atomic physics effects on iec ion radial flow...various atomic physics processes play a role •it...

JFS / GAE 2006Fusion Technology Institute

John F. Santarius & Gilbert A. Emmert

Fusion Technology InstituteUniversity of Wisconsin

Japan-US Workshop on IEC FusionKansai University, Osaka

10-12 May 2006

Atomic Physics Effects on Convergent, Spherically Symmetric Ion Flow


Outline

One-population model

Source region model

Multiple-population model

Child-Langmuir equation


IEC Model

Anode grid

φ = 0

neutral gas

Vacuum wall

Source region

Intergrid region

Ion trajectory

Flux Γ0, Energy E0

Cathode grid

φ = -V0


Charge Exchange Produces Fast Atoms and Cascading Down of the Ion Energy

Tota

l Ion

Ene

rgy

Radius, r

qφ(r)

cathode anode

×

×

Ions

Atoms

CX event×

×


Ionization Produces Cold Ionswithout Loss of Energy

Radius, r

qφ(r)

cathode anode

×

×

Ions

Ionization event×

Tota

l Ion

Ene

rgy


Charge Exchange “Attenuates” Initial Ion Current as Ions Oscillate Radially

Frac

tion

of o

rigin

al c

urre

nt

Region of low ion energy, where

charge exchange has little effect on

the ion distribution.

Region where charge exchange creates a “2nd

generation,” low-energy ion

grid

Pass 1

Pass 2

Pass 3Pass 4

0

0.2

0.8

0.6

0.4

1

Radus (cm)0 5 10 15 20 25


Assumptions of the One-Population Model

• Background D2 gas• Deuterium ions (no molecular ions)

Collisionless motion except for charge exchange and ion impact ionization interactions

• Fast deuterium atoms Collisionless motion

• Prescribed electrostatic potential profileChild-Langmuir or vacuum in intergrid regionFlat in the cathode region

• Spherical symmetry – ignore stalk and defocusing• No electrons


Multiple Ion Passes Are Modeled Using anIntegral Equation Formalism

• Cold ion source function = S(r)• Attenuation function = g(r, r′ )

• Ion flux dΓ(r) at r due to ions born at r′

• Sum over all generations of cold ions and all ion passes

• A(r) = cold ion source due to ions from the anode

( ) ( ) ( ) rdrSrrgrrdr ′′′′=Γ ,22

( )( )

′′′′−=′ ∫

′r

r cxg rdrVnrrg σexp),(

gas density charge exchange cross-section

( ) ( ) ( ) ( )∫ ′′′+=anode

,

rrdrSrrKrArS


Kernel Relates the Source at One Radiusto the Source at Another Radius

( ) ( )( )( ) ( )

( )( )rgT

rrgrg

Trrg

rrrrEnrrK

cpc

cpc

totg ′−′′

+′

′′=′

2

2

2

2

1,

,,, σ

sum over passesgas densitytotal cross-section cathode transparency

grid

complete pass probability

S(r′ )S(r)r r′ radius


The Volterra Equation Gets Solved byFinite Difference Methods

• Set up a mesh in the intergrid region (the Volterra equation is only defined there).

• Calculate the attenuation coefficients in the intergrid region numerically and in the cathode region analytically.

• The ion current Γ0 leaving the anode is unknown experimentally and is adjusted to match the calculated cathode current with the measured value.

• The model reproduces the general trends of neutron production rate with changes in cathode current, cathode voltage, and gas pressure.

• The calculated neutron production rates are close to the measured values at low voltage and about a factor of 4 low at high voltage.


Cold Ion Source Function Can be Significantly Larger than the Ion Source at the Anode

1.E+17

1.E+18

1.E+19

1.E+20

1.E+21

0 0.05 0.1 0.15 0.2 0.25 0.3Radius (m)

S(r)

, (m

-3 s

-1)

A(r)S(r)

Deuterium fuel

Vc = 166 kV

p = 2 mtorr

Ic = 68 mA


Average Ion Energy Can Be SubstantiallyBelow the Anode-Cathode Voltage Difference

0

2

4

6

8

10

12

14

0 50 100 150 200Energy (keV)

f(E)*

10-1

6 (m

-2s-1

eV-1

)

InwardOutward

Deuterium fuel

Vc = 166 keV

p = 2 mTorr

Ic = 68 mA

• Energy Spectrum at Cathode


Single-Pass Ions at Low PressurePossess Nearly Their Full Energy at the Cathode

0.00E+00

5.00E+15

1.00E+16

1.50E+16

2.00E+16

2.50E+16

3.00E+16

3.50E+16

4.00E+16

0 10 20 30 40 50 60 70Energy (keV)

