We investigate the dynamics of longitudinally flat bunches created with a second harmonic cavity in a high energy collider. We study Landau damping in a second harmonic cavity with analytical and numerical methods. The latter include particle tracking and evolution of the phase space density. The results are interpreted in the context of possible application to the LHC. A possible path to a luminosity upgrade at the LHC is through the creation of longitudinally flat bunches. They can increase the luminosity roughly by 40% when the beam intensities are at the beam-beam limit. Lower momentum spread which can reduce backgrounds and make collimation easier as well lower peak fields which can mitigate electron cloud effects are other advantages. Use of a second harmonic rf system is a frequently studied method to create such flat bunches. Here we consider some aspects of longitudinal dynamics of these bunches in the LHC at top energy. First we consider intensity limits set by the loss of Landau damping against rigid dipole oscillations. Next we describe numerical simulations using both particle tracking and evolution of the phase space density. These simulations address the consequences of driving a bunch at a frequency that corresponds to the maximum of the synchrotron frequency

/Fermilab

Bhat, Chandra

Kim, Hyung Jin

Ostiguy, Jean-Francois

Sen, Tanaji

UNT Digital Library

DYNAMICS OF FLAT BUNCHES WITH SECOND HARMONIC RF∗T. Sen† , C.M. Bhat, H.J. Kim, J-F. Ostiguy, FNAL, Batavia, IL 60510, USAAbstractWe investigate the dynamics of longitudinally flatbunches created with a second harmonic cavity in a highenergy collider. We study Landau damping in a second har-monic cavity with analytical and numerical methods. Thelatter include particle tracking and evolution of the phasespace density. The results are interpreted in the context ofpossible application to the LHC.INTRODUCTIONA possible path to a luminosity upgrade at the LHC isthrough the creation of longitudinally flat bunches. Theycan increase the luminosity roughly by 40% when the beamintensities are at the beam-beam limit. Lower momentumspread which can reduce backgrounds and make collima-tion easier as well lower peak fields which can mitigateelectron cloud effects are other advantages. Use of a sec-ond harmonic rf system is a frequently studied method tocreate such flat bunches. Here we consider some aspects oflongitudinal dynamics of these bunches in the LHC at topenergy. First we consider intensity limits set by the loss ofLandau damping against rigid dipole oscillations. Next wedescribe numerical simulations using both particle trackingand evolution of the phase space density. These simulationsaddress the consequences of driving a bunch at a frequencythat corresponds to the maximum of the synchrotron fre-quency.RF WITH TWO HARMONICSWe choose the voltage to be of the formV (φ) = VRF [sinφ+ k sinn(φ− φs)] (1)where k is the ratio of the higher harmonic (=n) voltageto the voltage of the fundamental harmonic. The energygain per turn of the synchronous particle is the same as ifonly the fundamental harmonic cavity were present. Thepotential function is U(φ) =∫ φ0V (φ′)dφ′. The ratio ofthe stationary bucket area in the second harmonic systemrelative to that of the single harmonic system isAbuck(k, φs = π)Abuck(k = 0, φs = π)=12[√1 + 2k +Arcsinh(√2k)√2k](2)This ratio increases monotonically with k, e.g. at k = 1/2,the ratio is 1.15. There is no change in the bucket accep-tance for a stationary second harmonic bucket.Beam Distribution and Beam Induced VoltageConsider a general binomial distribution ρ(W,φ) ∝[Hb −H ]p where H is the Hamiltonian and Hb is its value∗This work partially supported by the US Department of Energythrough the US LHC Accelerator Research Program (LARP).