- Email: [email protected]
Quadrupole magnet field mapping for FRIB
Nuclear Instruments and Methods in Physics Research B 317 (2013) 271–273
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Quadrupole magnet field mapping for FRIB M. Portillo a,⇑, A.M. Amthor a,1, S. Chouhan a, K. Cooper b, A. Gehring b, M. Hausmann a, S. Hitchcock b, J. Kwarsick b, S. Manikonda c,2, C. Sumithrarachchi b a
Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA c Argonne National Laboratory, Argonne, IL, USA b
a r t i c l e
i n f o
Article history: Received 26 February 2013 Accepted 17 July 2013 Available online 21 August 2013 Keywords: Fragment mass separator Fringing field Mapping
a b s t r a c t Extensive magnetic field map measurements have been done on a newly built superconducting quadrupole triplet with sextupole and octupole coils nested within every quadrupole. The magnetic field multipole composition and fringe field distributions have been analyzed and an improved parameterization of the field has been developed within the beam transport simulation framework. Parameter fits yielding standard deviations as low as 0.3% between measured and modeled values are reported here. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The fragment separator being designed for the Facility for Rare Isotope Beams (FRIB) will purify and transport rare isotope beams at the highest quality possible. It will be optimized for in-flight production from beams of up to 400 kW on target at energies of 200 MeV/u for uranium, and higher for lower mass products [1]. Separator acceptances as high as 40 mrad in angle and 5% in magnetic rigidity require magnets with large bores that are also compact and can have sextupole and octupole fields nested for correction of higher order aberrations. The operating range of some of the quadrupoles can require pole-tip fields where iron becomes saturated. The models used to simulate beam transport need to account for the changes in the shape of the field over the entire operating range. Furthermore, rare isotope beam quality and transport efficiency is directly related to the field quality of all magnets. It is important to maintain optimal accuracy in the field description to be used by the beam optics model to predict the properties of the beam and necessary field strengths in the fragment separator. All of the magnets that will make up the last sections of the FRIB separator are currently in use in the A1900 separator for the delivery of rare-isotope beams at the National Superconducting Cyclotron Laboratory (NSCL). Recently, a new A1900 triplet was constructed and provided the rare opportunity to measure field maps and test the accuracy of a mapper that had been used in ⇑ Corresponding author. Tel.: +1 5179087440; fax: +1 5173535967. 1 2
E-mail address: [email protected] (M. Portillo). Department of Physics and Astronomy, Bucknell University, Lewisburg, PA, USA. Advanced Magnet Lab, Palm Bay, FL, USA.
0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.07.036
the past for commissioning. It is useful to further our knowledge of field quality, especially since newer magnet types are going to be built for the next separator. The new measurements and results carried out for this study are described here. 2. Triplet and mapper setup There are five types of quadrupoles that make up the A1900 triplets. We refer to them as types QA through QE as done in [2], which also provides more details about their design and specifications. The triplet in this study is of the type QB–QC–QB, of which four others are in operation in the A1900. QB and QC quadrupoles are the only types having nested sextupole and octupole coils, which were designed for use as higher order correctors [3]. Their cross sections along the transverse plane are the same, with a pole-tip radius at 15 cm designed to have a maximum operating field of 2.37 T. At 71.5 cm the pole-tip length of QC is just over twice that of QB. The pole-tip length of QB is 32.5 cm long, which is just over two times the pole-tip radius. All three quadrupoles are mounted such that the distance between the far edges of the pole-tips is 186 cm. The nested coils lie inside the pole-tip radius and are separated by about 3 cm from the innermost warm-bore tube. The beam travels through this tube, which has an inner radius of 10 cm. The field mapper consists of a guide tube that mounts inside the warm-bore tube. Inside the guide tube a cart rolls on wheels and carries three Hall probes positioned to measure the radial component of the field along the same azimuthal angle h but varying radii. The probes lie at radii r1 ¼ 36:7 mm, r2 ¼ 59:0 mm and r3 ¼ 81:2 mm. A commercially available Hall probe system was
272
M. Portillo et al. / Nuclear Instruments and Methods in Physics Research B 317 (2013) 271–273
used which has a control system that reads back the temperature at each probe and also applies temperature compensation to the field values. The cart is pulled along the tube by a chain that is moved by a computer controlled stepper motor with rotary encoder reference. The full length of travel of the cart is 250 cm. The mapper tube and cart can make a full rotation about the beam axis, while a digital protractor is used to record the orientation of the tube.
