We present a model of dynamics to account for the possible impact of space charge on the equilibrium emittances in storage rings and apply the model to study the current design of the International Linear Collider (ILC) damping rings

Oide, Katsunobu

Venturini, Marco

Wolski, Andy

eScholarship - University of California

Space Charge and Equilibrium Emittances in Damping Rings

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SPACE CHARGE AND EQUILIBRIUM EMITTANCESIN DAMPING RINGS ∗M. Venturini† , LBNL, Berkeley CA 94720 , USAK. Oide, KEK, Oho, Tsukuba, Ibaraki 305-0801, JapanA. Wolski, Univ. of Liverpool and Cockcroft Institute, Daresbury, UKAbstractWe present a model of dynamics to account for the pos-sible impact of space charge on the equilibrium emittancesin storage rings and apply the model to study the currentdesign of the International Linear Collider (ILC) dampingrings.INTRODUCTIONDirect space charge effects have the potential to be ofsome relevance in the ILC damping rings because of longring circumference and small transverse emittance. Spacecharge was extensively studied in the reference lattices con-sidered for the baseline recommendation [1]. It was foundthat at 5 GeV the ‘dogbone’ lattices with about 16-17 kmlength could be vulnerable to space charge but some ef-fects were also noticed in the shorter lattices, especiallywhen lattice errors were considered. In those studies aweak-strong model was used to determine rms emittancedegradation by tracking macroparticles over a number ofturns comparable to the damping time. We started from abeam with the design emittances, and looked for the ex-citation of dangerous resonances. Radiation effects werenot included during tracking and the reference values forthe equilibrium emittances (which determined the initialbeam sizes at the start of tracking), were those estimatedin the absence of any collective effect. Far from the harm-ful resonances identified in that study, the question can beraised of the possible impact of space charge on the beamequilibrium and in particular the vertical emittance. In lep-ton storage rings the beam equilibrium is determined bythe balance between radiation damping and quantum ex-citations. In a non-ideal lattice, space charge can modifythex/y coupling and alter the determination of the verticalequilibrium emittance – horizontally we do not expect anynoticeable effect as the emittance is much larger and space-charge effects much smaller than in the vertical direction.In [1] a preliminary application of the envelope formalism[2] was carried out by K. Oide by making a simple linearapproximation of the transverse space charge force validin a small neighborhood of the bunch center. Sizeable ef-fects were found for the dogbone lattices. In this paperwe revisit the model yielding that result and argue that the∗Work supported by the Department of Energy Contract no. DE-AC02-05CH11231†mventurini@lbnl.govapproximation there employed would overestimate the im-pact of space charge on the beam equilibrium. We base ourcontention on Sacherer’s observation [4] that under certainconditions, which appear to be naturally met in dampingrings operating far from resonances, the evolution of thebeam envelopes in the presence of space charge can indeedbe reproduced using a linear approximation of the spacecharge forces but with a value for the effective charge den-sity used to estimate those forces that is only a fraction ofthe charge density of the actual bunches.MODEL OF BEAM DYNAMICSWe consider a model of beam dynamics described by theVlasov-Fokker-Planck equation∂f∂s+ [f, H] =∂∂zi(Aijzjf) +12Bij∂2∂zi∂zjf, (1)wherez = (x, px, y, py, z, δ) is the vector of the dynami-cal variables,f the single-bunch beam distribution in phasespace. The RHS of the above equation describes the dy-namical effect of radiation emission with thes−dependentmatricesA andB modelling radiation damping and quan-tum excitation respectively. In the LHS,[· ·] are the Pois-son brackets andH the Hamiltonian generating the sym-plectic part of the dynamics (q, m are the particle chargeand mass;γ0 the relativistic factor):H =12(p2x + kxx2 + p2y + kyy2 + p2z + kzz2) + αψ. (2)We assume the linear approximation for the external forces.The self-force is described by the potentialψ giving rise tothe electric fieldE = ∇ψ. The coefficientα = q/mv20γ30multiplying ψ in (2) accounts for the partial cancellation ofthe transverse self-force in the ultra-relativistic regime thatis of interest here. In the same regime the self-force in thelab frame appears mostly transverse and we will neglect itslongitudinal component in the equations of motion. As forthe transverse components of the electric field at locationzalong the bunch, they can be determined as if the bunch wasinfinitely long and longitudinally uniform with transversedensity equal to the density atz.First consider the case where space-charge forces