† tsen@fnal.govat the bunch boundary. The line density is the projec-tion on the φ axis, λ(φ) = λ0 [U(φ)− U(φ2)]p+1/2 Aspecial case of this distribution is the elliptic distributionρ(W,φ) ∝ √Hb −H which was first considered by Hof-mann and Pedersen [1]. This has the special feature thatthe line density is proportional to the potential. Let φ1, φ2denote the phases at the ends of the bunch. Then the linecharge density isλ(φ) = −Nbf[cosφ2 − cosφ+ (φ2 − φ) sinφs+kn(cosn(φ2 − φs)− cosn(φ− φs))] (3)where the function f(φ1, φ2) is determined by the normal-ization condition∫dφλ(φ) = Nb. For a full stationarybucket f(0, 2π) = −2π(1 + k/2).Assuming that the effective coupling impedance ismostly reactive, Zeff (ω)/n = i[ω0L − g0Z0/(2βγ2)]where n = ω/ω0. For a circular beam of radius a in acircular beam pipe of radius b, g = 1 + 2 ln(b/a). Includ-ing the induced voltage, the total voltage isVt(φ) = V (φ) +2πh2Ib, avIm(Zeff/n)VRF f(φ1, φ2)[V (φ)− V (φs)](4)The relative change in the total focusing voltage is given bythe factor kt is kt = (Vt(φ)− V (φs))/(V (φ)− V (φs)) =1 + 2πh2Ib,avIm(Zeff/n)/(VRF f(φ1, φ2)). The new po-tential function is Ut(φ) = ktU(φ). while the net area andsynchrotron frequency are reduced by√kt.LANDAU DAMPING THRESHOLDThe maximum intensity that can be accelerated is givenby the condition that the beam induced voltage reduces thebucket area to zero.Nb,max = − 1eω0h2VRFIm(Zeff/n)f(φT , φu) (5)where φT , φu are the bucket endpoints. The frequency ofcoherent dipole oscillations is found to beωc = ω0√h|η|2πeVRFβ2Es√g(φ1, φ2)|f(φ1, φ2) | (6)where g(φ1, φ2) ≡ 1/(V 2RF )∫ φ2φ1(∂U/∂φ)2dφ. Let themaximum synchrotron frequency ωs be ωmaxs . The con-dition for the loss of Landau damping against rigid dipoleoscillations is that ωc ≥ ωmaxs , the threshold intensity forthis isNb,LandauNb,max= [f(φ1, φ2)f(φT , φu)− sgn(f(φ1, φ2))f(φT , φu)g(φ1, φ2)(ω̄maxs )2](7)TUPD067 Proceedings of IPAC’10, Kyoto, Japan207805 Beam Dynamics and Electromagnetic FieldsD05 Instabilities - Processes, Impedances, Countermeasures-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0  50  100  150  200  250  300  350Line density (scaled)Phase [degrees]k=-0.5k=0.5k=1.0Figure 1: The scaled line density λ/(Nb/2π) of a bunchfilling a stationary bucket with 3 values of k. The flattestprofile is at k = 1/2. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 180  200  220  240  260  280  300  320  340  360Scaled synchrotron frequencyPhase [degrees]k = -0.5k = -0.4k = 0.0k = 0.4k = 0.5k = 0.6k = 0.7k = 1.0Figure 2: Scaled synchrotron frequency ω̄s with a 2nd har-monic rf in a stationary bucket with −1/2 < k < 1.where sgn is the sign function. The synchrotron frequencyin the second harmonic system scaled by the frequency atsmall amplitude in a single harmonic system is obtainedfromω̄s(φb)−1=∫ φbφadφ√|(cosφb − cosφ)[1 + k(cosφb + cosφ)]|(8)Figure 2 shows the scaled synchrotron frequency ω̄ s.When k ≤ 0, the frequency decreases monotonically withamplitude. When k ≤ 0.5 the frequency at the center dropswith increasing k as√1− 2k and vanishes when k = 1/2.For larger k, the frequency at the center is zero due to theinner separatrix which has an unstable fixed point at π. Thefrequency has a local maxima at the stable fixed points.Another maximum exists in the range 295-320 degrees fork > 1/2. The maximum frequency fmaxs for k = 1/2 is17.915 Hz for LHC parameters. The inner separatrix alsoleads to a sharp cusp in the frequency at a phase amplitudewhere the inner separatrix crosses the phase axis.We now calculate the threshold at which Landau damp-ing is