Impact of the one-parameter approximation on the shape

of optically-thick lines

COST-529, Meeting at Mierlo, March 2006

D. Karabourniotis

University of Crete

GREECE

Plasma spectroscopy: an open problem

Diagnosing high-pressure discharges by optically-thick lines is an asymptotic process: starting somewhere get a first picture, which gradually refines

A correct interpretation of diagnostics results need an understanding of the plasma as it helps to improve this understanding

Diagnostics is yet an open problem due to a lack of fundamental physical knowledge (understanding)

In this sense plasma spectroscopy is yet an open problem

Outline √ Expression of line intensity and in terms of reduced functions for the plasma structure and the line profile

√ One-Parameter Approximation (OPA) for the source function

√ Validity of OPA to represent emissivity in case of position dependent line profile

√ Numerical examples of line shapes and optical-depth profiles

√ Experimental line shapes and optical-depth profiles

√ Numerical tests for the determination of the inhomogeneity parameter using the OPA model

Intensity of a spectral line

Symmetric plasma layer

0

I J

+xo-xo 0x

Iν

Emissivity

Density of the Upper Density of the Lower

3020

20 0 u

lu l

h gx x

gcJ n n

Planck law

0 0

2

0

e , cosh 1 ,2

x x

x

x x x dx dx

x L x U x 0l lL x n x n 0u uU x n x n

0

0

,,

,x

L x Q xx

L x Q x dx

0, , ,0Q x P x P

,P x : Position-dependent line profile

Line emissivity in terms of x

1 1

00 0, ,s L x Q x dx L x Q s x dx

Case of the Lorentz profile

2 2

0

1 ( ),

( ) ( )

xP x

x x

( ) ( )x x c

2 20

, ( ),

,0 ( ) ( )

P x xQ x

P w x x

0( ) ( )x x 0 0w

Relative Lorentz profile

0 ( 0)x

Line emissivity in terms of y

1

2

0 0

e , cosh 1 ,2

y

x y y dy dy

1

0

, , ,y Q y y dyQ

0

0

0

x

x

x

L x dx

L x dxy

+xo-xo 0 x

Iν1 02y

, ,Q y Q x

, ,y dy x dx

Self-reversed lines

00d

at sd

Condition for reversal permits determination of τs=τ(ν0+s)

Ks=Κ(ν0+s) becomes a function of Λ(y) and Q (s0 ,y)

Emissivity at the line maximum

ν0ν0+s ν

Bleu wingRed wing IM

Im

One-Parameter Approximation (OPA)

1y y

y L y U y

0 0

0 0

x x

L x dx U x dx

α (alpha) = inhomogeneity parameter

Validity of the OPA to represent Ks

Case:

• Position independent line profile, P(λ,χ)=P(λ)

• Position dependent line profile, P(λ,χ)

Atomic collision broadening

Electronic collision broadening

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 (OPA) for representing Ks when P(λ,χ)=P(λ) better than 3%

Karabourniotis, van der Mullen (2005)

Κi/Κd~1.03

Κi/Κd<1.003

δ=c → 1

2

0

e cosh 12

y y dy

2 4 6 8

0.47

0.48

0.49

0.5

0.51

0.52

Atomic collision broadening

0

0.7

0( )x T x T

Decreasing L(x)

Parabolic T(x), α=1.62

s0=s/δ0

Κs

(i)

(d)

δ = c

12.60 Without shift :

0.73

2 4 6 8

0.46

0.48

0.5

0.52

0.54

0.7

0( )x T x T

Decreasing L(x)

Parabolic T(x), α=1.62

0.73

s0=s/δ0

Κs

δ = c

With shift :

(i)

(d)

2 4 6 8

0.26

0.27

0.28

0.29

0.3

2 4 6 8

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.73

Κs

s0=s/δ0

α=2.64

α=6

δ = c

δ = c

Hollow L(x)

Increasing L(x)

(i)(d)

(i)

(d)

2 4 6 8

1.01

1.02

1.03

1.04

1.05

Atomic collision broadening

Κi/Κ

d

s0=s/δ0

Decreasing L(x), α=1.62

Hollow L(x), α=2.64

0.7

0( )x T x T

Parabolic T(x)

5 12

00

1 1( ) exp

2

Ix T x T

k T x T

1.74, 6I eV

0 2 3 4 5 6 70.94

0.95

0.96

0.97

0.98

0.99

1

Electronic collision broadening

s0=s/δ0

Decreasing L(x) α=1.62

Κi/Κ

d

-4 -2 2 4 6

5

10

15

20

25

Atomic collision broadening

τ(ν)

-10 -5 5 10

0.1

0.2

0.3

0.4

0.5

w=(ν-ν0)/δ0→

Κ(ν)

