Nonlinear fields arising from eddy currents in the vac-uum chamber and remanent fields in the magnets of the CERN SPS vary with time and with the acceleration cycle. We describe a procedure of constructing a nonlinear op-tics model for the SPS, by considering sextupolar, octupo-lar, and decapolar field errors in the dipole and quadrupole magnets, respectively, whose strengths are adjusted so as to best reproduce the measured nonlinear chromaticities up to third order in the momentum deviation. Applying this procedure to SPS chromaticity measurements taken at 26 GeV/c, we have obtained a refined optics model. The tune shifts with the transverse amplitude predicted by this model are consistent with a direct measurement

Arduini, Gianluigi

Collier, Paul

Faus-Golfe, A

Zimmermann, Frank

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN-SL DIVISIONCERN-SL-2002- 036 (AP)Modelling Nonlinear Optics in the CERN SPSF. Zimmermann, G. Arduini,  A. Faus-Golfe*, P. Collier  Nonlinear fields arising from eddy currents in the vac-uum chamber and remanentfields in the magnets of the CERN SPS vary with time and with the accelerationcycle.  We describe a procedure of constructing a nonlinear op-tics model for theSPS, by considering sextupolar, octupo-lar, and decapolar field errors in the dipoleand quadrupole magnets, respectively, whose strengths are adjusted so as to bestreproduce the measured nonlinear chromaticities up to third order in themomentum deviation. Applying this procedure to SPS chromaticity measurementstaken at 26 GeV/c, we have obtained a refined optics model. The tune shifts withthe transverse amplitude predicted by this model are consistent with a directmeasurement.8th European Particle Accelerator Conference, 3-7 June 2002Paris, FranceGeneva, Switzerland21 June, 2002SL/DIV Reps*IFIC, Valencia, Spain;Modelling Nonlinear Optics in the CERN SPSA. Faus-Golfe, IFIC, Valencia, Spain;G. Arduini, P. Collier, F. Zimmermann, CERN, Geneva, SwitzerlandAbstractNonlinear fields arising from eddy currents in the vac-uum chamber and remanent fields in the magnets of theCERN SPS vary with time and with the acceleration cycle.We describe a procedure of constructing a nonlinear op-tics model for the SPS, by considering sextupolar, octupo-lar, and decapolar field errors in the dipole and quadrupolemagnets, respectively, whose strengths are adjusted so asto best reproduce the measured nonlinear chromaticities upto third order in the momentum deviation. Applying thisprocedure to SPS chromaticity measurements taken at 26GeV/c, we have obtained a refined optics model. The tuneshifts with the transverse amplitude predicted by this modelare consistent with a direct measurement.1 INTRODUCTIONThe nonlinear chromaticity and amplitude-dependentbetatron tune shifts in the CERN SPS were measured dur-ing a series of machine studies in the years 2000 and 2001[1, 2]. This study was motivated by two observations: (1)The measured detuning with amplitude significantly variedfrom year-to-year and also after changes to the SPS super-cycle. These often required re-optimization of sextupolesand Landau-damping octupoles. (2) Measurements of thenonlinear chromaticity revealed a large cubic component.The changes were attributed to time-dependent varia-tions of the remanent fields in the SPS dipoles and possiblyquadrupoles. The SPS accommodates two types of dipoles,called MBB and MBA with different chamber heights andaspect ratio (the full width of MBA chambers is 152 mm,and the full height 34.5 mm; for the MBB chambers thecorresponding numbers are 129 mm, and 48.5 mm).We developed a procedure where we fit normal sextupoleand decapole errors b3 and b5 to the two