lost in the LHC assuming Zind/n = 0.1 Ohms. Theleft figure in Fig 3 shows the coherent and maximum inco-herent synchrotron frequencies including the effect of theinductive impedance and space charge as a function of thevoltage ratio k with bunch intensity Nb = 1011. The maxi-mum incoherent frequency is above the coherent frequencyexcept in the range 0.55 < k < 0.67. showing that Landaudamping would be lost at this intensity and these k val-ues. The right plot in Fig 3 shows the threshold intensity at 16 18 20 22 24 26 28 30 0  0.2  0.4  0.6  0.8  1Synchrotron Frequencies [Hz]kBunch intensity = 1011Zind = 0.1 Ohms IncoherentCoherent 0 2e+12 4e+12 6e+12 8e+12 1e+13 1.2e+13 0  0.2  0.4  0.6  0.8  1Landau damping threshold intensitykZind = 0.1 Ohms2 eVsec3 eVsec4 eVsecFigure 3: Left: Maximum incoherent and coherent fre-quency vs the voltage ratio k for nominal LHC parameters.Right: Intensity at which Landau damping is lost vs k atdifferent emittances.which Landau damping is lost as a function of k for differ-ent emittances. The threshold is above the design intensityof 1011 until k = 0.54 but Landau damping is lost even atzero intensity when 0.56 ≤ k ≤ 0.67. At k = 0.5 andεL = 3 ev-sec, the threshold is 3.6×1011 which is beyonddesign intensities at the LHC.NUMERICAL SIMULATIONSExperiments in the SPS with a fourth harmonic rf sys-tem showed a strong coherent response in the beam trans-fer function at a frequency corresponding to f maxs [2]. Weconsider the beam response to an external forced oscillatingat this frequency. We use two different simulation methods:a) particle tracking with the code ESME [3], b) evolutionof the density with a code developed for this purpose. Thesimulations reported here were done with k = 0.5.ESME SimulationsThe multi-particle longitudinal beam dynamics codeESME [3] is used. The bucket is populated according to theHoffman-Pedersen distribution with about 35,000 macro-particles. The impedance and the space charge forces areturned on adiabatically and an additional 80 synchrotronperiods are allowed to reach equilibrium. Next an externalrf drive at a frequency= 17.915 Hz and 0.1 MV amplitude isturned on. This amplitude may be too large but it is aboutthe same as the induced voltage at the bunch edges. Thebunch evolution is monitored for the next 60 seconds.The centroid is observed to oscillate (see Fig 4) at thedrive frequency but the amplitude modulates at a beat fre-quency equal to the difference in frequencies between thedrive and the coherent oscillation. From the beat frequency(0.319 Hz), we find the coherent frequency in this case tobe 17.606 Hz compared with the estimate of 17.876 Hz us-ing Eq (6). This small discrepancy (1.5%) can howeveraffect the prediction of when Landau damping may be lost.The rms emittance growth corresponding to three differentbunch lengths are also shown in Figure 4. The emittanceincrease is largest for the bunch (1.01 eV-sec) whose phaseextent just matches the phase location (297 degrees) ofωmaxs . The increase is smaller for the longer bunch whichextends up to 330 degrees. After reaching a maximum, theemittance oscillates with a frequency equal to twice thebeat frequency of the centroid oscillations before settlingProceedings of IPAC’10, Kyoto, Japan TUPD06705 Beam Dynamics and Electromagnetic FieldsD05 Instabilities - Processes, Impedances, Countermeasures 2079-25-20-15-10-5 0 5 10 15 20 25 5  5.5  6  6.5  7  7.5  8  8.5  9  9.5  10Thetabar (deg)Time [secs]1.01 eVs1.65 eVs 0.9 0.95 1 1.05 1.1 1.15 1.2 4  6  8  10  12  14  16  18  20RMS-Emit/RMS-Emit-InitTime [secs]1.65 eVs1.01 eVs0.53 eVsFigure 4: ESME simulations when the beam is driven at afrequency fmaxs . Left: