- Email: [email protected]
Simulation study of beam extraction from a synchrotron using colored noise with digital filter
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 608 (2009) 37–41
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Simulation study of beam extraction from a synchrotron using colored noise with digital filter Tetsuya Nakanishi , Kohei Tsuruha College of Industrial Technology, Nihon University, 1-2-1 Izumicho, Narashino, Chiba 275-8575, Japan
a r t i c l e in fo
abstract
Article history: Received 21 February 2009 Received in revised form 15 June 2009 Accepted 18 June 2009 Available online 25 June 2009
A simulation program is developed for a slow-extraction method using a fast Q magnet (FQ) and an RFknockout. In this extraction method, after the separatrix is produced with excitation of sextupole magnets, a required quantity of circulating beam is extracted by shrinking the separatrix with excitation of the FQ. Then the emittance of circulating beam is diffused to an original size with the RF-knockout. This process is repeated with required timing until the entire circulating beam is completely extracted. An algorithm using a digital filter and white noise is proposed for a colored noise as a signal source for the RF-knockout. Spill structures with the present computing method were similar to the results obtained using a conventional algorithm with the sum of cosines of many frequency components. The results are also in agreement with the experimental results using the HIMAC synchrotron. The computing time of colored noise for simulation of 106 turns was 0.5 h for the filter method and 5.0 h for the conventional one. It is indicated that a colored noise with a wider frequency bandwidth gives a better spill structure that smoothly increases with time. & 2009 Elsevier B.V. All rights reserved.
Keywords: Slow beam extraction Synchrotron Simulation
1. Introduction In the case of spot scanning irradiation for cancer therapy application, a fast control of beam extraction from the heavy ion synchrotron is a key function [1]. A beam-extraction method based on transverse beam heating (RF-knockout method) has been developed as a useful one, which was originally operated under a constant separatrix [2,3]. Based on the progress of the above method, we proposed another method based on controlling a quadruple field of fast response, assisted by the RF-knockout (QAR method) [4]. The operational principle of this idea is as follows: (1) particles of a circulating beam are diffused by the RF-knockout to just the boundary of transverse separatrix under a resonant condition, (2) the separatrix size is shrunk with the excitation of a fast Q magnet (FQ) to a certain size, and the particles outside the separatrix are extracted, (3) the FQ is turned off, and (4) the above process is repeated until the entire circulating beam is completely extracted. The QAR method has the following characteristics: (1) it can precisely extract particles prescribed at the required timing because the extraction period is controlled by only the FQ and (2) it can reduce the cost of power supplies of main magnets because it does not require a very low current ripple, such as
Corresponding author. Tel.:+81 47474 2375; fax:+81 47474 2399.
E-mail address: [email protected] (T. Nakanishi). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.06.041
the 106 order, by widening the original separatrix area more than an emittance of circulating beam but enough. A proof-of-principle experiment of this method has been carried out using the HIMAC synchrotron successfully [4]. Characteristics of extracted beam were also measured [5]. To carry out a study for the practical use of this method, its simulation program is developed as the next step. In this work, an algorithm for the kick angle of the RF-knockout with a colored noise signal is proposed using a digital filter method and white noise. In the present paper, the simulation method is described, and the calculated results are compared with the experimental results and with ones using the conventional algorithm given by the sum of cosines of many frequency components. Characteristics of the extracted beam are also studied on the dependence of frequency band of colored noise.
2. Simulation method 2.1. Algorithms for the production of kick angle of RF-knockout with a colored noise signal A digital filter is recently well used in a digital signal processing field. When the kth input signal is expressed with xk, an output yk of a Finite Impulse Response (FIR) digital filter is
ARTICLE IN PRESS 38
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150
given by the latest and previous input values multiplied by the filter coefficient hn [6]: yk ¼
Nh X
hn xkn ;
ð1Þ
where Nh+1 is the total number of filter coefficients in the FIR system and hn is given by the next equation for a band-pass filter and hn ¼
2
pm
[arb.]
