# Self-Reversed scandium lines as a diagnostic tool for multicomponent HID lamps.

1 INTRODUCTION

High-pressure discharge lamps (HID) are still state of the art in professional lighting. But even after decades of research the understanding of the physics inside the lamp is still challenging [Araoud 2010, Brok 2006, Dabringhausen 2005, Dreeben 2011, Hartel 1999, Wei[beta] 2005], where the examination of a multi component plasma is a much more complex task compared to a 'simple' mercury HID lamp, which is the 'white mouse' of HID lamp research.

A detailed characterization of the plasma is desirable in many applications where the diagnostics of temperature and partial pressure is an inevitable precondition. Optically thin lines can be utilized for this purpose by evaluating the emission coefficient [epsilon] of the given line, by performing a Boltzmann plot or by fitting the spectral line shape [van den Hoek 1983, Lochte-Holtgreven 1995]. However, these methods can be disadvantageous for several reasons that may limit or hamper their applicability: The evaluation of temperatures from emission coefficients already requires information about particle densities; the Boltzmann plot strongly depends on the excitation level distribution of the selected species as well as on the accuracy of their transition probabilities; and a spectral line fit is a sophisticated task requiring a couple of input parameters.

In this paper a lamp with a complex fill composition, consisting of scandium and sodium in addition to mercury and argon as start gas, is examined by means of optical emission spectroscopy. This is motivated by the following facts: Scandium as fill material is of particular importance for the general lighting markets in North America as well as for the automotive lighting industry. Furthermore, mercury containing plasma light sources are intended to be replaced by environment-friendly mercury-free alternatives excluding Hg lines as a diagnostic tool. In [Born and Stosser 2007] and [Kettlitz and others 2007] mercury-free automotive headlight lamps containing scandium were investigated, where plasma temperatures were obtained by numerical plasma simulations in [Born 2007] and by fitting the spectral line shape of optically thick sodium lines in [Kettlitz and others 2007].

The purpose of this work is to analyze the capability of self-reversed scandium lines to determine the radial temperature profile using Bartels method [Bartels 1950a, Bartels 1950b, Schneidenbach and Franke 2008].

Self-reversed spectral lines are observed in spatially inhomogeneous optically thick plasmas. Due to strong self-absorption in the center of a spectral line a minimum of the line contour is formed, where no radiation of the hot arc center can escape. However, in the line wings, where the optical depth is lower, radiation of the hot arc center can escape forming self-reversal maxima. Bartels [Bartels 1950a, Bartels 1950b] as well as Cowan and Dieke [Cowan and Dieke 1948] found that the radiance of the self-reversal maxima of optically thick spectral lines is related to the maximum temperature along a line of sight in a plasma. Local thermodynamic equilibrium (LTE) usually is assumed for the plasma.

Since the analysis of self-reversed lines is well established for mercury [Karabourniotis and others 1982, Schneidenbach and Franke 2008], the self-reversed Hg I lines at 435.8 nm and 546.1 nm were chosen as reference lines in this work. It is shown that self-reversed scandium lines are a proper diagnostic tool for temperature determination in HID lamps.

The measurements performed in this work with the associated experimental setup are described in Section 2. Resulting temperature profiles obtained by scandium lines are compared with mercury reference line in Sec. 3 where also the partial pressures are obtained. Finally, the work is summarized in Sec. 4.

2 SETUP AND MEASUREMENTS

As an example of a scandium containing HID lamp a commercial multicomponent metal halide lamp (Sylvania M400) was investigated. The inner discharge tube consists of quartz. Exact inner vessel dimensions were obtained by a X-ray system (VISCOM X8060) which scanned the whole lamp with submillimeter resolution as shown in Fig. 1. The inner diameter was determined to be 19.4 mm and the electrode gap was 36.9 mm. The fill consists of several mbar argon as start gas, mercury as the main component and additives of sodium and scandium iodides.

The lamp was operated vertically by a power amplifier (FM Elektronik FM1295) with a 50 Hz sine current at an rms value of 3.26 A. Stationary conditions were established about 5 minutes after ignition at a power level of 400 [+ or -] 2W and an rms voltage of 123 V. Electrical data were measured by an oscilloscope (Tektronix TDS784, 1 Gs/s).

