Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at...
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![Page 1: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/1.jpg)
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
![Page 2: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/2.jpg)
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
![Page 3: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/3.jpg)
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
![Page 4: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/4.jpg)
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
![Page 5: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/5.jpg)
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
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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
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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
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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
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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
![Page 10: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/10.jpg)
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
![Page 11: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/11.jpg)
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
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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 :
![Page 13: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/13.jpg)
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)
![Page 14: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/14.jpg)
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)
![Page 15: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/15.jpg)
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)
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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
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-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)
![Page 18: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/18.jpg)
-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→
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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)
![Page 20: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/20.jpg)
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%
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-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
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-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
![Page 23: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/23.jpg)
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
![Page 24: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/24.jpg)
-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
![Page 25: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/25.jpg)
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
![Page 26: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/26.jpg)
-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
![Page 27: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/27.jpg)
-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
![Page 28: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/28.jpg)
-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
![Page 29: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/29.jpg)
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
![Page 30: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/30.jpg)
-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
![Page 31: Impact of the one-parameter approximation on the shape of optically-thick lines COST-529, Meeting at Mierlo, March 2006 D. Karabourniotis University of.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d2a5503460f949fe9a4/html5/thumbnails/31.jpg)
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.
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Sechin, Starostin et al, JQSRT 58, 887 (1997)
“Resonance radiation transfer in dense media”
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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
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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