3. Measurements In order to map any field setting of a quadrupole field, scanning was done in either a mode of rotation (h) or axial (z) motion. The beam travels in the positive z direction and in Cartesian coordinates, the mapper y-axis points against the direction of gravity and the x axis coincides with r at h ¼ 0. The magnets were mapped using both modes during different combinations of coil currents. In rotation mode, the entrance, center and exit positions of each magnet were scanned at 5° steps, and in total 34 scans were accumulated. In axial mode, we performed 133 scans with at least two angle orientations (done during separate scans) for most quadrupole settings. Scans at up to 10 angles were done for a few select settings. The two outer short quadrupoles, QB1 and QB2 (in order experienced by the beam) could be excited simultaneously without having overlapping fringe fields; hence, both were measured in a single axial scan. For pure quadrupole excitations, axial scans were measured for currents varying by steps of 10% of the maximum allowed quadrupole current for each magnet. The maximum is 70 A for QB types and 80 A for QC. Field distributions along z measured at r1 for quadrupole excitations of QB1 and QC are shown in Fig. 1, where z = 0 is taken to be the center of each respective magnet. Using both modes and employing a laser tracker, the accuracy of the mapper setup was investigated by measuring the position of a target centered on the mapper cart. From the measurements we evaluated the standard deviations r in all three dimensions of the cart position, relative to the nominal beam axis. The results show that during axial scanning, the cart moves with rx = 0.04 mm and ry = 0.07 mm. The difference between the tracker z position and encoder reference yielded a standard deviation of rz = 0.24 mm, and is mainly attributable to the flexibility in the drive chain pulling the cart. During rotation scans the center
position deviated with rz = 0.09 mm and in roughly circular motion radially with a deviation of rr = 0.24 mm. These values are averaged over four scans taken at evenly spaced z-positions along the tube. We find that reducing rr would have the greatest impact on increasing the accuracy of the mapper device. 4. Analysis method and results Rotation scans have been analyzed by Fourier analysis in order to decompose the field at a given z. The parametric model adopted for this analysis assumes the radial component of the field in cylindrical coordinates is given by [4],
Br ðr; h; zÞ ¼
1 X
½Br;n sinðnhÞ þ Ar;n cosðnhÞ
ð1Þ
n¼1
We refer to Br;n as the normal term and Ar;n as the skew term, and both are functions of r and z. The normal term is expanded in r as given by,
Br;n ðr; zÞ ¼
2mþn1 1 X r bn;m ðzÞ r 0 m¼0
ð2Þ
A similar expansion using an;m terms also applies for the skew component. The pole-tip radius of the quadrupoles has been chosen as the reference radius r 0 . For the analysis of z scans, the field calculated by the transport code COSY INFINITY [5] has been fitted to the data by optimizing the effective length, pole-tip field, and the coefficients of Enge functions [6]. These parameters are optimized to minimize the standard deviation of the residual (difference between measured and calculated) values with the use of a simplex algorithm. The standard deviation used here is evaluated from the sum of squares of the residuals divided by the field value at the center of the magnet, ðBmeasured Bmodel Þ=Bmodel ðz ¼ 0Þ, where we treat z ¼ 0 as the center for each magnet. This normalization allows the standard deviation to be independent of field excitation and radial position of any probe. The fields are evaluated to 5th order at every measured point using an algorithm adopted from Manikonda [7]. In the case of the quadrupole, this means that the expansion in r goes up to m ¼ 2 in Eq. (2). Since a few thousand points are involved and a few hundred iterations are required in the fitting algorithm, the computation time on 64 bit Linux machines was typically about 2 h for a fit that simultaneously included scans at two angles. One clear disadvantage using this technique is how computationally expensive the evaluation can become as more scans are integrated into the fit. 4.1. Quadrupole analysis
Fig. 1. Hall probe field at r 1 and 45° for QB1 and QC over the allowed operating range for quadrupole excitation currents.