arecompletely negligible, in which caseH is a quadratic formH = (z, SHz)/2, with SH being a symmetric matrix. Be-cause the external forces are purely linear, it can be shownthat study of Eq. (1) can be replaced with that of the equa-tions for the first and second moments off . Followingconventional notation we write the second moments as theentries of the symmetric6×6 ‘sigma’ matrixΣij = 〈zizj〉.These obey the differential equations [2]Σ′ = LΣ + ΣLT + B, (3)with L = JSH − A (whereJ is the fundamental sym-plectic matrix) and the prime′ indicating differentiationwith respect tos. Devising an integration method for theseequations is straightforward. A simple first-order schemeconsists of introducing a splitting between non-diffusive(LΣ + ΣLT ) and diffusive (B) parts in the RHS of (3)and appling the latter in the kick approximation. If the dif-fusive kicks are separated by the small interval∆s integrat-ing through a kick ats yieldsΣ(s+) = Σ(s−) + ∆sB(s).Between these kicks propagation of theΣ-matrix under thedeterministic part of the dynamics can be done exactly interms of the transfer map (including damping)M(s′, s)with s → s′ = s + ∆sΣ(s′) = M(s′, s)Σ(s)MT (s′, s). (4)Because of damping, solutions of (3) starting from arbi-trary initial conditions will relax to the equilibrium beamenvelopesΣeq of a gaussian distribution in phase space.The corresponding rms (eigen-)emittances can then be de-termined as the imaginary part of the eigenvalues of thematrixΣJ [3]. As the dynamics is not symplectic the emit-tances so defined will not be exact invariant through thelattice but violation of the invariance is small. Althoughstraightforward this method is not very efficient because ofthe slow relaxation to equilibrium. Alternately, the equilib-rium beam envelopes or the emittances can be determinedat once using the method developed in [2]. If we define thetransformed diffusion matrixB(s) =∫ s+CsdτM(s + C, τ)B(τ)MT (s + C, τ) (5)whereC is the machine circumference, the beam envelopeat equilibrium is then given by solving forΣ the followingalgebraic linear equationΣ(s) = M(s)Σ(s)MT (s) + B(s). (6)When space-charge effects are present but are small, asis the case for the ILC damping rings, it seems reason-able to predict that an equilibrium may still exist and thatthe form of the beam density at equilibrium will remainclose to gaussian. If this is the case we can make use ofSacherer’s observation [4]that for beams with charge den-sity displaying elliptical symmetry (including in particu-lar gaussian beams) the envelope equations in the presenceof space charge form a closed set of equations providedthat the beam rms emittance is conserved (or known a pri-ori). To illustrate Sacherer’s statement consider the en-velope equations for the horizontal motion in differentialform〈x2〉′ = 2〈xpx〉, (7)〈xpx〉′ = 〈p2x〉 − kx〈x2〉+ α〈xEx〉, (8)〈p2x〉′ = −2kx〈xpx〉+ 2〈pxEx〉. (9)Suppose we write the beam distribution in phase space asf = fz(z, δ)f⊥(x, px, y, py). In the ultrarelativistic regimethe transverse components of the electric field obey the 2DPoisson equation∂Ex∂x+∂Ey∂y= qNλρε0(10)whereλ(z) =∫dδfz is the longitudinal linear density andρ(x, y) the transverse density with an elliptic symmetryi.e.ρ = ρ(x2/a2 + y2/b2), normalized to unity. Sacherershowed that for these beams the expectation value〈xEx〉5only depends on the rms transverse sizesσ2x = 〈x2〉 andσ2y = 〈y2〉:〈xEx〉5 = qNλ4πε0σxσx + σy. (11)In the above〈·〉5 refers to the average taken over all phasespace except thez direction. If we consider a density gaus-sian inz, λ(z) = exp(−z2/2σ2z)/√2πσz and average overthe longitudinal variable as well we find for the last term inEq. (8)〈xEx〉 = qN4πε01√21√2πσzσxσx + σy, (12)with a similar expression holding in the vertical plane.Now, assuming rms emittance conservation we can use theequationε2x = 〈p2x〉〈x2〉 − 〈xpx〉2 to find 〈p2x〉 in terms of〈x2〉 and〈xpx〉, thus replacing Eq. (9), and are left with aclosed set of equations for the second moments. Next, ob-serve that Eq. (12) is equivalent to the expression we wouldobtain if we used an effective purely linear transverse E-field of the formEeffx =qN4πε01√21√2πσzxσx(σx + σy). (13)Under Sacherer’s assumptions the evolution of the beamenvelopes described by Eqs. (7-9) is undistinguishablefrom that resulting from using this purely linear field –emittance conservation, in particular, is a direct conse-quence of the linearity of (13). As a result, Eq. (3) is stillv lid provided that we allow for the matrixL = L(Σ) adependence on the envelopes to account for space charge inaccordance to (13). The integration scheme outlined belowEq. (3) can be trivially extended to include space chargealso in the kick approximation. Eq. (6) for the equilibriumemittances is also still valid. However determining its solu-tion is more complicated as the transfer matricesM(s′, s)will also depend on the envelope matricesΣ turning (6)into a nonlinear