- Email: [email protected]
Precise isochronous field shimming using correlation matrix for compact cyclotrons
Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Precise isochronous field shimming using correlation matrix for compact cyclotrons B. Qin n, J. Yang, K.F. Liu, D.Z. Chen, D. Li, Y.Q. Xiong, T.Q. Yu, M.W. Fan College of Electrical and Electronics Engineering, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
a r t i c l e i n f o
abstract
Article history: Received 2 April 2012 Received in revised form 27 June 2012 Accepted 9 July 2012 Available online 20 July 2012
An isochronous field shimming method using pre-calculated correlation matrix was applied to the magnet shaping of a 10 MeV compact proton cyclotron. Measurement results demonstrate this method effective with high accuracy. Only after performing two iterative pole shimming procedures, 7 41 for history phase slip was achieved. As well as the shimming effect from measurement has a good match to the predicted value. Detailed method and experimental procedures are described. & 2012 Elsevier B.V. All rights reserved.
Keywords: Isochronous field shimming Correlation matrix Compact cyclotron
1. Introduction Compact cyclotrons produce proton beam with range below 30 MeV or other ion species with similar rigidity, which plays a significant role in isotopes production for medical and industrial applications [1,2]. In recent years, positron emission tomography (PET) using compact cyclotrons have a rapid growth. Most of this type of cyclotrons accelerate single ion species like H , and isochronous magnetic field can be implemented by pole shaping. With the aid of three-dimensional FEM (Finite Element Method) code on the calculation of the static magnetic field such as TOSCA [3], preliminary optimization on field isochronism of the magnet poles can be made. As well as beam dynamics such as working diagram of betatron tunes can be estimated with high precision, by applying numerical integration on field maps. From this point of view, an optimized magnet model that fulfills the requirements of isochronism and beam dynamics is available before construction. However, magnet shimming is essential due to the realistic isochronous field errors coming from the field mapping. Prior to the magnet shimming procedures, closed orbits and isochronous field error should be computed. Gordon’s algorithm and its extension [4,5] is commonly used in azimuthally varying field (AVF) cyclotrons. With the increase in computation speed, more direct method using numerical integration on field maps is used to find closed orbits and accurate gyration frequency as a function of the particle energy or pole radius, then isochronous field error can be derived [6,7].
n
Corresponding author. Tel.: þ86 27 87557634. E-mail address: [email protected] (B. Qin).
0168-9002/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.07.010
Different methods exist for iterative pole shimming procedures, such as the analytical algorithm [8], the hard-edge approximation [6], and the matrix method [9,10]. A gyration frequency based method was proposed for the criterion of the isochronous field formation [7], with the prerequisite that two initial limiting sector shapes should be provided for calculations. Another different solutions claimed to have smooth pole shape optimization by using a polynomial function of the radius to fit the pole edge, with the coefficients calculated by either the random search or matrix method [11]. In the previous study we proposed a shimming method based on the multiple linear regression model of the magnetic field, by calculating independent pole cutting effects along the pole radius [10]. Minor adjustment is suggested to reduce the oscillations from the least square fitting result, mean time, fast convergence has been demonstrated by numerical simulations. This method was validated on a 10 MeV PET cyclotron, with some improvements relating to the prediction accuracy on the field change due to the pole shimming. Good coincidence between the experimental data and the simulation results was shown, and only performing two iterative pole shimming procedures, 741 for the history phase was achieved.
