G.A. Emmert, J.F. Santarius and E.A. Aldersonkhachan/IEC2011/Presentations/Emmert.… · GAE & JFS 2011 Fusion Technology Institute, University of Wisconsin 8 Example Energy Spectra

Update on the VICTER Code for Modeling

Gridded, Spherically Symmetric IEC Devices*

Fusion Technology Institute

University of Wisconsin

*Research supported by the US Dept. of Energy

under grant DE-FG02-04ER54745

and the Grainger Foundation

G.A. Emmert, J.F. Santarius

and E.A. Alderson

US-Japan Workshop

Sydney, Australia

Dec. 7-8, 2011

GAE & JFS 2011 Fusion Technology Institute, University of Wisconsin 2

VICTER* Model

Anode grid

grounded

Neutral gas

Cathode grid at

potential –V0

Source region

Intergrid region

Ions enter from the

source region

X

X Fast neutrals

formed by charge

exchange and

dissociative

processes

Ions oscillate in the

potential well while

being attenuated

Ions formed in the

intergrid and cathode

regions by

ionization,

charge exchange,

and dissociative

processes

* Volterra Integral Code for Transport in Electrostatic Reactors


Basic Assumptions in VICTER

• Background D2 gas

• Spherical symmetry – ignore stalk, defocusing, and jets

• Prescribed electrostatic potential profile Child-Langmuir or vacuum potential in intergrid region Flat in the cathode region

• Deuterium (D+, D2+, and D3

+ ) ions enter from the source region

• D+, D2+ ions created in the intergrid and cathode regions by impact

ionization, charge exchange, and dissociation of fast ions colliding with the background D2 gas

• D- ions created by charge exchange processes

• Interactions occur without momentum transfer between nuclei; daughter products travel at the same speed as parent

• Collisionless ion motion between interactions


Atomic and Molecular Processes Included

D+ + D2 → D + D2+ charge exchange of D+

D+ + D2 → D+ + … stationary D+ production

D+ + D2 → D2+ + … stationary D2

+ production

D2+ + D2 → various products destruction of D2

+

D2+ + D2 → D+ + … fast D+ production

D2+ + D2 → D+ + … stationary D+ production

D2+ + D2 → D2

+ D2+ charge exchange of D2

+

D3+ + D2 → various products destruction of D3

+

D3+ + D2 → D+ + … fast D+ production

D3+ + D2 → D2

+ + … fast D2+ production

D3+ + D2 → D+ + … stationary D+ production

D3+ + D2 → D2

+ + … stationary D2+ production

Some of these processes are sums over various reaction channels.


Two Coupled Volterra Integral Equations

Determine the Source Functions, Si(r)

1,2 ,,2

1

anode

irdrSrrKrArSj

rjijii

Si(r) = number of ions born per unit volume per sec at radius r.

Ai(r) = slow ion source due to ions from source region

Sum over all generations of daughter ions and all ion passes

for D+ (i = 1) and D2+ (i = 2)


Kernel relates the Source at one Radius

to the Source at another Radius

K ij r, r ng ij E r, r r 2

r2

g j r, r Tc

2gcpj

r g j r, r

1 Tc

2gcpj r

r′ r

sum over passes gas density

cross-section for

producing i from j

cathode transparency

radius

grid

complete pass

probability

rS j rSi

Slow Source contribution:


The Ion Energy Spectra are Obtained

from the Source Functions

E e r r

2 2

2

( ) ( , )( , ) 4 4

1 ( )

in s ss s s

c cps

S r g r rF r E er eb h f r E e r

e T g r

r

2 22

2

2

( )( )4( , ) 4

( , ) 1 ( ) ( )

c cps c cpsout ss s

s c cps s

T g r T fS rerF r E eb h E e r

eg r r T g r f r

r

Inward traveling ions:

Outward traveling ions:

where

and s denotes the species (s = 1 (D+), s = 2 (D2+), and s = 3 (D3

+))


Example Energy Spectra of

D+ Ions Traveling Inward

100 kV, 100 mA, P=2 mTorr, rc=0.1 m, ra=0.25 m,

source D+:D2+:D3

+=0.06:0.23:0.71

D+ from

source region

Surviving D+

from source

region

Continuum

in energy of

D+ created

at rc < r < ra

Continuum-energy

D+ ions

Note: This figure’s z-axis is only

to f = 0.005

(The tops of the delta functions

are cut off.)

