Damping rate limitations for transverse dampers in large hadron colliders
DDAMPING RATE LIMITATIONS FOR TRANSVERSE DAMPERS IN LARGE HADRON COLLIDERS *
V. A. Lebedev †, Fermilab, Batavia, USA
Abstract
The paper focuses on two issues important for design and operation of bunch-by-bunch transverse damper in a very large hadron collider, where fast damping is required to suppress beam instabilities and noise induced emittance growth. The first issue is associated with kick variation along a bunch which affects the damping of head-tail modes. The second issue is associated with effect of damper noise on the instability threshold.
INTRODUCTION
An achievement of maximum luminosity in a collider requires large beam current and small emittance. In had-ron colliders of very large energy the collider size be-comes so large that the frequency of lowest betatron side-band approaches kHz range where spectral density of acoustic and magnetic field noise is unacceptably large. This noise drives the emittance growth resulting in fast luminosity decay. Effective suppression of this emittance growth may be achieved by fast transverse damping [1,2]. Fast emittance growth and its suppression by the damper was demonstrated at the LHC commissioning [3,4]. The required damper gain grows with the size of the collider and approaches few turns for a collider which will follow the LHC (like FCC). The instability suppression is typi-cally less demanding to damping rate but still it is another important reason for fast damping. There are many phenomena which limit the maximum damper gain [5]. Here we discuss two of them in details. (1) A damper gain increase results in a better suppression of zero head-tail mode. However, such increase may ex-cite higher head-tail modes, and thus make the bunch unstable. This effect is exacerbated by presence of non-zero chromaticity and wake-fields which destroy sym-metry of head-tail motion. As will be seen below an intro-duction of kick non-uniformity along the bunch may allow significant reduction of excitation of head-tail modes and, consequently, increases the beam stability margin. (2) Any practical damper has internal noise. Depending on damper design it is related to the thermal noise of its preamps and/or noise of digitization. This noise drives small amplitude beam motion which due to betatron fre-quency spread results in an emittance growth. The beta-tron motion non-linearity introduced for suppression of head-tail modes makes this noise-induced diffusion de-pending on a particle betatron amplitude. With time that changes the particle transverse distribution and, conse- quently, may result in a loss of Landau damping. This phenomenon was observed in the LHC where the beam could lose transverse stability minutes after bringing the beams to collisions without any visible changes in the machine. The effect was pronounced stronger in the case of external excitation of transverse motion [6,7]. The beam stability study based on the multiparticle tracking is reported in Ref. [7]. It showed that the latency of stability loss is related to changes in the distribution function in-duced by the damper noise. In this paper we consider a semi-analytical theory which attempts to show details of the process in a one-dimensional model. Below we assume that the damper is bunch-by-bunch type so that each bunch is damped separately.
DAMPING OF INTRABUNCH MOTION
For analysis of intrabunch motion we use the air-bag square-well (ABS) model initially suggested in Ref. [8] and actively used by A. Burov for analysis of bunch damping (see for example [9]). In this model the bunch is presented by two fluxes moving in opposite directions with particle reflection at the bucket boundaries. In difference to the linear longitu-dinal motion in the air-bag model [10] where the bunch density is picked at the bunch ends this model has a uni-form density distribution along bunch. Therefore, ABS model better suits for description of damper effect on damping of head-tail modes. In dimensionless variables the equations of motion for two fluxes are: ,2 2 .2 2 x x f qx x xs i i ix x f qx x xs i i i (1) where x and x are the transverse coordinates for the respective fluxes, / / s p p is the head-tail phase, is the tune chromaticity, s is the synchrotron tune, p / p represent the momentum deviations for parti-cles in the positive and negative fluxes, = s t is the dimensionless time, [0, ] s is the dimensionless longi-tudinal particle coordinate, q = sc / s is the space charge parameter, sc is the space charge tune shift, and f char-acterizes the forces coming from the damper and wake-fields. Following Ref. [9] we