- Email: [email protected]
Investigation for the vertical focusing enhancement of compact cyclotron by asymmetrical shimming bar
Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Investigation for the vertical focusing enhancement of compact cyclotron by asymmetrical shimming bar Tianjue Zhang ⇑, Chuan Wang, Junqing Zhong, Hongjuan Yao China Institute of Atomic Energy, Beijing 102413, PR China
a r t i c l e
i n f o
Article history: Available online 22 April 2011 Keywords: AVF cyclotron Vertical focusing enhancement Asymmetric shimming bar
a b s t r a c t CYCIAE-100, a 100 MeV H cyclotron under construction at China Institute of Atomic Energy (CIAE), is an AVF compact cyclotron. With energy above 70 MeV, the straight-edge sector magnet, instead of the spiral one normally used for AVF cyclotrons in this case, is still used to simplify the engineering procedures. The vertical focusing is likely not strong enough at the outer region and the Walkinshaw resonance may occur, if either the permeability decreasing tolerance in large scale pure iron castings and forgings, or the fabrication tolerance during the magnet construction, are seriously excessive. Theoretical investigation and numerical simulation results presented in this paper show that this kind of risk could be avoided by using a set of specially designed asymmetrical shimming bars between the pole edge and dummy Dee of the RF cavity at the outer region. In this way, the vertical focusing at outer radius region will increase substantially. This investigation provides a protective measure for the main magnet construction of CYCIAE-100. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction A 100 MeV H compact azimuthally varying field (AVF) cyclotron, CYCIAE-100, is under construction at China Institute of Atomic Energy (CIAE). It is widely accepted in cyclotron engineering that the spiral sector structure should be adopted to insure sufficient vertical focusing and to prevent nonlinear resonant if the final energy of the AVF machine exceeds 70 MeV [1]. CYCIAE-100 still adopts the straight-edge sector magnet to reduce the complexity in not only main magnet, but also in the RF, beam diagnostics and beam extraction system induced by the spiral structure. The loss of vertical focusing can be compensated by using a variable hill gap. The beam dynamics calculations indicate that the Walkinshaw resonance only occurs in the centre region, where a large radial gain per turn could be obtained and the resonance crossing is not dangerous [2]. The dangerous resonance crossing in large radius should be avoided. This work theoretically and numerically investigates the idea of installing specially designed asymmetrical shimming bars on the sides of sector magnets to prevent the vertical focusing loss caused by large degradation of magnetic material and out-of-tolerance. The results show that the risk of harmful resonance in large radius
can be successfully reduced by using these asymmetrical shimming bars. 2. The key factors on the vertical focusing of CYCIAE-100 2.1. The degradation of permeability The permeability of large pure iron castings and forgings used for poles and yokes may be different between design and engineering stages. This degradation in the permeability will affect the distribution of the magnetic field in the median plane, which consequently causes the loss of vertical focusing. So the effects of permeability degradation on the loss of vertical focusing have been carefully investigated in our previous design. We have built the finite element model of main magnet with the reference permeability value of ANSI 1008# iron. We have achieved the isochronous fields under several cases with different permeability, and investigated their corresponding betatron oscillations properties, respectively, as is shown in Fig. 1. Apparently, the case with the permeability uniquely reduced by 10% is out-of-tolerance, which means that the dangerous resonance cannot be avoided by conventional bar shimming. 2.2. Deviation of pole angle
⇑ Corresponding author. Tel./fax: +86 10 69357341. E-mail addresses: [email protected] (T. Zhang), [email protected] (C. Wang), [email protected] (J. Zhong), [email protected] (H. Yao). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.04.051
The vertical focusing in AVF cyclotrons could be approximately summarized as the following formula [3]:
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T. Zhang et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
0.7
Tune Diagram
0.6
Vr/2 of 100% permeability Vz of 100% permeability Vr/2 of 97% permeability Vz of 97% permeability Vr/2 of 90% permeability Vz of 90% permeability Vr/2 of 110% permeability Vz of 110% permeability Vr/2 of 105% permeability Vz of 105% permeability
0.5
0.4 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
R/m Fig. 1. Betatron oscillation variation caused by shift of permeability.
m2z ¼ l0re1 þ R þ
n2
1 2
1 02 þ u02 1 n 2 un C 2n þ C n C 0n 2ðn2 1Þ n2 1
2n2 1 C 02 2n2 ðn2 1Þ n
The 2 tan2 h item is the extra factor compared to the formula for straight-edge sector structure, and apparently, it is spiral angle related.
