Impact of the 50 Hz harmonics on the beam evolution of the Large Hadron Collider
Sofia Kostoglou, Gianluigi Arduini, Leandro Intelisano, Yannis Papaphilippou, Guido Sterbini
IImpact of the 50 Hz harmonics on the beam evolution of the Large Hadron Collider ∗ S. Kostoglou † CERN, Geneva 1211, SwitzerlandandNational Technical University of Athens, Athens 15780, Greece
G. Arduini, Y. Papaphilippou, and G. Sterbini
CERN, Geneva 1211, Switzerland
L. Intelisano
CERN, Geneva 1211, SwitzerlandandINFN, Sapienza Università di Roma, Rome 00185, Italy (Dated: September 8, 2020)Harmonics of the mains frequency (50 Hz) have been systematically observed in the transversebeam spectrum of the Large Hadron Collider (LHC) since the start of its operation in the form ofdipolar excitations. In the presence of strong non-linearities such as beam-beam interactions, asmany of these power supply ripple tones reside in the vicinity of the betatron tune they can increasethe tune diffusion of the particles in the distribution, leading to proton losses and eventually to asignificant reduction of the beam lifetime. The aim of this paper is to determine whether the 50 Hzharmonics have an impact on the beam performance of the LHC. A quantitative characterizationof the ripple spectrum present in the operation of the accelerator, together with an understandingof its source is an essential ingredient to also evaluate the impact of the 50 Hz harmonics on thefuture upgrade of the LHC, the High Luminosity LHC (HL-LHC). To this end, simulations withthe single-particle tracking code, SixTrack, are employed including a realistic ripple spectrum asextracted from experimental observations to quantify the impact of such effects in terms of tunediffusion, Dynamic Aperture and beam lifetime. The methods and results of the tracking studiesare reported and discussed in this paper.
I. INTRODUCTION
Power supply ripple at harmonics of the mains powerfrequency (50 Hz) has been systematically observed inthe transverse beam spectrum of the Large Hadron Col-lider (LHC) since the start of its operation. In a previouspaper [ ? ], we investigated the source of the perturbationbased on a systematic analysis of experimental observa-tions. It was clearly shown that the 50 Hz harmonics arenot an artifact of the instrumentation system but corre-spond to actual beam oscillations.Two regimes of interest have been identified in thetransverse beam spectrum: a series of 50 Hz harmon-ics extending up to approximately 3.6 kHz, referred to asthe low-frequency cluster , and a cluster of 50 Hz harmon-ics centered around the alias of the betatron frequency,i.e., f rev − f x , where f rev =11.245 kHz is the LHC revo-lution frequency and f x the betatron frequency, namelythe high-frequency cluster . Both clusters are the resultof dipolar beam excitations and it was demonstratedwith dedicated experiments that the source of the low- ∗ Research supported by the HL-LHC project. † sofi[email protected] frequency cluster are the eight Silicon Controlled Recti-fier (SCR) power supplies of the main dipoles [1, 2]. Ta-ble I presents the frequency and the amplitude of the 12most important 50 Hz harmonics in the transverse spec-trum of Beam 1 during collisions as observed in the Q7pickup of the transverse damper Observation Box (AD-TObsBox) [3–5]. The offsets induced in Beam 2 due tothe power supply ripple are also depicted, which are ap-proximately lower by a factor of two compared to Beam1. Based on the source, the 50 Hz harmonics are expectedto be present in the future operation of the accelerator.In the presence of strong non-linearities such as beam-beam interactions and fields of non-linear magnets, amodulation in the dipole strength due to power supplyripple results in a resonance condition [6, 7]: l · Q x + m · Q y + p · Q p = n (1)where l, m, p, n are integers, Q x and Q y are the horizon-tal and vertical tunes, respectively, and Q p is the powersupply ripple tune. Power supply ripple can impact thebeam performance by introducing resonances in additionto the ones excited due to the non-linearities of the lat-tice. These resonances can increase the tune diffusion ofthe particles in the distribution, which may lead to thereduction of the beam