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Orbit effects of the end field multipoles in superconducting magnets
Nuclear Instruments and Methods 192 (1982) 157-159 North-Holland Publishing Company
157
ORBIT EFFECTS OF THE END FIELD MULTIPOLES IN SUPERCONDUCTING MAGNETS * G. PARZEN
Brookhaven National Laboratory, Upton, New York, U.S.A. Received 9 June 1981 Two possible effects due to the end field multipoles in superconducting magnets are pointed out. One effect is the sagittaend-multipole effect in dipoles. A result of this is that the u-shift due to an end multipole is not cancelled by an equal and opposite correction multipole distributed over the central part of the dipole, which complicates the correction of the end multipole effects. A second effect is the aperture-higher-multipole effect. The presence of higher end field multipoles for which there are no correction coils may reduce the good field aperture. These two effects are illustrated by computing them for the case of the ISABELLE accelerator.
1. Introduction The end field multipoles are the field multipoles introduced by the ends of a magnet. In a superconducting magnet [ 1], the end field multipole can be considerably reduced by properly designing the ends. One approach is to properly space the turns at the ends of the magnet. A question that arises is whether such a careful design of the ends is required. This paper points out two possibly harmful effects of the end field multipoles which may make it necessary to reduce the end multipoles. One effect is the sagitta-end-multipole effect in the dipoles. In the dipoles, the central orbit of the particles may depart from the axis of the dipole by the orbit sagitta. A result of this is that the u-shift due to an end multipole, bn, is not cancelled by an equal and opposite correction multipole, -bn, in the central part of the dipole. This complicates the correction of the end multipoles using correction multipoles spread over the central part of the dipole, for not only is a correction multipole - b n required, but also the corrections bn_l, bn_2, etc. may be required. A second effect is the aperture - higher-multipole effect [2]. The presence of higher end multipoles, for which there are no correction coils, may limit the good field aperture by introducing a curvature in the working line, the dependence of the v-value on momentum. * Work performed under the auspices of the U.S. Department of Energy.
0029-554X/82/0000-0000/$02.75 © 1982 North-Holland
In this paper these two effects are examined for the end multipoles in the magnets of the ISABELLE accelerator and it is found that a design of the ends of the dipoles which reduces the end multipoles is required. If the ends are wound naturally, with no special spacers to reduce the end multipoles, it is found that correction of the effects due to the end field sectupole would require almost the entire capacity of the quadrupole trim coils provided in the present ISABELLE design. It is also found that the higher end multipoles for naturally wound ends would cause an appreciable reduction in the good field aperture.
2. Sagitta-end.multipole effect This effect may be illustrated by considering the effect of the end field sextupole. The v-shift as a function of momentum due to this field multipole is given by [21,
1 avdp) = G
1
f dSBo(1 + ~lp) 2Bob2(x s +Xp Ap/p) ,
(1) where the sextupole field is given by Bob2x 2, x s is the position of the central orbit relative to the axis of the dipole, Bo is the dipole field, and Xp Ap/p gives the position of the off momentum orbit relative to the central orbit. One can see from eq. (1), that the same integrated sextupole b2 = fds b2 will give different Av(p) depending on whether b: is concentrated at the ends,
G. Parzen /End field multipoles in superconducting magnets
158
or whether it is distributed uniformly over the central region. If we write, Avx(p) = AVo + AVlAp/p, then we will find that the linear part, AVl, is roughly the same for the central or end b : , but the constant term Avo is not the same. Thus equal and opposite integrated sextupoles in the ends and in the central region will cancel the linear term in Avx(p), but the remaining constant v-shift may be corrected using the quadrupole correction coils. For end multipotes, one finds, Av0 ~x~2 ½ [xs(L/2) + Xs(-L/2)] AVl oc ~2
Xp, center,
where x s is evaluated at the ends s =+-L/2, and Xp, center is Xp at the center of the dipole. For the central multipole, one finds, Avo ~ b-2~s. AVl ~ b2Xp,