n¼0
cosðmo0 TÞ sinðmob TÞðma0Þ;h0 ¼ 4fb Tðm ¼ 0Þ;m
¼ n Nh =2;o0 ¼ ðoH þ oL Þ=2;ob ¼ ðoH oL Þ=2;
dNrev ;i ¼ CyðNrev 1ÞNs þi ;
ð2Þ
where C is the amplitude coefficient, Nrev the turn number, and i the bin number. Such a subscript of the kick angle is used since the number of kick angles a revolution requires is the total bin number Ns. On the other hand, another method was proposed as below [7].
dNrev ;i
N qffiffiffiffiffiffiffiffiffiffi X i þ yk ;Dfk Sðfk ÞDfk cos 2pfk Nrev þ ¼2 Ns k¼1 ¼ fk fk1 ; fk ¼
fk þ fk1 ; 2
ð3Þ
where fk is the frequency component of a colored noise normalized by the revolution frequency, N the total number of frequency components, Sðfk Þ the amplitude coefficient of each frequency component, and yk the random phase between 0 and 2p. S is assumed as a constant for all frequency components in this simulation. In Fig. 1 are shown frequency spectra of serial values of the kick angles given by Eq. (2) (the filter method), where the lower and the higher cut-off frequencies were set to be 0.12 and 1.2, respectively. Fig. 1(a) depicts the result for Ns ¼ 40 (the number of bins) and Nh ¼ 6000 (the number of filter coefficients), and Fig. 1(b) for Ns ¼ 40 and Nh ¼ 10,000. Such kick angles were calculated using different random numbers of xk of 10,000 sets, and Fourier analyses were done for the 10,000 sets and averaged. The figures are the averaged spectra. The expected flat distributions obtained though the Gibbs phenomenon show a spectrum that has a ripple at the band edge, as seen in Fig. 1(a). It was confirmed that the beam simulations with Nh values 6000 and 10,000 gave similar results. The Fourier analysis for the kick angles by Eq. (3) (the cos method) is shown in Fig. 1(c). The frequency bandwidth was
0 0.0
0.5
1.0 f/f0
1.5
2.0
0.5
1.0 f/f0
1.5
2.0
0.5
1.0 f/f0
1.5
2.0
[arb.]
70
0 0.0 2
[arb.]
when n goes from 0 to Nh, m goes from Nh/2 to Nh/2. T the sampling period, which gives an interval between the input signals, fH (oH ¼ 2pfH) the upper cut-off frequency, and fL the lower one. Such a digital filter is used to produce the kick angle of the RFknockout with a colored noise signal as follows. Serial values of xk produced at random between –1 and 1 are equivalent to the digital white noise sampled with period T. Then serial values of yk are obtained from Eq. (1), which form a digital colored noise with bandwidth ranging from fL to fH. Hence the obtained serial values can be used as the kick angle by which the particles are perturbed at the RF-knockout in each turn when the sampling period is used as a revolution time in a synchrotron. In the beam simulation, since the coasting beam in a synchrotron is divided longitudinally into Ns bins, time interval between the bins is used as the sampling period. The beam in each bin has an equal emittance, and all particles in a bin are perturbed with the same kick angle. The parameters in Eq. (1) are as follows: fH and fL are normalized by the revolution frequency of synchrotron, and therefore T is given by the inverse of the number of bins, Ns. The kick angle, then, is obtained by
0
0.0
Fig. 1. Frequency spectra of the kick angles by colored noises produced by the filter method with (a) Nm ¼ 6000, Ns ¼ 40 and (b) Nm ¼ 10,000, Ns ¼ 40, and (c) by the cos method with N ¼ 20,000, Ns ¼ 40. f0 is the revolution frequency.
divided by 20,000 with a same width, which has an interval of the frequency components of 5.4 105 in units of tune. Random numbers of 10,000 sets were also used for the yk, and the results were averaged. The flat distribution was obtained, except the Gibbs phenomenon.