lifetime. The aim of the presentpaper is to determine whether the 50 Hz harmonics act a r X i v : . [ phy s i c s . acc - ph ] S e p TABLE I. The beam offsets observed in the Q7 pickup of the transverse damper for the 12 largest 50 Hz harmonics in thehorizontal spectrum of Beam 1 (B1) and 2 (B2) during collisions.Frequency (kHz) Cluster B1 offset (nm) B2 offset (nm) Frequency (kHz) Cluster B1 offset (nm) B2 offset (nm)7.6 High 166.2 76.51 7.7 High 109.75 51.381.2 Low 98.63 71.87 7.65 High 97.4 51.977.8 High 92.75 42.78 2.95 Low 91.62 32.212.35 Low 88.48 29.61 7.9 High 82.23 25.467.5 High 77.82 35.08 2.75 Low 67.43 31.777.85 High 67.37 18.86 0.6 Low 54.14 14.33 as a limiting factor to the beam performance of the LHC,as well as its future upgrade, the High-Luminosity LHC(HL-LHC) [8], using single-particle tracking simulationswith power supply ripple in the form of dipolar excita-tions. The impact of power supply ripple that results ina tune modulation is discussed in a previous paper [9].First, the motion of a particle in the presence of amodulated dipolar field error is described using a linearformalism (Section II). Second, single-tone dipolar ex-citations are considered in the tracking simulations todefine a minimum power supply ripple threshold that re-sults in a reduction of the Dynamic Aperture (DA) inthe presence of strong non-linearities such as long-rangeand head-on beam-beam encounters, chromatic arc sex-tupoles and Landau octupoles. Then, the impact of theharmonics on the beam performance in terms of tune dif-fusion, DA and lifetime is discussed in Sec. III by includ-ing in the simulations a power supply ripple spectrumas acquired from experimental observations. Finally, theimpact of controlled dipolar excitations on the beam life-time, conducted with the transverse damper, is describedin Section IV, which provides a tool for the validation ofthe DA simulations in the presence of power supply rip-ple.
II. LINEAR FORMALISM FOR A MODULATEDDIPOLAR FIELD ERROR
In a circular accelerator, the kick related to a modu-lated dipolar field error Θ p with a deflection θ p and afrequency of Q p oscillations per turn can be represented in normalized phase space as: ¯ P n = (cid:18) p ( n ) (cid:19) = (cid:18) √ βθ p cos (2 πQ p n ) (cid:19) , (2)where n is the turn considered and β is the β -function atthe location of the perturbation.In the linear approximation, considering only the hor-izontal motion and assuming that the source and the ob-servation point are situated in the same location, thevector representation of the position and momentum atturn N is: ¯ X N = (cid:18) ¯ x N ¯ x (cid:48) N (cid:19) = N (cid:88) n = −∞ M N − n ¯ P n , (3)where M is the linear rotation equal to: M N = (cid:18) cos(2 πQN ) sin(2 πQN ) − sin(2 πQN ) cos(2 πQN ) (cid:19) (4)with Q representing the machine betatron tune. Com-bining Eq. (2), (3) and (4) and assuming that the per-turbation is present from n → −∞ , yields: ¯ x N = (cid:112) βθ p N (cid:88) n = −∞ cos (2 πQ p n ) sin (2( N − n ) πQ ) , ¯ x (cid:48) N = (cid:112) βθ p N (cid:88) n = −∞ cos (2 πQ p n ) cos (2( N − n ) πQ ) . (5)In Eq. (5), using Euler’s formula and writing the sumexpressions in terms of geometric series yields: ¯ x N = (cid:112) βθ p lim k →−∞ (cid:16) cos(2 πQ p N ) sin(2 πQ )2(cos(2 πQ p ) − cos(2 πQ ))+ − cos(2 πQ p ( k − πQ ( k − N )) + cos(2 πQ p k ) sin(2 πQ ( k − − N ))2(cos(2 πQ p ) − cos(2 πQ )) (cid:17) , ¯ x (cid:48) N = (cid:112) βθ p lim k →−∞ (cid:16) cos(2 πQ p ( N + 1)) − cos(2 πQ p N ) cos(2 πQ )2(cos (2 πQ p ) − cos (2 πQ ))+ cos(2 πQ p ( k − πQ ( k − N )) + cos(2 πQ p k ) cos(2 πQ ( k − − N ))2(cos (2 πQ p ) − cos (2 πQ )) . (6)The limit of the terms in the numerator of Eq. (6) in-volving k in the argument of the trigonometric functionsis bounded but indeterminate. Using the Cesàro meanthe trigonometric products involving k as an argumentin Eq. (6) is equal to zero and the expressions for theposition and momentum simply become: ¯ x N = (cid:112) βθ p cos(2 πQ p N ) sin(2 πQ )2(cos (2 πQ p ) − cos (2 πQ )) , (7) ¯ x (cid:48) N = (cid:112) βθ p cos(2 πQ p N ) cos(2 πQ ) − cos(2 π ( N + 1) Q p )2(cos (2 πQ p ) − cos (2 πQ )) . In physical coordinates, the maximum offset computedfrom Eq. (7) is: | x max ,N | = (cid:12)(cid:12)(cid:12)(cid:12) βθ p sin (2 πQ )2 (cos (2 πQ p ) − cos(2 πQ )) (cid:12)(cid:12)(cid:12)(cid:12) . (8) III. SIMULATIONS OF 50 HZ HARMONICSWITH BEAM-BEAM INTERACTIONS
