The 3 km long TI 8 transfer line is used to transfer 450 GeV proton and ion beams from the SPS to LHC collider. As part of a detailed optics investigation program the chromaticity of the transfer line was measured. Kick response data of the transfer line was recorded for various extraction energy offsets in the SPS. The quadrupolar and sextupolar field errors (b2 and b3, respectively) over the whole transfer line dipoles, a systematic error of the main quadrupole strengths and the initial momentum error were estimated by a fit. Using the updated model, the chromaticity of the line was then calculated

Fartoukh, S

Fuchsberger, K

Goddard, B

Jones, R

Kain, V

Meddahi, M

Mertens, V

Wenninger, J

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DETERMINATION OF THE CHROMATICITY OF THE TI 8 TRANSFERLINE BASED ON KICK RESPONSE MEASUREMENTSK. Fuchsberger, S. Fartoukh, B. Goddard, R. Jones, V. Kain, M. Meddahi,V. Mertens, J. Wenninger, CERN, Geneva, SwitzerlandAbstractThe 3 km long TI 8 transfer line is used to transfer 450GeV proton and ion beams from the SPS to LHC collider.As part of a detailed optics investigation program the chro-maticity of the transfer line was measured. Kick responsedata of the transfer line was recorded for various extractionenergy offsets in the SPS. The quadrupolar and sextupolarfield errors (b2 and b3, respectively) over the whole trans-fer line dipoles, a systematic error of the main quadrupolestrengths and the initial momentum error were estimatedby a fit. Using the updated model, the chromaticity of theline was then calculated.MEASUREMENTSKick response measurements in 2006 revealed a visiblephase error in the vertical plane in the transfer line. Byfitting the strengths of the two main quadrupole families arelatively large detuning (6.5 permill) of the main verticalfocussing quadrupole strength was found, while the hori-zontal focussing quadrupole strength was in good consis-tency with the model (error of about 0.6 permill) [1]. Later,during the injection tests in 2008 [2], a strong dispersionmismatch with the onset around the junction between TI 8and the LHC was observed, as shown in Fig. 1.Figure 1: Horizontal dispersion of TI 8 and LHC sector 78.Bars represent the measured dispersion and the line respre-sent the dispersion calculated from the nominal model.Many possible sources for these inconsistencies were in-vestigated [3]. As part of the TI 8 optics investigation pro-gram also the determination of the quadrupolar and sex-tupolar field errors of the main bends in the line was sched-uled for further studies.The measurement procedure consisted of recording kick-response data for 4 correctors in each plane. The correctorswere excited by ±40μrad with respect to their nominalsetting. The measurement was repeated for 7 different val-ues of the initial energy offset δpp at the exit of the SPS (-2,-1, -0.5, 0.0, +0.5, +1 and +2 permill).Already from the raw measurement data it is clear thatthe horizontal response depends strongly on δpp (Fig. 2),while the vertical response does not (Fig. 3).(a) δpp= −0.002 (b) δpp= +0.002Figure 2: Horizontal response for one horizontal corrector.Bars represent the measured data, dots the nominal model.A clear dependence on δpp is visible, which is only partlyreproduced by the model.(a) δpp= −0.002 (b) δpp= +0.002Figure 3: Vertical response for one vertical corrector. Barsrepresent the measured data, dots the nominal model. Al-most no difference in the measured data is visible, whilethere should be a dependence on δpp according to the model.ANALYSIS PRINCIPLEThe data analysis for this paper was done using theAloha (Another linear optics helper application) software[4]. This software is a JAVA-reimplementation of the wellknown principle, initially implemented by James Safranekin the LOCO software [5]. The reimplementation led toa software which was available online in the control roomduring the LHC injection tests and commissioning. Alohais able to fit kick-response measurements and dispersionmeasurements. It has