- Email: [email protected]
Optimized lattice for the Collector Ring (CR)
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
Optimized lattice for the Collector Ring (CR) A. Dolinskii*, P. Beller, K. Beckert, B. Franzke, F. Nolden, M. Steck GSI, Planckstrasse 1, Darmstadt D-64291, Germany Available online 15 September 2004
Abstract The Collector Ring (CR) is designed for fast stochastic cooling of rare isotope beams (RIB) and, after changing the magnet polarity, of antiproton (pbar) beams. With the chosen layout of the CR lattice it is possible to find nearly optimal ring optics for both kinds of beams. This is necessary because of different ion velocity and different initial beam parameters. The RIB are injected at bE0:83 with transverse emittances e> of 200 mm mrad and a full momentum width ðdp=pÞFW of 3.5%, whereas the parameters for pbar beams are bE0:95; e> ¼ 240 mm mrad, and (dp=pÞFW ¼ 6%: The CR has to be operated also in the so-called isochronous mode as TOF mass spectrometer for short-lived secondary nuclei. The ring consists of two arcs and two long straight sections. Presently two solutions for the CR lattice are under consideration: a ‘‘symmetric ring’’ with identical lattice functions in the arcs and the ‘‘split ring’’ with different lattice functions in the arcs. Results of particle tracking simulations for both types of lattices, taking into account all relevant higher order field effects, will be presented and discussed in our contribution. r 2004 Elsevier B.V. All rights reserved. PACS: 29.20.Dh; 41.85.p; 41.85.Gy Keywords: Storage ring; Chromatic correction; Dynamic aperture
1. Introduction In the frame of the new future GSI project [1] the Collector Ring (CR) must provide efficient stochastic cooling of hot radioactive ion beams coming from the Super Fragment Separator (SFRS) with a momentum spread of 3.5% and a transverse emittance in both planes of 200 mm mrad. To minimize the cost of the project this ring should be used also for cooling of *Corresponding author. Tel.: +049-06159-712351; fax: +049-6159-712039. E-mail address: [email protected] (A. Dolinskii).
antiproton beams (pbar) with a momentum spread of 6% and transverse emittance of 240 mm mrad in both planes. Efficient chromatic correction is essential for a large off-momentum dynamic aperture. Therefore, the CR lattice requires strong sextupole correctors which increase the amplitude dependent and nonlinear chromatic aberration. These effects tend to reduce the dynamic aperture. Consequently, the ring must be optimized to achieve good properties for the stochastic cooling [2] and at the same time the maximum dynamic aperture. The improvement of dynamic aperture starts from optimization of linear optics and chromaticity correction
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.083
ARTICLE IN PRESS 484
A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
schemes. In the following sections, we review the modifications made to the initial lattice design and present tracking studies including the effect of magnetic and chromatic errors and the effect of lattice periodicity. All the tracking studies have been done using the MIRKO and MAD codes.
Table 1 Summary of major CR parameters Max.mag.rigidity, Br (Tm) Circumference, C (m) Symmetric ring Split ring
13 200.6 205.5
Lattice type
2. Lattice layout The present layout of the CR is the same as that given in Refs. [3,4], where the ring lattice has twofold symmetry and periodicity with two identical arcs and two long straight sections. Each arc in the new lattice has two identical symmetric cells, where there are also two identical cells with mirror symmetry. At present two optical configurations for the CR are considered. One of them is the ‘‘symmetric ring’’ (two arcs have the same lattice functions—the periodicity is 2) and another one is the ‘‘split ring’’ (each arc has different lattice functions—the periodicity is 1). Each straight section consists of two mirror symmetric matching cells. The length of the straight sections depends on the chosen ring symmetry. In the present design of the CR, the focusing strength of the quadrupoles is reduced (from 7.3 T/m in the previous design to 4.3 T/m), since the stronger focusing in the CR lattice increases the beam sensitivity to magnetic and chromatic errors and generates larger chromaticity. In the straight sections some quadrupole magnet positions are constrained to provide 3.2 m drift spaces for injection/extraction devices. To relax the cell optics it has been found advantageous to add sextupole fields to the quadrupoles by pole shaping. This results in a better separation of horizontal and vertical correction of the chromaticities and reduces the sextupole strengths. The new main parameters of the CR for both ‘‘symmetric’’ and ‘‘split’’ rings are given in Table 1.
