One of the main challenges when designing final focus systems of particle accelerators is maximising the beam stay clear in the strong quadrupole magnets of the inner triplet. Moreover it is desirable to keep the quadrupoles in the inner triplet as short as possible for space and costs reasons but also to reduce chromaticity and simplify corrections schemes. An algorithm that explores the triplet parameter space to optimise both these aspects was written. It uses thin lenses as a first approximation for a broad parameter scan and MADX for more precise calculations. The thin lens algorithm is significantly faster than a full scan using MADX and relatively precise at indicating the approximate area where the optimum solution lies

Abelleira, Jose

Cruz Alaniz, Emilia

Seryi, Andrei

Van Riesen-Haupt, Leon

Abelleira Fernandez, Jose L.

Oxford University Research Archive

A CODE FOR OPTIMISING TRIPLET LAYOUT ∗L. van Riesen-Haupt, A. Seryi, J. L. Abelleira, E. Cruz-Alaniz JohnAdams Institute, University of Oxford, OxfordAbstractOne of the main challenges when designing final focussystems of particle accelerators is maximising the beam stayclear in the strong quadrupole magnets of the inner triplet.Moreover it is desirable to keep the quadrupoles in the in-ner triplet as short as possible for space and costs reasonsbut also to reduce chromaticity and simplify correctionsschemes. An algorithm that explores the triplet parameterspace to optimise both these aspects was written. It usesthin lenses as a first approximation for a broad parameterscan and MADX for more precise calculations. The thinlens algorithm is significantly faster than a full scan usingMADX and relatively precise at indicating the approximatearea where the optimum solution lies.INTRODUCTIONIn order to maximise the luminosity of a particle collider,it is essential to minimise the beam size at the interactionpoint (IP) by focusing it using quadrupoles. For high energyproton colliders, the beams have a large rigidity, meaningthat high gradient quadrupoles are required for the focusing.The strength of these quadrupoles is limited by both magnettechnology as well as aperture available to the beam or beamstay clear (BSC), which is normally measured in standarddeviations, σ, from the peak of the particle distribution.On top of this, the total length of the focusing section isalso a major constraint. Not just for cost and space reasons ina circular collider but also because the longer off-momentumparticles are exposed to strong quadrupole fields the morethey tend to disperse from the main beam, thereby increas-ing the chromaticity. Further constraints include, amongothers, how much the beam deviates from the reference or-bit; the space, shielding and magnetic fields required bythe experiments in the interaction regions; as well as beam-beam effects that can also be manipulated to increase theluminosity.The goal of this research was to come up with an algo-rithm that scans a large parameter space to determine a layoutfor an triplet that best fulfill the constraints outlined above.Normally aperture simulations are computationally demand-ing so this algorithm uses the thin lens approximations toquickly and efficiently scan through a large parameter spaceand identify where more detailed scans should take place.This idea could be eventually be applied to a wider range ofcolliders.∗ Work supported by The European Circular Energy-Frontier Collider Study(EuroCirCol), EU’s Horizon 2020 grant No 654305.FAST THIN LENS CALCULATIONSSimplified Lens SolutionThe triplet can be modeled by two common transfer ma-trices – the one for a drift D of length l and the matrix of athin lens quadrupole Q of focal strength g = kL whereD =(1 l0 1), Q =(1 0g 1). (1)Thin lenses have been previously used in other algorithmsto approximate the quadrupoles and drifts required for a finalfocus telescope with set transfer properties [1]. One of themain methods used for this was developed by Alex Chao [2].However, this approach was different as it simplifies thesystem by realising that the exact magnifications of the finalfocus are irrelevant – the key property of an triplet is tofocus a particle originating at x = 0 with and arbitrary x ′ faraway from the triplet to x = 0 at the interaction point. Thisproperty is illustrated in Fig. 1.Figure 1: Schematic Layout of Triplet Modeled as ThinLensesBy considering the overall transfer matrix and initial andfinal canonical positions and momenta,(0x ′IP)=(m11 m12m21 m22) (0x ′exit), (2)one can derive that this property can be summarised asmx12 = my12 = 0. Matrix multiplication and analyticalmethods were used to derive formulas giving