Kinetics Effects in Multiple Intrabeam Scattering IBS'

1
Kinetics Effects in Multiple Intra-beam
Scattering (IBS).

• P.R.Zenkevich, A.E.Bolshakov, O.
  Boine-Frankenheim
• ITEP, Moscow, Russia
• GSI, Darmstadt, Germany
• The work is performed within framework of
  INTAS/GSI grant Advanced Beam Dynamics

2
Contents.

• Introduction.
• One event kinematics.
• IBS in infinite medium
• - Focker- Planck (FP) equation in
  momentum space.
• - Langevin map.
• -Binary collision map.
• IBS in circular accelerators
• - FP equation in momentum-coordinate
  space
• - Invariants evolution due to IBS.
• - FP equation in invariant space.
• - Longitudinal FP equation
  (semi-Gaussian model). .
• - MOCAC code and its applications.
• Summary.

3
Introduction.

• IBS includes
• 1) Multiple IBS.
• 2) Single-event IBS (Touschek effect). The
  effect can be included in multi-particle codes by
  straight-forward way. It is out of frame of this
  review.
• Fundamental accelerator papers
• Bjorken, Mtingwa (BM)Piwinski, Martini
  (PM).
• Results analysis of one-event
  kinematics and r. m. s. invariants evolution.
  Main difference in BM theory Coulomb logarithm
  is constant, in PM Coulomb logarithm is dependent
  on momentum.
• Gaussian model
• theory of rms invariants evolution in
  PM and BM theories is based on assumption that
  the beam has Gaussian distribution on all degrees
  of freedom.
• The reason IBS results in beam
  maxwellization!
• Creation of numerical codes for rms invariants
  evolution
• - Mohl and Giannini, Katayama and Rao
  and so on.
• - BETACOOL (Meshkovs
  groupZenkevich). The code includes additional
  effects electron cooling, Beam-Target
  Interaction (BTI) and so on.

4
Kinetic approach.

• Why we need kinetic description?
• Solution of kinetic equation is not Gaussian
  with account of boundary conditions (particle
  losses).
• Other effects (for example, e-cooling) produce
  Non-Gaussian tails.
• How to investigate these effects?
• The simplest way to solve Focker-Planck
  (FP) equation.
• One-dimensional Focker-Planck equations.
• - Spherical symmetry (Globenko, ITEP,
  1970).
• - One-dimensional longitudinal equation
  (Lebedev, Burov, Boine-Frenkenheim).
• Three dimensional codes.
• - Monte Carlo Code (MOCAC) code
  (Zenkevich, Bolshakov) (IBSE-coolBTI).
• Six-dmensional codes.
• PTARGET code (Dolinsky) (IBSE-coolBTI).

5
One-event kinematics.

• Let us introduce vector of the particle
  dimensionless momentum
• Let for test particle
• for field particle
• Here ,
• Then

6
Moments evolution 1.

• Averaging on azimuthal angle we find that for one
  collision event
• Rutheford cross-section
• Scattering probability

7
Moments evolution 2.

• Average friction force due to multiple IBS
• Average diffusion coefficient
• Here Coulomb logarithm

8
Fokker-Planck equation in momentum space
(infinite medium).

• Evolution of the distribution function in
  infinite medium is described by following FP
  equation (for constant Coulomb logarithm)
• If we neglect weak dependence of logarithm on
  momentum
• Here
• This equation should be solved with initial
  condition (initial distribution function and its
  derivative should be smooth)

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Langevin map 1.

• Let at time t distribution in phase space is
  defined by
• Then the friction force and diffusion
  coefficients are
• To simulate the particle evolution let us change
  of the test particle momentum is
• Here the random diffusion kick should satisfy to
  conditions

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Langevin map 2

• We see that Langevin map includes the following
  steps
• 1. Calculation the friction force and
  diffusion coefficients for the test particle.
• 2. Calculation of the momentum change due to
  friction.
• 3. Random choice of the the momentum change due
  to diffusion.
• 4. Repetition of the process for all
  particles.
• Energy conservation is absent at the second
  order on time step. It can result in unphysical
  growth of six-dimensional emittance!
• However, we have algorithm which provides energy
  conservation
• BINARY
  COLLISION MAP!