2.2. Beam-tracking method and the main parameters Beam tracking was performed using the transfer matrix where thin-lens approximation was used for sextupole magnets and an RF-knockout. The data of kick angles of the RF-knockout were computed for all turns before the start of beam tracking. The HIMAC synchrotron has two sets of sextupole magnets as the separatrix exciter [8]. Divergence kicks of the sextupole are given by
Dx0 ¼ 12K2 ðx2 y2 Þ; Dy0 ¼ lK2 xy
ð4Þ
ARTICLE IN PRESS T. Nakanishi, K. Tsuruha / Nuclear Instruments and Methods in Physics Research A 608 (2009) 37–41
where l is the length of sextupole magnet and K2 ¼ B00 /Br. Though the HIMAC synchrotron has other sextupole magnets for chromaticity correction, they were neglected since the simulation was performed with RF-off condition and then the relative momentum spread DP/P (1s) was 4 104, which could be neglected in the simulation. A quadruple magnet for tune correction was used as the FQ. In the simulation, the K2-values were increased linearly during 105 turns (60 ms for the HIMAC synchrotron) and then kept constant. After that, the K-value of FQ was increased linearly in the period of 11 ms to extract the beam, and then the RF-knockout was fired for 11 ms to diffuse a circulating beam. The K-value was decreased to zero in the 0.3 ms period. The time chart of this operation is shown in Fig. 2. Duty factors of the FQ and the RFknockout were set to 50% in the beam experiment; however, the RF-knockout was fired 0.1 ms before the FQ coil current reached the peak value. Such operation condition is included in the simulation. A sawtooth wave for the FQ coil current was used to find a particle density distribution diffused by the RF-knockout. The separatrix phase space area shrinks with increasing K-value, and the ratio of shrunk separatrix area is approximately proportional to the change of the K-value in small range. A spill structure computed, therefore, corresponds to the particle density distribution around the separatrix. An initial particle distribution was assumed to be Gaussian, and the emittance (one sigma) was assumed to be 6p mm mrad. The particle distribution at the beginning is shown in Fig. 3(a). Particles in the inner part of Gaussian distribution were removed in this work, since those particles could not be extracted even after the maximum turn number of 4.5 105 (0.28 s). This resulted in a shorter computing time and increase in particle density near the separatrix boundary. Fig. 3(b) shows the particle distribution after the sextupole magnets were excited. Other parameters were also matched to the ones used in the beam experiment, in which
FQ RFKO
Carbon ions were accelerated up to 400 MeV/u with a repetition period of 3.3 s and extracted during the flattop of 2 s [4]. The bare tunes (nx,ny) were 3.685 and 3.13, respectively. The separatrix was shrunk to 63.2% of the initial area at the maximum K-value of FQ. A frequency bandwidth of colored noise was set to 0.12–1.2, since the frequency characteristic of all-pass network in the RFknockout system had a flat distribution from 0.2 MHz to around 2 MHz and a revolution frequency of 1.653 MHz.
3. Simulation results and comparison with experimental results Spill structures computed with the filter method by Eq. (2) are shown in Fig. 4, where each figure corresponds to 4 times extraction. The parameters Nh and Ns were 6000 and 40, respectively. Spike just before the spill falling to zero is due to the overlap of FQ and the RF-knockout operations. In the figure, the shape of the spill indicates that the particles diffused with the RF-knockout because the separatrix shrunk to the same size at each extraction and particles outside the separatrix were extracted. The width of spill can be changed by controlling the amplitude of colored noise, and in the simulation the amplitude was chosen so that the width of spill has approximately the same value of the experimental result. This was around 2 ms, which corresponds to the beam emittance of 6%. The number of particles in a spill corresponds to around 0.1% of the accelerated particles. An experimental result is shown in Fig. 5, where data were taken with a digital oscilloscope. From the top to the bottom, the current of FQ, the RF-knockout signal and spill structures are illustrated. Fig. 6 shows comparison between the results of experiment and simulation. It is found that the simulation can reproduce spill structures similar to the experimental results. On the other hand, the simulation and experimental results indicate that the spill structures vary a little at every extraction and have
300
11ms
11ms
[arb.]
0.3ms
200 100 0 0.175
0.1ms t Fig. 2. Time chart of the FQ and the RF-knockout.
39
0.185
0.195
0.205
0.215
0.225
0.235
[s] Fig. 4. Spill structures calculated with the filter method.
Fig. 3. (a) Initial particle distribution in horizontal phase space and (b) after the separtrix was produced.