Radially resolved spectral measurements were performed in the middle of the lamp, where high spectral resolution is necessary in order to detect and resolve the self-reversal line structure. The wavelength difference between the two reversal maxima of the mercury line at 435.8 nm in Fig. 3 is about 0.05 nm which demands a spectral resolution well below this value. For this purpose the setup shown in Fig. 2 was used. It consists of an Echelle spectrometer system (Solar TII MSE750+MS7501) combined with an intensified CCD camera (Proscan IHSS 1024-25, 1024 x 1024 pixel). Such a spectrometer concept allows the predispersion of a wavelength interval by the Czerny-Turner type spectrometer and a further dispersion by the following Echelle type spectrometer. Considering the apparatus function of this system a spectral resolution of about 0.002 nm full width at half maximum (FWHM) was achieved. Since the lamp was driven by a 50 Hz sine current all measurements were synchronized to the current maximum with an exposure time of 2.5 ms. Absolute intensity calibration was performed with a tungsten ribbon lamp which was included into the setup.

The mercury lines at 435.8 nm and 546.1 nm were chosen for reference, because they are isolated and not disturbed by neighboring lines. Fig. 3 shows the 435.8 nm line at different radial positions. The self-reversal is characterized by the double peak structure with minimum intensity near the central wavelength. The two self-reversal maxima used for temperature evaluation at lower and higher wavelengths around the minimum will be referred to as the 'red wing' and 'blue wing' respectively. For scandium three lines at 473.8, 474.1 and 474.4 nm were found to be suitable for evaluation as they show a well developed self-reversal at the given plasma conditions. Even though there are much more scandium lines with a self-reversal, especially around 400 nm, these lines are either too weak or disturbed by neighboring lines. Intensity profiles of the three scandium lines at different radial positions are shown in Fig. 4.

Additionally, optically thin lines were measured for subsequent partial pressure calculation. All analyzed lines with their atomic data are listed in Table 1.

3 RESULTS AND DISCUSSION

Since the radiance of the self-reversal maxima is crucial for temperature evaluation with Bartels method, care must be taken in the determination of that radiance value. Experimental uncertainties may arise from absolute intensity calibration, from background radiation as well as from limited spectral resolution.

But firstly it must be noticed that radiation is lost by reflection at each interface were the refractive index is changing [Seehawer 1973, Wendt 2008]. Reflection losses depend on the angle of incidence and the optical depth. In the optically thick limit follows the reflection for perpendicular incidence directly from Fresnel equation:

[phi] = [(n-1)/(n+1).sup.2] (1)

where [phi] is the reflection coefficient for one interface and n the refractive index of quartz. In case of a burner with an outer bulb there are four interfaces between quartz and vacuum and ambient air, respectively. Hence, for a refractive index of quartz n = 1.46 the radiance of self-reversal maxima is reduced by around 14 percent, which must be considered before evaluation.

Furthermore, as can be seen from Fig. 4, the red line wing of the Sc I line at 474.1 nm is running into the blue wing of the Sc I line at 474.4 nm. By extrapolation one obtains a background contribution to the blue self-reversal radiance of the Sc I line at 474.4 nm of around 800 W [m.sup.-2] [nm.sup.-1] [sr.sup.-1] which amounts to 14 percent of the self-reversal radiance. However, one has to consider that the plasma at wavelengths around a self-reversal maximum is not optically thin but characterized by an optical depth of at least [iota]=2. Hence the background contribution is reduced by a factor of [e.sup.-2] to 108 W [m.sup.-2] [nm.sup.-1] [sr.sup.-1], amounting to less than 2 percent of the self-reversal radiance.

To resolve self-reversal line contours usually one needs sufficient high spectral resolutions. But even with the setup used in this work deconvolution of spectral lines has been performed with apparatus profile to check if there was some effect of spectral resolution on the self-reversal radiance. Finally is was found that deconvolution leads to self-reversal radiances that are 10 percent higher in case of Hg lines and 20 percent higher in case of Sc lines, where the uncertainty of this correction is still around 30 percent, because deconvolution is an ill-posed process amplifying statistical noise.