introduce the new transverse coordinate: , , 0 ,, , 0 . i si s e x sx e x s (2) Here we also introduced the phase describing the syn-chrotron motion so that , [ , ] s . Performing substitutions we can reduce two equations in Eq. (1) to ___________________________________________ * Work supported by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy † [email protected] ne: i s x x fe q x xi (3) The force coming from the wake is determined by the following equation: ( ) ( ) ( ) ( ) . f s W s s x s x s ds (4) In this paper we consider two wake-functions: the con-stant wake – ( ) ( ) W s W s , (5) and the resistive wall wake – ( ) / 4 ( ) / W s W s s . (6) The coefficient in the resistive wake definition was cho-sen so that for the uniform bunch displacement the force at the bunch tail would be equal for both wakes. We assume that the force coming from the damper is determined by the following equation: ( ) cos ( / 2 )2( ) ( ) cos ( / 2 ) . k kp p
Gf s i k sx s x s k s ds (7) Here k p and p determine the sensitivity of damper pickup to a particle position along the bunch, and k k and k de-termine dependence of the kick on the longitudinal coor-dinate along the bunch. In the absence of space charge, damping and wakes the solutions of Eq. (3) are: ( , ) . inn n n x x a e (8) In the first order of perturbation theory when only a damper is present (no wakes and space charge) we obtain the growth rate: n n n p p k kn d Ga R k R kda (9) where ( , ) cos cos ,2 , . in x x x x dR k e k nx k p (10) As one can see from Eq. (9) all modes are damped (have negative growth rates) if k p = k k and p = k . In the general case we look for a solution in the form: m m . n N imn nmm N x e A e (11) where N m determines how many harmonics approximate the exact solution. Substituting this equation into Eq. (3), using definitions of Eqs. (4) and (7), multiplying obtained equation by e - in and integrating we obtain a system of 2 N m +1 linear equations. The eigen-values and eigen-vectors of this matrix equation yield complex frequencies for each mode and its structure ( x n ( )). To warrant a solu-tion accuracy, the 161 modes (±80) were used. After find-ing the eigen-vectors the modes were ordered in ascend-ing order of imaginary part of n (tune shift). First, we consider the instability in the absence of damper and the space charge. Calculations show that for = 0 the transverse mode coupling instability threshold is: W = W th W = W thr = 0 and = -2. For = 0 (strong head-tail case) and the wake twice above thresh-old only 0-th and 1-st modes are coupled making only one mode unstable. As one can see from the bottom plot many modes became unstable for = -2. Although growth rates for both wakes (step-like and resistive wall) are close the tune shifts of the modes are significantly larger for the resistive wall wake. Figure 1: Dependence of growth rate, Re( n ), on the mode coherent frequency, Im( n ), for different modes and the wake amplitude twice above threshold; top – = 0, bot-tom – = -2; red dots – step-like wake, blue circles – resistive wall wake. Further we characterize damping by the growth rate of the most unstable mode. Typically, it is the mode for which n . Figure 2 shows the growth rate of the most unstable mode on the head-tail phase, , for different damper gains when both pickup and kicker have flat re-sponses ( k p = k k =0). One can see in the top plot that there is no instability for G = 0 and = 0 as should be expected elow the instability threshold. However, for G = 0 the beam is unstable for any other (non-zero) head-tail phase. An increase of the damper gain reduces the growth rate for the most unstable mode everywhere except close vi-cinity of = 0. Optimal damping is achieved at G [0.5, 1.4] for both wakes. Further increase of the gain does not improve beam stability. For the wake twice above the threshold the beam is unstable for all . Note that the considered model does not have Landau damping (discussed below) which stabilizes the beam if the growth rate is sufficiently small and these calculations do not show actual stability thresholds. Note also that the oscillations in the growth rates with are related to switching from one to another most unstable mode, so that one period represents the growth rate for one mode. Figure 2: Dependence of the growth rate of the most un-stable mode on for different damper gains ( G = 0, 1, 2, 4, 6, 9, 15) for wake amplitudes twice below (top) and twice above the threshold; the step-like wake. Insets show dependences near = 0. Now we consider how changes in the response func-tions of pickup and kicker affect the beam stability. Figure 3 presents