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Gas temperature determination from doppler-broadened spectral lines with self-absorption
Physiea 124C (1984) 85--90 North-Holland, Amsterdam
GAS T E M P E R A T U R E D E T E R M I N A T I O N F R O M D O P P L E R - B R O A D E N E D SPECTRAL LINES WITH S E L F - A B S O R P T I O N
J.F. B E H N K E and H. SCHEIBNER Sektion Physik/Elektronik, Ernst-Moritz-Arndt-Universitiit, Greifswald, German Democratic Republic C.J. TIMMERMANS
Physics Department, Eindhoven University of Technology, Eindhoven, The Netherlands Received 16 June 1983 Revised 30 September 1983
The gas temperature determination from spectral line profiles recorded by means of a Fabry-Perot interferometer requires the deconvolution of the profiles into the influence of the apparatus profile, the natural line broadening and, if the optical thickness of the spectral line is not negligible, self-absorption. In this paper it is considered that self-absorption is important in a lot o f practical cases. Folding integrals for the line profiles are calculated with self-absorption taken into account for optical thickness, values kE from 0.1 up to 10. The theoretical results are compared with experiments in a hollow cathode arc discharge.
1. Introduction
The interferometric determination of the gas temperature in a discharge plasma from the Doppler broadening of the spectral lines requires the deconvolution of the recorded line profiles from the apparatus profile of the spectrometer used. For the apparatus function of a FabryPerot interferometer, the Voigt profile with a Gaussian and a Lorentzian part is a sufficient approximation. Both portions can be found experimentally by the registration of a laser line profile with the same experimental setup. For the deconvolution of the recorded profiles of Doppler broadened spectral lines originating from an optical thin sheath some methods exist [1, 2]. But unfortunately under this condition the line intensities are very often too low for quantitative interferometric investigations. To get a good signal to noise ratio measurements on spectral lines of high intensity are favoured with the consequence that the line profiles are influenced by self-absorption. The neglect of this fact could simulate too large line widths and lead to too high estimates of gas temperatures.
The deconvolution of a line profile distorted by self-absorption requires knowledge of the optical thickness of the spectral line under the given discharge conditions and, additionally, knowledge of the parameters of the apparatus profile. If the half width at half maximum (HWHM) of the apparatus profile (A)t,) is negligible with respect to the H W H M (AADA) of the line influenced by self-absorption, the Doppler H W H M (AAD) follows at known optical thickness kE of the emitted spectral line from the relation given by Krebs [3] as A~DA =
AAD
{In kE-- In (In 1+ex~2-t ~- kvE~ ! 1 In 2
1/2
(1)
In discharge plasmas with a relatively low gas temperature the H W H M of the apparatus profile is of the order of the Doppler HWHM. Under these circumstances the deconvolution is only possible by means of calculated integrals, which result from the folding of an absorption profile and the apparatus profile. In special cases, where the natural line width is
0378-4363/84/$03.00 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
86
J.F. B e h n k e et al. / Gas temperature determination
not negligible, its influence is included in the Lorentzian part of the apparatus profile, because the HWHM of two Lorentzian profiles are to be added in the folding process. The intensity distribution of a spectral line resulting from the superposition of two independent distributions Ao(x) and V~(x) is calculated by the folding integral
Usually the Voigt profile Va(x~) of the apparatus function is calculated by means of the folding integral of a Gauss function 1 G.(xl) = X/-~B~2exp(- (x1//3.2)2)
(4) with I ~ G~(xl) dxl = 1
H(x) = ~:~ I~(x)Va(x - y) dy,
(2) and a Lorentz function
where I~(x) is the Doppler profile modified by self-absorption and V.(x) is a Voigt profile representing the apparatus function.