The uncertainty by absolute radiance calibration with a tungsten ribbon lamp accounts to 10 percent. Hence the total experimental radiance uncertainty of self-reversal maxima is estimated to be less than 18 percent in case of Sc lines and less than 16 percent in case of Hg lines.

The maximum plasma temperature along a line-of-sight is given by the equation [Bartels 1950]:

T = ([E.sub.u] - [E.sub.l])/[K.sub.B]/In(2h[c.sub.0.sup.5]/[[lambda].sub.0.sup.5]) + In(MY) - In([L.sub.max]) (2)

where [E.sub.u] and [E.sub.l] are the upper and lower energy of the levels involved in the transition, [k.sub.B] is the Boltzmann constant, h the Planck constant, [c.sub.0] the velocity of light in vacuum, [[lambda].sub.0] the center wavelength of the transition, [L.sub.max] the radiance of the self-reversal maximum. All quantities must be given in SI units. The Bartels functions M and Y are defined by the following equations in the most simple approach of the Bartels method:

M = [square root of ([E.sub.l]/[E.sub.u])]

Y = 0.736 + 0.264[p.sup.2]

p = 6/[pi]arctan ([M.sup.2]/ [square root of (1+[2M.sup.2]) (3)

where p is called inhomogeneity parameter assuming a parabolic temperature profile. This approach is of limited applicability. In more general approaches of Bartels method M would be a function of the spatial plasma temperature distribution and the plasma composition. In this case equations would have to be solved iteratively, like it was done here. For the lines evaluated here it was found that iteration yields almost the same temperatures as the most simple approach of Bartels method (Differences are less than 1 percent). But this is not a common behavior and usually it is necessary to apply the more general approaches of Bartels method as discussed in [Franke and Schneidenbach 2007, Schneidenbach and Franke 2008].

Without going into too much detail the Bartels functions can be given by these equations in their general form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Y = [2(1-p).sup.-1/2][(1 - [square root of (p)])/[(1 + [square root of (p)])].sup.1/(2[square root of (p)])] (4)

where [n.sub.u,l] are the population densities of the upper and the lower level, respectively, and [g.sub.u,l] are the corresponding statistical weights. [T.sub.m] represents the maximum temperature along the line-of-sight coordinate x at a given side-on position. Cylinder symmetry of the problem is assumed. The analytic approximation of Y in (4) has a higher accuracy of about 0.4 percent compared to that originally given by Bartels in (3). In M enters the so-called far line wing approximation, which requires a well developed self-reversal, allowing to separate the profile function into a product of a frequency dependent and a position dependent term. [PSI](x) represents that position dependent term which equals the profile width in case of a pure Lorentz line profile. Further approximations applied throughout this work are the assumption of a Boltzmann distribution of the excited level population and the ideal gas law. In this case all Bartels functions still depend on the side-on position and an iterative scheme has to be applied to yield plasma temperatures. It has to be noticed that ionization as well as radial cataphoresis and axial segregation violate the applicability of the ideal gas law, limiting the accuracy of the Bartels method. This is of particular interest, because a significant ionization and spatial demixing of additives in metal-halide lamps is expected [Dakin 1989, Hashiguchi and others 2002]. However, it reveals that the method is not sensitive to moderate deviations from spatially uniform partial pressures of additives [Funk and others 1970], as long as the main contribution to the Bartels functions arise by the integrals from regions around the discharge axis [Franke and Schneidenbach 2007]. The reasonable agreement of plasma temperatures obtained from self-reversed mercury lines and scandium lines can be motivated by this fact. A more detailed, quantitative analysis of this topic requires plasma composition and radiative transfer calculations which are beyond the scope of this paper.

