impact of the one-parameter approximation on the shape of optically-thick lines cost-529, meeting at...
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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