Cooling channel with Li lenses

1
V.Balbekov, August 2006

• Cooling channel with Li lenses
• (or supersolenoids)
• (beam dynamics problems)
• V. Balbekov, Aug.14, 2006
• Ultimate performances of Li lens cooling
  channel
• Short-periodic channel
• Long-periodic channel
• Ring cooler with Li lens
• Curved Li lens

2
V.Balbekov, Aug. 2006
Ultimate performances of Li lens cooling
channel Li lens is a current-carrying Li rod
3
V.Balbekov, Aug. 2006
4
V.Balbekov, Aug. 2006
5
V.Balbekov, August 2006

• Idealized multi-lens cooling channel
• In an optimized multi-lens cooling channel, the
  lens field should increase, and the lens length
  should decrease by the same factor step by
  step.
• If all the lenses have the same length, a
  constant increase/decrease factor has to be
  applied.
• Exponential transverse cooling is achieved by
  this, in spite of scattering.
• Transverse decrement in the lens is about 0.66
  / ß4? per meter.
• Longitudinal increment in the lens is about
  1.3 / ß4?3 per meter.

6
V.Balbekov, August 2006

• Example 7 x 154 cm Li lens channel
• The lens field increases, and the radius
  decreases by factor 1.6 / cell.
• Muon momentum drops from 317 MeV/c to 211
  MeV/c in any lens.
• Beta function, equilibrium transverse emittance
  and beam radius are given at
• 211 MeV/c.

7
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• 7x154 cm channel
• simulation
• Only lenses are simulated.
• Ideal matrices are used for the matching.
• ---------------------------------
• Initial gt final emittance
• Trans 6.1 gt 0.22 mm
• (final equilib. 0.12 mm)
• Decrement 0.31 / m
• Long 3.0 gt 22 mm
• Increment 0.18 / m
• 6D 110 gt 1.0 mm3
• Decrement 0.44 / m
• Good agreement with analytical predictions

8
V.Balbekov, August 2006

• Ultimate performances summary
• In an optimized multi-lens cooling channel, the
  lens field should increase, and the length --
  decrease step by step.
• Required relations rmaxBmax const 7.5
  T-m, Jlens const 375 kA
• At these conditions, beta- function and
  equilibrium emittance are
• 20 T field is believed to be achievable,
  resulting in
• ß 1 cm, eeq 90
  mm-mrad at p 200 MeV/c.
• Transverse emittance decreases in the lens with
  decrement 0.66/ß4? per m
• Longitudinal emittance increases in the lens
  with increment 1.3/ß4?3 per m

9
V.Balbekov, August 2006

• Chromatic aberrations in matching sections is the
  main problem
• 90o section with focusing gradient G2 matches
  regions of high gradient
• G1 (low ß) and low gradient G3 (high ß), if
  (G2)2 G1G3
• However, it is possible for reference particle
  only otherwise the phase ellipse rotates on
  angle.
• ?f f ?p/p
• Low phase advance per cell f
• mitigates the violation
• and makes the problem easier.

10
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Short period linear cooling channel lattice
• To reach maximal energy acceptance, betatron
  phase advance should be 90o in each part ½
  Li lens, short solenoid, ½ long solenoid.
• Identical cells are used in this simulation.
  The cell parameters

11
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Short period linear cooling channel ß-function
• ß-function vs distance
  Central
  ß-function vs energy
• 
  
  (dashed spur of the tr. matrix)
• Linear resonances at phase advances 2p/cell and
  3p/cell are suppressed by special choice of the
  lens and solenoid parameters (p/2 phase
  advances per unit are important).
• Resulting ß lt 5 cm at 190 MeV lt E lt 340 MeV.

12
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Short period linear cooling channel simulation
• 
  
• 
  Transverse phase
  space
• 
  
  Initial
  Final
• 
  Longitudinal
  phase space
• 
  Initial
  Final
  

13
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Short period linear cooling channel summary
• The channel has large longitudinal acceptance
  (?p/p 20) and good transmission (97 without
  decay).
• Simulated emittance almost achieves the
  theoretical limit (0.52 / 0.46 mm).
• However, transverse cooling factor is small
  (3 on 100 m).
• A channel with decreasing ß-function is
  required to reach more cooling.
• However, a contraction of Li lenses is
  required also to maintain phase advance p
  on lens (L lens 3ß).
• Required number of the lenses and cells
  increases also.
• No ideas how to apply this scheme to really
  high field lenses.
• No ideas how to introduce emittance exchange.

14
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Long period linear cooling channel
• Idea (Jim Norem, 1998?)
• 3600 synchrotron phase advance should be provided
  in each matching section. Then average energy of
  any particle approximately coincides with
  synchronous energy independently on the
  synchrotron amplitude, a property which can
  mitigate chromatic effects.
• However
• The matching sections should be rather long to
  provide so large phase advance.
• Therefore, energy growth in the section should
  be large as well to provide a high cooling
  decrement.
• Li lenses should be long enough to assure
  corresponding energy loss.
• Rather high average energy is needed because
  the chromatic effects are
• proportional to ?p/p ? ?E/E.

