Edward A. Startsev, Ronald C. Davidson and Hong Qin

1. The Heavy Ion Fusion Virtual National Laboratory Simulation Studies of Temperature Anisotropy Instability in Intense Charged Particle Beams for IBX Parameters Edward A. Startsev, Ronald C. Davidson and Hong Qin Princeton Plasma Physics Laboratory Presented at APS DPP2003 Meeting October 30, 2003 Albuquerque, NM

2. The Integrated Beam Experiment (IBX) is a proof-of-principal experiment for heavy ion fusion designed to test source-to-target beam physics using a single beam of ions of duration 0.2 - 1.5 , accelerated to energies ~ 5-10 MeV, and driver-scale normalized perviance in the range . An important physics issue to be addressed by IBX is the effect of longitudinal-transverse coupling on the beam transport and focusibility of the driver. Our previous numerical and theoretical studies of intense charged particle beams with large temperature anisotropy [E. A. Startsev, R. C. Davidson and H. Qin, PRSTAB, 6, 084401, 2003] demonstrated that a fast, electrostatic, Harris-like instability may develop. This paper reports the results of recent simulations of the temperature anisotropy instability using the Beam Equilibrium Stability Transport (BEST) code for IBX parameters. Abstract

3. Outline • Short introduction to HIF(Heavy Ion Fusion) (driver specific). • Why studying temperature anisotropy is important for HIF? • What is the origin of the temperature anisotropy in particle beams? • What are the mechanisms that provide beam temperature • equipartitioning? • 3D delta-f PIC Simulations of the temperature anisotropy instability. • Theory of the temperature anisotropy instability. • Simulation of instability for several IBX design parameter sets. • Conclusions and outlook.

4. Heavy Ion Fusion Requirements • Illustrative parameters: • Total Beam Energy 5 MJ • Focal spot radius ~ 3 mm • Ion range ~ 0.1 g/cm (1 mm in typical materials) • Pulse duration ~10 ns • Peak power ~ 400 TW • Ion Energy ~10 GeV • Current on target ~ 40 kA (total) • Ion mass ~ 200 amu 2

5. HIF driver

6. 12 m, 36 hlp 6 MeV Compress x 250  25 ns The IBX is a Single-Beam Short-Pulse Source-to-Focus Experiment 2 m 250 ns 23 m, 48 hlp Shaping/ matching section Velocity tilt section Injector Accelerator Dv sect Drift Compression/ Bend section 1.7 MeV 6 MeV Ion: K+ 1 beamline Pulse duration: 250 ns -> 25 ns Final energy: 6 MeV Total half-lattice periods: 98 (of constant length) Total length: 42 m Cost: ~ 60 - 80 M$ Final Focus 5 m 25 ns

7. Integrated Beam Experiment (IBX) Components INJECTOR: Ion K+ or Ar+ Energy 1.0 - 1.7 MeV Current 0.2 - 0.7 A Pulse Length (FWHM) 0.25 -3 ms ACCELERATOR: Half Lattice Period (HLP) Length 30 cm Number of HLPs ~ 40 - 100 Final Energy 5 - 10 MeV BEND or DRIFT COMPRESSION: Number of HLPs ~ 30 Compression Ratio 10 Final Beam Pulse Length 25 ns PLASMA NEUTRALIZATION & CHAMBER FINAL FOCUS: Number of HLPs 10 Beam Pulse Length 25 ns Final Perveance ~ 1x10-3 Final Space Charge Density 0.7 – 2 mC/m

8. Alternating Gradient Focusing Systems S is Lattice period =Beam velocity =charge of the beam particle =Mass of the beam particle

9. Smooth focusing approximation

10. Beams with strong self-fields • Cold beam has a uniform density and • Force balance condition in equilibrium is beam intensity parameter beam plasma frequency squared • For warm beam the focusing has to balance self-charge force AND thermal pressure.

11. Spot size requirements Drift compression gives a factor of 15: Need to have beam with

12. Spot size requirements spot size Need to have beam with

13. For example: Temperature anisotropies develop naturally in accelerators • For a beam of charged particles with charge q accelerated through a voltage V • Energy spread is constant during acceleration, which gives • Transverse temperature remains relatively unaffected by acceleration.

14. Transverse emittance(phase volume) and can increase due to nonlinearities • Transverse emittance (entropy, phase volume) growth is cased by nonlinear • external focusing forces by nonlinear space-charge forces, instabilities or collisions. • Emittance (entropy, phase volume) grows as beam relaxes toward a final • stationary (equilibrium) state.

15. Example of transverse emmitance growth x px/pz -0.002 0 .002 -0.01 0 .01 -0.01 0 .01 End of injector End of matching section 50 lattice periods (100 quads) past matching section

16. Intrabeam collisions: Possible equilibration mechanisms • Collective instabilities: fast, with growth rate of a fraction of plasma frequency • Electrostatic: Harris-like instability where oscillations in magnetic • fields are replaced by betatron oscillations • Electromagnetic: Weibel-like instability • For HIF: • Mean free path in the Lab frame

17. Internal resonances are responsible for coupling of the transverse and longitudinal degrees of freedom • Particle motion inside the field of the wave • The force seen by the particle is modulated by • unperturbed periodic particle motion: • When resonanse condition is satisfied, and particle exchanges energy with the wave. • The wave provides coupling of the transverse and longitudinal degrees • of freedom which may lead to instability.