0.245
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Fig. 5. Experimental result. From the top, the FQ current (2 V/div), the RF-knockout signal (5 V/div), spill structures (0.5 V/div), and other shot of the spill (0.5 V/div). The time scale is 10 ms/div.
200
240 exp
[arb.]
cal
100
[arb.]
160 0 0.175
0.185
0.195
0.205
80
0.215
0.225
0.235
0.245
[s] Fig. 8. Spill structures calculated with the filter method with bandwidth 0.12–5.
0 0.200
0.202 [s]
0.204
300
300
200
exp [arb.]
cal
[arb.]
200
100
100
0 0.178
0 0.221
0.180 [s]
0.182
[arb.]
300 200 100
0.185
0.195
0.205
0.225
0.227
[s]
Fig. 6. Spill structures magnified for similar results in the simulation (a, c) and the experiment (b, d). The time scale is 2 ms/div.
0 0.175
0.223
0.215 [s]
0.225
0.235
Fig. 7. Spill structures calculated with the cos method.
0.245
Fig. 9. Magnified spill structure in Fig. 8. The jagged pattern may be attributed to particles not sufficient for the simulation.
sometimes a steep increase as shown in Fig. 6(b). Fig. 7 is a result with the cos method given by Eq. (3), and the result is similar to that of the filter method. In Fig. 8 are shown spill structures computed with the filter method when the upper cut-off frequency was widened to 5, where Nh and Ns are kept at the same value. It is found that variation of spill structures at every extraction was smaller, and moreover the structure increases smoothly with time as shown in Fig. 9. This is because the colored noise becomes almost a white noise when we increase the upper cut-off frequency. Computing time with the filter method was 0.5 h for the kick angle calculation and 17.6 h for the particle tracking, where the numbers of revolutions and particles were 106 turns (0.6 s for
ARTICLE IN PRESS T. Nakanishi, K. Tsuruha / Nuclear Instruments and Methods in Physics Research A 608 (2009) 37–41
the HIMAC synchrotron) and 2 105, respectively. The other parameters were the same as described in the above simulation condition. The kick angles were computed for the interval in which the RF-knockout was operated. Computing times with the cos method were 5.0 h for kick angle calculation and 17.6 h for particle tracking, respectively. Such a long time for the kick angle calculation is due to the calculation of Cosine term 20,000 times for all kick angles (Eq. (3)). On the other hand, there are only simple multiplications in the filter method (Eq. (1)) since the coefficient hn including cosine is computed only at the start of the simulation at first. The computing times of particle tracking are same because of the same calculation. The difference of computing times for kick angles of two methods becomes large when we increase the revolution turns.
4. Conclusions The simulation program for the QAR method has been developed, and the validity was confirmed by comparing the simulated results with the experimental ones. The algorithm for the kick angles calculation by a colored noise has been proposed using the digital filter method, and it was proved that thusobtained results are similar to the ones of the conventional Cos method. The computing time of the proposed algorithm is much less than for the conventional method. It was found from the
41
simulation that a colored noise with a wider bandwidth gives a good spill structure that smoothly increases with time.