2. Shimming method using correlation matrix 2.1. Estimation of shimming values by using the least square fitting on the correlation matrix For a given three-dimensional or two-dimensional mid-plane magnetic field distribution, either from simulated field maps or
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B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
from measurement data, one can calculate gyration frequency fp(r) as a function of the pole radius r (transformed from particle energy), using equilibrium orbit codes which is based on the numerical integration on field maps, like PTP [12]. Then the isochronous field error DBðrÞ can be evaluated using the following equation:
DBðrÞ BðrÞBiso ðrÞ ¼ BðrÞ
g2 ðrÞ Df ðrÞ 1 þ g2 ðrÞ Df ðrÞ
ð1Þ
where Biso ðrÞ is the estimated isochronous field at radius r, B(r) is the calculated or measured azimuthally average field at radius r, gðrÞ ¼ 1 þ Ek ðrÞ=E0 . Df ðrÞ is the gyration frequency error defined by Df ðrÞ ðf p ðrÞf 0 Þ=f 0 , with f0 being the designed ion orbital frequency. In order to obtain a precise estimation on the shimming value from DBðrÞ which takes into account the radial fringe effect of the cutting patches, multiple linear regression can be employed with two assumptions: (1) the magnetic field change due to a cutting patch is proportional to the patch area and (2) the accumulated magnetic field change is equal to the superposition of the effects due to independent cutting patches along the radius. These have been validated by analytical formulas and numerical simulation [10], with the prerequisite that the cutting thickness is small enough compared to the pole circumference. First we define the magnetic field change vector y ¼ ½y1 ,y2 , . . . , yn T , in which yk ¼ Bi ðr k ÞBi1 ðr k Þ corresponds to the field change at radius rk from two adjacent iterations i1 and i. y can be calculated by the following equation: y¼X b
ð2Þ
where b ¼ ½b1 , b2 , . . . , bm T denotes the shimming vector with bj corresponding to the normalized cutting thickness at radius rj. X is a n m correlation regressor matrix which is pre-calculated from m set magnet models. For each model, small change in sector shape is applied using a unit shimming vector bunit,j ¼ ½0, . . . , bj ¼ 1,0T ðj ¼ 1, . . . ,mÞ, and field change yj is extracted by comparing the modified model with the original model. From Eq. (2), one obtains X j ¼ ½X 1j ,X 2j , . . . ,X nj T ¼ ½yj1 ,yj2 , . . . ,yjn T :
ð3Þ
The same procedure is performed on j ¼ 1, . . . ,m, then we have the full correlation matrix X. Although the data is retrieved from simulation models with TOSCA code, it can be applied to the realistic pole shaping because the discrepancy between the TOSCA calculation and the measured field is small, especially for prediction of the field change. This will be demonstrated by our experimental data. For a isochronous field error yiso , the least square solution of Eq. (2) which minimizes the residuals e ¼ JX byJ is
b ¼ ðX T XÞ1 X T yiso :
shows that the radial space Dr has a significant influence on these oscillations. By numerical comparison, we chose Dr ¼ 20 mm for good balance between the smooth in b and enough accuracy of the estimation. The basic requirement for the shimming processes is that b Z0 should be fulfilled when yiso Z0. Even with a good selection of Dr, this condition is not always true for the least square fit solution, then some adjustment on the shimming vector should be performed. For example, if b l o 0, it will be reset to zero, and the value of b l1 and b l þ 1 is decreased to compensate the accumulated shimming effect. By replacing the original b with this improved vector bimp , the final predicted shimming effect ypred can be calculated using ypred ¼ X bimp , again by Eq. (2).
2.2.2. Improvements on the prediction precision of the shimming effect and reduction in the computation time For estimation of b by Eq. (4), Dr ¼ 20 mm was selected. As shown in Fig. 1, 11 models with a 20 mm 2 mm triangle area cutting corresponding to central cutting radii from 7 cm to 27 cm with Dr ¼ 20 mm were calculated and compared to the original one, and the correlation matrix can be derived. However, in order to have more precise prediction on the shimming effect, smaller step size for the correlation matrix X is preferred. The blue dotted line in Fig. 2 shows the raw shimming effect retrieved from 11 TOSCA models, with Gaussian-like distribution and decreasing peak field due to the smaller ratio compared to the circumference along the radii. By performing the spline interpolation on the raw data, we get 27 data sets corresponding to the shimming effect from r ¼ 6 cm to r ¼ 32 cm with Dr ¼ 10 mm, and a fine prediction correlation matrix X pred with dimension ðn 27Þ can be calculated. As discussed in the experimental results, the shimming effect using X pred has a very good match to the field mapping data, which guarantees the shimming procedures within the control. Since this matrix method heavily relies on the three-dimensional FEM field computation, techniques to reduce computation time and manual operation are studied. Scripts written by Python are employed to facilitate automated modeling, calculation and field acquisition based on the original magnet model, and interpolation helps to reduce the number of calculation models. As well as an interface was built with functions including
ð4Þ
2.2. Improvements of the fundamental matrix method The solution given by the least square fit provides a good estimation of the magnet pole shimming. However, some factors related to the correlation matrix need to be considered to avoid oscillation of b , and the final shimming effect should be evaluated with a more precise way in order to assure the safety of the shimming procedures. 2.2.1. Radial space for calculation of the correlation matrix and minor adjustment on the shimming vector Oscillations of the least square fit solution will cause unsmooth pole edge, and more important, the negative elements are not acceptable during practical shimming procedures. The study
Fig. 1. Pole cutting scheme for calculating field change compared to the original model.
