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Spectroscopic study of the argon lines emitted by a stabilized arc and an rf plasma: deviations from local thermal equilibrium (LTE)
Spectmchtmm Acfa, Voi Printed I” Great Bntain
488.
No. 5. pp
711-722.
1993
0584-8547193 $4 00 + .W @ 1993 Pergamon Press Ltd
Spectroscopic study of the argon lines emitted by a stabilized arc and an rf plasma: deviations from local thermal ~uilibrium (LTE) S. VACQIJIB and A. GLEIZES Laboratoire
de D&charges dam les Gaz (URA 277) Universitt France
Paul Sabatier, 31062 Toulouse CBdex,
and M. SABSABI
and M. I. BOULOS
Centre de Recherche en Technologie des Plasmas (CRTP), University of Sherbrooke, Quebec, Canada JlK 2Rl
Sherbrooke,
(Received 22 July 1992; accepted 8 December 1992) Abstract-The fines emitted by argon plasmas at atmospheric pressure along a diameter and a chord of a wall-stabilized arc and those of an rf plasma were studied experimentally and theoretically. The line profiles, calculated taking into account all the causes of broadening and shift, were compared to the experimental profiles. It was shown that: (a) the non-absorbed or partially absorbed lines present an intensity peak situated between the nominal wavelength of the line and the line with the greatest shift on the arc axis; (b) the strongly absorbed lines present self-reversal. The absorption m~imum presents a red shift with respect to the nominal wavelength of the line; and (c) total local thermal equilibrium is not reached in the peripheral regions of the arc where the 4s levels are overpopulated owing to radiative cascades. This leads to an increase in the absorption of the transitions going towards the 4s levels.
1. I~ODU~IO~ THE TECHNOLOGY based on the use of thermal plasmas produced by torches and transferred arcs is steadily being developed. These types of plasmas operate most frequently in an argon atmosphere and are increasingly used in metal-working industries. The development of the processes requires long and costly testing programmes and the di~culties inherent in creating all the possible working conditions in the laboratory have led to the design of more and more sophisticated mathematical models. In many cases, these models assume the existence of local thermodynamic equilibrium (LTE). Validation of the models usually involves the measurement of parameters such as temperature and particle density obtained from the spectroscopic study of the lines emitted by the arc. The determination of the local intensity of the lines requires Abel inversion of the experimentally obtained integrated profile (Fig. 1). To enable the inversion to be carried out, two conditions must be strictly observed:
Fig. 1. Section of the discharge: J(r) is the local intensity; f(x) is the total emission integrated along a chord. 711
712
-
Argon
Argon
1
2
Fig. 2. Arc chambers: (1) wall-stabilized arc; (2) rf plasma torch.
(i) the plasma must be symmetrical with respect to the axis of obse~ation; and (ii) the line must not be self-absorbed. With electric arcs and in particular in wall-stabilized arcs, the first condition is reached. In rf plasmas this is not always the case since inside the coil, asymmetry of the emission can be observed [l]. The second condition depends on the line being studied, some lines being strongly self-absorbed. Absorption depends on the thickness of the plasma. In our exprimental conditions (see Section 2.2), the temperature profile is relatively flat with a minimum on the axis. In a wall-stabilized arc the temperature is greatest on the axis and decreases rapidly towards the walls. For a given temperature at the axis, absorption will be greater in rf plasmas. Thus, Abel inversion applied to a strongly absorbed line, such as that at 811.53 nm (transition 3p54p-3p54s), leads to erroneous results [2]. Also, the lines are broadened and subjected to shift owing to the Stark effect. The signal emitted along a chord or diameter of the discharge is the sum of the emissions arising from regions with different values of temperature, T, and/or electron number density, n,. The result is generally a line that is broadened and asymmetric with an emission ma~mum situated at a value of the wavelength X that depends on the structure of the discharge. The shift, therefore, cannot be considered to depend solely on the characteristics of the atomic transition lines [3]. This led us to study the line profiles of argon, along a diameter, emitted by two types of thermal discharge with very different diameters: a wall-stabilized arc of 6 mm in diameter; and an rf plasma torch of 50 mm in diameter. The experimental study allowed us to define the characteristics of the lines and the infhrence of the thickness of the plasma. From the temperature profiies T(r) and assuming total LTE, the profile of the lines J(h) can be calculated for each point of the plasma. It is therefore also possible to calculate the profile of the line emitted by a diameter (or a chord) of the discharge. Comparing the experimental profiles and the calculated profiles checks the validity of the hypothesis that assumes total LTE.
2. EXPERIMENTAL 2.1. Wall-stabilized arc The arc used in this study was of the cascade type (Fig. 2). The chamber was composed of a stack of hollow water-cooled discs. The central hole forming the chamber had a diameter of 6 mm. Sightings were made laterally between the two central discs. All the experiments were performed at atmospheric pressure. Arcing was initiated by means of a tungsten rod passing through one of the electrodes. The distance between the electrodes was 7 cm. The maximum current output of the power supply was about 100 A.
Spectroscopic study of Ar lines
713
2.2. High-frequency plasma The essential features of the standard-design induction plasma torch used in this work are shown in Fig. 2. The plasma was generated using a radio-frequency (rf) power supply with a nominal oscillator frequency of 3 MHz and a maximum plate power of 25 kW. Two gas streams were introduced into the torch. The sheath gas, QS, was of pure argon, 72 1 min-‘; the plasma gas, Q2, was also of pure argon, 43 1 min -I. No central gas, Q1. was introduced into the torch in the present measurements. The plasma was confined in a 50 mm i.d. water-cooled quartz tube. The windings were 20 mm apart and the outside diameter of the coil that they formed was 68 mm (Fig. 2). The operating pressure was one atmosphere. 2.3. Optical system A 20 cm focal length converging lens formed a 1:l scale image of the discharge on the inlet slit of the monochromator. Two iris diaphragms were placed at the focal points of the lens. Radial exploration of the discharge was carried out by side-on sighting moving the arc chamber perpendicularly to the optical axis of the system. Under our experimental conditions, the apparatus function was measured using a low-pressure argon spectral lamp (its lines broadened by the Doppler effect), with the inlet and outlet slits set at about 20 km: the apparatus width was found to be around 0.025 nm. For the high-frequency plasma, the proper choice of the focal length of a lens pair allowed the formation of a l/10 reduced image of the plasma on the plane of the monochromator entrance slit. A pair of parallel mirrors was used for vertical displacement to change the scanning level of the plasma. This could not, however, be freely chosen owing to the obstruction of the line of sight by the induction coil. The level at which measurements were carried out (Fig. 2) was 45 mm downstream of the coil. The output spectrum was sampled by an array of photodiodes laid out along the axis of wavelength dispersion. Each photodiode received a given wavelength of the spectrum, and the largest spectral interval resolved was about 0.02 nm. In our experimental conditions, with an entrance slit of around 20 km, the apparatus function was found to be roughly 0.035 nm with a low-pressure argon lamp.