Acknowledgements The authors would like to express their thanks to Drs. K. Noda and T. Furukawa at National Institute of Radiological Sciences for their useful discussion and continuous support of the beam experiment, and they are grateful to the members of Accelerator Engineering Corporation (AEC) for their skilful operation of the HIMAC accelerator complex. They are also indebted to Dr. T. Katayama at Gesellschaft fur Schwerionenforschung for his invaluable discussion. References [1] E. Pedroni, et al., Proceedings of the NIRS International Workshop on Heavy Charged Particle Therapy and Related Subjects (1991) 94. [2] T. Furukawa, K. Noda, Nucl. Instr. and Meth. A 489 (2002) 59. [3] K. Saito, et al., in: Proceedings of the EPAC08, pp. 1827–1829. [4] T. Nakanishi, T. Furukawa, K. Yoshida, K. Noda, Nucl. Instr. and Meth. A 553 (2005) 400. [5] T. Nakanishi, T. Furukawa, K. Noda, Nucl. Instr. and Meth. B 266 (2008) 2169. [6] S. Nakamura, Digital Filter, TokyoDenkiDaigakuShuppankyoku, Tokyo, 2006 p. 74 (in Japanese). [7] K. Hiramoto, M. Nishi, Nucl. Instr. and Meth. A 322 (1992) 154. [8] K. Noda, et al., Nucl. Instr. and Meth. A 374 (1996) 269.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Simulation study of beam extraction from a synchrotron using colored noise with digital filter Tetsuya Nakanishi , Kohei Tsuruha College of Industrial Technology, Nihon University, 1-2-1 Izumicho, Narashino, Chiba 275-8575, Japan
a r t i c l e in fo
abstract
Article history: Received 21 February 2009 Received in revised form 15 June 2009 Accepted 18 June 2009 Available online 25 June 2009
A simulation program is developed for a slow-extraction method using a fast Q magnet (FQ) and an RFknockout. In this extraction method, after the separatrix is produced with excitation of sextupole magnets, a required quantity of circulating beam is extracted by shrinking the separatrix with excitation of the FQ. Then the emittance of circulating beam is diffused to an original size with the RF-knockout. This process is repeated with required timing until the entire circulating beam is completely extracted. An algorithm using a digital filter and white noise is proposed for a colored noise as a signal source for the RF-knockout. Spill structures with the present computing method were similar to the results obtained using a conventional algorithm with the sum of cosines of many frequency components. The results are also in agreement with the experimental results using the HIMAC synchrotron. The computing time of colored noise for simulation of 106 turns was 0.5 h for the filter method and 5.0 h for the conventional one. It is indicated that a colored noise with a wider frequency bandwidth gives a better spill structure that smoothly increases with time. & 2009 Elsevier B.V. All rights reserved.
Keywords: Slow beam extraction Synchrotron Simulation
1. Introduction In the case of spot scanning irradiation for cancer therapy application, a fast control of beam extraction from the heavy ion synchrotron is a key function [1]. A beam-extraction method based on transverse beam heating (RF-knockout method) has been developed as a useful one, which was originally operated under a constant separatrix [2,3]. Based on the progress of the above method, we proposed another method based on controlling a quadruple field of fast response, assisted by the RF-knockout (QAR method) [4]. The operational principle of this idea is as follows: (1) particles of a circulating beam are diffused by the RF-knockout to just the boundary of transverse separatrix under a resonant condition, (2) the separatrix size is shrunk with the excitation of a fast Q magnet (FQ) to a certain size, and the particles outside the separatrix are extracted, (3) the FQ is turned off, and (4) the above process is repeated until the entire circulating beam is completely extracted. The QAR method has the following characteristics: (1) it can precisely extract particles prescribed at the required timing because the extraction period is controlled by only the FQ and (2) it can reduce the cost of power supplies of main magnets because it does not require a very low current ripple, such as
Corresponding author. Tel.:+81 47474 2375; fax:+81 47474 2399.
E-mail address: [email protected] (T. Nakanishi). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.06.041
the 106 order, by widening the original separatrix area more than an emittance of circulating beam but enough. A proof-of-principle experiment of this method has been carried out using the HIMAC synchrotron successfully [4]. Characteristics of extracted beam were also measured [5]. To carry out a study for the practical use of this method, its simulation program is developed as the next step. In this work, an algorithm for the kick angle of the RF-knockout with a colored noise signal is proposed using a digital filter method and white noise. In the present paper, the simulation method is described, and the calculated results are compared with the experimental results and with ones using the conventional algorithm given by the sum of cosines of many frequency components. Characteristics of the extracted beam are also studied on the dependence of frequency band of colored noise.