types of dipolemagnets and octupole field components b4 to the focusingand defocusing quadrupoles so as to reproduce the mea-sured nonlinear chromaticities through third order. Thenonlinear SPS model so obtained can be used, e.g., to pre-dict the detuning with amplitude, whose direct measure-ment probes the validity of the model.2 MEASUREMENTSOptics data were taken with a 48-bunch LHC-type beamon October 18–20, 2000, and July 17–19, 2001. The num-ber of protons per bunch varied from 1.1−1.8×10 10. Thesynchrotron tune was about 250 Hz (2 MV rf voltage at 200MHz) and the Landau damping octupoles were switchedoff, unless noted otherwise.2.1 ChromaticityTypical measurements of horizontal and vertical chro-maticity are shown in Figs. 1 and 2. The beam momentumwas varied by changing the rf frequency. Its change ∆δis related to the average shift in the horizontal closed orbit∆R [m] via ∆δ ≈ 0.4∆R. Note that the δ (≡ ∆p/p) val-ues in the experimental curves were inferred from the shiftin the horizontal BPM readings. The discrete sampling ofthe orbit at the BPMs gives rise to a correction factor 0.8with respect to the average orbit change, which is includedon the horizontal axis.y = -4.56x3 + 0.4069x2 + 0.0338x + 0.6218R2 = 0.90.62000.62050.62100.62150.62200.62250.62300.62350.62400.62450.6250-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10Q*dp/0.8QhFigure 1: Measured horizontal tune versus product of mo-mentum offset and tune, Qδ, on 19th July 2001 (series 2).y = 3.6849x3 + 0.1451x2 + 0.0333x + 0.5826R2 = 0.96610.57800.58000.58200.58400.58600.58800.5900-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10Q*dp/0.8QvFigure 2: Measured vertical tune versus product of momen-tum offset and tune, Qδ, on 19th July 2001 (series 2).The measurement reveals a significant third-order chro-maticity. From a fit of the measured Qx and Qy vs. δp/p,we obtain the linear, quadratic and cubic components of thetwo chromaticities, asQx,y = Q0x,y +Q′x,yδ +Q′′x,y2δ2 +Q′′′x,y6δ3 .Table 1 lists the values obtained for the year 2001.MDs Q0x Q′x 1/2 Q′′x 1/6 Q′′′x R2xQ0y Q′y 1/2 Q′′y 1/6 Q′′′y R2y17/07 0.623 4.5 500 −156000 0.980.582 1.75 −313 156000 0.9818/07 0.622 1.085 480 −167000 0.87(Series 1) 0.583 1.937 −17 48000 0.9918/07 0.622 0.992 420 −146000 0.89(Series 2) 0.583 2.027 −100 −49000 0.9718/07 0.622 1.358 423 −245000 0.88(Series 3) 0.581 3.987 343 −86000 0.9819/07 0.622 1.424 584 −218000 0.91(Series 1) 0.580 0.934 −38 185000 0.9319/07 0.622 1.125 451 −168000 0.90(Series 2) 0.583 1.106 160 135000 0.97Table 1: Linear and nonlinear chromaticity components ob-tained from a polynomial fit of the measured betatron tuneas a function of rf frequency.2.2 DetuningThe horizontal detuning with amplitude was measuredon 25th July for three different octupole settings, namelyfor (−2,0), (0,0), (2,0), where the pairs of strengths quotedare in units of [m−4] and refer to the horizontal and verticaloctupoles, respectively. The measurement of Qx vs. Ax isdepicted in Fig. 3 By determining the parabolic dependence12 (∂Q2x/∂A2x) of each data set using a fit, the optimum hor-izontal octupole setting, for zero detuning, was inferred tobe roughly −0.875 m−4. An analogous measurement wasperformed in the vertical plane [2].0.610.6150.620.6250.630.6350.640 0.1 0.2 0.3 0.4 0.5 0.6Oscillation amplitude [a.u.]Qh(0  0) (+2 0) (-2 0)Figure 3: Measured horizontal tune versus amplitude in[a.u.] on 25th July 2001, for various settings of the horizon-tal octupoles in [m−4]. The vertical octupoles were heldconstant at 0.0 m−4. Note that 0.1 [a.u.] on the horizontalscale corresponds to 3.652 mm in the horizontal pick-up(βx = 100 m).Note that the geometric beam emittance is about