B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
131
Table 1 Main parameters of the magnet.
Fig. 2. Shimming effect calculated from TOSCA models and its fine interpolation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Parameter
Value
Ion source Number of sectors Open angle of the sector Pole radius Gap size of hill/valley Hill/valley magnetic field Extraction energy Extraction radius Radio frequency Harmonic number Dee voltage Energy gain per turn
Internal PIG source 4 501 33 cm 24 mm/96 mm 2.1 T/1.0 T 10 MeV 28 cm 100.8 MHz 4 40 kV 0.12 MeV
The magnet pole used high quality steel DT8 produced by TISCO Co. with very low carbon content less than 0.02%. To achieve good mechanical and magnetic performance with high uniformity, the forging process, heat treatment, and annealing process aiming to eliminate internal stresses were performed on the original ingot steel before the magnet machining. 3.2. Field measurement setup
Fig. 3. Main magnet of CYCHU-10 cyclotron.
Since CYCHU-10 adopt internal ion source for compactness, Polar coordinate mapping system cannot be used. A Cartesian coordinate field mapping system was designed and assembled [15], with the schematic view shown in Fig. 4. The design of field mapping platform for KIRAMS-13 cyclotron was referred [16]. Compared to the double platforms using synchronization control on serving motors, we employed single platform which is located on one side of the cyclotron magnet. Both mechanical and control apparatus are simplified without synchronization problem. To enhance the system stability, a reference base plane using marble material was adopted. However, it is difficult to apply the ‘flying mode’ which has a rapid mapping speed, due to the instability during probe’s moving in this mode. The Group 3 DTM-151 Teslameter with MPT-141 Hall probe is used to measure Bz. Using servomotors combined with the optical linear encoder system, the position precision 7 13 mm is achieved for the mapping system, and step size 5 mm on both x and y directions is adopted for field mapping. 3.3. Shimming procedures
estimation/adjustment of the shimming vector, evaluation of the shimming effect, and analysis of the beam dynamics.
3. Experimental results 3.1. Magnet description The proposed shimming method was applied to a 10 MeV H PET cyclotron CYCHU-10 [13]. The main magnet shown in Fig. 3 was constructed, with parameters listed in Table 1. To increase the average magnetic field for compactness, a small-valley-gap scheme was adopted, which achieves 1.6 T azimuthal average field. To estimate the H dissociation due to the Lorentz stripping, simplified formulas described in Ref. [14] are used. The electric field experienced by H in its rest frame is given by Erf ¼ 3gbBz with unit MV/cm, where b and g are the relativistic factors. For 10 MeV H , Erf ¼ 0:72 MV=cm is far below the critical limit 2 MV/cm, hence the fractional beam loss caused by the Lorentz stripping can be neglected.
The shimming procedure is done by machining the edge of four poles symmetrically using high precision CNC (Computer Numerical Control). As well as the first field harmonics of the magnetic field is well controlled within 4 Gs for various radii during this shimming procedure. The distribution of Bz from first field mapping is shown in Fig. 5, with the azimuthal average field error compared to the TOSCA model shown in Fig. 6. Since the average field is around 1.6 T, the relative error along the radius is within 71%. The BH curve used for TOSCA calculation is measured from the pole materials, however, discrepancy between the TOSCA model and measured field exists due to modeling details, machining/ assembling errors, measurement error on BH curve, etc. We should investigate if we could trust the estimation result which uses correlation matrix calculated from TOSCA models. The first trial shimming was performed by using the small cutting values far from the isochronism limit. As shown in Fig. 7, this first shimming vector ranging from r¼ 18 cm to r ¼28 cm, corrected only a small part of the isochronous field error (dot line) for validation. By comparing the estimated field change using the fine
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B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
Fig. 4. Schematic view of the Cartesian coordinate field mapping system. a: alignment support; b: marble reference plane; c: x-axis plane; d: y-axis plane; e: Hall probe; h: servomotor; i: ball screw; j: guide rail.