L.(xl) =
~,1
2
1
2
7/" Xl + / 3 a l
(5)
with ~_+?L.(xl) dxl = 1 2. The calculation of the folding integrals
according to the expression
The spectral profile of a Doppler broadened spectral line in emission or absorption is given by the expression
V.(xx, o~,) = ~:= G.(y)L~(x~- y) dy (6) with f+: V~(Xl, a~) dxl = 1,
1
IG(xl) = ~/-~/32kES(k~) Xl 2
x { 1 - e x p [ - k E exp(-(~22 ) )]}
(3)
where /3a2 is the half of the 1/e-width of the Gaussian part of the apparatus function,/3al the half width at half maximum of the Lorentzian one, and eta =/3~1//3~2 characterizes their ratio.
with f® Io(xl) dxl = 1, J-® where k~ kE(O)/E k~(O) =
S(kE)
the the the the
optical thickness of the line, absorption coefficient in the line centre, geometrical sheath th!ckness, Ladenburg self-absorption function [4], which is defined by the expression:
S(kE) = 1 + ~ (-1)' ,=1
k i
E (i + 1)! X//--~-'
the frequency difference relative to the central frequency v0, the wavelength of the line centre, h0 /32 = llAo(ik T/m ) m the half of the 1/e-width of the Doppler profile in s-1, m the atom mass, the Boltzmann constant, k T the gas temperature in K. X I = p - - b'0
J.F. Behnke et al.
Gas temperature determination
The calculation of the folding integral (2) supposes the knowledge of the Voigt functions (6), which depend on the parameter a~ and are presented usually in tabulated form in the literature. For the numerical solution of the integral (2) it is profitable to fold the profiles (3) and (5) in a first step to
3. Results and tables
The integral (8) was calculated for the following set of parameters: kE = 0.01, 0.1, 0.5, 2, 3, 3, 5, 7, 10 a = 0,0.1, 0.2, 0.3, 0.4, 0.5 y
Va(Xb a, kE) = f~. IG(y, kE)La(xl - y, a) dy
H(x,, a, y, kE) = J_~ V~(y, a, kE)G.(x~ - y, y) d y . (8) If /32 is the half of the 1/e-width of the Doppler broadened spectral line the spectral functions (3), eqs. (5) and (4) may be expressed by means of the parameters and
Y =/3-2//32
and the substitutions of the variables U = Y//32
v = xl//32
and
through the following formulas: 1
IG(v, kE) = ~/-~kES(kE) {1 - exp[- kE exp(" v2)]}, (3a)
La(U - v, a) = rr(a2 + a(u - v)2) ' Ga(u - v, Y)= ~---~--~exp- [ ( ~ - ~ ) 2 ]
(5a)
.
0,0.4,0.8, 1.2.
(7)
and then the profiles (7) and (4) in a second step to
a =/3adf12
87
(4a)
The folding integrals (7) and (8) were calculated numerically in dependence on the parameters a, ~" and kE for an interval 0 ~< v ~< v0 with a stepwidth Av = 0.1. The largest value v0 for which the integrals were calculated, resulted from the condition H(vo) < 0.03.
To analyze the recorded spectral line profiles according to the presented procedure we need the H W H M v0.5 of the function H(v, a, y, kE) as well as the reduced v coordinates h, = v,/vo.5, which are attached to the function values nH ( v . . . . ). For n <0.5 the reduced coordinate h, = v,/vo.5 decreases with increasing optical thickness kE. Consequently the optical thickness may be evaluated in principle through measurements of the reduced coordinates h,. This procedure suffers a restriction because the values v,/vo.5 may be determined exactly for n > 0.2 indeed, but their dependence on kE is not pronounced. On the other hand, the uncertainty in the determination of VdVo.5 increases immensely for n < 0.05. Therefore the evaluation of the recorded spectral line profiles should be undertaken by means of the reduced values h0.2, h0A and h0.05 of the normalized profile H(v, a, y, kE). Table I contains the calculated values v0.5 and additionally h0.8, h0.7, h0.2, h0.1 and h0.05 for the above given set of the parameters a, y and kE. Fig. 1 illustrates exemplarily the successive folding processes according to (8) with the special expressions (3a), (4a) and (5a) for the parameter triplet a = 0.3, y = 0.4, kE = 2. For kE--->0 the parameter values v0.5 and h, given in table I approach the results in ref. [2]. For the following process we suppose the knowledge of the parameters flal and /3,2 which characterize the apparatus profile. From the measurements of the 50%, 20%, 10% and 5% width of the Fabry-Perot spectrogram of the self-absorbed spectral line, the optical depth ks and the Doppler half width v0.5 can be defined. This can be done by spline interpolation with the help of a computer. But it can also be done in a
J.F. Behnke et al. / Oo.s temperature determination
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J.F. Behnke et al.
