Investigation of a muon cooling channel with Li lenses and high field solenoids

1. Investigation of a muon cooling channel with Li lenses and high field solenoids V. Balbekov, MCTF meeting 03/18/2010 Li lenses are used for ionization cooling High field solenoids (up to 50 T) are used for adiabatic matching RF 200 MHz, about 16 MeV/m is applied 150 m long channel including 15 Li lenses of 1 m length is investigated No emittance exchange 1

2. Some preliminary notes The main problem of Li lenses channel is matching section between the lenses. It occurs because of unavoidably huge difference of beta-functions in the lenses (typically 1 cm) and in a transport part (solenoid) (~1 m). Corresponding matching section brings large chromaticity and really can’t provide a matching of a beam with reasonable momentum spread. Using of a high field (25-50 T) solenoid could facilitate the problem because its own beta function is several cm. It is important that transition from high field to a modest one occurs adiabatically, in practice (it is more difficulty to make an abrupt transition). Similar adiabatic solenoid channel has been proposed by R. Palmer for final cooling. I am going to use these features for adiabatic matching of Li lenses. 2

3. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Schematic and idea of the channel (V.Balbekov, MCD Workshop, Dec. 2009) Main parts of the channel: Blue – solenoid coils Red -- Li lenses Green – 200 MHz cavities Transition from high to low field region is adiabatic itself. Li lens should provide the adiavatic transition from (+) field to (-) one Li rod should have a special form to provide the adiabaticity (example) With solenoid field B and lens gradient G, beta – function is: Adiabaticity condition (Palmer) Gray – fringe regions where a special study is needed ? 3

4. Only the lenses are simulated. Ideal matrix is used instead of matching sections. The following basic parameters are taken for this investigation Channel length 150 m Solenoid cell length 10 m Maximal solenoid field 50 T Number of Li lenses 15 Li lens length 1 m Li lenses gradient 27-108 T/cm Linacs 14 x 8 m + 2 x 4 m Accelerating frequency 200 MHz Linac accelerating gradient 15.85 MV/m, 127 MV/linac Reference energy rate 11.255 MeV/m, 90 MeV/linac Synchronous phase 45.2° 02c_rod.pdf 4

5. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Solenoid field Blue: high-field solenoid coils, right-hand part Pink: a half of Li lens (schematically). Volume of diameter 30 cm is provided for it Axial field in the solenoid axes. Maximal value 50 T Transport 4 Tesla solenoid has inner radius 60 cm. 02c_rod.pdf Other coils were considered as well (shape, size, etc.). In all the cases, transition from high to low field is adiabatic (at proper current density), and required beta-function is provided by optimization of the Li lens. Conclusion: design of this assembly is an engineering problem, primarily. 5

6. Only the lenses are simulated. Ideal matrix is used instead of matching sections. 02c_rod.pdf Gradients of Li lenses (Tesla/cm) are presented as functions of longitudinal coordinate Z measured from the lens center. 15 lenses are shown: lower – 1st lens, higher – 15th one. The transition region 7.5 – 10 cm long is applied to get linearly decreasing beta-function. 6

7. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Beta-functions at 250 MeV/c vs longitudinal coordinate Z 02c_rod.pdf Left: global picture (1/2 of cell). Beta-function does not depend on cell number at Z>50 cm, that is out the lens. Zoom of central part is shown in the right-hand plot. Beta-functions in the Li lenses. Lower curve – 1st lens, higher – 15th one, In the transition region dβ/dz = 0.25. 7

8. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Effective solenoid fields vs longitudinal coordinate at 250 MeV/c. These fields would provide the same beta-function as real solenoid+Li lenses. 02c_rod.pdf It is seen that the focusing is provided mostly by Li lenses. In particular, effective field of15th lens is four times more then actual one. It means that contributions of the lens and the solenoid are related as 3:1. 8

9. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Reference momentum and simulated average beam momentum vs longitudinal coordinate. 02c_rod.pdf The reference momentum oscillates from 200 to 300 MeV/c. Rise – acceleration by linacs, fast decrease – deceleration by Li lenses. One can see small coherent oscillations of the beam. Actually, longitudinal phase space is very complicated – a fact which requires a special consideration. 9

10. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Transverse cooling (rms emittance vs length). Global view: normalized rms emittance, beam radius, and transverse momentum as functions of longitudinal coordinate Transverse emittance: scans around several last lenses 02c_rod.pdf Beam size oscillations are observed from the outset. It means that all the field transitions are not sufficiently adiabatic. In the first half of the channel, it does not causes a visible emittance growth. However, the emittance growth is observed around the last lenses. It constrains the emittance decrease in 15th lens, though the cooling continues in its body In this simulation: initial / final emittances are 1000 / 86 μm. 10

11. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Scans of the beam radius Scans of transverse beam momentum 02c_rod.pdf Transverse beam size beating is ~±6% in the beginning of 1st lens. Then it decrease – almost to 0 in 5th lens. Then it increase again achieving ±10% in the beginning of 15th lens. Systematic growth of the beam emittance is observed just after 5th lens. I can’t explain this fact. 11

12. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Adiabaticity violation in the lens boundaries and the beam beating inside the lenses are observed at dβ/ds = 0.25. It results in additional beam heating and restricts achievable beam emittance. A decrease of dβ/ds looks reasonably – at least in the latest lenses. However, it requires an expansion of transition areas at the lenses ends, which were up to 10 cm (20% of full lens length). It causes an increase of average beta functions of the lenses – an effect which overrides the adiabaticity improvement . As a result, a positive effect is not reached in this way. 02c_rod.pdf 12

13. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Transverse merit-factors and transmission vs length. characterizes growth of phase density characterizes growth of luminosity Particles loss is 23% at L=150 m (10% decay loss). Merit factor 1 (phase density) is 9.0 at L=150 m. Merit factor 2 (luminosity) reaches maximum 7.5 at L=120 m 13

14. Parameters of Li lenses The parameters are given for central part of the lenses (~80 cm long) with constant gradient. Radius of this part is 4σmax. Most of the lenses have surface field about 20 T. Probably, gradients and field of first pair can be decreased without loss of efficiency (additional optimization is needed). The last pair should be excluded because of very low efficiency. 14

15. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Longitudinal motion Longitudinal emittance, rms energy spread, and transmission vs longitudinal coordinate Z. Longituginal phase space at Z = 0, 50, 100, and 150 m Horizontal – ct (cm), vertical – ΔE (MeV) 15

16. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Longitudinal motion -- some conclusions Longitudinal emittance very fast increases causing the particles loss. Two factors seems to be the most important: straggling and dependence of time of flight on betatron amplitude. The factors work cumulatively: exclusion of any of them strongly decreases the emittance growth rate. 02c_rod.pdf 16

17. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Is it possible to reach less transverse emittance at less beam momentum? Less momentum means less beta function and equilibrium emittance. It is important on the latest cells, because beam emittance is much more of the equilibrium one in the beginning. Therefore I consider the same channel at decreasing middle momentum 02c_rod.pdf 17

18. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Reference momentum and average beam momentum vs longitudinal coordinate. Acceleration 11.1 MeV/m by 200 MHz RF 02c_rod.pdf Initial reference momentum oscillates with amplitude 50 MeV/c. Middle momentum drops from 250 MeV/c to 210 MeV/c . 18

19. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Transverse cooling (rms emittance vs length). Global view: normalized rms emittance, beam radius, and transverse momentum as functions of longitudinal coordinate Transverse emittance: scans around several last lenses 02c_rod.pdf The plots look very much likely to previous ones. However, final emittance is less: 86 μm → 73 μm (factor 0.85) It is very nearly the final momenta ratio: 210 / 250 ≈ 0.84 19

20. Only the lenses are simulated. Ideal matrix is used instead of matching sections. However, the decrease of transverse emittance does not result in increase if merit factors, because of more particles loss. Merit factors and transmission are plotted below. Lelt – old results at final momentum 250 MeV/c, right – new ones at 210 MeV/c. Particles loss rise steeply in seconf half of the channel resulting in decrease of the merit factors. 20

21. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Longitudinal motion is the primary source of the particles loss Longitudinal emittance, rms energy spread, and transmission vs longitudinal coordinate Z. Longituginal phase space at Z = 0, 50, 100, and 150 m Horizontal – ct (cm), vertical – ΔE (MeV) Long bunch tails appear at last cestions of the channel. These particles are out the separatix an are lost, infact. 21

22. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Longitudinal motion (cont’d) Longitudinal emittance very fast increases causing the particles loss. Two factors seems to be the most important: straggling and dependence of time of flight on betatron amplitude. The factors work cumulatively: exclusion of any of them strongly decreases the emittance growth rate. These effects create the main obstacles to use a lower beam momentum – an action which would be very desirable for transverse cooling. The momentum lowering significantly decreases RF separatrix, i.e. momentum acceptance of the channel (probably, slope of dE/dx curve is not so important). As a result, transmission, merit factors, etc. essentially drop at lower momentum 02c_rod.pdf 16

23. Only the lenses are simulated. Ideal matrix is used instead of matching sections. Why the solenoid field flip is needed? Transverse phase space is shown before the channel entry and after its exit Field flip Li lens X-Px projection X-Py projection With field flip: Perfect (upright) phase ellipsoid. Minimal problems of matching with subsequent device (e.g. linac). Without field flip but with Li lens: a little distortion of phase ellipsoid. Probably, not great growth of effective emittance. Solenoid without field flip: outgoing beam has large radius and angle spread at strong X-Py and Y-Px correlations. Problematic matching. Yes Yes 02c_rod.pdf No Yes No No 17

24. Conclusion • It is shown that high field solenoid can be used for adiabatic matching of Li lenses. • Alternate solenoids is a preferable choice because it providing a perfect upright 4D ellipsoid in transverse phase space after the cooling. • High gradient RF system can be used for compensation of ionization energy loss, though substantial growth of longitudinal emittance occurs (200 MHz / 16 MeV/m accelerator is actually used at simulations). • Achievable transverse emittance is about 80 μm at the solenoid field 50 T and middle beam momentum 250 MeV/c. • Less emittance can be reached at lower momentum, but this does not increase merit factor because of higher particles loss at longitudinal motion. • The problem could be solved by using an induction linac, but similar channel would be longer in order of value. 18

25. Next step I am going to consider similar (adiabatic) channel at less solenoid field because 50 T may be extremely high. Preliminary result is very expected: transverse emittance 0.16 mm is reached with 26 T solenoid, that is achievable emittance is inversely proportional to the field. 18