3. THEORETICAL STUDY Knowing the local values of the temperature, T, and the electron density, n,, the profile of a line J(h) can be calculated. Note that the line will be broadened and will undergo a shift. Each mechanism of line broadening depends on the type of interaction involved and will give a profile that is: (a) Gaussian through the Doppler effect; (b) Lorentzian through the Stark effect; and (c) Lorentzian through interaction with neutral particles. Convolution of a Gaussian profile with a Lorentzian profile gives a Voigt profile, which can be written in a normalized way and in frequency units:
with: a = g
(ln2)“* D
b
=
(v
-
vo SD
+
A>
ln2 J
(3)
where 26, is the total half-width of the Doppler profile, vO is the frequency at the nominal value of the line peak, 26r is the sum of the half-widths due to the effect of collision (Stark and neutral van der Waals broadening), A is the line shift, and y is the mute variable that corresponds to the exponent in the Gaussian profile initially defined by: Y=
7
(ln2)“*.
714
S. VAcQulB et al.
The local monochromatic radiation emission coefficient J(X), at a distance r from the axis, is given (in Watts per cm3 and per steradian) by the expression:
J(X) =
&A,, N&A).
(4)
The term A,, is the probability of spontaneous emission of the emitter level m towards level rz. P(A) is the shape factor defined by Eqn (1). Thus, the profile of the line is defined; N, is the density of the emitting level that, assuming LTE, depends on the pressure and temperature. Figure 1 represents the scheme of a section of the discharge. The intensity emitted by a chord is the sum of the local emissivities along the chord through the inhomogeneities of the plasma. If we call k(r,h) the coefficient of local absorption, it can be shown that the intensity 1(x,X) emitted by a chord is expressed as a function of J and of k [4] by the following expression:
Z(x,A) =
x R
1
rdr
(r2- x2)l12
J(r,A) and k(r’,A) are the implicit functions of the excited atom densities and are therefore implicit functions of the temperature. Under the influence of charged particles, the lines are shifted by a value A calculated from the results of GRIEM [5,61. The values A0 and A, given by GRIEM lead to results that differ from the experimental values determined with the wall-stabilized arc in particular for the line at 811.5 nm. A good agreement is found by multiplying the values of A0 and 6, by a factor of 0.65. So, assuming the existence of LTE, if the temperature profile is known, the intensity I(X) emitted by a chord can be calculated. However, the excited atoms on the levels involved in the transition in question are not necessarily in thermodynamic equilibrium at the electron temperature of the medium. This is often the case, with argon, for the four 4s levels (two metastable levels and two resonant levels that can be overpopulated) [7]. This gives rise to a greater absorption of the transitions leading to the 4s levels than that predicted assuming complete LTE. To account for these deviations from LTE, either the absorption coefficient k(A) of the transition or the density of the excited atoms on the metastable or resonant level must be known. The total absorption coefficient is given by:
k(A)dh=$,,,,iV,, x where f,,,,, is the oscillator force on absorption of the transition, and N, the density of excited atoms on the lower level n of the transition m + n. We also have: k(A) = KP(A). Knowing N, and P(A) we can therefore by a diameter using Eqn (5).
reconstruct
(7)
the profile of the line emitted
Spectroscopic
715
study of Ar lines
II
0
5
10
15
20
r (mm)
Fig. 3. Temperature
profiles (1) for the arc and (2) for the rf plasma.
4. METHODOF CALCULATION To determine the temperature, we used the classic Boltzmann plots method. Two groups of lines were measured. The first includes seven lines arising from the 3p55p-3~~4s transitions of argon (415.8, 418.1, 420, 425.9, 426.6, 427.2 and 430 nm). These lines are not at all absorbed by the plasma. The second group is composed of four lines coming from the 3p54p-3p54s transitions. Some of the lines from these transitions can be strongly absorbed. We therefore chose four lines with negligible absorption: 696.5, 706.7, 727.3 and 738.4 nm. The local value of the intensity of each line was obtained after Abel inversion of the integrated profile. The method does not allow temperatures lower than 8500 K to be obtained without making large errors. We therefore calculated the profile for our experimental conditions using an energy balance model (of the Elenbaas-Heller type) for the stabilized arc [8] and a more complex model for the rf plasma [9]. A good agreement was found between calculation and experiment. Below 8500 K, the agreement was good with the results published in the literature [9, lo]. We were thus able to extrapolate our experimental values to 7000 K (Fig. 3). It is very possible that in the argon plasma, LTE is only partial and that radiative cascades overpopulate the 4s level when the electron density is insufficient for collisional equilibrium. In order to account for any deviation from LTE, the real densities of the excited atoms on the 4s levels must be introduced into the calculation. We used values obtained a few years ago under strictly identical experimental conditions with a wall-stabilized arc [ll]. They were determined by absorption spectroscopy using the continuum of a high-pressure xenon lamp in a side-on observation and with Abel inversion. The details of the method and the results obtained are fully described in Ref. [ll]. In Fig. 4, for a given temperature profile, we compare the theoretical values of the number density of 1~~ (the metastable level) for LTE with the experimental values. On the axis there is good agreement. At the edges, the LTE value rapidly tends to zero whereas the experimental value remains at a density of around 2 X lOI* cmw3 (measurements for r > 3 mm correspond to the diffusion region between the discs). We used these non-equilibrium experimental values in the calculation. The experimental set-up used did not allow the same measurements to be made with the rf plasma. So, for each value of the temperature, T, in the rf plasma, we used the corresponding value of the 1s5 level obtained with the wall-stabilized arc. Henceforward, the figures obtained are called “experimental values” indifferently for the stabilized arc and for the rf plasma, knowing however that for the rf plasma it is a justified analogy since the time-characteristic for diffusion losses is much larger than that for collisions. It can therefore be assumed that losses through diffusion are negligible
716
S. VacourB et al. 20
I
0
1
I
I
2
3
3
r(mm)
Fig. 4. Comparison between theoretical and experimental values of the metastable atoms (Is5 level) along a radius of the wall-stabilized arc.
compared to collisional losses and that it is the local temperature and electron density that govern the metastable atom densities more than the gradients. The line profiles at LTE were calculated assuming that the whole set of levels was in equilibrium for the temperature of the medium. To take deviation from LTE into account. two methods of calculation were used. In the first, we assumed that the upper level of the transition in question was in equilibrium with the metastable level. The intensity profile J(X) of the line at a given point in the plasma was calculated for a Voigt profile (Eqn 1) using Kirchhoff’s law: J(X) = B, k(h).