Gas temperature determination
d
1.0
i
I
P,(v)
'
I
1
OG(v)
• exp(-v 2 )
2
,~tv, ktl " "l"exPl-ZexP(~2))
3
1-eXl~.-2l V=(v,oL = 0.3, kt = 2 }
4
H (v,d*, = 0.3, ~ =0.t,, kt=2 )
2)1/2/3~ 1) through ,(o)= -o.5
'
,,(,) AvR~=/p 2 , it follows that
Avg ) = (In 2) 1/2AVRec I~(0)
(12)
v0.5
yielding the second step for c~(u and T o). This iterative procedure results after i iterations in approximate values
0.5
2W2 /3.1 a ci) = (In , Av---T~ ) }
89
and
3'(i) =
a(i)/3aa
/3al"
(13)
4
In the first step the measured value of AVR,~ is substituted for Av~) and the values k~ ) and vgi.~ are then calculated. These values yield a new approximate value I
i 2
-
v
3
Fig. 1. N o r m a l i z e d profiles Pi(v), for the illustration of the stepwise d e c o n v o l u t i o n p r o c e d u r e beginning at the recorded profile (4) u p to the p u r e D o p p l e r profile (1) (condition of the normalization: P / ( 0 ) = 1). U s e d p a r a m e t e r s a = 0.3, -y = 0.4 and kE = 2.
A v O+') = (In 2) '/2 AVR~
,~(i) , ~0.5
which allows to calculate the Doppler temperature, T(i+l)_ A2 m
- i-~ more approximate way, by using the data in table I. The starting procedure is as follows: Measure the half width half maximum (HWHM) of the recorded spectral line Ava=, which relates to /3t2°), the first estimate of the Doppler width, through AVR~ = (In 2~1/2/3 (°) 1 2 = 0.833/3(2°)
(9)
Use as starting values for the reduced coordinates a and 3' respectively _/3al_ 2W2 flax ato)_/3(2°)- (In , AvR~'
(10)
v c°)-/3' - On 2) 1/2 p,2 a¢o)/3., -/3pa~,~.~ = /3a--S"
(11)
In this way from these values a ~°), 3,(0) and the measured reduced coordinates h~ )= vdvo.5 approximate values of k(~) and gc0)0.5are calculated. With the value of vto°) 5, which is related to the better e s t i m a t e of the half Doppler width Aug)= (In
(14)
[ a ~ +']2
(15)
This procedure converges and is interrupted, if the temperatures found in two successive iterative steps differ sufficiently little. 4. A p p l i c a t i o n
For the purpose of the determination of the neutral gas temperature on the axis of a hollow cathode arc discharge on argon Fabry-Perot spectrograms of the intensive, but noticeably self-absorbed argon line A = 6 9 6 . 5 n m were recorded. The temperature is uniform along the line of observation within the discharge region and no contribution from collisional or Stark broadening affects the line profiles. The temperature values resulting from the "10%-method" are presented in table II together with further parameters characterizing the discharge. Results obtained on the weakly selfabsorbed argon line A = 714.7 nm [kE(714.7 nm) 0.1ke(696.5 nm)] are shown in the final column for the purpose of comparison.