Tracking simulations are performed using the single-particle symplectic code SixTrack [10, 11]. A distributionof particles is tracked in the element-by-element LHC andHL-LHC lattice at top energy, including important non-linearities such as head-on and long-range beam-beamencounters, sextupoles for chromaticity control and Lan-dau octupoles for the mitigation of collective instabilities.For the LHC case, the chromaticity in the horizontal andvertical plane is Q (cid:48) =15 and the amplitude detuning coef-ficients α n n = ∂Q n /∂ (2 J n ) with n , n ∈ { x, y } are: α xx = 16 . × m − , α xy = − . × m − ,α yy = 16 . × m − (9)as computed from the PTC-normal module of the MAD-X code [12].The beam-beam interactions are simulated with theweak-strong approximation. The maximum duration ofthe tracking is turns that corresponds to approxi-mately 90 seconds of beam collisions in operation. Themain parameters used in simulations for LHC and HL-LHC are presented in Table II. For the LHC, the simu-lated conditions refer to the start of the collisions. Thesimulated conditions in the HL-LHC case correspond tothe nominal operational scenario at the end of the β ∗ -leveling, taking place after several hours from the startof the collisions [13].In a previous paper [ ? ], it was demonstrated that, asfar as the low-frequency cluster is concerned, the powersupply ripple is distributed in all eight sectors. An ac-curate representation of the power supply ripple propa-gation across the chains of the LHC dipoles requires amodel of the transfer function as a transmission line forall the spectral components in the low-frequency cluster,similarly to the studies performed for the SPS [14]. Fur-thermore, as the exact mechanism of the high-frequency cluster is not yet clearly identified, an accurate transferfunction is not known at the moment.To simplify these studies, the distributed power supplyripple is projected to a single location with an equivalentimpact on the beam’s motion. Specifically, a horizontalmodulated dipolar source is included at the location ofthe Q7 pickup of the transverse damper, which coincideswith the observation point. To simulate the dipolar ex-citation, the strength of a horizontal kicker is modulatedwith a sinusoidal function: ∆ k ( t ) = θ p cos (2 πf p t ) , (10)where t is the time, θ p the deflection, f p = Q p · f rev thepower supply ripple frequency and f rev the revolutionfrequency listed in Table II. In this way, the contributionof all the dipoles is represented by a single, equivalentkick and the offsets observed in the LHC spectrum arereproduced in the simulations. The maximum offset in-duced on the particle’s motion is computed from Eq. (8)with the horizontal β -function listed in Table II.The simulations are then repeated for the HL-LHCcase. The need to perform projections for the HL-LHC isjustified by the fact that no modifications are envisagedfor the power supplies of the main dipoles. Consequently,based on the source, the 50 Hz harmonic are expected toalso be present in the HL-LHC era. In the following sec-tions, the HL-LHC studies are based on the power supplyripple spectrum acquired experimentally from LHC, al-though it is expected that the foreseen upgrade of thetransverse damper system will lead to a more efficientsuppression of the harmonics. A. Impact of single-tone ripple spectrum on theDynamic Aperture
As a first step, we consider individual tones in thepower supply ripple spectrum for increasing values of thedeflection θ p . For each study, a different combinationof the frequency f p and the amplitude of the excitation θ p is selected. For each case, the minimum DA in theinitial configuration space, i.e., the initial horizontal x and vertical y displacements is compared to the valuederived in the absence of power supply ripple. In thesesimulations, the tune modulation due to the coupling ofthe synchrotron oscillations to the betatron motion for achromaticity and a relative momentum deviation listedin Table II is also included.The scan in the ripple parameter space ( f p , θ p ) is per-formed to define the most dangerous tones of the lowand high-frequency cluster, i.e., the frequencies that, fora constant excitation amplitude, have a maximum impacton the DA. Then, the offset induced in the beam motionat the location of observation point is estimated fromEq. (8) with the parameters f p , θ p and the β -function ofTable II. A threshold for the minimum offset at the Q7pickup that leads to a reduction of DA is determined. TABLE II. The LHC and HL-LHC parameters at top energy used in the tracking simulations. The LHC parameters correspondto the start of collisions, while the HL-LHC parameters refer to the end of the β ∗ -leveling as envisaged in the nominal scenario[13].Parameters (unit) LHC (values) HL-LHC (values)Beam energy (TeV) 6.5 7Bunch spacing (ns) 25 25RMS bunch length (cm) 7.5 7.5Bunch population (protons) . × . × Beam-beam parameter ξ x,y × − × − Horizontal tune Q x a Vertical tune Q y a µ m rad ) 2.0 2.5Horizontal and vertical chromaticity 15 15Octupole current (A) 550 -300IP1/5 Half crossing angle ( µ rad ) 160 250Horizontal and vertical IP1/5 β ∗ (cm) 30 15Total RF voltage (MV) 12 16Synchrotron frequency (Hz) 23.8 23.8Revolution frequency (kHz) 11.245 11.245Relative momentum deviation δp/p × − × − Horizontal β -function at Q7 pickup (m) 105.4 102.3 a Nominal/optimized working point.
FIG. 1. The ripple frequency as a function of the horizontaloffset in the Q7 pickup, as computed from the deflection andEq. (8). A color code is assigned to the reduction of the min-imum DA from the value computed in the absence of powersupply ripple.
Figure 1 presents the ripple frequency as a functionof the offset at the observation point. The harmonics ofthe low and the high-frequency cluster that reside in thevicinity of f x and f rev − f x have been selected for theanalysis. A color code is assigned to the reduction of theminimum DA to illustrate the regimes with a negligible(blue) and significant (yellow) impact.From the scan, it is evident that the ripple frequencieswith the largest impact are the ones that reside in theproximity of the tune and its alias. For the LHC, an offsetthreshold of . µ m is defined, while this limit reduces to . µ m for the HL-LHC. For comparison, the maximumexcitation observed experimentally due to the 50 Hz lines is approximately 0.16 µ m as presented in Table I. B. Frequency Map Analysis with a realistic powersupply ripple spectrum
A more significant impact on the beam performanceis anticipated in the presence of multiple ripple tones,similar to the experimental observations of the 50 Hzharmonics, than considering individual tones due to theresonance overlap [15, 16]. To this end, a realistic powersupply ripple spectrum must be included in the simula-tions. Similarly to Table I, the offsets and the frequenciesof the 40 largest 50 Hz harmonics are extracted from thehorizontal beam spectrum as observed during operation.For each frequency, the equivalent kick at the location ofthe Q7 pickup is computed from Eq. (8) and it used asan input for the power supply ripple simulations.Figure 2 shows the horizontal spectrum of Beam 1 fromthe experimental observations (black) centered aroundthe low (left) and high (right) frequency cluster. A singleparticle is tracked in the LHC lattice including the mostimportant 50 Hz harmonics. The output of the simula-tions (green) is also illustrated in Fig. 2. The comparisonof the two is a sanity check illustrating the good agree-ment between the simulated and the experimental beamspectrum. A similar agreement (between simulated andexpected beam spectrum) is found for the HL-LHC case,were the equivalent kicks have been recomputed due toa small variation of the β -function as shown in Table II.The studies are organized as follows. First, a study inthe absence of power supply ripple is performed that isused as a baseline. Second, the largest 50 Hz harmon-ics of the low-frequency cluster are considered. Then, a FIG. 2. The spectrum centered around the low (left) andhigh (right) frequency cluster as acquired experimentally fromthe horizontal plane of Beam 1 (black) and from trackingsimulations (green). separate study is conducted including only the most im-portant harmonics of the