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Neat Bunching Scheme with Amplitude Modulation
Nuclear Physics B (Proc. Suppl.) 149 (2005) 268–270 www.elsevierphysics.com
Neat Bunching Scheme with Amplitude Modulation Y.Iwashitaa a
Institute for Chemical Research, Kyoto University, Gokanosho, Uji, Kyoto, Japan
A bunching scheme for a long bunch with large energy spread is discussed. This neat bunching scheme is based on fast amplitude modulation by use of beating between frequencies. Tracking simulation results starting from a pion production target in a 16T magnetic field show the feasibility of a velocity compliant bunching scheme that uses time of flight information of the muons. Such a scheme can preserve the longitudinal emittance of the beam, and thus the energy spread can be kept small.
1. INTRODUCTION
2. CAPTURE/DECAY/DRIFT
Muons are obtained from decay of pions emitted from a production target irradiated by a proton beam. The distribution of pions as the secondary product is rather wide in six-dimensional coordinates except for time if the bunch length of the primary proton beam is short enough. When the primary beam has a very short bunch length, a long drift section after the production target results in a good correlation between the TOF (time of flight) and muon energy, where the bunch can become as long as 20m. Then the large energy spread can be reduced by so-called phase rotation, which decelerates early coming fast muons and accelerates late slow muons. This operation has to be completed within the muon lifetime of 2.2µs, which requires rather high electric field gradient, say 0.5MV/m. This high electric field gradient must decrease in a long time period of about 100ns, corresponding to the bunch length (∼20m), and thus the frequency becomes as low as 5MHz. The high gradient, however, makes it difficult to use magnetic material as an inductive load to decrease the frequency. An air core cavity, however, would have a radius of more than a few meters for this low frequency. In order to obviate the need for such huge low frequency cavities and to make use of higher frequency cavities, two high frequency bunching schemes were proposed [1–7]. This paper shows the result of 3D numerical simulations based on the amplitude modulation scheme that uses beating of frequencies.
Because of the rest energy difference between pion and muon, the velocity of a muon is not the same as that of its mother pion, and thus, the longitudinal distribution of muons becomes worse even if the pions have zero transverse emittance. In order to estimate the time spread of the muons, a simulation was performed in a realistic configuration. Figure 1 shows the assumed pion capture and decay/drift sections and the magnetic field distribution for confinement of the muons along the axis: the production target is in a 16T magnetic field that drops to 1T within 5m of the target. Then the magnetic field level is kept constant towards the end. The discussions hereafter are based on this configuration. A simulation whose initial pion distribution has a spread in energy only shows that the time spread caused from the decay process is about 10ns for 100MeV muons (see Fig. 2). This restricts the bunching frequency not more than 40MHz.
0920-5632/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2005.05.042
Capture&Decay section Adiabatic expansion Production target Drift section
B 16T 1T 5m
Buncher cavities (~40MHz) (Phase Rotator)
100m
Figure 1. Magnetic field distribution.
200m
Z
269
Y. Iwashita / Nuclear Physics B (Proc. Suppl.) 149 (2005) 268–270 MeV
MeV
µ+
start
160
at 200m
at buncher (100m)
140
50~150MeV
kick
120 100
99~101MeV
kick
80
ONAXIS start from π+
kick
60 100 80 60 40 20 0 300
y = m1*exp(-((M0-m2)/m3)^ 2) Value Error m1 73.211 4.0002 m2 357.74 0.43056 m3 9.6492 0.60885 2 3031.8 NA R 0.93806 NA
Mu+ (2ns bin) ModBunA5m 99-101MeV @90m 320
340
360
380
400 ToF [ns]
Figure 2. Time spread of a muon bunch becomes 10ns after 90m drift even if the transverse emittance is zero. Top: Longitudinal phase distribution at 90m. Bottom: time spread of the muons with energy range of 99∼101MeV.
3. VELOCITY COMPLIANT BUNCHER Figure 3 shows the proposed bunching scheme. A 100MeV muon bunch with an energy spread of ±50% starts at t=0ns to travel 100m and its length is elongated to about 100ns. Then the bunch is divided into sections at even intervals, and the phase distribution of each period is flipped by a kick from an RF electric field. After one more 100m drift, each phase space distribution stands up right. The bunching system has to handle a time dependent central energy and to change its amplitude rapidly. Although a sawtooth waveform is preferable for 100% bunching, just a single component sine wave with a rapid amplitude change will do the job. Such a rapid amplitude modulation can be achieved by the superposition of a few frequency components. For example, the function shape of the amplitude(envelope) can be expressed by Venvelope (t) = −5.17 cos(0.127 + 4.7124 107 t) + 15 sin(0.648 − 1.5708 107 t) [MV].
kick 40
t=0 ns
400
500
700
900 ns
Figure 3. Longitudinal phase space manipulation for bunching.