(8)
L?, is given by Planck’s law and is written, as a function of the wavelength,
A, by:
2 ~
[exp (hc/hkT)-11-l
(9)
where the terms have their usual meaning. Here, k(h) is obtained from Eqns (6) and (7). The metastable atom density, N, (Eqn 6), was obtained experimentally. It is clear that if the metastable
level is overpopulated,
the intensity of the line,
J(X), will be
overestimated. So, the total intensity integrated along a diameter (EqnA5) will be the upper limit. In the second method, the starting level was assumed to be in LTE and, for the final level, we used the experimentally determined value. This is the more realistic procedure. The second method was generally used, except for the strongly absorbed 811.5 nm line where the two methods were compared. As mentioned earlier, the widths of the apparatus function were determined experimentally: there was no apparent dissymmetry. Theoretically, with the photomultiplier, the function should be a triangle; with the diode array, it is more complex. It was demonstrated that no detectable error was incurred on the final result when it was assumed that the curves were Gaussian in shape on the condition that the same width at half height was taken. It was therefore considered in the calculations that the apparatus functions were Gaussian.
5. RESULTS The results presented concern three characteristic lines: (i) the 420 nm line (3p55p-3p54s transition), which is almost totally unabsorbed; (ii) the 763.5 nm line
Spectroscopic
r
420
0675
study of Ar lines
717
nm 0
ExperImental _--
L
420
0
420.
I
420.2
420.3
h (nm)
Fig. 5. Comparison between the experimental and the theoretical profiles of the 420 nm line: (1) nominal wavelength of the line; (2) maximum of the line emitted by a diameter; and (3) maximum of the line on the axis.
(3p54p-3~~4s transition), which is only slightly reabsorbed; and (iii) the 811.5 nm line (3~~4~7-3~~4~transition), which is very strongly reabsorbed. Figure 5 presents the theoretical profile given by a diameter of the stabilized arc for a line that is not self-absorbed (420 nm) of the transition 3p55p-3~~4s. It can be seen that for a value of 11000 K on the temperature axis, the profile of a line emitted by a diameter is no longer symmetrical. The maximum is situated at A, which is between A,,, the nominal wavelength of the undisturbed line, and A,, + Ah,,, AA,, being the value of the maximum shift of the line on the axis of the arc (AA,ax = 0.0475 nm). The shift observed for a line emitted by a diameter can therefore not be taken as a measurement of the characteristic shift of the line for the hottest regions of the arc. The profile obtained is in very good agreement with the experimental profiles recorded with either of the set-ups used. Figure 5 also gives the experimental profile of the 420 nm line obtained with the rf plasma at atmospheric pressure. As absorption is negligible, the general shape of the line profile emitted by a diameter is not dependent on the thickness of the plasma but on the temperature maximum, which is identical in the arc and in the plasma. The strong lines corresponding to the transitions 3p54p-3~~4s are absorbed. So, the shape of the line emitted by the plasma depends on the structure of the discharge. Figure 6 gives the theoretical profile of the 763.5 nm line emitted by a diameter of the stabilized arc. This line corresponds to the transition between level 2ps of the 3p54p levels and the metastable Is5 level of the 3p54s configuration. The same phenomenon is observed as that seen with the non-absorbed line, i.e. the emission peak occurs between the nominal wavelength of the line (A = 763.525 nm) and the maximum wavelength of the line with the greatest shift on the axis of the arc, which corresponds to T = 11000 K. No anomalies owing to absorption are noted in this theoretical result. Figure 6 also represents the experimental profile of the line obtained along a diameter of the stabilized arc. Near the peak, in the blue part, a small disturbance, due to absorption, is observed in contrast to the theoretical profile. This disturbance is in the vicinity of the nominal wavelength of the line. OLSEN [12], who observed a similar phenomenon on this line, suggested that it could be caused by reabsorption in the plasma. The poor agreement in Fig. 6 between the theoretical curve (taking
S. VACQW? et al.
718
163.521 nm
763.4
763.5
763 6
It (nm)
Fig. 6. Comparison between the experimental and the theoretical profile of the 763.5 nm line, emitted by a diameter of the wall-stabilized arc.
deviation from LTE into account) and the experimental results strongly indicate the existence of another reason. We had also observed this phenomenon in the past but did not find a satisfactory interpretation [13]. With the rf plasma, and with a power input of 22 kW, the line emitted by a diameter is dissymmetrical, shifted, broadened and clearly shows absorption near the peak on the blue part (Fig. 7). The line emitted by a chord at 10 mm from the axis presents the same characteristics with a stronger absorption than in the previous example. At 15 mm from the axis, the line is much less intense, not shifted, and very slightly dissymmetrical. Figure 7 also represents the calculated profile of the 763.5 nm line obtained without assuming the existence of LTE for the Is5 level for which we used the experimentally determined densities. With a Gaussian apparatus function of 0.035 nm, we observed the same shape as that of the experimental profile with a disturbance on the blue part near the peak. The uncertainties on the line broadening and on the line shift and also the errors on the temperature profiles and on the atom densities explain the difference between the half-widths of the experimental and the theoretical profiles.
163 4
763.5
163 6
k (nm)
Fig. 7. Comparison
between the experimental and the theoretical profiles of the 763.5 nm line, emitted by a diameter of the rf plasma.
719
Spectroscopic study of Ar lines
. I
I XII
81 1.6
5 h
(nm)
Fig. 8. Calculated profiles of the 811.5 nm line for the wall-stabilized arc without apparatus function convolution. Curve 1: LTE hypothesis. Curve 2: Kirchhoff’s law and experimental value of metastable atom density N,,,. Curve 3: partial LTE for the upper level of the transition (2p,) and experimental value for N,,,.