high-frequency cluster. Last,both regimes are included in the simulations.Particles extending up to 6 σ in the initial configu-ration space are tracked for turns in the LHC andHL-LHC lattice. The particles are placed off-momentumas shown in Table II, without however considering theimpact of synchrotron oscillations. A Frequency MapAnalysis (FMA) is performed for each study [17–20]. Theturn-by-turn data are divided into two groups containingthe first and last 2000 turns, respectively. The tune ofeach particle is computed for each time interval using theNumerical Analysis of Fundamental Frequencies (NAFF)algorithm [21–23]. Comparing the variation of the tune ofeach particle across the two time spans reveals informa-tion concerning its tune diffusion rate D that is definedas: D = (cid:113) ( Q x − Q x ) + ( Q y − Q y ) , (11)where Q j , Q j with j ∈ { x, y } denote the tunes of eachparticle in the first and second time interval, respectively.Figure 3 illustrates the frequency maps (left panel) andthe initial configuration space (right panel) for the fourstudies in the LHC. For the frequency map, the particletunes of the second interval are plotted and a color codeis assigned to logarithm of the tune diffusion rate D todistinguish the stable particles (blue) from the ones withan important variation of their tune (red) due to the res-onances. The gray lines denote the nominal resonances,i.e., the resonances that intrinsically arise from the non-linear fields such as non-linear magnets and beam-beameffects without power supply ripple.In the absence of power supply ripple (Fig. 3a), animportant impact is observed due to the third-order res- onance (3 · Q y = Q x = Q p . (12)As the power supply ripple is injected in the horizontalplane, these resonances appear as vertical lines in thetune domain in a location equal to the excitation fre-quency (black). Due to the coupling of the transverseplanes, the strongest dipolar excitations are also visiblein the vertical plane, appearing as horizontal lines in thefrequency maps.The excitations of the high-frequency cluster appearas aliases (blue). For instance, the excitation at 7.8 kHzin Fig. 3c is located at f rev − . . For the high-frequency cluster (Fig. 3c), the most critical resonancesare: Q x = 1 − Q p . (13)For the 50 Hz harmonics in the vicinity of the beta-tron tune and its alias these additional resonances arelocated inside the beam’s footprint. As clearly shown inthe x − y plane (right panel), the existence of such reso-nances impacts both the core and the tails of the distri-bution. Reviewing the impact of both clusters (Fig. 3d)indicates that the main contributors to the increase oftune diffusion rate are the spectral components in thehigh-frequency cluster. Similar results are obtained forthe HL-LHC case. C. Intensity evolution simulations
Quantifying the impact of the power supply rippleand the non-linearities on the intensity evolution requirestracking a distribution with realistic profiles in all planesthat are similar to the ones observed experimentally. Ifimpacted by resonances, particles at the tails of the distri-bution close to the limit of DA diffuse and will eventuallybe lost. Therefore, a detailed representation of the tailsof the distribution is needed along with a realistic longi-tudinal distribution that extends over the whole bucketheight.The initial conditions form a 4D round distribution,which is sampled from a uniform distribution, extendingup to 7 σ both in the initial configuration space, as well asin the initial phase space. In the longitudinal plane, therelative momentum deviation of the particles is a uniformdistribution that extends up to the limit of the bucketheight (not the value presented in Table II). a) b)c) d) FIG. 3. The frequency maps (left) and the initial configuration space (right) color-coded with the logarithm of the tunediffusion rate (Eq. (11)) for the LHC simulation parameters of Table II (a) in the absence of power supply ripple , (b) withthe 50 Hz harmonics of the low and (c) high-frequency cluster and (d) combining both regimes. The gray lines represent thenominal resonances and the black and blue lines illustrate the resonances (Eq. (12) and (13)) excited due to power supplyripple.
