Simulations to investigate the effect of circuit

imperfections on the slow extraction process

Wouter Van De Pontseele

Supervisor: Matthew A. FraserCo-supervisor F. Velotti

CERN, Geneva, Switzerland

September 26, 2016

Abstract

The Super Proton Synchrotron (SPS) is a 400 GeV proton acceleratorat CERN. Its main purpose is the last step in pre-acceleration of particlesbefore injecting them into the Large Hadron Collider. Besides that, itis used for experiments directly connected to the SPS. For these fixedtarget experiments, a steady slow extraction of the SPS beam is necessaryover thousands of turns. The SPS slow extraction process makes use ofthe third-integer resonance in combination with an electrostatic septum.The stability of the extracted intensity and the losses at the septum areinfluenced by the imperfections of the current circuit that drives the SPSquadrupoles. This report summarizes the studies that were carried outwith MAD-X to simulate these imperfections.

1 Introduction

The Super Proton Synchrotron (SPS) is the second largest particle acceleratorat CERN. Nowadays, it is used as the last pre-accelerating stage before enteringthe Large Hadron Collider. Nonetheless, there are experiments connected to theSPS itself, COMPASS for example. Here, a steady beam intensity during thespill is important. This is achieved by making use of the third integer resonancein the SPS. The concept of accelerator resonances is based on the tune Q: thenumber of betatron oscillations per revolution in a circular accelerator. For slowextraction purposes in the SPS, a resonant tune of 26 + 2/3 was chosen. Theamplitude of resonant particles in phase space will grow rapidly due to repetitivekicks every three revolutions. The machine is configured in a way that resonantparticles will be extracted at a series of 5 electrostatic wire septa, called the ZSsepta. The extraction bump at the ZS septa is visualised in figure 1.

The chromaticity, Q′, is defined as the variation of the betatron tune Q with

1

Figure 1: The effect of activating the extraction bump in the SPS Long StraightSection 2 (LSS2).

the relative momentum deviation:

Q′ =dQ

dδ.

Often, the relative chromaticity ξ = Q′/Q is quoted. Therefore, each particlehas a certain momentum deviation which corresponds to a given tune value. Bysweeping the tune, the whole range of particle momenta can be extracted in acontrolled way.

The most obvious way to sweep the horizontal tune is by altering the current,and therefore the strength, of the focussing and defocussing quadrupoles. In SPSthe current is changed in an approximately linear way to produce a steady beamextraction matched to the momentum deviation distribution with a duration ofapproximately 4.8 seconds.

This report summarises the effect of imperfections in the current circuit duringthe slow extraction process at the SPS.

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2 The simulation framework

All simulations in this study were carried out with MAD-X. MAD, MethodicalAccelerator Design, is a project developed by the CERN Accelerator BeamPhysics Group and is the standard to describe particle accelerators, simulatebeam dynamics and optimize beam optics. I will outline the procedure followedto obtain the simulation results in this section. Note that all results in thisreport make use of the measured orbit by F. Velotti. For this purpose, a set ofvirtual kickers is added to the SPS sequence.

The basic SPS configuration as defined in cern.ch/eng/sps/2015/ is loaded.Then, the elements are sliced and converted into thin optical elements to fa-cilitate the particle tracking. In the next step, the quadrupole and sextupolestrengths are matched to the horizontal and vertical tune and chromaticity asmeasured in the machine for the central trajectory. Once the SPS is set-upthe extraction bump is turned on. After this is done, the Twiss parametersare written to a file. As a last step, misalignments of certain elements can beadded.

The changing current during the tune sweep is implemented as a change inquadrupole strength in the simulation. Although the linear assumption isgood in the case of a slowly varying tune, it does not include any dampingof the magnetic field caused by the shielding of the vacuum chamber in whichthe beam circulates. Eddy currents in the vacuum chamber act to damp thechange in quadrupole strength. Literature suggests that for increasing frequen-cies, the damping increases rapidly. The relation between the current and thequadrupole field strength could be improved by using a transfer function de-scribing the frequency dependence of the damping, which is beyond the scopeof this study [1].

To study the rate of extracted particles and losses in the ring, I extensively usedthe particle tracking functions in MAD-X. Based on the Twiss parameters, adistribution of initial coordinates for the particles at a fixed point in the SPScan be generated. Figure 2 gives a few examples for simulated beams. Forthis study, the most general case of a Gaussian beam profile with a uniformlydistributed energy spread centred on 400 GeV was used.