Under pure quadrupole coil excitation, the decomposed Br;n and Ar;n components from the Fourier analysis are determined from numerical integration of the scans in h. For all three quadrupoles, the scans were done at the center and edge positions of the yoke while under 50 A excitation. At all three positions, the ratio of all other Br;n terms, up to n ¼ 10, are found to typically be below the estimated error in the analysis. We estimate that the error in the Fourier analysis up to about n ¼ 4 is approximately 0.5%. The error seems to gradually increase for higher n components, mostly due to numerical integration error when measuring at 5° steps. This determination was done after correcting for offsets that were extracted from the results of all pure quadrupole excitations. Offsets due to imperfections in the alignment and absolute field reading of each probe could be estimated from the Fourier analysis. Absolute field offsets show up in the monopole term Ar;0 , which should be zero. To minimize the chance of introducing errors, we
M. Portillo et al. / Nuclear Instruments and Methods in Physics Research B 317 (2013) 271–273
determined these from h scans with all currents off. For all other Ar;n terms the magnitudes are susceptible to imperfections in the probe alignment. Based on the mechanical properties and doing sensitivity studies, the effects are mostly attributed to misalignment in h for each probe, relative to its mounting position. The offsets are determined by minimizing the Ar;2 for cases of pure quadrupole excitation. Once these corrections had been applied, the results show that the fields from all three quadrupoles are aligned within approximately 0.2° of each other. Results from the analysis of the z scans support those of the Fourier analysis mention above. In cases where up to 10 angles had been measured, it was possible to let the fit search for multipole impurities up to 5th order. Relative to the fitted quadrupole strength, all the other multipole components were found to have strengths consistent with zero within the uncertainties. In the case of the short quadrupoles, the entrance and exit fringe fields overlap significantly, as can be seen in Fig. 1. This effect is more pronounced at higher fields, as the effective length of the quadrupole decreases with increasing field. At higher fields this can be observed also for QC, albeit to a lesser extent than for QB. This overlapping of the fringe fields also causes a peak in the residuals at z = 0 in Fig. 2 if the entrance and exit Enge functions are fitted separately. This is overcome by fitting instead the product of the two Enge functions using an algorithm developed by Suzuki [8]. As shown by the bottom plot, this dropped the deviation by about 0.1% and resulted in typical standard deviations in the range of 0.3–0.7% of the peak field. In the case of QB, the overlapping of the fringe fields also leads to the nonlinear terms in Eq. (2) becoming noticeable even at z = 0. As a result, the field at the pole-tip radius, which is used in COSY INFINITY and other ion-optics codes to parameterize the strength of the quadrupole, can no longer be determined accurately by a linear extrapolation of the measured fields versus radius. In the case of QB, the linear approximation of the pole tip field deviates by about 1% from the result obtained by fitting the entire data set, which takes into account the nonlinear terms b2;1 and b2;2 .
4.2. Results for sextupole and octupole excitations The pure sextupole excitations were measured at 36 A, close to the maximum allowed current. The results of the Fourier analysis
273
on the rotation scans exhibit a Br;1 (dipole) component that is about equal to the Br;3 (sextupole) component at r1 . All other components are below the measurable accuracy. The induced dipole component is caused by a mismatch between the sextupole geometry and that of the quadrupole iron yoke. As the quadrupole excitation saturates the iron, this dipole term is expected to subside. To obtain the best accuracy when fitting the z scan data, the fits included scans that were measured at all six peak Br orientations of the sextupole (i.e. h ¼ 0 ; 60 ; . . . ; 300 ). This set of angles is sufficient, considering that all other harmonic components were much less significant, based on the Fourier analysis of the h scans. The option in COSY INFINITY used for the multipole normally does not include the n ¼ 1 field; hence, we have implemented an algorithm that allows for this. The resulting r values from the residuals were about 2%. This is about four times greater than those obtained for the quadrupole, which results from having less uniformity along both z and h. This is reasonable considering that the yoke is not shaped to help enhance the uniformity of the sextupole field and is also farther away from the coils. The pure octupole excitations were measured at 18 A, close to the maximum allowed current. Other than the n ¼ 4 component, no other appreciable ones were observed above 0.5% for all three magnets; hence, it was not crucial to scan at all peak Br orientations of the octupole (i.e. h=22.5°, 67.5°, . . ., 337.5°). The r values obtained for the octupole excitations are about 7%. The return yoke is furthest from these coils, hence it is the excitation with the least chance of having enhanced uniformity by the return yoke. One of the notable qualities found by this study is that for all three multipole excitations, the effective lengths and pole-tip fields between the two QB quadrupoles were within 0.5% of each other. In this respect, the two are identical within the accuracy of the mapper. 