To reduce the statistical variations, × particlesare tracked in the LHC lattice including synchrotron os-cillations. In the post-processing, a weight is assigned toeach particle according to its initial amplitude, as com-puted from the Probability Density Function (PDF) ofthe final distribution that needs to be simulated. Theweight of each particle is: w = (cid:81) j =1 PDF ( r j , σ j ) (cid:80) i w i , (14)where j iterates over the three planes, r j = x j + p j with x j , p j the initial normalized position and momentum co-ordinates, σ j represents the RMS beam size in the j -planeand the denominator denotes the normalization with thesum of the weights of all the particles.Experimental observations have shown that the tails ofthe LHC bunch profiles are overpopulated in the trans-verse plane and underpopulated in the longitudinal planecompared to a normal distribution [24, 25]. For an ac-curate description of the bunch profiles, the PDF of theq-Gaussian distribution is employed [26–28]: TABLE III. The parameters of the q-Gaussian PDF shown inEq. (15) for the longitudinal and transverse plane [24, 25].Parameters Longitudinal plane Transverse plane q σ G /σ qG PDF ( r, σ ) = (cid:40) e − r σ √ πσ q = 1 √ q − ( q − ) (cid:18) ( q − r σ +1 (cid:19) − q √ πσ Γ ( − q − q − ) 1 < q < √ − q Γ ( + − q ) (cid:18) ( q − r σ +1 (cid:19) − q √ πσ Γ ( − q ) q < (15)The parameter q and the ratio of the RMS beam sizeof the Gaussian ( q =1) to the q-Gaussian distribution σ G /σ qG are presented in Table III for the longitudinaland the transverse plane (assuming the same profiles forthe horizontal and vertical plane).Furthermore, a mechanical aperture is defined in thepost-processing at 6.6 σ to simulate the effect of the pri-mary collimators on the transverse plane based on the FIG. 4. Intensity evolution without power supply ripple(black), considering only the low (blue) or high (orange) fre-quency cluster and including the 50 Hz harmonics in bothregimes (red). settings of the 2018 operation. Particles beyond thisthreshold are considered lost and their weight is set tozero.Figure 4 presents the intensity evolution without powersupply ripple (black), including the 50 Hz harmonics ofthe low (blue) or high (orange) frequency cluster andconsidering both regimes (red). The results indicate thatthe presence of the 50 Hz harmonics leads to a reductionof the beam intensity, which is already visible with atracking equivalent to 90 seconds of beam collisions. Theresults also show that, for the time span under consider-ation, the high-frequency cluster is the main contributorto the decrease of the beam intensity. A similar effect isobserved when considering the same power supply ripplespectrum for the HL-LHC case with the high-frequencycluster acting as the main contributor.To quantify the impact of the 50 Hz harmonics on thebeam performance, the lifetime τ is computed from theintensity evolution I as: I ( t ) = I · e − tτ , (16)where I is the initial intensity and t the time. An ap-proximation of the instantaneous lifetime is estimated byfitting the exponential decay of the intensity with a slid-ing window, with each step consisting of a few thousandturns.Figure 5 illustrates the intensity evolution in the ab-sence of power supply ripple (black), including the powersupply ripple spectrum from the experimental observa-tions of Beam 1 (blue) and 2 (red). The green line indi-cates the fits of Eq. (16) to compute the lifetime evolu-tion.As shown in Table I, the fact that the power supplyripple spectrum of Beam 2 is lower by approximately afactor of two compared to Beam 1 results in an asym-metry of the intensity evolution between the two beams.Starting from τ = 28 . h in the absence of power supply FIG. 5. Intensity evolution in the absence of power supplyripple (black), including the power supply ripple spectrum ofBeam 1 (blue) and 2 (red). The green line indicates the fitsof the exponential decay of the intensity to compute lifetime(Eq. (16)). ripple, the lifetime reduces to τ = 27 . h when includingthe ripple spectrum of Beam 2 in the simulations and τ = 22 . h when considering the spectrum of Beam 1.During the operation of the accelerator in run 2 (2015-2018), it has been observed that the lifetime of Beam 1was systematically lower than the lifetime of Beam 2 [29].It is the first time that simulations reveal that, amongstother mechanisms, the 50 Hz harmonics can contributeto this effect. IV. SIMULATION BENCHMARK WITHCONTROLLED EXCITATIONS
