Presented for LBNL Center for Beam Physics

11/29/2012

Mikhail Dorf

Lawrence Livermore National Laboratory, Livermore, CA 94550

Heavy Ion Fusion Science

Virtual National Laboratory

Modeling of Intense Ion Beam Transport and

Focusing for High Energy Density Physics Applications

In collaboration with

PPPL: R. Davidson, I. Kaganovich, E. Startsev, H. Qin, E. Gilson

LLNL/LBNL : A. Friedman, R. Cohen, D. Grote, S. Lund, P. Seidl, S. Lidia, J. Barnard, J. Kwan

IAP RAS (Russia): V. Zorin, A. Sidorov, V. Skalyga, and the ECR team

LLNL-PRES-605078

This research was supported by USDOE contracts DE-AC52-07NA27344 and AC02-76CH-O3073; Russian Foundation for Basic Research (RFBR), grant n0.

11-02-97056-r_povolzhie_a; Federal Targeted Program "Scientific and Educational Personnel of the Innovative Russia" for 2009-2013;

Ion-Beam-Driven High Energy Density Physics

High production efficiency and repetition rate

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

High efficiency of energy delivery and deposition

Why ion beams?

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 A

τb~ 1 ns

Eb ~ 0.1-1 MeV

Ions: K+, Li+

foil Presently accessible ρ-T

regime

ρ~1 g/cm3, T~1 eV

2

Outline

Modeling of gas-dynamic ECR plasma ion source

Studies of nonlinear dynamics and collective processes in intense

ion beams in periodic focusing accelerators and transport systems

ion

source

acceleration

and transport

neutralized

compression

final

focusing

target

Studies of neutralized ion beam transport along a weak

solenoidal magnetic field

Studies of collective ion beam focusing by a weak solenoidal

magnetic field

This presentation reviews studies of transport properties of an

intense ion beam propagating in an ion driver

3

Ion driver

• Ion beam extraction

• Emittance measurements

• Equilibrium (matched) beam distributions

• Halo production by a beam mismatch

• Enhanced self-focusing

• Electromagnetic wave excitation

• Collective focusing magnetic lens

• Collective focusing for NDCX final focus

Neutralized Drift Compression Experiment (NDCX)

Beam parameters at the target plane

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

4

8T

Target Final Focus

Solenoid

Schematic of the NDCX-I experimental setup (LBNL) The heavy ion driver for Warm Dense Matter Experiments

I. Modeling of the ion beam extraction from a gas-

dynamic ECR plasma ion source

5 5

30kV 0 kV

Pla

sm

a

Ion beam

R-Z PIC (WARP) simulations

R (m)

Z (m)

References

[1] M. Dorf et al., J. Appl. Phys. 102, 054504 (2007).

[2] M. Dorf et al., Phys. Plasmas, 15, 093501 (2008).

[3] M. Dorf et al., submitted to Nucl. Instr. Meth. Res. A (2012).

[4] A. Sidorov, M. Dorf, et al., Rev. Sci. Instrum. 79, 02A317 (2008).

Gas-Dynamic ECR Ion Source (SMIS 37)

6

Extracted beam

I~150 mA

εN~0.9 π∙mm∙mrad

j~1A/cm2

N+ N++ N+++

<Zi>~2

ECR dischargebeam

Puller

FC

L~30 cm

Magnetic

trap (2T)

Lp~25 cm

Plasma electrode

(U~20-60 kV)

Microwave

radiation

(37.5 GHz,

100 kW)

n~1013 cm-3 Te~50 eV

6

High current / Low Charge State

ion sources

Multi-charged / Low current ion

sources

Gas-dynamic ECR ion source

Vacuum arc sources (MEVVA), RF (multicusp) ECR ion sources (classical) Zi >> 1

Moderate charge-state (Zi>1) High beam current

SMIS 37

Build and operated

IAP RAS (Russia)

Gas-dynamic ECR source looks promising as a source for HIF driver

7

Principles of the Ion Beam Extraction from Plasma

i

eeff

effBM

TZenZj 061.0

Bohm current Child-Langmuir current

2

0

2/3

72.1d

U

A

Zj

i

eff

CL

d0

Plasma-vacuum surface (meniscus) evolves in order to match jB ≈ jCL

S0 0

Uext U=0

7

d>d0

S>S0

S<S0

d<d0

Low-density plasma (jB<j0CL) Overdense plasma (jB>j0CL)

WARP code WARP code

jb / jCL ≈ 0.5 jb / jCL ≈ 9

Single-aperture extraction: Experiment and Simulations

Experiment WARP code

0

0.5

1

1.5

2

2.5

3

3.5

15 25 35 45 55

IFC (mA)

