Study on the fringe field and the field interference effects of quadrupoles in the Rapid Cycling Synchrotron of the China Spallation Neutron Source

Nuclear Inst. and Methods in Physics Research, A 927 (2019) 424–428

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Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima

Study on the fringe field and the field interference effects of quadrupoles in the Rapid Cycling Synchrotron of the China Spallation Neutron Source Jianliang Chen a,b,c , Shouyan Xu a,b , Xiaohan Lu a,b , Yuwen An a,b , Yong Li a,b , Sheng Wang a,b,c ,∗ a

Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Dongguan Neutron Science Center, Dongguan 523803, China c University of Chinese Academy of Sciences, Beijing 100049, China b

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Keywords: CSNS/RCS Quadrupoles Fringe field effects Fringe field interference effects

ABSTRACT The Rapid Cycling Synchrotron (RCS) of the China Spallation Neutron Source (CSNS) accelerates beam from 80 MeV to 1.6 GeV to strike a solid metal target to produce spallation neutrons with a repetition rate of 25 Hz. In this kind of high bunch intensity synchrotrons, to suppress the space-charge effect, the large acceptance is always required. For CSNS, with the design acceptance of 540𝜋 mm⋅mrad, the apertures of quadrupoles are quite large, which are comparable to the effective magnet lengths. In this case, the fringe field effect cannot be neglected. For some quadrupoles, the interval between the quadrupole and its neighbor magnet is comparable to the aperture of quadrupole, and the effect of field interference need to be considered. The fringe field and field interference effects have been studied, and by including these effects, the lattice was re-matched based on slicing model of quadrupoles to include these effects. After the re-matched lattice was applied to the beam commissioning of the RCS of CSNS, the measured twiss parameters are very close to the design parameters.

1. Introduction

1.2. Beam commissioning with design lattice

1.1. Introduction of CSNS/RCS

The lattice design of CSNS/RCS is based on hard edge model of quadrupoles, and the nominal tunes (𝜈𝑥 /𝜈𝑦 ) are 4.86/4.78 which stay away from fourth-order resonance lines. In the first phase of the beam commissioning, the RCS was operated at Direct Current (DC) mode like a storage ring without acceleration and the online model of quadrupoles is based on hard edge model. As shown in Fig. 1, the measured tunes in the horizontal and vertical plane were 4.761 and 4.589 respectively, which seriously deviated from the nominal values of 4.86 and 4.78. Such large tune deviation is unacceptable for high intensity beam commissioning.

The China Spallation Neutron Source (CSNS) is an accelerator-based science facility. CSNS is designed to accelerate proton beam pulses to 1.6 GeV for striking a solid metal target to produce spallation neutrons. CSNS accelerators consist of an 80 MeV Linac and a Rapid Cycling Synchrotron (RCS). CSNS/RCS accumulates and accelerates protons from 80 MeV to 1.6 GeV with a repetition rate of 25 Hz. The CSNS/RCS consists of 24 dipoles, 48 quadrupoles, 16 sextupoles and 32 dipole correctors. The lattice is based on triplet cells with four-fold symmetry structure [1–3]. For CSNS/RCS, the tune shift due to the space charge is 0.3 [4]. In this kind of high bunch intensity synchrotron, to suppress the space-charge effect, the large acceptance is always required. With the design acceptance of 540𝜋 mm mrad, the apertures of quadrupoles are quite large, which are comparable to the effective lengths, as listed in Table 1. In this case, the fringe field effect cannot be neglected. For some quadrupoles, since the interval between the quadrupole and its neighbor magnet is comparable to the aperture of quadrupole, the effect of field interference, which results in the reduction of integral field, need to be considered [5,6]. ∗

2. The effects of fringe field and interference of quadrupoles in CSNS/RCS 2.1. Fringe field distributions of quadrupoles in CSNS/RCS There are 4 types of quadrupoles in the RCS. The magnetic field distribution along the beam trajectory of the quadrupoles were measured by using the Hall probe system [5]. The Hall probe system consists of the marble platform, air-floatation equipment, one-Axis hall sensor, digital teslameter/gauss meter and the UMAC hardware. The position accuracy is better than ± 10 μm and an ambient temperature

Corresponding author at: Dongguan Neutron Science Center, Dongguan 523803, China. E-mail address: [email protected] (S. Wang).

https://doi.org/10.1016/j.nima.2019.03.002 Received 23 November 2018; Received in revised form 22 February 2019; Accepted 3 March 2019 Available online 7 March 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.