2. Simulation method 2.1. Algorithms for the production of kick angle of RF-knockout with a colored noise signal A digital filter is recently well used in a digital signal processing field. When the kth input signal is expressed with xk, an output yk of a Finite Impulse Response (FIR) digital filter is
ARTICLE IN PRESS 38
T. Nakanishi, K. Tsuruha / Nuclear Instruments and Methods in Physics Research A 608 (2009) 37–41
150
given by the latest and previous input values multiplied by the filter coefficient hn [6]: yk ¼
Nh X
hn xkn ;
ð1Þ
where Nh+1 is the total number of filter coefficients in the FIR system and hn is given by the next equation for a band-pass filter and hn ¼
2
pm
[arb.]
n¼0
cosðmo0 TÞ sinðmob TÞðma0Þ;h0 ¼ 4fb Tðm ¼ 0Þ;m
¼ n Nh =2;o0 ¼ ðoH þ oL Þ=2;ob ¼ ðoH oL Þ=2;
dNrev ;i ¼ CyðNrev 1ÞNs þi ;
ð2Þ
where C is the amplitude coefficient, Nrev the turn number, and i the bin number. Such a subscript of the kick angle is used since the number of kick angles a revolution requires is the total bin number Ns. On the other hand, another method was proposed as below [7].
dNrev ;i
N qffiffiffiffiffiffiffiffiffiffi X i þ yk ;Dfk Sðfk ÞDfk cos 2pfk Nrev þ ¼2 Ns k¼1 ¼ fk fk1 ; fk ¼
fk þ fk1 ; 2
ð3Þ
where fk is the frequency component of a colored noise normalized by the revolution frequency, N the total number of frequency components, Sðfk Þ the amplitude coefficient of each frequency component, and yk the random phase between 0 and 2p. S is assumed as a constant for all frequency components in this simulation. In Fig. 1 are shown frequency spectra of serial values of the kick angles given by Eq. (2) (the filter method), where the lower and the higher cut-off frequencies were set to be 0.12 and 1.2, respectively. Fig. 1(a) depicts the result for Ns ¼ 40 (the number of bins) and Nh ¼ 6000 (the number of filter coefficients), and Fig. 1(b) for Ns ¼ 40 and Nh ¼ 10,000. Such kick angles were calculated using different random numbers of xk of 10,000 sets, and Fourier analyses were done for the 10,000 sets and averaged. The figures are the averaged spectra. The expected flat distributions obtained though the Gibbs phenomenon show a spectrum that has a ripple at the band edge, as seen in Fig. 1(a). It was confirmed that the beam simulations with Nh values 6000 and 10,000 gave similar results. The Fourier analysis for the kick angles by Eq. (3) (the cos method) is shown in Fig. 1(c). The frequency bandwidth was
0 0.0
0.5
1.0 f/f0
1.5
2.0
0.5
1.0 f/f0
1.5
2.0
0.5
1.0 f/f0
1.5
2.0
[arb.]
70
0 0.0 2
[arb.]
when n goes from 0 to Nh, m goes from Nh/2 to Nh/2. T the sampling period, which gives an interval between the input signals, fH (oH ¼ 2pfH) the upper cut-off frequency, and fL the lower one. Such a digital filter is used to produce the kick angle of the RFknockout with a colored noise signal as follows. Serial values of xk produced at random between –1 and 1 are equivalent to the digital white noise sampled with period T. Then serial values of yk are obtained from Eq. (1), which form a digital colored noise with bandwidth ranging from fL to fH. Hence the obtained serial values can be used as the kick angle by which the particles are perturbed at the RF-knockout in each turn when the sampling period is used as a revolution time in a synchrotron. In the beam simulation, since the coasting beam in a synchrotron is divided longitudinally into Ns bins, time interval between the bins is used as the sampling period. The beam in each bin has an equal emittance, and all particles in a bin are perturbed with the same kick angle. The parameters in Eq. (1) are as follows: fH and fL are normalized by the revolution frequency of synchrotron, and therefore T is given by the inverse of the number of bins, Ns. The kick angle, then, is obtained by
0
0.0
Fig. 1. Frequency spectra of the kick angles by colored noises produced by the filter method with (a) Nm ¼ 6000, Ns ¼ 40 and (b) Nm ¼ 10,000, Ns ¼ 40, and (c) by the cos method with N ¼ 20,000, Ns ¼ 40. f0 is the revolution frequency.
divided by 20,000 with a same width, which has an interval of the frequency components of 5.4 105 in units of tune. Random numbers of 10,000 sets were also used for the yk, and the results were averaged. The flat distribution was obtained, except the Gibbs phenomenon.