Bobnxn where Bo is the main dipole field. The end field sextupole present in the ISABELLE dipoles is b2 = 1.49 × 10 -4 cm -2 for naturally wound ends. When Avx(P) and Avy(p) are computed using eq. (1), with this end sextupole present and an equal and opposite sextupole over the center o f the dipole, it is found that Avx(P ) = 0.48, AVy(p)=- 0.47, a constant v-shift, independent of momentum. This v-shift could be corrected by the quadrupole correction coils provided in ISABELLE. However, the quadrupole correction required would use up essentially all the capacity presently planned for the ISABELLE quadrupole correction coils. In the same way, because of the s a g i t t a - e n d effect, the v-shift caused by an end field decapole cannot be corrected by just an equal and opposite decapole distributed over the central region of the magnet. The v-shift as a function o f momentum due to a decapole field is given by 1 f ds
1
center,
X 4Bob4 (Xs3 + 3x2sXp Ap/p + 3xsXg(Ap/p) 2 where x-s = ( l / L ) f ds x s, the average sagitta over the dipole. The computed end multipoles for the ISABELLE dipoles and quadrupoles are given in table 1 for the case where the ends are wound naturally, with no special spacers to reduce the end multipoles. The results listed in table 1 are computed results. The multipoles listed are ( I / L ) f ds bn over the dipoles, where L is the dipole length and the multipole field is given by
Table 1 End field multipoles for ISABELLE dipoles and quadrupoles with naturally wound ends, where the ends are not spaced to reduce the end multipoles. The b n listed are the equivalent central multipoles which distributed over the length of the magnet would give the same integrated effect. The multipole field is given by Bobnxn, where Bo is the dipole field. The bn are computed results. Dipoles
Quadrupoles
n
bn (cm -n)
n
bn (cm -n)
0 2 4 6 8 10 12
3.62 X 10 -2 -1.49 × 10 -4 -1.16 × 10 -6 -1.28×10 -8 -1.69X10 -1° -2.57 x 10 -12 ~ , . t 2 × 10 -14
1 5 9 13 17
7.18 -5.50 -9.07 -2.5 -7.6
x 10 -3 × 10 -7 × 10 -11 ×10 -14 ×10 -18
+ X~(Ap/p) 3} ,
(2)
where the decapole field is given by Bob4×4. One can see from eq. (2) that a decapole concentrated at the ends does not give rise to the same Avx(p) as a decapole distributed over the central region of the magnet. A decapole correction can be used to cancel the (Ap/p) 3 term in Avx(P) due to an end decapole. The quadratic, linear and constant terms in Avx(p) would then have to be corrected if necessary by the octupote, sextupole and quadrupole correction coils. In the case of ISABELLE, the end field decapole present in the dipole for naturally wound ends is b4 = 1.16 × 10 -6 cm -4. When Avx(P ) and AVy(p) are computed with the end decapole present, and an equal and opposite decapole over the center of the dipole, one finds that the remaining v-shift is largely quadratic in Ap/p, and Av x = 0.077, Avy = 0.05 for Ap/p =0.01. This u-shift can be corrected by the octupole correction coils in ISABELLE, and would require an appreciable fraction of the octupole correction provided. 3. The aperture-higher-multipole e f f e c t The presence of higher end field multipoles for which there are no correction coils can limit the good
G. Parzen /End field multipoles in superconducting magnets
f\
/ /
\/NATURAL ~ ' ~ END - -
3
--
MULTIPOLES
Z~ U x / I 0 -
\
2
2 --
\ \ \ I
\
i \
N
I
I --15
--I
Io
-5
END MULTIPOLES
.5
A
Aplp(%)
I 15
-2
-3
Fig. 1. Plot of Avx(P) due to the higher end multipoles versus Ap/p, showing the effect of the end field multipoles due to naturally wound ends in the dipoles.
field aperture by introducing a curvature in the working line, Vx(p) and Vy(p).This effect was studied
159
for the central multipoles in a previous paper [2] where the details for computing Vx(P) due to the higher multipoles, and the criteria used for the good field aperture are presented. For the case of the ISABELLE accelerator, the curvature in the working line introduced by the higher end field multipoles can make control of the transverse instability difficult. For naturally wound ends in the dipole, the end field b6 computed to be present is b 6 = -1.28 × 10 -a cm -6. The effect of this b6 on the working line is shown in fig. 1, where AVx(P) due only to the higher end field multipoles is plotted against Ap/p for naturally wound ends in the dipoles and quadrupoles. Mso plotted in fig. 1 is AVx(p) for the case where the ends in the dipoles are spaced to reduce the end field multipoles using the criterion given in ref. 2. The good field aperture is reduced by the presence of the higher end field multipoles from Ap/p = 2% to Ap/p = 1.5%.
References [1] A.D. Mclnturff, W.B. Sampson, K.E. Robins, P.F. Dahl, R. Damm, D. Kassner, J. Kaugerts and C. Lasky, Proc. of 1976 Applied Superconducting Conference (1976). [2] G. Parzen, Trans. of the 1981 IEEE Particle Accelerator Conference (to be published).
