's equations. On the right hand side of Maxwell's equation, the expression of the beam current density j (t,θ) and therefore of the current which will be noted S//(t,θ) (in Ampere) is required. Machine physicists are used to visualizing the beam current by looking with an oscilloscope at the signal drawn from longitudinal PU electrodes. We will assume a perfect longitudinal PU electrode with infinite bandwidth located at position θ around the ring. We will also assume that a single test particle rotates in the machine. At time t = 0 the fictive reference is at θ = 0. The PU is located at position θ as shown in Fig. 3.