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(

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

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. 14. We conclude with the horizontal closed orbit induced by the dipole fringe fields : it does not exceed 10~*m. The plot reflects the periodicity of the closed orbit, as expected from the 4-fold periodicity of the defects (the fringe fields). 5 CONCLUSION

A thorough, well behaved (symplectic) ray-tracing through the 4-periodic LHC ring in the presence of the dipole fringe fields (White Book data) and quadrupole fringe fields (CEA- Saclay data), at injection conditions, show that these have no effect on the chromaticity, anaharmonicity and other relevant machine parameters. The only sensitive effect is, as expected, a slight drift of the tunes ( vx/vz: 0.2800/0.2944 -> 0.2836/0.3057), when the fringe fields are switched on)which can be as usual fixed by slightly re-tuning the dipole field.

REFERENCES

1. A. Garren, X. Luo, F. Meot, W. Scandale, The LHC Lattice, Version 4, LHC Note, CERN SL/95-06 (AP), CERN, 2 March 1995.

2. The « Pink Book », Design study of the LHC, Report CERN 91-03, 2 May 1991, CERN, Geneva.

3. F. Meot and S. Valero, Zgoubi users ’ guide, Note CEA/DSMZLNS/GT/93-12, CEA- Saclay, 1993.

4. G. Leleux, Complements sur la physique des accelerateurs, DEA de physique et technologie des grands instruments, rapport CEA/DSM/LNS/86-101, Mars 1986 (page 117). 5 6 7 8 9 10

5. H. A. Enge, Deflecting magnets, in Focusing of charged particles, vol. II, A. Septier ed., Academic Press, New-York and London, 1967.

6. Data provided by S. Russenchuk, CERN/AT/MA, CERN, 1994.

7. The « White Book », Large Hadron Collider, the acceleratot project, CERN/AC/93-03, 8 Nov. 1993.

8. Data from the DAPNIA Department, CEA/DSM, Saclay, 1994.

9. H. Grote, F. C. Iselin, The MAD program V8.10, CERN/SL/90-13 (AP), January 1993.

10. G. Leleux, cours de l’INSTN de mathematiques pour les accelerateurs, Laboratoire National Satume, CEA-Saclay, 1970. FIGURE CAPTIONS

—> —> FIGURE 1 : Geometrical principle of the step by step ray-tracing. R = position, u = normalized velocity, ds = integration step size from Mo to Mi.

FIGURE 2 : Optical functions in the arc cell (left) and odd type DR. (right) - injection conditions.

FIGURE 3 : Fringe fields in the dipole (left ; « White Book » design) and quadrupole (right ; CEA-Saclay design). Solid line : the analytical model (eq. 4) used in the ray-tracing.

FIGURE 4 : Horizontal phase-space, 70000 turn in the fringe field free machine,

— = 10-4, 100 and 200-4^-;— = 10"4—Observation point : S17LO. TV TV TV TV

FIGURE 5 . Horizontal and vertical fractional tunes, from Fourier analysis of the ray-tracing schemed in Fig. 4 : whatever sxz / tv , vxj vz = 0.2800/0.2944.

FIGURE 6 : No fringe fields, horizontal phase - space at S17LO, with -210"3 < dp /p< 210"3 and 4Vp - ±10^, sx and sz {(shJ .

FIGURE 7 : Horizontal fractional tunes, depending an 5p/p, after Fourier analysis of the horizontal motions of fig. 6. Similar results are obtained with the vertical motion. The chromaticities vx/vz = —102/—102 follow.

FIGURE 8 : Fringe fields on ; horizontal (left) and vertical (right) phase-spaces at S17LO, for -210-3 < Sp/p < 210~3, including 4?/p = ±10^ ; sx and . Note the high degree of precision of the ray-tracing, e. g., the z motion is resolved at much better than 10~7 m/lO”7 rad .

FIGURE 9 : Vertical fractional tunes, depending on Sp/p, after Fourier analysis of the vertical motions of fig. 8 (right). Similar results are obtained for the horizontal motion. From the tune values at Sp/p ~ PICT4, the chromaticities vx/vz = —102/—102 come out.

FIGURE 10 : Horizontal (left) and vertical (right) tunes v.s. dp/p. The tunes at dp/p = 0 are vx/vz = 0.2836/0.3057, to be compared to the fringe field free case, vx/vz = 0.2800/0.2944 (Fig- 5).

FIGURE 11 : Horizontal (left) and vertical (right) tune derivatives dv/dp/p , v.s. dp/p. The chromaticities are vx'/ vz ' = —102/—102, identical to the fringe field free case (fig. 7). FIGURE 12 : Dispersion function r|x(left) and its derivative dr| x/ds (right) v.s. dp/p, at S17LO.

