2004

Simulationsrechnungen zum Niederenergie-Ionenspeicher

Dr.-Ing. Martin Droba

2004

2.

Inhalt:

• Magnetische toroidale Systeme

• Geometrie

• Magnetfeld

• Teilchensimulation

• Ausblick

2004

3.

Magnetische Systeme mit toroidaler Symmetrie

• Gyrationsbewegung

• Drift

BgradF

eRmvF

FFF

BF

B

rc

Bc

rr

rr

rrr

rr

⋅−=

⋅=

+=

×

∇

∇

µ

/2

||

mqBg /=ω

B

Ladungstrennung aufgrund der Drift

-

2004

4.

Theorie

Driftgeschwindigkeit und Gyrationsfrequenz bei einer Feldkrümmung:

Bei konstanter Steigung a=Bz/Blong in z-Richtung

qBmvrm

qBvv

qBR

mv ggd /,,

2

2

2

||0 ⊥⊥ ==

+= ω

2

0201,

1

1a

avv ggdd −=

−= ωω

2004

5.

Geometrie

Parameter:

h = 0.6 m

r = 0.25 m

R = 1 m

Steigung:a) Linear

b) cos-Funktion

°= 72ϕB

r

R

h

ϕ

2004

6.

Teilchensimulation

• Feldberechnung – Biot-Savart Gesetz

• Bewegungsgleichung

∫′−

′−×=

3

)(

4 rr

rrldIB rr

vrrr

π

µ

Bvqtd

rdm

rrr

×=2

2

Input

Position Feld

Graph

2004

7.

Numerische Methoden

• Numerische Integration

• FDTD – Methode

„leap-frog“ Verfahren

=∆

+∆

−∆

+++

t

z

m

qB

t

y

m

qB

t

x nny

nnz

n

22

1

,

1

,2

1

t

z

m

qB

t

y

m

qB

t

x

t

x nny

nnz

nn

∆+

∆−

∆−

∆−−−

22

2 1

,

1

,2

1

2

n

n-1

n+1

zeit

xx yy zzBxBx ByBy BzBz

2004

8.

Magnetfeld

• Symmetrisch

• 2% Schwankung auf der Achse

• 1/R – Abhängigkeit in radialer Richtung

0 2 4 6 8 10 12 144,7x100

4,8x100

4,8x100

4,9x100

4,9x100

5,0x100

5,0x100

5,1x100

Bs

[T]

s [m]-2 0 2 4 6 8 10 12 14

4,925x100

4,938x100

4,950x100

4,963x100

4,975x100

4,988x100

5,000x100

5,013x100

5,025x100

Bs

[T]

s [m]

0,00 0,05 0,10 0,15 0,204,0x100

4,2x100

4,4x100

4,6x100

4,8x100

5,0x100

8ring500 windings1000 steps...@s=2m phi=pi/2 (horizontal plane)cos function in z plane

Bs

[T]

r[m]

2004

9.

Rotation-Symmetrie

Drehung der magnetischen Flächen

2004

10.

Einzelnteilchenbewegung

0,0 1,0x10-5 2,0x10-5 3,0x10-5 4,0x10-5

0,92

0,94

0,96

0,98

1,00

1,02

1,04

1,06

1,08

1,10

1,12

Field as a function of particle position

ma

gne

tic fi

eld

B [T

]

time [s]

geometrische Achse

Teilchenbahn

FxB Drift

FxB Drift

B-Feld

• Kurzzeit-Simulation

(1-10 Umläufe)

• Stabile Bewegung

• Übereinstimmung mit der Theorie

2004

11.

Einschussebene

-0,15 -0,10 -0,05 0,00 0,05 0,10 0,15

-0,10

-0,05

0,05

0,10

0,15

S

vlong/vtrans = const

W = 150 keV

B = 5 T

R = 1 m

R1=0.25m

geometrische Achse

2004

12.

Einschusswinkel

vlong/vtrans = const

W = 150 keV

B = 5 T

R = 1 m

R1=0.25m

0,00

0,02

0,04

0,06

-0,04 -0,02 0,00 0,02 0,04

Einschusswinkel: 0°15°30°45°60°

Energie W = 150 keV

Einschusswinkel

geometrische Achse

2004

13.

Ausblick

• Mehrteilchensimulation

• Raumladung

• Form der Magnetspulen

• Stabilität

2004

14.

New parameter set

• B=1T, r=0.25m, R=0.5m

• Electrons Protons

HzmqBg

111076.1/ ⋅==ω HzmqBg

7106.9/ ⋅==ω

][107.5/12

mvvqBmrg ⊥−

⊥ ⋅⋅=⋅= ][1004.1/8

mvvqBmrg ⊥−

⊥ ⋅⋅=⋅=

+

−

⋅=

+= ⊥

−⊥

21

107.5

2

22

||

2

2

1222

||0

vv

c

v

vv

qBR

mvd

+⋅=

+= ⊥−⊥

21004.1

2

2

2

||

8

2

2

||0

vv

vv

qBR

mvd

[ ]BEB

vE ×=2

1

2004

15.

