1. Collective Focusing of a Neutralized Intense Ion Beam Propagating Along a Weak Solenodial Magnetic Field M. Dorf (LLNL) In collaboration with I. Kaganovich, E. Startsev, and R. C. Davidson (PPPL) LLNL- PRES-635456 Heavy Ion Fusion Science Virtual National Laboratory DOE Plasma Science Center, Teleseminar, April 19, 2012 This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344, and by the Princeton Plasma Physics Laboratory under contract AC02-76CH-O3073

2. Motivation: Controlled Fusion 2 W. Sharp et al, http://hif.lbl.gov/tutorial/assets/fallback/index.html

3. Inertial Confinement Fusion: Lasers Versus Ion Beams High production efficiency and repetition rate High efficiency of energy delivery and deposition Why ion beams? 3 Present approach – Laser Driven ICF Promising alternative – Heavy Ion Fusion National Ignition Facility (LLNL, Livermore, CA)

4. Ion-Beam-Driven High Energy Density Physics Heavy Ion Fusion (Future) Warm Dense Matter Physics (Present) D-T I total ~100 kA τ b ~ 10 ns E b ~ 10 GeV Atomic mass ~ 200 corresponds to the interiors of giant planets and low-mass stars I b ~1-10 A τ b ~ 1 ns E b ~ 0.1-1 MeV Ions: K +, Li + foil Presently accessible ρ-T regime ρ~1 g/cm 3, T~0.1 ÷ 1 eV 4 ion source acceleration and transport neutralized compression final focusing target Block scheme of an ion driver for high energy density physics ρ~1000 g/cm 3 T~ 10 KeV

5. Neutralized Drift Compression Experiment (NDCX) Heavy ion driver for Warm Dense Matter Experiments 5 Built and operated at the Lawrence Berkeley National Laboratory

6. Neutralized Drift Compression Experiment (NDCX) Schematic of the NDCX-I experimental setup (LBNL) Beam parameters at the target plane Weak fringe magnetic fields (~100 G) penetrate deeply into the background plasma K + @ 300 keV (β b =0.004) I b ~2 A, r b < 5 mm (n b ~10 11 cm -3 ) upgrade NDCX-II Li + @ 3MeV (β b =0.03) I b ~30 A, r b ~1 mm (n b ~6∙10 12 cm -3 ) T target ~0.1 eV T target ~1 eV 6 8T Fringe magnetic fields TargetFinal Focus Solenoid

7. Outline I. Enhanced self-focusing of an ion beam propagating through a background plasma along a weak (~100 G) solenodial magnetic field Collective focusing with B 0 ~ 1 kG is equivalent to standard magnetic focusing with B 0 ~10 T important for the design of a heavy-ion driver (e.g. NDCX neutralized drift section) II. Collective Focusing (Robertson) Lens can be utilized in ion beam self-pinch transport applications (e.g. HIF drivers) 7 can be used for the ion beam final focus (e.g. NDCX-I, II) perhaps can be utilized for collimation of laser generated proton beams

8. I. Ion Beam Propagation through a Neutralizing Background Plasma Along a Solenoidal Magnetic Field ion beam plasma B0B0 e-e- vbvb 8

9. Magnetic Self-Pinching (B 0 =0) The ion beam space-charge is typically well-neutralized beam e-e- e-e- e-e- e-e- 9 Inductive field accelerates electrons What about ion beam current? r z λ=c/ω pe collisionless electron skin depth (n p ~10 11 cm -3 → λ~1.7 cm) defines the characteristic length scale for screening current (or magnetic-field) perturbations in a cold plasma (“inductive” analog of the Debye length) Electron radial force balance Current neutralization Magnetic pinching is dominant

10. Enhanced Collective Self-Focusing (B 0 ~100 G) There is a significant enhancement of the ion beam self-focusing effect in the presence of a weak solenoidal magnetic field (for r b <<c/ω pe ) Radial displacement of background electrons is accompanied by an azimuthal rotation Strong radial electric field is produced to balance the magnetic V×B force acting on the electrons for 10 This radial electric field provides enhanced ion focusing M. Dorf et al, PRL 103, 075003 (2009).

11. Enhanced Self-Focusing is Demonstrated in Simulations Gaussian beam: r b =0.55c/ω pe, L b =3.4r b, β=0.05, n b =0.14n p, n p =10 10 cm -3 Central beam slice Beam density profile Analytic LSP (PIC) R (cm) B ext =0 Magnetic self-pinching Collective self-focusing F r /Z b e (V/cm) Radial focusing force B ext =300 G The enhanced focusing is provided by a strong radial self-electric field Influence of the plasma-induced collective focusing on the ion beam dynamics in Radial electric field B ext =300 G LSP (PIC) ω ce /2β b ω pe =9.35 NDCX-I is negligible NDCX-II is comparable to the final focusing of an 8 T short solenoid 11

12. e-e- ++ + ++ Veφ<0Veφ<0 e-e- -- - -- F V×B -eE r Veφ>0Veφ>0 δr>0 Radial electric field is defocusing Paramagnetic plasma response Radial electric field is focusing Diamagnetic plasma response Weak magnetic field ( ω ce < 2 β b ω pe ) δr<0 B0B0 B0B0 Moderate magnetic field ( ω ce > 2 β b ω pe ) Beam charge is overcompensatedBeam charge is under-neutralized Local Plasma Response is Drastically Different for ω ce > 2 β b ω pe and ω ce < 2 β b ω pe resonant excitation of large-amplitude whistler waves ω ce =2β b ω pe n p =10 11 cm -3, β b =0.05 B 0 =100G F V×B -eE r 12 M. Dorf et al, PoP 19, 056704 (2012).