dependences of the growth rate of the most unstable mode on for different damper responses. As one can see for negative an increase of k p = k k from 0 to 1 reduces the growth rate of most unstable mode by about 2 times. One can also see from the top plot that there is an area near = 0 where all modes are stable. Variations of p and k and making k p and k k different did not exhibit stability improvement. Figure 3: Dependence of the growth rate of the most un-stable mode on for different damper responses for the cases of the beam intensity twice less (top) or twice more (bottom) than the strong head-tail threshold; the step-like wake. Figure 4: Dependence of the growth rate of the most un-stable mode on for different kicker responses: red lines - k k = 1.5, blue lines - k k = 0; top two lines - W is twice above threshold, bottom two lines - W is twice below threshold; for all curves: k p = 1.5, p = k = q =0; the resistive wall wake. All calculations were also repeated for the resistive wall wake and for different space charge parameter q . The results show that there is a reduction of the growth rate of most unstable mode by about two times for k p = k k k p = k k = 0. Similar improvement hap-pens in transition from k p k k = 0 to k p = k k
1. In the present LHC damper the pickup response to particle position is harmonic at 400 MHz frequency. The bunch length of 18 cm (~2 ) corresponds k p k k = 0) and as can be seen in Figure 4 that negatively affects the beam stability. Thus, making the kicker waveform as a few-periods 400 MHz sinusoid (short enough to avoid overlapping of signals of different bunches) would reduce the excitation of head-tail modes by factor of ~2. EFFECT OF DAMPER NOISE ON THE INSTABILITY THRESHOLD
For a continuous beam and the smooth lattice approxi-mation the equation of a particle motion under external force ( ) i t
F t F e is:
22 2 20 0 0 0 i i lat i c x Q Q x Q Q x F (12) Here i enumerates particles, is the circular frequency of particle revolution, Q is the small amplitude betatron tune, ( , ) i i i lat lat x y Q Q J J is the tune shift of particle betatron motion due to lattice non-linearity for a particle with betatron actions i x J and i y J , / (4 ) c cw Q Q ig is the coherent tune shift which includes the tune shifts due to ring impedance, Q cw , and due to transverse damp-er with damping rate per turn equal to g /2. Following the standard recipe [10, 11] we obtain the beam response to an external perturbation: ( ) .( ) x RF (13) Here x is the Fourier harmonic of beam centroid deter-mined as ( ) ( ) / , N ii x t x t N ( ) / ( , ) 0 x x yx lat x y J dJ dJfR dJ Q J J i (14) is the response function in the absence of particle interac-tion, f = f ( J x , J y ) is the particle distribution function nor-malized so that ( , ) 1 x y x y f J J dJ dJ , ( ) 1 ( ) c Q R (15) is the beam permeability, = - n is the frequency deviation from n -th betatron sideband, n = ( n - Q ) , i ( ) nm s Q n mQ , where Q s is the synchrotron tune. Second, we need to account that a damper kick may excite multiple synchrotron-betatron modes. That is accounted by coefficients w m . Consequently, Eq. (13) is modified to the following form: ( )( ) nm nmm nm nm Rx w F . (16) Here nm = nm , and in Eq. (15) we need to account that the coherent tune shifts are different for each mode nm c c Q Q so that: ( ) 1 ( ) nm nm nm c nm Q R , (17) where R ( ) is still determined by Eq. (14). Eq. (16) determines the amplitude of particle motion for a given synchro-betatron mode. For small amplitude excitation each synchro-betatron mode is excited inde-pendently and to obtain the total motion in the bunch one needs to sum motions of all modes. The instability boundary ( i.e. maximum coherent tune shift nm c Q for a given mode is determined by the condi-tion when with growth nm c Q the beam permeability ap-proaches zero the first time at any possible detuning. That corresponds to the solution of equation, ( ) 0 nm , (18) for real , which determines the stability boundary in the complex plane of Q c . As follows from Eq. (13) the beam response of stable beam for a given mode is ampli-fied by 1/| nm ( nm )| times. Damper noise drives the transverse beam motion which due to spread in the betatron tunes results in an emittance growth. In the absence of particle interaction and active damping the emittance growth rate is [1]: ,4 kick nn d Pdt (19) where kick is the horizontal beta-function at the kicker location, and P ( ) is the spectral density of kicker angu-lar noise normalized so that the rms value of the kicks is: ( ) . P d Taking Eq. (16) into account we can rewrite Eq. (19) in the following form: , ( , ) x y x y x y d D J J f J J dJ dJdt . (20) Here