5. Summary The magnetic fields of a superconducting quadrupole triplet with nested sextupole and octupole coils have been mapped and the results have been used to parameterize the fields in the beam transport code. The method of fitting the measured and modeled fields can reduce the amount of mapping necessary, but requires some assumptions and can require a fair amount of computation. Further knowledge and experience has been gained that is valuable for the production of future magnets that are needed for the FRIB separator. Acknowledgments This work is supported by the U.S. Department of Energy under Cooperative Agreement DE-SC0000661 and Contract No. DE-AC0206CH11357. The authors are grateful to Georg Bollen, Dave Morrissey, Jerry Nolen, Jr., and Brad Sherrill for advice on the subject. Special thanks go to our RIKEN collaborators Toshiyuki Kubo, Hiroshi Suzuki and Hiroyuki Takeda for many useful discussions and exchange of ideas on the topic. References
Fig. 2. Plot of residual for one of the QB quadrupoles at 70 A pure quadrupole excitation. The standard deviation and fitted pole-tip field are shown in each plot.
[1] R. York et al., Proceedings of PAC2011, New York, NY, USA, 2011, pp. 2561– 2565. [2] A. Zeller et al., Adv. Cryog. Eng. 43A (1998) 245–252. [3] X. Wu et al., Proceedings of PAC97 Conference, Vancouver, Canada, 1997, pp. 198–200. [4] J. Napolitano, T. Hunter, Nucl. Instrum. Methods Phys. Res. A 301 (1991) 465– 472. [5] K. Makino, M. Berz, Nucl. Instrum. Methods Phys. Res. A 558 (2005) 346–350. [6] H. Enge, Rev. Sci. Instrum. 35 (1964) 278–287. [7] S.L. Manikonda, Private communications (July 2010). [8] H. Suzuki, Private communications (August 2012).
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Quadrupole magnet field mapping for FRIB M. Portillo a,⇑, A.M. Amthor a,1, S. Chouhan a, K. Cooper b, A. Gehring b, M. Hausmann a, S. Hitchcock b, J. Kwarsick b, S. Manikonda c,2, C. Sumithrarachchi b a
Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA c Argonne National Laboratory, Argonne, IL, USA b
a r t i c l e
i n f o
Article history: Received 26 February 2013 Accepted 17 July 2013 Available online 21 August 2013 Keywords: Fragment mass separator Fringing field Mapping
a b s t r a c t Extensive magnetic field map measurements have been done on a newly built superconducting quadrupole triplet with sextupole and octupole coils nested within every quadrupole. The magnetic field multipole composition and fringe field distributions have been analyzed and an improved parameterization of the field has been developed within the beam transport simulation framework. Parameter fits yielding standard deviations as low as 0.3% between measured and modeled values are reported here. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The fragment separator being designed for the Facility for Rare Isotope Beams (FRIB) will purify and transport rare isotope beams at the highest quality possible. It will be optimized for in-flight production from beams of up to 400 kW on target at energies of 200 MeV/u for uranium, and higher for lower mass products [1]. Separator acceptances as high as 40 mrad in angle and 5% in magnetic rigidity require magnets with large bores that are also compact and can have sextupole and octupole fields nested for correction of higher order aberrations. The operating range of some of the quadrupoles can require pole-tip fields where iron becomes saturated. The models used to simulate beam transport need to account for the changes in the shape of the field over the entire operating range. Furthermore, rare isotope beam quality and transport efficiency is directly related to the field quality of all magnets. It is important to maintain optimal accuracy in the field description to be used by the beam optics model to predict the properties of the beam and necessary field strengths in the fragment separator. All of the magnets that will make up the last sections of the FRIB separator are currently in use in the A1900 separator for the delivery of rare-isotope beams at the National Superconducting Cyclotron Laboratory (NSCL). Recently, a new A1900 triplet was constructed and provided the rare opportunity to measure field maps and test the accuracy of a mapper that had been used in ⇑ Corresponding author. Tel.: +1 5179087440; fax: +1 5173535967. 1 2
E-mail address: [email protected] (M. Portillo). Department of Physics and Astronomy, Bucknell University, Lewisburg, PA, USA. Advanced Magnet Lab, Palm Bay, FL, USA.
