Heavy Ion Fusion Science
Virtual National Laboratory
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)
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
LLNL- PRES-635456
Motivation: Controlled Fusion
W. Sharp et al, http://hif.lbl.gov/tutorial/assets/fallback/index.html
2
Inertial Confinement Fusion:
Lasers Versus Ion Beams
Present approach – Laser Driven ICF
Promising alternative – Heavy Ion Fusion
Why ion beams?
High production efficiency and repetition rate
High efficiency of energy delivery and deposition
3
National Ignition Facility
(LLNL, Livermore, CA)
Ion-Beam-Driven High Energy Density Physics
Heavy Ion Fusion (Future)
Warm Dense Matter Physics (Present)
foil Presently accessible ρ-T
Itotal~100 kA
D-T
ρ~1000 g/cm3
T~ 10 KeV
τb~ 10 ns
Ib~1-10 A
τb~ 1 ns
Eb ~ 10 GeV
Eb ~ 0.1-1 MeV
Atomic mass ~ 200
Ions: K+, Li+
regime
ρ~1 g/cm3, T~0.1 ÷ 1 eV
corresponds to the
interiors of giant planets
and low-mass stars
Block scheme of an ion driver for high energy density physics
ion
source
4
acceleration
and transport
neutralized
compression
final
focusing
target
Neutralized Drift Compression Experiment (NDCX)
Heavy ion driver for Warm Dense Matter Experiments
Built and operated at the Lawrence Berkeley National Laboratory
5
Neutralized Drift Compression Experiment (NDCX)
Schematic of the NDCX-I experimental setup (LBNL)
Final Focus Target
Solenoid
8T
Fringe magnetic fields
Beam parameters at the target plane
K+ @ 300 keV (βb=0.004)
upgrade
Li+ @ 3MeV (βb=0.03)
Ib~2 A , rb < 5 mm (nb~1011 cm-3)
NDCX-II
Ib~30 A , rb~1 mm (nb~6∙1012 cm-3)
Ttarget ~0.1 eV
6
Ttarget ~1 eV
Weak fringe magnetic fields (~100 G) penetrate deeply into the
background plasma
Outline
I. Enhanced self-focusing of an ion beam propagating through a
background plasma along a weak (~100 G) solenodial magnetic field
plasma
ion beam
vb
e-
important for the design of a heavy-ion
driver (e.g. NDCX neutralized drift section)
can be utilized in ion beam self-pinch
transport applications (e.g. HIF drivers)
B0
II. Collective Focusing (Robertson) Lens
neutralized ion beam
e-, i+
magnetic lens
can be used for the ion beam final
focus (e.g. NDCX-I, II)
B0
perhaps can be utilized for collimation
of laser generated proton beams
Collective focusing with B0 ~ 1 kG is equivalent
to standard magnetic focusing with B0 ~10 T
7
I. Ion Beam Propagation through a
Neutralizing Background Plasma Along a
Solenoidal Magnetic Field
plasma
ion beam
B0
8
vb
e-
Magnetic Self-Pinching (B0=0)
The ion beam space-charge is typically well-neutralized
What about ion beam current?
e-e
r
beam
z

Bself
t
ee-

Bself
Ez
Inductive field
accelerates electrons
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)
Electron radial force balance
(a) rb<c/ωpe
(b) rb>c/ωpe
jb

Er  Bself
Vez c
Current neutralization
c/ωpe
|je|
rb
r
current is not-neutralized (locally)
maximum (jbxBself) self-pinching
9
jb=|je|
Vez Vb ~ Zb nb ne  1
Magnetic pinching is dominant
current is well-neutralized (locally)
negligible self-pinching

