Exploring and optimizing Adiabatic Buncher and Phase Rotator for Neutrino Factory in COSY Infinity

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Exploring and optimizing Adiabatic Buncher and
Phase Rotator for Neutrino Factory in COSY
Infinity

• A.Poklonskiy (SPbSU, MSU), D.Neuffer (FNAL)
• C.Johnstone (FNAL), M.Berz (MSU), K.Makino (MSU)

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Goal Problems

• Goal
• Build neutrino factory to study different
  neutrino-related things and/or muon collider to
  collide
• Problems
• Muons are short-living particles ?? compact
  lattice, fast beam gymnastics
• Muons are produced with large initial momentum
  spread ? cooling
• Energy spread is large ? energy spread reduction
  before cooling
• Some desired beam manipulations requires new
  types of field configuration ? development of
  such new elements
• All these ? small muons production rate (lt0.2)
  andPRICE!

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RD goal affordable ?e, ?? -Factory

• Improve from baseline
• Collection
• Induction Linac ?high-frequency buncher
• Cooling
• Linear Cooling ? Ring Coolers
• Acceleration
• RLA ? non-scaling FFAG

? ? e ne ??
? ? e n? ?e
and/or
The Neutrino Factory and Muon Collider
Collaboration http//www.cap.bnl.gov/mumu/mu_home_
page.html
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RF Cavity and Solenoid in Pictures
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Adiabatic buncher (????) Rotator (David
Neuffer)

• Drift (90m)
• ??? decay, beam develops ???? correlation
• Buncher (60m) (333Mhz?200MHz, 0?4.8MV/m)
• Forms beam into string of bunches
• ???? Rotator (12m) (200MHz, 10 MV/m)
• Lines bunches into equal energies
• Cooler (50m long) (200 MHz)
• Fixed frequency transverse cooling system

Replaces Induction Linacs with medium-frequency
RF (200MHz)
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Longitudinal Motion (2D simulations)
Buncher
Drift
?
?

(???E) rotator
Cooler
?
?
System would capture both signs (?, ?-)
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Key Parameters

• Drift
• Length LD
• Buncher
• Length LB
• RF Gradients EB
• Final RF frequency ?RF (LD, LB, ?RF (LD LB)
  ?(1/?) ?RF)
• Phase Rotator
• Length L?R
• Vernier offset, spacing N?R, ?V
• RF gradients E?R

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Lattice Variations

• Shorter bunch trains (for ring cooler, more ?s
  lost)?
• Longer bunch trains (more ?s survived)?
• Different final frequencies? (200,88,44Mhz)
• Number of different RF frequencies and gradients
  in buncher and rotator (6010)?
• Different central energies (200MeV, 280MeV,
  optimal)?
• Matching into cooling channel, accelerator
• Transverse focusing (150m B1.25T solenoidal
  field or)?
• Mixed buncher-rotator?
• Cost/perfomance optimum?

OPTIMIZATION IS NEEDED
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COSY Infinity Simulations

• COSY Infinity code (M. Berz, K. Makino, et al.)
• Where M map of the equations of motion (flow),
  obtained as a set of DA vectors (Taylor
  expansions of final coordinates in terms of
  initial coordinates)
• uses DA methods to compute maps to arbitrary
  order
• own programming language allows complicated
  optimization scenarios writing
• internal optimization routines and interface to
  add more
• provides DA framework which could significantly
  simplify use of gradient optimization methods
• model is simple now, but much more complicated
  in future and COSY has large library of standard
  lattice elements

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Big Problem

• Use of Taylor series leads to tricky way of
  handling beams with large coordinate spread (and
  that is exactly the case) Relative coordinates
  should be lt 0.5 (empirical fact)

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Straightforward division
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Equations of Longitudinal Motion
synchronous particle
equations in deviations from synchronous particle
nonlinear oscillator
COSY Infinity uses similar coordinates
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Consequences of Equations of Motion

• Existence of stable regions, where we have
  oscillatory motion and unstable regions, and,
  therefore existence of separatrix (depends on
  frequency, RF gradient, synch. phase, etc ).
  Stable area is called the bucket.

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Beam Evolution
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Clever Division

• Use central energies as centres of boxes, use RF
  period as ranges for the box
• Add jumping between intervals after each step.
  We change buckets and particles could be lost in
  one bucket and re-captured in another

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Still a Problem

• 50 central energies x 60 RF cavities in Buncher
  300 maps
• We are using DA arithmetic, everything is a
  DA-vector, including elementary functions (sin),
  so we need relatively high expansion order. Use 2
  times more intervals 5th order leads to natural
  advantage in buncher maybe.
• Use COSYs ability to generate parameter-dependent
  maps with ease and special law of bunches
  central energies distribution (smaller energies
  tends to be closer to each other)
• 40-50 maps ? 12-15 maps
• Potential calculation time reduction.
  Implementing.
• 6000 particles, 50 central energies, 70 RFs
• 1st order 0 hrs 10 min
• 7th order 8 hrs some mins

OPTIMIZATION?
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Sin Taylor expansion
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Different Order Simualtions
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Different Order Simualtions
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Conclusions

• Model is implemented in COSY Infinity and checked
  for consistency with other codes
• Some removal of the obstacles is done
• Brute-force optimization still seems to be
  infeasible. Looking for some more sophisticated
  method.

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Yet Unanswered Questions

• Should longitudinal motion be studied separately,
  or should it be included on the very early
  stages?
• Are there any map-dependent criterias which could
  be used for map-based optimization?

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Second Optimization Approach

• From synchronism condition one could derive
  following relation for kinetic energies of synch
  particles

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Final Kinetic Energy Relation

• From the rotator concept one could derive amount
  of energy gained by n-th synchronous particle in
  RF
• So for final energy n-th particle has after the
  rotator consists of m RFs we have

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Evolution of central energies shape T(n,m)
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Energies shape in buncher and amount of kick they
get in rotator
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Energy Shape Evolution in Rotator
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Objective Functions

• The idea of the whole structure is to reduce
  overall beam energy spread and to put particles
  energies around some central energy. So we have
  general objective function
• First, we can set and get

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Different optimized paremters (n vs T_fin)
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Different optimized paremters (T_0 vs T_fin)
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Evolution of central energies shape (unoptimized)
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Evolution of central energies shape (optimized)
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Evolution of central energies shape (optimized)
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Different optimized paremters (T_0 vs T_fin)
energies distribution
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Objective Functions

• For calculating we can use particles
  energies distribution

n energy particles -----------------
----------------------------- -12 963.96
1023 17.050000 -11 510.85 692
11.533333 -10 374.64 537
8.950000 -9 302.98 412
6.866667
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Different optimized paremters (n vs T_fin)
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Summary

• Model of buncher and phase rotator was written in
  COSY Infinity
• Simulations of particle dynamics in lattice with
  different orders and different initial
  distributions were performed
• Comparisons with previous simulations (David
  Neuffers code, ICOOL, others) shows good
  agreement
• Several variations of lattice parameters were
  studied
• Model of lattice optimization using control
  theory is proposed
• Model of central energies distribution is
  developed. Some results for parameters have been
  obtained

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Future Plans

• Finish central energies optimizations, try
  changing more parameters, check optimized
  parameters for whole distribution (COSY, ICOOL?)
• Develop some criteria for simultaneous
  optimization of central energies and energies of
  all paritcles in a beam or use control theory
  approach for the whole longitudinal motion
  optimization
• Study transverse motion and particles loss
  because of decay and aperture, final emittance
  cut
• Different lattices for different cooling
  sections/targets/whatever proposed (project is on
  RD status)