ð1-1Þ 3.2. The theoretical analysis of asymmetrical shimming bars design
where, l0re1 correction factor and Cn the amplitude of nth harmonic. For cyclotrons with four sectors, as CYCIAE-100, since the 4th harmonic is the major factor contributing to the vertical betatron oscillation frequency, only the n = 4 case is considered. According to the above formula, assuming that the pole angle deviates from its designed value by 5.00°, 1.00°, 0.50°, 0.10°, 0.00°, 0.10°, 0.50°, 1.00°, 5.00°, the vertical betatron oscillation frequency will vary by 0.0976, 0.0192, 0.0095, 0.0018, 0.00, 0.0021, 0.0098, 0.0193, 0.0944, respectively, maintaining the same magnetic rigidity and using the hard boundary assumption during calculation. The results indicate that the variation of the vertical focusing is tolerable if the pole angle deviation is less than 0.10° and the vertical focusing will be close to resonance zone if the pole angle deviation is around 0.50°. If the pole angle deviation is over 1.00°, the dangerous resonance will occur in the current design. Although the permeability degradation of 5% and angle deviation of 0.50° will be reached during fabrication and installation, according to the craft of smelting in industry, the possibility of out-of-tolerance still exists. 3. The theoretical basis of asymmetrical shimming bars design 3.1. The vertical focusing of cyclotrons with spiral sector structure Charged particles in the straight-edge sector structure will be affected by the Thomas force. Two more forces will be induced in spiral sector structure, i.e. the Kerst force and the Laslett force, which will increase the vertical focusing. The vertical betatron oscillation frequency mz and spiral angle n show the following relation [4,5]:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mz ¼
2 1 2 þ f 2 ð1 þ 2tan2 nÞ 0
ð2-1Þ
Enlightened by the spiral sector structure, the asymmetrical shimming bars are designed in a way that provides extra vertical focusing force on our straight sector structure design. We have already had a vertical focusing mz0 curve by using conventional symmetrical shimming bars in our previous design. Based on that, the goal is set to increase mz0 mzideal in large radius to prevent dangerous resonances. According to the above formula, it can be derived that:
m2zideal m2z0 ¼ F 2 ðtan2 ðn0 þ nreal Þ tan2 n0 Þ
ð2-2Þ
Here, F2 is the square of flutter and n0 is the nominal spiral angle in conventional symmetrical shimming bars design which is obtained by computation of particle orbit using functional inverse (on n) of formula (2-1) after obtaining vz. Then we can get the real spiral angle nreal by solving the Eq. (2-2).As piece-wised line segments to represent the shape of a shimming bar in finite element model and polar coordinates with their origin lying in the center of the cyclotron were used, we can define the rn to be the radius of end point of the nth line segment and hn its angle in the specified polar ihnþ1 ih ne n using simple coordinates. The spiral angle nn ¼ arg rnþ1 e rn eihr n Euler’s complex number formula and the geometric relation can be then obtained (see Fig. 2).Then, we can get, r nþ1 sinðhnþ1 hn Þ tan nn ¼ rnþ1 rn ¼ Kn cosðh nþ1 hn Þ 1 rn
ð2-3Þ
As the relation 90° > hn+1 > hn > 0 holds for the sides from which the particles leave the magnet pole (the A sides), it can be derived that sin (hn1 hn) > 0. If we define xn = cos(hn1 hn) as relation rn xn > rnþ1 > 0 holds, then we get the value of xn.
xn ¼
rn K 2n rnþ1 þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 K 2n þ 1 r2 n K 2n nþ1
1 þ K 2n
ð2-4Þ
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T. Zhang et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
Fig. 2. Sketch map of angular and radius coordinates of shimming bars in polar coordinates.
In this way, hn can be derived when specifying the hn+1. Using a similar procedure and the values of hn and nn1, hn1can be obtained. Then, we can recursively obtain the angular coordinates hn2, . . ., h0 of the shimming bars. In order to maintain the isochronous field, the angle of both sides (A and B sides) in any radius should not be changed, and then the angular coordinates of the other sides from which the particles go in towards the magnet pole (the B side) are obtained. Theoretically, the vertical frequency will be increased by:
Dmz ¼ mzideal mz0
ð2-5Þ
4. The numerical calculation of asymmetrical shimming bars for CIAE-100 The challenge for the asymmetrical shimming bar concerns the limited space between the magnet pole and the RF liner. Fortunately, we have adequate space for asymmetric shimming bars in large radius in the current design of the CYCIAE-100 cyclotron, as is shown in Fig. 3.