x,y ≈140 nm. The rms beam sizes at the pick up (βx,y ≈ 100m, Dx ≈ 3 m, σp/p ≈ 1.1 × 10−3) are σx ≈ 5.2 mmand σy ≈ 3.8 mm. Thus, the maximum amplitude in Fig. 3corresponds to about 3.5σx. Note that for amplitudes be-low about 2σ a refined analysis of the detuning would berequired, e.g., using the formalism developed in Ref. [3],according to which the measured detuning∆Q is enhancedapproximately by the factor (1 + 1/(2Z 2))2, where Z de-notes the kick amplitude in units of σ. At 2σ the enhance-ment is 28%. At 1σ it amounts to a factor of 2.6. For 3.5σthe correction can be neglected.3 OPTICS MODELThe nonlinear model is constructed as follows. For eachdata set, we first set the tunes to the measured values Q0x,yby adjusting the main quadrupole families. The changesare a fraction of per mill.In order to obtain the measured chromaticities we in-sert a b3 component at the center of each of the two kindsof dipole and we match its value to obtain the best possi-ble agreement with the measured linear chromaticity (Q ′x,yfrom Table 1). The resulting values for the b3 componentsare summarised in Table 2, where the parameter b3a refersto MBA and b3b to MBB. The values for b3b are fairly re-producible with a variation of about± 20%, but its averageis about three times the value measured in the year 2000,which may be related to the different cycle. The value ofb3a shows a scatter by up to a factor of 2. However onaverage it appears to be closer to the value found in 2000.Also, we note that b3a and b3b are of opposite sign for allthe measurements.The variation of the sextupole component in the dipoleswith beam energy contains a field-proportional term, a con-stant remanent field, and an eddy current term (∝ B˙). Thelast one is zero in our case. Using the historical mag-net data in [4, 5] at 26 GeV/c we would expect b3a ≈1.05 × 10−3m−2 and b3b ≈ 1.8 × 10−4m−2. The valueof b3a agrees well with our fit. The expected value of b3b isalmost zero, while the fit yields a large negative value. Thismight be the consequence of the different cycles consideredfor our measurements and for the historical data. The varia-tion in b3b observed between 2000 and 2001 seems to indi-cate that this component, b3b, is more sensitive to remanentfields.Next we add a b4 component at the center of each kind ofquadrupole and match its value to reproduce the measuredsecond order chromaticity (Q ′′x,y from Table 1). The result-ing values are summarised in table 2. The multipole com-ponent b4f refers to the horizontally focusing quadrupoles(QF1.F , QF1A.F , QF2.F and QF2A.F ) and b4d to thehorizontally defocusing quadrupoles (QD.F andQDA.F )We observe that b4f is reproducible and even comparableto the value found in 2000. On the other hand, there is alarge scatter in the value of b4d, which may indicate that itis not well constrained by the fit.Adding the same b5 component for the two kinds ofdipole [1], we match its value to get the best possible agree-ment with the measured third order chromaticity (Q ′′′x,yfrom Table 1). Also these results are summarised in Ta-ble 2. The maximum spread in b5 is less than 50%. Theaverage is about twice the value fitted in 2000, which againcould be a feature of the different SPS cycle.element dipoles quadrupoles dipolesMDs b3a b4f b5b3b b4dunits 10−3[m−2] 10−1[m−3] [m−4]18/10/2000 1.366 0.808 -5.833−0.826 −2.55117/07/2001 1.024 0.974 −17.956−2.774 2.47018/07/2001 0.985 1.056 −16.779(Series 1) −3.188 −0.49118/07/2001 1.662 0.884 −14.399(Series 2) −3.567 0.70318/07/2001 1.665 1.080 −25.417(Series 3) −3.360 −4.35719/07/2001 0.815 1.305 −24.936(Series 1) −2.452 −0.50919/07/2001 0.759 1.056 −19.304(Series 2) −2.466 −2.422Table 2: Matched