The betatron coupling introduced by the experimental solenoids in the LHC is small at injection and negligible at collision energy. We present a study of these effects and look at possible corrections. Additionally we report about the implementation of solenoids in the MAD-X program. A thin solenoid version is also made available for tracking purposes

Burkhardt, H

Koschik, A

Risselada, Thys

Schmidt, F

CERN Document Server

A. Koschik, H. Burkhardt, T. Risselada, F. Schmidt EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsON THE IMPLEMENTATION OF EXPERIMENTAL SOLENOIDS INMAD-X AND THEIR EFFECT ON COUPLING IN THE LHCThe betatron coupling introduced by the experimental solenoids in the LHC is small at injection andnegligible at collision energy. We present a study of these effects and look at possible corrections.Additionally we report about the implementation of solenoids in the MAD-X program. A thin solenoidversion is also made available for tracking purposes.Large Hadron Collider ProjectAbstractLHC Project Report 929CERN,CH-1211 Geneva 23,SwitzerlandGeneva, June 2006CERN, Geneva, SwitzerlandPresented atEPAC'06, Edinburgh, UK, June 26-30, 2006ON THE IMPLEMENTATION OF EXPERIMENTAL SOLENOIDS INMAD-X AND THEIR EFFECT ON COUPLING IN THE LHCA. Koschik, H. Burkhardt, T. Risselada, F. Schmidt, CERN, Geneva, SwitzerlandAbstractThe betatron coupling introduced by the experimentalsolenoids in the LHC is small at injection and negligibleat collision energy. We present a study of these effects andlook at possible corrections. Additionally we report aboutthe implementation of solenoids in the MAD-X program.A thin solenoid version is also made available for trackingpurposes.INTRODUCTIONThe transverse coupling introduced by an experimentalsolenoid in the LHC can be quantified [1] by the complexcoupling coefficient c,c∓sol = −i4πBslBρ(√β∗yβ∗x±√β∗xβ∗y), (1)where Bsl is the integrated solenoid field strength and Bρthe beam rigidity. β∗z=x|y denotes the beta function valueat the center of the solenoid, which is the IP (interactionpoint). The LHC is designed to operate with round beams,hence β∗x = β∗y and the sum coupling resonance (c+sol = 0)is not exited. At injection energy (450 GeV), where theeffect is most pronounced, this coefficient amounts toc−CMS, 450 GeV = −i2πBslBρ= −0.0053 i, (2)for the strongest experimental solenoid (CMS-magnet).This is small and other sources of coupling, in particularthe a2 errors in the main dipoles are expected to give acoupling coefficient which is more than one order of mag-nitude larger. A global coupling correction for the wholemachine is planned [2]. The solenoidal effects are too smallto justify dedicated magnets for local solenoid compensa-tion. Still, it may be desirable to allow to adjust couplingseparately for each solenoid which may be turned on or off.An optics design code such as MADX [3] can be usedto study the coupling effects. Several recent developmentsin MADX concerning solenoids have been done, in par-ticular tracking with solenoids can be performed and theuse of PTC allows to compute the coupled lattice functionsdefined by Ripken [4]. Implementation details regardingsolenoids will be discussed in the last section.Relevant LHC (beam) parameters are summarized inTab. 1, values referring to the experimental solenoids aregiven in Tab. 2. A schematic layout of the IP5 region(CMS) and its optics is depicted in Fig. 1.Table 1: LHC general parametersParameter Inj. Coll.Momentum p [GeV/c] 450 7000Trans. norm. emittance εN [μm rad] 3.5 3.75Horizontal Tune Qx 64.28 64.31Vertical Tune Qy 59.31 59.32Max. β H/V (cell) βmax. [m] 177/180Min. β H/V (cell) βmin. [m] 30/30Average β (= R/Q) 〈β〉 [m] 66/72Max. Dispersion H/V (cell) Dmax. [m] 2.018/0.0RMS beam size IP5 σrms [μm] 375.2 16.7Half crossing angle IP1/IP5 [μrad] ± 160 ± 142.5Half parallel separation IP1/IP5 [mm] ± 2.5 0.0Plane of crossing IP1 verticalPlane of crossing IP5 horizontalβ at IP1/IP5 β∗ [m] 17 0.556100. 6400. 6700. 7000. 7300.Momentum offset =    0.00 %s (m)IP5 (CMS) - LHC 450GeV0.050.100.150.200.250.300.