FIGURE 13 : Horizontal phase-space, from the ray-tracing of five particles on x, z invariants ranging in —— : 10^ —> 200-^-. n tv The smear of the invariant is negligible I 7v) < 10'4(sxz / tv} ]. The Fourier analysis

gives zero anharmonicities : the tune values are vx/vz = 0.2836/0.3057, whatewer sx, / n.

FIGURE 14 : Non-zero horizontal closed orbit along the machine as induced by the fringe fields. The horizontal axis represents the pick-up number (From the MAD files). The closed orbit excursion does not exceed 10"4 m. o

—^ ^ FIGURE 1 : Geometrical principle of the step by step ray-tracing. R = position, u = normalized velocity, ds - integration step size from M0 to Mi.

O0OE+OO

600 X (rn) 800 60 (in) 80

FIGURE 2 : Optical functions in the arc cell (left) and odd type IR (right) - injection conditions. FIELD CD GRADIENT (relative)

S (cin)

FIGURE 3 : Fringe fields in the dipole (left ; « White Book » design) and quadrupole (right ; CEA-Saclay design). Solid fine : the analytical model (eq. 4) used in the ray-tracing.

X (cm)

FIGURE 4 : Horizontal phase-space, 70000 FIGURE 5 : Horizontal and vertical turn in the fringe field free machine, fractional tunes, from Fourier analysis of the — = 10”4, 100 and 200-^;-^= 10"4^ ray-tracing schemed in Fig. 4 : 7t 7t n n whatever sX 2 In, vxjvz = 0.2800/0.2944. Observation point : S17LO. -......

" X' (mrad)

; '

1 ' i " 1 1 1 1 1 ' 1 1 1 1 ' ' ■ 1 1 1 I

■ ^ X (cm)

-.4 •.2 0.0 0.2 0.4

FIGURE 6 : No fringe fields, horizontal phase - FIGURE 7 : Horizontal fractional tunes, space at S17LO, with -210~3 < Sp /p < 210"3 depending an Sp/p, after Fourier analysis of the horizontal motions of fig. 6. Similar results are and 5p{p = +10"4, sx and sz {{sinj. obtained with the vertical motion. The chromaticities v'x/v'z - -102/-102 follow.

......

X' (mrad) - ^

0.02 K . '

...... \: : \ ' \

i x ; /' ;-i - Z' (mrad) '' : /

■ ^ X (cm) Z (cm) <4'~ •d 7. l .... 1 .... 1...... - -.001 ■.0005 0.0 0.0005 0.001

FIGURE 8 : Fringe fields on ; horizontal (left) and vertical (right) phase-spaces at S17LO, for -210™3 < 4?/p < 210"3, including cp/p - ±1(X ; sx and sz«smj - Note the high degree of precision of the ray-tracing, e. g., the z motion is resolved at much better than 10-7m/l0 _7 rad . 0.1 0.2 0.3 0.4 0.5

FIGURE 9 : Vertical fractional tunes, depending on Sp/p, after Fourier analysis of the vertical motions of fig. 8 (right). Similar results are obtained for the horizontal motion. From the tune values at dp/p - F1CT4, the chromaticities vx/vz = -102/-102 come out.

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 dp/p dp/p

FIGURE 10 : Horizontal (left) and vertical (right) tunes v.s. dp/p. The tunes at dp/p = 0 are vx/vz =0.2836/0.3057, to be compared to the fringe field free case, vx/vz = 0.2800/0.2944 (Fig. 5). -101,5 -101.8

.AD' -4--- _ -102

-102.2 102.5

-103 -102.6

■103.5

-104 -103.2 Chromaticity, z

-104.5 -103.. -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0,0015 0.002 dp/p dp/p

FIGURE 11 : Horizontal (left) and vertical (right) tune derivatives dv/dp/p, v.s. dp/p. The chromaticities are vx'/ vz' = -102/-102 , identical to the fringe field free case (fig. 7).

0.02953 2.188 ’RayTracing' 0.02952 ‘MAD’ --f— - 2.186 0.02951

2.184 0.0295

2.182 0.02949

0.02948

0.02947 2.178 0.02946 Dispersion derivalive, atS17LO

2,176 0.02945 Dispersion at S17LO 2.174 0,02944 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002

FIGURE 12 : Dispersion function % (left) and its derivative dr| x/ds (right) v.s. dp/p, at S17LO. X' (mrad)

i / /

X (cm) -0.3

FIGURE 13 : Horizontal phase-space, from the ray-tracing of five particles on x, z invariants ranging in —: 10^ —> 200—^. 7t K The smear of the invariant is negligible [ a(^sx z /tt) < I0’4(sXtZ / ;r) ]. The Fourier analysis gives zero anhannonicities : the tune values are vx/vz = 0.2836/0.3057, whatewer sx: / n. 0.01

X (cm)

0.0075

0.005

0.0025

-.0025

-.005

-j0075

i I i i i M i i i I i i i i I i i j____ l I i i i I i -0.01 _ I_i 50 100 150 200 250 300 350 Pick-up number