Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 1

Introduction

Progress of CR modeling for molecules relevant to fusionD. Wünderlich, U. Fantz

424 426 428 430 432 434

λ [nm]

Hγ

424 426 428 430 432 434

λ [nm]

Hγ

426 427 428 429 430 431 432

CH (A2∆→X2Π)

λ [nm]

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 2

Molecules in fusion experiments

Wall materials• ITER: Be, W and C

• ASDEX Upgrade: W

• ITER like wall of JET: Be, W

• Boronization of the walls (impurities, recycling)

Low temperatures in the plasma edge ⇒ formation of molecules

Atom: Discrete linesCarbon

Beryllium

Tungsten

Recycling at the wall H2, D2, T2, HD, HT, DT

Plasma wall interactionCH, CD, CT, C2

BeH, BeD, BeTBH, BD, BT

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 3

CR models for molecules

Split-up of electronic energy levels due to vibrational and rotational excitation )''J,'J(A)'J(n ''n'n

''v'v'n'v ⋅=ε

..

n, v, J

n’, v’, J’

n’’, v’’, J’’

Ion

Radiation

Ground state

Excited state .

...

...

...

...

CR models for moleculesmuch more complex than for atoms

Vibrationally and rotationally resolved cross sections and transition probabilities

• Coupling with molecules, atoms or ions (e.g. dissociative excitation or recombination)

• Isotope effect

• Low intensity of single lines ⇒ no relevance of optical thickness

• Thermalization ⇒ Rotational levels neglected in most CR models

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 4

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

X1Σ+g

Singlet system

Pot

entia

l ene

rgy

[eV

]

Internuclear distance [Å]

H+2

EF1Σ+g

B1Σ+u

C1Πu

GK1Σ+g

B'1Σ+uJ1∆g

D1Πu

I1Πg

HH1Σ+g

c3Πu

b3Σ+u

a3Σ+g

e3Σ+u

j3∆g

h3Σ+gi3Πg

d3Πu

g3Σ+g

Potential curves of H2

Born-Oppenheimer approximation Electronic and nuclear motion can be separated due to mass ratio

⇒ Separate treatment of electronic, vibrational (and rotational) excitation

Potential curves: eigenvalues of electronic wave functions

)(),(),( RrRrR Ψ⋅Υ=Φ

Repulsive state

v=0 metastable

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 5

TraDiMo for BH

0 1 2 3 4 5 6 7

0

0.2

0.4

0.6

0.8

1.0

01

234567

B1Σ+→X1Σ+

FCF

v(B1 Σ

+ )

v(X 1Σ +)

0 1 2 3 4 5 6 7

0

0.2

0.4

0.6

0.8

1.0

01

234567

A1Π→X1Σ+

FCF

v(A1 Π)

v(a 3Π

u )

1 2 3 4 5

0

1

2

3

4

5

6

7

A1Π

BH

Ene

rgy

[104 c

m-1]

Internuclear distance [Å]

X1Σ+

B1Σ+

Potential curve of B1

state: double well

2'FCF ΨΨ=

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 6

TraDiMo for C2

0 1 2 3 4 5 6 7

0

10

20

30

40

50

60

70

01

234567

Mulliken band: D1Σ+u→X1Σ+

g

Aik [1

06 s-1

]

v(D1 Π

+u)

v(X 1Σ +

g )

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

7

01

234567

Swan band: d3Πg→a3Πu

Aik [1

06 s-1

]

v(d3 Π g

)

v(a 3Π

u )

1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

11

d3Πg

c3Σ+u

b3Σ-g

a3Πu

D1Σ+u

C1Πg

B'1Σ+g

A1Πu

C2

Ene

rgy

[104 c

m-1]

Internuclear distance [Å]

X1Σ+g

B1∆g

2el''v'v ''M'A ΨΨ∝

Singlet system andtriplet sytem

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 7

TraDiMo for CH

1 2 3 4 5

0

1

2

3

4

5

B2Σ-

C2Σ+g

A2∆

CH

Ene

rgy

[104 c

m-1]

Internuclear distance [Å]

X2Π

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

02

46

8B2Σ−→X2Π

Aik [1

06 s-1

]

v(B2 Σ

− )v(X 2

Π)

0 1 2 3 4 5 6 7 8 9

0

0.5

1.0

1.5

2.0

02

46

8

Aik [1

06 s-1

]

v(A2 ∆)

v(X 2Π)

Gerö band: A2∆→X2Π

0 1 2 3 4 5 6 7 8 9

02

4

6

8

10

12

02

46

8C2Σ+→X2Π

Aik [1

06 s-1

]

v(C2 Σ

+ )

v(X 2Π)

Different shape of potential curves

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 8

Excitation rate coefficients: forbidden transitions

More accurate data for forbidden transitions highly desirable

Gryzinski methodBased on Franck Condon Factors, low accuracy

• In principle very simple method

• Some obscurities, like definition of “next allowed level” (difficult in molecules)