interactive access to the MADXmodel of the machine via a slim JAVA-API for MADX.Therefore a large variety of possible fit parameters is avail-able.Proceedings of PAC09, Vancouver, BC, Canada TH6PFP038Beam Dynamics and Electromagnetic FieldsD01 - Beam Optics - Lattices, Correction Schemes, Transport 3787Fitting procedureA fit in Aloha varies a set of Nf parameters cl,1 ≤ l ≤ Nf , of the model such that the differences betweenthe measured and calculated observables are minimized.Concerning the analysis in this paper the observables arethe elements of the response matrix R, whose elements aredefined byRij =uiδj, (1)where ui is the position (x or y) at the ith monitor and δj isthe kick of the jth corrector. The difference between mea-surement and model is expressed by the difference vectorV whose elements are defined byVk = Rdiﬀij :=Rmeasij −Rmodelijσi. (2)Rmodel is the response matrix calculated from the modeland Rmeas is the measured one. σi denotes the rms noiseof the ith BPM. The fit minimizes the quadratic norm of V .To approach the minimal solution the problem is lin-earized. As a first step the system of linear equationsSΔc = V (3)is solved in a least square sense to obtain a vector c repre-senting the parameter changes. This is achieved with leastsquare algorithms like SVD or MICADO. The sensity ma-trix S is defined bySkl = −∂Vk∂cl=1σi∂Rmodelij∂cl. (4)These sensity-matrix elements are numerically approxi-mated by the local fit gradient,Skl =Rmodelij (cl + δcl)−Rmodelij (cl)δcl σi=ΔRij(cl)σi. (5)The parameter increment δcl has to be chosen carefully foreach parameter. After determining Δc from Eq. (3) a newset of parameter-values can be calculated,cl′ = cl + Δcl. (6)The values cl ′ are then applied to the model and the proce-dure is iterated until the parameters are stable.To estimate the quality of a fit the rms of the differenceof the response-matrix elements with respect to the monitornoise is used:Δrms =√1N − 1∑i,jRdiﬀij δj . (7)Rdiﬀij is defined in Eq. (2), δj is the kick of corrector j, Nis the number of all valid elements of the response-matrix.This value will usually be larger than one and will convergeto one when the fit error is of the same magnitude as theBPM noise. This would correspond to a perfect fit in oursense.MODEL OF ΔppDEPENDENCEThe influence of systematic quadrupolar and sextupolarfield errors in the main bends of the transfer line on theresponse matrix can be expressed byΔRij = Aij(ΔKK− Δpp)+ Bijb2 + Cijb3Δpp. (8)ΔKK denotes a systematic error of the main quadrupolestrengths with respect to the nominal settings. b2 and b3denote the systematic relative quadrupolar and sextupolarfield errors in units of 10−4 with respect to the main fieldof the bend. These were implemented directly in theMADXmodel of the transfer line by the use ofkn−1 =bnRn−1refαl10−4(n− 1)!, (9)where kn denote the multipole strengths applied to themain bends inMADX, α is the bending angle of the magnet,l its length and Rref is a reference radius which is definedfor the main bends in the transfer line as Rref = 0.025 m.The “true” momentum mismatch Δpp is given byΔpp=δpp+Δp0p, (10)where δpp denotes the trimmed momentum offset andΔp0pan a priori unknown initial momentum error.There are four degrees of freedom which have to be de-termined by the fit algorithm: Δp0p ,ΔKK , b2 and b3. The fac-tors Aij , Bij and Cij in Eq. (8) correspond to the fit gradi-ents which are calculated implicitely from Eq. (5). Bij andCij act with the same sign as Aij in the horizontal planeand with the opposite sign than Aij in the vertical one.FIT RESULTSA special fit method was implemented to combine themeasurements with different momentum offsets δpp intoone sensity matrix. This allowed to fit for all four parame-ters simultaneously, converging after 6 iterations tob2 = 0.85, b3 = −5.06,Δp0p= 7.51× 10−4, ΔKK= 5.61× 10−3. (11)The fit error as defined by Eq. (7) results in Δrms = 1.99,which is small compared to the initial values between 5 and8 for the different measurements.The b3 was very well reproduced by various fit variantswhile the other three parameters can only be determinedup to a constant ε since they are not completely decoupled.The fit procedure is insensitive to the transformationΔp0p→ Δp0p+ ε,ΔKK→ ΔKK− ε,b2 → b2 + 2b3Rref〈Dx〉 ε, b3 → b3.