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, DOE Plasma Science Center, Teleseminar, April 19, 20122012

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

Motivation: Controlled Fusion

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

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)

Ion-Beam-Driven High Energy Density Physics

Heavy Ion Fusion (Future) Warm Dense Matter Physics (Present)

D-T

Itotal~100 kA

τb~ 10 ns

Eb ~ 10 GeV

Atomic mass ~ 200

corresponds to the interiors of giant planets

and low-mass stars

Ib~1-10 Aτb~ 1 ns

Eb ~ 0.1-1 MeV

Ions: K+, Li+

foil Presently accessible ρ-T regime

ρ~1 g/cm3, 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/cm3

T~ 10 KeV

Neutralized Drift Compression Experiment (NDCX)

Heavy ion driver for Warm Dense Matter Experiments

5

Built and operated at the Lawrence Berkeley National Laboratory

Neutralized Drift Compression Experiment (NDCX)Schematic of the NDCX-I experimental setup

(LBNL)

Beam parameters at the target planeBeam parameters at the target plane

Weak fringe magnetic fields (~100 G) penetrate deeply into the background plasma

K+ @ 300 keV (βb=0.004)

Ib~2 A , rb < 5 mm (nb~1011 cm-3)

upgrade

NDCX-II

Li+ @ 3MeV (βb=0.03)

Ib~30 A , rb~1 mm (nb~6∙1012 cm-3)

Ttarget ~0.1 eV Ttarget ~1 eV

6

8T

Fringe magnetic fields

TargetFinal Focus Solenoid

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 B0 ~ 1 kG is equivalent to standard magnetic focusing with

B0 ~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

neutralized ion beam

e-, i+

magnetic lens

B0

ion beam

plasma

B0

e-vb

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

I. Ion Beam Propagation through a Neutralizing Background Plasma

Along a Solenoidal Magnetic Field

ion beam

plasma

B0

e-vb

8

Magnetic Self-Pinching (B0=0) The ion beam space-charge is typically well-neutralized

beam

e-

e-

e-

e-

selfB

9

t

Bself

zE Inductive field accelerates electrons

What about ion beam current?

r

z

λ=c/ωpe collisionless electron skin depth (np~1011 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)

cVBE ezselfr

1~ ebbbez nnZVV

cVBE bselfr

Electron radial force balance

Current neutralization

Magnetic pinching is dominant

jb=|je|

rb

current is well-neutralized (locally) negligible self-pinching

(b) rb>c/ωpe

jb

|je|c/ωpe

current is not-neutralized (locally) maximum (jbxBself) self-pinching

(a) rb<c/ωpe

r

Enhanced Collective Self-Focusing (B0~100 G)

There is a significant enhancement of the ion beam self-focusing effect in the presence of a weak solenoidal magnetic field (for rb<<c/ωpe)

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

dr

dn

nVmZF b

pbebsf

122

2

2

1pe

ce

ce

bb

Vr

10

This radial electric field provides enhanced ion focusing

M. Dorf et al, PRL 103, 075003 (2009).

constrAc

ermP ee V

02B

rA

Enhanced Self-Focusing is Demonstrated in SimulationsGaussian beam: rb=0.55c/ωpe, Lb=3.4rb, β=0.05, nb=0.14np, np=1010 cm-3

-60

-40

-20

0

20

40

60

0 2 4 6 8

Central beam slice

Beam density profile

Analytic

LSP (PIC)

R (cm)

Bext=0Magnetic self-pinching Collective self-focusing

Fr/Zbe (V/cm)

Radial focusing force

Bext=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 inInfluence of the plasma-induced collective focusing on the ion beam dynamics in

Radial electric field

Bext=300 G LSP (PIC)

ωce/2βbωpe=9.35

NDCX-I is negligible

NDCX-II is comparable to the final focusing of an 8 T short solenoid

11

e-

+ +++ +

Veφ<0

z

e-

- --- - FV×B

-eEr

Veφ>0

δr>0

z

Radial electric field is defocusing

Paramagnetic plasma response

Radial electric field is focusing

Diamagnetic plasma response

Weak magnetic field (ωce<2βbωpe)

δr<0

B0B0

Moderate magnetic field (ωce>2βbωpe)

Beam charge is overcompensated Beam charge is under-neutralized

Local Plasma Response is Drastically Different for ωce>2βbωpe and ωce<2βbωpe

resonant excitation of large-amplitude whistler waves

ωce=2βbωpe

np=1011 cm-3, βb=0.05 B0=100G

FV×B

-eEr

12M. Dorf et al, PoP 19, 056704 (2012).