Centroid motion for two differentemittances. Note the beating motion in both cases. Right:Emittance growth (normalized) for different emittances.Figure 5: Mountain range view of the line density. Top:ESME simulation with εL = 1.01eV-sec. Bottom: Vlasovsimulation for a longer bunch. Note the development ofshoulders in both profiles.down to an equilibrium value. Neither growth nor oscil-lations are seen for the shortest bunch length whose phaseextent (270 degrees) does not reach the location of ωmaxs .Figure 5 shows a mountain range plot of the bunch profilefor the bunch with largest emittance growth. A shoulderdevelops in the profile, similar to that seen in the SPS ex-periments [2].Vlasov SimulationsParticle tracking, while conceptually simple, does not re-solve well the tails of the beam distribution where there arefew particles. An alternative is to adopt the Eulerian pointof view, i.e. focus on specific locations in phase space astime evolves or equivalently solve the Vlasov equation. InFigure 6: Vlasov simulation of the phase space density for abunch with εL = 1 ev-sec and phase extent to 295 degrees..Left: initial, Right: final distribution after 45 secs developsa halo.contrast to particle tracking, all locations in phase space aremonitored in time and treated on equal footing. Rather thansolving the Vlasov partial differential equation, we fol-low the approach in reference [4] where a semi-Lagrangiantechnique is used. Symplectic maps are used to ensurelong-term preservation of the phase space density. Thedensity distribution is interpolated on a 2D regular rectan-gular grid using C′ third order Lagrange polynomials. Ateach step in time, the density is updated using the prescrip-tion F (q, p, tn) = F (M−1(q, p, tn−1)) where M−1 rep-resents the inverse of the forward map from tn−1 to tn.The maps are implemented as localized kicks and are ap-plied every turn. Currently, the code can model arbitraryrf cavity waveforms, linear or non linear phase slippage aswell as longitudinal impedances and space charge effects.In a typical simulation over 0.5× 106 turns, on a grid with104 third order cells, we observe less than 10−3 variationin the integral of the phase space density.Results from this Vlasov code are similar to those fromESME. Centroid oscillations show the same beating periodand emittance growth is also about the same. Fig 5 showsthe mountain range profile of a bunch. Fig 6 shows an ex-ample of the phase space distribution at the start and end ofabout 1000 synchrotron periods. Here the bunch extends tothe location of ωmaxs and develops a halo due to the excita-tion. Bunches longer than 340 degrees or shorter than 270degrees experience very little emittance growth.To summarize, we have found that voltage ratios be-tween 0.55-0.68 may not be suitable if Landau dampingis to be preserved in a second harmonic cavity. Numer-ical simulations have shown that bunches with phase ex-tent shorter or much longer than the location of the max-imum synchrotron frequency experience very little emit-tance growth when driven at this frequency.We thank Jim MacLachlan for help with ESME.REFERENCES[1] A. Hofmann and F. Pedersen, IEEE Trans. Nuc. Sci. NS-26,3526 (1979).[2] E. Shaposhnikova et al., PAC 2005, pg 2309 (2005).[3] J. A. MacLachlan, HEACC98, 184, (1998).[4] R. Warnock & J. Ellison, in The Physics of High BrightnessBeams (World Scientific, Singapore, 2000).TUPD067 Proceedings of IPAC’10, Kyoto, Japan208005 Beam Dynamics and Electromagnetic FieldsD05 Instabilities - Processes, Impedances, Countermeasures

Dynamics of Flat Bunches with Second Harmonic RF

Abstract

Similar works

Full text

Available Versions

UNT Digital Library