A highly effective collimation scheme is required in the LHC to limit heating of the vacuum chamber and superconducting magnets by protons either uncaptured at injection or scattered from the collision points. The proposed system would consist of one set of primary collimators followed by three sets of secondary collimators downstream to clean up protons scattered from the primaries. Each set of collimators would consist of four pairs of jaws - horizontal, vertical, and 45 o and 135 o skew. A study is reported of the optimization of the longitudinal positions of these jaws with the aim of minimizing the maximum betatron amplitudes of protons surviving the collimation system. This is performed using an analytical representation of the action of the jaws and is confirmed by tracking. Significant improvement can be obtained by omitting inactive jaws and adding skew jaws

Craddock, M K

Jeanneret, J B

Kaltchev, D I

Servranckx, R V

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsLarge Hadron Collider Project LHC Project Report 37Optimization of Collimator Jaw Locations for the LHCD.I.Kaltchev,z, M.K.Craddock,x, R.V.Servranckx, J.B.Jeannerety,{AbstractA highly eective collimation scheme is required in the LHC to limit heating of the vacuumchamber and superconducting magnets by protons either uncaptured at injection or scatteredby non-linear phenomena. The proposed system would consist of one set of primary collimatorsfollowed by three sets of secondary collimators downstream to clean up protons scattered fromthe primaries. Each set of collimators would consist of four pairs of jaws - horizontal, vertical,and 45and 135skew. A study is reported of the optimization of the longitudinal positions ofthese jaws with the aim of minimizing the maximum betatron amplitudes of protons survivingthe collimation system. This is performed using an analytical representation of the action of thejaws and is conrmed by tracking. Signicant improvement can be obtained by omitting inactivejaws and adding skew jaws.Paper presented at the 5th European Particle Accelerator Conference, EPAC96,Sitges, Spain, 10-14 June 1996.TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T2A3yCERN{SL DivisionzOn leave from INRNE, Soa, Bulgaria, kaltchev@decu10.triumf.caxAlso at Physics Dept, UBC, Vancouver, Canada.{jbj@mail.cern.chAdministrative SecretariatLHC DivisionCERNCH-1211 Geneva 23SwitzerlandGeneva, 29 July 19961 IntroductionEcient collimation in LHC requires a two-stage collimation system: a primary col-limator shaping the beam by limiting the maximum betatron amplitudes and secondarycollimators trimming the secondary particles produced by elastic nuclear and electromag-netic interactions in the primary collimator surfaces (so-called secondary beam halo) [1].The lattice of the IR3 straight section, where the betatronic cleaning will be done, andthe collimator locations must be appropriately chosen to minimize the maximum betatronamplitude of uncaptured halo particles (escaping all secondary collimators).In initial calculations [2], [3] of the maximum extent of the secondary halo theshapes of both primary and secondary collimators were assumed approximately elliptical(circular in normalized transverse coordinates). In reality each collimator will be made ofseveral sets of at jaws. For example, a set of four pairs of jaws { horizontal, vertical, and45and 135skew { clustered at the same longitudinal coordinate form a regular octagonwhich does not deviate much from the inscribed circle.In practice the jaws must be separated longitudinally { an additional degree offreedom which may be utilized to achieve better collimation { a deeper cut into the halo.We describe an algorithm [4] allowing us to nd the exact limits of the secondary haloin such a system of separated primary and secondary jaws, distributed along an arbitrarylattice. The code also provides automatic minimization of the maximum secondary haloamplitude.2 Collimator design code DJ (Distribution of Jaws)2.1 General DescriptionThe approximations used are the same as in [2]:{ the primary jaws are assumed \pure scatterers"- scattered particles are producedalong the line dening the boundary of the jaw in the transverse plane.{ the secondary jaws are assumed \black absorbers" - if a particle touches a secondaryjaw it is considered lost.The geometric representation of a pair of opposing jaws (POJ) in normalized transversecoordinates (X;Y ) at longitudinal position s is a pair of parallel lines:jX cosk+ Y sinkj = n: (1)Here the angle kbetween the POJ and the Y axis and the aperture n (in units ofr.m.s. beam size) at which the POJ is set, take discrete sets of valuesn =(6 for primary POJ7 for secondary POJk= (k   1)=N; k = 1; 2; 3; : : : ; N: (2)For N = 4 (2) represents vertical (k=1), horizontal (k=3) and skew POJ (k=2,4); usinglarge N allowed us to describe circular collimators and reproduce the results given in [2].As input, DJ takes an initial distribution of POJ from a jaw-position table, containingfor each POJ the horizontal betatron phase advance ( x) corresponding to its positionin the lattice and its type (primary or secondary and angle k). The user also provides atable of IR3 lattice functions in MAD OPTICS, or DIMAD-output format.DJ performs several kinds of calculations:11. for a xed jaw-position table it nds the maximum values of the amplitudes {horizontal (Ax=pX2+X02), vertical (Ay=pY2+ Y02) and combined (A =qA2x+A2y) of halo particles escaping all secondary POJ;2. it minimizes the maximum combined amplitude Amaxby distributing the secondaryPOJ longitudinally, thus creating new jaw-position tables;3. it tracks particles starting from the primary POJ to nd the amplitude distributionof the secondary halo.2.2 Basic algorithm { nding the maximum halo amplitude for a givenlongitudinal distribution of jawsWe consider particles generated at the points P = (X;Y ) on the perimeter of theoctagon dened by the jaws of the primary collimator (see Figure 1) and seek to determinethe range of initial angles in the X0Y0plane that survives the secondary collimators.For this purpose, each secondary collimator jaw is imaged in the