the strength ofg1 and g2 to meet this requirement for fixed lengths and g3.Figure of MeritApart from simplifying the triplet solution, one key ad-vantage of the thin lens matrices is that it allows one to trackthe position of particles in every lens. In order to track a 1σparticle, one can set its initial position to be xIP = 0 andx ′IP =√ε/β∗, where ε is the beam emittance and β∗ is the βfunction at the IP. This position can then be used to estimatethe BSC, N , in each quadrupole usingN =rσ=rx. (3)Proceedings of IPAC2017, Copenhagen, Denmark TUPVA04301 Circular and Linear CollidersA01 Hadron CollidersISBN 978-3-95450-182-32163 Copyright©2017CC-BY-3.0andbytherespectiveauthorsWhere x is the position in each lens and r is the radius ofthe magnet aperture. The radius of the aperture depends onthe field gradient k required by this lens and is limited bythe maximum magnetic field, ΔB that can be achieved withthe available technology and the finite length of the realisticquadrupole, lq . A simple approximation for the radius isr =ΔBlqgBρ, (4)where Bρ is the beam rigidity and g is the focal strength.Next one can substitute Eq. 4 into 3 and ignore to the con-stants to get a figure of merit (FOM) proportional to theBSC, which we will define asFOM =lqgx. (5)FOM ScansAll of the above can be brought together to create a codethat varies g3 and the drift lengths between the lenses anduses the thin lens solution to work out the strengths of theother two lenses. For each layout it can then work out thetrajectory of a 1σ particle in the horizontal and vertical plane.Next lq for each quadrupole can be computed from the spaceavailable between the lenses.Using this, one can work out the FOM from Eq. 5 for eachconfiguration to get an estimate of the beam stay clear ateach configuration. Since this entire process is done usingsimple mathematical operations, one can work out the FOMfor a large amount of parameters in a very short time.PRECISE SCANSIn order to compute the accurate BSC of a triplet,one needs to trace the beam size through realistic finitequadrupoles. This can be done using simulation programssuch as MADX [3] and is computationally much more in-tensive than the thin lens FOM. Therefore, this scan can belimited to a small range around the optimum point identifiedby the thin lens FOM.MADX Aperture ModuleThe first step required in the finite lens analysis is to workout the length of the individual quadrupoles. Next, the exactstrengths of the finite quadrupoles have to be computed. Thethin lens strength g = kL can be used as a first approxima-tion, however the MADX matching module has to be usedto work out the exact strengths.In order to test the accuracy of the analytical method, aFOM scan was performed to find a maximum FOM triplet.Next PyMADX [4] was used to write a script to comparethe predicted and matched strengths of the quadrupoles intriplets with the spatial configuration of the maximum FOMtriplet with different outer quadrupole strengths. The scriptwrote and executed MADX files of triplets, matched themand recorded the strengths of the quadrupoles. The matchedstrengths and predicted strengths were then plotted and areshown in Fig. 2.Figure 2: Predicted and Matched Quadrupole Strengthsfor Maximum FOM Configurations with Different StrengthOuter Quadrupole.From Fig. 2 it can be seen that whilst there is a dis-crepancy between the actual and predicted strengths of thequadrupoles, these two quantities follow similar patterns.Next, the new strength can be used to work out the availablecoil aperture using Eq. 4 and the MADX aperture modulecan work out the beam stay clear at every point, returning theminimum. An example plot for the maximum FOM tripletwas produced using the aperture module is shown in Fig. 3.Figure 3: Example Plot Showing β Functions and BeamStay Clear for Maximum FOM Triplet.Comparison of Thin and Precise ScanAgain, PyMADX was used to write a python scriptthat iterated through several different triplet configurations.The script produced MADX scripts for each configuration,matched them and ran the aperture module. The minimumbeam stay clear was recorded for each configuration andis plotted in Fig. 4. For comparison, the FOM was calcu-lated for each of these configurations and is also shown inTUPVA043 Proceedings of IPAC2017, Copenhagen, DenmarkISBN 978-3-95450-182-32164Copyright©2017CC-BY-3.0andbytherespectiveauthors01 Circular and Linear CollidersA01 Hadron CollidersFig. 