multipole components for the nonlinearchromaticity measurements in 2000 [1] and 2001.To simulate the detuning with amplitude for the differentLandau damping octupole settings we followed the sameprocedure as described above in order to match the tunesand the linear chromaticity (by adjusting b3a and b3b). Rec-ognizing the spread in the fitted higher-order nonlinear fieldcomponents, for computation in MAD we then assumed thevalues b4f , b4d and b5 for a typical case in Table 2. Tak-ing the nonlinear chromaticity components from 19 th July2001 (series 1), we calculated with MAD the horizontaltune versus the horizontal amplitude for the same octupolesettings as used in the measurements. The results are dis-played in Fig. 4. Figures 3 and 4 show good agreementbetween the measured and fitted detuning. A similar agree-ment was obtained in the vertical plane [2].4 CONCLUSIONSWe fitted nonlinear field errors up to decapolar compo-nents in the SPS dipole and quadrupole magnets so as toreproduce the nonlinear chromaticity measured at 26 GeV.We find that at this energy the sextupolar fields b3 insidethe two types of SPS dipoles need to have opposite signto explain the measurements, which is roughly consistentwith expectation. The values of these b3 components inthe main dipoles are in general quite reproducible with theexception of those measured on 18/07/2001 (b 3a namely).Nevertheless, the quality of the fit to the data is not partic-0.610.6150.620.6250.630.6350.640 0.001 0.002 0.003 0.004 0.005 0.006Qxxstart[m]Octupole scans - horizontal detuning with dp/p = 0.0KLOD: 0.0, KLOF:  0.0KLOD: 0.0, KLOF:  2.0KLOD: 0.0, KLOF: - 2.0Figure 4: Predicted horizontal tune versus amplitude fordifferent octupole settings, after fitting to linear chromatic-ity measured on 25th July and to nonlinear chromaticityfrom 19th July 2001 (series 1). Note that 1 mm at the start-ing point corresponds to 0.971 mm in the pick-up.ularly good, especially for the horizontal tune dependence,as indicated by the low value ofR2x in Table 1. To match themeasured second order chromaticities an octupolar com-ponent was introduced in each kind of quadrupole magnet.While the fitted b4 component for the focusing quadrupolesis fairly reproducible from measurement to measurement,this is much less the case for the horizontally defocusingquadrupoles, where the fitted b4 values show a large vari-ation and even a change in sign. As a last step, we deter-mined the decapole component in the two dipole familieswhich would yield the measured third order chromaticities.The b5 value is similar for all the measurements, varyingby about±25%. Using the fitted values of the b3, b4 and b5field components in dipoles and quadrupoles, we then pre-dicted the amplitude-dependent detuning. The result is ingood agreement with the observations, for various settingsof the Landau-damping octupoles. In particular, the settingof the Landau-damping octupoles resulting in zero detun-ing inferred from the model agrees with that determinedin a direct measurement. Therefore, it may be possible tooptimize the octupole settings by fitting a measurement ofthe nonlinear chromaticity. The latter could be obtainedquickly and for the entire cycle at once [6].5 REFERENCES[1] G. Arduini, F. Zimmermann, A. Faus-Golfe, SL Note 2001-030 MD (2001).[2] G. Arduini, P. Collier, F. Zimmermann, A. Faus-Golfe, SLNote 2001-049 MD (2001).[3] R.E. Meller, et al., SSC-N-360 (1987).[4] V. Hatton, A. Riche, A. Swift, Lab II/MA-Int.75-2 (1975).[5] M. Cornacchia, Lab. II-DI-PA/Int. 75-8 (1975).[6] J. Wenninger, private communication (2001).

Modelling Nonlinear Optics in the CERN SPS

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