J.F. B e h n k e et al. / Gas temperature determination
90
Table II Neutral gas t e m p e r a t u r e T in the axis of an argon h o l l o w c a t h o d e arc discharge (gas pressure p = 40 Pa, tube radius r = 3.5 cm) A = 696.5 n m A = 714.7 n m I(A)
AvR~(MHz)
VO.I/VO.5 ot
y
kE
T(K)
T(K)
15 20 25 30 40
915 915 931 999 965
1.835 1.848 1.851 1.815 1.868
0.345 0.336 0.326 0.307 0.295
2.23 1.94 1.82 2.06 1.30
590 620 670 750 810
660 720 760 800 800
0,309 0.301 0.292 0.275 0.265
Table III C o n v e r g e n c e of the iteration prokedure (discharge parameters: p = 40Pa, I = 40 A, r --- 3.5 cm, argon) A = 696.5 n m
A =- 714.7 n m
i
ot0)
k O)
v~!5
r0+l)(K)
a (i)
kE
~o.5"(°)
T0+I)(K)
0 1 2 3 4 5 6
0.190 0.237 0.254 0.260 0.263 0.264 0.265
0.687 1.011 1.178 1.250 1.281 1.294 1.299
1.042 1.114 1.144 1.156 1.162 1.164 1.165
1008 881 836 818 811 808 806
0.220 0.263 0.272 0.274 0.274
0.130 0.130 0.130 0.130 0.130
0.994 1.028 1.036 1.038 1.038
870 813 801 798 797
/3al = 220 MHz, AVR~ = 965 M H z fla2 = 245 MHz,/32 = 828 M H z
The convergence of the described iteration procedure is demonstrated for an example with the discharge parameters I = 40 A, p = 40 Pa in table IlI. We conclude that with the described method a good estimate of the discharge temperature can be made using self-absorbed lines. The approximative method can b e easily adapted to analysis by a computer*.
Acknowledgements We are indebted to Dr. W. Schielke of the * This c o m p u t e r program is available via C.J. T i m m e r m a n s , Physical D e p a r t m e n t , E i n d h o v e n University of Techn'ology, Eindhoven, T h e Netherlands.
/3al = 220 MHz, AvR~ = 1000 M H z /3~2 = 247 M H z , 132 = 804 M H z
Ernst-Moritz-Arndt Universit~it for the measurement performed on his hollow cathode arc. Also we are grateful to Mr. J.J. Bleize of the Eindhoven University for his indispensable technical assistance.
References [1] G. Elste, Z. Astrophys. 33 (1953) 39. [2] J.D. Davies and J.M. V a u g h a n , Astrophys. J137 (1963) 1302. [3] K. K r e b s , Z. Phys. 101 (1936) 604. [4] R. L a d e n b u r g and S. Levy, Z. Phys. 65 (1930) 189; A.C.G. Mitchell and M.W. Z e m a n s k y , R e s o n a n c e Radiation a n d E x c i t e d A t o m s (Cambridge University Press, 1971).
GAS T E M P E R A T U R E D E T E R M I N A T I O N F R O M D O P P L E R - B R O A D E N E D SPECTRAL LINES WITH S E L F - A B S O R P T I O N
J.F. B E H N K E and H. SCHEIBNER Sektion Physik/Elektronik, Ernst-Moritz-Arndt-Universitiit, Greifswald, German Democratic Republic C.J. TIMMERMANS
Physics Department, Eindhoven University of Technology, Eindhoven, The Netherlands Received 16 June 1983 Revised 30 September 1983
The gas temperature determination from spectral line profiles recorded by means of a Fabry-Perot interferometer requires the deconvolution of the profiles into the influence of the apparatus profile, the natural line broadening and, if the optical thickness of the spectral line is not negligible, self-absorption. In this paper it is considered that self-absorption is important in a lot o f practical cases. Folding integrals for the line profiles are calculated with self-absorption taken into account for optical thickness, values kE from 0.1 up to 10. The theoretical results are compared with experiments in a hollow cathode arc discharge.
1. Introduction
The interferometric determination of the gas temperature in a discharge plasma from the Doppler broadening of the spectral lines requires the deconvolution of the recorded line profiles from the apparatus profile of the spectrometer used. For the apparatus function of a FabryPerot interferometer, the Voigt profile with a Gaussian and a Lorentzian part is a sufficient approximation. Both portions can be found experimentally by the registration of a laser line profile with the same experimental setup. For the deconvolution of the recorded profiles of Doppler broadened spectral lines originating from an optical thin sheath some methods exist [1, 2]. But unfortunately under this condition the line intensities are very often too low for quantitative interferometric investigations. To get a good signal to noise ratio measurements on spectral lines of high intensity are favoured with the consequence that the line profiles are influenced by self-absorption. The neglect of this fact could simulate too large line widths and lead to too high estimates of gas temperatures.
