copyright, 1996 © dale carnegie & associates, inc. beyond piwinski & bjorken-mtingwa: ibs...

Copyright, 1996 © Dale Carnegie & Associates, Inc.

Beyond Piwinski & Bjorken-Mtingwa: IBS theories, codes, and

benchmarking

Jie Wei

Brookhaven National Laboratory, USA ([email protected])

Institute of High Energy Physics, China ([email protected])

Mini Workshop IBS07

August 28 - 29, 2007

August 29, 2007 Wei 2

Outline• Introduction

– Answer to Swapan’s questions: IBS mechanism as we understood now

– IBS examples: beyond Piwinski & Bjorken-Mtingwa

• Intra-beam scattering theories (sampled) – Scaling on beam emittance growth– Fokker-Planck approach for arbitrary distributions– Molecular dynamics method for particle-particle interaction

• Benchmarking experiments in RHIC– rms beam emittance growths in three directions– Beam loss at tail & de-bunching– Beam distribution evolution: Gaussian-like vs. hollow beams

• SummaryAcknowledgements: M. Blaskiewicz, A. Fedotov, W. Fischer, R.

Connolly, X.-P. Li, N. Malitsky, H. Okamoto, G. Parzen, T. Satogata, A.M. Sessler, S. Tepikian …


Answer to Swapan’s IBS questions

Assumption: no radiation damping/quantum effects

• Why in a Liouville system there is emittance growth?– In the rest frame of the reference particle, the Hamiltonian

system is explicitly time-dependent» There is no constant of motion in the system» There possibly exist linear/non-linear resonances

between the driving lattice frequency and particle motion frequency (betatron tune modified by Coulomb interaction – phonon bands)

– For an “ideal” uniform lattice (time-independent Hamiltonian), there exist IBS growths if it is above transition energy

» Opposite signs between transverse and longitudinal terms in the Hamiltonian

» There exists a constant of motion for the 6-D phase space, but each 2-D phase space is not conserved

– No growth exists for an “ideal” uniform lattice below transition


A view from the beam rest frame• Particle motion in the beam rest frame

• Intra-beam Coulomb scattering among particles of the same bunch

•J. Wei, X-P. Li, and A. M. Sessler, Formal report BNL-52381 (June 1993)•J. Wei, “General relativity derivation of beam rest-frame Hamiltonian”, PAC’01, 1678-1680 (2001)•“Handbook of accelerator physics and engineering: a compilation of formulae and data”, Section of “Crystalline Beams”, edited by A. Chao and M. Tigner, World Scientific, Singapore, 1998. •J. Wei, X-P. Li, and A. M. Sessler, Phys. Rev. Lett., Vol. 73, pp. 3089-3092 (1994)•J. Wei, H. Okamoto, and A.M. Sessler, Phys. Rev. Lett., Vol. 80, pp. 2606-2609 (1998)


Transformed rest-frame Hamiltonian

• Observe particle motion in the rest frame of the beam

• Transformed Hamiltonian

• Coulomb potential (now non-relativistic)

• Time-dependent Hamiltonian in beam rest frame

),,(2

)(1

2

)(

22

)(

2,,,,,, 2

22222

zyxVPFyKPxKP

PzPyPxH Czzyyxx

zyx

2

1

t

zF

)('''

)('''2

2

straightsDDD

bendsDDDDFz

j

jjj

Czzyyxx

V222

1

Wei, March 16, 2004 6

Below transition: positive-mass regime• In the ideal case of uniform focusing, the Hamiltonian

is positive definite– There exists an equilibrium state– In the equilibrium state, the beam has equal temperature

in all three directions (isotropic in the velocity space)

• In general, the Hamiltonian is time-dependent; system is not conserved (AG focusing)

• Quasi-equilibrium state: approaching equilibrium yet still allows growth in beam size

p

y

y

x

x

Wei, March 16, 2004 7

Above transition: negative-mass regime• The Hamiltonian is NOT positive definite in any case

– There usually exists no equilibrium state– All beam dimension can grow– Asymptotic relation exists between different dimension

• Typically vertical dimension grows only through transverse coupling

222ppx D

Wei, March 16, 2004 8

Phonon spectrum of a super-cold beam

• Ring lattice periodicity: 8

• Horizontal tune 2.07; vertical tune 1.38

• Energy = 1.1

• Max. phonon frequency: 3

• Max. allowed tune <2.83;

• Needed for cooling <2

•X.-P. Li, H. Enokizono, H. Okamoto, Y. Yuri, A.M. Sessler, and J. Wei, “Phonon spectrum and maintenance condition of crystalline beams”, Phys. Rev. ST-Accel. Beams, Vol. 9, 034201 (2006).


