Radiation modeling for optically thick plasmas

Application to plasma diagnostics

D. Karabourniotis

University of Crete

GREECE

Plasma Light Model-Inventory workshop, Madeira, April 2005

Outline • General expression of spectral intensity Iλ from a plasma layer

• Εxpression of Iλ with constant line width

• Validity of calculating emissivity by means of a simple empirical radiation model based on the “inhomogeneity parameter”

• Numerical validation of a method for determining the “inhomogeneity parameter” from line-reversal

• Application to the temperature determination in a Hg-NaI lamp

General expression of side-on intensity

The equation of radiation transfer

Expression for the side-on intensity

Because of the plasma symmetry

I r dr r I dr

expR

R r

RI r r dr dr

0 0

2 exp2

coshR r

I r r dr dr

Expression of Iλ in terms of emissivity Kλ

η=n/g,

2

50

020

u

l

rhcr

I K

exp ulu l E kT

0

00

0

0

exp2

,, cosh

2 ,2

,

rR

R

R

L r Q r drU r Q r dr

L r Q r drp

L r Q r dr

K

Relative distribution functions

• For the absorbing atoms:

• For the emitting atoms:

• For the line profile:

• Column density:

0l lL r r

0u uU r r

,

0,0,

P r

PrQ

002 0,0lp C P

0

2R

l l r dr

Expression of Kλ in terms of the optical coordinate Y

• optical coordinate: 0

R R

rY r L r dr L r dr

2 00 1

0

1

0

,2 cosh 1

2 ,,

YQ Y dY

p Y dYQ Y dY

e Q YK

Y r U Y r L Y r

Assumption: • Expression of emissivity Kλ

• Condition for reversal at line peak,

,P r P

1

0

exp cosh 12 2

Y Y dYK

0 s 0dK d

Ks=K(λο±s) at the line peaks becomes a function of Λ(Y)

A simple empirical plasma model for line self-absorption

• Source function:

• Alpha: inhomogeneity parameter, with

1Y

0 0

R RL r dr U r dr

Ks=K(λο±s) becomes a function of alpha, α

Relationship between Ks and α

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Em

issiv

ity a

t m

axim

um

, K

s

Inhomogeneity parameter, alpha

Accuracy of the one-parameter approach for representing Ks better than 3%

Karabourniotis, van der Mullen (2004)

How the alpha can be determined from line contours?

• Contour of a self-reversed line and definitions

• δ: half-width of a Lorentzian line profile P(λ)

|s|/δ

Imax

Imin

s0

(λ- λ0)/δ0

Construction of a discharge model

• Absorbing atoms:

• Emitting atoms:

• Source function:

2 2

1 expa c b cr rL rR R

U r r L r

0

0

exp 1ulT rE TkT T r

Numerical experiment Inputs and outputs

rL rUIN

PU

TS

:O

UT

PU

TS

α(y)

total optical depth at λο along a plasma diameter

0 :

τ0(y)τs(y)

Ks(y)K0(y)

0 0 1ss y y y

L(r) U(r) τ0

Optical depth at λ0

s 0

max min

K y /K y =

I y /I y

Example: Decreasing L(r): a = 10, b = 20, c = 0.5; Parabolic T(r): T0=6000 K, Tw=1000 K; Eul = 3 eV

L

Λ

U

Input: τ0 =10 at y=0

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Inflexion point

Ks/

K0

y/R

-0.2 -0.1 0.0 0.1 0.2 0.30.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

log(

I max

/Im

in)

log(s0)

Dexp=1.05

At y/R=0log(Imax/Imin)exp=0.336

Inflexion point

y=0

Experimental data –contour characteristics

alpha

1.5

1.7 1.6

alpha

1.5

1.7

1.6

Dexp

log(Imax/Imin)exp

Contour characteristics calculated from the one-parameter approach for the source function

Inputs:α = 1.62Ks = 0.520

Results:α = 1.64Ks = 0.514

Karabourniotis, ICPIG-2005

Electron temperature in a 12-atm, 150-W Hg-NaI “standard” lamp

0 1 2 3 4 5 6 7 8 9 100.0

0.5

1.0

1.5

2.0

Imin

5461 Åy=0 mm

Ab

solu

te in

ten

sity

, I(

1013 W

m-3sr

-1)

Relative wavelength distance, (Å)

Imax

s

0 2 4 6 8 10 120

2

4

6

8

10

12

Relative wavelength distance, (Å)

s

Imin

Imax

5890 Åy=0 mm

Ab

so

lute

in

ten

sity,

I(1

012 W

m-3sr-1

)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

Ab

so

lute

in

ten

sity,

I(1

01

3 Wm

-3sr-1

)

Relative wavelength distance, (Å)

4047 Åy=0 mm

Imax

Imin

s

2.8 2.9 3.0 3.1 3.2 3.30.00

0.04

0.08

0.12

0.16

5461 Å

D=0.427, ΔD=0.037At y=0: α=1.213, Ks=0.752

log

(Im

ax/

Imin

)

log(s), s in mÅ

y=0

y=0.5

y=1

y=1.25y=1.5

1.10 1.15 1.20 1.25 1.30 1.35

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22lo

g(Im

ax/I

min

)

log(s), s in mÅ

D=0.596 ΔD=0.018

α=1.276, Ks=0.701

4047 Å

-0.5 0.0 0.5 1.04000

4400

4800

5200

5600

6000

Te

mp

era

ture

, (K

)

Radial distance, r(mm)

5461 4047 Te

T(5461)

T(4047)

Telectron

Telectron ≡ T(63P2,63P0)

Karabourniotis, Drakakis, Skouritakis, ICPIG-2005

Summary

• The emissivity at the peak of a self-reversed line is readily obtained if the inhomogeneity parameter (alpha) is known.

• The alpha-value can be deduced from the line contours. This was proved through plasma numerical-simulation.

• The distribution and the electron temperature were determined in 12-atm Hg-NaI lamp.