U (kV)

d0=3 mm

d0=7 mm

np0=2.4×1012 cm-3, Ti=10 eV

np0=2.8×1012 cm-3, Ti=10 eV

8

φp+U

φp+U

φp+

U

U

U

N+ N2+ N3+

0 V

0 V

Boltzman

Electrons

Vs

Extracting

electrode (puller)

Secondary

electrons

(not included in

the simulations)ion

Fara

day C

up

IFC

IPe-

Schematic of WARP simulations Extracted beam current

The results of the numerical simulations are in very good agreement

with the experimental measurements

Plasma sheath model

• Ions are injected with the Bohm velocity

• Boltzman electrons with Te= 50 eV are assumed

• Possion equation is used for potential variations

• N+:N++:N+++ obtained from measurments

• np0 and Ti are chosen to match the measurements

Extraction aperture - 1 mm

d0

M. Dorf et al. submitted to NIMA (2012)

Noise Suppression and Stabilization of an Ion Beam

Extracted from Overdense Plasma

Beam formation in the plateau regime

significantly enhances the brightness (BPlateau~16Bregime of maximum extracted current)

provides stable extraction despite large plasma fluctuations

9

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.2 0.4 0.6 0.8

Plasma density, 1013cm-3

IFC (mA)

Large fluctuations

Larg

e n

ois

eLarge fluctuations

Low

nois

e

Plateau

WARP Simulations

0

0.4

0.8

1.2

1.6

2

0 200 400 600 800 1000 1200

Jpuller+JBCF, mA/cm2

IFC (mA)Experimental dataIllustrative cartoon

M. Dorf et al., J. Appl. Phys., 102, 054504 (2007).

It is of particular practical importance to suppress noise in the extracted

beam current provided by fluctuations in the plasma density

Measurements of the Transverse Beam Emittance

Images of

plasma

discharge

Scintillator plate image Beam phase-space measurements

using the pepper-pot method

Beam phase-space reconstruction

Multiaperture plasma electrode

2 cm

Ø=3mm

The diagnostics demonstrates high values of the ion beam transverse temperature (~10 eV)

A. Sidorov, M. Dorf, V. Zorin, et al.,

Rev. Sci. Instrum. 79, 02A317 (2008).

Schematic of the diagnostics setup

10 10

Similar assumption of high ion temperature (~10 eV) was required in numerical modeling to match the experiments

A reduced analytical model was developed to explain such high ion temperatures M. Dorf et al, Phys. Plasmas, 15, 093501 (2008).

II. Studies of nonlinear transverse beam dynamics in

periodic focusing transport systems

11 11

References

[1] M. Dorf et al., Phys. Plasmas, 18, 043109 (2011).

[2] M. Dorf et al., Phys. Plasmas, 16, 123107 (2009).

[3] M. Dorf et al., Phys. Rev. ST AB 9, 034202 (2006).

[4] M. Dorf et al., PAC Proceeding 2007; 2009.

Periodic Focusing Lattices

ˆ ˆ( ) ( )( )

ˆ ˆ( ) ( )( )

q q x y

foc

q q x y

B x B z y x

F x z xe ye

e e

Alternating-gradient quadrupole lattice Periodic-focusing solenoidal lattice

z

Smooth-focusing approximation

Single-particle motion (no space-charge forces)

rsffoc rF e

1ˆ ˆ ˆ( ) ( )( ) ( )

2

ˆ ˆ( ) ( )( )

sol z x y z z

foc

sol s x y

B x B z y x B z

F x z xe ye

e e e

12

SSmooth focusing

(betatron) period,

2π/κsf

Vacuum phase advance

σvac = κsfS/2π

In the presence of space charge, the phase advance, σ, is depressed,

σ<σvac

σ/σvac→1 emittance dominated beam

σ/σvac→0 space-charge dominated beam

z

x

Equilibrium (Matched) Beam Distribution

Smooth-focusing approximation

ˆ ˆfoc

sf sf x yF x y e e

Beam Equilibrium

The transverse Hamiltonian

(particle energy) is not conserved The transverse Hamiltonian is an

invariant of beam particles motion

Oscillating focusing field

0 2 2 21 1

2 2sfH x y r r

Kinetic

energy

Applied focusing

potential

Self-

potential

000

Hff bb

Equilibrium (matched) beam envelope

Szz latticelattice

How to find (or load) a matched-beam quasi-equilibrium?