4. A low resolution for this was chosen since the MADXmatching and aperture calculations take long to compute.Figure 4: FOM from Thin Lens (Left) and BSC (right) forDifferent Configurations of Triplet - Yellow Corresponds toa Large FOM or BSC Whilst Blue Corresponds to a LowOne.From Fig. 4 one can see that whilst the maximum pointsidentified by the FOM and precise scans are not found at thesame exact points in parameter space, the thin lens FOMprovides a good approximation of the general behaviour ofthe BSC. Moreover, the two points are not very far apart andone can select a sensible margin around the FOM maximumfor where precise scans could be performed. An exampleof such a margin is shown by the region surrounded by ared box in both plots. One can optimise the size of thesemargins by monitoring the BSC scans and adjusting them ifneeded.APPLICATIONSFCC-hhThe triplet optimisation code was initially written withthe aim of optimising an alternative final focus of the FutureCircular hadron-hadron Collider (FCC-hh) [5]. The uniqueproblem of the FCC-hh is that large amount of collisiondebris requires the triplet quadrupoles to have a significantamount of shielding. Therefore, the shielding was addedwhen performing the precise aperture scans.The code was further modified to constrain the lengthof the first and third quadrupoles to be identical and letall three quadrupoles have the same coil aperture to easemanufacturing. Moreover, the aperture radius was workedout using an empirical formula based on data presented bythe FCC magnet team [6]. The code was used to iterate overtriplet of increasing lengths, finding the first one that offereda BSC of 15.5σ. An outline of this algorithm is shown inFig. 5.It is very important to have just enough shielding to protectthe triplet from debris but not unnecessarily limit the BSC.Therefore, once an adequate triplet was found using thealgorithm, radiation studies were performed to see if moreor less shielding would be required [7]. If the shielding hadto be changed, this was done and the algorithm would be runagain. This was repeated until a short triplet with enoughBSC and shielding was found.Figure 5: Outline of Triplet Optimisation Code used for anAlternative Triplet in FCC-hh.Future Application and VariationsBy modifying the optimisation algorithm in a similar wayas for the FCC-hh, one could use it to optimise the triplet ofother high energy proton machines such as the High EnergyLHC. Depending on the stage and nature of the acceleratordesign one can add or relax constrains.The code could also be changed to optimise other proper-ties such as the dynamic aperture instead or on top of BSC.In this case the magnets could be simplified to thin lens again.If that representation would not be sufficient, the thin lenssolution can be used as a good starting point for matchingfinite lenses.CONCLUSIONThe optimisation of triplets was tackled by approximat-ing the quadrupoles as thin lenses and simplifying the focalproperties of an triplet. This made it possible to analyti-cally work out the strengths for a thin lens triplet, whichcan also be used as a starting point when matching finitelenses. A thin lens FOM was developed that shows a similarbehaviour as more precise scans. This allows one to greatlydecrease the parameter space range over which precise scansshould be performed in order to find an optimum triplet.This method has been applied to the FCC-hh and possiblefurther application have also been discussed.Proceedings of IPAC2017, Copenhagen, Denmark TUPVA04301 Circular and Linear CollidersA01 Hadron CollidersISBN 978-3-95450-182-32165 Copyright©2017CC-BY-3.0andbytherespectiveauthorsREFERENCES[1] B. W. Zotter, “Design Considerations for a Chromatically Cor-rected Final Focus System for TeV Colliders”, CLIC-Note-64(1988).[2] Y. Chao, et.al., “Solution of a Three-thin-lens system with arbi-trary transfer properties”, SLAC, Stanford University, Stanford,USA, October 1992, SLAC-PUB-5834 (1992).[3] “Methodical Accelerator Design - X”, http://cern.ch/madx.[4] L. Nevay, S. Boogert, “pymadx”,http://www.pp.rhul.ac.uk/bdsim/manual/pymadx.html[5] L. van Riesen-Haupt, “Exploring the Triplet ParameterSpace to Optimise the Final Focus of the FCC-hh”, pre-sented at IPAC’17, Copenhagen, Denmark, May 2017, paperTUPVA041, this conference.[6] “FCC-hh magnet-beam dynamics coordination meeting 01”CERN, Genevan Switzerland (2017).[7] J. L. Abelleira, L. van Riesen-Haupt, “Energy Deposition Stud-ies for the new Compact FCC-hh Final-Focus System”, pre-sented at IPAC’17, Copenhagen, Denmark, May 2017, paperTUPVA037, this conference.TUPVA043 Proceedings of IPAC2017, Copenhagen, DenmarkISBN 978-3-95450-182-32166Copyright©2017CC-BY-3.0andbytherespectiveauthors01 Circular and Linear CollidersA01 Hadron Colliders

A code for optimising triplet layout

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