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On the Effects of Fringe Fields in the Lhc Ring 06 oo 2s 36 2— ON THE EFFECTS OF FRINGE FIELDS IN THE LHC RING F. MEOT CFA DSM UFCA GT Lahoratoire National Saturne Groupe Theorie F-91191 (iif-sur- Yvette Cedex, France LHC95. International Workshop on single particle effects in Large Hadron Colliders, Montreux, Switzerland, October 1995. Rapport Interne CEA/DSM/GECA/GT/95- 10 December 1995 lt//l Centre National de la Recherche Scientifique cea Commissariat a I’Energie Atomique 36oo 6 Z- ON THE EFFECTS OF FRINGE FIELDS IN THE LHC RING F. MEOT Laboratoire National Saturne Groupe Theorie F-91191 Gif-sur-Yvette Cedex, France LHC95. International Workshop on single particle effects in Large Hadron Colliders, Montrenx, Switzerland, October 1995. The effects of the dipole and quadrupole fringe fields on such machine parameters as chromaticity, anharmonicity, closed orbit, etc. are investigated by stepwise ray-tracing in the Version 4 of the LHC ring. First the ray-tracing methodis described, then follows an overview of the relevant LHC characteristics and in particular the fringe field data and the corresponding field models. Some key points concerning the application to the ray-tracing in the LHC ringare emphasized. Then follows a detailed study of the machine parameters. 1 INTRODUCTION The effects of the dipole and quadrupole fringe fields on such machine parameters as chromaticity, anharmonicity, closed orbit, etc., are investigated by stepwise ray-tracing in the Version 4 of the LHC ring U). First the ray-tracing method is described, then follows an overview of the relevant LHC characteristics and in particular the fringe field data and the corresponding field models. Some key points concerning the application to the tracking in the LHC ring are emphasized. Then follows a detailed study of the machine parameters. 2 THE RAY-TRACING METHOD 2.1 Resolution of the equation of motion^ The Lorentz equation which governs the dynamic of a particle of charge q and mass m in a —^ magnetic field B, dv m— = q (1) —> is solved by stepwise Taylor expansions of the position R and normalized velocity u, —>—>—> —> QS R(MX) = R(M0) + u(M0)ds+ .....+ u (M0)— 6 ! (2) = zr(MJ + «(MJc6+..... +w ""(Mo)— where ds is the step size between the two successive points M0 and (fig. 1), —^ ^ —>* —> J —> —>■ —> —^ zz = v/ v, tih = vdt, u - d iijds and mv = mvii = qBp n . The derivatives of u are obtained by successive differentiatious, u = ux b, n =\^u x bj = ux b+ ux b , etc., that involve the derivatives of b - b /Bp up to the fourth order. 2.2 The magnetic field B and its derivatives A unique generic model for the magnetic field in the dipoles and quadrupoles of the ring is used, namely, the scalar potential 3,45 sin ogfrx*2 +zl)q Z (3) 4qq\(n + q)! m-0 m\(ii - m)\ where x, z, are the transverse coordinates (orthogonal to the magnet axis), a„>0(j)is the longitudinal form factor for the field at x = z = 0, and its 2q-th order derivative w. r. t. the longitudinal coordinate s. In case saveral harmonics are present in a magnet, the superposition theorem is applied. The model of longitudinal form factor used is5). 1 an,oO^O _ 1 + exp[P(<f)] (4) P(d) - C0 + Q — + C2(^—j +. where d is the distance to the effective field boundary, and X, C0-C5 are determined from a matching with numerical fringe field data. 3 LHC CHARACTERISTICS The ring of concern is the Version 4, characterized as summarized below 13. Total circumference 26658.863 m Cell length 106.478 m Cell curvature 30.478 10'8 rad Number of cells per arc 23 Arc length 2446 m Arc curvature 700.997 10'3 rad Odd insertion length13 885.631 m Even insertion length 887.084 m Nominal/maximum energy 7/7.54 TeV Field at nominal/maximum energy 8.35/9T Dipole length 14.2 m Dipole bending radius 2795.447 m Quadrupole length 3 m Gradient for 90° at nominal energy 210 T/m The typical optical functions are given in fig. 2 (injection conditions in the interaction regions). The total machine tunes are vx/vz = 66.28/66.32. The four-fold superperiodicity machine is considered in this study. The normalized emittance is 7£/7t = 3.75 10"6 m.rad. The fringe fields involved^, for the dipole magnets (« White book » design 73 ) and the quadruples (CEA - Saclay 8)) are shown in fig. 3, which also gives the fitting analytical model (eq. 4) superimposed. The values of the corresponding coefficients X, C0-C5 are as follows. For the dipole a = 0.112m,Q = 0.1553,Q = 3.875g,Q = -2.3622,Q = 2.9782,Q = 12.6o4,Q = 15.026 and also Ki.Gap = Pint.Gap = 2.1211 10"2 m (For respectively the Transport and MAD9) codes field fall-off models). For the quadrupole ^ = 0.056m, C„ = -0.01097, Q = 5.4648, Q = 0.9968, Q = 1.5688, Q = -5.6716, C, = 18.506 These values also assure the exact magnetic lengths, in particular 14.2 m for the main dipole and 3 m for the main quadrupole. 