Decreasing L(x), Parabolic T(x), s0 = 4, η = 0.73 (α=1.62)

-10 -5 5 10

0.025

0.05

0.075

0.1

0.125

0.15

-4 -2 2 4 6

2

4

6

8

10

12

14

Increasing L(x), Parabolic T(x), s0 = 4, η=0.3 (α = 4.1)

τ(ν)

Κ(ν)

w=(ν-ν0)/δ0→

Electronic collision broadening

-2 -1 1 2 3 4

10

20

30

40

50

60

-10 -5 5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Decreasing L(x), Parabolic T(x), s0 = 4, η =1.74 (α=1.62)

w=(ν-ν0)/δ0→

τ(ν)

Κ(ν)

Experimental I(λ) and τ(λ)

exp

I I

I

M/Chr

L

Spher. Mir.

Neutr. Filter Choper

Uncertainty:

Neutral-filter absorbance 90%

τNF=4.5, Δτ/τ=4.6%

-2 -1 0 1 2 3 40

1

2

3

4

5

0

1

2

3

4

5

6

7

Inte

nsi

ty (

a.u

.)

Δλ(Å)

Hg-5461

Optic

al d

epth

PHILIPS: R=6 mm, Ig=18 mm, 7.14 mg Hg, 100mbar Ar/Kr,

150 W, 2.7A, P~3 atm

Karabourniotis, Drakakis, Palladas XX ICPIG, 1991

Hg- 5461

-4 -2 0 2 4 6 8 10 120

2

4

6

8

10

0

1

2

3

4

5

6

Inte

nsi

ty (

a.u

.)

Δλ(Å)

Tl-5350

Op

tica

l de

pth

OSRAM: R=9 mm, Ig=48 mm, 60mg Hg, 6 mg TlI, 30 mb Ar

300 W, 2.8A, P~6atm

Tl- 5350

Drakakis, Palladas, Karabourniotis J. Phys. D 1992

5880 5885 5890 5895 5900 5905 5910

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5In

ten

sity (

a.u

.)

Wavlength (Å)

Na D-lines

PHILIPS: R=6 mm, Ig=18 mm, 5 mg Hg,1.87 mg NaI, 100mbar Ar/Kr,

150 W, 3.65 A, P~2.8 atm

Emission line

-10 -5 0 5 10 15 20 25

0

1

2

3

4

5

6

Op

tica

l de

pth

Δλ(Å)

5890 5896

Na D-lines

Experimental observations

√ Optical depth at the line center, τ0, one order of magnitude lower than the calculated one on the basis of the classical theories

√ Intensity at the line minimum one to two orders of magnitude higher than the calculated one on the basis of the classical theories

-0.4 -0.2 0 0.2 0.4

0.5

1

1.5

2

2.5

Hg-5461, LTE, δ = c, Lin.(1) T(x), α=2.23lo

g(I

M/I m

)

log(so)

20 0 1s s 15

4.2

Determination of alpha (α)

23

2.1s

0

7.4

-0.4 -0.2 0 0.2 0.4

0.5

1

1.5

2

2.5

log(IM/Im)

D

0

log log

log logM m M md I I d I I

Dd s d s

7.40

6

-0.4 -0.2 0 0.2

0.1

0.2

0.3

0.4

0.5

log(so)

Na-5890, Hollow L(x), δ = c, Para. T(x), α = 2.18

0.2 0.4 0.6 0.8 1

1

2

3

4

r→

Llo

g(I

M/I

m)

60 2.38s

0.05 0.1 0.15 0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

log(IM/Im)

D6

0

-0.4 -0.2 0 0.2 0.4 0.6 0.8

2

4

6

8

10

Tl-5350

Increasing L(x), δ = c, Constr. T(x), α = 18

log(so)

log

(IM/I

m)

220

2.00s

Conclusions

•For optical depths <12 the Ks-value is affected from the radial change of P(λ,x) by less than 2%.

•In order to determine Ks one needs to know only the α-value instead of the exact plasma structure.

•The difference in Ks using the OPA is less than 5%

•The measurements give optical depths at the line center less than ~6.

•The determination of alpha is proved to be possible at these low optical depths using the OPA model.

Sechin, Starostin et al, JQSRT 58, 887 (1997)

“Resonance radiation transfer in dense media”

584 586 588 590 592 594

0

1

2

3

4

5

Inte

nsity

(a.

u.)

λ (Å)

Auto-lamp

Very high-pressure

P: 20-40atm

5894

Na D-lines

→D-line

Emission line

20 Å

----------------

Philips-Dusseldorf

Hg-5461, LTE, δ = constant, Para(2) T(x), α=1.23

-0.4 -0.2 0 0.2 0.4

0.1

0.2

0.3

0.4

log

(IM

/I m)

log(so)

4.5

3.2s

0