11
Collision Map 1

• Let us choose t he scattering angle according to
  expression
• Then work of the friction force and increase of
  moments because of the diffusion terms coincide
  with the corresponding exact values
• Azimuthal angle is defined by random choice on
  interval .
• We have invented this map and included it in
  code named MOCAC (MOnte-Carlo code). However,
  this idea was suggested earlier by T. Takizuka
  and Hirotada Abe (1977).

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Collision Map 2

• From electrostatic analogy we know, that for code
  smoothening the minimal distance between
  particles is limited.
• Let , then
• Computational parameters of the code are
  .
• Number of collisions for each step is equal to
  where N is number of
  macro-particles.
• The different algorithm (which is really used for
  calculations in MOCAC code and Takizuka-Abe
  paper) assumes random choice of one partner in
  each step. For same error we should choose time
  step in order to
  .

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FP equation in momentum-coordinate space

• FP equation in indefinite medium can be
  generalized for the beam by straight-forward way.
  Then
• Initial condition
• Boundary condition
• We can use same methods for numerical solution as
  for infinite medium (for example, collision map).
• However, particles oscillate in transverse
  direction therefore time step should satisfy to
  the condition
• To diminish computer time up to reasonable limit
  we should use simplified models or different
  approaches.

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Longitudinal FP equation 1

• Let us introduce following assumptions
• 1)coasting beam
• 2) the distributions on transverse degrees
  of freedom are Gaussian ones with equal r.m.s
  values of transverse momentum
• 3) dispersion function is equal to zero
  such assumption is acceptable if we are working
  far below critical energy.
• Then
• Averaging on ,
  we can derive one-dimensional (longitudinal) FP
  equation

,
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Longitudinal FP equation 2

• Here the friction force and diffusion
  coefficients are
• The kernels are

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Dependence of kernels for the friction force and
diffusion coefficients on parameter blue
curve 1, red curve 2 and green curve
3.
17
Invariant space 1

• Linear transverse particle motion is described by
  conservation of Courant-Snyder invariant
• Here are Twiss
  functions depending on longitudinal variable s
  for horizontal motion
• here D and are dispersion function
  and its derivative
• for vertical motion
• For coasting beams (CB) we can use as
  longitudinal invariant or momentum
• deviation , or its squared value

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Invariant space 2.

• The last form is more symmetrical In linear
  approximation for bunched beam (BB) the invariant
  can be written as follows
• Here
• -distance from bunch
  center,
• Let us introduce invariant vector with
  components
• and phase vector

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Evolution of Invariants and their Moments 1

• In scattering event the coordinates does not
  change we have
• Here and are
  defined by
• Let us define operator
• For m1,3
• For m2

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Evolution of Invariants and their Moments 2

• For the second order moments of the invariants
  with m1,3 we find
• For the second order of invariant with m2
• Here the coefficients
• For the invariant with m1,2

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Evolution of Invariants and their Moments 3

• Calculation of evolution of invariants moments
• -Expression of momentae with account
  of (locality condition) for field particle
  through its invariant and coordinates
• -Change of variables in integrals over
  local distribution function
• - Transfer for test particle from
  variables to variables
• using the expressions
• - Averaging on invariants and phases
  of test particle and on variable s.

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Evolution of Invariants and their Moments 4

• Then we obtain
• Here kernels have form of four-dimensional
  integrals on phases and longitudinal variable s.
  Example (for m1,3)
• Here
• Function
• For m1,3

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Focker- Planck equation in invariant space 1.

• FP equation is
• The distribution on phases is uniform on
  interval
• This equation should be solved with following
  initial and boundary conditions
• Initial distribution function should be smooth
  with its first derivative.
• Absorbing wall boundary condition
• Zero flux boundary condition
• Here flux

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Application of Langevin method for solution of FP
equation in invariant space 1.

• If we know the kernels we can use for numerical
  solution Langevin method. Let
• Then
• Change of invariant
• Here change of the invariant due to friction is
  defined by

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Application of Langevin method for solution of FP
equation in invariant space 2.

• Statistical correlation there is only for
  longitudinal and horizontal motion therefore all
  diagonal elements are equal to zero, besides
  m1,2. Then diffusion vector can be found by
  random choice from the distribution
• here
• The last step - comparison of new values of the
  invariants with boundary conditions.
• If the particle is outside the
  absorbing wall it is considered as lost
• If one of the positive components
  becomes negative (transfer through reflecting
  wall) its value is changed on opposite one.

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MOCAC code and its applications1.