3. Chromatic correction Chromaticity correction is an essential part of the CR design. The CR lattice is rather dense
Max. energy, E (MeV/u) Hor. acceptance (mm mrad) Ver. acceptance (mm mrad) Mom. acceptance, Dp=p (%) Betatron tunes, Qx =Qy Symmetric ring Split ring Transition, gtr Symmetric ring Split ring Local transition, gpk =gkp a Symmetric ring Split ring
pbar 3000 240 240 6
FODO RIB 740 200 200 3.5
4.62/4.19 5.64/4.42
3.36/2.88 5.14/3.81
4.48 4.82
2.7 3.4
3.96/5.01 3.93/5.85
2.32/3.08 2.30/5.85
a gpk is local gamma transition between pick-up and kicker and gkp —between kicker and pick-up.
with focusing elements. Since the momentum spread of the particles is relatively large (max 6%) each individual quadrupole gives an essential contribution to the chromatic effect of the ring. Calculations show that in order to obtain ideally zero chromaticity one has to apply a large number of sextupoles magnets (ideally the same number as the quadrupole magnets in arcs) and additionally to use strong sextupolar fields for correction of Q0 : Since we have restrictions with respect to the positioning of the large number of laminated sextupole magnets it is proposed to include the sextupole components in the profile of the quadrupole magnets. Additionally to have flexibility in correction at different optics eight separated sextupole magnets must be installed. The value of the sextupole components integrated in the quadrupole magnets can be estimated from the following. In the frame of the thin lens approximation the focal strength of a quadrupole is f 1 ¼ kx L
ð1Þ
ARTICLE IN PRESS A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
where L is the length of the quadrupole, kx is the quadrupole strength e dBy : kx ¼ cp dx
ð2Þ
The general magnetic field equation including only the most commonly used upright multipoles elements is given by By ¼ By0 þ Gx þ þ
1 0 2 G ðx y2 Þ 2
1 00 3 G ðx 3xy2 Þ þ ? 6
ð3Þ
ð4Þ
d ¼ Dp=p: Taking into account Eqs. (2)–(4) and considering that the coordinate for non-monochromatic particle is x ¼ Dd (D is the dispersion function within a quadrupole) the focal strength can be written as f 1 ¼
eðG þ G 0 DdÞL : p0 cð1 þ dÞ
ð6Þ
Solving this equation one gets a condition for the ratio of sextupole and quadrupole components within a quadrupole G 0 Dð1 d þ d2 Þ ¼ Gð2 dÞ:
Optics
G0 =G
pbar (sym. ring) RIB (sym. ring) pbar (split ring) RIB (split ring)
0.51; 0.51; 0.52; 0.25;
0.75 0.75 0.57; 0.71; 0.75; 1.00; 1.71 0.33; 0.42; 1.10; 1.24; 1.71
G0 2 E : G D
ð8Þ
As was mentioned before, the CR should operate at three different optics modes, where the shape of the dispersion function is different. That means one has to find the optimal values of the ratio G0 /G for all optics and in combination with two additional lumped sextupole families to reach chromaticity close to zero. In Table 2 the values of the G0 /G for the CR are provided. It is seen that for the symmetric ring one needs two types of quadrupoles with sextupole component. In the case of the split ring the number of such quadrupoles is larger by a factor of 3 and for different optics one needs different ratio of the G 0 =G: Therefore for the split ring the sextupole field must be variable from pbar to rare isotope beam (RIB) optics.