13. Numerical Simulations Demonstrate Qualitatively Different Local Plasma Responses r b =0.55c/ω pe, l b =3.4r b, β=0.05, n b =0.14n p, n p =10 10 cm -3 LSP (PIC) B 0 =25 G (ω ce /β b ω pe =1.56)B 0 =300 G (ω ce /β b ω pe =18.7) Radial electric field LSP (PIC) Radial electric field Focusing electric fieldDefocusing electric field Diamagnetic response (δB z <0) ω ce >2β b ω pe ω ce <2β b ω pe Paramagnetic response (δB z >0) 13

14. Whistler Wave Excitation PIC (LSP)Semi-analyticAnalytic 1 Schematic of the detected signal Signal amplitude ω ce / 2β b ω pe Magnetic pick-up loop Numerical integration (FFT) assuming weak dissipation Analytical treatment of poles in the complex plane Analytic theory is in very good agreement with the PIC simulations Wave-field excitations can be used for diagnostic purposes ω ce /2β b ω pe Strong wave excitation occurs at ω ce /2β b ω pe (supported by PIC simulations) beam velocity=phase velocity=group velocity Enhanced self-focusing (local fields) Strong wave-field excitation β b =0.33, l b =10r b, r b =0.9∙c/ω pe, n b =0.05n p, B ext =1600G, n p =2.4∙10 11 cm -3 Transverse magnetic field 14 M. Dorf et al, PoP 17, 023103 (2010).

15. II. The Use of Weak Magnetic Fields for Final Beam Focusing Schematic of the present NDCX-I final focus section drift section (8 Tesla) FFS Challenges: Can a weak magnetic lens be used for tight final beam focusing? Operate 8 T final focus solenoid Fill 8 T solenoid with a background plasma 15

16. Magnetic Lens 16 charged particle beam q, v b magnetic lens B0B0 BrBr vbvb v b × B r provides azimuthal rotation (v φ ) v φ × B 0 provides radial focusing -v φ Conservation of canonical azimutal (angular) momentum Equation of motion Cyclotron frequency centrifugal force V×B Lorentz force

17. Collective Focusing Lens ei+ei L e ~0.01 mmL i ~ 10 mL c =(L e L i ) 1/2 ~ 1 cm B=500 G L=10 cm Electron beam focusing Ion beam focusing Neutralized beam focusing The use of a collective focusing lens reduces the required magnetic field by (m i /m e ) 1/2 B=500 G L=10 cm β=0.01 *S. Robertson, PRL 48, 149 (1982) 17 neutralized ion beam e -, i + magnetic lens B0B0 Collective Focusing Concept* Collective Focusing Lens

18. Collective Focusing Lens (Cont’d) Strong ambipolar electric field is produced to balance the magnetic V×B force acting on the co-moving electrons Centrifugal force V×B magnetic force Electric force Conditions for collective focusing Electrons traversing the region of magnetic fringe fields acquire an azimuthal velocity of 1. Quasi-neutrality 2. Small magnetic field perturbations 3. No pre-formed plasma inside the lens 18

19. The collective focusing concept can be utilized for final ion beam focusing in NDCX. No need to fill the final focus solenoid (FFS) with a neutralizing plasma Magnetic field of the FFS can be decreased from 8 T to ~700 G Collective Focusing Lens for a Heavy Ion Driver Final Focus Schematic of the present NDCX-I final focus section The ion beam can extract neutralizing electrons from the drift section filled with plasma. drift section (FFS) FFS 19

20. Numerical Simulations Demonstrate Tight Collective Final Focus for NDCX-I Schematic of the NDCX-I simulation R-Z PIC (LSP) Beam injection parameters: K + @ 320 keV, r b =1.6 cm, I b =27 mA T b =0.094 eV, Plasma parameters (for the simulation II) 35266271281 Plasma Final Focus Solenoid (700 G) 2 cm beam Conductivity model kinetic 251 Tilt gap z=8 -11 0 I simulationII simulation z (cm) n p =10 11 cm -3, T e =3 eV 280281282283 0 0.05 0.1 0.15 0.2 z (cm) r (cm) n b (cm -3 ) Beam density @ focal plane 6×10 12 cm -3 2595256525762585 0.0 0.5 1.0 1.5 2.0 time (ns) I b (A) Beam current @ focal plane 20 E r (kV/cm) Radial electric field inside the solenoid Analytic LSP (PIC) r (cm) M. Dorf et al, PoP 18, 033106 (2011)

21. Conclusions Even a weak magnetic field (several hundred gauss) can have a significant influence on neutralized ion beam transport. ion beam plasma B0B0 e-e- neutralized ion beam e -, i + magnetic lens B0B0 Self-focusing force can be significantly enhanced by application of a weak magnetic field (~100 G) for r b <<c/ω pe. For a given focal length the magnetic field required for a neutralized beam is smaller by a factor of (m e /m i ) 1/2. Can be important for the design of a heavy-ion driver (e.g. NDCX) Can be utilized for self-pinch ion beam transport applications Can be utilized for the ion beam final focus (e.g. NDCX-I, II) Perhaps could be utilized for collimation of laser generated proton beams

22. Enhanced Self-Focusing VS Collective Lens Enhanced self-focusing ion beam plasma B0B0 e-e- neutralized ion beam e -, i + magnetic lens B0B0 Collective Lens ω ce >>2β b ω pe Conditions: no plasma inside the lens In the limit of r b ~r ge and Z b n b ~n e → F self ~F col Conditions: n p =10 11 cm -3, β b =0.05 B 0 >100 G r b >1 cm 22 Linear force (important for beam focusing) Non-linear force can effectively balance p~T b n (important for self-pinch transport)