For discussion of uncertainties of temperature evaluation the most simple approach is still followed. In this case equations (2) and (3) lead to a very simple and direct representation of the plasma temperature as a function of the self-reversal radiance for the Hg I 435.8 nm line:

T = 3.3005 x [10.sup.4]/36.562 + In(0.7062) - In([L.sub.max]). (5)

An equivalent equation can be given for the Sc I 473.8 nm line:

T = 3.0369 x [10.sup.4]/36.1463 + In(0.4777) - In([L.sub.max]). (6)

Error propagation now can tell us that even if the radiance of the self-reversal maximum is known with an uncertainty of 20 percent the plasma temperature can be given with an accuracy better than 3 percent. Furthermore it can bee seen from the equations above that only well-known atomic constants are needed for temperature evaluation. There is no need to know the radiators partial pressure or transition probabilities, which are often given with significant uncertainties.

In Table 2 results of temperature evaluation with Hg and Sc lines are shown. Axis temperatures determined with Hg and Sc lines are around the mean value of 5210 K.

Due to the fact that self-reversal vanishes around radial positions r/R=0.3 radial temperatures can be given up to this point and have to be extrapolated to an expected wall temperature of around 1000 K.

Figure 5 shows the measured temperature profile of the Hg I 435.8 nm line with a parabolic extrapolation to the wall, whereas Fig. 6 depicts a more detailed view of all measured temperature profiles near the arc center. It can be stated that temperatures determined with Bartels method from self-reversed Sc lines agree well within experimental uncertainties compared to temperatures obtained from self-reversed Hg lines.

This indicates that all these Sc lines are suitable for temperature determination in HID lamps with Bartels method. This is of particular interest, because the lower levels of the considered Sc lines are around 1.4 eV, whereas Bartels required in his theory a lower level energy of [E.sub.l] [much greater than] 1 eV. This condition is sufficiently fulfilled and proven for self-reversed Hg lines (546.1 nm & 435.8 nm), where the lower levels have energies higher than 4.8 eV, but not for the discussed Sc lines. But even for the Tl I line at 535.0 nm with a lower energy level of 0.966 eV the Bartels method was already found to work well [Funk and others 1970]. Hence it can be concluded that the Sc lines at 473.8, 474.1 and 474.4 nm are indeed well suited for temperature determination with Bartels method.

The most challenging restriction is to provide a sufficient spectral resolution for proper evaluation of self-reversal maxima which should be better than 0.01 nm. A more detailed analysis of Bartels method applied to scandium lines requires elaborate plasma composition and radiative transfer calculations [Franke and Schneidenbach 2007, Schneidenbach and Franke 2008] which are beyond the scope of this work.

Once the plasma temperature is known, partial pressures of the radiating species can be obtained from optically thin lines, applying Abel inversion to their side-on radiance. For LTE conditions the emission coefficient [epsilon] can be given as a function of the plasma temperature T by:

[epsilon](T) = hc/4[pi] x [A.sub.ul][g.sub.u]/[[lambda].sub.0] x p/Q(T)[[kappa].sub.B]T x exp x (-[E.sub.u]/[[kappa].sub.B]T) (7)

with the transition probability [A.sub.ul], the statistical weight of the excited (emitting) state [g.sub.u] and its energy [E.sub.u], the emission wavelength [[lambda].sub.0], the radiators partial pressure p, the Boltzmann constant [k.sub.B] and the partition function Q(T) for the given species.

Table 3 summarizes the partial pressures obtained for a mean central arc temperature of 5210 K, where error propagation yields:

[delta]p/p = [sigma]T/T[absolute value of [R.sub.u]/[[kappa].sub.B]T - 1] + [delta][epsilon]/[epsilon]. (8)

Therefore, the relative uncertainty in partial pressure determination scales with upper level energy [E.sub.u]. Hence, assuming 10 percent uncertainty in the emission coefficient and 3 percent uncertainty in plasma temperature (as discussed above), the uncertainty in partial pressure ranges from 20 percent to 66 percent for upper level energies from 2 eV to 9 eV. Keeping this in mind, different partial pressures obtained from single lines agree within experimental uncertainties.