J. Chen, S. Xu, X. Lu et al.

Nuclear Inst. and Methods in Physics Research, A 927 (2019) 424–428

simulated field distributions along the beam trajectory of 222Q with and without interference are plotted in Fig. 4. The measurement results show that the normalized integral gradient reduction of 222Q due to the interference with adjacent sextupoles can get up to 2.3% [5]. 2.2. The effects of the fringe field and interference The slicing model was adopted to analyze the fringe field effects of quadrupoles in CSNS/RCS [7]. The larger the number of slices is, the more accurate the calculation results are. As shown in Fig. 5, the calculation results converge with the increase of the number of slices. However, the number of slices cannot be too large. Otherwise the magnetic field data processing will be particularly time-consuming. Considering the aperture of different types of quadrupoles, the number of slices for 206Q and 222Q was chosen as 9, while it was 11 for 253Q and 272Q. The effects of fringe field were calculated on the basis of the measured field distributions of quadrupoles introduced in Section 2.1. The comparison of the tunes between the measured results and the calculated results based on the hard edge model and slicing model of quadrupoles are shown in Table 2. The calculated tunes of CSNS/RCS based on slicing model were 4.783/4.573. Compared with the nominal tunes 4.86/4.78, the tune-shifts induced by fringe field effects can get down 0.077/0.207 respectively. Furthermore, considering interference effects of all quadrupoles, the calculated tunes were 4.759/4.604, which seriously deviated from the nominal values of 4.860/4.780 and were very close to the measured values of 4.761/4.589. To include the fringe field and interference effects, the lattice was re-matched to the nominal tunes 4.860/4.780 based on slicing model of quadrupoles.

Fig. 1. The measured tunes of the design lattice.

Table 1 Main parameters of the quadrupoles. Types

Aperture (mm)

Yoke length (mm)

Effective length (mm)

Aspect ratio

206Q 222Q 253Q 206Q

206 222 253 272

320 370 510 810

410 450 620 900

0.502 0.493 0.408 0.302

3. Application in CSNS/RCS commissioning The re-matched lattice based on slicing model of quadrupoles was applied to CSNS/RCS beam commissioning. The beam was accumulated successfully and the beam transmission rate reached 100% at DC mode. The measured twiss parameters were measured and compared with the calculated results based on hard edge model and slicing model respectively.

Table 2 The measured tunes and the calculated values based on slicing model. Nominal tunes

Measured tunes

Calculated tunes based on slicing model

4.860/4.780

4.761/4.589

Only fringe field effects Interference effects included 4.783/4.573

3.1. Tunes

4.759/4.604

By performing Fast Fourier Transform (FFT) to Turn-By-Turn (TBT) data of Beam Position Monitor (BPM), the measured tunes were obtained. As shown in Fig. 6, the measured tunes in the horizontal and vertical plane were 4.855 and 4.783 respectively at DC mode, which were very close to the calculated values of 4.860 and 4.780 based on slicing model, but deviated seriously from the calculated values of 4.936 and 5.005 based on hard edge model.

Table 3 The RMS deviations of orbit response between measurement and the calculated values (Unit: mm/mrad). Model

R1DH01

R3DH04

R2DV08

R4DV11

Slicing model Hard edge model

0.68 6.83

0.59 6.65

0.66 43.51

0.48 59.16

3.2. Orbit response matrices ◦ C.

is controlled at 22 ± 2 Since the field on axis in a quadrupole is zero, the measurements are carried out on two lines (x = ± 50 mm, y = 0 mm) with a step size of 5 mm in z direction. The magnetic field distribution along the beam trajectory was calculated according to the magnetic field measured results on two lines at x = ± 50 mm. The values of magnetic field of 206Q, 222Q, 253Q, and 272Q are measured at 1103A, 1100A, 1270A and 1356A respectively, as shown in Fig. 2. The core-to-core distances between 222Q and two neighboring sextupoles are 200 mm/275 mm, which are comparable to the aperture of 222Q. The layout of the 222Q and neighboring sextupoles is shown in Fig. 3. The field distributions with the fringe field interference were simulated by using the electromagnetic software OPERA. By using the software OPERA, the integral gradient reduction induced by the fringe field interference was also calculated. The simulation results by using the software OPERA were checked by using the rotating coil system. The calculated reduction of the integrated field strength agrees well with the measurement results by using the rotating coil system. The