4 RAY-TRACING Two means are used simultaneously in order to evaluate the machine parameters namely, on one hand multiturn ray-tracing (unless specified, of the order of 2000 full 8-octants machine turns for one tune value) followed by regular Fourier analysis to get the tunes and elliptical fit to get the optical functions, smear, etc. , on the other hand, one-turn first order mapping followed by beam matrix computation (by identification with cos p. I + sin \x J). In both cases the symplecticity is thoroughly checked, in terms of the smear by a calculation of the dispersion &(sxz / in the first case, in terms of the second order symplectic conditions (determinants, etc.10)) in the second case. From both survey means, the chromaticity dv/dp/p , anharmonicity dv/de/n, the (3 functions and their derivatives w.r.t. dp/p, the horizontal closed orbit, the dispersion functions p x and rj'x and other relevant machine parameters can be evaluated. The results of the study are displayed in figs. 4-14 as follows. Fig. 4 : a. preliminary test of the symplectic behaviour in the absence of fringe fields (first order machine, sharp edge magnet models) and of the conformity with the MAD results : three particles on the respective invariants — =10—L , 100 and 200 —while TC TV TV \ £^=10->i2L mj 8.33 10 9m.rad are ray-traced over 7xl0 4 turns. The observation 7V TV \ 7T point is at the azimut of the so-called (cf. the MAD filesr)) S17LO straight section at the end of the odd type arc. The most relevant results are vx /v, = 0.2800/0.2944 (Fig. 5) exactly as expected (note that vz = 0.2944 instead of 0.32 due to the absence of the longitudinal field component at the dipole ends). Fig. 6 and Fig. 7 : Still in the absence of fringe fields, the horizontal phase-space diagram provides an efficient check of the symplectic behaviour of the integration for ofF-momemtum particles (-210 3 <Sp!p<+2\0 3, step 0.510 3), launched on very small x, z, invariants f b, x,z // a ^ By the mean time, it is verified that the dispersion r|x and its derivative d%/ds are in agreement with the MAD results, and the chromaticity as well. Namely, from Fourier analysis as schemed in fig. 7 it comes out = —102/—102 . From now-on, the fringe fields are switched on. Fig. 8 : Phase-space plots ( = 2000 machine turns per particle) at S17LO. This clearly reveals, for instance, the dfijdp/p effects, however not sensitively different of the fringe field free case. Fig. 9 : Machine tunes, depending on dp/p with the following results : vx/vz = 0.2836/0.3057, and the chromaticities vx/vz = -102/-102 . The comparison between ray-tracing and mapping is synthesized in the Table below, which shows the quite satisfying agreement between both means. Sp/p Vx Vz (10-3) TRACK ' / '. MAP . TRACK MAP -2 0.48834 0.48881 0.51050 0.51052 -1-5 0.43700 0.45915 -1 0.38576 0.38577 0.40790 0.40790 -0-5 0.33464 0.35678 -10-4(10-3) 0.29381^ (0.28463f 0.315943} (0.306762)4) 0 0.28362 0.28361 0.30575 0.305746 +10-4 (10-5) 0.273421} (0.28259f 0.295552) (0.304723)4) 0-5 0.23268 0.25480 1 0.18177 0.18177 0.20391 0.203908 1-5 0.13080 0.15298 2 0.07937 0.07932 0.10176 0.10177 l) => Avx/Splp = -101.96 ;2) => Avx/Sp/p = -101.95 3) => A vJSpjp = -101.94 ; 4) A vjdpjp - -101.96 Fig. 10 : a synthesis of the momentum detunings v.s. Sp/p Fig. 11 : chromaticities v.s. 5p/p Fig. 12. The dispersion function p x and its derivative dpjds v.s. 8p/p Fig. 13. We now proceed on studies relevant with the determination of the machine anharmonicities dv x,jds xz Ik, in the presence of fringe fields. Five particles are launched on horizontal and/or vertical invariants ranging from 10-4 g^/# to 200 einjln. The smear appears to be negligible [o-(sx, / tt)«(sxz / tt) ] and the tunes stay constant, namely vx/vz = 0.2836/0.3057, whatever sX)Z/tt . From this it can be concluded that the non linearities introduced by the fringe fields have but negligible effect on the tunes, whatever the amplitude. Fig.
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