157
ORBIT EFFECTS OF THE END FIELD MULTIPOLES IN SUPERCONDUCTING MAGNETS * G. PARZEN
Brookhaven National Laboratory, Upton, New York, U.S.A. Received 9 June 1981 Two possible effects due to the end field multipoles in superconducting magnets are pointed out. One effect is the sagittaend-multipole effect in dipoles. A result of this is that the u-shift due to an end multipole is not cancelled by an equal and opposite correction multipole distributed over the central part of the dipole, which complicates the correction of the end multipole effects. A second effect is the aperture-higher-multipole effect. The presence of higher end field multipoles for which there are no correction coils may reduce the good field aperture. These two effects are illustrated by computing them for the case of the ISABELLE accelerator.
1. Introduction The end field multipoles are the field multipoles introduced by the ends of a magnet. In a superconducting magnet [ 1], the end field multipole can be considerably reduced by properly designing the ends. One approach is to properly space the turns at the ends of the magnet. A question that arises is whether such a careful design of the ends is required. This paper points out two possibly harmful effects of the end field multipoles which may make it necessary to reduce the end multipoles. One effect is the sagitta-end-multipole effect in the dipoles. In the dipoles, the central orbit of the particles may depart from the axis of the dipole by the orbit sagitta. A result of this is that the u-shift due to an end multipole, bn, is not cancelled by an equal and opposite correction multipole, -bn, in the central part of the dipole. This complicates the correction of the end multipoles using correction multipoles spread over the central part of the dipole, for not only is a correction multipole - b n required, but also the corrections bn_l, bn_2, etc. may be required. A second effect is the aperture - higher-multipole effect [2]. The presence of higher end multipoles, for which there are no correction coils, may limit the good field aperture by introducing a curvature in the working line, the dependence of the v-value on momentum. * Work performed under the auspices of the U.S. Department of Energy.
0029-554X/82/0000-0000/$02.75 © 1982 North-Holland
In this paper these two effects are examined for the end multipoles in the magnets of the ISABELLE accelerator and it is found that a design of the ends of the dipoles which reduces the end multipoles is required. If the ends are wound naturally, with no special spacers to reduce the end multipoles, it is found that correction of the effects due to the end field sectupole would require almost the entire capacity of the quadrupole trim coils provided in the present ISABELLE design. It is also found that the higher end multipoles for naturally wound ends would cause an appreciable reduction in the good field aperture.
2. Sagitta-end.multipole effect This effect may be illustrated by considering the effect of the end field sextupole. The v-shift as a function of momentum due to this field multipole is given by [21,
1 avdp) = G
1
f dSBo(1 + ~lp) 2Bob2(x s +Xp Ap/p) ,
(1) where the sextupole field is given by Bob2x 2, x s is the position of the central orbit relative to the axis of the dipole, Bo is the dipole field, and Xp Ap/p gives the position of the off momentum orbit relative to the central orbit. One can see from eq. (1), that the same integrated sextupole b2 = fds b2 will give different Av(p) depending on whether b: is concentrated at the ends,
G. Parzen /End field multipoles in superconducting magnets
158
or whether it is distributed uniformly over the central region. If we write, Avx(p) = AVo + AVlAp/p, then we will find that the linear part, AVl, is roughly the same for the central or end b : , but the constant term Avo is not the same. Thus equal and opposite integrated sextupoles in the ends and in the central region will cancel the linear term in Avx(p), but the remaining constant v-shift may be corrected using the quadrupole correction coils. For end multipotes, one finds, Av0 ~x~2 ½ [xs(L/2) + Xs(-L/2)] AVl oc ~2
Xp, center,
where x s is evaluated at the ends s =+-L/2, and Xp, center is Xp at the center of the dipole. For the central multipole, one finds, Avo ~ b-2~s. AVl ~ b2Xp,