Fig. 5. Measured vertical field (Bz) distribution.
Fig. 6. Azimuthal average field error between measurement and TOSCA model.
prediction matrix X pred with the measured field change, we found very good coincidence with error less than 5 Gs, which proves the accuracy of the correlation matrix acquired from TOSCA models. Fig. 8 illustrates the second formal shimming procedure. First we got the least square solution for the shimming vector with Dr ¼ 2 cm, then the real shimming values are decided from this solution. The pole end part is modified to keep smooth on the pole shape. By using X pred , the predicted field change is controlled within the range of the isochronous field error. As shown, the measured field change has a good match to the predicted one, except for small discrepancy located at the central region and the pole end. After this shimming, the frequency error per turn is controlled within 70.05% and the history phase slip is within 741 (see Fig. 9), which fulfills the strict requirements for the isochronism. Working diagram of transverse betatron tunes calculated from the TOSCA model and field maps from the final measurement is shown in Fig. 10. Even with very strong negative field gradient at the edge of the magnet pole, the harmful nonlinear resonance nr 2nz ¼ 0 is avoided during extraction.
B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
133
Fig. 7. First trial shimming. For the isochronous field error calculated from the first field mapping result, a small part was corrected for validation of the correlation matrix acquired from TOSCA models.
R7 R8 R9
Fig. 8. Second formal shimming procedure.
4. Conclusions Both numerical and experimental data proved both the effectiveness and the high precision results of the method based on the correlation matrix for the isochronous magnet pole shimming calculation. Even the correlation matrix was established on the TOSCA magnet models with measured BH curve data, and absolute
error exists between the calculation and mapping field result, the prediction of the field change in terms of the shimming effect still shows good coincidence during the shimming procedures. With this proposed method, very few iterations during magnet pole shimming are required, as well as good field isochronism with very small history phase slip can be achieved. Pre-calculation of the correlation matrix using FEM code is essential and time-consuming,
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B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
but automatic manipulation covering modeling and data process with scripting language can ease these procedures.
Acknowledgment This work was supported by National Nature Science Foundation of China (10905025). We would like to thank Prof. T.J. Zhang and Dr. J.Q. Zhong in CIAE, for their helps in BH data measurement and useful discussions on magnet shimming procedures. References
Fig. 9. (Top) Gyration frequency error, (Bottom) history phase slip. Calculated from the field mapping result after second shimming, with f RF ¼ 100:84 MHz and energy gain per turn DE ¼ 120 keV.
Fig. 10. Working diagram of betatron tunes.