• Idea of code is change of kernel calculation by
  successive application of the binary collision
  map in momentum space.
• Algorithm steps
• 1. Random choice of macro-particle phases
  for given set of macro-particles invariants.
• 2. Calculation of momentae and coordinates
  of macro-particles.
• 3. Computation of macro-particles
  distribution on the space cells.
• 4. Application of collision map and each
  macro-particle using local ensemble for each
  cell.
• 5. Calculation of new set of macro-particles
  invariants.

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MOCAC code and its applications2.

• This operation can be considered as collective
  map in invariant space each particle is test
  and field particle simultaneously.
• The map is repeated through the time interval
• If the magnetic lattice is non-uniform, it is
  presented as set of discrete points corresponding
  different longitudinal coordinates. Each point is
  characterized by its set of Twiss parameters and
  dispersion function. The points are distributed
  uniformly on the lattice period.
• The map is made through each time interval in new
  point of period (averaging on the lattice).

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The list of code parameters
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Code validation.Dependence of r. m. s. momentum
spread on time

• Smooth model of TWAC ring with zero dispersion.
• Beam and ring parameters kind of ions
• T620 MeV/u, ?1.66,
• ?ring 251.0 m , Q9.3.
• Code computational parameters Npart 100000, ,
  Ngrid 3030, ?t 0.01 sec, , ?max1.0.

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Code validation.Dependence of beam invariant
oscillations on time.

• Smooth model of TWAC ring with zero dispersion.
• Beam invariant

32
Code validation.Dependence of r. m. s. momentum
spread on time

• Smooth model of TWAC ring with non-zero
  dispersion (D0.461)
• Beam and ring parameters
• kind of ions
• T620 MeV/u, ?1.66, ?0.0116203, ?ring
  251.0 m, Q9.3.
• code computational parameters
• Ngrid 3030 (blue curve)
• and 55(red curve), Npart 100000, ?t
  0.01 sec, , ?max1.0

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Code validation.Dependence of beam invariant on
time

• Smooth model of TWAC ring with non-zero
  dispersion (D0.461)
• code computational parameters
• Ngrid 3030 (blue curve) and 55 (red
  curve)
• We see regular growth of invariant deviation for
  small number of grid points!

34
Code validation.Dependence of r. m. s. momentum
spread on time

• Smooth model of TWAC ring with non-zero
  dispersion (D0.461) and high ion energy (not
  very far from critical one).
• Beam and ring parameters
• kind of ions
• T3800 MeV/u, ?5.05
• ?0.0116203, ?ring 251.0m, Q9.3.
• code computational parameters
• Ngrid 3030 (blue curve)
• and 55(red curve),
• Npart 100000, ?t 10 sec, , ?max1.0
• We see regular growth for red curve (small number
  of points in the grid)!

35
Code validation.Dependence of beam invariant on
time

• Smooth model of TWAC ring with non-zero
  dispersion (D0.461) and high ion energy (not
  very far from critical one).
• Beam and ring parameters
• kind of ions
• T3800 MeV/u, ?5.05
• ?0.0116203, ?ring 251.0m, Q9.3.
• code computational parameters
• Ngrid 3030 (blue curve)
• and 55(red curve),
• Npart 100000, ?t 10 sec, , ?max1.0
• We see regular growth for red curve (small number
  of points in the grid)!

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Comparison with ESR experiments ( )
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Numerical modeling of multi-turn injection in
TWAC 1.

• Fig. 7. Dependence of r. m. s. momentum spread
  on time for real option of TWAC lattice and
  multi-turn charge-exchange injection.
• Beam parameters kind of ions,
  T620 MeV/u, booster frequency1Hz, number of
  injected particles
• Npart per booster cycle,
• charge-exchange target Au
• g/cm2.
• We see increase of longitudinal temperature due
  to IBS maxwellization.

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Numerical modeling of multi-turn injection in
TWAC 2.
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Summary

• Using semi-Gaussian model it is derived
  longitudinal integro-differential FP equation,
  which can be solved using grid or macro-particle
  method.
• For matched beam IBS is described by
  three-dimensional integro-differential FP
  equation for invariant-vector.
• It is developed multi-particle code MOCAC, which
  allows us to solve FP equation using binary
  collisions method.
• Validation of the code has shown its long term
  stability for appropriate choice of numerical
  parameters.
• Future plans
• Development of new IBS collective maps and
  creation of map library.
• Investigations of convergence and benchmarking.
• Comparison with the experiment.
• Application to GSI project.