(12)TH6PFP038 Proceedings of PAC09, Vancouver, BC, Canada3788Beam Dynamics and Electromagnetic FieldsD01 - Beam Optics - Lattices, Correction Schemes, TransportHere 〈Dx〉 denotes the average horizontal dispersion overall the bends with errors.Applying the values of Eq. (11) to the model results in avery good agreement between model and measurement asdemonstrated in Fig. 4 and Fig. 5. The effect on the disper-sion is not evident and has to be verified by further mea-surements.(a) δpp= −0.002 (b) δpp= +0.002Figure 4: Horizontal response for the same corrector as inFig. 2 with the values of Eq. (11) applied to the model. Barsrepresent the measured data, dots the updated model.(a) δpp= −0.002 (b) δpp= +0.002Figure 5: Vertical response for the same corrector as inFig. 3 with the values of Eq. (11) applied to the model. Barsrepresent the measured data, dots the updated model.CHROMATIC BEHAVIOURFigure 6 shows the dependence of the phase on δpp atthe TI 8-LHC junction for the nominal model. The sloperepresents the natural chromaticity of the line which issimilar for both planes and can be read from the plot asQ′x = −13.71 and Q′y = −14.74.Figure 6: Chromaticity derived from the nominal model.The influence of the b2 and b3 on the horizontal and ver-tical phases μx and μy is given byΔμx = Q′x(Δpp− ΔKK)+ A2xb2 + A3xb3Δpp, (13a)Δμy = Q′y(Δpp− ΔKK)−A2yb2 −A3yb3 Δpp. (13b)A2x, A2y , A3x and A3y are positive factors which are cal-culated from the model. Therefore b2 and b3 act on the twoplanes with the opposite signs while the natural chromatic-ities (Q′x, Q′y), which are negative quantities, andΔKK acton both planes with the same sign.Figure 7 shows the dependence of the phase for themodel with the values of Eq. (11). The b3 in the line ap-proximately compensates the vertical natural chromaticityand results in a total vertical chromaticity of ξy = +3.28and total horizontal chromaticity of ξx = −34.11 which isabout twice the natural one. This explains the phase behav-ior already qualitatively demonstrated by Fig. 2 and Fig. 3.Figure 7: The chromaticity derived from the model withthe values of Eq. (11).SUMMARY AND OUTLOOKAsymmetrical phase dependence on δpp was qualitativelyobserved for the two transverse planes in the TI 8 transferline. By fits to several off-momentum kick response mea-surements clearly an unexpectedly high systematic sex-tupolar error (b3) was identified (about 6 units of 10−4 atRref = 25 mm). From the resulting model the chromaticityof the line was extracted which explains very well the δppdependence of the measured data. The natural chromatic-ity of the line is approximately compensated by the b3 inthe vertical plane and doubled in the horizontal plane. It isforeseen to repeat these off-momentum measurements withdifferent time constants related to the ramp to check the na-ture of this b3 (geometric or dynamic).REFERENCES[1] J. Wenninger, “Study of the TI 8 optics and beam stabilitybased on beam trajectories”, CERN AB-Note-2006-021 OP.[2] M. Lamont et al., “the LHC Injection Tests”, LHC Perfor-mance Note 001.[3] I. Agapov et al., “TI 8 transfer line optics studies”, LHC Per-formance Note 004.[4] K. Fuchsberger, “Aloha - Optics studies by combined kick-response and dispersion fits”, CERN BE-Note-2009-020 OP.[5] J. Safranek, “Experimental Determination of Storage RingOptics Using Orbit Response Measurements”, Nucl. Instr.Meth. A, 388, 1997, p 27-36.Proceedings of PAC09, Vancouver, BC, Canada TH6PFP038Beam Dynamics and Electromagnetic FieldsD01 - Beam Optics - Lattices, Correction Schemes, Transport 3789

Determination of the Chromaticity of the TI 8 Transfer Line Based on Kick Response Measurements

Determination of the Chromaticity of the TI 8 Transfer Line Based on Kick Response Measurements

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