The plasma temperatures of around 5210 K found here reasonably agree with values given by [Dakin 1989] who obtained a plasma temperature of about 5000K in the middle of the discharge. The Hg partial pressure given in this work is around 6.4 bar whereas [Dakin 1989] obtained a Hg partial pressure of about 4.7 bar which is within the experimental uncertainty. However, the partial pressures of additives (Na, Sc) given here with mean values of around 0.55 mbar for scandium and 0.30 mbar for sodium significantly deviate from the values given by [Dakin 1989] but they are in the same order of magnitude. This is acceptable because partial pressures of additives are sensitive to cold spot temperatures which may differ comparing our measurements with [Dakin 1989].

4 SUMMARY

Temperature determination utilizing self-absorbed spectral lines is well established for mercury high-pressure discharge lamps. If the self-reversed Hg lines are not superimposed by spectral lines of additives they are well suited for temperature measurements in multicomponent HID lamps, too. In this work it is demonstrated that also self-reversed Sc lines can be applied for plasma temperature determination. The three scandium lines analyzed here were found to be suitable for evaluation with the Bartels method, hence extending diagnostic options. The experimental uncertainty is found to be better than 3 percent for plasma temperatures. The most challenging requirement for the proposed method is a sufficient spectral resolution to resolve the self-reversed line structure.

[c] 2013 The Illuminating Engineering Society of North America

doi: 10.1582/LEUKOS.2013.09.03.002

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NIST Atomic Spectra Database (ver. 4.1.0), [Online]. Technical report, National Institute of Standards and Technology, Gaithersburg (MD). Available: http://physics.nist.gov/asd [2012, July 10].

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Jacob Zalach (1) and Steffen Franke PhD

(1.) INP Greifswald, Felix-Hausdorff-Str. 2, 17489 Greifswald, Germany *Corresponding author: Jacob Zalach, Email: zalach@inp-greifswald.de

High-pressure discharge lamps (HID) are still state of the art in professional lighting. But even after decades of research the understanding of the physics inside the lamp is still challenging [Araoud 2010, Brok 2006, Dabringhausen 2005, Dreeben 2011, Hartel 1999, Wei[beta] 2005], where the examination of a multi component plasma is a much more complex task compared to a 'simple' mercury HID lamp, which is the 'white mouse' of HID lamp research.

A detailed characterization of the plasma is desirable in many applications where the diagnostics of temperature and partial pressure is an inevitable precondition. Optically thin lines can be utilized for this purpose by evaluating the emission coefficient [epsilon] of the given line, by performing a Boltzmann plot or by fitting the spectral line shape [van den Hoek 1983, Lochte-Holtgreven 1995]. However, these methods can be disadvantageous for several reasons that may limit or hamper their applicability: The evaluation of temperatures from emission coefficients already requires information about particle densities; the Boltzmann plot strongly depends on the excitation level distribution of the selected species as well as on the accuracy of their transition probabilities; and a spectral line fit is a sophisticated task requiring a couple of input parameters.

In this paper a lamp with a complex fill composition, consisting of scandium and sodium in addition to mercury and argon as start gas, is examined by means of optical emission spectroscopy. This is motivated by the following facts: Scandium as fill material is of particular importance for the general lighting markets in North America as well as for the automotive lighting industry. Furthermore, mercury containing plasma light sources are intended to be replaced by environment-friendly mercury-free alternatives excluding Hg lines as a diagnostic tool. In [Born and Stosser 2007] and [Kettlitz and others 2007] mercury-free automotive headlight lamps containing scandium were investigated, where plasma temperatures were obtained by numerical plasma simulations in [Born 2007] and by fitting the spectral line shape of optically thick sodium lines in [Kettlitz and others 2007].

The purpose of this work is to analyze the capability of self-reversed scandium lines to determine the radial temperature profile using Bartels method [Bartels 1950a, Bartels 1950b, Schneidenbach and Franke 2008].

Self-reversed spectral lines are observed in spatially inhomogeneous optically thick plasmas. Due to strong self-absorption in the center of a spectral line a minimum of the line contour is formed, where no radiation of the hot arc center can escape. However, in the line wings, where the optical depth is lower, radiation of the hot arc center can escape forming self-reversal maxima. Bartels [Bartels 1950a, Bartels 1950b] as well as Cowan and Dieke [Cowan and Dieke 1948] found that the radiance of the self-reversal maxima of optically thick spectral lines is related to the maximum temperature along a line of sight in a plasma. Local thermodynamic equilibrium (LTE) usually is assumed for the plasma.