IBS theories (samples)• Gaussian beam rms growth rates calculation

– A. Piwinski (1974); J.D. Bjorken/S.K. Mtingwa (1983); M. Martini (1984) – growth rates formulae & integral for general lattices

– G. Parzen (1987); J. Wei (1993) – scaling laws & asymptotic rules

– A. Fedotov, J. Wei (2004) – quantitative comparison between models

• Bi-Gaussian beam: beam spread with dense core under cooling

– G. Parzen (2004) – estimate of IBS growth for e-cooled beam

• Beam profile evolution: beam loss and beam shape study– J. Wei, A.G. Ruggiero (1990) Fokker-Planck approach– Used in RHIC design to predict beam de-bunching loss

• Particle-by-particle molecular-dynamics simulation– J. Wei, X.P. Li, A.M. Sessler (1993) – crystalline beam

formation and heating due to Coulomb interactions


IBS examples: beyond Piwinski & Bj-M

• Limited phase space, significant beam loss– Relativistic Heavy Ion Collider (RHIC), overwhelming IBS

effects due to high charge state of ions: Z4/A2 scaling– 10-hour store of gold beam

» Emittance grows by more than a factor of 4» Beam loss of about 40% escaping RF bucket (de-

bunching) » Luminosity decrease by a factor of 10 from start to end

• Low temperature, high particle density “crystalline” state

– Usually IBS heating rate increases as the 6-D bunch emittance reduces

– What happens when the emittances are so small that the beam starts to “crystallize”?


Design goals: 10 hour store, heavy ion species from p to Au


Au-Au luminosity limit: intra-beam scattering

Intensity loss (~40%)

Luminosity loss

• Luminosity loss – frequent refill– Transverse

emittance growth

– Longitudinal growth & beam loss due to RF voltage limitation

• De-bunching & physics background – beam gap cleaning

Time (~5 hour per fill)


IBS beam experiment diagnostics• Transverse

– Ionization profile monitor

– Simultaneous measurement of emittance on different bunches

– Constant improvements over electron-cloud interference

• Longitudinal– Wall current

monitor– Measurement of

intensity & profile

Vertical emittance growth (~30%) [norm. 95% 10-6 m rad]

DC beam intensity (aperture)

Bunched beam intensity (IBS; 20%)

Time (~ 70 minutes)

Time (~ 60 minutes)


Multi-layer beam simulated in actual ring

• Characteristic distance:

– (1 -- 100 m)

• Typical (lab frame) inter-particle distance:

• Highest density:

31

22

20

r

3/216.1 eff 2222 ,min xyeff

20

232

2

rave

ave


Closed orbit + phonon modes


Molecular dynamics calculation of growth• Finest level, particle-on-particle interaction

• Predicts a growth-rate turn-over when the beam is cooled towards the crystalline state


rms beam size growth• Assuming an unbounded Gaussian beam

• Proportional to Proportional to 6-D phase-space density

• Analytic expression for FODO lattice; integral formula for actual lattice

• Inadequate when beam loss occurs / for non-Gaussian beams

2

4

AZ

•G. Parzen, Nucl. Instru. Methods, A 256 231 (1987); EPAC’88 821 (1988)•J. Wei, “Evolution of hadron beams under intra-beam scattering”, PAC’03, 3653-3655 (1993)•J. Wei and G. Parzen, “Intra-beam scattering scaling for very large hadron colliders”, PAC’01, 42-44 (2001)•J. Wei, “Synchrotrons and accumulators for high-intensity proton beams”, Reviews of Modern Physics, Vol. 75, No. 4, 1383 - 1432 (2003)


Fokker-Planck approach• Start from general 6D F-P

eq.

• Starting from Rutherford scattering cross section between any two phase space location in the beam rest frame

• In terms of lab frame quantities

• Action angle variables•R. Cohen, L. Spitzer, P. McRoutly, Phys. Rev. 80, 230 (1950); … •J. Wei and A.R. Ruggiero, BNL report 45269, Note AD/RHIC-81, (1990)•J. Wei and A.G. Ruggiero, PAC’91 1869-1871 (1991)•J. Wei, “Stochastic Cooling and Intra-Beam Scattering in RHIC”, Proc. Workshop on Beam Cooling and Related Topics, Montreux, Switzerland, 132-136 (1993, CERN 94-03)


F-P for a bunch in a single-harmonic bucket (non-linear RF force)• Evolution of particle distribution in phase space

– IBS growth typically much slower than synchrotron/betatron oscillation period -- averaging over phase angles

– For RHIC, averaging over transverse directions: time dependent transverse Gaussian – arbitrary longitudinal distribution