As the 811.5 nm line is strongly reabsorbed we shall study the calculation of the reconstruction of its profile in more detail. Figure 8 represents the theoretical profile of the line emitted by a diameter of the stabilized arc with the three hypotheses used. Curve 1 assumes LTE; curve 2 was calculated with a Voigt profile using Kirchhoff’s law and the experimental value for level 1~~; curve 3 was calculated with a Voigt profile but Kirchhoff’s law was not used: level 2ps was assumed to be in equilibrium and level Is, at its experimentally determined density. Strong reabsorption is observed at a wavelength close to the nominal wavelength of the line. In order to be able to compare these curves to the experimental ones, the profile must be convoluted with the apparatus function, which has a half-width of 0.025 nm. The results obtained are presented in Fig. 9. Curve 1 represents the profile of the line assuming LTE. Taking the apparatus function into account causes the disappearance of the modification due to absorption on the blue wing near the peak. A good agreement is seen between the curve obtained by calculation after convolution of the apparatus function and the experimental curve when the 2pg level is not considered to be in equilibrium with the lsg level (curve 3 of Fig. 7). Figure 10 represents the 811.55 nm line profiles obtained experimentally on the rf torch at 100 kPa for a power of 22 kW at a distance z of 45 mm and for 0 and 10 mm from the axis. Self-reversal of the line can be seen. The absorption maximum is not situated exactly at the nominal wavelength of the line (here, it is observed at 811.537 nm). The line emitted at 10 mm from the axis presents an absorption dip greater than that observed across a diameter. The chord is shorter than the diameter but emission is stronger there (Fig. 3). However, absorption is more intense and the central dip in the line is deeper. The broadening of the lines due to the Stark effect is greater for the 420 nm line than for the 763.5 nm line or the 811.5 nm line. For T = 10 000 K, the broadening Ahs due to the Stark effect is 3.3 X lo-* nm for the 420 nm line and 1.6 x lo-* nm for the 763.5 nm line and the 811.5 nm line. So, the 763.5 nm line, being only slightly absorbed, its half-width is smaller than that of the 420 nm line. On the other hand, the 811.5 nm line, being strongly absorbed, its apparent half-width is greater than the
S. VACQUI~et
120
al.
I XII
I
-
811 6
5
h (nm) Fig. 9. Comparison between experimental and calculated profiles of the 811.5 nm line for the wall-stabilized arc after apparatus function convolution. Curve 1: LTE hypothesis. Curve 2: curve 3 of Fig. 8 after convolution, Curve 3: experimental profile.
8114
811
5
XII
6
XII
7
h(nm)
Fig. 10. Experimental
profiles of the 811.5 nm line emitted by a diameter (x = 0) and by a chord (x = 10 mm) of the rf plasma.
half-width of the 420 nm line (and of course than the half-width of the 763.5 nm line). This is clearly visible in the experimental results. In Fig. 11 we have presented the theoretical profiles of the 811.5 nm line along a diameter and along a chord (at 10 mm) for a half-width of the apparatus function of 0.035 nm and assuming that LTE is reached in the plasma. Absorption greatly reduces the intensity of the line peak but is only apparent on the profile through a slight dip near the summit. Comparison with Fig. 10 shows therefore that if LTE is assumed to exist, the profiles calculated are very different to those obtained experimentally. On
Spectroscopic
8114
721
study of Ar lines
811
5
811.6
811.7
h(nm)
Fig. 11. Calculated profiles of the 811.5 nm line emitted by a diameter (X = 0) and by a chord (x = 10 mm) of the rf plasma assuming LTE.
reconstructing the profile with levels 2p9 and lss, good agreement with experimental profiles is obtained. Figure 12 represents the same profiles for 0 and 10 mm from the axis assuming the 2p, level to be in Saha equilibrium with the electrons and taking the experimentally determined density for the Is5 level. It can be seen that for an instrument function of 0.035 nm, agreement with experimental results is good. Here again, the absorption maximum is seen to be shifted with respect to the nominal wavelength of the line. The results show that the peripheral regions of the arc and of the rf plasma are not in LTE. The metastable overpopulation leads to line profiles emitted across a diameter or a chord that are very different to those that would be found if LTE were reached.
-*= X= ,___
811.4
811.5
811.6
811
7
h(nm)
Fig. 12. Calculated protiles of the 811.5 nm line emitted by a diameter (x = 0) and by a chord (X = 10 mm) of the rf plasma with the experimental values of metastable atom density determined in the arc.
722
S.
VACQ&et al.
6. CONCLUSION The present study has enabled comparisons to be made between the profiles of the argon lines emitted by the plasma of a wall-stabilized arc and that of an rf plasma across a diameter and across a chord. It also enabled us to compare the experimentally obtained profiles with theoretical profiles calculated firstly by assuming LTE and secondly by incorporating experimental values. We can thus show that: (a) the peak of the non-absorbed lines is situated between the nominal value of the line and the maximum wavelength of the line that has undergone the greatest shift (where T is maximum). The line profiles in the arc and in the rf plasma are identical; (b) the weakly reabsorbed lines present the same characteristics, absorption introducing a disturbance in the blue part; and (c) in the case of self-reversed lines, the absorption maximum is shifted towards the blue with respect to the position of the maximum Stark shift. Considering the greater diameter of the plasma, the lines undergo stronger absorption. This gives rise to a greater width at apparent mid-height (0.08 nm in the arc, 0.12 nm in the torch). These results show the influence of the various processes of broadening and confirm that the argon plasma is not in total LTE in the peripheral zones of the discharge where metastable atoms are overpopulated and lead to an increase in the absorption of the transitions towards the 4s levels. REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13]
L. Bydder and G. P. Miller, Specfrochim. Acta 43B, 1431 (1988). Uchida, K. Tanabe, Y. Nojiri, H. Haraguchi and K. Fuwa, Spectrochim. Acta 36B, 711 (1981). Kato, H. Fukushima and T. Kakajima, Specrrochim. Acta 39B, 979 (1984). Vacquit, J. Bacri, M. Capderou, J. P. Dinguirard and A. M. Gomes, J. Quant. Spectrosc. Radiat. Transfer 13, 1333 (1973). H. R. Griem, Phys. Rev. 128, 515 (1962). H. R. Griem, Spectral Line Broadening by Plasma. Academic Press, New York, London (1974). J. Bacri, S. VacquiC, A. M. Gomes and M. F. Habibi, XI ht. Conf. on Ionization Phenomena in Gases, p. 415, Prague (1973). A. Gleizes, H. Kafrouni, H. Dang Due and V. C. Maury, J. Phys. D: Appl. Phys. 15, 1031 (1982). J. Mostaghimi, P. Proulx and M. Boulos, Numerical Heat Transfer 8, 187 (1985). R. J. Giannaris and F. P. Incropera, J. Quark Spectrosc. Radial. Transfer 11, 298 (1971). J. Bacri, A. Gleizes and A. M. Gomes, J. Quant. Spectrosc. Radiat. Transfer 15, 7 (1975). H. N. Olsen, J. Quant. Spectrosc. Radiat. Transfer 3, 67 (1963). A. Gleizes, Thtse de 3” cycle no 1542, University Paul Sabatier, Toulouse, France (1974). E. H. K. S.