Fig. 3. Sketch map of the distance between shimming bar edge and RF liner.
Fig. 4. Projection view of shimming bars, RF liner and magnet poles on cyclotron middle plane.
The projections of asymmetric shimming bars on the cyclotron middle plane, as well as the projections of magnet pole, RF liner and symmetric shimming bars are sketched in Fig. 4.
4.1. The techniques used in numerical calculation Due to the asymmetry induced by the differences of coordinates between Side A and Side B, the one sixteenth model and our symmetric orbit code CYCIAE2000 in the previous main magnet design are not adequate [6]. As an alternative, the one eighth model with a similar mesh size is used as well as the orbit code Cyclop [5]. Our previous design indicates that the different element type and the different mesh size have non-ignorable effect on the betatron oscillation frequency [6], and consequently simulation has been performed using one eighth model with reasonable mesh size.
4.2. The numerical results of asymmetric shimming bars design By means of the one eighth model, the isochronous field map was firstly obtained using symmetric shimming and the betatron oscillation frequency were calculated. As shown in Fig. 5 the mz curve drops at large radius around extraction area even without degradation of permeability. During the design of CYCIAE-100, we should consider to prevent dangerous resonance in critical situation, as described in the previous section, and to suppress the drops of mz around extraction area. Because of the space restriction between RF system and the edge of shimming bars, we use asymmetric shimming method from r = 155 cm to the final radius and have designed a serial of proper spiral angles as is shown in Table 1. Using these spiral angle values and the formula in previous section, we could get the profile of the asymmetric shimming bars. After a few numerical shimming iterations by adding or subtracting the same value of shims in both A and B sides of asymmetric shimming bars, we could obtain the isochronous field map, and consequently the vertical focusing as shown in Fig. 5. The vertical focusing has significantly increased at radii larger than 155 cm with the asymmetrical shimming bars. This shows the possibility to avoid the Walkinshaw resonance, even in cases of permeability degradation, by the use of asymmetrical shimming bars.
T. Zhang et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
2971
Fig. 5. Comparison of betatron oscillations frequency between symmetric and asymmetric shimming.
Table 1 Spiral angles and radius of the asymmetric shimming bars for CYCIAE-100. R/cm n/deg R/cm n/deg
183 1.6 166 1.24
180 1.11 164 1.39
177 1.25 162.5 1.698
174 1.34 159 1.71
171 1.6 157 1.4
169 1.45 155 0.72
5. Conclusions Both theoretical analysis and numerical evaluation results prove that the asymmetric shimming bars design can significantly increase the vertical focusing. However, the increase of vertical focusing is not exactly the same as described in formula (2-5), due to the fact that the same value of shims have been added or subtracted in both A and B sides of asymmetric shimming bars to obtain the isochronous field map. Based on the above asymmetric shimming bars design, 16 spare symmetric shimming bars were changed to asymmetric ones for
the case that asymmetric shimming is needed when encountering a large degradation of permeability or out-of-tolerance in the engineering stage of the CYCIAE-100 project. References [1] J.L. Xie, Accelerators and Scientific Innovation, Tsinghua University Press, Beijing, 2000, p. 82 (in Chinese). [2] T.-J. Zhang, Z.-G. Li, J.-Q. Zhong, S.-Z. An, H.-J. Yao, B. Ji, S.-M. Wei, F.-P. Guan, J.-J. Yang, Y.-J. Bi, X.-L. Jia, C. Wang, Physics design of CYCAIE-100, Chin. Phys. C. 33 (S2) (2009) 33–38. [3] H.L. Hagedoorn, N.F. Verster, Orbits in an AVF cyclotron, Nucl. Instr. Meth. 18, 19 (1962) 201–228. [4] S. Orass, Techniques Applied to the Design of the TRIUMF Magnet Poles, TRIUMF Design Note: TRI-70-4. [5] M.M. Gordon, Computation of closed orbits and basic focusing properties for sector-focused cyclotrons and the design of cyclops, Part. Accelerators 16 (1984) 39–62. [6] J. Zhong, T. Zhang, J. Yang, C. Chu, 3D finite element analysis method for main magnet design of 100 MeV compact cyclotron, At. Energ. Sci. Technol. 39 (No. 2) (2005) 110–113 (in Chinese).