plane of the primarycollimator. The particles escaping all secondary jaws lie inside a polygon in the X0Y0plane as depicted in Fig. 1 (down). Jaws whose lines lie outside the polygon are inactive.For each point P of the octagon, the program computes Ax, Ayand A for each vertex ofthe polygon and chooses the maximum values^Ax,^Ayand^A associated with one of thevertices.Finally:1. the maximum halo amplitude is determined:Amax= maxP^A ; P 2 all primary POJ ;2. the vertex and the secondary jaws associated with Amaxare identied.As the point P = (X,Y) moves along each side of the octagon the point (^Ax;^Ay)describes the limits of the secondary halo in the amplitude plane (Figure 2).Figure 1: (a) - Normalized coordinate space (above) and angle space (below) at the longitudinalposition sprimaryof the primary POJ; (b) M secondary POJ. TPis a linear mapping (origin shiftplus scaling). For each point P on the primary POJ: 1) each pair of parallel lines (stripe) incoordinate space is mapped into a stripe in angle space; 2) the overlap region of all stripes formsa polygon (shaded).22.3 Minimization of AmaxAmaxis minimized by changing the longitudinal location of pairs of secondary jaws,thus changing the position and orientation of the associated lines in the X0Y0plane.Although the Simplex and Newton methods minimized Amaxsuccessfully the al-gorithm nally chosen for DJ involves only the two maximum-amplitude POJ at eachiteration:1) one of the two maximum-amplitude POJ is shifted by a step ds in the appropriatedirection to decrease Amax. Note that this may change the two maximum-amplitude POJthemselves.2) if no decrease of Amaxis achieved after all possible combinations are tried, thestep ds is halved.Figure 2: Secondary halo images in the amplitude plane before (left) and after (right) mini-mization for the lattice of IR3 in [1]The procedure converges (asymptotically) to a POJ distribution with lower Amax(Fig. 2).Minimization is done in several stages: at each stage a new primary POJ and suf-cient number of secondary POJs are added so as to decrease Amaxto some low targetvalue (for example 8). Secondary POJ which are inactive (i.e. do not change Amax) areremoved after each stage.2.4 TrackingA large number of particles is generated with initial coordinates taken from thesame set of points P used in the mapping calculations described above. For each point Pthe initial angles are uniformly distributed within a cone, which should be a pessimisticassumption.3 Applications to IR3 of LHCA preliminary study has been made of several IR3 lattices and the following featureswere found favourable:{ varying tune split x- y{ high phase advance.The following results have been obtained for IR3 lattice [3] with tune advance 2.2across the insertion.If the primary jaws are all at the same location (x= y= 0), then the sec-ondary POJ positions after minimization are not far from the three optimum phases(opt, =2 and    opt, where opt= arccos(6=7)) predicted by the circular-collimatormodel for a lattice with equal phase advances x= y. This is because 1) a regular3octagon does not deviate much from its inscribed circle; 2) xand ydo not dier bymore than 0.2 anywhere in the collimation section (due to the high beta values). Startingfrom Amax= 9.85 for three octagonal secondary collimators located at the theoreticallyoptimum positions, optimization using DJ reduces Amaxto 9.5 (Fig. 3).Given a long enough system and a sucient number of secondaries, it should bepossible to bring the maximum extent of the halo to its theoretical limit Amax= 7. Using16 instead of 12 secondary POJ we were able to obtain Amax= 8.4 if the primary POJ werefree to move, and 8.6 if they were restricted to locations of maximum beta (Figure 4). Tond the optimum number of secondary POJ of each type the iterative process explainedin the previous section was used.Figure 3: DJ minimization result for primary jaws (thicker lines) all at the same location:(above) distribution of the 12 secondary jaws, represented by the uprights of the H; (left) haloimages and (right) amplitude distribution of the surviving secondary halo obtained by tracking.Figure 4: DJ minimization result for distributed primary jaws (thicker lines): (above) distri-bution of the 16 secondary jaws, (left) halo images and (right) amplitude distribution of thesurviving secondary halo obtained by tracking.It will be noted that the optimized distribution of the 16 POJ is very dierentfrom that in the previous case (Fig. 3), where the sets of four dierently oriented POJwere clustered near the theoretical optimum positions for circular collimators and no tune4split. Here 6 sets of skew POJ are required, with 45and 135POJ always located closetogether, while only 2 sets of horizontal and vertical POJ seem to be needed, and at quiteseparate locations. The basis for this distribution remains to be determined.Other questions to be studied include how the collimation eciency is aected bychanges in tune or POJ location. This implies accurate simulation of scattering in thejaws coupled with tracking around the ring and will require other computer codes.The optimised use of 16 separate pairs of secondary jaws instead of the three 8-jawtanks proposed in [1] oers an improvement in the maximum amplitude of the secondaryhalo by A=1.5 . This substantial gain must be compared to the eective secondaryaperture of 10 expected at injection into the LHC. Further studies, looking for opticsoering the best maximum amplitudes, might oer even better results with the samenumber of pairs of jaws.References[1] The Large Hadron Collider, Conceptual Design, CERN/AC/95-05(LHC),1995.[2] T.Trenkler, J.Jeanneret, The principles of two stage betatron and momentum collima-tion in particle accelerators, Particle Accelerators, 50, 287 (1995); see also the paperscited there.[3] T.Risselada, Optical Requirements for an LHC Cleaning insertion with Elliptical Col-limators, SL Note 95-67, 1995.[4] D.Kaltchev, Collimator Design Code DJ (Distribution of Jaws), TRIUMF Note, 1996(in preparation).5

Optimization of Collimator Jaw Locations for the LHC

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