Experiments and Theory of non-neutral plasma

• W. Clark (Maxwell Lab., California, USA, 1975)

- relativistic electrons, densities ~ 1010 cm-3, potential well ~ 300 kV

- toroidal magnetic field, 7.8 kG, rise time 2 - 3.5 ms, + vertical field for stabilizing

- R=0.5 m, r=0.081 m, Al shell 2.1 cm, vacuum 10-9 Torr

- Diocotron modes

- 2 probes

1. Circular stain-less disk mounted flush with the vacuum wall, but electrically insulated from it…terminated 50 Ohm cable to osciloscop…imaging charges detection

2. High impedance voltage probe….0.5 mm wire covered with an insulated glass sheath except for the tip which is exposed to the electron cloud, measuring potentials up to ~ 100 kV

- oscillation after 200 µs ion resonance instability ?

BRrQf 0

38/ επ≈

2004

16.

• A. Mohri (Nagoya University, Japan, 1975)

- 2 stage Marx generator (100 kV, 5 kJ)

- Magnetic field ~ 7 kG, + vertical magnetic field

- Lifetime of the beam current fluctuated from run to run…longest ~ 20 µs, beam was not hollow

- Vacuum 6x10-7 Torr, Icurrent=300 A, density ~ 8x108 cm-3, potential well ~ 1.1x104 V

- Radial electric field ~ 4x103 V/cm

- Dump not observed in high gas pressure ~ 0.2 Torr

- Current at the onset of the dump was nearly constant over a wide range of pressure

- Ion resonance instability ? (Buneman, Levy, Daugherty)

- Low ion density azimuthal wave mode l=1

- higher vertical field displacement to major axis excentricity diocotron oscillations hard-x-ray bursts second instability

ieEiccEee meZnBen 0

222

1

22

0 2/,2

1

4

12/ εεω =ΩΩ=Ω−

Ω+Ω≅=

2004

17.

• Puravi Zaveri (Bhat, India,1991)

- Electrons should be lost due to curvature drift

- But strong ExB drift ->rotation overcomes curvature drift

- Electrostatic hoop forces tend to expand ring

- Equilibrium -> closer to the inboard conducting shell

- Strong toroidicity->strong distortion of the surfaces ->large elipticity and

triangularity

- B <40,150>G

- Pressure 4x10-7 Torr, I=150 mA, density=108cm-3,confinement time 2-2.5 ms

- Theory->Only 3µs to reach the chamber wall due to (grad B) drift

- Diocotron instability

- Charge injection 15µs, grad B drift ~ 106 m/s, ExB drift ~ 108 m/s

- Single particle to collective behaviour (Icritical=1 mA)

2004

18.

• T.S.Pedersen (Colombia University, New York, USA,2002)

-Diocotron modes can be stabilized either by Landau damping

-or by magnetic shear

-electrical current -> change in the magnetic field

• K. Avinash ( Bhat, India,1991)

- drift surfaces shifted toward the major axis ~ r/R(ωp/ωc)2

- in the absence of the conducting shell, the force along major radius could be balanced by an externally applied electric field

- question: as the walls of vessels are never perfect conducting, what happens to the shifted equilibrium in the presence of finite resistivity of the walls? resistivity destroy imaging charges grow of diocotron modes instability

( )2

2

///

≈

c

Bec

annBB

ωδ

)2/(,)/(,1/)/(2

0

2

0 eBeeDBD mBnneTnnR εελιλ ==<<⋅

2004

19.

• M. R. Stoneking ( Lawrence University, Wisconsin, USA, 2001)

- electron plasma, partial torus, horizontal electric field E=5-10 V/cm (larger than

by Bhattacharyya ~ 1.4 V/cm for the same charge ~ 8.1 nC ~5x1010 electrons)

- Vacuum chamber 3mm Al, square poloidal cross-section, R=0.5m

- Electron poloidal ExB drift frequency 240 kHz

- B = 196 G, typically energy W=100-200 eV

- Coils 24 evenly spaced bundles of 4 turns each, connected in series

- 2 interleaved sets (12 bundles each) are wound with opposing helicity net

toroidal current = 0

- Electron source – 0.5mm tungsten spiral – 22 turns - I=10.2A - T=1900K

- Pressure 10-6 Torr, I=150 mA, density=3.1x106cm-3,confinement time 0.1 ms

- At small electric field no evidence of traping

- Some evidence of low frequency oscilation ion resonance possible

diocotron modes modification with horizontal electric field

2004

20.

Magnetic coordinates for stellarator fields

• X.Bonnin (Greifswald, Germany, 2003)

- Boozer-like coordinates field line trace

- Definition of flux surfaces

- Rotational transformation ι

- Toroidal and poloidal currents flow

• F.Alladio (Frascati, Italy, 1995)

-

- covariant representation (assumption of scalar pressure MHD equilibrium j flows along flux surfaces )

- Magnetic field on flux surface

- Contravariant representation

∫= ldBrr

0χ

ldBdrr

=0χ

TB ψβχ ∇+∇=rrr

0

0=∇⋅ Tj ψrr

0||χ∇=rr

B

0θψ ∇×∇=rrr

TB