ð5Þ
The quadrupole is achromatic (the focal strength does not depend on the particle momentum deviation) if there is a solution of d 1 ðf Þ ¼ f 1 : dd
Table 2 Sextupole components in the quadrupoles of the CR
matic if
where G is the quadrupole field component, G 0 the sextupole component, G 00 the octupole component. Let us consider the particle motion in the horizontal midplane (y=0) with field components up to sextupole. For the particles with some momentum error (non-monochromatic particles) one can write 1 1 ¼ ; cp cp0 ð1 þ dÞ
485
ð7Þ
Usually, Dp=p51 (for CR Dp=pmax ¼ 70:03), therefore a quadrupole lens is almost achro-
4. Dynamic aperture The required dynamic aperture is determined by the available mechanical aperture. In our study we calculate the dynamic aperture for both symmetric and split rings each of them at the RIB and pbar optics. The main issue is to determine how sensitive is the CR to the nonlinear field errors. If the dynamic aperture is rather low compared to the physical aperture, a nonlinear corrector system that increases the dynamic aperture must be developed. It is important to maximize first the dynamic aperture for the ideal lattice without any magnetic or misalignment errors where there are only
ARTICLE IN PRESS 486
A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
chromaticity sextupoles. Maximizing the error free dynamic aperture necessarily involves optimization of linear optics and the chromaticity correction system, minimization of chromatic and high-order effects and optimizing betatron tunes. To improve the dynamic aperture of the CR one has to install additional sextupoles in the long straight sections with zero dispersion function. The phase advance between sextupoles and the first quadrupole in the
arc must be about of p=3: For the symmetric ring one family of the sextupoles is enough to maximize the dynamic aperture. In the case of the split ring one has to use two families of such sextupoles. In Fig. 1 the dynamic aperture of the CR in the horizontal plane calculated by the MIRKO code with only chromaticity sextupoles is shown for the ‘‘symmetric ring’’. In comparison to the previous calculation given in Ref. [4] the reduction of the quadrupole strength by a factor of about 2 results in an increase of the dynamic aperture at least by factor of 2–4. The same increase of the dynamic aperture was observed also in the vertical plane. Then simulations have been done with high-order reasonable field imperfections (up to 9th order) by using the MAD code. Since there is no information about the quality of the CR magnets, the field errors are assumed
Vertical aperture, Y [cm]
50
With only chromaticity sextupoles (SC) With CS & systematic errors With CS & random errors With CS & systematic & random errors Physical aperture
40
30
20
10
0 0
10
20
Vertical aperture, Y [cm]
40
Fig. 1. The dynamic aperture in the horizontal phase space with only chromaticity sextupoles for the symmetric ring with: (a) RIB optic; (b) pbar optic.
40
50
With only chromaticity sextupoles (SC) With SC& systematic errors With SC& random errors With SC& systematic & random errors Physical aperture
35 30 25 20 15 10 5 0 −40
(b)
30
Horizontal aperture, X [cm]
(a)
−30
−20
−10
0
10
20
30
40
Horizontal aperture, X [cm]
Fig. 2. The dynamic aperture for the symmetric ring with: (a) RIB optic; (b) pbar optic.
ARTICLE IN PRESS A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
Vertical aperture, Y [cm]
20
With only chromaticity sextupoles (SC) With SC & systematic & random errors Physical aperture
15
10
5
0
−20
−15
(a)
−10
−5
Vertical aperture, Y [cm]
20
5
10
15
20
With only chromaticity sextupoles (SC) With CS & systematic & random errors Physical aperture
15
10
5
0
(b)
0
Horizontal aperture, X [cm]
−20
−15
−10
−5
0
5
10
15
487
B00sx Leff ¼ 1:5 102 T=m and B00sk Leff ¼ 0:5 102 T=m). The CR dipoles are 2.2 m long and the dispersion and beta functions vary rapidly along the length. Hence it is incorrect to treat each magnet as a thick element with a single thin lens nonlinearity at the centre. Magnets need to be divided into several slices. In tracking simulations the dipole magnets are split into 10 pieces and the quadrupoles are split into 5 pieces. The dynamic aperture was calculated at the symmetry point of the straight section, where bx ¼ 4 m; by ¼ 12 m: The number of tracking turns is 1000. The results of calculations made with the MAD code are shown in Figs. 2 and 3. The simulations show that the dynamic aperture of the CR is dominated by systematic multipoles in the dipoles and sextupole components in the quadrupoles. That means the lowest order allowed harmonics—sextupole and decapole—must be corrected by small correction coils at the end of each CR dipole. This problem is especially relevant for the split ring, where the dynamic aperture is much lower than the physical aperture.
20
Horizontal aperture, X [cm]
Fig. 3. The dynamic aperture for the split ring with: (a) RIB optic; (b) pbar optic.
arbitrarily taking into account typical values of the existing magnets (for example the sextupole and skew sextupoles components were chosen as the same as for dipole magnets of the SIS18, where
References [1] An international accelerator facility for beams of ions and antiprotons, Conceptual Design Report, GSI-Darmstadt, November 2001 (see www.gsi.de). [2] F. Nolden, et al., Proceedings of the Particle Accelerator Conference, Chicago, 2001, p. 569. [3] B. Franzke, et al., Proceedings of the Seventh European Particle Conference, Vienna, 2000, p. 536. [4] A. Dolinskii, et al., Proceedings of the Eighth European Particle Conference, Paris, 2002, p. 572.
Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
Optimized lattice for the Collector Ring (CR) A. Dolinskii*, P. Beller, K. Beckert, B. Franzke, F. Nolden, M. Steck GSI, Planckstrasse 1, Darmstadt D-64291, Germany Available online 15 September 2004
Abstract The Collector Ring (CR) is designed for fast stochastic cooling of rare isotope beams (RIB) and, after changing the magnet polarity, of antiproton (pbar) beams. With the chosen layout of the CR lattice it is possible to find nearly optimal ring optics for both kinds of beams. This is necessary because of different ion velocity and different initial beam parameters. The RIB are injected at bE0:83 with transverse emittances e> of 200 mm mrad and a full momentum width ðdp=pÞFW of 3.5%, whereas the parameters for pbar beams are bE0:95; e> ¼ 240 mm mrad, and (dp=pÞFW ¼ 6%: The CR has to be operated also in the so-called isochronous mode as TOF mass spectrometer for short-lived secondary nuclei. The ring consists of two arcs and two long straight sections. Presently two solutions for the CR lattice are under consideration: a ‘‘symmetric ring’’ with identical lattice functions in the arcs and the ‘‘split ring’’ with different lattice functions in the arcs. Results of particle tracking simulations for both types of lattices, taking into account all relevant higher order field effects, will be presented and discussed in our contribution. r 2004 Elsevier B.V. All rights reserved. PACS: 29.20.Dh; 41.85.p; 41.85.Gy Keywords: Storage ring; Chromatic correction; Dynamic aperture
1. Introduction In the frame of the new future GSI project [1] the Collector Ring (CR) must provide efficient stochastic cooling of hot radioactive ion beams coming from the Super Fragment Separator (SFRS) with a momentum spread of 3.5% and a transverse emittance in both planes of 200 mm mrad. To minimize the cost of the project this ring should be used also for cooling of *Corresponding author. Tel.: +049-06159-712351; fax: +049-6159-712039. E-mail address: [email protected] (A. Dolinskii).
antiproton beams (pbar) with a momentum spread of 6% and transverse emittance of 240 mm mrad in both planes. Efficient chromatic correction is essential for a large off-momentum dynamic aperture. Therefore, the CR lattice requires strong sextupole correctors which increase the amplitude dependent and nonlinear chromatic aberration. These effects tend to reduce the dynamic aperture. Consequently, the ring must be optimized to achieve good properties for the stochastic cooling [2] and at the same time the maximum dynamic aperture. The improvement of dynamic aperture starts from optimization of linear optics and chromaticity correction
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.083
ARTICLE IN PRESS 484
A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
schemes. In the following sections, we review the modifications made to the initial lattice design and present tracking studies including the effect of magnetic and chromatic errors and the effect of lattice periodicity. All the tracking studies have been done using the MIRKO and MAD codes.
Table 1 Summary of major CR parameters Max.mag.rigidity, Br (Tm) Circumference, C (m) Symmetric ring Split ring
13 200.6 205.5
Lattice type
2. Lattice layout The present layout of the CR is the same as that given in Refs. [3,4], where the ring lattice has twofold symmetry and periodicity with two identical arcs and two long straight sections. Each arc in the new lattice has two identical symmetric cells, where there are also two identical cells with mirror symmetry. At present two optical configurations for the CR are considered. One of them is the ‘‘symmetric ring’’ (two arcs have the same lattice functions—the periodicity is 2) and another one is the ‘‘split ring’’ (each arc has different lattice functions—the periodicity is 1). Each straight section consists of two mirror symmetric matching cells. The length of the straight sections depends on the chosen ring symmetry. In the present design of the CR, the focusing strength of the quadrupoles is reduced (from 7.3 T/m in the previous design to 4.3 T/m), since the stronger focusing in the CR lattice increases the beam sensitivity to magnetic and chromatic errors and generates larger chromaticity. In the straight sections some quadrupole magnet positions are constrained to provide 3.2 m drift spaces for injection/extraction devices. To relax the cell optics it has been found advantageous to add sextupole fields to the quadrupoles by pole shaping. This results in a better separation of horizontal and vertical correction of the chromaticities and reduces the sextupole strengths. The new main parameters of the CR for both ‘‘symmetric’’ and ‘‘split’’ rings are given in Table 1.