equation. Finally, one can make the inter-esting observation that compared with the linearized fieldclose to the center of a gaussian bunch, the effective field(13) turns out to be1/23/2 smaller (for the sameN ).APPLICATION TO THE ILC DR’SWe applied these ideas to study the equilibrium emit-tances for the current design of the 6.7 km ILC dampingrings. To this end we added further functionality to theMaryLie/Impact (MLI) [5] code, previously used in thestudy [1], to include treatment of synchrotron radiation. Westudied a non ideal lattice with random vertical displace-ment of the sextupoles to create a finite vertical emittanceclose to theεy = 2 pm target in the design specifications.In the absence of space charge we varied the varianceσsxtof the sextupole displacement and determined the depen-dence of the vertical equilibrium emittance. For eachσsxtwe created 1000 lattice error realizations. The5% − 95%percentile ranges of the resultingεy are reported in Fig. 1 asvertical bars, together with the expected average accordingto theory (dashed line) [6].0 20 40 60 80 100 120 140Σsxt HΜmL0246810ΕyHpmLFigure 1: Equilibrium vertical emittance as a function of arandom rms vertical displacement of the sextupoles. Boxesare the average values over the5%− 95% percentile rangeof the emittances found; the dots the average over the fullrange. Space charge not included.For the case of a non-ideal lattice with space chargeperturbed with a single vertically displaced sextupole wecalculated the equilibrium vertical emittance by numeri-cal solving Eq. (3) (‘slow method’, boxes in Fig. 2) andby solving the envelope equation (6) for equilibrium (‘fastmethod’, dots in Fig. (2). To solve (6) we adopted a sim-plified iteration scheme expected to converge to the actualsolution only in the limit of vanishing space-charge. Com-parison between the results obtained with the two methodsprovides a range where the fast method can be used, seeFig 2. Finally in Fig. 3 we show an example of frequencydistribution of vertical emittances over 200 lattice error re-alizations with and without space charge (obtained usingthe fast method). A large value of the bunch population(N = 5.6× 1010 vs. theN = 2× 1010 design value) wasdeliberately used here to enhance the effect.Our conclusion on the basis of the model discussed inthis paper is that the impact of space charge on the equilib-rium emittances far from resonances for the current designof the 6.7 km ILC damping rings should be extremely mild.Only by increasing the bunch population and experiment-ing with emittances somewhat smaller than design specifi-cation (hence boosting space charge) we could find a sig-nificant effect.2 4 6 8 10 12 14Nbunch Hunits of 1010L22.22.42.6ΕyHpmL’fast method’’slow method’Figure 2: Equilibrium vertical emittancevs. bunch popula-tion.1 2 3 4 5 6Εy HpmL0.20.40.60.811.2<Εy> = 1.37 pmN=5.6´1010Σsxt = 60 Μm1 2 3 4 5 60.20.40.60.811.2<Εy> = 1.02 pmno space chargeΣsxt = 60 ΜmFigure 3: Frequency plot of equilibrium vertical emittancewith and without space charge.REFERENCES[1] M. Venturini and K. Oide, LBNL Report, LBNL-59511,Berkeley (2006); J. Gao, S. Guiducci, and A. Wolski Edi-tors, Recommendations for the ILC Damping Rings BaselineConfiguration, LBNL Report LBNL-59449, Berkeley (2006).[2] F. Ruggiero, E. Picasso, L.A. Radicati, Annals of Physics,197, 396-438 (1990); K. Ohmi, K. Hirata, and K. Oide, Phys.Rev. E49,751 (1994).[3] A. Wolski, Phys. Rev. ST Accel. Beams 9, 024001 (2006).[4] F. Sacherer, IEEE Trans. Nucl. Sci. NS-18, 1105 (1971).[5] A. Dragt,et al., MARYLIE Manual, University of Maryland,College Park, MD (2004).[6] T. O Raubenheimer, SLAC-PUB-4937 Rev., SLAC, (1991).This document was prepared as an account of worksponsored by the United States Government. Whilethis document is believed to contain correctinformation, neither the United States Governmentnor any agency thereof, nor The Regents of theUniversity of California, nor any of theiremployees, makes any warranty, express or implied,or assumes any legal responsibility for theaccuracy, completeness,or usefulness of anyinformation, apparatus, product, or processdisclosed, or represents that its use wouldnot infringe privately owned rights. Referenceherein to any specific commercial product,process, or service by its trade name, trademark,manufacturer, or otherwise, does not necessarilyconstitute or imply its endorsement,recommendation, or favoring by the United StatesGovernment or any agency thereof, or The Regentsof the University of California. The views andopinions of authors expressed herein do notnecessarily state or reflect those of theUnited States Government or any agency thereof,or The Regents of the University of California.Ernest Orlando Lawrence Berkeley NationalLaboratory is an equal opportunity employer.

Space Charge and Equilibrium Emittances in Damping Rings

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