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Effects of Potential Energy Spread on Particle Dynamics in Magnetic Proceedings of FEL2014, Basel, Switzerland THP032 EFFECTS OF POTENTIAL ENERGY SPREAD ON PARTICLE DYNAMICS IN MAGNETIC BENDING SYSTEMS ∗ R. Li , Jefferson Lab, Newport News, VA 23606, USA Abstract transverse dynamics have been analyzed earlier [5–7], and Understanding CSR effects for the generation and trans- it is found that their harmful effects related to the potent port of high brightness electron beams is crucial for designs feature of strong transverse dependence are cancelled. After of modern FELs. Most studies of CSR effects focus on the the cancellation, the transverse dynamics of particles is per- impacts of the longitudinal CSR wakefield. In this study, turbed by the remaining driving factors such as the effective we investigate the impact of the initial retarded potential longitudinal CSR force and the centrifugal force related to energy of particles, due to bunch collective interaction, on particles’ initial potential energy. the transverse dynamics of particles on a curved orbit. It is In this study, we present the role of potential energy in shown that as part of the remnants of the CSR cancellation the particle transverse dynamics after the cancellation effect effect when both the longitudinal and transverse CSR forces is taken into account. We will show that the initial slice are taken into account, this initial potential energy at the potential energy spread of a bunch, which we call pseudo entrance of a bending system acts as a pseudo kinetic en- slice energy spread, is indistinguishable from the usual slice ergy, or pseudo energy in short, because its effect on particle kinetic energy spread in its perturbation to the transverse par- optics through dispersion and momentum compaction is in- ticle optics via both dispersion and momentum compaction. distinguishable from effect of the usual kinetic energy offset This effect is measurable only when the peak current of from the design energy. Our estimation indicates that the the bunch is high enough such that the pseudo slice energy resulting effect of pseudo energy spread can be measurable spread is appreciable compared to the slice kinetic energy only when the peak current of the bunch is high enough such spread. The implication of this study on simulations and that the slice pseudo energy spread is appreciable compared experiments of CSR effects will be discussed. to the slice kinetic energy spread. The implication of this . study on simulations and experiments of CSR effects will be discussed. ROLE OF POTENTIAL ENERGY IN BUNCH TRANSVERSE DYNAMICS INTRODUCTION In this section the CSR cancellation effect is briefly re- When a high brightness electron beam is transported viewed. We show how a centrifugal force term, which is through a curved orbit in a bending system, the particle related to the initial potential energy of particles, emerges dynamics is perturbed by the coherent synchrotron radiation as one of the remnant of the cancellation. We also discuss (CSR) forces, or the collective Lorentz force as a result of the role of initial potential energy in transverse particle dy- the Lienard-Wiechert fields generated by particles in the namics. bunch. The longitudinal CSR interaction takes place when Consider an ultrarelativistic electron bunch moving on the fields generated by source particles at bunch tail over- a circular orbit with design radius R and design energy take the motion of the test particles [1] at bunch head and 2 E0 = γ0mc . Let x = r R be the radial offset of par- cause changes of kinetic energy for the head particles. For ticles from the design orbit.− The single particle optics is parameters currently used in most machine designs and op- determined by the configuration of the external magnetic erations, the approximation of the longitudinal CSR force fields, while for a bunch with high peak current, this design by that calculated using 1D rigid-line bunch model [2] often optics will be perturbed by the Lorentz force Fcol result- gives good description of the observed CSR effects [3]. ing from the collective electromagnetic interaction amongst In addition to the longitudinal CSR force, the transverse particles within the bunch. For transverse dynamics, such CSR force [4] can directly perturb transverse particle dy- perturbation is expressed in terms of the first order equation namics. This force features energy independence and, due to divergent contribution from nearby-particle interaction, has d2 x x ∆E Fcol strong nonlinear dependence on the transverse (and longitu- + = + x , (1) c2dt2 R2 RE E dinal) positions of particles inside the bunch. Meanwhile, 0 0 the potential energy change, as a result of both longitudinal col with Fx being the radial component of the collective and radial CSR or Coulomb forces, can cause change of Fcol = cole cole ∆ = Lorentz force Fs s + Fx r , and E E E0 kinetic energy of the particles and impact transverse particle being the deviation of the kinetic energy from the design− dynamics via dispersion. The joint effects of both the trans- energy. The existence and effect of transverse CSR force verse CSR force and the kinetic energy change on bunch col Fx were first pointed out by Talman [4] when he analyzed ∗ col Work supported by Jefferson Science Associates, LLC under U.S. DOE Fx for space charge interaction of a bunch on a circular Contract No. DE-AC05-06OR23177. orbit using Lienard-Wiechert fields. His study shows that ISBN 978-3-95450-133-5 Electron Bunch Generation and Manipulation 779 Copyright © 2014 CC-BY-3.0 and by the respective authors THP032 Proceedings of FEL2014, Basel, Switzerland eff as a result of logarithmic divergence of nearby-particle in- with the longitudinal effective force term Fv defined as teraction, this force is dominated by a term with the potent ∂Φcol ∂Acol undesirable feature of strong nonlinear dependence over par- Feff = e β . (7) ticles’ transverse position within the bunch. Besides its role v c∂t − · c∂t ! in the second driving term on the RHS of Eq. (1), the CSR (r v ) force also causes change of particle’s kinetic energy For a bunch with phase space density distribution f , ,t , the retarded potentials (Φcol,Acol) in the above expression t are given by ∆E(t) = ∆E (t = 0) + Fcol vdt′, (2) Z0 · f (r′,v′,t′) Φcol(r,t) = e dr′dv′, which impacts the beam optics via dispersion as depicted Z r r′ ′| −′ |′ ′ by the first driving term in Eq. (1). As an example, the case v (r v ) Acol r = f , ,t r′ v′ col = col = ( ,t) e ′ d d (8) of zero bunch charge corresponds to Fx Fs 0 and Z r r ∆E = E(0) E , when Eqs. (1) and (2) are reduced to equa- | − | − 0 ′ ′ tions for single particle dynamics. For the case of 1D CSR for t = t r r /c, and with Ne electrons in the bunch, − | − | col = model as used in ELEGANT simulation, one has (1) Fx 0 col r v r v = and (2) Fs in Eq. (2) is obtained for a 1D bunch moving f ( , ,t)d d Ne . (9) along the design orbit with its line-charge density distribu- Z tion obtained by projecting the actual 3D bunch distribution col The advantage of Eqs. (3)-(7) is that the potent term in Fx (at the time of force calculation) onto the design orbit, with of Eqs. (1) and (3), contributed from the divergent local the assumption that this 1D projected distribution has been interactions, is now cleanly represented by the centrifugal frozen as it is for all retarded times. In general, however, the space charge force FCSCF in Eq. (4), leaving the remaining kinetic energy can be changed by the usual longitudinal CSR eff effective radial force Fx free from the energy-independent force acting on the bunch as well as by the potential energy local divergence of the order of FCSCF. Similarly, all the change caused by various ways of particle-bunch interac- local-interaction contributions to ∆E(t) are summarized by tion. Examples of such particle-bunch interaction include e[Φcol(t) Φcol(0)] in Eq. (6), leaving the remaining con- (1) betatron motion in the potential well set by the radial − − eff tributions from the effective longitudinal force Fv also free CSR force for a coasting beam [8], (2) noninertial space from the energy-independent local divergence. Substitut- charge force related to the radiative part of the longitudinal ing Eqs. (3) and (6) into Eq. (1), one obtains the transverse Lienard-Wiechert electrical field experienced by off-axis dynamical equation particles interacting with a line bunch on a circular orbit [9], 2 (3) longitudinal space-charge interaction for a converging d x x δE (0) + = + Gˆ cor (10) bunch on a straight section right before the bunch entering c2dt2 R2 R into the last dipole of a bunch compression chicane [10]. In total all these examples, the collective-interaction-induced poten- with δE (0) the relative energy deviation from design tial energy eΦcol shares the same feature of strong nonlinear energy dependence of particles’ transverse position as that shown (0) E0 in the Talman’s force. δE (0) = E − , for (0) = E(0) + Eφ (0) (11) E E The relation between the two driving terms in Eq. (1) 0 col ∆ becomes clear when both Fx and E for the test particle with col are written in terms of retarded potentials [5]. The Taman’s Eφ (0) = eΦ (0), (12) force is written as and Gˆ cor contains all the terms related to the interaction col = eff CSCF Lagrangian [7] that has negligible energy-independent local- Fx Fx + F , (3) interaction contributions CSCF with the centrifugal space charge force F term and the t eff ˆ cor = 1 1 eff ′ ′ eff effective tranverse force Fx term defined respectively as G Fv (t )dt + Fx + Gres (13) E0 R Z0 ! col e βs A FCSCF = s , (4) with the residual of cancellation r Φcol Acol dAcol Φcol( ) col( ) eff = ∂ ∂ x t βs As t Fx e β e .
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