3. Chromatic correction Chromaticity correction is an essential part of the CR design. The CR lattice is rather dense
Max. energy, E (MeV/u) Hor. acceptance (mm mrad) Ver. acceptance (mm mrad) Mom. acceptance, Dp=p (%) Betatron tunes, Qx =Qy Symmetric ring Split ring Transition, gtr Symmetric ring Split ring Local transition, gpk =gkp a Symmetric ring Split ring
pbar 3000 240 240 6
FODO RIB 740 200 200 3.5
4.62/4.19 5.64/4.42
3.36/2.88 5.14/3.81
4.48 4.82
2.7 3.4
3.96/5.01 3.93/5.85
2.32/3.08 2.30/5.85
a gpk is local gamma transition between pick-up and kicker and gkp —between kicker and pick-up.
with focusing elements. Since the momentum spread of the particles is relatively large (max 6%) each individual quadrupole gives an essential contribution to the chromatic effect of the ring. Calculations show that in order to obtain ideally zero chromaticity one has to apply a large number of sextupoles magnets (ideally the same number as the quadrupole magnets in arcs) and additionally to use strong sextupolar fields for correction of Q0 : Since we have restrictions with respect to the positioning of the large number of laminated sextupole magnets it is proposed to include the sextupole components in the profile of the quadrupole magnets. Additionally to have flexibility in correction at different optics eight separated sextupole magnets must be installed. The value of the sextupole components integrated in the quadrupole magnets can be estimated from the following. In the frame of the thin lens approximation the focal strength of a quadrupole is f 1 ¼ kx L
ð1Þ
ARTICLE IN PRESS A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
where L is the length of the quadrupole, kx is the quadrupole strength e dBy : kx ¼ cp dx
ð2Þ
The general magnetic field equation including only the most commonly used upright multipoles elements is given by By ¼ By0 þ Gx þ þ
1 0 2 G ðx y2 Þ 2
1 00 3 G ðx 3xy2 Þ þ ? 6
ð3Þ
ð4Þ
d ¼ Dp=p: Taking into account Eqs. (2)–(4) and considering that the coordinate for non-monochromatic particle is x ¼ Dd (D is the dispersion function within a quadrupole) the focal strength can be written as f 1 ¼
eðG þ G 0 DdÞL : p0 cð1 þ dÞ
ð6Þ
Solving this equation one gets a condition for the ratio of sextupole and quadrupole components within a quadrupole G 0 Dð1 d þ d2 Þ ¼ Gð2 dÞ:
Optics
G0 =G
pbar (sym. ring) RIB (sym. ring) pbar (split ring) RIB (split ring)
0.51; 0.51; 0.52; 0.25;
0.75 0.75 0.57; 0.71; 0.75; 1.00; 1.71 0.33; 0.42; 1.10; 1.24; 1.71
G0 2 E : G D
ð8Þ
As was mentioned before, the CR should operate at three different optics modes, where the shape of the dispersion function is different. That means one has to find the optimal values of the ratio G0 /G for all optics and in combination with two additional lumped sextupole families to reach chromaticity close to zero. In Table 2 the values of the G0 /G for the CR are provided. It is seen that for the symmetric ring one needs two types of quadrupoles with sextupole component. In the case of the split ring the number of such quadrupoles is larger by a factor of 3 and for different optics one needs different ratio of the G 0 =G: Therefore for the split ring the sextupole field must be variable from pbar to rare isotope beam (RIB) optics.