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Investigation for the vertical focusing enhancement of compact cyclotron by asymmetrical shimming bar Tianjue Zhang ⇑, Chuan Wang, Junqing Zhong, Hongjuan Yao China Institute of Atomic Energy, Beijing 102413, PR China
a r t i c l e
i n f o
Article history: Available online 22 April 2011 Keywords: AVF cyclotron Vertical focusing enhancement Asymmetric shimming bar
a b s t r a c t CYCIAE-100, a 100 MeV H cyclotron under construction at China Institute of Atomic Energy (CIAE), is an AVF compact cyclotron. With energy above 70 MeV, the straight-edge sector magnet, instead of the spiral one normally used for AVF cyclotrons in this case, is still used to simplify the engineering procedures. The vertical focusing is likely not strong enough at the outer region and the Walkinshaw resonance may occur, if either the permeability decreasing tolerance in large scale pure iron castings and forgings, or the fabrication tolerance during the magnet construction, are seriously excessive. Theoretical investigation and numerical simulation results presented in this paper show that this kind of risk could be avoided by using a set of specially designed asymmetrical shimming bars between the pole edge and dummy Dee of the RF cavity at the outer region. In this way, the vertical focusing at outer radius region will increase substantially. This investigation provides a protective measure for the main magnet construction of CYCIAE-100. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction A 100 MeV H compact azimuthally varying field (AVF) cyclotron, CYCIAE-100, is under construction at China Institute of Atomic Energy (CIAE). It is widely accepted in cyclotron engineering that the spiral sector structure should be adopted to insure sufficient vertical focusing and to prevent nonlinear resonant if the final energy of the AVF machine exceeds 70 MeV [1]. CYCIAE-100 still adopts the straight-edge sector magnet to reduce the complexity in not only main magnet, but also in the RF, beam diagnostics and beam extraction system induced by the spiral structure. The loss of vertical focusing can be compensated by using a variable hill gap. The beam dynamics calculations indicate that the Walkinshaw resonance only occurs in the centre region, where a large radial gain per turn could be obtained and the resonance crossing is not dangerous [2]. The dangerous resonance crossing in large radius should be avoided. This work theoretically and numerically investigates the idea of installing specially designed asymmetrical shimming bars on the sides of sector magnets to prevent the vertical focusing loss caused by large degradation of magnetic material and out-of-tolerance. The results show that the risk of harmful resonance in large radius
can be successfully reduced by using these asymmetrical shimming bars. 2. The key factors on the vertical focusing of CYCIAE-100 2.1. The degradation of permeability The permeability of large pure iron castings and forgings used for poles and yokes may be different between design and engineering stages. This degradation in the permeability will affect the distribution of the magnetic field in the median plane, which consequently causes the loss of vertical focusing. So the effects of permeability degradation on the loss of vertical focusing have been carefully investigated in our previous design. We have built the finite element model of main magnet with the reference permeability value of ANSI 1008# iron. We have achieved the isochronous fields under several cases with different permeability, and investigated their corresponding betatron oscillations properties, respectively, as is shown in Fig. 1. Apparently, the case with the permeability uniquely reduced by 10% is out-of-tolerance, which means that the dangerous resonance cannot be avoided by conventional bar shimming. 2.2. Deviation of pole angle
⇑ Corresponding author. Tel./fax: +86 10 69357341. E-mail addresses: [email protected] (T. Zhang), [email protected] (C. Wang), [email protected] (J. Zhong), [email protected] (H. Yao). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.04.051
The vertical focusing in AVF cyclotrons could be approximately summarized as the following formula [3]:
2969
T. Zhang et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
0.7
Tune Diagram
0.6
Vr/2 of 100% permeability Vz of 100% permeability Vr/2 of 97% permeability Vz of 97% permeability Vr/2 of 90% permeability Vz of 90% permeability Vr/2 of 110% permeability Vz of 110% permeability Vr/2 of 105% permeability Vz of 105% permeability
0.5
0.4 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
R/m Fig. 1. Betatron oscillation variation caused by shift of permeability.
m2z ¼ l0re1 þ R þ
n2
1 2
1 02 þ u02 1 n 2 un C 2n þ C n C 0n 2ðn2 1Þ n2 1
2n2 1 C 02 2n2 ðn2 1Þ n
The 2 tan2 h item is the extra factor compared to the formula for straight-edge sector structure, and apparently, it is spiral angle related.