• Kinematical drift (heat transfer) and diffusion


Fokker-Planck approach on density evolution


Counter-measure example: stochastic cooling

Key for bunched-beam stochastic cooling in a collider:

eliminate coherent spikes at GHz range that may saturate the cooling system

IBS among gold ions in RHIC may diffuse possible soliton mechanism(M. Brennan, M. Blaskiewicz, et al …)


Molecular dynamics approaches• Use beam rest frame:

– Non-relativistic motion of particles

– Easy to adopt the molecular dynamics methods

– Crystallization: zero temperature

• Derivation of equations of motion:

– Use general relativity formalism -- EOM in tensor forms

– Find the coordinate system transformation

– Transform the EOM from lab frame to the beam rest frame

– Use Molecular Dynamics methods

•J. Wei, “General relativity derivation of beam rest-frame Hamiltonian”, Proc. Particle Accelerator Conference, Chicago, 1678-1680 (2001)•J. Wei, X.-P. Li, A.M. Sessler, BNL Report 52381 (1993); PAC’93, 3527 (1993)


Beam rest-frame equations of motion• Bending section:

• Straight section:

– Quads, skew quads, sextupoles, RF cavity, …– Coulomb force:

iCzyxi VxzxPPPH ,2222

2

1

2

1

siC

szyxi

UV

xyxn

xynyxn

PPPH

,

2321

221222 3622

1

21

,

222,

ijjjjjiC zzyyxxV


Molecular dynamics methods• MD cells with longitudinal periodic condition

• Long-range coulomb force -- Ewald-type summation to enhance computational efficiency, considering beam image charge

• Integrate EOM with 4th order Runge-Kutta algorithm, potential by 15th order Gauss-Laguerre (later improved to be symplectic)

• Start with a random distribution, and simulate actual cooling process and heating process (or perform artificial ``periodic cooling’’ by imposing periodic condition & drift correction for the ground state)

CLb

Ldk

k

LkJLkz

Lrxx ijij

ijji

/log2

12exp

1/2/2cosh41,

0

0


What are included in study? What are not?• Included:

– Charged particles in a storage ring

– Relativistic effects– Intrabeam scattering– Space charge– Actual magnets and

cavities (all order magnets)– Beam cooling methods– Image charge– Magnetic errors,

imperfections, nonlinearities

– Neutralization (gas, collision events)

– Beam-beam model

• Not included:– Beam radiation– Quantum effects


Beam cooling methods

• Stochastic cooling– 3D but slow; optical range?

• Electron cooling– 3D, relatively fast

• Laser cooling– Fast, but mostly

longitudinal (1D only); specific atoms

– Possible for couple 3D cooling

• Radiation cooling– electron; easily

stimulated?– Tapered cooling: cooling for

the same angular velocity


Benchmarking experiments in RHIC• Verify the growth of rms beam sizes under IBS

– Early theories by Piwinski, Bjorken/Mtingwa– Detailed lattice implementation by Martini– Asymptotic behavior analysis/approximation by Parzen– Approximation model by Wei– Recent compilation by the Russian collaborators /

Fedotov

• Verify the beam de-bunching behavior under IBS– Predictions by Wei using the Fokker-Planck approach

• Verify the longitudinal bunch profile evolution under IBS

– Predictions by Wei using the Fokker-Planck approach– Similar approach used in stochastic cooling

analysis/predictions

•J. Wei, A. Fedotov, W. Fischer, N. Malitsky, G. Parzen, and J. Qiang, “Intra-beam scattering theory and RHIC experiments”, ICFA Advanced Beam Dynamics Workshop on High Intensity Particle Accelerators, Bensheim, Germany, AIP Conference Proceedings (2004)


Beam experiment observables• rms beam sizes (bunch length, transverse emittances)

– Transverse emittance from ionization profile monitor; longitudinal bunch length from wall current monitor

– Need to single out IBS from other processes -- “turn-off” beam-beam, tune kicker, Landau cavity, dual RF, RF noise …

– Need to calibrate Ionization Profile Monitor readings & transverse coupling conditions

• Beam loss– Wall current monitor and DCCT readings– Need to turn off the secondary harmonic RF system (200

MHz), using 28 MHz RF system alone

• Beam profile (longitudinal)– Wall current monitor readings – Hollow bunch created by RF phase jump, versus

Gaussian-like bunch -- profile evolution comparison– Asymptotic beam shape observation