The simulation is batched to speed up the calculations and is defined by anumber of input parameters through a python script:

• Number of revolutions

• Number of batches

• Number of particles per batch

• The input function that changes the quadrupole strength each turn

The output of the simulation consists of tracking files which log the character-istics at certain SPS elements during the tracking and losses files which containthe information of the particle at extraction. In the simulation, extraction isdefined as exceeding the aperture at the first ZS septum.

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Initial distributions

Figure 2: Some examples of possible particle distributions in the SPS simula-tions.

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3 Analysis & Results

The goal of this section is to see how sensitive the simulated extraction processis to different distortions of the focussing quadrupole current. To visualise theeffect of different simulation conditions on the output, the same parameters werecompared for every run and are visualised in Figure 3.

Figure 3: Visualization of the output of the simulated slow extraction process.

The first row illustrates the extraction rate. The peak during the first fewhundred turns is expected. The tune is matched to the particle with the lowestmomentum. Particles that are slightly off resonance, but have a high enoughamplitude in phase space, are also extracted.. After these few hundred turns,the extraction rate should be constant in the ideal case.

The second row gives the position in phase space of a particle at the upstreamaperture of the first septum at the point of extraction. This distribution ischaracterised with the emittance:

ε2ZS =< (x− x̄)2 >< (x′ − x̄′)2 > − < (x− x̄)(x′ − x̄′) >2 .

Particles can hit the septum from both sides. Particles that crash on the insideof the septum appear on the right side of the black line because they will passthe upstream aperture and will be stopped at the more stringent downstreamaperture. The yellow region in the plot represents particles that exceeded theupstream aperture of the septum and crash onto the outside of the septum.This fraction is calculated as the sum of three effects:

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Figure 4: Examples of particle trajectories in the electric field of the septum.

• The septum itself has a certain angle towards the centre of the beam, thisis of the order of 4 mm for every of the five 3.13 m septa.

• The particle has an initial angle towards the septum.

• To the extraction side of the septum, there is a homogeneous electric fieldof 200 kV over 20 mm. The deflection due to this field can be calculatedfrom the Lorentz force:

dp

dt= e · ( ~E + ~v × ~B)

In the case considered here, there is only an electric field , in a goodapproximation perpendicular to the particle during the extraction. Therelativistic factor, γ can also be taken out of the derivative since the par-ticle velocity is constant and almost equal to the speed of light, therefore:

mγvxdt

== mγvd tan(θ)

dt= mγv2

d tan(θ)

ds= eE (1)

tan(θ) =eE

γmc2· s. (2)

Where θ is the angle between the incoming direction and the directionafter a travelling a distance z in the field. Since tan(θ) = dx/ds, we find:

x =eE

2γmc2· s2 (3)

Note that γmc2 equals the total beam energy, 400 GeV in our case.

The addition of these three affects on the trajectory is illustrated for a fewrandom start conditions in Figure 4.

The third row in the figure reflects the changing beam intensity during slowextraction. A little more than one percent of the simulated beam survives theresonance and stays in the beam.

In the following subsections, changes will be made to this ideal case. Firstthe influence of the septum thickness and the number of turns on the losses isdiscussed. In the second subsection a ripple is added to the current. The last twosubsections study realistic cases of the quadrupole current imperfections.

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3.1 Linear tune sweep

Figure 5: Comparison of extraction simulated during 34095 turns and 204570turns.

To decrease computation times, only one sixth of the extraction process wassimulated, the effects of this are shown in Figure 5.

The most striking difference is the increased angular spread at the septum. Thiswas expected due to the movement in time of the beam, which is caused by anon-zero dispersion and feed-down effects from the non-zero closed orbit. In thefuture, this effect could be compensated by adjusting the extraction bump overtime during the extraction process. Since this study focusses on relative effects,simulating one sixth of the process is a good approximation.

Table 1 summarizes some cross-checks concerning the losses at the septum inthe case of a 34095 turn simulation. The rate of losses is independent of theextraction peak at the start. On the contrary, the uncertainty on the positionand the thickness of the wire septum are important for the losses. One can alsonote that in this simplified simulation approximately 90% of the particles thathit the septum do it in the first part. These results are all heavily dependenton the exact septum geometry and further studies are required to make firmconclusions.