488.
No. 5. pp
711-722.
1993
0584-8547193 $4 00 + .W @ 1993 Pergamon Press Ltd
Spectroscopic study of the argon lines emitted by a stabilized arc and an rf plasma: deviations from local thermal ~uilibrium (LTE) S. VACQIJIB and A. GLEIZES Laboratoire
de D&charges dam les Gaz (URA 277) Universitt France
Paul Sabatier, 31062 Toulouse CBdex,
and M. SABSABI
and M. I. BOULOS
Centre de Recherche en Technologie des Plasmas (CRTP), University of Sherbrooke, Quebec, Canada JlK 2Rl
Sherbrooke,
(Received 22 July 1992; accepted 8 December 1992) Abstract-The fines emitted by argon plasmas at atmospheric pressure along a diameter and a chord of a wall-stabilized arc and those of an rf plasma were studied experimentally and theoretically. The line profiles, calculated taking into account all the causes of broadening and shift, were compared to the experimental profiles. It was shown that: (a) the non-absorbed or partially absorbed lines present an intensity peak situated between the nominal wavelength of the line and the line with the greatest shift on the arc axis; (b) the strongly absorbed lines present self-reversal. The absorption m~imum presents a red shift with respect to the nominal wavelength of the line; and (c) total local thermal equilibrium is not reached in the peripheral regions of the arc where the 4s levels are overpopulated owing to radiative cascades. This leads to an increase in the absorption of the transitions going towards the 4s levels.
1. I~ODU~IO~ THE TECHNOLOGY based on the use of thermal plasmas produced by torches and transferred arcs is steadily being developed. These types of plasmas operate most frequently in an argon atmosphere and are increasingly used in metal-working industries. The development of the processes requires long and costly testing programmes and the di~culties inherent in creating all the possible working conditions in the laboratory have led to the design of more and more sophisticated mathematical models. In many cases, these models assume the existence of local thermodynamic equilibrium (LTE). Validation of the models usually involves the measurement of parameters such as temperature and particle density obtained from the spectroscopic study of the lines emitted by the arc. The determination of the local intensity of the lines requires Abel inversion of the experimentally obtained integrated profile (Fig. 1). To enable the inversion to be carried out, two conditions must be strictly observed:
Fig. 1. Section of the discharge: J(r) is the local intensity; f(x) is the total emission integrated along a chord. 711
712
-
Argon
Argon
1
2
Fig. 2. Arc chambers: (1) wall-stabilized arc; (2) rf plasma torch.
(i) the plasma must be symmetrical with respect to the axis of obse~ation; and (ii) the line must not be self-absorbed. With electric arcs and in particular in wall-stabilized arcs, the first condition is reached. In rf plasmas this is not always the case since inside the coil, asymmetry of the emission can be observed [l]. The second condition depends on the line being studied, some lines being strongly self-absorbed. Absorption depends on the thickness of the plasma. In our exprimental conditions (see Section 2.2), the temperature profile is relatively flat with a minimum on the axis. In a wall-stabilized arc the temperature is greatest on the axis and decreases rapidly towards the walls. For a given temperature at the axis, absorption will be greater in rf plasmas. Thus, Abel inversion applied to a strongly absorbed line, such as that at 811.53 nm (transition 3p54p-3p54s), leads to erroneous results [2]. Also, the lines are broadened and subjected to shift owing to the Stark effect. The signal emitted along a chord or diameter of the discharge is the sum of the emissions arising from regions with different values of temperature, T, and/or electron number density, n,. The result is generally a line that is broadened and asymmetric with an emission ma~mum situated at a value of the wavelength X that depends on the structure of the discharge. The shift, therefore, cannot be considered to depend solely on the characteristics of the atomic transition lines [3]. This led us to study the line profiles of argon, along a diameter, emitted by two types of thermal discharge with very different diameters: a wall-stabilized arc of 6 mm in diameter; and an rf plasma torch of 50 mm in diameter. The experimental study allowed us to define the characteristics of the lines and the infhrence of the thickness of the plasma. From the temperature profiies T(r) and assuming total LTE, the profile of the lines J(h) can be calculated for each point of the plasma. It is therefore also possible to calculate the profile of the line emitted by a diameter (or a chord) of the discharge. Comparing the experimental profiles and the calculated profiles checks the validity of the hypothesis that assumes total LTE.
2. EXPERIMENTAL 2.1. Wall-stabilized arc The arc used in this study was of the cascade type (Fig. 2). The chamber was composed of a stack of hollow water-cooled discs. The central hole forming the chamber had a diameter of 6 mm. Sightings were made laterally between the two central discs. All the experiments were performed at atmospheric pressure. Arcing was initiated by means of a tungsten rod passing through one of the electrodes. The distance between the electrodes was 7 cm. The maximum current output of the power supply was about 100 A.
Spectroscopic study of Ar lines
713
2.2. High-frequency plasma The essential features of the standard-design induction plasma torch used in this work are shown in Fig. 2. The plasma was generated using a radio-frequency (rf) power supply with a nominal oscillator frequency of 3 MHz and a maximum plate power of 25 kW. Two gas streams were introduced into the torch. The sheath gas, QS, was of pure argon, 72 1 min-‘; the plasma gas, Q2, was also of pure argon, 43 1 min -I. No central gas, Q1. was introduced into the torch in the present measurements. The plasma was confined in a 50 mm i.d. water-cooled quartz tube. The windings were 20 mm apart and the outside diameter of the coil that they formed was 68 mm (Fig. 2). The operating pressure was one atmosphere. 2.3. Optical system A 20 cm focal length converging lens formed a 1:l scale image of the discharge on the inlet slit of the monochromator. Two iris diaphragms were placed at the focal points of the lens. Radial exploration of the discharge was carried out by side-on sighting moving the arc chamber perpendicularly to the optical axis of the system. Under our experimental conditions, the apparatus function was measured using a low-pressure argon spectral lamp (its lines broadened by the Doppler effect), with the inlet and outlet slits set at about 20 km: the apparatus width was found to be around 0.025 nm. For the high-frequency plasma, the proper choice of the focal length of a lens pair allowed the formation of a l/10 reduced image of the plasma on the plane of the monochromator entrance slit. A pair of parallel mirrors was used for vertical displacement to change the scanning level of the plasma. This could not, however, be freely chosen owing to the obstruction of the line of sight by the induction coil. The level at which measurements were carried out (Fig. 2) was 45 mm downstream of the coil. The output spectrum was sampled by an array of photodiodes laid out along the axis of wavelength dispersion. Each photodiode received a given wavelength of the spectrum, and the largest spectral interval resolved was about 0.02 nm. In our experimental conditions, with an entrance slit of around 20 km, the apparatus function was found to be roughly 0.035 nm with a low-pressure argon lamp.