The simplest buncher voltage V (t) is a product of the Venvelope (t) and sine wave: V (t) = Venvelope (t) sin(2π 40 106 t + α),
(1)
where α is a minor phase adjustment. V (t) can be reduced into four components by trigonometric reduction. These four components are mapped to four cavities with corrections: the frequencies are adjusted from the original ones in inverse proportion to their distances from the production target. Suppose each cavity is 3m long and their centers are located at 92.5, 97.5, 102.5, 107.5m from the production target, the waveforms are expressed by V1 (t) = 5.17 sin(21.167 − 2π 35.1351 106 t)
(2)
V2 (t) = 5.17 sin(−16.370 − 2π 4.8718 106 t) V3 (t) = 15 sin(4.873 + 2π 41.4634 106 t)
(3) (4)
V4 (t) = 15 sin(−9.275 + 2π 34.8837 106 t)
(5)
Figure 4 shows the results of 3D tracking simulation, where the initial pions are given by MARS calculations of 100k protons on a Tungsten target of 20cm length with a diameter of 1cm. The actual cavity length may have to be longer than 3m and/or the number of cavities may need to be increased to achieve the voltage. In this simulation, only pions and muons with both polarities are tracked. The top figure shows only positive muons for the clarity, where the muons
270
Y. Iwashita / Nuclear Physics B (Proc. Suppl.) 149 (2005) 268–270
4. DISCUSSION
160
µ+
140 120 100 80 60 40
800
900
1000 MuMu+
800
850 900 ToF [ns]
950 1000
2000 1500 1000 500 0 700
1400 1200 1000 800 600 400 200 0 0
750
Mu+
Mu-
As can be seen, the energy spread is kept small within each time slice and the total voltage is fairly small (about 40MV in total). Once the muons are bunched, we can use higher frequencies to manipulate their longitudinal phase space distribution. One method to reduce the energy spread is explained in [1,3,5]. The use of more cavities distributed along the beam line will increase the bunching efficiency, while the RF power and cavity costs will also increase. A frequency of 20MHz may not be high enough to achieve a gradient of 0.5MV/m. A scheme to shorten the bunch length has to be devised for higher bunching frequencies. Isochronous arc transport at the decay section may be useful for this purpose. Another possibility is to collect the higher energy part of the pions/muons and to elongate the resulting short muon bunch with a non-isochronous bend at the drift section. REFERENCES
10
20 30 40 Mod(ToF [ns], 50)
50
Figure 4. 3D simulation results. Top: phase space distribution of positive muons. Center: projected histogram for muons with both charges. Bottom: population of muons in 50ns period.
are bunched in 50ns periods. The bottom projected histogram shows both polarities of muons, bunched in out of phase with each other. The histogram over a 50ns period is shown at the bottom. 0.16 µ/p are bunched into 1/3 of 50ns with this simple four-cavity configuration. Thus a bunching system located 100m from the production target can bunch a muon beam of 100MeV ± 50% with 40MHz RF, and a 20MHz bunched muon beam is obtained at 200m.
1. Y. Iwashita: Another Phase Rotator (Energy Compression Linac) MHIC, OMHIC, MHOMIC and CPEC, KEK Int’l Workshop on High Intensity Muon Sources, 1-4 Dec, 1999, World Scientific, Singapore, pp.307-314 2. Y. Iwashita: High gradient air core cavity for long bunch, NIM A, 472/3, pp.645-651 (2001) 3. Y. Iwashita and A. Morita: High gradient cavities for long bunch muon beams, Proc. PAC2001, 18-22 June 2001, pp.960-2 4. D.Neuffer and A.V.Ginneken: HighFrequency Adiabatic Buncher, NuFact’01, May 24-30, 2001, Tsukuba, Japan, http://psux1.kek.jp/ nufact01/ 5. Y. Iwashita: Distributed modulating buncher for the phase rotation of muons, NIM A, 503, pp.312-317 (2003) 6. D. Neuffer: ”High-Frequency” Buncher and Phase Rotation, NuFact03, AIP conference proceedings, Vol. 721, p.407-412 7. Y. Iwashita: Velocity Compliant Bunching Scheme with Amplitude Modulation, NuFact03, June 5-11, 2003, New York, http://www.cap.bnl.gov/nufact03/WG3/
Neat Bunching Scheme with Amplitude Modulation Y.Iwashitaa a
Institute for Chemical Research, Kyoto University, Gokanosho, Uji, Kyoto, Japan
A bunching scheme for a long bunch with large energy spread is discussed. This neat bunching scheme is based on fast amplitude modulation by use of beating between frequencies. Tracking simulation results starting from a pion production target in a 16T magnetic field show the feasibility of a velocity compliant bunching scheme that uses time of flight information of the muons. Such a scheme can preserve the longitudinal emittance of the beam, and thus the energy spread can be kept small.