Numerical Simulations Demonstrate Qualitatively Different Local Plasma

Responses

rb=0.55c/ωpe, lb=3.4rb, β=0.05, nb=0.14np, np=1010 cm-3

LSP (PIC)

B0=25 G (ωce/βbωpe=1.56) B0=300 G (ωce/βbωpe=18.7)

Radial electric fieldLSP (PIC)

Radial electric field

Focusing electric field Defocusing electric field

Diamagnetic response (δBz<0)

ωce>2βbωpe ωce<2βbωpe

Paramagnetic response (δBz>0)

13

Whistler Wave Excitation

PIC (LSP) Semi-analyticAnalytic

1

Schematic of the detected signal

Sign

al a

mpl

itud

e

ωωcece//22ββbbωωpepe

Magnetic pick-up loopNumerical 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

Strong wave excitation occurs at ωωcece/2/2ββbbωωpe pe (supported by PIC simulations)

beam velocity=phase velocity=group velocity

Enhanced self-focusing (local fields)

Strong wave-field excitation

βb=0.33, lb=10rb, rb=0.9∙c/ωpe, nb=0.05np, Bext=1600G, np=2.4∙1011cm-3

Transverse magnetic field

14M. Dorf et al, PoP 17, 023103 (2010).

II. The Use of Weak Magnetic Fields for Final

Beam FocusingSchematic 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

Magnetic Lens

16

charged particle beam

q, vb

magnetic lens

B0Br

vb

vb × Br

provides azimuthal rotation (vφ)

v φ × B0

provides radial focusing

-vφ

Conservation of canonical azimutal (angular) momentum

Equation of motion

constrAc

qmrP v

02B

rA

c

r

2v

mc

qBc

0

Cyclotron frequency

rFc

Bq

rm

dt

rdm 0

2

2

2 vv

centrifugal force

V×B Lorentz force

4

2c

r

mrF

Collective Focusing Lens

e i+ei

Le ~0.01 mm Li ~ 10 m Lc=(LeLi)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 (mi/me)1/2

B=500 G L=10 cm

B=500 G L=10 cm

β=0.01 β=0.01 β=0.01

*S. Robertson, PRL 48, 149 (1982)17

neutralized ion beam

e-, i+

magnetic lens

B0

Collective Focusing Concept*

Collective Focusing Lens

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

r

VmBV

c

eeE e

eer

2

0

Electrons traversing the region of magnetic fringe fields

acquire an azimuthal velocity of

e

rmE eer 4

2

1. Quasi-neutrality

2. Small magnetic field perturbations

2

rV ee

3. No pre-formed plasma inside the lens

18

2epe

peb cr 2

cm

eB

ee

0

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

Numerical Simulations Demonstrate Tight Collective Final Focus for NDCX-I

Schematic of the NDCX-I simulation R-Z PIC (LSP)Schematic of the NDCX-I simulation R-Z PIC (LSP)Beam injection parameters:

K+ @ 320 keV, rb=1.6 cm, Ib=27 mA Tb=0.094 eV,

Plasma parameters (for the simulation II)

35 266 271 281

Plasma

Final Focus Solenoid (700 G)

2 cm

beam

Conductivity model

kine

tic251

Tilt gap z=8 -11

0

I simulation II simulation

z (cm)np=1011 cm-3, Te=3 eV

280 281 282 2830

0.05

0.1

0.15

0.2

z (cm)

r (c

m)

nb (cm-3)

Beam density @ focal plane

6×1012 cm-3

25952565 2576 25850.0

0.5

1.0

1.5

2.0

time (ns)

I b (

A)

Beam current @ focal plane

20

-10

-8

-6

-4

-2

0

2

4

0 0.4 0.8 1.2

Er (k

V/c

m)

Radial electric field inside the solenoid

Analytic

LSP (PIC)

r (cm)

M. Dorf et al, PoP 18, 033106 (2011)

Conclusions

Even a weak magnetic field (several hundred gauss) can have

a significant influence on neutralized ion beam transport.

ion beam

plasma

B0

e-

neutralized ion beam

e-, i+

magnetic lens

B0

Self-focusing force can be significantly enhanced by application

of a weak magnetic field (~100 G) for rb<<c/ωpe.

For a given focal length the magnetic field required for a neutralized beam

is smaller by a factor of (me/mi)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

Enhanced Self-Focusing VS Collective Lens

Enhanced self-focusing

ion beam

plasma

B0

e-

neutralized ion beam

e-, i+

magnetic lens

B0

Collective Lens

ωce>>2βbωpe

2 2 1 bself b e b

e

dnF Z m V

n dr

4col i e i

rF m

Conditions:

no plasma inside the lens

In the limit of rb~rge and Zbnb~ne → Fself~Fcol

2

2

1pe

ce

ce

bgeb

Vrr

Conditions:np=1011 cm-3, βb=0.05

B0>100 G

rb>1 cm

22

Linear force (important for beam focusing)

Non-linear force can effectively balance p~Tb n (important for self-pinch transport)

2epe

peb cr 2