The deconvolution of a line profile distorted by self-absorption requires knowledge of the optical thickness of the spectral line under the given discharge conditions and, additionally, knowledge of the parameters of the apparatus profile. If the half width at half maximum (HWHM) of the apparatus profile (A)t,) is negligible with respect to the H W H M (AADA) of the line influenced by self-absorption, the Doppler H W H M (AAD) follows at known optical thickness kE of the emitted spectral line from the relation given by Krebs [3] as A~DA =
AAD
{In kE-- In (In 1+ex~2-t ~- kvE~ ! 1 In 2
1/2
(1)
In discharge plasmas with a relatively low gas temperature the H W H M of the apparatus profile is of the order of the Doppler HWHM. Under these circumstances the deconvolution is only possible by means of calculated integrals, which result from the folding of an absorption profile and the apparatus profile. In special cases, where the natural line width is
0378-4363/84/$03.00 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
86
J.F. B e h n k e et al. / Gas temperature determination
not negligible, its influence is included in the Lorentzian part of the apparatus profile, because the HWHM of two Lorentzian profiles are to be added in the folding process. The intensity distribution of a spectral line resulting from the superposition of two independent distributions Ao(x) and V~(x) is calculated by the folding integral
Usually the Voigt profile Va(x~) of the apparatus function is calculated by means of the folding integral of a Gauss function 1 G.(xl) = X/-~B~2exp(- (x1//3.2)2)
(4) with I ~ G~(xl) dxl = 1
H(x) = ~:~ I~(x)Va(x - y) dy,
(2) and a Lorentz function
where I~(x) is the Doppler profile modified by self-absorption and V.(x) is a Voigt profile representing the apparatus function.
L.(xl) =
~,1
2
1
2
7/" Xl + / 3 a l
(5)
with ~_+?L.(xl) dxl = 1 2. The calculation of the folding integrals
according to the expression
The spectral profile of a Doppler broadened spectral line in emission or absorption is given by the expression
V.(xx, o~,) = ~:= G.(y)L~(x~- y) dy (6) with f+: V~(Xl, a~) dxl = 1,
1
IG(xl) = ~/-~/32kES(k~) Xl 2
x { 1 - e x p [ - k E exp(-(~22 ) )]}
(3)
where /3a2 is the half of the 1/e-width of the Gaussian part of the apparatus function,/3al the half width at half maximum of the Lorentzian one, and eta =/3~1//3~2 characterizes their ratio.
with f® Io(xl) dxl = 1, J-® where k~ kE(O)/E k~(O) =
S(kE)
the the the the
optical thickness of the line, absorption coefficient in the line centre, geometrical sheath th!ckness, Ladenburg self-absorption function [4], which is defined by the expression:
S(kE) = 1 + ~ (-1)' ,=1
k i
E (i + 1)! X//--~-'
the frequency difference relative to the central frequency v0, the wavelength of the line centre, h0 /32 = llAo(ik T/m ) m the half of the 1/e-width of the Doppler profile in s-1, m the atom mass, the Boltzmann constant, k T the gas temperature in K. X I = p - - b'0
J.F. Behnke et al.
Gas temperature determination
The calculation of the folding integral (2) supposes the knowledge of the Voigt functions (6), which depend on the parameter a~ and are presented usually in tabulated form in the literature. For the numerical solution of the integral (2) it is profitable to fold the profiles (3) and (5) in a first step to
3. Results and tables
The integral (8) was calculated for the following set of parameters: kE = 0.01, 0.1, 0.5, 2, 3, 3, 5, 7, 10 a = 0,0.1, 0.2, 0.3, 0.4, 0.5 y
Va(Xb a, kE) = f~. IG(y, kE)La(xl - y, a) dy
H(x,, a, y, kE) = J_~ V~(y, a, kE)G.(x~ - y, y) d y . (8) If /32 is the half of the 1/e-width of the Doppler broadened spectral line the spectral functions (3), eqs. (5) and (4) may be expressed by means of the parameters and
Y =/3-2//32
and the substitutions of the variables U = Y//32
v = xl//32
and
through the following formulas: 1
IG(v, kE) = ~/-~kES(kE) {1 - exp[- kE exp(" v2)]}, (3a)
La(U - v, a) = rr(a2 + a(u - v)2) ' Ga(u - v, Y)= ~---~--~exp- [ ( ~ - ~ ) 2 ]
(5a)
.