1. INTRODUCTION
2. CAPTURE/DECAY/DRIFT
Muons are obtained from decay of pions emitted from a production target irradiated by a proton beam. The distribution of pions as the secondary product is rather wide in six-dimensional coordinates except for time if the bunch length of the primary proton beam is short enough. When the primary beam has a very short bunch length, a long drift section after the production target results in a good correlation between the TOF (time of flight) and muon energy, where the bunch can become as long as 20m. Then the large energy spread can be reduced by so-called phase rotation, which decelerates early coming fast muons and accelerates late slow muons. This operation has to be completed within the muon lifetime of 2.2µs, which requires rather high electric field gradient, say 0.5MV/m. This high electric field gradient must decrease in a long time period of about 100ns, corresponding to the bunch length (∼20m), and thus the frequency becomes as low as 5MHz. The high gradient, however, makes it difficult to use magnetic material as an inductive load to decrease the frequency. An air core cavity, however, would have a radius of more than a few meters for this low frequency. In order to obviate the need for such huge low frequency cavities and to make use of higher frequency cavities, two high frequency bunching schemes were proposed [1–7]. This paper shows the result of 3D numerical simulations based on the amplitude modulation scheme that uses beating of frequencies.
Because of the rest energy difference between pion and muon, the velocity of a muon is not the same as that of its mother pion, and thus, the longitudinal distribution of muons becomes worse even if the pions have zero transverse emittance. In order to estimate the time spread of the muons, a simulation was performed in a realistic configuration. Figure 1 shows the assumed pion capture and decay/drift sections and the magnetic field distribution for confinement of the muons along the axis: the production target is in a 16T magnetic field that drops to 1T within 5m of the target. Then the magnetic field level is kept constant towards the end. The discussions hereafter are based on this configuration. A simulation whose initial pion distribution has a spread in energy only shows that the time spread caused from the decay process is about 10ns for 100MeV muons (see Fig. 2). This restricts the bunching frequency not more than 40MHz.
0920-5632/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2005.05.042
Capture&Decay section Adiabatic expansion Production target Drift section
B 16T 1T 5m
Buncher cavities (~40MHz) (Phase Rotator)
100m
Figure 1. Magnetic field distribution.
200m
Z
269
Y. Iwashita / Nuclear Physics B (Proc. Suppl.) 149 (2005) 268–270 MeV
MeV
µ+
start
160
at 200m
at buncher (100m)
140
50~150MeV
kick
120 100
99~101MeV
kick
80
ONAXIS start from π+
kick
60 100 80 60 40 20 0 300
y = m1*exp(-((M0-m2)/m3)^ 2) Value Error m1 73.211 4.0002 m2 357.74 0.43056 m3 9.6492 0.60885 2 3031.8 NA R 0.93806 NA
Mu+ (2ns bin) ModBunA5m 99-101MeV @90m 320
340
360
380
400 ToF [ns]
Figure 2. Time spread of a muon bunch becomes 10ns after 90m drift even if the transverse emittance is zero. Top: Longitudinal phase distribution at 90m. Bottom: time spread of the muons with energy range of 99∼101MeV.
3. VELOCITY COMPLIANT BUNCHER Figure 3 shows the proposed bunching scheme. A 100MeV muon bunch with an energy spread of ±50% starts at t=0ns to travel 100m and its length is elongated to about 100ns. Then the bunch is divided into sections at even intervals, and the phase distribution of each period is flipped by a kick from an RF electric field. After one more 100m drift, each phase space distribution stands up right. The bunching system has to handle a time dependent central energy and to change its amplitude rapidly. Although a sawtooth waveform is preferable for 100% bunching, just a single component sine wave with a rapid amplitude change will do the job. Such a rapid amplitude modulation can be achieved by the superposition of a few frequency components. For example, the function shape of the amplitude(envelope) can be expressed by Venvelope (t) = −5.17 cos(0.127 + 4.7124 107 t) + 15 sin(0.648 − 1.5708 107 t) [MV].
kick 40
t=0 ns
400
500
700
900 ns
Figure 3. Longitudinal phase space manipulation for bunching.