During the latest LHC run in 2018, controlled dipolarexcitations were applied on the beam using the transversedamper kicker. The goal of the experiment was to studythe impact of dipolar excitations at various frequenciesand amplitudes and to validate our simulation frameworkin a controlled manner. The experiments were performedat injection energy.Experimentally, some of the excitations led to a sig-nificant reduction of the beam lifetime. To retrieve theinitial deflection applied from the transverse damper, theoffset and the frequency are extracted from the calibratedADTObsBox beam spectrum. For instance, Fig. 6 showsthe horizontal spectrum of Beam 1 during a controlledexcitation at 2.5 kHz (green star-shaped marker) for asingle bunch. The gray vertical lines illustrate the multi-ples of 50 Hz. Then, using the offset and the frequency,the equivalent kick is computed from Eq. 8. This proce-dure is repeated for all the excitations applied during theexperiments.The impact of the excitations on the beam lifetime iscompared against the DA scans computed with trackingstudies in the presence of dipolar power supply ripple.
FIG. 6. The horizontal spectrum of a single bunch (black)during a controlled dipolar excitation at 2.5 kHz (greenmarker) performed using the transverse damper. The verticallines represent the multiples of 50 Hz.TABLE IV. The LHC parameters at injection energy used inthe simulations with dipolar power supply ripple.Parameters (unit) ValuesBeam energy (GeV) 450Bunch spacing (ns) 25RMS bunch length (cm) 13RF voltage (MV) 8Betatron tunes ( Q x , Q y ) (0.275, 0.295)H/V normalized emittance ( µ m rad ) 2.5H/V chromaticity 15Octupole current (A) 20Bunch population (protons) . × Horizontal β -function at Q7 (m) 130.9 The important parameters of the tracking simulations atinjection energy are summarized in Table IV.In each study, a different combination of the excitationfrequency and amplitude is selected. In particular, con-sidering a constant excitation frequency, the value of thedeflection is increased and the minimum DA is computedfor each case. A ripple amplitude threshold is defined asa function of the excitation frequency. Beyond this limit,a reduction of DA is expected.Figure 7 presents the frequency of the excitation as afunction of the deflection. A color code is assigned to theminimum DA to distinguish the regime where the powersupply ripple has no significant impact (blue) from theone where a significant reduction of DA (red) is observedin the simulations.In Fig. 7, the star-shaped markers denote the exper-imental excitation kicks and frequencies. A color codeis assigned to the markers that allows distinguishing theexcitations that had no impact on lifetime experimen-tally (blue) from those that lead to a lifetime drop (red).Although an excitation at 8.1 kHz was performed exper-imentally, the position measurements at this time werenot stored and a star-shaped marker is not included.
FIG. 7. The frequency of the excitation as a function ofthe deflection. A color code is assigned to the minimum DAcomputed with tracking simulations including power supplyripple. The star-shaped markers present the equivalent kicks,as computed from the beam spectrum and Eq. (8) during thecontrolled excitations with the transverse damper. The redmarkers indicate that the excitation had an impact on thebeam lifetime. The blue markers denote the excitations thatdid not affect the beam lifetime.