3. THEORETICAL STUDY Knowing the local values of the temperature, T, and the electron density, n,, the profile of a line J(h) can be calculated. Note that the line will be broadened and will undergo a shift. Each mechanism of line broadening depends on the type of interaction involved and will give a profile that is: (a) Gaussian through the Doppler effect; (b) Lorentzian through the Stark effect; and (c) Lorentzian through interaction with neutral particles. Convolution of a Gaussian profile with a Lorentzian profile gives a Voigt profile, which can be written in a normalized way and in frequency units:
with: a = g
(ln2)“* D
b
=
(v
-
vo SD
+
A>
ln2 J
(3)
where 26, is the total half-width of the Doppler profile, vO is the frequency at the nominal value of the line peak, 26r is the sum of the half-widths due to the effect of collision (Stark and neutral van der Waals broadening), A is the line shift, and y is the mute variable that corresponds to the exponent in the Gaussian profile initially defined by: Y=
7
(ln2)“*.
714
S. VAcQulB et al.
The local monochromatic radiation emission coefficient J(X), at a distance r from the axis, is given (in Watts per cm3 and per steradian) by the expression:
J(X) =
&A,, N&A).
(4)
The term A,, is the probability of spontaneous emission of the emitter level m towards level rz. P(A) is the shape factor defined by Eqn (1). Thus, the profile of the line is defined; N, is the density of the emitting level that, assuming LTE, depends on the pressure and temperature. Figure 1 represents the scheme of a section of the discharge. The intensity emitted by a chord is the sum of the local emissivities along the chord through the inhomogeneities of the plasma. If we call k(r,h) the coefficient of local absorption, it can be shown that the intensity 1(x,X) emitted by a chord is expressed as a function of J and of k [4] by the following expression:
Z(x,A) =
x R
1
rdr
(r2- x2)l12
J(r,A) and k(r’,A) are the implicit functions of the excited atom densities and are therefore implicit functions of the temperature. Under the influence of charged particles, the lines are shifted by a value A calculated from the results of GRIEM [5,61. The values A0 and A, given by GRIEM lead to results that differ from the experimental values determined with the wall-stabilized arc in particular for the line at 811.5 nm. A good agreement is found by multiplying the values of A0 and 6, by a factor of 0.65. So, assuming the existence of LTE, if the temperature profile is known, the intensity I(X) emitted by a chord can be calculated. However, the excited atoms on the levels involved in the transition in question are not necessarily in thermodynamic equilibrium at the electron temperature of the medium. This is often the case, with argon, for the four 4s levels (two metastable levels and two resonant levels that can be overpopulated) [7]. This gives rise to a greater absorption of the transitions leading to the 4s levels than that predicted assuming complete LTE. To account for these deviations from LTE, either the absorption coefficient k(A) of the transition or the density of the excited atoms on the metastable or resonant level must be known. The total absorption coefficient is given by:
k(A)dh=$,,,,iV,, x where f,,,,, is the oscillator force on absorption of the transition, and N, the density of excited atoms on the lower level n of the transition m + n. We also have: k(A) = KP(A). Knowing N, and P(A) we can therefore by a diameter using Eqn (5).
reconstruct
(7)
the profile of the line emitted
Spectroscopic
715
study of Ar lines
II
0
5
10
15
20
r (mm)
Fig. 3. Temperature
profiles (1) for the arc and (2) for the rf plasma.
4. METHODOF CALCULATION To determine the temperature, we used the classic Boltzmann plots method. Two groups of lines were measured. The first includes seven lines arising from the 3p55p-3~~4s transitions of argon (415.8, 418.1, 420, 425.9, 426.6, 427.2 and 430 nm). These lines are not at all absorbed by the plasma. The second group is composed of four lines coming from the 3p54p-3p54s transitions. Some of the lines from these transitions can be strongly absorbed. We therefore chose four lines with negligible absorption: 696.5, 706.7, 727.3 and 738.4 nm. The local value of the intensity of each line was obtained after Abel inversion of the integrated profile. The method does not allow temperatures lower than 8500 K to be obtained without making large errors. We therefore calculated the profile for our experimental conditions using an energy balance model (of the Elenbaas-Heller type) for the stabilized arc [8] and a more complex model for the rf plasma [9]. A good agreement was found between calculation and experiment. Below 8500 K, the agreement was good with the results published in the literature [9, lo]. We were thus able to extrapolate our experimental values to 7000 K (Fig. 3). It is very possible that in the argon plasma, LTE is only partial and that radiative cascades overpopulate the 4s level when the electron density is insufficient for collisional equilibrium. In order to account for any deviation from LTE, the real densities of the excited atoms on the 4s levels must be introduced into the calculation. We used values obtained a few years ago under strictly identical experimental conditions with a wall-stabilized arc [ll]. They were determined by absorption spectroscopy using the continuum of a high-pressure xenon lamp in a side-on observation and with Abel inversion. The details of the method and the results obtained are fully described in Ref. [ll]. In Fig. 4, for a given temperature profile, we compare the theoretical values of the number density of 1~~ (the metastable level) for LTE with the experimental values. On the axis there is good agreement. At the edges, the LTE value rapidly tends to zero whereas the experimental value remains at a density of around 2 X lOI* cmw3 (measurements for r > 3 mm correspond to the diffusion region between the discs). We used these non-equilibrium experimental values in the calculation. The experimental set-up used did not allow the same measurements to be made with the rf plasma. So, for each value of the temperature, T, in the rf plasma, we used the corresponding value of the 1s5 level obtained with the wall-stabilized arc. Henceforward, the figures obtained are called “experimental values” indifferently for the stabilized arc and for the rf plasma, knowing however that for the rf plasma it is a justified analogy since the time-characteristic for diffusion losses is much larger than that for collisions. It can therefore be assumed that losses through diffusion are negligible
716
S. VacourB et al. 20
I
0
1
I
I
2
3
3
r(mm)
Fig. 4. Comparison between theoretical and experimental values of the metastable atoms (Is5 level) along a radius of the wall-stabilized arc.
compared to collisional losses and that it is the local temperature and electron density that govern the metastable atom densities more than the gradients. The line profiles at LTE were calculated assuming that the whole set of levels was in equilibrium for the temperature of the medium. To take deviation from LTE into account. two methods of calculation were used. In the first, we assumed that the upper level of the transition in question was in equilibrium with the metastable level. The intensity profile J(X) of the line at a given point in the plasma was calculated for a Voigt profile (Eqn 1) using Kirchhoff’s law: J(X) = B, k(h).