3.2 Superposing a ripple on the current

The results in this section are obtained by perturbing the linear quadrupolecurrent with a sine ripple. The ripple is characterised by its period and ampli-tude. The natural units in the simulation are number of turns for the periodand quadrupole magnet strength as amplitude. These can be easily converted

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Table 1: Losses at the septa in the case of a linear tune sweep in different cases.

TolerancesLosses at first septum Losses at total septum

Without cut 1000 turns cut Without cut 1000 turns cut

100 µm 3.71% 3.72% 3.99% 4.01%200 µm 4.85% 4.85% 5.15% 5.15%300 µm 6.00% 6.00% 6.33% 6.33%

to the current amplitude if a linear transfer function is assumed. The periodcan also be converted to the corresponding frequency. Some examples of theeffect of the ripple on the extraction are given in Figure 6.

Figure 6: Examples of current ripples, the characteristics are determined by thename: number of turns corresponding to a period, amplitude in units of 10−8 inquadrupole strength and number of thousand particles used in the simulation.

Figure 7 summarises the effect of current ripples on the emittance of the ex-tracted beam. The results may be severely overestimated in the case of higherfrequencies due to the damping in the transfer function between the currentripple and the actual quadrupole strength. Therefore, the magnetic field seenby the beam will be lower and one should rather interpret the results in Figure 7as relative tune ripple.

Figure 8 shows a similar result concerning the losses at the septum in the caseof the linear geometry. A slight tendency that some ripples could lower thelosses is visible. This tendency is probably correlated with the emittance, asis investigated in Figure 9, and is probably dependent on the choice of angularalignment between the beam and ZS in the simulation. Further error studiesare required to confirm this behaviour.

In the case of a linear tune sweep it was mentioned that approximately 90%of the losses were in the first part of the septum, this percentage is not signif-icantly altered by introducing the current ripple, as can be seen in Figure 10.

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∆II

1e 4 emmitance at ZS upstream

1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6Relative emittance (units 2.91e-8)

Figure 7: Interpolation of the effect of quadrupole strength ripples on the emit-tance at the first septum aperture.

The variations in the plot are of the same order of magnitude as the expectedfluctuations introduced by Poisson statistics.

3.3 Effects of glitches on the focusing quadrupole cur-rent

The principal incentive of this study are the occurrences of current glitches in thepower circuit of the focussing quadrupoles. An example from the ELOGBOOKis given in Figure 11.

The current glitches were modelled by a damped sine wave based on a measure-ment sample provided by TE-EPC. As glitches occur in various forms, threedifferent cases were simulated and compared. The focussing quadrupole currentwith the glitch perturbation is visualised in Figure 12.

The effect on the outcome is shown in Figure 13. As expected from studying theripples, a glitch causes the emittance at the septum to blow up and results ina peak in the extracted beam intensity. Nevertheless, concerning the activationof the septum, no significant increase in losses is seen.

3.4 Comparison with quadrupole current data

Even without the glitches, there are still current imperfections on a much smallerscale. To study them, a real current sample was used. After baseline subtrac-

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0 100 200 300 400 500 600 700Hz

0.0

0.2

0.4

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1.0

∆II

1e 4 combined losses at all ZS septa (18.77m)

2.1 2.4 2.7 3.0 3.3 3.6 3.9 4.2 4.5Losses in percent

Figure 8: Interpolation of the effect of quadrupole strength ripples on the lossesat the five septa combined.

tion, a fast Fourier analysis was done to transform the time signal into a powerspectrum. This is presented in Figure 14. The peaks in the right column givesus an idea about the frequencies that are important in the current noise. Peaksaround multiples of 50 are expected due to the frequency of the original alternat-ing current. The 50 Hz, 100 Hz, 150 Hz, and more recently, 300 Hz components,can be suppressed by injecting current on the servo-quads in anti-phase to sta-bilise the extracted beam intensity. To mimic this effect, in the last row of thepicture, these frequencies are subtracted from the spectrum.

To study this effect in the simulation, a part of the ripple was interpolated andadded as a perturbation on the linear current with different levels of compensa-tion. The results are given in Figure 15.