βx(m),βy(m)-0.20.00.20.50.81.01.21.51.82.02.2Dx(m),Dy(m)β x β yDx DyFigure 1: LHC IP5 (CMS) layout and optics.COUPLING CORRECTIONThere are different strategies to compensate coupling inan accelerator. A solenoid modifies the transverse oscilla-tion modes and rotates the beams by an angle θ = Bsl2Bρ . Acompensation should eliminate or minimize these effects.An obvious solution is the use of anti-solenoids left andright of the main solenoids. This is not practical for highenergy machines with very large solenoids like the LHC.Another standard technique is based on the use of skewquadrupoles. This resonance method [1] uses Hamiltonianformalism and treats the coupling fields as a perturbationof the uncoupled optics. The skew quadrupoles are placedand adjusted such that the main resonance coupling termsdisappear.MQSXMQSX XSOLFigure 2: LHC IP5 layout. CMS solenoid (XSOL) and nearby skew quadrupoles (MQSX)LHC AND SOLENOIDSIn the LHC a global coupling correction scheme will beused to minimize the impact of field imperfections and tilterrors on the beam quality [2]. We now look at the effectof the strongest LHC experimental solenoid (CMS, IP5) onbeam optics at 450 GeV.At injection energy (450 GeV) beams will generally beseparated by means of separation bumps. Additionally, thebeams cross from inside to outside and vice versa at eachIP. The solenoid field slightly tilts the crossing plane. Thehalf separation of the two beams is decreased by 15μm,which is less than 1 % of the nominal separation. If noseparation bumps are used, e.g. in an early collision runat 450 GeV, the solenoid will introduce a separation of thisorder. This can easily be corrected by adjusting the separa-tion bumps.Table 2: Parameters of experimental solenoids in LHCProperty/Experiment ATLAS CMSMagnetic induction at IP [T] 2.0 4.0Coil length [m] 5.3 12.5The β-beating which is induced by the CMS solenoid isshown for the LHC machine in Fig. 3. The peak β-beating(Δβ/β)peak = 0.1% is well below and within the ac-cepted margin [5] of 21 %.Equally the dispersion beating is well within the bud-get of 30 %. To analyze we use the normalized dispersionfunction Dx|y,N = Dx|y/√βx|y,ΔDx(s)√βx(s)/Dx,qf√βx,qfand ΔDy(s)√βy(s)/Dy,qd√βy,qd, (3)where Dx,qf = 2.1 m, Dy,qd = 16 cm and βx,qf =βy,qd = 180 m. The peak dispersion beating expressed inthese quantities amounts to Dx,N/Dx,qf,N = 0.2% andDy,N/Dy,qd,N = 2.5%.To summarize, we can say that the impact on beam opticsis small and will hardly be visible in standard operation.We note, that the local solenoid compensation used inLEP with two skew quadrupole pairs relied on the symme-try of the optical functions around the IP [6]. This schemecannot simply be employed for the antisymmetric LHC op-tics.XYs [m]0 5000 10000 15000 20000 25000  [%]yβ/ yβ Δ,  xβ/xβ Δ-0.08-0.06-0.04-0.0200.020.040.060.080.1XYs [m]0 5000 10000 15000 20000 25000  [%]y,qd,N/Dy,N DΔ,  x,qf,N/Dx,N DΔ-2-1012Figure 3: Induced β- and dispersion-beating by the CMSsolenoid in the LHC at 450 GeV.SOLENOIDS AND MAD-XFollowing a request on the last MADX day [7], we im-plemented a thin solenoid in the tracking module as wellas in the twiss module. A thick version was available al-ready, however since the tracking module exclusively oper-ates with thin elements, this became necessary.The implementation follows closely the formulae givenin [8], where the canonical equations of motion are deriveddirectly from the Hamiltonian. We only report the relevantformulae here and note some important facts:• The hard-edge model is used. Fringe fields of thesolenoid are taken into account, however they are fi-nite ’hard-edge’ steps of the B-field;• the solution of the equations of motion are obtainedfrom the expanded Hamiltonian, and it that sense thesolutions or not ’exact’;• and although this is a thin element, both the normal-ized magnetic strength ks and its product ks · L withthe length L have to be known and used in the equa-tions. This makes the solenoid different from e.g. themultipoles.If zi, zf denote initial and final (canonical) coordinates,where z ∈ [x, px, y, py, σ, pσ], we can write the solution ofthe equations of motion in a solenoid of length L = Δs inthe thin-lens approximation asxf = xi · cosΔΘ + yi · sinΔΘ,pfx = pˆfx · cosΔΘ + pˆfy · sinΔΘ,yf = −xi · sinΔΘ + yi · cosΔΘ,pfy = −pˆfx · sinΔΘ + pˆfy · cosΔΘ,σf = σˆf +{xi · pˆfy + yi · pˆfx} · H(s0) ·Δs[1 + f(piσ)]2 · f ′(piσ),pfσ = piσ,(4)where the intermediate variables zˆ are given bypˆfx = pix −xi[1 + f(piσ)]·H(s0)2 ·Δs,pˆfy = piy −yi[1 + f(piσ)]·H(s0)2 ·Δs,σˆf = σi − f′(piσ)[1 + f(piσ)]2 ·H(s0)2 ·Δs ·12{(xi)2 + (yi)2}.