(8)
L?, is given by Planck’s law and is written, as a function of the wavelength,
A, by:
2 ~
[exp (hc/hkT)-11-l
(9)
where the terms have their usual meaning. Here, k(h) is obtained from Eqns (6) and (7). The metastable atom density, N, (Eqn 6), was obtained experimentally. It is clear that if the metastable
level is overpopulated,
the intensity of the line,
J(X), will be
overestimated. So, the total intensity integrated along a diameter (EqnA5) will be the upper limit. In the second method, the starting level was assumed to be in LTE and, for the final level, we used the experimentally determined value. This is the more realistic procedure. The second method was generally used, except for the strongly absorbed 811.5 nm line where the two methods were compared. As mentioned earlier, the widths of the apparatus function were determined experimentally: there was no apparent dissymmetry. Theoretically, with the photomultiplier, the function should be a triangle; with the diode array, it is more complex. It was demonstrated that no detectable error was incurred on the final result when it was assumed that the curves were Gaussian in shape on the condition that the same width at half height was taken. It was therefore considered in the calculations that the apparatus functions were Gaussian.
5. RESULTS The results presented concern three characteristic lines: (i) the 420 nm line (3p55p-3p54s transition), which is almost totally unabsorbed; (ii) the 763.5 nm line
Spectroscopic
r
420
0675
study of Ar lines
717
nm 0
ExperImental _--
L
420
0
420.
I
420.2
420.3
h (nm)
Fig. 5. Comparison between the experimental and the theoretical profiles of the 420 nm line: (1) nominal wavelength of the line; (2) maximum of the line emitted by a diameter; and (3) maximum of the line on the axis.
(3p54p-3~~4s transition), which is only slightly reabsorbed; and (iii) the 811.5 nm line (3~~4~7-3~~4~transition), which is very strongly reabsorbed. Figure 5 presents the theoretical profile given by a diameter of the stabilized arc for a line that is not self-absorbed (420 nm) of the transition 3p55p-3~~4s. It can be seen that for a value of 11000 K on the temperature axis, the profile of a line emitted by a diameter is no longer symmetrical. The maximum is situated at A, which is between A,,, the nominal wavelength of the undisturbed line, and A,, + Ah,,, AA,, being the value of the maximum shift of the line on the axis of the arc (AA,ax = 0.0475 nm). The shift observed for a line emitted by a diameter can therefore not be taken as a measurement of the characteristic shift of the line for the hottest regions of the arc. The profile obtained is in very good agreement with the experimental profiles recorded with either of the set-ups used. Figure 5 also gives the experimental profile of the 420 nm line obtained with the rf plasma at atmospheric pressure. As absorption is negligible, the general shape of the line profile emitted by a diameter is not dependent on the thickness of the plasma but on the temperature maximum, which is identical in the arc and in the plasma. The strong lines corresponding to the transitions 3p54p-3~~4s are absorbed. So, the shape of the line emitted by the plasma depends on the structure of the discharge. Figure 6 gives the theoretical profile of the 763.5 nm line emitted by a diameter of the stabilized arc. This line corresponds to the transition between level 2ps of the 3p54p levels and the metastable Is5 level of the 3p54s configuration. The same phenomenon is observed as that seen with the non-absorbed line, i.e. the emission peak occurs between the nominal wavelength of the line (A = 763.525 nm) and the maximum wavelength of the line with the greatest shift on the axis of the arc, which corresponds to T = 11000 K. No anomalies owing to absorption are noted in this theoretical result. Figure 6 also represents the experimental profile of the line obtained along a diameter of the stabilized arc. Near the peak, in the blue part, a small disturbance, due to absorption, is observed in contrast to the theoretical profile. This disturbance is in the vicinity of the nominal wavelength of the line. OLSEN [12], who observed a similar phenomenon on this line, suggested that it could be caused by reabsorption in the plasma. The poor agreement in Fig. 6 between the theoretical curve (taking
S. VACQW? et al.
718
163.521 nm
763.4
763.5
763 6
It (nm)
Fig. 6. Comparison between the experimental and the theoretical profile of the 763.5 nm line, emitted by a diameter of the wall-stabilized arc.
deviation from LTE into account) and the experimental results strongly indicate the existence of another reason. We had also observed this phenomenon in the past but did not find a satisfactory interpretation [13]. With the rf plasma, and with a power input of 22 kW, the line emitted by a diameter is dissymmetrical, shifted, broadened and clearly shows absorption near the peak on the blue part (Fig. 7). The line emitted by a chord at 10 mm from the axis presents the same characteristics with a stronger absorption than in the previous example. At 15 mm from the axis, the line is much less intense, not shifted, and very slightly dissymmetrical. Figure 7 also represents the calculated profile of the 763.5 nm line obtained without assuming the existence of LTE for the Is5 level for which we used the experimentally determined densities. With a Gaussian apparatus function of 0.035 nm, we observed the same shape as that of the experimental profile with a disturbance on the blue part near the peak. The uncertainties on the line broadening and on the line shift and also the errors on the temperature profiles and on the atom densities explain the difference between the half-widths of the experimental and the theoretical profiles.
163 4
763.5
163 6
k (nm)
Fig. 7. Comparison
between the experimental and the theoretical profiles of the 763.5 nm line, emitted by a diameter of the rf plasma.
719
Spectroscopic study of Ar lines
. I
I XII
81 1.6
5 h
(nm)
Fig. 8. Calculated profiles of the 811.5 nm line for the wall-stabilized arc without apparatus function convolution. Curve 1: LTE hypothesis. Curve 2: Kirchhoff’s law and experimental value of metastable atom density N,,,. Curve 3: partial LTE for the upper level of the transition (2p,) and experimental value for N,,,.