15
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Long period linear cooling channel schematic
• 
  
• 
  

16
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Long period linear cooling channel simulation
• Transverse emittance from 1 mm to 0.27 mm
  (equilibrium 0.12 mm).
• The emittance increases in the matching sections
  because of chromatic and nonlinear aberrations.
• Longitudinal emittance decreases in the matching
  sections because of scrapping.
• 
  
• 
  

17
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Long period linear cooling channel summary
• The idea to suppress chromatic effects at 3600
  synchrotron phase advance/cell works.
• However, the suppression is not perfect,
  especially at lower energy.
• Because of this, as well as because of
  nonlinear effects, achievable emittance
  considerably more of the theoretical limit 0.27
  / 0.12 mm (compare
  0.52 / 0.46 mm in the short period cooler).
• At the same reasons, significant non-decay
  particles loss occurs, mainly at large energy
  deviation and respectively large chromatic
  aberrations.
• The particles energy should be rather high to
  mitigate these effects. It partly depresses the
  cooling resulting in more final emittance.
• Rather large field of matching solenoids is
  required. Probably, short low gradient Li lenses
  (broadening of the edges) can be used instead.
• Bent solenoid added for emittance exchange
  makes worse the chromaticity suppression. The
  investigation is required.

18
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Ring cooler with Li lens (Y.Fukui, D.Cline,
  A.Garren, H.Kirk, 2003)
  --------------------------------------------------
  ------------------------------------------------
• Ring 37.5 m,
  
• p 250 MeV/c, ?E 30 MeV/turn

Straight sections simulation
6 m long, ßmax2 m
Li Lens ß 1 cm,
Bmax 833 T?

efin 0.33 mm, eeq 0.08mm
19
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Ring cooler with Li lens (contd)
• --------------------------------------------------
  ------------
• Ring cooler simulation
• Transverse emittances
  Longitudinal emittance and transmission

20
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Curved Li lens
• T. Vsevolozhskaja, 1998. Theoretical analysis
  of motion in curved Li lens. Estimation of the
  dispersion.
• Y.Fukui, D.Cline, A.Garren, H.Kirk, 2003.
  Simulation of toy model
• R 16 cm, rmax1 cm, ß 1 cm
• Equilibrium vertical emittance 80 mm-mrad almost
  coincides with theoretical value
• 85 mm-mrad. Probably, horizontal emittance is
  more (100 mm-mrad) because of
• dispersion.

21
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Summary
• At an ideal matching, Li lens cooling channel
  can provide transverse rms emittance of muon beam
  less of 100 mm-mrad at a reasonable field
  strength.
• For this, field of the lenses should increase,
  and their radius decrease step by step
  complying with the condition Bmaxrmax const
  7-8 T-cm.
• Then, an achievable beta-function is ß
  (2.5p)1/2 /Bmax (cm, MeV/c, T), which is
  about 1 cm at p 200 MeV/c and reasonable
  Bmax 20 T, rmax 0.375 cm.
• Therefore, achievable transverse rms emittance
  is e 0.0085 ß 0.013p1/2 /Bmax that is
  0.009 cm 90 mm-mrad.
• A solenoid based cooling channel has
  considerably worse parameters at the same field
  and H2 absorber ß 2p/3B 7 cm, e
  0.004 ß 0.03 cm 300 mm-mrad. 67 T field is
  required to reach 90 mm-mrad.
• --------------------------------------------------
  --------------------------------------------------
  ----------
• However, a satisfactory design of the matching
  sections is not reached yet. The main problem is
  suppression of chromatic effects in a cell with
  strongly modulated beta-function.

22
V.Balbekov, August 2006
Only the lenses are simulated. Ideal matrix is
used instead of matching sections.

• Summary (contd)
• Short-period cooling channel with betatron
  phase advance about 3p /cell provides rather good
  matching and transmission. Achievable emittance
  is close to the analytically predicted
  equilibrium limit. However, very short Li lenses
  would be required to reach extremely small
  beta-function because of the relation L 3ß.
  Is it possible to break up this coupling?
• Long-period cooling channel with synchrotron
  phase advance 2p /cell is free from this
  drawback. However, it can be made approximately
  achromatic only at higher energy. Even so, the
  matching is imperfect, resulting in large
  particles loss and twice more achievable
  emittance in comparison with theoretical limit.
• A satisfactory method for emittance exchange is
  not proposed yet for the Li lens cooler. Attempts
  to insert Li lenses into a ring cooler are
  unsuccessful. In any case, a possibility to
  increase the lens field as the beam emittance
  decreases cannot be applied to a ring cooler. A
  curved Li lens looks as a very complicated method
  to create a dispersion. No ideas to use curved Li
  lens without additional wedge absorber.
  
• --------------------------------------------------
  --------------------------------------------------
  ------------
• In conclusion Similar problems are unavoidable
  in any cooler with high modulated beta-function
  (Li lens, supersolenoid, parametric resonance )