ð1-1Þ 3.2. The theoretical analysis of asymmetrical shimming bars design
where, l0re1 correction factor and Cn the amplitude of nth harmonic. For cyclotrons with four sectors, as CYCIAE-100, since the 4th harmonic is the major factor contributing to the vertical betatron oscillation frequency, only the n = 4 case is considered. According to the above formula, assuming that the pole angle deviates from its designed value by 5.00°, 1.00°, 0.50°, 0.10°, 0.00°, 0.10°, 0.50°, 1.00°, 5.00°, the vertical betatron oscillation frequency will vary by 0.0976, 0.0192, 0.0095, 0.0018, 0.00, 0.0021, 0.0098, 0.0193, 0.0944, respectively, maintaining the same magnetic rigidity and using the hard boundary assumption during calculation. The results indicate that the variation of the vertical focusing is tolerable if the pole angle deviation is less than 0.10° and the vertical focusing will be close to resonance zone if the pole angle deviation is around 0.50°. If the pole angle deviation is over 1.00°, the dangerous resonance will occur in the current design. Although the permeability degradation of 5% and angle deviation of 0.50° will be reached during fabrication and installation, according to the craft of smelting in industry, the possibility of out-of-tolerance still exists. 3. The theoretical basis of asymmetrical shimming bars design 3.1. The vertical focusing of cyclotrons with spiral sector structure Charged particles in the straight-edge sector structure will be affected by the Thomas force. Two more forces will be induced in spiral sector structure, i.e. the Kerst force and the Laslett force, which will increase the vertical focusing. The vertical betatron oscillation frequency mz and spiral angle n show the following relation [4,5]:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mz ¼
2 1 2 þ f 2 ð1 þ 2tan2 nÞ 0
ð2-1Þ
Enlightened by the spiral sector structure, the asymmetrical shimming bars are designed in a way that provides extra vertical focusing force on our straight sector structure design. We have already had a vertical focusing mz0 curve by using conventional symmetrical shimming bars in our previous design. Based on that, the goal is set to increase mz0 mzideal in large radius to prevent dangerous resonances. According to the above formula, it can be derived that:
m2zideal m2z0 ¼ F 2 ðtan2 ðn0 þ nreal Þ tan2 n0 Þ
ð2-2Þ
Here, F2 is the square of flutter and n0 is the nominal spiral angle in conventional symmetrical shimming bars design which is obtained by computation of particle orbit using functional inverse (on n) of formula (2-1) after obtaining vz. Then we can get the real spiral angle nreal by solving the Eq. (2-2).As piece-wised line segments to represent the shape of a shimming bar in finite element model and polar coordinates with their origin lying in the center of the cyclotron were used, we can define the rn to be the radius of end point of the nth line segment and hn its angle in the specified polar ihnþ1 ih ne n using simple coordinates. The spiral angle nn ¼ arg rnþ1 e rn eihr n Euler’s complex number formula and the geometric relation can be then obtained (see Fig. 2).Then, we can get, r nþ1 sinðhnþ1 hn Þ tan nn ¼ rnþ1 rn ¼ Kn cosðh nþ1 hn Þ 1 rn
ð2-3Þ
As the relation 90° > hn+1 > hn > 0 holds for the sides from which the particles leave the magnet pole (the A sides), it can be derived that sin (hn1 hn) > 0. If we define xn = cos(hn1 hn) as relation rn xn > rnþ1 > 0 holds, then we get the value of xn.
xn ¼
rn K 2n rnþ1 þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 K 2n þ 1 r2 n K 2n nþ1
1 þ K 2n
ð2-4Þ
2970
T. Zhang et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
Fig. 2. Sketch map of angular and radius coordinates of shimming bars in polar coordinates.
In this way, hn can be derived when specifying the hn+1. Using a similar procedure and the values of hn and nn1, hn1can be obtained. Then, we can recursively obtain the angular coordinates hn2, . . ., h0 of the shimming bars. In order to maintain the isochronous field, the angle of both sides (A and B sides) in any radius should not be changed, and then the angular coordinates of the other sides from which the particles go in towards the magnet pole (the B side) are obtained. Theoretically, the vertical frequency will be increased by:
Dmz ¼ mzideal mz0
ð2-5Þ
4. The numerical calculation of asymmetrical shimming bars for CIAE-100 The challenge for the asymmetrical shimming bar concerns the limited space between the magnet pole and the RF liner. Fortunately, we have adequate space for asymmetric shimming bars in large radius in the current design of the CYCIAE-100 cyclotron, as is shown in Fig. 3.