• Calculations for H2, C2, CH

1 20 40 60 80 10010-28

10-26

10-24

10-22

10-20

Gryzinskicalculations

v'=1

v'=4

v'=2

v'=3

σ [m

2 ]

Ee [eV]

v'=0

C2: X1Σ+

g(v=0)→d3Πg(v')

v=0

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 9

Excitation rate coefficients: allowed transitions

Impact parameter methodBased on transition probabilities, very accurate

• IPProg: simple tool to calculate rate coefficients

• Good agreement with Born-Bethecalculations*

• Calculations for H2, C2, CH, BH

Sufficient data basis for allowed transitions

*: R. Celiberto et al, Plasma Phys. Control. Fusion 51, 2009, 085012

1 2 3 4 5 6 7 8 9 1010-18

10-17

10-16

10-15

10-14

10-13

0-0

CH: X2Π - C2Σ+

CH: X2Π - B2Σ-

1-1

1-0

0-0

Rat

e co

effic

ient

[m3 /s

]

Electron temperature [eV]

0-1 (*10)

1-0

0-0

1-1

0-0

1-0

0-1

CH: X2Π - A2∆

BH: X1Σ+ - A1Π

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 10

Yacora: CR model for CH

0

1

2

3

4

5

6

C2Σ+

B2Σ−

A2∆

E [e

V]

X2Π

Collisions, IPProg Radiation, TraDiMo Ionization, Gryzinski

Yacora CH

10-6

10-4

10-2

100

10-9 10-7 10-5 10-3 10-1 10110-10

10-8

10-6

10-4

A2Σ+(v=0)B2Σ−(v=0)

v=5

v=4

v=3

v=2

v=1

A2∆(v=0)

v=8v=7

v=9

v=6

v=5

v=1

v=4

v=2

v=3

Rel

ativ

e po

pula

tion

v=0

X2Π(v)

v=2

v=1

Simulated time [s]

Rel

ativ

e Po

pula

tion

Gerö band (A2∆→X2Π) at 431 nm

• Temporal evolution of population densities

• Different typical time scales, interplayof electronic and vibrational population

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 11

Yacora: CR model for C2

0

1

2

3

4

5

6

d3Πg

c3Σ+u b3Σ−

g

C1Πg

B'1Σ+g

A1Πu

a3Πu

E [e

V]

X1Σ+g

Collisions, IPProg Radiation, TraDiMo Collisions, Gryzinski Ionization, Gryzinski

D1Σ+u

B1∆g

Yacora C2

0.0

0.2

0.4

0.6

0.8

1.0

1015 1016 1017 1018 1019 10200.0

0.2

0.4

0.6

0.8

1.0 depopulation

population

electron impactionization

electron impactde-excitation

electron impactde-excitation

electron impactexcitation

electron impactexcitation

Spontaneous emission

Spontaneous emission

Rel

ativ

e co

ntrib

utio

n

A1Πu(v=0)

ne [m-3]

Rel

ativ

e co

ntrib

utio

n

Swan band (d3Πg→a3Πu) at 516 nm

Mulliken band (D1Σu→X1Σg) at 231 nm++

• Strong influence of plasma parameters on relevance of excitation and de-excitation processes

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 12

Application of CR models: effective rate coefficients

0.0

0.2

0.4

0.6

0.8

1015 1016 1017 1018 1019 10200.0

0.5

1.0

1.5

2.0

Corona case

Effe

ctiv

e em

issi

on ra

te c

oeffi

cien

t [10

14 m

3 /s]

Gerö: A2∆→X2Π (0-0)

B2Σ−→X2Π (0-0)

CH

Corona case

Electron density [m-3]

Swan: d3Πg→a3Πu (0-0)

Mulliken: D1Σ+u→X1Σ+

g (0-0)

C2

10-10

10-9

10-8

10-7

10-6

Den

sity

ratio

: n(d

3 Πu)/n

0

H2: d3Πu

D2

measuredH2

Miles/Sawada

Janev

1 2 3 4 510-11

10-10

10-9

10-8

10-7

D2

Den

sity

ratio

n(I

1 Πg)/n

0Electron temperature [eV]

H2: I1Πg

measured

Janev

Miles/Sawada

H2

H2: drastically improved agreement of measured and calculated population densities by exchange of input data

CH/C2: Deviation from corona model in a wide parameter range, application of the CR model mandatory

Molecular CR Modeling for Fusion, Ringberg, 6. October 2009D. Wünderlich 13

Overview and conclusions

Electron impact cross sectionsIPProg, Gryzinski method

FCF, Transition probabilitiesTraDiMo

Potential curvesLiterature

CR modelYacora

C2, CH, BH, BeHH2

C2, CH, CD, CT, BH, BeHH2, D2, T2, HD, DT

C2, CH, BH, BeHH2

C2, CHH2

(some transitions)

• Proved set of codes for generating input data and CR models

• Foundation for CR modeling of fusion relevant diatomic molecules

• Additional effort necessary (e.g. replace Gryzinski cross sections)