(5)The normalized solenoid strength ks = e·Bp0·c is related toH(s0) byH(s0) =12ep0 · c · Bs(0, 0, s0) =12· ks. (6)H(s0) is not the Hamiltonian, but an abbreviation for theexpression defined above, see also [8, (2.23e, p.9)]. Ac-cordingly the quantity ΔΘ is then related to ksi = ks · L(= ks ·Δs to conform to the notation in [8]) and given byH(s0) ·Δs = 12ksi,ΔΘ =H(s0) ·Δs[1 + f(piσ)].(7)RIPKEN [8] uses the function f(pσ) to stress the factthat the relative momentum deviation Δp/p0 depends onthe longitudinal canonical variable pσ. It is given byf(pσ) =√1 + 2pσ + β2p2σ − 1,f ′(pσ) =1 + β2pσ√1 + 2pσ + β2p2σ.(8)There is an important difference in the set of canonicalvariables used in MADX and by RIPKEN [8], but notingthatpσ =pTβ,σ = β · T,(9)where (T, pT ) are the longitudinal variables used inMADX and β is the relativistic quantity of the beam,we can completely integrate the formulae from [8] intoMADX.The transverse transport map can also be written in ma-trix form, where we use K = ks/2.Mthin, sol =2664cos ΔΘ 0 sinΔΘ 00 cos ΔΘ 0 sinΔΘ− sinΔΘ 0 cos ΔΘ 00 − sinΔΘ 0 cos ΔΘ3775| {z }pure rotation·2666641 0 0 0− K2L1+f(piσ)1 0 00 0 1 00 0 − K2L1+f(piσ)0377775| {z }focusing in both planes(10)The last decomposition of the thin-lens solenoid trans-fer matrix shows, that it can be separated into a pure rota-tional part, which refers to the solenoid body without anyedge effects and a focusing part, which give the effect of thefringe fields. It should be also apparent, that slicing a thicksolenoid into several thin ones and replacing the effect ofthe thick solenoid byMthick, sol ≈ MDrift·Mthin, sol·MDrift . . .MDrift·Mthin, sol·MDriftdoes in general not give the correct edge focusing effect.However, it can be shown numerically that such a slicingconverges to the thick lens solution.It should be noted that the Hamiltonian for the solenoidcan also be solved exactly, see e.g. [9], however, the so-lution is not symplectic. Therefore the solution from theexpanded Hamiltonian is used.CONCLUSION AND OUTLOOKSolenoid coupling effects in the LHC are small andglobal compensation at injection should be sufficient. Wediscussed the size of the effects and propose a global com-pensation which allows to correct each solenoid individu-ally. We also reported on the status and implementation ofsolenoids in MADX.ACKNOWLEDGEMENTSThe authors would like to thank J.Jowett and S.Fartoukhfor helpful discussions and gratefully acknowledge valu-able input from all MADX users.REFERENCES[1] G. Guignard, “The general theory of all sum and differenceresonances in a three-dimensional magnetic field in a syn-chrotron”, (CERN-76-06), 23 Mar 1976.[2] J.P. Koutchouk, “Correction of the systematic and randombetatron coupling in LHC with an application to LHC version2”, (SL-Note-94-101-AP; LHC-Note-306), 1 Jan 1994.[3] MAD-X web page, “http://mad.web.cern.ch/mad/”.[4] F. Willeke and G. Ripken, “Methods of beam optics”, AIP184:758–819, 1989, DESY 88-114.[5] S. Fartoukh and O. Bruning, “Field quality specification forthe LHC main dipole magnets”, LHC-Project-Report-501.[6] G. Guignard, “Revised schemes of skew quadrupoles forsolenoid compensation in LEP”, CERN-LEP-70/75, 1978.[7] MAD-X Day, 23 Sep 2005,“http://mad.web.cern.ch/mad/mad-X day.html”.[8] G. Ripken and F. Schmidt, “A symplectic six-dimensionalthin lens formalism for tracking”, CERN-SL-95-12-AP;DESY 95-063.[9] E. Forest, “Beam dynamics: A new attitude and framework”,Amsterdam, Netherlands: Harwood (1998) 463 p.

On The Implementation Of Experimental Solenoids In MAD-X And Their Effect On Coupling In The LHC

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