Bobnxn where Bo is the main dipole field. The end field sextupole present in the ISABELLE dipoles is b2 = 1.49 × 10 -4 cm -2 for naturally wound ends. When Avx(P) and Avy(p) are computed using eq. (1), with this end sextupole present and an equal and opposite sextupole over the center o f the dipole, it is found that Avx(P ) = 0.48, AVy(p)=- 0.47, a constant v-shift, independent of momentum. This v-shift could be corrected by the quadrupole correction coils provided in ISABELLE. However, the quadrupole correction required would use up essentially all the capacity presently planned for the ISABELLE quadrupole correction coils. In the same way, because of the s a g i t t a - e n d effect, the v-shift caused by an end field decapole cannot be corrected by just an equal and opposite decapole distributed over the central region of the magnet. The v-shift as a function o f momentum due to a decapole field is given by 1 f ds
1
center,
X 4Bob4 (Xs3 + 3x2sXp Ap/p + 3xsXg(Ap/p) 2 where x-s = ( l / L ) f ds x s, the average sagitta over the dipole. The computed end multipoles for the ISABELLE dipoles and quadrupoles are given in table 1 for the case where the ends are wound naturally, with no special spacers to reduce the end multipoles. The results listed in table 1 are computed results. The multipoles listed are ( I / L ) f ds bn over the dipoles, where L is the dipole length and the multipole field is given by
Table 1 End field multipoles for ISABELLE dipoles and quadrupoles with naturally wound ends, where the ends are not spaced to reduce the end multipoles. The b n listed are the equivalent central multipoles which distributed over the length of the magnet would give the same integrated effect. The multipole field is given by Bobnxn, where Bo is the dipole field. The bn are computed results. Dipoles
Quadrupoles
n
bn (cm -n)
n
bn (cm -n)
0 2 4 6 8 10 12
3.62 X 10 -2 -1.49 × 10 -4 -1.16 × 10 -6 -1.28×10 -8 -1.69X10 -1° -2.57 x 10 -12 ~ , . t 2 × 10 -14
1 5 9 13 17
7.18 -5.50 -9.07 -2.5 -7.6
x 10 -3 × 10 -7 × 10 -11 ×10 -14 ×10 -18
+ X~(Ap/p) 3} ,
(2)
where the decapole field is given by Bob4×4. One can see from eq. (2) that a decapole concentrated at the ends does not give rise to the same Avx(p) as a decapole distributed over the central region of the magnet. A decapole correction can be used to cancel the (Ap/p) 3 term in Avx(P) due to an end decapole. The quadratic, linear and constant terms in Avx(p) would then have to be corrected if necessary by the octupote, sextupole and quadrupole correction coils. In the case of ISABELLE, the end field decapole present in the dipole for naturally wound ends is b4 = 1.16 × 10 -6 cm -4. When Avx(P ) and AVy(p) are computed with the end decapole present, and an equal and opposite decapole over the center of the dipole, one finds that the remaining v-shift is largely quadratic in Ap/p, and Av x = 0.077, Avy = 0.05 for Ap/p =0.01. This u-shift can be corrected by the octupole correction coils in ISABELLE, and would require an appreciable fraction of the octupole correction provided. 3. The aperture-higher-multipole e f f e c t The presence of higher end field multipoles for which there are no correction coils can limit the good
G. Parzen /End field multipoles in superconducting magnets
f\
/ /
\/NATURAL ~ ' ~ END - -
3
--
MULTIPOLES
Z~ U x / I 0 -
\
2
2 --
\ \ \ I
\
i \
N
I
I --15
--I
Io
-5
END MULTIPOLES
.5
A
Aplp(%)
I 15
-2
-3
Fig. 1. Plot of Avx(P) due to the higher end multipoles versus Ap/p, showing the effect of the end field multipoles due to naturally wound ends in the dipoles.
field aperture by introducing a curvature in the working line, Vx(p) and Vy(p).This effect was studied
159
for the central multipoles in a previous paper [2] where the details for computing Vx(P) due to the higher multipoles, and the criteria used for the good field aperture are presented. For the case of the ISABELLE accelerator, the curvature in the working line introduced by the higher end field multipoles can make control of the transverse instability difficult. For naturally wound ends in the dipole, the end field b6 computed to be present is b 6 = -1.28 × 10 -a cm -6. The effect of this b6 on the working line is shown in fig. 1, where AVx(P) due only to the higher end field multipoles is plotted against Ap/p for naturally wound ends in the dipoles and quadrupoles. Mso plotted in fig. 1 is AVx(p) for the case where the ends in the dipoles are spaced to reduce the end field multipoles using the criterion given in ref. 2. The good field aperture is reduced by the presence of the higher end field multipoles from Ap/p = 2% to Ap/p = 1.5%.
References [1] A.D. Mclnturff, W.B. Sampson, K.E. Robins, P.F. Dahl, R. Damm, D. Kassner, J. Kaugerts and C. Lasky, Proc. of 1976 Applied Superconducting Conference (1976). [2] G. Parzen, Trans. of the 1981 IEEE Particle Accelerator Conference (to be published).