2.2. Beam-tracking method and the main parameters Beam tracking was performed using the transfer matrix where thin-lens approximation was used for sextupole magnets and an RF-knockout. The data of kick angles of the RF-knockout were computed for all turns before the start of beam tracking. The HIMAC synchrotron has two sets of sextupole magnets as the separatrix exciter [8]. Divergence kicks of the sextupole are given by
Dx0 ¼ 12K2 ðx2 y2 Þ; Dy0 ¼ lK2 xy
ð4Þ
ARTICLE IN PRESS T. Nakanishi, K. Tsuruha / Nuclear Instruments and Methods in Physics Research A 608 (2009) 37–41
where l is the length of sextupole magnet and K2 ¼ B00 /Br. Though the HIMAC synchrotron has other sextupole magnets for chromaticity correction, they were neglected since the simulation was performed with RF-off condition and then the relative momentum spread DP/P (1s) was 4 104, which could be neglected in the simulation. A quadruple magnet for tune correction was used as the FQ. In the simulation, the K2-values were increased linearly during 105 turns (60 ms for the HIMAC synchrotron) and then kept constant. After that, the K-value of FQ was increased linearly in the period of 11 ms to extract the beam, and then the RF-knockout was fired for 11 ms to diffuse a circulating beam. The K-value was decreased to zero in the 0.3 ms period. The time chart of this operation is shown in Fig. 2. Duty factors of the FQ and the RFknockout were set to 50% in the beam experiment; however, the RF-knockout was fired 0.1 ms before the FQ coil current reached the peak value. Such operation condition is included in the simulation. A sawtooth wave for the FQ coil current was used to find a particle density distribution diffused by the RF-knockout. The separatrix phase space area shrinks with increasing K-value, and the ratio of shrunk separatrix area is approximately proportional to the change of the K-value in small range. A spill structure computed, therefore, corresponds to the particle density distribution around the separatrix. An initial particle distribution was assumed to be Gaussian, and the emittance (one sigma) was assumed to be 6p mm mrad. The particle distribution at the beginning is shown in Fig. 3(a). Particles in the inner part of Gaussian distribution were removed in this work, since those particles could not be extracted even after the maximum turn number of 4.5 105 (0.28 s). This resulted in a shorter computing time and increase in particle density near the separatrix boundary. Fig. 3(b) shows the particle distribution after the sextupole magnets were excited. Other parameters were also matched to the ones used in the beam experiment, in which
FQ RFKO
Carbon ions were accelerated up to 400 MeV/u with a repetition period of 3.3 s and extracted during the flattop of 2 s [4]. The bare tunes (nx,ny) were 3.685 and 3.13, respectively. The separatrix was shrunk to 63.2% of the initial area at the maximum K-value of FQ. A frequency bandwidth of colored noise was set to 0.12–1.2, since the frequency characteristic of all-pass network in the RFknockout system had a flat distribution from 0.2 MHz to around 2 MHz and a revolution frequency of 1.653 MHz.
3. Simulation results and comparison with experimental results Spill structures computed with the filter method by Eq. (2) are shown in Fig. 4, where each figure corresponds to 4 times extraction. The parameters Nh and Ns were 6000 and 40, respectively. Spike just before the spill falling to zero is due to the overlap of FQ and the RF-knockout operations. In the figure, the shape of the spill indicates that the particles diffused with the RF-knockout because the separatrix shrunk to the same size at each extraction and particles outside the separatrix were extracted. The width of spill can be changed by controlling the amplitude of colored noise, and in the simulation the amplitude was chosen so that the width of spill has approximately the same value of the experimental result. This was around 2 ms, which corresponds to the beam emittance of 6%. The number of particles in a spill corresponds to around 0.1% of the accelerated particles. An experimental result is shown in Fig. 5, where data were taken with a digital oscilloscope. From the top to the bottom, the current of FQ, the RF-knockout signal and spill structures are illustrated. Fig. 6 shows comparison between the results of experiment and simulation. It is found that the simulation can reproduce spill structures similar to the experimental results. On the other hand, the simulation and experimental results indicate that the spill structures vary a little at every extraction and have
300
11ms
11ms
[arb.]
0.3ms
200 100 0 0.175
0.1ms t Fig. 2. Time chart of the FQ and the RF-knockout.