The orbit responses, which are the beam orbit deviations when the beam is kicked by steering magnets, are determined absolutely by the first order beam optics. There are 16 horizontal and 16 vertical correctors, and 32 horizontal and 32 vertical BPMs in CSNS/RCS. The responses of BPMs to correctors were measured at DC mode and compared with the calculated values based on hard edge model and slicing model. The calculated orbit responses were obtained by using Accelerator Toolbox (AT) [8]. Fig. 7 shows the comparison of responses of BPMs to the horizontal correctors of R1DH01 and R3DH04 and the vertical correctors of R2DV08 and R4DV11 between the measured results and the calculated values. Table 3 shows the RMS deviations of orbit response between the measurement results and the calculated values, which are expressed as √ 𝑋𝑅𝑀𝑆 = 425

∑𝑁 𝑖=1

𝑁

𝑥2𝑖

(1)

J. Chen, S. Xu, X. Lu et al.

Nuclear Inst. and Methods in Physics Research, A 927 (2019) 424–428

Fig. 2. Field distributions of four types of quadrupoles. (The horizontal coordinate represents the longitudinal distance s from the center of the quadrupole, and the vertical coordinate represents the measured magnetic field.)

Fig. 3. The layout of the 222Q and neighboring sextupoles.

Fig. 5. The influence of the number of slices on the calculated tunes.

Fig. 4. Normalized Field distributions of 222Q with and without field interference. (The horizontal coordinate represents the longitudinal distance s from the center of the quadrupole, and the vertical coordinate represents the normalized field gradient.)

slicing model are much smaller than the results between measurement and the calculated values based on hard edge model. It is obvious that the measured orbit responses are very close to the calculated values based on slicing model, and seriously deviate from the values based on hard edge model, especially for the vertical correctors of R2DV08 and R4DV11.

where 𝑋𝑖 represents the orbit deviation at the 𝑖-th BPM between measurement and calculated values based on slicing model or hard edge model, 𝑁 represents the number of BPMs. The RMS deviations of orbit response between measurement and the calculated values based on 426

J. Chen, S. Xu, X. Lu et al.

Nuclear Inst. and Methods in Physics Research, A 927 (2019) 424–428

Fig. 6. The measured tunes of re-matched lattice.

3.3. Fudge factors of the quadrupoles The fudge factor [9] of a quadrupole is defined by 𝐹 =

𝐺𝑓 𝑖𝑡 𝐺𝑑𝑒𝑠𝑖𝑔𝑛

(2)

where 𝐺𝑓 𝑖𝑡 is the quadrupole gradient value predicted by LOCO (Linear Optics from Closed Orbit) [10], and 𝐺𝑑𝑒𝑠𝑖𝑔𝑛 is the theoretical quadrupole gradient value from the designed model. LOCO technique is a powerful tool to tune linear optics in the ring. The fudge factor can be used to indicate the difference between the design model and the model of the real machine. The fudge factor was analyzed based both on hard edge model of design lattice and slicing model of re-matched lattice. The normalized magnetic field gradient is defined by 𝑘(𝑠) =

1 𝐺(𝑠) 𝐵𝜌

Fig. 7. The measured orbit response and the calculated values based on hard edge model and slicing model.

(3)

where 𝐺(𝑠) is magnetic field gradient, and 𝐵𝜌 is magnetic rigidity. The distribution of the normalized magnetic field gradient is independent of exciting current. During the process of analyzing fudge factor based on slicing model, the magnetic field gradients of all slices of the same quadrupole were adjusted simultaneously to match the experimental orbit response matrices. In other words, the quadrupole gradient of every slice was changed with the same ratio as to the central slice. The fudge factor, analyzed based on slicing model, can be defined by 𝐹𝐿 =

𝐺𝐿𝑓 𝑖𝑡 𝐺𝐿𝑟𝑒−𝑚𝑎𝑡𝑐ℎ𝑒𝑑

(4)

where 𝐺𝐿𝑓 𝑖𝑡 is the integral field of the quadrupole predicted by LOCO, and 𝐺𝐿𝑟𝑒−𝑚𝑎𝑡𝑐ℎ𝑒𝑑 the theoretical integral field of the quadrupole from the re-matched model based on the slicing model. As shown in Fig. 8, the analyzed fudge factors based on slicing model of re-matched lattice were less than 1%, while the analyzed fudge factors based on hard edge model of design lattice were up to 4%. As a result, the slicing model, considering both the fringe field effects and interference effects, is much more close to the model of real machine than the hard edge model.

Fig. 8. The fudge factors of quadrupoles based on hard edge model of design lattice and slicing model of re-matched lattice.

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J. Chen, S. Xu, X. Lu et al.