Fig. 3. Sketch map of the distance between shimming bar edge and RF liner.
Fig. 4. Projection view of shimming bars, RF liner and magnet poles on cyclotron middle plane.
The projections of asymmetric shimming bars on the cyclotron middle plane, as well as the projections of magnet pole, RF liner and symmetric shimming bars are sketched in Fig. 4.
4.1. The techniques used in numerical calculation Due to the asymmetry induced by the differences of coordinates between Side A and Side B, the one sixteenth model and our symmetric orbit code CYCIAE2000 in the previous main magnet design are not adequate [6]. As an alternative, the one eighth model with a similar mesh size is used as well as the orbit code Cyclop [5]. Our previous design indicates that the different element type and the different mesh size have non-ignorable effect on the betatron oscillation frequency [6], and consequently simulation has been performed using one eighth model with reasonable mesh size.
4.2. The numerical results of asymmetric shimming bars design By means of the one eighth model, the isochronous field map was firstly obtained using symmetric shimming and the betatron oscillation frequency were calculated. As shown in Fig. 5 the mz curve drops at large radius around extraction area even without degradation of permeability. During the design of CYCIAE-100, we should consider to prevent dangerous resonance in critical situation, as described in the previous section, and to suppress the drops of mz around extraction area. Because of the space restriction between RF system and the edge of shimming bars, we use asymmetric shimming method from r = 155 cm to the final radius and have designed a serial of proper spiral angles as is shown in Table 1. Using these spiral angle values and the formula in previous section, we could get the profile of the asymmetric shimming bars. After a few numerical shimming iterations by adding or subtracting the same value of shims in both A and B sides of asymmetric shimming bars, we could obtain the isochronous field map, and consequently the vertical focusing as shown in Fig. 5. The vertical focusing has significantly increased at radii larger than 155 cm with the asymmetrical shimming bars. This shows the possibility to avoid the Walkinshaw resonance, even in cases of permeability degradation, by the use of asymmetrical shimming bars.
T. Zhang et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 2968–2971
2971
Fig. 5. Comparison of betatron oscillations frequency between symmetric and asymmetric shimming.
Table 1 Spiral angles and radius of the asymmetric shimming bars for CYCIAE-100. R/cm n/deg R/cm n/deg
183 1.6 166 1.24
180 1.11 164 1.39
177 1.25 162.5 1.698
174 1.34 159 1.71
171 1.6 157 1.4
169 1.45 155 0.72
5. Conclusions Both theoretical analysis and numerical evaluation results prove that the asymmetric shimming bars design can significantly increase the vertical focusing. However, the increase of vertical focusing is not exactly the same as described in formula (2-5), due to the fact that the same value of shims have been added or subtracted in both A and B sides of asymmetric shimming bars to obtain the isochronous field map. Based on the above asymmetric shimming bars design, 16 spare symmetric shimming bars were changed to asymmetric ones for
the case that asymmetric shimming is needed when encountering a large degradation of permeability or out-of-tolerance in the engineering stage of the CYCIAE-100 project. References [1] J.L. Xie, Accelerators and Scientific Innovation, Tsinghua University Press, Beijing, 2000, p. 82 (in Chinese). [2] T.-J. Zhang, Z.-G. Li, J.-Q. Zhong, S.-Z. An, H.-J. Yao, B. Ji, S.-M. Wei, F.-P. Guan, J.-J. Yang, Y.-J. Bi, X.-L. Jia, C. Wang, Physics design of CYCAIE-100, Chin. Phys. C. 33 (S2) (2009) 33–38. [3] H.L. Hagedoorn, N.F. Verster, Orbits in an AVF cyclotron, Nucl. Instr. Meth. 18, 19 (1962) 201–228. [4] S. Orass, Techniques Applied to the Design of the TRIUMF Magnet Poles, TRIUMF Design Note: TRI-70-4. [5] M.M. Gordon, Computation of closed orbits and basic focusing properties for sector-focused cyclotrons and the design of cyclops, Part. Accelerators 16 (1984) 39–62. [6] J. Zhong, T. Zhang, J. Yang, C. Chu, 3D finite element analysis method for main magnet design of 100 MeV compact cyclotron, At. Energ. Sci. Technol. 39 (No. 2) (2005) 110–113 (in Chinese).