Quadrupole lattice

X, Y beam

envelope

z

S

Szfzf bb

z

Rb

Matched beam

envelope

Nonlinear effects of the intense

beam self-fields provide a significant

challenge for analytical studies

1

3

ydxdfr b

13

Adiabatic Formation of a Matched Beam Distribution

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 10 20 30 40 50 60

0

rms

b

X

X

Evolution of the beam envelope

σ/σvac=0.26, σvac=660

s/S

2 1 q

q sf x y q x yF V s x y V s s x y e e e e

Smooth- focusing

equilibrium with Quasiequilibirum

matched to a

quadrupole lattice

Adiabatic turn-off of the

uniform focusing field Adiabatic turn-on of the

oscillating focusing field

The average (smooth-focusing) effects of the total

focusing field is maintained fixed

Quiescent beam propagation over several hundred lattice periods has been

demonstrated for a broad range of beam intensities and vacuum phase advances

V(s)

s

1

0

Transition function

Quadrupole lattice (WARP simulations)

Dorf et al., Phys. Plasmas 16, 123107 (2009) 1

4

THfb

0exp

FFT of the beam envelope time evolution

Mismatch

modes

~6 orders of magnitude

14

Self-Similar Beam Density Evolution

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5

Quadrupole lattice σ/σvac=0.26, σvac=660 (WARP simulations)

0bx R

( )x a s

Regularly normalized coordinates

Scaled coordinates

-1.0

-0.5

0.0

0.5

1.0

1.5

56 57 58 59 60

Self-similar beam density evolution has been demonstrated for a broad range of

beam intensities. The self-similarity feature becomes less accurate for σvac~800. Dorf et al., Phys. Plasmas 16, 123107 (2009)

0

2

0

)()(

b

b

bn

n

R

sbsa

Beam

density

0b

b

n

n

Beam

density

Evolution of the beam envelope

0

rms

b

X

X

Lattice

function

Number of periods

κq(s)

sXsa rms2

sYsb rms2

Contour plots of the beam density (σvac=450)

1

5 15

ω1

ω2

Due to the nonlinear effects of

the beam self-fields particles

with higher energy move with

higher frequency

Rb

z

Matched

beam radius

Mismatch

oscillations

ωpart~ ωmis/2

Parametric resonant

interaction

Smooth-focusing approximation

Halo Production by a Beam Mismatch

Resonance

(O point)

Separatrix X point

R

VR

Radial Phase Space

core

halo

Energy transfer Collective motion Individual motion

(each particle is an oscillator)

Beam Halo

Cause degradation of the beam quality

Can provide activation of the chamber wall

One of the important problems:

How to quantitatively define

beam halo?

16

16

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1 2 3 4 5

Spectral Method for Halo Particle Definition

Bump-on-tail structure for ωβ>ωm/2 can be

attributed to beam halo particles

Smooth-focusing model (WARP simulations)

Beam is an ensemble of betatron oscillators

Initial

(matched)

distribution

Final

(mismatched)

distribution

Snapshot

Poincare plots

Radial phase-space

core halo

halo

halo core

FFT of a particle trajectory

Betatron

frequency

distribution

(a.u.)

R

R

R

R

qx

qx

25.0vac ωm/2

halo

Consider a mismatched intense beam with δRb/Rb=0.12

ωq=κsfVb smooth-focusing frequency

ωm frequency of mismatch oscillations

Beam

frame

17 17

M. Dorf et al Phys. Plasmas, 18, 043109 (2011).

Extension of the Spectral Method to the Case of a

Quadrupole Lattice

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Quadrupole lattice (WARP simulations)

Betatron

frequency

distribution

(a.u.)

qx

X

X

X

X

25.0vac

Bump-on-tail structure for ωβ>ωodd/2 can be

attributed to beam halo particles

ωodd/2 ωeven/2

Symmetric (even) and quadrupole (odd)

modes are simultaneously excited

halo

initial (matched)

final

(mismatched)

Transverse phase-space

Halo

removal

18 18 M. Dorf et al Phys. Plasmas, 18, 043109 (2011).

017vac

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0

Smooth-focusing approximation. Intense beam with σ/σvac=0.25 (WARP simulations)

We introduce a mismatch by an instantaneous increase in the smooth-focusing frequency

ωqf=1.4ωqi

Beam betatron frequency distribution

ω/ωqf

1.0

1.2

1.4

1.6

1.8

2.0

0 1000 2000 3000

Normalized emittance

ωm/2

ωqi t/2π

Most of the halo is generated relatively fast

Core relaxation process takes much more time

Core relaxation process also leads to emittace growth

Emittance after the

halo removal

Frequency distribution @ t=617*2π/ωqi

t=3140*2π/ωqi

f

i

Nu

mb

er o

f p

arti

cles

(ar

bit

rary

un

its)

Mismatch Relaxation in Intense Beams

19 19

III. Intense Ion Beam Transport through a

Background Plasma

Weak fringe magnetic fields (~100 G) penetrate deep into background plasma.

Can a weak solenoidal magnetic field has a significant influence on intense ion

propagation through a background plasma?