As the 811.5 nm line is strongly reabsorbed we shall study the calculation of the reconstruction of its profile in more detail. Figure 8 represents the theoretical profile of the line emitted by a diameter of the stabilized arc with the three hypotheses used. Curve 1 assumes LTE; curve 2 was calculated with a Voigt profile using Kirchhoff’s law and the experimental value for level 1~~; curve 3 was calculated with a Voigt profile but Kirchhoff’s law was not used: level 2ps was assumed to be in equilibrium and level Is, at its experimentally determined density. Strong reabsorption is observed at a wavelength close to the nominal wavelength of the line. In order to be able to compare these curves to the experimental ones, the profile must be convoluted with the apparatus function, which has a half-width of 0.025 nm. The results obtained are presented in Fig. 9. Curve 1 represents the profile of the line assuming LTE. Taking the apparatus function into account causes the disappearance of the modification due to absorption on the blue wing near the peak. A good agreement is seen between the curve obtained by calculation after convolution of the apparatus function and the experimental curve when the 2pg level is not considered to be in equilibrium with the lsg level (curve 3 of Fig. 7). Figure 10 represents the 811.55 nm line profiles obtained experimentally on the rf torch at 100 kPa for a power of 22 kW at a distance z of 45 mm and for 0 and 10 mm from the axis. Self-reversal of the line can be seen. The absorption maximum is not situated exactly at the nominal wavelength of the line (here, it is observed at 811.537 nm). The line emitted at 10 mm from the axis presents an absorption dip greater than that observed across a diameter. The chord is shorter than the diameter but emission is stronger there (Fig. 3). However, absorption is more intense and the central dip in the line is deeper. The broadening of the lines due to the Stark effect is greater for the 420 nm line than for the 763.5 nm line or the 811.5 nm line. For T = 10 000 K, the broadening Ahs due to the Stark effect is 3.3 X lo-* nm for the 420 nm line and 1.6 x lo-* nm for the 763.5 nm line and the 811.5 nm line. So, the 763.5 nm line, being only slightly absorbed, its half-width is smaller than that of the 420 nm line. On the other hand, the 811.5 nm line, being strongly absorbed, its apparent half-width is greater than the
S. VACQUI~et
120
al.
I XII
I
-
811 6
5
h (nm) Fig. 9. Comparison between experimental and calculated profiles of the 811.5 nm line for the wall-stabilized arc after apparatus function convolution. Curve 1: LTE hypothesis. Curve 2: curve 3 of Fig. 8 after convolution, Curve 3: experimental profile.
8114
811
5
XII
6
XII
7
h(nm)
Fig. 10. Experimental
profiles of the 811.5 nm line emitted by a diameter (x = 0) and by a chord (x = 10 mm) of the rf plasma.
half-width of the 420 nm line (and of course than the half-width of the 763.5 nm line). This is clearly visible in the experimental results. In Fig. 11 we have presented the theoretical profiles of the 811.5 nm line along a diameter and along a chord (at 10 mm) for a half-width of the apparatus function of 0.035 nm and assuming that LTE is reached in the plasma. Absorption greatly reduces the intensity of the line peak but is only apparent on the profile through a slight dip near the summit. Comparison with Fig. 10 shows therefore that if LTE is assumed to exist, the profiles calculated are very different to those obtained experimentally. On
Spectroscopic
8114
721
study of Ar lines
811
5
811.6
811.7
h(nm)
Fig. 11. Calculated profiles of the 811.5 nm line emitted by a diameter (X = 0) and by a chord (x = 10 mm) of the rf plasma assuming LTE.
reconstructing the profile with levels 2p9 and lss, good agreement with experimental profiles is obtained. Figure 12 represents the same profiles for 0 and 10 mm from the axis assuming the 2p, level to be in Saha equilibrium with the electrons and taking the experimentally determined density for the Is5 level. It can be seen that for an instrument function of 0.035 nm, agreement with experimental results is good. Here again, the absorption maximum is seen to be shifted with respect to the nominal wavelength of the line. The results show that the peripheral regions of the arc and of the rf plasma are not in LTE. The metastable overpopulation leads to line profiles emitted across a diameter or a chord that are very different to those that would be found if LTE were reached.
-*= X= ,___
811.4
811.5
811.6
811
7
h(nm)
Fig. 12. Calculated protiles of the 811.5 nm line emitted by a diameter (x = 0) and by a chord (X = 10 mm) of the rf plasma with the experimental values of metastable atom density determined in the arc.
722
S.
VACQ&et al.
6. CONCLUSION The present study has enabled comparisons to be made between the profiles of the argon lines emitted by the plasma of a wall-stabilized arc and that of an rf plasma across a diameter and across a chord. It also enabled us to compare the experimentally obtained profiles with theoretical profiles calculated firstly by assuming LTE and secondly by incorporating experimental values. We can thus show that: (a) the peak of the non-absorbed lines is situated between the nominal value of the line and the maximum wavelength of the line that has undergone the greatest shift (where T is maximum). The line profiles in the arc and in the rf plasma are identical; (b) the weakly reabsorbed lines present the same characteristics, absorption introducing a disturbance in the blue part; and (c) in the case of self-reversed lines, the absorption maximum is shifted towards the blue with respect to the position of the maximum Stark shift. Considering the greater diameter of the plasma, the lines undergo stronger absorption. This gives rise to a greater width at apparent mid-height (0.08 nm in the arc, 0.12 nm in the torch). These results show the influence of the various processes of broadening and confirm that the argon plasma is not in total LTE in the peripheral zones of the discharge where metastable atoms are overpopulated and lead to an increase in the absorption of the transitions towards the 4s levels. REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13]
L. Bydder and G. P. Miller, Specfrochim. Acta 43B, 1431 (1988). Uchida, K. Tanabe, Y. Nojiri, H. Haraguchi and K. Fuwa, Spectrochim. Acta 36B, 711 (1981). Kato, H. Fukushima and T. Kakajima, Specrrochim. Acta 39B, 979 (1984). Vacquit, J. Bacri, M. Capderou, J. P. Dinguirard and A. M. Gomes, J. Quant. Spectrosc. Radiat. Transfer 13, 1333 (1973). H. R. Griem, Phys. Rev. 128, 515 (1962). H. R. Griem, Spectral Line Broadening by Plasma. Academic Press, New York, London (1974). J. Bacri, S. VacquiC, A. M. Gomes and M. F. Habibi, XI ht. Conf. on Ionization Phenomena in Gases, p. 415, Prague (1973). A. Gleizes, H. Kafrouni, H. Dang Due and V. C. Maury, J. Phys. D: Appl. Phys. 15, 1031 (1982). J. Mostaghimi, P. Proulx and M. Boulos, Numerical Heat Transfer 8, 187 (1985). R. J. Giannaris and F. P. Incropera, J. Quark Spectrosc. Radial. Transfer 11, 298 (1971). J. Bacri, A. Gleizes and A. M. Gomes, J. Quant. Spectrosc. Radiat. Transfer 15, 7 (1975). H. N. Olsen, J. Quant. Spectrosc. Radiat. Transfer 3, 67 (1963). A. Gleizes, Thtse de 3” cycle no 1542, University Paul Sabatier, Toulouse, France (1974). E. H. K. S.