0,0.4,0.8, 1.2.
(7)
and then the profiles (7) and (4) in a second step to
a =/3adf12
87
(4a)
The folding integrals (7) and (8) were calculated numerically in dependence on the parameters a, ~" and kE for an interval 0 ~< v ~< v0 with a stepwidth Av = 0.1. The largest value v0 for which the integrals were calculated, resulted from the condition H(vo) < 0.03.
To analyze the recorded spectral line profiles according to the presented procedure we need the H W H M v0.5 of the function H(v, a, y, kE) as well as the reduced v coordinates h, = v,/vo.5, which are attached to the function values nH ( v . . . . ). For n <0.5 the reduced coordinate h, = v,/vo.5 decreases with increasing optical thickness kE. Consequently the optical thickness may be evaluated in principle through measurements of the reduced coordinates h,. This procedure suffers a restriction because the values v,/vo.5 may be determined exactly for n > 0.2 indeed, but their dependence on kE is not pronounced. On the other hand, the uncertainty in the determination of VdVo.5 increases immensely for n < 0.05. Therefore the evaluation of the recorded spectral line profiles should be undertaken by means of the reduced values h0.2, h0A and h0.05 of the normalized profile H(v, a, y, kE). Table I contains the calculated values v0.5 and additionally h0.8, h0.7, h0.2, h0.1 and h0.05 for the above given set of the parameters a, y and kE. Fig. 1 illustrates exemplarily the successive folding processes according to (8) with the special expressions (3a), (4a) and (5a) for the parameter triplet a = 0.3, y = 0.4, kE = 2. For kE--->0 the parameter values v0.5 and h, given in table I approach the results in ref. [2]. For the following process we suppose the knowledge of the parameters flal and /3,2 which characterize the apparatus profile. From the measurements of the 50%, 20%, 10% and 5% width of the Fabry-Perot spectrogram of the self-absorbed spectral line, the optical depth ks and the Doppler half width v0.5 can be defined. This can be done by spline interpolation with the help of a computer. But it can also be done in a
J.F. Behnke et al. / Oo.s temperature determination
88
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J.F. Behnke et al.
Gas temperature determination
d
1.0
i
I
P,(v)
'
I
1
OG(v)
• exp(-v 2 )
2
,~tv, ktl " "l"exPl-ZexP(~2))
3
1-eXl~.-2l V=(v,oL = 0.3, kt = 2 }
4
H (v,d*, = 0.3, ~ =0.t,, kt=2 )
2)1/2/3~ 1) through ,(o)= -o.5
'
,,(,) AvR~=/p 2 , it follows that
Avg ) = (In 2) 1/2AVRec I~(0)
(12)
v0.5
yielding the second step for c~(u and T o). This iterative procedure results after i iterations in approximate values
0.5
2W2 /3.1 a ci) = (In , Av---T~ ) }
89
and
3'(i) =
a(i)/3aa
/3al"
(13)
4
In the first step the measured value of AVR,~ is substituted for Av~) and the values k~ ) and vgi.~ are then calculated. These values yield a new approximate value I
i 2
-
v
3
Fig. 1. N o r m a l i z e d profiles Pi(v), for the illustration of the stepwise d e c o n v o l u t i o n p r o c e d u r e beginning at the recorded profile (4) u p to the p u r e D o p p l e r profile (1) (condition of the normalization: P / ( 0 ) = 1). U s e d p a r a m e t e r s a = 0.3, -y = 0.4 and kE = 2.