220 2, 0 , ,4 ( ( , )) m nkickx y n m nm lat x y w PD J J Q J J (21) and we accounted that the spectral density of kicker noise does not change across one synchro-betatron sideband, noises at different frequencies do not correlate, and only resonant frequencies drive the emittance growth. It is straightforward to find the emittance growth for the case of zero chromaticity, when only zero’s synchro-betatron mode is excited. Assuming strong damping, n cw g Q , octupole non-linearity in the horizontal plane only, ( ) lat x xx x Q J a J , and Gaussian distribution, / ( ) / x a J Jx a f J e J , we obtain:
16 , .4 e 0 nkick xx xn an
P JD J y Jx dxg x y i (22) Here xx a a J is the rms frequency tune spread, and g n is the damper gain at the n -th betatron sideband. Sub-stituting diffusion of Eq. (22) into Eq. (20) and perform-ing numerical integration one obtains a perfect coinci-dence with the result obtained in Ref. [1]:
16 , 4 .4 kick n nn n d P gdt g (23) Note that Eq. (21) is applicable in the general case while Eq. (23) in the case of zero chromaticity and far away from the instability threshold. Note also that the deriva-tion of Eq. (23) in Ref. [1] does not actually determine the tune relative to which is computed. This question is addressed by Eq. (21). To find a change in the instability threshold related to a change in the distribution we need to investigate the dis-tribution function evolution. Considering that the kicks are small and uncorrelated; and, consequently, the process is very slow relative to the betatron motion the evolution can be described by the diffusion equation. In the general case of uncoupled betatron motion the diffusion in the 2D-space of actions is described by the following diffu-sion equation: ( , ) ( , ) x x x y y y x yx x y y f f fJ D J J J D J Jt J J J J . (24) Here the diffusion in the horizontal plane is determined by Eq. (21). The vertical plane diffusion is obtained by changing corresponding indices. Figure 5: Dependencies of mode magnitudes, | X n | | x n + x n |, along the bunch for the parameters of the LHC damper: = 1, k p = 1.5, k k = p = k = q =0, W = 2 W thr for the resistive wall wake. Numbers show the mode num-bers. In the presence of impedance and chromaticity each kicker kick excites multiple head-tail modes. Only few of them are damped by the damper. Figure 5 shows shapes of few lowest head-tail modes for the damper model de-scribed in the previous section for the LHC parameters. As one can see all of them have significant variations along the bunch while the kicker kick is the same for all particles. Therefore, each kick in addition to the zero mode excites other modes. To find corresponding contri-butions we equalize the kick dependence along the bunch and weighted sum of the mode amplitudes: ˆcos / 2 ( ) . k k m mm k i w x (25) where x m ( ) is determined by Eq. (11) and is additionally normalized so that x m ( /2) = 1. The solution of this equa-tion yields coefficients ˆ m w . To obtain coefficients w m which determine relative excitation for different head-tail modes we additionally need to account how a given mode with amplitude ˆ m w contributes to the emittance growth. That yields:
22 2 ˆ( ) m m m w x w . Figure 6 shows ˆ m w for the modes presented in Figure 5. One can see that the mode zero has the largest contribution, ˆ w , and the only one which has significant damping. Figure 6: Dependences on the head-tail mode number for ˆ n w (red circles) and the damping rate (blue dots). To demonstrate an effect of damper noise on the beam stability boundary we initially assume that only one of the head-tail modes is near the threshold and it dominates the emittance growth. Applicability of this assumption we will discuss later. We also assume that the focusing non-linearity is in one plane only. That allows us to consider a one-dimensional problem. Then, from Eqs. (21) and (24) we obtain a simplified diffusion equation:
1ˆ ˆ( ) , ( ) .( ) x x xx x x f fJ D J D JJ J J (26) Here we transited to the dimensionless variables so that the action J x is measured in units of rms action J a , and time is chosen to make the diffusion coefficient equal to the one in the absence of beam interaction. We also took into account that the diffusion is proportional 1/| | at the esonance frequency which is directly related to the action as lat xx x Q a J . That yields the univocal dependence of beam permeability on the action: max ( ) 1 .0 Jc x xx xx x x x