ð5Þ
The quadrupole is achromatic (the focal strength does not depend on the particle momentum deviation) if there is a solution of d 1 ðf Þ ¼ f 1 : dd
Table 2 Sextupole components in the quadrupoles of the CR
matic if
where G is the quadrupole field component, G 0 the sextupole component, G 00 the octupole component. Let us consider the particle motion in the horizontal midplane (y=0) with field components up to sextupole. For the particles with some momentum error (non-monochromatic particles) one can write 1 1 ¼ ; cp cp0 ð1 þ dÞ
485
ð7Þ
Usually, Dp=p51 (for CR Dp=pmax ¼ 70:03), therefore a quadrupole lens is almost achro-
4. Dynamic aperture The required dynamic aperture is determined by the available mechanical aperture. In our study we calculate the dynamic aperture for both symmetric and split rings each of them at the RIB and pbar optics. The main issue is to determine how sensitive is the CR to the nonlinear field errors. If the dynamic aperture is rather low compared to the physical aperture, a nonlinear corrector system that increases the dynamic aperture must be developed. It is important to maximize first the dynamic aperture for the ideal lattice without any magnetic or misalignment errors where there are only
ARTICLE IN PRESS 486
A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
chromaticity sextupoles. Maximizing the error free dynamic aperture necessarily involves optimization of linear optics and the chromaticity correction system, minimization of chromatic and high-order effects and optimizing betatron tunes. To improve the dynamic aperture of the CR one has to install additional sextupoles in the long straight sections with zero dispersion function. The phase advance between sextupoles and the first quadrupole in the
arc must be about of p=3: For the symmetric ring one family of the sextupoles is enough to maximize the dynamic aperture. In the case of the split ring one has to use two families of such sextupoles. In Fig. 1 the dynamic aperture of the CR in the horizontal plane calculated by the MIRKO code with only chromaticity sextupoles is shown for the ‘‘symmetric ring’’. In comparison to the previous calculation given in Ref. [4] the reduction of the quadrupole strength by a factor of about 2 results in an increase of the dynamic aperture at least by factor of 2–4. The same increase of the dynamic aperture was observed also in the vertical plane. Then simulations have been done with high-order reasonable field imperfections (up to 9th order) by using the MAD code. Since there is no information about the quality of the CR magnets, the field errors are assumed
Vertical aperture, Y [cm]
50
With only chromaticity sextupoles (SC) With CS & systematic errors With CS & random errors With CS & systematic & random errors Physical aperture
40
30
20
10
0 0
10
20
Vertical aperture, Y [cm]
40
Fig. 1. The dynamic aperture in the horizontal phase space with only chromaticity sextupoles for the symmetric ring with: (a) RIB optic; (b) pbar optic.
40
50
With only chromaticity sextupoles (SC) With SC& systematic errors With SC& random errors With SC& systematic & random errors Physical aperture
35 30 25 20 15 10 5 0 −40
(b)
30
Horizontal aperture, X [cm]
(a)
−30
−20
−10
0
10
20
30
40
Horizontal aperture, X [cm]
Fig. 2. The dynamic aperture for the symmetric ring with: (a) RIB optic; (b) pbar optic.
ARTICLE IN PRESS A. Dolinskii et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 483–487
Vertical aperture, Y [cm]
20
With only chromaticity sextupoles (SC) With SC & systematic & random errors Physical aperture
15
10
5
0
−20
−15
(a)
−10
−5
Vertical aperture, Y [cm]
20
5
10
15
20
With only chromaticity sextupoles (SC) With CS & systematic & random errors Physical aperture
15
10
5
0
(b)
0
Horizontal aperture, X [cm]
−20
−15
−10
−5
0
5
10
15
487
B00sx Leff ¼ 1:5 102 T=m and B00sk Leff ¼ 0:5 102 T=m). The CR dipoles are 2.2 m long and the dispersion and beta functions vary rapidly along the length. Hence it is incorrect to treat each magnet as a thick element with a single thin lens nonlinearity at the centre. Magnets need to be divided into several slices. In tracking simulations the dipole magnets are split into 10 pieces and the quadrupoles are split into 5 pieces. The dynamic aperture was calculated at the symmetry point of the straight section, where bx ¼ 4 m; by ¼ 12 m: The number of tracking turns is 1000. The results of calculations made with the MAD code are shown in Figs. 2 and 3. The simulations show that the dynamic aperture of the CR is dominated by systematic multipoles in the dipoles and sextupole components in the quadrupoles. That means the lowest order allowed harmonics—sextupole and decapole—must be corrected by small correction coils at the end of each CR dipole. This problem is especially relevant for the split ring, where the dynamic aperture is much lower than the physical aperture.
20
Horizontal aperture, X [cm]
Fig. 3. The dynamic aperture for the split ring with: (a) RIB optic; (b) pbar optic.
arbitrarily taking into account typical values of the existing magnets (for example the sextupole and skew sextupoles components were chosen as the same as for dipole magnets of the SIS18, where
References [1] An international accelerator facility for beams of ions and antiprotons, Conceptual Design Report, GSI-Darmstadt, November 2001 (see www.gsi.de). [2] F. Nolden, et al., Proceedings of the Particle Accelerator Conference, Chicago, 2001, p. 569. [3] B. Franzke, et al., Proceedings of the Seventh European Particle Conference, Vienna, 2000, p. 536. [4] A. Dolinskii, et al., Proceedings of the Eighth European Particle Conference, Paris, 2002, p. 572.