A v O+') = (In 2) '/2 AVR~
,~(i) , ~0.5
which allows to calculate the Doppler temperature, T(i+l)_ A2 m
- i-~ more approximate way, by using the data in table I. The starting procedure is as follows: Measure the half width half maximum (HWHM) of the recorded spectral line Ava=, which relates to /3t2°), the first estimate of the Doppler width, through AVR~ = (In 2~1/2/3 (°) 1 2 = 0.833/3(2°)
(9)
Use as starting values for the reduced coordinates a and 3' respectively _/3al_ 2W2 flax ato)_/3(2°)- (In , AvR~'
(10)
v c°)-/3' - On 2) 1/2 p,2 a¢o)/3., -/3pa~,~.~ = /3a--S"
(11)
In this way from these values a ~°), 3,(0) and the measured reduced coordinates h~ )= vdvo.5 approximate values of k(~) and gc0)0.5are calculated. With the value of vto°) 5, which is related to the better e s t i m a t e of the half Doppler width Aug)= (In
(14)
[ a ~ +']2
(15)
This procedure converges and is interrupted, if the temperatures found in two successive iterative steps differ sufficiently little. 4. A p p l i c a t i o n
For the purpose of the determination of the neutral gas temperature on the axis of a hollow cathode arc discharge on argon Fabry-Perot spectrograms of the intensive, but noticeably self-absorbed argon line A = 6 9 6 . 5 n m were recorded. The temperature is uniform along the line of observation within the discharge region and no contribution from collisional or Stark broadening affects the line profiles. The temperature values resulting from the "10%-method" are presented in table II together with further parameters characterizing the discharge. Results obtained on the weakly selfabsorbed argon line A = 714.7 nm [kE(714.7 nm) 0.1ke(696.5 nm)] are shown in the final column for the purpose of comparison.
J.F. B e h n k e et al. / Gas temperature determination
90
Table II Neutral gas t e m p e r a t u r e T in the axis of an argon h o l l o w c a t h o d e arc discharge (gas pressure p = 40 Pa, tube radius r = 3.5 cm) A = 696.5 n m A = 714.7 n m I(A)
AvR~(MHz)
VO.I/VO.5 ot
y
kE
T(K)
T(K)
15 20 25 30 40
915 915 931 999 965
1.835 1.848 1.851 1.815 1.868
0.345 0.336 0.326 0.307 0.295
2.23 1.94 1.82 2.06 1.30
590 620 670 750 810
660 720 760 800 800
0,309 0.301 0.292 0.275 0.265
Table III C o n v e r g e n c e of the iteration prokedure (discharge parameters: p = 40Pa, I = 40 A, r --- 3.5 cm, argon) A = 696.5 n m
A =- 714.7 n m
i
ot0)
k O)
v~!5
r0+l)(K)
a (i)
kE
~o.5"(°)
T0+I)(K)
0 1 2 3 4 5 6
0.190 0.237 0.254 0.260 0.263 0.264 0.265
0.687 1.011 1.178 1.250 1.281 1.294 1.299
1.042 1.114 1.144 1.156 1.162 1.164 1.165
1008 881 836 818 811 808 806
0.220 0.263 0.272 0.274 0.274
0.130 0.130 0.130 0.130 0.130
0.994 1.028 1.036 1.038 1.038
870 813 801 798 797
/3al = 220 MHz, AVR~ = 965 M H z fla2 = 245 MHz,/32 = 828 M H z
The convergence of the described iteration procedure is demonstrated for an example with the discharge parameters I = 40 A, p = 40 Pa in table IlI. We conclude that with the described method a good estimate of the discharge temperature can be made using self-absorbed lines. The approximative method can b e easily adapted to analysis by a computer*.
Acknowledgements We are indebted to Dr. W. Schielke of the * This c o m p u t e r program is available via C.J. T i m m e r m a n s , Physical D e p a r t m e n t , E i n d h o v e n University of Techn'ology, Eindhoven, T h e Netherlands.
/3al = 220 MHz, AvR~ = 1000 M H z /3~2 = 247 M H z , 132 = 804 M H z
Ernst-Moritz-Arndt Universit~it for the measurement performed on his hollow cathode arc. Also we are grateful to Mr. J.J. Bleize of the Eindhoven University for his indispensable technical assistance.
References [1] G. Elste, Z. Astrophys. 33 (1953) 39. [2] J.D. Davies and J.M. V a u g h a n , Astrophys. J137 (1963) 1302. [3] K. K r e b s , Z. Phys. 101 (1936) 604. [4] R. L a d e n b u r g and S. Levy, Z. Phys. 65 (1930) 189; A.C.G. Mitchell and M.W. Z e m a n s k y , R e s o n a n c e Radiation a n d E x c i t e d A t o m s (Cambridge University Press, 1971).