39
0.185
0.195
0.205
0.215
0.225
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[s] Fig. 4. Spill structures calculated with the filter method.
Fig. 3. (a) Initial particle distribution in horizontal phase space and (b) after the separtrix was produced.
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Fig. 5. Experimental result. From the top, the FQ current (2 V/div), the RF-knockout signal (5 V/div), spill structures (0.5 V/div), and other shot of the spill (0.5 V/div). The time scale is 10 ms/div.
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[s] Fig. 8. Spill structures calculated with the filter method with bandwidth 0.12–5.
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Fig. 6. Spill structures magnified for similar results in the simulation (a, c) and the experiment (b, d). The time scale is 2 ms/div.
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Fig. 9. Magnified spill structure in Fig. 8. The jagged pattern may be attributed to particles not sufficient for the simulation.
sometimes a steep increase as shown in Fig. 6(b). Fig. 7 is a result with the cos method given by Eq. (3), and the result is similar to that of the filter method. In Fig. 8 are shown spill structures computed with the filter method when the upper cut-off frequency was widened to 5, where Nh and Ns are kept at the same value. It is found that variation of spill structures at every extraction was smaller, and moreover the structure increases smoothly with time as shown in Fig. 9. This is because the colored noise becomes almost a white noise when we increase the upper cut-off frequency. Computing time with the filter method was 0.5 h for the kick angle calculation and 17.6 h for the particle tracking, where the numbers of revolutions and particles were 106 turns (0.6 s for
ARTICLE IN PRESS T. Nakanishi, K. Tsuruha / Nuclear Instruments and Methods in Physics Research A 608 (2009) 37–41
the HIMAC synchrotron) and 2 105, respectively. The other parameters were the same as described in the above simulation condition. The kick angles were computed for the interval in which the RF-knockout was operated. Computing times with the cos method were 5.0 h for kick angle calculation and 17.6 h for particle tracking, respectively. Such a long time for the kick angle calculation is due to the calculation of Cosine term 20,000 times for all kick angles (Eq. (3)). On the other hand, there are only simple multiplications in the filter method (Eq. (1)) since the coefficient hn including cosine is computed only at the start of the simulation at first. The computing times of particle tracking are same because of the same calculation. The difference of computing times for kick angles of two methods becomes large when we increase the revolution turns.
4. Conclusions The simulation program for the QAR method has been developed, and the validity was confirmed by comparing the simulated results with the experimental ones. The algorithm for the kick angles calculation by a colored noise has been proposed using the digital filter method, and it was proved that thusobtained results are similar to the ones of the conventional Cos method. The computing time of the proposed algorithm is much less than for the conventional method. It was found from the
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simulation that a colored noise with a wider bandwidth gives a good spill structure that smoothly increases with time.
Acknowledgements The authors would like to express their thanks to Drs. K. Noda and T. Furukawa at National Institute of Radiological Sciences for their useful discussion and continuous support of the beam experiment, and they are grateful to the members of Accelerator Engineering Corporation (AEC) for their skilful operation of the HIMAC accelerator complex. They are also indebted to Dr. T. Katayama at Gesellschaft fur Schwerionenforschung for his invaluable discussion. References [1] E. Pedroni, et al., Proceedings of the NIRS International Workshop on Heavy Charged Particle Therapy and Related Subjects (1991) 94. [2] T. Furukawa, K. Noda, Nucl. Instr. and Meth. A 489 (2002) 59. [3] K. Saito, et al., in: Proceedings of the EPAC08, pp. 1827–1829. [4] T. Nakanishi, T. Furukawa, K. Yoshida, K. Noda, Nucl. Instr. and Meth. A 553 (2005) 400. [5] T. Nakanishi, T. Furukawa, K. Noda, Nucl. Instr. and Meth. B 266 (2008) 2169. [6] S. Nakamura, Digital Filter, TokyoDenkiDaigakuShuppankyoku, Tokyo, 2006 p. 74 (in Japanese). [7] K. Hiramoto, M. Nishi, Nucl. Instr. and Meth. A 322 (1992) 154. [8] K. Noda, et al., Nucl. Instr. and Meth. A 374 (1996) 269.