[1] B.F. Milton, Commercial compact cyclotrons in the 90’s, in: Proceedings of the 14th International Conference on Cyclotrons and Their Applications, Cape Town, South Africa, October 8–13, 1995, CONF-951039-4 (online) /http:// legacyweb.triumf.ca/people/milton/papers/tr30/cyc95.pdfS. [2] Y. Jongen, Review of compact commercial accelerator products and applications, in: PAC 1997, Vancouver, May 1997, pp. 3770–3774. [3] Opera-3D User Guide, Vector Fields Limited, England. [4] M.M. Gordan, Particle Accelerators 13 (1983) 67. [5] C. Baumgarten, Nuclear Instruments and Methods in Physics Research Section A 647 (1) (2011) 31. [6] B. Qin, et al., Nuclear Instruments and Methods in Physics Research Section B 261 (1–2) (2007) 56. [7] S. C´irkovic´, et al., IEEE Transactions on Nuclear Science NS-56 (5) (2009) 2821. [8] A. Papash, T. Zhang, Field shimming of commercial cyclotrons, in: Proceedings of 17th International Conference on Cyclotrons and Their Applications, 2004, Tokyo, Japan. [9] W. Kleeven, et al., Magnetic field calculation and shimming of the selfextraction cyclotron, in: Proceedings of 16th International Conference on Cyclotrons and Their Applications, East Lansing, USA, 2001, pp. 348–350. [10] B. Qin, et al., Nuclear Instruments and Methods in Physics Research Section A 620 (2010) 121. [11] P. Sing Babu, Nuclear Instruments and Methods in Physics Research Section A 624 (2010) 560. [12] B. Qin, et al., Design study of compact cyclotron magnet in virtual prototyping environment, in: PAC 2007, Albuquerque, NM, 2007, pp. 3354–3356. [13] B. Qin, et al., Main magnet and central region design for a 10 MeV PET cyclotron CYCHU-10, in: PAC 09, Vancouver, May 2009, pp. 184–186. [14] U. Gastaldi, M. Placentino, Nuclear Instruments and Methods in Physics Research Section A 451 (2000) 318. [15] J. Yang, et al., Magnetic field measurement system for CYCHU-10, in: PAC 09, Vancouver, May 2009, pp. 181–183. [16] K.H. Park, et al., Nuclear Instruments and Methods in Physics Research Section A 545 (2005) 533.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Precise isochronous field shimming using correlation matrix for compact cyclotrons B. Qin n, J. Yang, K.F. Liu, D.Z. Chen, D. Li, Y.Q. Xiong, T.Q. Yu, M.W. Fan College of Electrical and Electronics Engineering, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
a r t i c l e i n f o
abstract
Article history: Received 2 April 2012 Received in revised form 27 June 2012 Accepted 9 July 2012 Available online 20 July 2012
An isochronous field shimming method using pre-calculated correlation matrix was applied to the magnet shaping of a 10 MeV compact proton cyclotron. Measurement results demonstrate this method effective with high accuracy. Only after performing two iterative pole shimming procedures, 7 41 for history phase slip was achieved. As well as the shimming effect from measurement has a good match to the predicted value. Detailed method and experimental procedures are described. & 2012 Elsevier B.V. All rights reserved.
Keywords: Isochronous field shimming Correlation matrix Compact cyclotron
1. Introduction Compact cyclotrons produce proton beam with range below 30 MeV or other ion species with similar rigidity, which plays a significant role in isotopes production for medical and industrial applications [1,2]. In recent years, positron emission tomography (PET) using compact cyclotrons have a rapid growth. Most of this type of cyclotrons accelerate single ion species like H , and isochronous magnetic field can be implemented by pole shaping. With the aid of three-dimensional FEM (Finite Element Method) code on the calculation of the static magnetic field such as TOSCA [3], preliminary optimization on field isochronism of the magnet poles can be made. As well as beam dynamics such as working diagram of betatron tunes can be estimated with high precision, by applying numerical integration on field maps. From this point of view, an optimized magnet model that fulfills the requirements of isochronism and beam dynamics is available before construction. However, magnet shimming is essential due to the realistic isochronous field errors coming from the field mapping. Prior to the magnet shimming procedures, closed orbits and isochronous field error should be computed. Gordon’s algorithm and its extension [4,5] is commonly used in azimuthally varying field (AVF) cyclotrons. With the increase in computation speed, more direct method using numerical integration on field maps is used to find closed orbits and accurate gyration frequency as a function of the particle energy or pole radius, then isochronous field error can be derived [6,7].
n
Corresponding author. Tel.: þ86 27 87557634. E-mail address: [email protected] (B. Qin).
0168-9002/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.07.010
Different methods exist for iterative pole shimming procedures, such as the analytical algorithm [8], the hard-edge approximation [6], and the matrix method [9,10]. A gyration frequency based method was proposed for the criterion of the isochronous field formation [7], with the prerequisite that two initial limiting sector shapes should be provided for calculations. Another different solutions claimed to have smooth pole shape optimization by using a polynomial function of the radius to fit the pole edge, with the coefficients calculated by either the random search or matrix method [11]. In the previous study we proposed a shimming method based on the multiple linear regression model of the magnetic field, by calculating independent pole cutting effects along the pole radius [10]. Minor adjustment is suggested to reduce the oscillations from the least square fitting result, mean time, fast convergence has been demonstrated by numerical simulations. This method was validated on a 10 MeV PET cyclotron, with some improvements relating to the prediction accuracy on the field change due to the pole shimming. Good coincidence between the experimental data and the simulation results was shown, and only performing two iterative pole shimming procedures, 741 for the history phase was achieved.