D+ from full-energy

D2+ and D3

+

passing through

cathode region


The Ions Produce Fast Atoms and Molecules by

Charge Exchange and Dissociative Processes

3

, ,

1

, ,in out in out

a s g sfa

s

S r E dE F r E n E dE

, ,

, ,, ,in out in out

a m a mF r E S r Er

Define a fast neutral atom source function:

Fast Atom and Molecule Energy Spectra is

gotten from solving:


Formation of Fast Neutral Atoms

and Molecules Included

D+ + D2 → fast D + …

D2+ + D2 → fast D + …

D2+ + D2 → fast D2 + …

D3+ + D2 → fast D + …

D3+ + D2 → fast D2 + …


Typical D Atom Energy Spectra

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

0 20 40 60 80

Sp

ectr

a (

A/m

2)

Energy (keV)

70 kV, 30 mA, 1.25 mTorr

at center

at cathode


Dave Boris and David Donovan Developed a

Low-Noise, Charged-Particle Detection System

1 MeV

tritons

70 kV, 30 mA, 1.25 mTorr 3 MeV

protons

•Examining either side of the double-

peaked spectra can yield center-of-mass

energy of the deuterium reactants

IEC

Isometric

FIDO: Fusion Ion

DOppler Shift

Diagnostic


FIDO diagnostic


Using VICTER to Simulate FIDO

Integrating over the detection cone and the solid angle

for protons to reach the detector gives

2 2

2 20

1

4

sin , ,cos cos

fido p p g f CM

cLi Lo

i o lab

p i p o

S E dE n E

V Vdr d F r E F r E dE

V V

Sfido(Ep) = number of protons detected at energy Ep per unit

energy per unit time.


Simulating the Proton Energy Spectra

0

50

100

150

200

250

2600 2800 3000 3200 3400

Cou

nts

(arb

itra

ry s

cale

)

Proton Energy (keV)

Experiment

Code

70 kV

30 mA

1.25 mTorr

3.8


0

50

100

150

200

250

2600 2800 3000 3200 3400

Cou

nts

(arb

itra

ry s

cale

)

Proton Energy (keV)

Experiment

Code

Simulating the Proton Energy Spectra

1

70 kV

30 mA

1.25 mTorr


“Line – Averaged” Deuteron Energy Spectra

To infer the “line – averaged” F(E) from the experimental

proton energy spectra, have to assume parallel or antiparallel

fusion events

CM

p

CM

p

lab VVV

70 kV, 30 mA,1.25 mTorr

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

10 20 30 40 50 60 70 80

F(E

) (a

rb

itrary

un

its)

Energy in lab frame (keV)

Exp.

Code


Time Of Flight (TOF) Diagnostic is an

Advancement on the FIDO concept

• Initiated by Boris and developed by Donovan

• 2 identical FIDO setups on opposite sides of HOMER

• Direct line of sight created through both arms and center of chamber

18


Donovan’s Time of Flight Diagnostic -

Comparison of Neutron Production Profile

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

1.E+11

0 10 20 30 40 50Neu

tron

so

urc

e d

ensi

ty

(arb

itrary

sca

le)

Radius (cm)

code

experiment

60 kV, 30 mA, 2 mTorr

anode radius


Negative Ions are a Recent Addition to VICTER

Negative Ion Processes included:

D+ + D2 → D- + 2D+

D2+ + D2 → D- + D+ + D2

+

D3+ + D2 → D- + 2D2

+

D + D2 → D- + D2+

D + D2 → D- + 2D + + e-

D2 + D2 → D- + D+ + D2

D- + D2 → D + D2 + e- (stripping )


Modeling Negative Ions

, ,, ,in out in out

N g species

species

S r E dE n F r T dT Energy spectra of positive

ions and neutrals

Cross section for producing

negative ions

, , , , ,in out

N NF r E S r E p r r E dr Survival probability

Negative ion source function

The negative ion energy spectra is then:

total energy of a negative ion


Survival Probability

Probability of a negative ion born at rʹ with total energy E

and reaching r is

The integral is along the path of the negative ion from the birth

point rʹ to r. There are three kinds of paths:

1. Purely outward motion

2. Inward, pass through the center to become outward

3. Inward, reflect at the turning point to become outward

See Alderson’s talk for details and experimental comparison

, , exp g stripp r r E n r dr


Summary and Conclusions

• The VICTER code can calculate the detailed energy

spectra of the various ion and neutral particle species as

a function of radius.

• Negative ions have added to the code.

• Comparison with experimental results:

Numerical energy spectra are in approximate agreement with

experimental results, except

Experimental energy spectra does not show the predicted

discrete spectra.

Calculated neutron production profile is more peaked than

seen experimentally.

Thank you for

your attention.


Theoretical Neutron Production Rate is in Reasonable

Agreement with Experimental Results

70 kV, 30 mA,1.25 mTorr, rc=0.1 m, ra=0.2 m

source D+:D2+:D3

+=0.06:0.23:0.71

• NB: need to include cold

ion recombination with

cold electrons to make

agreement “reasonable”


Proton Energy in CM and Lab Frames

CMV

CMV

pCMV

pCMV

CMV

CMV

center of cathode

center of cathode

Proton energy

upshifted

Proton energy

downshifted

pCM

plab VV

pCM

plab VV


Molecular Ions are Attenuated by Dissociation

and Charge Exchange with D2 Gas T

ota

l Io

n E

ner

gy

Radius, r

qf(r)

cathode anode

×

D2+, D2

D3+

D+, D

D+, D2+

event

Dissociation products formed with finite kinetic energy

turning points

Ions created with zero kinetic energy

Documents

G.A. Emmert, J.F. Santarius and E.A. Aldersonkhachan/IEC2011/Presentations/Emmert.… · GAE & JFS 2011 Fusion Technology Institute, University of Wisconsin 8 Example Energy Spectra