Since the analysis of self-reversed lines is well established for mercury [Karabourniotis and others 1982, Schneidenbach and Franke 2008], the self-reversed Hg I lines at 435.8 nm and 546.1 nm were chosen as reference lines in this work. It is shown that self-reversed scandium lines are a proper diagnostic tool for temperature determination in HID lamps.

The measurements performed in this work with the associated experimental setup are described in Section 2. Resulting temperature profiles obtained by scandium lines are compared with mercury reference line in Sec. 3 where also the partial pressures are obtained. Finally, the work is summarized in Sec. 4.

2 SETUP AND MEASUREMENTS

As an example of a scandium containing HID lamp a commercial multicomponent metal halide lamp (Sylvania M400) was investigated. The inner discharge tube consists of quartz. Exact inner vessel dimensions were obtained by a X-ray system (VISCOM X8060) which scanned the whole lamp with submillimeter resolution as shown in Fig. 1. The inner diameter was determined to be 19.4 mm and the electrode gap was 36.9 mm. The fill consists of several mbar argon as start gas, mercury as the main component and additives of sodium and scandium iodides.

The lamp was operated vertically by a power amplifier (FM Elektronik FM1295) with a 50 Hz sine current at an rms value of 3.26 A. Stationary conditions were established about 5 minutes after ignition at a power level of 400 [+ or -] 2W and an rms voltage of 123 V. Electrical data were measured by an oscilloscope (Tektronix TDS784, 1 Gs/s).

Radially resolved spectral measurements were performed in the middle of the lamp, where high spectral resolution is necessary in order to detect and resolve the self-reversal line structure. The wavelength difference between the two reversal maxima of the mercury line at 435.8 nm in Fig. 3 is about 0.05 nm which demands a spectral resolution well below this value. For this purpose the setup shown in Fig. 2 was used. It consists of an Echelle spectrometer system (Solar TII MSE750+MS7501) combined with an intensified CCD camera (Proscan IHSS 1024-25, 1024 x 1024 pixel). Such a spectrometer concept allows the predispersion of a wavelength interval by the Czerny-Turner type spectrometer and a further dispersion by the following Echelle type spectrometer. Considering the apparatus function of this system a spectral resolution of about 0.002 nm full width at half maximum (FWHM) was achieved. Since the lamp was driven by a 50 Hz sine current all measurements were synchronized to the current maximum with an exposure time of 2.5 ms. Absolute intensity calibration was performed with a tungsten ribbon lamp which was included into the setup.

The mercury lines at 435.8 nm and 546.1 nm were chosen for reference, because they are isolated and not disturbed by neighboring lines. Fig. 3 shows the 435.8 nm line at different radial positions. The self-reversal is characterized by the double peak structure with minimum intensity near the central wavelength. The two self-reversal maxima used for temperature evaluation at lower and higher wavelengths around the minimum will be referred to as the 'red wing' and 'blue wing' respectively. For scandium three lines at 473.8, 474.1 and 474.4 nm were found to be suitable for evaluation as they show a well developed self-reversal at the given plasma conditions. Even though there are much more scandium lines with a self-reversal, especially around 400 nm, these lines are either too weak or disturbed by neighboring lines. Intensity profiles of the three scandium lines at different radial positions are shown in Fig. 4.

Additionally, optically thin lines were measured for subsequent partial pressure calculation. All analyzed lines with their atomic data are listed in Table 1.

