1. Muon cooling with Li lenses and high field solenoids V. Balbekov, MAP Winter Meeting 02/28-03/04, 2011 OUTLINE  Introduction: why the combination of Li lenses and HF solenoids is needed?  Schematic and main parts of the channel ● Li lenses for cooling ● HF solenoids for adiabatic matching ● RF 200 MHz or less for acceleration ● No emittance exchange  Cooling simulation ● 250 MeV/c ● 200 MeV/c ● 130 MeV/c  Comparison and conclusion 1

2.  The main problem of a channel with Li lenses is their matching.  The reason is a huge difference of beta-functions in the lenses (typically 1 cm) and in the matching part (e.g. transport solenoid, β ~1 m).  Similar channel unavoidably has large chromaticity and really can’t provide a deep cooling of a beam with reasonably large momentum spread.  However, high field solenoids (40-50 T) can solve the problem because its own beta function is comparable with Li lens beta (several cm).  Therefore smooth (adiabatic) transition from Li lenses to HF solenoids is possible, as well as transition from the high field solenoid to low field (transport) one.  The adiabatic HF solenoid channel has been proposed by R. Palmer for final cooling, but addition of Li lenses could to amend its performances.  Optimization of momentum and other parameters of the channel is aimed in this talk. 2 Introduction: why Li lenses and HF solenoids? Introduction: why Li lenses and HF solenoids?

3. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 3 The channel includes: Red -- Li lenses for cooling. 8-12 different lenses can be applied in the channel. Blue – Solenoid coils and field for adiabatic matching. Maximal field 50 T, transport solenoid 4 T. Field flip Is used for a value transverse cooling. It is applied inside the Li lenses and does not violate the adiabaticity. Green – RF cavities (linac) of 100 – 200 MHz, 10 – 12 MeV/m. Low frequency version (induction linac) is also investigated for comparison. Schematic of the channel Schematic of the channel × 4 - 6 depending on: -- beam momentum; -- accelerating frequency; -- field/gradient strength; -- initial/final emittances; -- etc. ×

4. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. High-field part of the solenoid coil is shown (right -hand part). Transport solenoid has inner radius 60 cm and thickness 1 cm. Axial field in the solenoid axes. Maximal value 50 T Transport solenoid provides 4 T field It is not an engineering proposal. The geometry is used to get Maxwellian field map with required characteristics. There are a lot of other versions which assure the requirements and can provide adiabaticity of the movement.. 4 Solenoid coils and field Solenoid coils and field

5. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Li rods should have a special form of ends for a smooth transmission to solenoids. Relatively low-gradient end bells should be enough long to provide the adiabaticity, but rather short to give a modest contribution to the scattering due to larger beta-function. High field matching solenoid allows to conform both requirements 5 Li lens: schematic and gradient Li lens: schematic and gradient The rod is sketched in the picture. Transition region is 5 – 10 cm long. Gray ends are fringe field regions where a special study is required.. The lens gradients and the solenoid field in the central region (250 MeV/c channel with 12 Li lenses is presented as an example)..

6. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 6 Beta functions of above mentioned 250 MeV/c cannel are presented. Maximal derivative in the lens is It satisfy the adiabaticity condition (R.Palmer) (maximal derivative in the solenoid 0.16) Beta-function at 250 MeV/c Beta-function at 250 MeV/c Minimal beta-function provided by the solenoid is 2.6 cm (42 cm in the transport solenoid). Li lens provide beta-functions 1.4 - 0.8 cm It means that the Li lenses are 2--3 times more effective then the solenoids. With solenoid field B and lens gradient G, beta – function is calculated by the formula

7. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf Channel length 120 m Number of Li lenses 12 Lattice cell length 10 m Maximal solenoid field 50 T Li lens length 1 m Li lenses gradient 34 -- 95 T/cm Linacs 11 x 8 m + 2 x 4 m Accelerating frequency 200 MHz Linac accelerating gradient 15.7 MV/m, 126 MV/linac Reference energy rate 11.1 MeV/m, 89 MeV/linac Synchronous phase 45° Initial transverse emittance 0.4 mm Initial longitudinal emittance 1.0 mm 7 Cooler parameters at average beam momentum 250 MeV/c Cooler parameters at average beam momentum 250 MeV/c

8. # R (mm) G (T/cm) Bmax (T) J (kA) 1 5.28 33.8 17.9 472 2 4.47 36.7 16.4 368 3 3.91 40.0 15.6 305 4 3.52 43.5 15.3 269 5 3.15 47.5 15.0 236 6 2.89 51.9 15.0 217 7 2.72 56.9 15.5 210 8 2.55 62.7 16.1 205 9 2.40 69.1 16.6 199 10 2.25 76.6 17.2 194 11 2.17 85.1 18.5 201 12 2.08 94.9 19.7 205 8 Parameters of the lenses are given for central parts (84--90 cm long) where their gradients are constant. Radii of these parts is 3.75 σ. The lenses have surface field less of 20 T. Beta-functions in centers of the lenses are presented at 250 MeV/c Li lenses parameters at 250 MeV/c Li lenses parameters at 250 MeV/c Beta (cm) 1.57 1.51 1.44 1.38 1.32 1.27 1.21 1.15 1.10 1.04 0.99 0.94

9. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf 9 Reference momentum ranges from 200 to 300 MeV/c Transverse emittance decreases from 400 μm to 85 μm Longitudtinal emittance increases from 1 mm to 9 mm Transmission 84% with decay Particles loss 16% = 7.5% (decay) + 7.5%(aperture) + 1% (longitudinal motion) Cooling simulation at 250 MeV/c Cooling simulation at 250 MeV/c Transmission w/o and with decay x 5 Trans. phase space (arbitrary units) Reference energy (mc 2 ) Long. phase space (arb. length units) Longitudinal emittance x 2 (cm) Transverse emittance x 10 (mm)

10. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 10 Left-hand plot demonstrates efficiency of the lenses. (scan of the beam transverse emittance is presented against the coordinate and the lens number). The first lens reduces emittance almost by 30%, the last – less of 5%. 250 MeV/c (continued) 250 MeV/c (continued) Transverse space. Red – beginning, blue -- end Longitudinal space. Blue – beginning, red -- end Bottom plots represent phase space of the beam in the beginning and in the end of the channel

11. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf  The channel provides transverse emittance cooling from 400 μm to 85 μm (cooling factor 4.7)  Transmission 84% is achievable. 16% loss including 7% decay loss  Longitudinal emittance increases from 1 mm to 9 mm (but the outgoing bunch has a long non-Gaussian tail).  6D emittance decreases from 0.16 mm 3 to 0.065 mm 3 (cooling factor 2.5) 11 Summary 250 MeV/c Summary 250 MeV/c Initial transverse emittance ~1 mm can be used with 2-3 additional lenses. Then about the same transverse emittance is achievable, but longitudinal emittance increases roughly 1 mm / lens (V.B. MCTF meeting 03/18/2010).

12. 02c_rod.pdf Decrease of the beam momentum is a straightforward way to reach less transverse beam emitance. Cooling simulation of 200 MeV/c beam is considered in this section. Minimal changes of the lattice are made: -- number of lenses is decreased to10, -- their parameters are given in the table 12 Cooler with average beam momentum 200 MeV/c # R (mm) G (T/cm) Bmax (T) J (kA) 1 5.38 34.1 18.4 493 2 4.27 38.2 16.3 348 3 3.60 42.8 15.4 277 4 3.08 48.2 14.9 229 5 2.65 54.3 14.4 190 6 2.40 61.5 14.7 177 7 2.25 70.1 15.8 178 8 2.07 80.3 16.6 172 9 1.93 92.6 17.9 173 10 1.83107.6 19.7 180 Beta (cm) 1.40 1.32 1.25 1.18 1.11 1.04 0.95 0.91 0.85 0.79 The most changes concern RF because 200 MHz system is unable to keep the beam. 100 MHz RF system is used in this simulation. Two alternatives are considered: ultimate voltage 17 MV/m and modest one 10 MV/m

13. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf 13 Cooling simulation at 200 MeV/c (17 MV/m) Transmission w/o and with decay x 5 Trans. phase space (arbitrary units) Longitudinal emittance (cm) Reference energy (mc 2 ) Long. phase space (arb. length units) Transverse emittance x 10 (mm) Reference momentum ranges from 143 to 258 MeV/c Transverse emittance decreases from 400 μm to 63 μm Longitudinal emittance increases from 1 mm to 26 mm Transmission 79% with decay Particles loss 21% = 8% (decay) + 6%(aperture) + 7% (longitudinal motion) 10 linacs 16.8 MV/m x × 8 m provide 134 MV/linac, 1340 MV in all. At 45º synchronous phase, acceleration is 11.9 MeV/m × 8 m = 95 MeV/linac, 950 MeV in all. Total length of the channel is 100 m

14. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf 14 Cooling simulation at 200 MeV/c (10 MV/m) Emittances are almost the same as in previous example (62-63 μm, 25-26 mm) However, a lot of particles come to a stop in 5 th – 10 th rods Therefore transmission decreases from 79% to 52% (with decay) Approximate loss distributions: 48% = 11% (decay) + 3%(aperture) + 34% (long. motion) 10 linacs 10.3 MV/m × 13 m provide 134 MV/linac, 1340 MV in all. At 45º synchronous phase, acceleration is 7.3 MeV/m × 13 m = 95 MeV/linac, 950 MeV in all. The cell length 15 m, total length of the channel is 150 m

15. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf  Lower accelerating frequency is required to confine lower momentum bunch.  The following parameters are achievable with 100 MHz / 17 MV/m RF, total length of the channel 100m:  Transverse emittance decreases from 400 μm to 63 μm (cooling factor 6.3)  Longitudinal emittance increases from 1 mm to 26 mm (with non-Gaussian tail).  6D emittance decreases from 0.16 mm 3 to 0.10 mm 3 (cooling factor 1.6)  Transmission 79%. Loss 21% including 8% decay loss  However, the transmission falls with less accelerating gradient: 52% at 10 MV/m  Less stability region of longitudinal motion is the main constraining factor.  Decay increases from 8% to 11% at the channel lengthening from 100 m to 150 m. 15 Summary 200 MeV/c Summary 200 MeV/c

16. 02c_rod.pdf 16 Cooler with average beam momentum 130 MeV/c # R (mm) G (T/cm) Bmax (T) J (kA) 1 6.18 30.0 18.6 573 2 4.67 34.0 15.9 371 3 3.63 38.9 14.1 256 4 2.98 44.2 13.2 196 5 2.62 50.4 13.2 172 6 2.35 57.8 13.6 160 7 2.16 66.4 14.3 155 8 2.00 76.5 15.3 153 9 1.97 89.1 16.6 155 10 1.77104.0 18.4 162 Beta (cm) 1.20 1.13 1.11 0.99 0.93 0.87 0.81 0.75 0.70 0.65 50 cm Li lenses are used in this channel. Their parameters are presented in the table. The solenoid coil is slightly modified to move the field maxima in the lens edges (±25 cm). A drastic decrease of accelerating frequency is required to keep the beam at so low average momentum. 5 MHz RF is actually applied at simulation. Two linacs with different accelerating gradients are investigated. Length of the transport solenoid is changed to provide needed room for the linac.

17. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf 17 Cooling simulation at 130 MeV/c (6.9 MeV/m) Transmission w/o and with decay x 5 Trans. phase space (arbitrary units) Longitudinal emittance /10 (cm) Reference energy (mc 2 ) Long. phase space (arb. length units) Transverse emittance x 10 (mm) Reference momentum ranges from 90 to 160 MeV/c After 9 th lens: Transverse emittance decreases from 400 μm to 51 μm Longitudinal emittance increases from 1 mm to 270 mm Transmission 79% with decay (like 200 MeV/c, 12 MeV/m version) Particles loss 21% = 10% (decay) + 5%(aperture) + 6% (longitudinal motion) 10 linacs 6.9 MeV/m x × 8.5 m provide energy gain 58.7 MeV/linac, 587 MeV in all. Total length of the channel is 100 m

18. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf 18 Cooling simulation at 130 MeV/c (1 MeV/m) Trans. phase space (arbitrary units) Longitudinal emittance /2 (cm) Transmission w/o and with decay x 5 Reference energy (mc 2 ) Long. phase space (arb. length units) Transverse emittance x 10 (mm) Reference momentum ranges from 90 to 160 MeV/c After 8 th lens: Transverse emittance decreases from 400 μm to 52 μm Longitudinal emittance increases from 1 mm to 420 mm Particles loss 55% (40% decay loss) 10 linacs 1.003 MeV/m x × 58.5 m provide 58.7 MeV/linac, 587 MeV in all. Linac length 58.5 m, cell length 60 m, channel length 600 m

19. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf  In principle, transverse emittance 50 μm is achievable at 130 MeV/c average momentum. (20% less then in 200 MeV/c channel)  However, a drastic decrease of accelerating frequency is required to avoid unacceptable particles loss because of coming to stop in the Li rods.  Accelerating gradient should be rather high (5-7 MeV/m) to hold acceptable length of the channel and moderate loss of particles.  The following parameters are obtained by simulation with a realistic voltage 1 MeV/m:  The cooler length 480 m, 8 Li lenses  Transverse emittance decreases from 400 μm to 52 μm (cooling factor 7.7).  Longitudinal emittance increases from 1 mm to 420 mm.  6D emittance increases from 0.16 mm 3 to 1.1 mm 3  Transmission 45%. Loss 55% including 40% decay loss. Unacceptable 19 Summary 130 MeV/c Summary 130 MeV/c

20. Only the lenses are simulated.Ideal matrix is used instead of matching sections. Only the lenses are simulated.Ideal matrix is used instead of matching sections. 02c_rod.pdf 20 70 cm Longitudinal phase space Longitudinal phase space 140 cm 2000 cm 106 MeV  Growth of the beam energy spread is the most constraining factor. It is caused by: Dependence dE/dx(E); straggling, and dependence of time of flight on betatron amplitude.  Longitudinal phase space is shown at 3 average central momenta (blue – start, red – end) 250 MeV/c, 200 MHz, 11 MeV/m 200 MeV/c, 100 MHz, 12 MeV/m 130 MeV/c, 5 MHz, 7 MeV/m  Resulting energy spread is about the same in considerer cases, because the same transverse cooling factors is aimed.  The spread does not allow to keep central kinetic energy below ~40 MeV, momentum 90 MeV/c.  Accelerating wavelength should be more at lower momentum to keep the beam with such energy spread.  Therefore final longitudinal emittance is about proportional to the wavelength (9; 26; 420 mm)

21.  High field solenoid can be used for adiabatic matching of Li lenses.  Cooling channels with 50 T solenoids and up to 20 T surface field lenses are investigated.  Achievable transverse emittance is 50 -- 85 μm in dependence on the beam momentum.  Less transverse emittance is achievable at lower momentum; however, it requires lower accelerating frequency.  Both technical and fundamental problems appear at lower RF (longer channel, more decay loss, larger longitudinal emittance).  The option: 200 MeV/c, 100 MHz, 10-12 MeV/m looks preferably providing: transverse emittance ~65 μm, longitudinal ~2.5 cm, transmission ~80%. 21 Conclusion Conclusion