2. Shimming method using correlation matrix 2.1. Estimation of shimming values by using the least square fitting on the correlation matrix For a given three-dimensional or two-dimensional mid-plane magnetic field distribution, either from simulated field maps or
130
B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
from measurement data, one can calculate gyration frequency fp(r) as a function of the pole radius r (transformed from particle energy), using equilibrium orbit codes which is based on the numerical integration on field maps, like PTP [12]. Then the isochronous field error DBðrÞ can be evaluated using the following equation:
DBðrÞ BðrÞBiso ðrÞ ¼ BðrÞ
g2 ðrÞ Df ðrÞ 1 þ g2 ðrÞ Df ðrÞ
ð1Þ
where Biso ðrÞ is the estimated isochronous field at radius r, B(r) is the calculated or measured azimuthally average field at radius r, gðrÞ ¼ 1 þ Ek ðrÞ=E0 . Df ðrÞ is the gyration frequency error defined by Df ðrÞ ðf p ðrÞf 0 Þ=f 0 , with f0 being the designed ion orbital frequency. In order to obtain a precise estimation on the shimming value from DBðrÞ which takes into account the radial fringe effect of the cutting patches, multiple linear regression can be employed with two assumptions: (1) the magnetic field change due to a cutting patch is proportional to the patch area and (2) the accumulated magnetic field change is equal to the superposition of the effects due to independent cutting patches along the radius. These have been validated by analytical formulas and numerical simulation [10], with the prerequisite that the cutting thickness is small enough compared to the pole circumference. First we define the magnetic field change vector y ¼ ½y1 ,y2 , . . . , yn T , in which yk ¼ Bi ðr k ÞBi1 ðr k Þ corresponds to the field change at radius rk from two adjacent iterations i1 and i. y can be calculated by the following equation: y¼X b
ð2Þ
where b ¼ ½b1 , b2 , . . . , bm T denotes the shimming vector with bj corresponding to the normalized cutting thickness at radius rj. X is a n m correlation regressor matrix which is pre-calculated from m set magnet models. For each model, small change in sector shape is applied using a unit shimming vector bunit,j ¼ ½0, . . . , bj ¼ 1,0T ðj ¼ 1, . . . ,mÞ, and field change yj is extracted by comparing the modified model with the original model. From Eq. (2), one obtains X j ¼ ½X 1j ,X 2j , . . . ,X nj T ¼ ½yj1 ,yj2 , . . . ,yjn T :
ð3Þ
The same procedure is performed on j ¼ 1, . . . ,m, then we have the full correlation matrix X. Although the data is retrieved from simulation models with TOSCA code, it can be applied to the realistic pole shaping because the discrepancy between the TOSCA calculation and the measured field is small, especially for prediction of the field change. This will be demonstrated by our experimental data. For a isochronous field error yiso , the least square solution of Eq. (2) which minimizes the residuals e ¼ JX byJ is
b ¼ ðX T XÞ1 X T yiso :
shows that the radial space Dr has a significant influence on these oscillations. By numerical comparison, we chose Dr ¼ 20 mm for good balance between the smooth in b and enough accuracy of the estimation. The basic requirement for the shimming processes is that b Z0 should be fulfilled when yiso Z0. Even with a good selection of Dr, this condition is not always true for the least square fit solution, then some adjustment on the shimming vector should be performed. For example, if b l o 0, it will be reset to zero, and the value of b l1 and b l þ 1 is decreased to compensate the accumulated shimming effect. By replacing the original b with this improved vector bimp , the final predicted shimming effect ypred can be calculated using ypred ¼ X bimp , again by Eq. (2).