The comparison between experimental observationsand the power supply ripple threshold defined by DA sim-ulations yields a fairly good agreement between the twofor the majority of the excitations, taking into accountthe simplicity of the machine model and the absence ofeffects such as linear and non-linear imperfections in thesimulations. This comparison provides a validation ofour simulation framework including power supply ripple.The method to benchmark simulations and experimentalfindings presented in this section is not only limited tothe studies of the 50 Hz harmonics but can be used tovalidate the tracking results for different types of powersupply ripple and noise effects.
V. CONCLUSIONS
The purpose of the current paper was to determinewhether the 50 Hz harmonics perturbation is a mecha-nism that can impact the beam performance during theLHC operation. To this end, single-particle tracking sim-ulations were performed in the LHC and HL-LHC lat-tice including important non-linear fields such as head-onand long-range beam-beam interactions, chromatic sex-tupoles and Landau octupoles.Including a realistic power supply ripple spectrum inthe simulations shows that the 50 Hz harmonics increasethe tune diffusion of the particles through the excitationof additional resonances, a mechanism that was clearlyshown with frequency maps. The high-frequency clusteris identified as the main contributor.From simulations that correspond to 90 seconds ofbeam collisions, the increase of the tune diffusion dueto the 50 Hz harmonics results in a lifetime reduction of21% with respect to the lifetime in the absence of powersupply ripple. Based on these results, it is concluded thatthe 50 Hz harmonics have an impact on the beam perfor-mance during the LHC operation. Mitigation measuresshould be incorporated in the future operation of the ac-celerator to suppress the high-frequency cluster from thebeam motion.Due to the asymmetry of the power supply ripple spec-trum between Beam 1 and 2 by a factor of approximatelytwo, the simulations illustrate a clear discrepancy in theintensity evolution of the two beams. Including realisticbunch profiles, the estimated lifetime is τ = 22 . h and τ = 27 . h for Beam 1 and 2, respectively. An importantlifetime asymmetry between the two beams has been ob-served since the beginning of run 2 and it is the first timethat tracking simulations show that power supply ripplecan contribute to this effect.In the context of this study, a general simulationframework has been developed. This paper illustrateda method to define an acceptable power supply ripplethreshold for operation through DA scans. The resultsof these scans were bench-marked against experimentalobservations with controlled excitations using the trans-verse damper. Finally, a method to determine the in-tensity evolution with weighted distributions and realis-tic bunch profiles has been demonstrated. The analysismethods presented in this paper can be applied to studiesof other types of power supply ripple and noise effects. ACKNOWLEDGMENTS
The authors gratefully acknowledge H. Bartosik,M. C. Bastos, O. S. Brüning, R. T. Garcia, M. Mar-tino and S. Papadopoulou for valuable suggestions anddiscussions on this work. We would like to thank D. Val-uch and M. Soderen for all transverse damper relatedmeasurements and experiments.
Appendix A: Benchmark of analytical formalism fora modulated dipolar field error with trackingsimulations
A comparison between the results of tracking simu-lations and Eq. (8) is performed as a sanity check. Asingle particle is tracked in the LHC lattice with Six-Track, in the presence of a dipolar modulation describedby Eq.(10). The amplitude of the kick is 1 nrad and thefrequency varies across the studies. The offset is com-puted from the particle’s spectrum for each study and isthen compared to the analytical formula (Eq. (8)).Figure 8 illustrates the offset as a function of the fre-quency computed analytically (black) and from simula-tions (blue) and a very good agreement is found betweenthe two. For a constant excitation amplitude, a reso-nant behavior is expected as the frequency approaches
FIG. 8. The offset as a function of the dipolar excitationfrequency with θ p = 1 nrad as computed from the closedform of Eq. (8) (black) and from simulations (blue). to k · f rev ± f x , where k is an integer.As a reference, the maximum offset observed in thebeam spectrum is equal to . × − σ for a normalizedemittance of (cid:15) n = 2 µ m rad , a beam energy of 6.5 TeVand a β − function equal to 105 m. Considering a sin-gle dipolar excitation at the location of the observationpoint, the equivalent kick, as computed from Eq. (8), is θ p = 0 .
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