Er  Bself
Vb c
Enhanced Collective Self-Focusing (B0~100 G)
e
P  me rVe  rA  const
c
r
A  B0
2
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
This radial electric field provides enhanced ion
focusing
1 dnb
Fsf  Z meV
n p dr
2
b
for rb 
2
b
Vb
ce
ce2
1 2
 pe
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)
10
M. Dorf et al, PRL 103, 075003 (2009).
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
Radial electric field
Radial focusing force
60
Central
40
beam
slice 20
Bext=300 G
Beam density profile
LSP (PIC)
ωce/2βbωpe=9.35
R (cm)
0
0
Fr/Zbe
(V/cm)
2
4
6
8
-20
Analytic
-40
LSP (PIC)
-60
Bext=0
Bext=300 G
Magnetic self-pinching
Collective self-focusing
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
NDCX-I is negligible
NDCX-II is comparable to the final focusing of an 8 T short solenoid
11
Local Plasma Response is Drastically Different for
ωce>2βbωpe and ωce<2βbωpe
Moderate magnetic field
(ωce>2βbωpe)
-eEr
-
- FV×B
- -
e-
Weak magnetic field
(ωce<2βbωpe)
FV×B
δr>0
ˆ
zˆ
+
+
+ -eEr
+ +
B0
Veφ<0
Veφ>0
e-
δr<0
ˆ
zˆ
B0
Beam charge is overcompensated
Beam charge is under-neutralized
Radial electric field is focusing
Radial electric field is defocusing
Diamagnetic plasma response
Paramagnetic plasma response
ωce=2βbωpe
resonant excitation of large-amplitude whistler waves
np=1011 cm-3, βb=0.05
12
M. Dorf et al, PoP 19, 056704 (2012).
B0=100G
Numerical Simulations Demonstrate Qualitatively
Different Local Plasma Responses
ωce>2βbωpe
ωce<2βbωpe
B0=300 G (ωce/βbωpe=18.7)
B0=25 G (ωce/βbωpe=1.56)
Radial electric field
Radial electric field
LSP (PIC)
Focusing electric field
Diamagnetic response (δBz<0)
LSP (PIC)
Defocusing electric field
Paramagnetic response (δBz>0)
rb=0.55c/ωpe, lb=3.4rb, β=0.05, nb=0.14np, np=1010 cm-3
13
Whistler Wave Excitation
Transverse magnetic field
PIC (LSP)
Analytic
Analytical treatment of poles in
the complex plane
Magnetic pick-up loop
Signal amplitude
Schematic of the detected signal
Strong wave-field
excitation
Enhanced self-focusing
(local fields)
1
Semi-analytic
ωce/2βbωpe
beam velocity=phase velocity=group velocity
14
Numerical integration (FFT)
assuming weak dissipation
βb=0.33, lb=10rb, rb=0.9∙c/ωpe, nb=0.05np, Bext=1600G, np=2.4∙1011cm-3
Analytic theory is in very good
agreement with the PIC simulations
Strong wave excitation occurs at
ωce/2βbωpe (supported by PIC simulations)
Wave-field excitations can be used for
diagnostic purposes
M. Dorf et al, PoP 17, 023103 (2010).
II. The Use of Weak Magnetic
Fields for Final Beam Focusing
Schematic of the present NDCX-I final focus section
FFS
drift section
(8 Tesla)
Challenges:
Operate 8 T final focus solenoid
Fill 8 T solenoid with a background plasma
Can a weak magnetic lens be used for tight final beam focusing?
15
Magnetic Lens
magnetic lens
charged particle beam
q, vb
Br
-vφ
B0
vb
vb × Br
provides azimuthal
rotation (vφ)
Conservation of
canonical azimutal
(angular) momentum
Equation of motion
16
q
P  mr v  rA  const
c
r
A  B0
2
v2
v B
d 2r
m 2  m  q  0  Fr
dt
r
c
centrifugal
V×B
force Lorentz force
v φ × B0
provides radial
focusing
r
v   c
2
m rc2
Fr  
4
Cyclotron frequency
c 
qB0
mc
Collective Focusing Lens
Collective Focusing Concept*
Electron beam
focusing
e
Neutralized
beam focusing
Ion beam
focusing
i
β=0.01
i+e
β=0.01
β=0.01
Le ~0.01 mm
Lc=(LeLi)1/2 ~ 1 cm
Li ~ 10 m
B=500 G
L=10 cm
B=500 G
L=10 cm
B=500 G
L=10 cm
Collective Focusing Lens
neutralized ion beam
e-, i+
magnetic lens
B0
The use of a collective focusing lens reduces the
required magnetic field by (mi/me)1/2
17
*S. Robertson, PRL 48, 149 (1982)
Collective Focusing Lens (Cont’d)
Electrons traversing the region of
magnetic fringe fields acquire an
azimuthal velocity of
Ve   e
r
2
e 
eB0
me c
Strong ambipolar electric field is produced to balance the magnetic V×B force
acting on the co-moving electrons
2
Ve
e
eEr  Ve B0  me
c
r
Electric
force
V×B magnetic
force
Centrifugal
force
E r  me  e2
r
4e
Conditions for collective focusing
1. Quasi-neutrality
 pe   e
2. Small magnetic
field perturbations
rb  2 c  pe
18
2
3. No pre-formed plasma inside the lens
Collective Focusing Lens for a Heavy Ion Driver Final Focus
Schematic of the present NDCX-I final focus section
drift section
FFS
(FFS)
The ion beam can extract neutralizing electrons from the drift section filled with plasma.
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
19
Numerical Simulations Demonstrate Tight Collective Final Focus for NDCX-I
Schematic of the NDCX-I simulation R-Z PIC (LSP)
Beam injection parameters:
Plasma
35
2 cm
beam
0
Conductivity
model
251 266
I simulation
271
281
Plasma parameters
(for the simulation II)
z (cm)
np=1011 cm-3, Te=3 eV
II simulation
Beam density
@ focal plane
Radial electric field
inside the solenoid
nb (cm-3)
cm-3
Er (kV/cm)
2
0.1
0.05
2.0
0
1.5
-2
-4
LSP (PIC)
-6
Analytic
-8
0
280
281
282
283
z (cm)
M. Dorf et al, PoP 18, 033106 (2011)
-10
0
0.4
0.8
r (cm)
1.2
Ib (A)
6×1012
0.15
20
Beam current
@ focal plane
4
0.2
r (cm)
K+ @ 320 keV, rb=1.6 cm,
Ib=27 mA Tb=0.094 eV,
Final Focus Solenoid
(700 G)
kinetic
Tilt gap
z=8 -11
1.0
0.5
0.0
2565
2576
2585
time (ns)
2595
Conclusions
Even a weak magnetic field (several hundred gauss) can have a
significant influence on neutralized ion beam transport.
plasma
neutralized ion beam
ion beam
e-
e-, i+
magnetic lens
B0
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 the ion beam
final focus (e.g. NDCX-I, II)
Can be utilized for self-pinch ion beam
transport applications
Perhaps could be utilized for collimation
of laser generated proton beams
Enhanced Self-Focusing VS Collective Lens
Enhanced self-focusing
plasma
Collective Lens
magnetic lens
neutralized ion beam
e-
ion beam
e-, i+
B0
B0
Fself  Z b2 meVb2
Non-linear force
can effectively balance p~Tb n
(important for self-pinch transport)
np=1011 cm-3, βb=0.05
ωce>>2βbωpe
Vb
ce
1
Linear force
(important for beam focusing)
 pe   e
Conditions:
rb  rge 
r
Fcol   mi  e i
4
1 dnb
ne dr
Conditions:
rb  2 c  pe
B0>100 G
ce2
2
 pe
rb>1 cm
no plasma inside the lens
In the limit of rb~rge and Zbnb~ne → Fself~Fcol
22
2