Q J dJfJ a dJ J J i (27) where J max is determined by the ring acceptance. The solution of Eq. (26) with beam permeability of Eq. (27) was carried out numerically. Т he action space was binned into boxes with boundaries at J n = n J , n [0, N max ], so that f n J is the probability to find a particle in n -th box and f n is the distribution function in the center of the box bounded by J n and J n +1 . An integration of Eq. (26)over J through one box yields the particle flux through the boundary between boxes n and n +1:
11 1 1 1 1 n n nn n n n nJ J f ffJ D J J DJ J . (28) Consequently, the change in the distribution is: ( ) ( ) ( ) ( ) , . n k n k n k n k k k f t f t t t t t t t (29) Time step t was chosen so that to be well below the instability threshold of the difference scheme, which is determined by: max 4 / 1. n nn S D J t J
Figure 7: Ratios of coherent tune shits to the synchrotron tune for different modes for parameters of Figure 5. Black line presents the stability boundary for Gaussian beam with non-linearity parameter a xx chosen so that the most unstable mode (marked by blue circle) would be 25% below stability threshold. For a harmonic perturbation f cos( J ) and S <<
1 this difference scheme yields good approximation for small . However, it reduces damping at the highest frequency of / (2 ) max J by ( /2) times. Note that a usage of implicit methods typically applied to the diffusion equa-tion solving is limited by two circumstances. First, a computation of diffusion at any point in the action space uses the entire particle distribution and therefore compu-tation of distribution at next point in time requires inver-sion of N max N max matrix instead of three-diagonal matrix for the case of implicit scheme. Second, as will be shown below, the instability is developing at high frequencies. That requires small steps in time. To accelerate computation of the integral in Eq. (27) it was reduced to a matrix multiplication so that the vector of beam permeability is equal to: ε Rf . (30) Here the vectors n and f f n determine the beam permeability and the distribution function. The elements of matrix R are determined by integration Eq. (27) be-tween nearby actions J n using Tailor expansion of f . Nu-merical tests verified that Eq. (30) results in good approx-imation of integral (27) in the absence of discontinuities in the distribution. Figure 8: Dependence of dimensionless diffusion (top) and distribution function (bottom) on the action for dif-ferent times, t ; a xx = 0.02, Q c = (-12.6+3.1 i )10 -3 . Red curve in the bottom plot shows the initial distribution (left scale) and other curves changes of the distribution multi-plied by 100 (right scale). Simulations showed that loss of stability due to distri-bution evolution under kicker noise strongly depends on the phase of the coherent tune r = arg( Q n ). Figure 7 presents the dimensionless coherent tune shifts (ratio of coherent tune shifts to the synchrotron tune) for different head tail modes for the parameters of Figure 5. The stabil-ity boundary was chosen to be 25% above most unstable mode for which r = 168 o (Re( Q n )/ Im( Q n ) = -4.7). The distance from the stability boundary to the next mode closest to the boundary is about twice larger, and conse-quently its effect on the diffusion is 4 times smaller. Fig-ure 8 shows a typical example of the evolution for initial-ly Gaussian distribution. The figure also shows the corre-ponding diffusion. The value of Q c / a xx was chosen so that the beam would be 25% below instability threshold (see Figure 7). In all simulations (as well as in Figure 8) it has been clearly seen that the instability, if happens, de-velops at the highest possible wave-number determined by J . An increase of N max decreases J and the span in the distribution where the instability is initially devel-oped. However, the location of the instability position in the action did not depend on N max . To explain the results of the simulations we consider the following model. We assume that the instability is developed at a small area near the action J r . In this area we look for a solution in the following form: ( , ) ( ) ( ) cos( ) x x f J t f J f t J , (31) where we assume the wave-number, , being very large, and the perturbation f ( t ) f to be much smaller than the initial distribution f ( J x ). A perturbation in the distribution results in a perturbation in the response function. Substi-tuting the perturbation of Eq. (31) into Eq. (14) we obtain a perturbation of response function: sin( )( ) ,0 x xx x i J JdJR R J fa J J i (32) where we accounted that the resonance frequency is xx x a J . For large the major contribution to the inte-gral comes from the area near J x . That allows us to extend the integration to - . Then, the integration becomes straight forward. It results in: ( ) . x i Jxxx JR e f ta (33) Using Eqs. (15) and (26), we obtain the diffusion:
1ˆ ,1 ( )2Re , x rc i Jr cr r D D DQ R RJ QD D e f (34) where r c r Q R is the beam permeability for unper-turbed beam computed at the resonant tune / = J r a xx , r r D is the corresponding diffusion, and in obtain-ing the second equality we used the Tailor expansion and replaced J x by J r in the non-oscillating term. As one can see a harmonic perturbation of the distribution results in a harmonic perturbation of the diffusion. Taking into account that we consider only small aria in the action space in vicinity of J r and very large wave-number (see the definition below) we can replace J x inside / x J in Eq. (26) by J r . That yields: . r rx x f f J D D f fJ J (35) Accounting that the unperturbed function satisfies the following equation: r rx x f fJ DJ J (36) and leaving only linear terms in Eq. (35) we obtain a linear differential equation for the perturbation r rx x x ff fJ D DJ J J . (37) We look for a solution in the following form: cos( v ) f fe I . (38) Substituting it into Eq. (37), assuming initial Gaussian distribution x J f e , and using Eq. (34) we obtain the damping rate as a function of J r : . r Jr r cr xx
J D e A AJ Q
A a (39) For large the last term can be neglected. Thus, for the Gaussian distribution the stability area for given Q c is determined by following equation,
21 Im 0 r Jr c cxx r
J Q Qea , (40) Figure 9: Stability diagram computed with accounting noise driven diffusion (blue curve) and without it (red curve.) Figure 10: Dependence of the resonant action and the loss in stability on the angle of the coherent tune shift in the complex plane. hich must be satisfied for all J r . Figure 9 presents the stability diagrams computed with the help of Eqs. (18) (red curve) and (40) (blue curve). One can see that the kicker noise results in significant reduction of the stability boundary. However, this reduction is negligible in vicinity of arg( Q c ) o . We will call the action J r at which the left-hand side in Eq. (40) approaches zero the first time the resonant action. It shows where instability develops when the beam is approaching to the instability boundary. Figure 10 shows how this resonant action depends on the angle of the coherent tune shift in the complex plane. The figure also shows the ratio of stability boundary sizes (ratio of | Q c | for given r = arg(| Q c |) for curves present-ed in Figure 9). Numerical simulations verified the reduc-tion of the stability boundary presented in Figure 9 and 10 and the location of the resonant action. Taking into account that the considered above instabil-ity develops at high frequency and the resonant actions of different head-tail modes are different, we, in the first approximation, can neglect mutual interaction of different modes. That results in that the considered above model should be applicable to the situation when multiple modes are close to the instability boundary. If required it is straightforward to extend this model to multiple modes introducing summation of different modes in Eq. (34). CONCLUSIONS
An introduction of harmonic variation in the kicker waveform looks as a promising method for an increase of stability boundary for the LHC. Such a kicker does not work well for suppression of emittance growth due to injection errors. Therefore, the existing low frequency kicker should be retained and used for damping injection errors. A new kicker operating at 400 MHz base frequen-cy could be used for the rest of the accelerating cycle and in the collisions. The power and space required for this new kicker are determined by the BPM noise and are well within the reach. The considered above mechanism for reduction of the stability boundary points out underlying reasons behind the observations of transverse beam stability loss in the LHC. We need to note that in this model we neglected other diffusion mechanisms which affect the evolution of the distribution. In normal operating conditions the intra-beam scattering is the major diffusion mechanism. It counteracts the effects introduced by the damper noise and therefore a reduction of stability boundary due to kicker noise should be somewhat smaller. An additional noise used in the LHC experiments reduced relative effect of the IBS driven diffusion with subsequent reduction of the stability boundary observed in the experiments [6].
ACKNOWLEDGMENTS
The author would like to thank A. Burov, E. Metral and X. Buffat for help in editing of this paper and many useful discussions on its subject.
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