0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.07.036
the past for commissioning. It is useful to further our knowledge of field quality, especially since newer magnet types are going to be built for the next separator. The new measurements and results carried out for this study are described here. 2. Triplet and mapper setup There are five types of quadrupoles that make up the A1900 triplets. We refer to them as types QA through QE as done in [2], which also provides more details about their design and specifications. The triplet in this study is of the type QB–QC–QB, of which four others are in operation in the A1900. QB and QC quadrupoles are the only types having nested sextupole and octupole coils, which were designed for use as higher order correctors [3]. Their cross sections along the transverse plane are the same, with a pole-tip radius at 15 cm designed to have a maximum operating field of 2.37 T. At 71.5 cm the pole-tip length of QC is just over twice that of QB. The pole-tip length of QB is 32.5 cm long, which is just over two times the pole-tip radius. All three quadrupoles are mounted such that the distance between the far edges of the pole-tips is 186 cm. The nested coils lie inside the pole-tip radius and are separated by about 3 cm from the innermost warm-bore tube. The beam travels through this tube, which has an inner radius of 10 cm. The field mapper consists of a guide tube that mounts inside the warm-bore tube. Inside the guide tube a cart rolls on wheels and carries three Hall probes positioned to measure the radial component of the field along the same azimuthal angle h but varying radii. The probes lie at radii r1 ¼ 36:7 mm, r2 ¼ 59:0 mm and r3 ¼ 81:2 mm. A commercially available Hall probe system was
272
M. Portillo et al. / Nuclear Instruments and Methods in Physics Research B 317 (2013) 271–273
used which has a control system that reads back the temperature at each probe and also applies temperature compensation to the field values. The cart is pulled along the tube by a chain that is moved by a computer controlled stepper motor with rotary encoder reference. The full length of travel of the cart is 250 cm. The mapper tube and cart can make a full rotation about the beam axis, while a digital protractor is used to record the orientation of the tube.
3. Measurements In order to map any field setting of a quadrupole field, scanning was done in either a mode of rotation (h) or axial (z) motion. The beam travels in the positive z direction and in Cartesian coordinates, the mapper y-axis points against the direction of gravity and the x axis coincides with r at h ¼ 0. The magnets were mapped using both modes during different combinations of coil currents. In rotation mode, the entrance, center and exit positions of each magnet were scanned at 5° steps, and in total 34 scans were accumulated. In axial mode, we performed 133 scans with at least two angle orientations (done during separate scans) for most quadrupole settings. Scans at up to 10 angles were done for a few select settings. The two outer short quadrupoles, QB1 and QB2 (in order experienced by the beam) could be excited simultaneously without having overlapping fringe fields; hence, both were measured in a single axial scan. For pure quadrupole excitations, axial scans were measured for currents varying by steps of 10% of the maximum allowed quadrupole current for each magnet. The maximum is 70 A for QB types and 80 A for QC. Field distributions along z measured at r1 for quadrupole excitations of QB1 and QC are shown in Fig. 1, where z = 0 is taken to be the center of each respective magnet. Using both modes and employing a laser tracker, the accuracy of the mapper setup was investigated by measuring the position of a target centered on the mapper cart. From the measurements we evaluated the standard deviations r in all three dimensions of the cart position, relative to the nominal beam axis. The results show that during axial scanning, the cart moves with rx = 0.04 mm and ry = 0.07 mm. The difference between the tracker z position and encoder reference yielded a standard deviation of rz = 0.24 mm, and is mainly attributable to the flexibility in the drive chain pulling the cart. During rotation scans the center
position deviated with rz = 0.09 mm and in roughly circular motion radially with a deviation of rr = 0.24 mm. These values are averaged over four scans taken at evenly spaced z-positions along the tube. We find that reducing rr would have the greatest impact on increasing the accuracy of the mapper device. 4. Analysis method and results Rotation scans have been analyzed by Fourier analysis in order to decompose the field at a given z. The parametric model adopted for this analysis assumes the radial component of the field in cylindrical coordinates is given by [4],
Br ðr; h; zÞ ¼
1 X
½Br;n sinðnhÞ þ Ar;n cosðnhÞ
ð1Þ
n¼1
We refer to Br;n as the normal term and Ar;n as the skew term, and both are functions of r and z. The normal term is expanded in r as given by,
Br;n ðr; zÞ ¼
2mþn1 1 X r bn;m ðzÞ r 0 m¼0