Dedicated IBS studies during year 2004• Several studies done in previous runs; latest beam

experiments: January - March, 2004

• Simultaneous IBS measurement under different intensities

– Each of the two rings contain 6 bunches of 3 intensities– Gaussian-like beam in one ring, longitudinal hollow beam

in the other


Comparison of Gaussian & hollow beams• Gold beam, store at 100 GeV/u with h=360 RF system;

no beam collision

• No Landau cavity, no dampers, no kickers

• Hollow beam in blue (RF phase jump), normal beam in yellow


Observation of emittance growths• Initial transverse emittance depends on intensity --

space-charge effects at Booster/AGS

• Emittance growth to be bench-marked with the theories


Observation of longitudinal beam loss• Distinctively different beam loss (de-bunching)

behavior

Hollow beam AC intensity

Gaussian beam intensities

Hollow beam DC intensity


Observation of beam profile evolution• Normal beam: Gaussian-like shape

• Hollow beam: reducing depth of the hole -> approaching Gaussian

normalhollow


Transverse emittance bench-marking

0 514.29 1028.57 1542.86 2057.14 2571.43 3085.71 36008

10

12

14

1616

8.0

EYexp kq 5

EYexp kq 11

EYsim121mtks 1

EYsim301mtks 1

36000 EYexp kq 0 1095 EYexp kq 0 1095 EYsim121mks 0 EYsim301mks 0

N=0.6*109

model (FODO cells)experiment

N=0.3*109

model (FODO cells)experiment

time [sec]

n95%

[mm mrad]

Vertical emittance

• Agreement satisfactory (dispersion uncertainty within 40%); uncertainty is in the coupling condition and actual machine dispersion


Longitudinal bench length bench-marking

0 600 1200 18006

7

8

9

10

1111.0

6

BLYexp121 kl 1

BLYsim121mks 1 12.7

18000 BLYexp121 kl 0 BLYsim121mks 0

• Agreement within 20%

bunch length (FWHM [ns])

time [s]


Beam de-bunching loss benchmarking• Run #4790, 30 minute WCM measurement vs. BBFP

code simulation


BBFP simulation of beam profiles• Good agreement obtained with codes BBFP (Bunched-

Beam Fokker-Planck solver)

• Details to be refined

normalhollow


BBFP calculation in the action space• Density projection in longitudinal action

• Convertible to the phase / momentum planes

normalhollow


• Strong dependence on average dispersion– Need an accurate estimate of dispersion and dispersion

wave– Asymptotically (online model & measurement comparison underway)

• Dependence on transverse coupling condition– Amount of coupling changes the relative transverse

growth– Actual dispersion wave, both in horizontal and vertical,

can enhance growth

• Calibration of IPM at store– Calibration was done only for lattice at injection– Possible IPM electronics degradation/peak suppression

may result in falsely large measured emittance value

Transverse emittance comparison issues

222ppx D


Transverse coupling dependence


Dispersion wave measurements

“ real” (MAD) lattice used in simulations

Online model & measured value(dispersion max location only)

1000 2000 3000 40001

1.25

1.5

1.75

22.0

1.0

Yhds1kd 1

Yhds2kd 1

400040 Yhds1kd 0 Yhds2kd 0


Complications from dual RF system

• Large rms bunch length (2ns) due to satellite beams

• Primary bucket bunch length (1ns) satisfies: 5 < bucket width


Example of RHIC ramp (year 2002)

Acceleration start

BLUE Fill 56 bunches

YELLOW Fill 56 bunches

Transition energy

Storage energy

Correction points (stepstones)Orbit – Tunes - Chromaticity

Bunched Yellow current

Total Blue currentBunched blue current

Total Yellow current


Discussions on RHIC measurements• Latest development demands improved IBS theories

– Conventional Gaussian-beam model predicts rms growths– Fokker-Planck approach predicts de-bunching loss– Molecular-dynamics approach predicts ultra-cold beam

behavior

• RHIC benchmarking experiments are promising– Agreement on rms beam size growth:

longitudinal within 20%; transverse within 40%– Agreement on de-bunching beam loss for both Gaussian &

hollow beams– Agreement on longitudinal profile for both Gaussian &

hollow beams

• Further RHIC studies are planned– More accurate dispersion model & measurement– IBS under different transverse coupling conditions– IPM device calibration & lattice (beta-function)

measurement– IBS study at injection


Summary • The mechanism of intra-beam scattering is well

understood.

• The theory of Piwinski & Bjorken-Mtingwa is usually good within a factor of 2 in growth rates under proper conditions (Gaussian distribution, coupling …)

• Several efforts were made as an extension or beyond these theories

– Approximate/analytical formulae and scaling laws– Fokker-Planck solver for the longitudinal phase space

(tail, loss, hollow bunch …)– Molecular dynamics method for ultra-low emittance

beams

• Benchmarking is satisfactory given measurement and machine uncertainties

copyright, 1996 © dale carnegie & associates, inc. beyond piwinski & bjorken-mtingwa: ibs...

Documents

beam rest frame slide

beam dimension

beam size slide

shape hollow beam

transition slide

transverse coupling

tepikian slide

hamiltonian system