2.2.2. Improvements on the prediction precision of the shimming effect and reduction in the computation time For estimation of b by Eq. (4), Dr ¼ 20 mm was selected. As shown in Fig. 1, 11 models with a 20 mm 2 mm triangle area cutting corresponding to central cutting radii from 7 cm to 27 cm with Dr ¼ 20 mm were calculated and compared to the original one, and the correlation matrix can be derived. However, in order to have more precise prediction on the shimming effect, smaller step size for the correlation matrix X is preferred. The blue dotted line in Fig. 2 shows the raw shimming effect retrieved from 11 TOSCA models, with Gaussian-like distribution and decreasing peak field due to the smaller ratio compared to the circumference along the radii. By performing the spline interpolation on the raw data, we get 27 data sets corresponding to the shimming effect from r ¼ 6 cm to r ¼ 32 cm with Dr ¼ 10 mm, and a fine prediction correlation matrix X pred with dimension ðn 27Þ can be calculated. As discussed in the experimental results, the shimming effect using X pred has a very good match to the field mapping data, which guarantees the shimming procedures within the control. Since this matrix method heavily relies on the three-dimensional FEM field computation, techniques to reduce computation time and manual operation are studied. Scripts written by Python are employed to facilitate automated modeling, calculation and field acquisition based on the original magnet model, and interpolation helps to reduce the number of calculation models. As well as an interface was built with functions including
ð4Þ
2.2. Improvements of the fundamental matrix method The solution given by the least square fit provides a good estimation of the magnet pole shimming. However, some factors related to the correlation matrix need to be considered to avoid oscillation of b , and the final shimming effect should be evaluated with a more precise way in order to assure the safety of the shimming procedures. 2.2.1. Radial space for calculation of the correlation matrix and minor adjustment on the shimming vector Oscillations of the least square fit solution will cause unsmooth pole edge, and more important, the negative elements are not acceptable during practical shimming procedures. The study
Fig. 1. Pole cutting scheme for calculating field change compared to the original model.
B. Qin et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 129–134
131
Table 1 Main parameters of the magnet.
Fig. 2. Shimming effect calculated from TOSCA models and its fine interpolation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Parameter
Value
Ion source Number of sectors Open angle of the sector Pole radius Gap size of hill/valley Hill/valley magnetic field Extraction energy Extraction radius Radio frequency Harmonic number Dee voltage Energy gain per turn
Internal PIG source 4 501 33 cm 24 mm/96 mm 2.1 T/1.0 T 10 MeV 28 cm 100.8 MHz 4 40 kV 0.12 MeV
The magnet pole used high quality steel DT8 produced by TISCO Co. with very low carbon content less than 0.02%. To achieve good mechanical and magnetic performance with high uniformity, the forging process, heat treatment, and annealing process aiming to eliminate internal stresses were performed on the original ingot steel before the magnet machining. 3.2. Field measurement setup
Fig. 3. Main magnet of CYCHU-10 cyclotron.
Since CYCHU-10 adopt internal ion source for compactness, Polar coordinate mapping system cannot be used. A Cartesian coordinate field mapping system was designed and assembled [15], with the schematic view shown in Fig. 4. The design of field mapping platform for KIRAMS-13 cyclotron was referred [16]. Compared to the double platforms using synchronization control on serving motors, we employed single platform which is located on one side of the cyclotron magnet. Both mechanical and control apparatus are simplified without synchronization problem. To enhance the system stability, a reference base plane using marble material was adopted. However, it is difficult to apply the ‘flying mode’ which has a rapid mapping speed, due to the instability during probe’s moving in this mode. The Group 3 DTM-151 Teslameter with MPT-141 Hall probe is used to measure Bz. Using servomotors combined with the optical linear encoder system, the position precision 7 13 mm is achieved for the mapping system, and step size 5 mm on both x and y directions is adopted for field mapping. 3.3. Shimming procedures
estimation/adjustment of the shimming vector, evaluation of the shimming effect, and analysis of the beam dynamics.