ð2Þ
A similar expansion using an;m terms also applies for the skew component. The pole-tip radius of the quadrupoles has been chosen as the reference radius r 0 . For the analysis of z scans, the field calculated by the transport code COSY INFINITY [5] has been fitted to the data by optimizing the effective length, pole-tip field, and the coefficients of Enge functions [6]. These parameters are optimized to minimize the standard deviation of the residual (difference between measured and calculated) values with the use of a simplex algorithm. The standard deviation used here is evaluated from the sum of squares of the residuals divided by the field value at the center of the magnet, ðBmeasured Bmodel Þ=Bmodel ðz ¼ 0Þ, where we treat z ¼ 0 as the center for each magnet. This normalization allows the standard deviation to be independent of field excitation and radial position of any probe. The fields are evaluated to 5th order at every measured point using an algorithm adopted from Manikonda [7]. In the case of the quadrupole, this means that the expansion in r goes up to m ¼ 2 in Eq. (2). Since a few thousand points are involved and a few hundred iterations are required in the fitting algorithm, the computation time on 64 bit Linux machines was typically about 2 h for a fit that simultaneously included scans at two angles. One clear disadvantage using this technique is how computationally expensive the evaluation can become as more scans are integrated into the fit. 4.1. Quadrupole analysis
Fig. 1. Hall probe field at r 1 and 45° for QB1 and QC over the allowed operating range for quadrupole excitation currents.
Under pure quadrupole coil excitation, the decomposed Br;n and Ar;n components from the Fourier analysis are determined from numerical integration of the scans in h. For all three quadrupoles, the scans were done at the center and edge positions of the yoke while under 50 A excitation. At all three positions, the ratio of all other Br;n terms, up to n ¼ 10, are found to typically be below the estimated error in the analysis. We estimate that the error in the Fourier analysis up to about n ¼ 4 is approximately 0.5%. The error seems to gradually increase for higher n components, mostly due to numerical integration error when measuring at 5° steps. This determination was done after correcting for offsets that were extracted from the results of all pure quadrupole excitations. Offsets due to imperfections in the alignment and absolute field reading of each probe could be estimated from the Fourier analysis. Absolute field offsets show up in the monopole term Ar;0 , which should be zero. To minimize the chance of introducing errors, we
M. Portillo et al. / Nuclear Instruments and Methods in Physics Research B 317 (2013) 271–273
determined these from h scans with all currents off. For all other Ar;n terms the magnitudes are susceptible to imperfections in the probe alignment. Based on the mechanical properties and doing sensitivity studies, the effects are mostly attributed to misalignment in h for each probe, relative to its mounting position. The offsets are determined by minimizing the Ar;2 for cases of pure quadrupole excitation. Once these corrections had been applied, the results show that the fields from all three quadrupoles are aligned within approximately 0.2° of each other. Results from the analysis of the z scans support those of the Fourier analysis mention above. In cases where up to 10 angles had been measured, it was possible to let the fit search for multipole impurities up to 5th order. Relative to the fitted quadrupole strength, all the other multipole components were found to have strengths consistent with zero within the uncertainties. In the case of the short quadrupoles, the entrance and exit fringe fields overlap significantly, as can be seen in Fig. 1. This effect is more pronounced at higher fields, as the effective length of the quadrupole decreases with increasing field. At higher fields this can be observed also for QC, albeit to a lesser extent than for QB. This overlapping of the fringe fields also causes a peak in the residuals at z = 0 in Fig. 2 if the entrance and exit Enge functions are fitted separately. This is overcome by fitting instead the product of the two Enge functions using an algorithm developed by Suzuki [8]. As shown by the bottom plot, this dropped the deviation by about 0.1% and resulted in typical standard deviations in the range of 0.3–0.7% of the peak field. In the case of QB, the overlapping of the fringe fields also leads to the nonlinear terms in Eq. (2) becoming noticeable even at z = 0. As a result, the field at the pole-tip radius, which is used in COSY INFINITY and other ion-optics codes to parameterize the strength of the quadrupole, can no longer be determined accurately by a linear extrapolation of the measured fields versus radius. In the case of QB, the linear approximation of the pole tip field deviates by about 1% from the result obtained by fitting the entire data set, which takes into account the nonlinear terms b2;1 and b2;2 .