18. Beam density nonuniformity makes analytical treatment complicated • For electrostatic perturbations of the form • Linearized Poison and Vlasov equations can be integrated along characteristics • Making expansion ,where are Bessel functions of order m, and • And using two-temperature Maxwellian, as unperturbed distribution • System of equations reduces to matrix dispersion equation

19. Beam density nonuniformity makes analytical treatment complicated • Dispersion matrix is • Phase integrals are calculated along the unperturbed particle trajectories.

20. where for m=0, and for m=1. Qualitative analysis of the dispersion equation • Dispersion equation is similar to two-stream dispersion equation • For m=0 • For m=1 • We can rewrite above equation as the same equation • There are two purely imaginary roots (one of them corresponds to instability) if • This is a threshold condition on beam intensity and normalized wavenumber • Therefore for m=1the unstable mode is purely growing • and for m=0

21. BEST Nonlinear F Simulation Code • 3D particle-in-cell simulation code with cylindrical geometry. • Multiple species. • Adiabatic field pusher for light particles (electrons). • Transition from delta-f to regular PIC code for large perturbations. • Simulation noise is reduced significantly when operating as delta-f. • Easily switched between linear and nonlinear operation. • Written in Fortran 90/95 and extensively object-oriented. • NetCDF data format for large-scale diagnostics and visualization.  • Achieved an average speed of 80 megaflops on IBM-SP (stage I) at NERSC. • The code has been parallelized using OpenMP and MPI BEST stands for “Beam Equilibrium, Stability and Transport code”

22. Solution to the nonlinear Vlasov-Maxwell equations are expressed as where are known equilibrium solutions. Description of the BEST Nonlinear F Simulation Code • Equations for the perturbed potentials are • Equation for the particle motion advance are

23. where • Time history of density perturbation . Simulations show fast instability Parameters of the simulation: Beam intensity parameter: • Parameters of the simulation: • The self-consistent equilibrium distribution functions is assumed to be:

24. The average longitudinal momentum distribution for a dominant initial perturbation with m=1. • Mode growth linearly with Re • The final width of the longitudinal velocity distribution • can be estimated as , where Saturation of the instability • Mode saturates via nonlinear frequency shift with • consequent formation of tails in axial momentum • space and Landau damping.

25. The growth rate for can be estimated from Instability Threshold • The instability is stabilized by formation of a tail in the longitudinal momentum • distribution and the consequent Landau damping of the wave excitations. where is the threshold temperature for stabilization. • Therefore:

26. Simulations of instability in IBX (I=0.692 A, emmitance=1 mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Unstable modes

27. Simulations of instability in IBX(I=0.692 A, emmitance=1mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Unstable modes

28. Simulations of instability in IBX (I=0.364 A, emmitance=1mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Unstable modes

29. Simulations of instability in IBX(I=0.364 A, emmitance=1mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Unstable modes

30. Simulations of instability in IBX (I=0.055 A, emmitance=1 mm x mrad) • Equilibrium radial density profile • Beam parameters Corresponds to: • Unstable modes

31. Simulations of instability in IBX(I=0.055 A, emmitance=1 mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Mode spectra

32. Simulations of instability in IBX (I=0.055 A, emmitance=0.39 mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Unstable modes

33. Simulations of instability in IBX (I=0.055 A, emmitance=0.39 mm x mrad) • Beam parameters • Equilibrium radial density profile Corresponds to: • Unstable modes

34. Will IBX be able to see the instability? • Full-scale version • Reduced version • IBX-Lite (no acceleration) • Low current, large-emmitance version • Low current version

35. This result is important for Heavy Ion Fusion driver where this ratio • needs to be kept small so that the beam could be focused • onto the small targets Conclusions • The 3D PIC (BEST) code, which implements the nonlinear scheme, • has been used to investigate the stability properties of intense charged particle • beams with large temperature anisotropy . • The simulation results clearly show that sufficiently intense beams with • are linearly unstable to short wavelength perturbations with • provided the temperature ratio is sufficiently small • The mode structure, growth rate and conditions for the onset of the • instability have been investigated with simulations and analytically. • In the nonlinear saturation stage, the total distribution function is still • far from equipartitioned

36. Conclusions and Outlook • In the present design versions, the IBX will be able to study the effects • of temperature anisotropy instability. • Depending on the design, 3 to 4 e-foldings for instability will be present, • which corresponds to 20—40 increase from the initial unstable mode amplitude. • The mode structures presented here can be used in further study of mode seeding • from the beam source. • We are planning to use WARP code to study effects of the acceleration and • alternating focusing on the dynamics of the instability and develop efficient tools • (criteria) for its measurement in IBX.