1. Copyright, 1996 © Dale Carnegie & Associates, Inc. Beyond Piwinski & Bjorken-Mtingwa: IBS theories, codes, and benchmarking Jie Wei Brookhaven National Laboratory, USA (jwei@bnl.gov) Institute of High Energy Physics, China (weijie@ihep.ac.cn) Mini Workshop IBS07 August 28 - 29, 2007

2. August 29, 2007 Wei 2 Outline Introduction –Answer to Swapans 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 Summary Acknowledgements: M. Blaskiewicz, A. Fedotov, W. Fischer, R. Connolly, X.-P. Li, N. Malitsky, H. Okamoto, G. Parzen, T. Satogata, A.M. Sessler, S. Tepikian …

3. August 29, 2007 Wei 3 Answer to Swapans 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

4. August 29, 2007 Wei 4 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, PAC01, 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)

5. August 29, 2007 Wei 5 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

6. 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

7. 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

8. 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).

9. August 29, 2007 Wei 9 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

10. August 29, 2007 Wei 10 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: Z 4 /A 2 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?

11. August 29, 2007 Wei 11 Design goals: 10 hour store, heavy ion species from p to Au

12. August 29, 2007 Wei 12 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)

13. August 29, 2007 Wei 13 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)

14. August 29, 2007 Wei 14 Multi-layer beam simulated in actual ring Characteristic distance: –(1 -- 100 m) Typical (lab frame) inter- particle distance: Highest density:

15. August 29, 2007 Wei 15 Closed orbit + phonon modes

16. August 29, 2007 Wei 16 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

17. August 29, 2007 Wei 17 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 G. Parzen, Nucl. Instru. Methods, A 256 231 (1987); EPAC88 821 (1988) J. Wei, Evolution of hadron beams under intra-beam scattering, PAC03, 3653-3655 (1993) J. Wei and G. Parzen, Intra-beam scattering scaling for very large hadron colliders, PAC01, 42-44 (2001) J. Wei, Synchrotrons and accumulators for high-intensity proton beams, Reviews of Modern Physics, Vol. 75, No. 4, 1383 - 1432 (2003)

18. August 29, 2007 Wei 18 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, PAC91 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)

19. August 29, 2007 Wei 19 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

20. August 29, 2007 Wei 20 Fokker-Planck approach on density evolution

21. August 29, 2007 Wei 21 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 …)

22. August 29, 2007 Wei 22 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); PAC93, 3527 (1993)

23. August 29, 2007 Wei 23 Beam rest-frame equations of motion Bending section: Straight section: –Quads, skew quads, sextupoles, RF cavity, … –Coulomb force:

24. August 29, 2007 Wei 24 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)

25. August 29, 2007 Wei 25 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

26. August 29, 2007 Wei 26 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

27. August 29, 2007 Wei 27 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)

28. August 29, 2007 Wei 28 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

29. August 29, 2007 Wei 29 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

30. August 29, 2007 Wei 30 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

31. August 29, 2007 Wei 31 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

32. August 29, 2007 Wei 32 Observation of longitudinal beam loss Distinctively different beam loss (de-bunching) behavior Hollow beam AC intensity Gaussian beam intensities Hollow beam DC intensity

33. August 29, 2007 Wei 33 Observation of beam profile evolution Normal beam: Gaussian-like shape Hollow beam: reducing depth of the hole -> approaching Gaussian normal hollow

34. August 29, 2007 Wei 34 Transverse emittance bench-marking N=0.6*10 9 model (FODO cells) experiment N=0.3*10 9 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

35. August 29, 2007 Wei 35 Longitudinal bench length bench-marking Agreement within 20% bunch length (FWHM [ns]) time [s]

36. August 29, 2007 Wei 36 Beam de-bunching loss benchmarking Run #4790, 30 minute WCM measurement vs. BBFP code simulation

37. August 29, 2007 Wei 37 BBFP simulation of beam profiles Good agreement obtained with codes BBFP (Bunched-Beam Fokker-Planck solver) Details to be refined normal hollow

38. August 29, 2007 Wei 38 BBFP calculation in the action space Density projection in longitudinal action Convertible to the phase / momentum planes normal hollow

39. August 29, 2007 Wei 39 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

40. August 29, 2007 Wei 40 Transverse coupling dependence

41. August 29, 2007 Wei 41 Dispersion wave measurements real (MAD) lattice used in simulations Online model & measured value (dispersion max location only)

42. August 29, 2007 Wei 42 Complications from dual RF system Large rms bunch length (2ns) due to satellite beams Primary bucket bunch length (1ns) satisfies: 5 < bucket width

43. August 29, 2007 Wei 43 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 current Bunched blue current Total Yellow current

44. August 29, 2007 Wei 44 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

45. August 29, 2007 Wei 45 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