4.2. Results for sextupole and octupole excitations The pure sextupole excitations were measured at 36 A, close to the maximum allowed current. The results of the Fourier analysis
273
on the rotation scans exhibit a Br;1 (dipole) component that is about equal to the Br;3 (sextupole) component at r1 . All other components are below the measurable accuracy. The induced dipole component is caused by a mismatch between the sextupole geometry and that of the quadrupole iron yoke. As the quadrupole excitation saturates the iron, this dipole term is expected to subside. To obtain the best accuracy when fitting the z scan data, the fits included scans that were measured at all six peak Br orientations of the sextupole (i.e. h ¼ 0 ; 60 ; . . . ; 300 ). This set of angles is sufficient, considering that all other harmonic components were much less significant, based on the Fourier analysis of the h scans. The option in COSY INFINITY used for the multipole normally does not include the n ¼ 1 field; hence, we have implemented an algorithm that allows for this. The resulting r values from the residuals were about 2%. This is about four times greater than those obtained for the quadrupole, which results from having less uniformity along both z and h. This is reasonable considering that the yoke is not shaped to help enhance the uniformity of the sextupole field and is also farther away from the coils. The pure octupole excitations were measured at 18 A, close to the maximum allowed current. Other than the n ¼ 4 component, no other appreciable ones were observed above 0.5% for all three magnets; hence, it was not crucial to scan at all peak Br orientations of the octupole (i.e. h=22.5°, 67.5°, . . ., 337.5°). The r values obtained for the octupole excitations are about 7%. The return yoke is furthest from these coils, hence it is the excitation with the least chance of having enhanced uniformity by the return yoke. One of the notable qualities found by this study is that for all three multipole excitations, the effective lengths and pole-tip fields between the two QB quadrupoles were within 0.5% of each other. In this respect, the two are identical within the accuracy of the mapper. 5. Summary The magnetic fields of a superconducting quadrupole triplet with nested sextupole and octupole coils have been mapped and the results have been used to parameterize the fields in the beam transport code. The method of fitting the measured and modeled fields can reduce the amount of mapping necessary, but requires some assumptions and can require a fair amount of computation. Further knowledge and experience has been gained that is valuable for the production of future magnets that are needed for the FRIB separator. Acknowledgments This work is supported by the U.S. Department of Energy under Cooperative Agreement DE-SC0000661 and Contract No. DE-AC0206CH11357. The authors are grateful to Georg Bollen, Dave Morrissey, Jerry Nolen, Jr., and Brad Sherrill for advice on the subject. Special thanks go to our RIKEN collaborators Toshiyuki Kubo, Hiroshi Suzuki and Hiroyuki Takeda for many useful discussions and exchange of ideas on the topic. References
Fig. 2. Plot of residual for one of the QB quadrupoles at 70 A pure quadrupole excitation. The standard deviation and fitted pole-tip field are shown in each plot.
[1] R. York et al., Proceedings of PAC2011, New York, NY, USA, 2011, pp. 2561– 2565. [2] A. Zeller et al., Adv. Cryog. Eng. 43A (1998) 245–252. [3] X. Wu et al., Proceedings of PAC97 Conference, Vancouver, Canada, 1997, pp. 198–200. [4] J. Napolitano, T. Hunter, Nucl. Instrum. Methods Phys. Res. A 301 (1991) 465– 472. [5] K. Makino, M. Berz, Nucl. Instrum. Methods Phys. Res. A 558 (2005) 346–350. [6] H. Enge, Rev. Sci. Instrum. 35 (1964) 278–287. [7] S.L. Manikonda, Private communications (July 2010). [8] H. Suzuki, Private communications (August 2012).