Radiation modeling for optically thick plasmas
description
Transcript of Radiation modeling for optically thick plasmas
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.