Ener

gy S

pect

rum

Inward

Helium fuel

Vc = 60 keV

p = 0.5 mtorr

Ic = 7.5 mA


Zero-D Model for the Source Region

• Treat the source region (outside the anode) using 0-D rate equations

• Atomic processes driven primarily by energetic electrons emitted from negatively biased (~ -200 V) filaments

• Goal is to calculate the mix of atomic and molecular ions entering the intergrid region (crossing through the anode)


Various Atomic Physics Processes Play a Role

• It turns out that only atomic processes involving D2 are significant. This is because D2 gas is, by far, the dominant species. Consequently, ionization of D2, dissociative ionization of D2, and interchange reactions are the dominant atomic processes.

• Processes included:Ionization, dissociation, interchange ( )Flow to wallsFlow through anode grid

DDDD 322 +→+ ++


Rate Equations for Source Region

+D ( ) ( )Vol

AACnVnnn

dtd wg

pp

+−= 11152011 2

1σ

( ) ( )Vol

AACnnnVnnn

dtd wg

pp

+−−= 2212021212021 2

1ασ+2D

( ) ( )Vol

AACnnnn

dtd wg +

−= 3312021231 21α+

3D

Ionization Interchange reactions

Flow to walls and grid

( ) iong IACnCnCne 2331221111 =++Ianode


Calculated Results for the Wisconsin IECSource Region at 2 mtorr

Species mix of current into intergrid region

80%

14%

6%

+3D+2D+D

Conclusion: we need to include molecular ions in the IEC model.


Many Reaction Chains Occur in the IEC Plasma

+2DFast +

3D Slow

+DFast +D Slow

+3D

E-accel

σ32f

σ31f

σ32s

σ31s

σ22s

σ21fs

σ12s

σ11s

σ21f

E-accel


We Simplify the Reaction Chains byAssuming Ions Are Always Born at Zero Energy

• Simplification: newly created fast ions are approximated as slow ions.

+2DFast +

3D Slow

+DFast +D Slow

+3D

E-accel

σ32s+ σ32f

σ22s

σ12s

σ11s

σ21s+ σ21fσ31s+ σ31f

σ31f

E-accel


Analysis Requires Coupled Integral Equations

• Note: only two species shown for simplicity.

( ) ( ) ( ) ( ) ( ) ( )drrSrrKdrrSrrKrArSb

r

b

r 2

121

1111 ,, ∫∫ ′+′+=

( ) ( ) ( ) ( ) ( ) ( )drrSrrKdrrSrrKrArSb

r

b

r 2

221

2122 ,, ∫∫ ′+′+=

Subscripts: 1 = D+ ions 2 = D2+ ions

Si(r) = number of ions of species i born at r per unit volume per unit time

Ai(r)= source at r of ions of species i produced by the D3+

ions from the source region


Kernels Include the Creation of the Same Species Plus Cross Terms for the Other Species

( ) ( ))(1

1),(

)(),(),(,

12

1

12

12

2

1111 rgTrrgrgT

rrgrrrrEnrrK

cpc

cpcsg ′−

′

′+′

′′=′ σ

( ) ( ))(1

1),(

)(),(),(,

22

1

22

22

2

2222 rgTrrgrgT

rrgrrrrEnrrK

cpc

cpcsg ′−

′

′+′

′′=′ σ

( ) ( ))(1

1),(

)(),(),(,

22

1

22

22

2

2112 rgTrrgrgT

rrgrrrrEnrrK

cpc

cpcg ′−

′

′+′

′′=′ σ

( ) ( ))(1

1),()(

),(),(,1

21

12

12

2

1221 rgTrrgrgT

rrgrrrrEnrrK

cpc

cpcsg ′−

′

′+′

′′=′ σ

… (for species 3)

fs 212121 σσ +=σwhere


Summary and Status

• The one-population model reproduces the general trends of neutron production rate with changes in cathode current, cathode voltage, and gas pressure.

• The calculated neutron production rates are close to the measured values at low voltage and about a factor of 4 low at high voltage.

• A source-region model indicates that molecular species are important, even at 2 mtorr; the amount of dissociation in transit is under investigation.

• Numerical solution of coupled Volterra equations is underway.

• Parametric variation cases will be run this Summer.

atomic physics effects on iec ion radial flow...various atomic physics processes play a role •it...

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