20 20

8T

Fringe magnetic fields

Target Final Focus

Solenoid

Schematic of the NDCX-I experiment

III. Ion Beam Propagation through a

Neutralizing Background Plasma Along a Solenoidal Magnetic Field

ion beam

plasma

B0

e- vb

21 21

References

[1] M. Dorf et al., Phys. Plasmas, 19, 056704 (2012) .

[2] M. Dorf et al., Phys. Plasmas, 17, 023103 (2010).

[3] M. Dorf et al., Phys. Rev. Lett., 103, 075003 (2009).

Enhanced Self-Focusing of an Intense Ion Beam Pulse

22

Weak magnetic field is applied Magnetic field is not applied

B0=0 B0~100 G

The ion beam space-charge is typically better compensated than the beam current

There is a significant enhancement of the ion beam self-focusing

effect in the presence of a weak solenoidal magnetic field.

e-

fM

fE

Veφ>0

δr>0

z

ωce>> ωpeβb

B0 beam

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 Net self-pinching force is produced due to the self-magnetic field

The total self-focusing force is given by

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

for

The effects of self-pinching is maximum for rb<<c/ωpe, and the focusing force is given by

pe

b

pe

ce

ce

b cr

V

2

2

1

drrnrn

Vmc

ZF b

p

be

pe

bsp

12

2

2

2

dr

dn

nVmZF b

p

bebsf

122

Fsf/Fsp~(c/ωperb)2>>1

zbeam

e- e-

e- e-

Self-pinching effect

selfB

22

Enhanced Self-Focusing is Demonstrated in Simulations

Gaussian 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=0

Magnetic 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 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

M. Dorf et al, PRL 103, 075003 (2009). 23

Electromagnetic Wave Excitation

PIC (LSP) Semi-analytic Analytic

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

Strong wave excitation occurs at

ωce/2βbωpe (supported by PIC simulations)

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

Transverse magnetic field (Whistler waves)

M. Dorf et al, Phys. Plasmas, 17, 023103 (2010).

IV. Collective Focusing Lens for Final Beam Focusing

25

Schematic of the present NDCX-I final focus section

drift section

(8 Tesla)

FFS

25

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

neutralized ion beam

e-, i+

magnetic lens

B0

IV. Collective Focusing of an Intense Neutralized Ion Beam Pulse

26 26

References

[1] M. Dorf et al., Phys. Plasmas, 18, 033106 (2011).

[2] M. Dorf et al., PAC Proceeding 2011.

Collective Focusing Lens Collective Focusing Concept (S. Robertson 1982, R. Kraft 1987)

27

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

The use of a collective focusing lens allows for decrease in the magnetic field by (mi/me)1/2

Conditions for collective focusing

r

VmBV

c

eeE

e

eer

2

0

2

rV cee From conservation of

canonical angular momentum

e

rmE eer

4

2The radial electric field is

given by

To maintain quasi-neutrality 2pe e

To assure small magnetic field

perturbations (due to the beam)

peb cr 2

27

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

B=500 G

L=10 cm

B=500 G

L=10 cm

β=0.01 β=0.01 β=0.01

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

Beam can drag neutralizing electrons from the drift section filled with plasma.

drift section

(FFS)

28

FFS

28

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, rb=1.6 cm,

Ib=27 mA Tb=0.094 eV,

Plasma parameters

(for the simulation II)

29

-10

-8

-6

-4

-2

0

2

4

0 0.4 0.8 1.2r (cm)

Er (k

V/c

m)

Radial electric field

inside the solenoid

Analytic

LSP (PIC)

35 266 271 281

Plasma

Final Focus Solenoid

(700 G)

2 c

m

beam

Conductivity

model

kin

etic

251

Tilt gap

z=8 -11

0

I simulation II simulation

z (cm)

np=1011 cm-3, Te=3 eV

M. Dorf et al., Phys. Plasmas, 18, 033106 (2011).

280 281 282 283 0

0.05

0.1

0.15

0.2

z (cm)

r (c

m)

nb (cm-3)

(a)

Beam density

@ focal plane

6×1012 cm-3

29

Conclusions

30

acceleration

and transport

neutralized

compression

final

focusing

• A numerical method for the adiabatic formation of a

quasi-equilibrium beam distribution matched to a

periodic focusing lattice was developed

• A spectral method for beam halo definition was proposed

• Enhanced self-focusing of an intense ion beam pulse

propagating through a magnetized plasma background

was found

• The feasibility of tight collective focusing of intense ion

beams for NDCX was demonstrated

Transport properties of intense ion beam pulse propagating in an ion

driver have been investigated, in particular:

30

ion

source

• Electromagnetic (whistler) wave excitation was analyzed

• Modeling of the ion beam extraction from the SMIS-37

ion source was performed

• Low-noise extraction regimes were identified

• Ion beam transverse emittance was analyzed