The simplest buncher voltage V (t) is a product of the Venvelope (t) and sine wave: V (t) = Venvelope (t) sin(2π 40 106 t + α),
(1)
where α is a minor phase adjustment. V (t) can be reduced into four components by trigonometric reduction. These four components are mapped to four cavities with corrections: the frequencies are adjusted from the original ones in inverse proportion to their distances from the production target. Suppose each cavity is 3m long and their centers are located at 92.5, 97.5, 102.5, 107.5m from the production target, the waveforms are expressed by V1 (t) = 5.17 sin(21.167 − 2π 35.1351 106 t)
(2)
V2 (t) = 5.17 sin(−16.370 − 2π 4.8718 106 t) V3 (t) = 15 sin(4.873 + 2π 41.4634 106 t)
(3) (4)
V4 (t) = 15 sin(−9.275 + 2π 34.8837 106 t)
(5)
Figure 4 shows the results of 3D tracking simulation, where the initial pions are given by MARS calculations of 100k protons on a Tungsten target of 20cm length with a diameter of 1cm. The actual cavity length may have to be longer than 3m and/or the number of cavities may need to be increased to achieve the voltage. In this simulation, only pions and muons with both polarities are tracked. The top figure shows only positive muons for the clarity, where the muons
270
Y. Iwashita / Nuclear Physics B (Proc. Suppl.) 149 (2005) 268–270
4. DISCUSSION
160
µ+
140 120 100 80 60 40
800
900
1000 MuMu+
800
850 900 ToF [ns]
950 1000
2000 1500 1000 500 0 700
1400 1200 1000 800 600 400 200 0 0
750
Mu+
Mu-
As can be seen, the energy spread is kept small within each time slice and the total voltage is fairly small (about 40MV in total). Once the muons are bunched, we can use higher frequencies to manipulate their longitudinal phase space distribution. One method to reduce the energy spread is explained in [1,3,5]. The use of more cavities distributed along the beam line will increase the bunching efficiency, while the RF power and cavity costs will also increase. A frequency of 20MHz may not be high enough to achieve a gradient of 0.5MV/m. A scheme to shorten the bunch length has to be devised for higher bunching frequencies. Isochronous arc transport at the decay section may be useful for this purpose. Another possibility is to collect the higher energy part of the pions/muons and to elongate the resulting short muon bunch with a non-isochronous bend at the drift section. REFERENCES
10
20 30 40 Mod(ToF [ns], 50)
50
Figure 4. 3D simulation results. Top: phase space distribution of positive muons. Center: projected histogram for muons with both charges. Bottom: population of muons in 50ns period.
are bunched in 50ns periods. The bottom projected histogram shows both polarities of muons, bunched in out of phase with each other. The histogram over a 50ns period is shown at the bottom. 0.16 µ/p are bunched into 1/3 of 50ns with this simple four-cavity configuration. Thus a bunching system located 100m from the production target can bunch a muon beam of 100MeV ± 50% with 40MHz RF, and a 20MHz bunched muon beam is obtained at 200m.
1. Y. Iwashita: Another Phase Rotator (Energy Compression Linac) MHIC, OMHIC, MHOMIC and CPEC, KEK Int’l Workshop on High Intensity Muon Sources, 1-4 Dec, 1999, World Scientific, Singapore, pp.307-314 2. Y. Iwashita: High gradient air core cavity for long bunch, NIM A, 472/3, pp.645-651 (2001) 3. Y. Iwashita and A. Morita: High gradient cavities for long bunch muon beams, Proc. PAC2001, 18-22 June 2001, pp.960-2 4. D.Neuffer and A.V.Ginneken: HighFrequency Adiabatic Buncher, NuFact’01, May 24-30, 2001, Tsukuba, Japan, http://psux1.kek.jp/ nufact01/ 5. Y. Iwashita: Distributed modulating buncher for the phase rotation of muons, NIM A, 503, pp.312-317 (2003) 6. D. Neuffer: ”High-Frequency” Buncher and Phase Rotation, NuFact03, AIP conference proceedings, Vol. 721, p.407-412 7. Y. Iwashita: Velocity Compliant Bunching Scheme with Amplitude Modulation, NuFact03, June 5-11, 2003, New York, http://www.cap.bnl.gov/nufact03/WG3/