3.5 Effect of misalignment of the first septum

As mentioned before, the thickness, exact position and geometry of the septum isessential to estimate the absolute losses. As an example two different alignmentsof the first 3.13 m septum are compared.

• First case: upper aperture at 67.9 mm, lower aperture at 63.8 mm (angleof -1.31 mrad)

• Second case: upper aperture at 67.7 mm, lower aperture at 63.34 mm (an-gle of -1.39 mrad)

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0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0relative emittance

1.5

2.0

2.5

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ent

loss

es

at

ZS

Figure 9: Scatter plot to visualise the hint that there is a weak negative corre-lation between emittance and losses at the septum.

In both cases, a thickness of 100 µm was used. Results are presented in Figure 16.In the first case, losses are higher, but there are only losses on the extraction sideof the septum. In the second case, a minority of the particles hits the septumon the beam side. Finding a septum geometry corresponding to the optimalbalance between the the losses on the inside and on the outside of the septumwill be the topic of a future project.

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0 100 200 300 400 500 600 700Hz

0.0

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0.4

0.6

0.8

1.0

∆II

1e 4 Fraction of losses at first septum

0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97Fraction of losses at first septum

Figure 10: Interpolation of the fraction of particles that hit the septum in thefirst part (3.13 m), compared to the whole septum of 18.77 m.

Figure 11: Example of the occurrence of a so called current glitch. The whiteline represents the current, the yellow line the beam intensity. The bumps inthe yellow line are caused by glitches in the current. The glitch in the currentis of the order of one thousandth and therefore not visible in this plot.

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0 5000 10000 15000 20000 25000 30000 35000Turns

0.4

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pole

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1e 5+1.464e 2

Damped Sine: ∆II

= 5. 6e− 4, 500 turns period, 500 turns damping

Damped Sine: ∆II

= 2. 8e− 4, 500 turns period, 1000 turns damping

Damped Sine: ∆II

= 2. 8e− 4, 500 turns period, 500 turns damping

Lineair tune sweep

Figure 12: Focussing quadrupole strength during the tune sweep to simulatethe current glitches.

Figure 13: The simulated outcome of different glitches at visualised in Figure 12.The difference between the glitch with the biggest amplitude and the others ismost likely underestimated because the amplitude exceeds the correspondingmomentum spread that was used in the 34095 turns case.

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Figure 14: Analysis of a current sample before it is send to the quadrupoles.In the last row, the 50 Hz, 100 Hz and 150 Hz frequencies are removed to mimicthe compensation process.

Figure 15: Comparison of adding a realistic current ripple to the linear sweep.The results are compared with a linear sweep in the left column. The three othercases differ in the frequencies that are suppressed in the ripple, as indicated inthe title of each column.

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Figure 16: Two different geometries of the septum are compared, a significantreduce in the losses is visible in the right column.

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4 Conclusion

The goal of slow extraction is to deliver a steady extracted beam to certainSPS experiments while minimizing the losses due to particles hitting the ex-traction septum. To obtain this result, a good understanding of the effects ofimperfections on the quadrupole current is essential. This report summarizesthe implications of these imperfections in various cases. It was found that theamount of losses is robust, even when current imperfections are pushed to ex-treme cases. The effect on the extracted beam intensity is more sensitive tothese imperfections, especially in the case of a current glitch. This study wasable to reproduce the measured effects of a real glitch which could help to putlimitations on the tolerable amplitudes and frequencies. In the future this workcould be extended by including a more advanced transfer function between thecurrent and the quadrupole strength. Another aspect that needs further in-vestigation is the effect of the exact septum alignment and thickness on thelosses.

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References

[1] A Dinius, L Jensen, and P Semanaz. Measurement of the transfer function ofthe main SPS Quadrupoles. Technical Report SL-Note-98-047-MD, CERN,Geneva, Jul 1998.

[2] Francesco Velotti, Wolfgang Bartmann, Thomas Bohl, Chiara Bracco, KarelCornelis, Matthew Fraser, Brennan Goddard, Verena Kain, and Linda Stoel.Characterisation of the SPS Slow-extraction Parameters. page THPOR055.4 p, 2016.

[3] Philipp Schicho. Optimizing Simulation Times of SPS Slow Extraction usingMAD-X. Jul 2015.

[4] Werner Herr and F Schmidt. A MAD-X Primer. (CERN-AB-2004-027-ABP):32 p, Jun 2004.

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