3. Experimental results 3.1. Magnet description The proposed shimming method was applied to a 10 MeV H PET cyclotron CYCHU-10 [13]. The main magnet shown in Fig. 3 was constructed, with parameters listed in Table 1. To increase the average magnetic field for compactness, a small-valley-gap scheme was adopted, which achieves 1.6 T azimuthal average field. To estimate the H dissociation due to the Lorentz stripping, simplified formulas described in Ref. [14] are used. The electric field experienced by H in its rest frame is given by Erf ¼ 3gbBz with unit MV/cm, where b and g are the relativistic factors. For 10 MeV H , Erf ¼ 0:72 MV=cm is far below the critical limit 2 MV/cm, hence the fractional beam loss caused by the Lorentz stripping can be neglected.
The shimming procedure is done by machining the edge of four poles symmetrically using high precision CNC (Computer Numerical Control). As well as the first field harmonics of the magnetic field is well controlled within 4 Gs for various radii during this shimming procedure. The distribution of Bz from first field mapping is shown in Fig. 5, with the azimuthal average field error compared to the TOSCA model shown in Fig. 6. Since the average field is around 1.6 T, the relative error along the radius is within 71%. The BH curve used for TOSCA calculation is measured from the pole materials, however, discrepancy between the TOSCA model and measured field exists due to modeling details, machining/ assembling errors, measurement error on BH curve, etc. We should investigate if we could trust the estimation result which uses correlation matrix calculated from TOSCA models. The first trial shimming was performed by using the small cutting values far from the isochronism limit. As shown in Fig. 7, this first shimming vector ranging from r¼ 18 cm to r ¼28 cm, corrected only a small part of the isochronous field error (dot line) for validation. By comparing the estimated field change using the fine
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Fig. 4. Schematic view of the Cartesian coordinate field mapping system. a: alignment support; b: marble reference plane; c: x-axis plane; d: y-axis plane; e: Hall probe; h: servomotor; i: ball screw; j: guide rail.
Fig. 5. Measured vertical field (Bz) distribution.
Fig. 6. Azimuthal average field error between measurement and TOSCA model.
prediction matrix X pred with the measured field change, we found very good coincidence with error less than 5 Gs, which proves the accuracy of the correlation matrix acquired from TOSCA models. Fig. 8 illustrates the second formal shimming procedure. First we got the least square solution for the shimming vector with Dr ¼ 2 cm, then the real shimming values are decided from this solution. The pole end part is modified to keep smooth on the pole shape. By using X pred , the predicted field change is controlled within the range of the isochronous field error. As shown, the measured field change has a good match to the predicted one, except for small discrepancy located at the central region and the pole end. After this shimming, the frequency error per turn is controlled within 70.05% and the history phase slip is within 741 (see Fig. 9), which fulfills the strict requirements for the isochronism. Working diagram of transverse betatron tunes calculated from the TOSCA model and field maps from the final measurement is shown in Fig. 10. Even with very strong negative field gradient at the edge of the magnet pole, the harmful nonlinear resonance nr 2nz ¼ 0 is avoided during extraction.
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Fig. 7. First trial shimming. For the isochronous field error calculated from the first field mapping result, a small part was corrected for validation of the correlation matrix acquired from TOSCA models.
R7 R8 R9
Fig. 8. Second formal shimming procedure.
4. Conclusions Both numerical and experimental data proved both the effectiveness and the high precision results of the method based on the correlation matrix for the isochronous magnet pole shimming calculation. Even the correlation matrix was established on the TOSCA magnet models with measured BH curve data, and absolute
error exists between the calculation and mapping field result, the prediction of the field change in terms of the shimming effect still shows good coincidence during the shimming procedures. With this proposed method, very few iterations during magnet pole shimming are required, as well as good field isochronism with very small history phase slip can be achieved. Pre-calculation of the correlation matrix using FEM code is essential and time-consuming,
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but automatic manipulation covering modeling and data process with scripting language can ease these procedures.
Acknowledgment This work was supported by National Nature Science Foundation of China (10905025). We would like to thank Prof. T.J. Zhang and Dr. J.Q. Zhong in CIAE, for their helps in BH data measurement and useful discussions on magnet shimming procedures. References
Fig. 9. (Top) Gyration frequency error, (Bottom) history phase slip. Calculated from the field mapping result after second shimming, with f RF ¼ 100:84 MHz and energy gain per turn DE ¼ 120 keV.
Fig. 10. Working diagram of betatron tunes.
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