PIC Simulations Epicyclic Helical Solenoid Demo

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PIC SimulationsEpicyclic Helical Solenoid Demo

• Andrei Afanasev
• Hampton University
• Low Emittance Muon Collider Workshop
• Fermilab, April 23, 2008

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Disclosure

• This is a simple demo of a beamline geometry that
  satisfies Derbenevs conditions for oscillating
  dispersion function
• Analytical and numerical calculations for the
  proposed implementation are in progress

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Helical Solenoid

• Proposed to use for 6D muon cooling with
  homogeneous absorber, see for details
• Derbenev, Johnson, Phys.Rev.ST Accel.Beams 8,
  041002 (2005)

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Straight Solenoid

• BzB0, BxBy0
• Dispersion const

XY-plane
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Helical Solenoid

• Derbenev, Johnson, Phys.Rev.ST Accel.Beams 8,
  041002 (2005)
• Dispersion const transverse-plane beam orbit is
  a circle

XY-plane
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Epicyclic Helical Solenoid

• Superimposed transverse magnetic fields with two
  spatial periods
• Oscillating dispersion function

XY-plane
k1-2k2 B12B2 EXAMPLE
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Epicyclic Helical Solenoid (cont.)

• Wave numbers k1,k2 may be opposite-sign or
  same-sign
• Commensurate to provide a closed orbit
• Transverse-plane beam orbits
• Same-sign-gt epitrochoid (planets trajectory in
  Ptolemys system)
• Opposite-sign -gt hypotrochoid (special case is an
  ellipse)

k1-k2 B12B2
k12k2 B12B2
k1-2k2 B12B2
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Oscillating Dispersion

• Derbenevs condition for nodes of dispersion
  function
• Assume k1-k2, then transverse-plane orbit is an
  ellipse k1kc/3, B24B1or k1kc/2, B29B1

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Twisted Undulator

• Another possibility to produce double-periodic
  transverse magnetic fields Smirnov,
  Rev.Sci.Inst. 72, 1649 (2001)

• We add logitudinal (solenoid) field
• Derive conditions for zeros of dispersion
  function

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Implementing in PIC

• Study different possibilities to generate
  double-periodic transverse magnetic field
  (epicyclic helical solenoid, twisted undulator,
  etc)
• Plan to develop an epicyclic helical solenoid as
  part of PIC cooling scheme
• (see also Derbenevs talk at this workshop)
• Elliptic option looks the simplest
• Numerical calculations/simulations underway

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Epicycles due to PtolemyAlmagest, Alexandria,
Egypt (A.D.150)