Nuclear Inst. and Methods in Physics Research, A 927 (2019) 424–428

Fig. 9 shows the comparison of beta functions between the measured results and the calculated values based on hard edge model and slicing model. Fig. 10 shows the comparison of the relative beta-beat between hard edge model and slicing model. The relative beta-beat in 𝑥-direction based on slicing model ranges from −6.09% to 8.46%, and that based on hard edge model ranges from −5.05% to 13.29%. The relative beta-beat in 𝑦-direction based on slicing model ranges from −9% to 9.99%, and that based on hard edge model ranges from −57.40% to 115%. The measured beta functions are much more close to the calculated values based on slicing model. 4. Conclusion Fig. 9. The measured beta functions and the calculated values based on hard edge model and slicing model.

To suppress the space-charge effect, CSNS/RCS employs large aperture quadrupoles to provide a large acceptance. The large aperture of quadrupoles cause serious fringe field effects. For some quadrupoles, since the interval between the quadrupole and its neighbor magnet is comparable to the aperture of quadrupole, and the effect of field interference need to be considered. The systematic study on the effects of fringe field and interference of quadrupoles in CSNS/RCS was carried out. The lattice of CSNS/RCS was re-matched and optimized considering the fringe field effects and fringe field interference. The re-matched lattice was applied to CSNS/RCS beam commissioning, and a series of optical measurements were performed. The measured results of optics agree very well with the calculated values. Acknowledgments The authors thank the colleagues from magnet group, the beam diagnostics group for their helps and contributions to this work, and colleagues from accelerator physics group for the helpful discussion.

Fig. 10. The comparison of the relative beta-beating between hard edge model and slicing model.

References

3.4. Betatron functions

[1] S.N. Fu, et al., Status of CSNS Project, IPAC13, Shanghai, China (2013), pp. 3995-3999. [2] J. Wei, et al., China spallation neutron source: design, R & D, and outlook, Nucl. Instrum. Methods Phys. Res. A 600 (1) (2009) 10–13. [3] S. Wang, et al., Introduction to the overall physics design of CSNS accelerators, Chin. Phys. C 33 (S2) (2009) 1–3. [4] S.Y. Xu, et al., Space Charge Effects for Different CSNS/RCS Working Points, (2013): WEPEA023. [5] L. Li, et al., Fringe field interference of neighbor magnets in china spallation neutron, Nucl. Instrum. Methods Phys. Res. A 840 (2016) 97–101. [6] M. Yang, et al., Magnetic fringe field interference between the quadrupole and corrector magnets in the CSNS/RCS, Nucl. Instrum. Methods Phys. Res. A 847 (2017) 29–33. [7] K.G. Steffen, High Energy Beam Optics, (Vol. 11), Interscience, New York, 1965. [8] A. Terebilo, Accelerator toolbox for MATLAB. In Talk presented at (No. SLAC-PUB-8732), 2001. [9] A. Morita, et al., Measurement and correction of on-and off-momentum beta functions at KEKB, Phys. Rev. Spec. Top. Accel. Beams 10 (7) (2007) 072801. [10] J. Safranek, Experimental determination of storage ring optics using orbit response measurements, Nucl. Instrum. Methods Phys. Res. A 388 (1–2) (1997) 27–36. [11] S.Y. Lee, Accelerator Physics, World Scientific Publishing Company, 2011, pp. 77–78.

Betatron functions of the lattice of CSNS/RCS were measured by using the Closed Orbit Distortion (COD) formula [11] √ 𝛽0 𝛽(𝑠) (5) 𝑢(𝑠) = 𝜃 cos(𝜋𝜈 − ||𝜓(𝑠) − 𝜓(𝑠0 )||) 2 sin 𝜋𝜈 where 𝑢(𝑠) is the Closed Orbit, 𝜃 is the closed orbit error which is generated by the corrector, 𝜈 is the betatron tune, 𝜓(𝑠), 𝜓0 are the betatron phase advance at 𝑠, 𝑠0 respectively, and 𝛽(𝑠), 𝛽0 are the values of the betatron functions at 𝑠, 𝑠0 respectively. 30 BPMs are installed inside of the corrector magnets, and the betatron phase advance between the corrector and corresponding BPM can be neglected. The betatron function at each BPM is given by 2𝛥𝑢𝑖 (𝑠) tan 𝜋𝜈 (6) 𝜃 where i represents the 𝑖th BPM, 𝜃 is the kicker angle of the corrector magnet , 𝛥𝑢𝑖 (𝑠) is the difference between two measured values with and without kicker of the corrector, 𝜈 is the measured tune, and 𝛽(𝑠) is the betatron function at each BPM. 𝛽(𝑠) =

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