TABLE 1. Atomic data used to calculate plasma temperatures and pressures. [[lambda].sub.0](mn) [E.sub.u] [g.sub.u] - [A.sub.ul] [E.sub.l](eV) [g.sub.l] ([s.sup.-1]) Hg I 435.8 7.730 - 4.886 3 - 3 5.57 x [10.sup.7] 546.1 7.730 - 5.460 3 - 5 4.87 x [10.sup.7] 576.9 8.852 - 6.703 5 - 3 2.36 x [10.sup.7] 579.1 8.844 - 6.703 5 - 3 2.14 x [10.sup.7] Sc I 473.8 4.049 - 1.432 4 - 6 8.8-[10.sup.9] 474.1 4.053 - 1.439 6 - 8 9.1-[10.sup.9] 474.4 4.060 - 1.447 8 - 10 9.8 x [10.sup.9] 364.7 5.386 - 1.986 4 - 6 3.66 x [10.sup.7] 404.4 5.034 - 1.969 8 - 8 2.32 x [10.sup.7] 407.5 5.028 - 1.986 6 - 4 3.18 x [10.sup.7] 435.2 5.142 - 2.294 4 - 2 1.66 x [10.sup.7] 546.8 4.268 - 2.001 4 - 6 1.07 x [10.sup.7] Na I 514.9 4.509 - 2.102 2 - 2 1.14 x [10.sup.6] 615.4 4.116 - 2.102 2 - 2 2.50 x [10.sup.6] 616.1 4.116 - 2.104 2 - 4 4.98 x [10.sup.6] [[lambda].sub.0] denotes the central line wavelength in nm, [A.sub.ul] the transition probability in [s.sup.-1], [E.sub.u] & [E.sub.l] the upper and lower levels in eV and [g.sub.u] & [g.sub.l] the statistical weights of the levels. The [A.sub.ul] value for the HgI 579.1 nm line is taken from [Hartel 1999]. Data for the other lines are available at the NIST Atomic Spectra Database [NIST 2012].

3 RESULTS AND DISCUSSION

Since the radiance of the self-reversal maxima is crucial for temperature evaluation with Bartels method, care must be taken in the determination of that radiance value. Experimental uncertainties may arise from absolute intensity calibration, from background radiation as well as from limited spectral resolution.

But firstly it must be noticed that radiation is lost by reflection at each interface were the refractive index is changing [Seehawer 1973, Wendt 2008]. Reflection losses depend on the angle of incidence and the optical depth. In the optically thick limit follows the reflection for perpendicular incidence directly from Fresnel equation:

[phi] = [(n-1)/(n+1).sup.2] (1)

where [phi] is the reflection coefficient for one interface and n the refractive index of quartz. In case of a burner with an outer bulb there are four interfaces between quartz and vacuum and ambient air, respectively. Hence, for a refractive index of quartz n = 1.46 the radiance of self-reversal maxima is reduced by around 14 percent, which must be considered before evaluation.

Furthermore, as can be seen from Fig. 4, the red line wing of the Sc I line at 474.1 nm is running into the blue wing of the Sc I line at 474.4 nm. By extrapolation one obtains a background contribution to the blue self-reversal radiance of the Sc I line at 474.4 nm of around 800 W [m.sup.-2] [nm.sup.-1] [sr.sup.-1] which amounts to 14 percent of the self-reversal radiance. However, one has to consider that the plasma at wavelengths around a self-reversal maximum is not optically thin but characterized by an optical depth of at least [iota]=2. Hence the background contribution is reduced by a factor of [e.sup.-2] to 108 W [m.sup.-2] [nm.sup.-1] [sr.sup.-1], amounting to less than 2 percent of the self-reversal radiance.

To resolve self-reversal line contours usually one needs sufficient high spectral resolutions. But even with the setup used in this work deconvolution of spectral lines has been performed with apparatus profile to check if there was some effect of spectral resolution on the self-reversal radiance. Finally is was found that deconvolution leads to self-reversal radiances that are 10 percent higher in case of Hg lines and 20 percent higher in case of Sc lines, where the uncertainty of this correction is still around 30 percent, because deconvolution is an ill-posed process amplifying statistical noise.

The uncertainty by absolute radiance calibration with a tungsten ribbon lamp accounts to 10 percent. Hence the total experimental radiance uncertainty of self-reversal maxima is estimated to be less than 18 percent in case of Sc lines and less than 16 percent in case of Hg lines.

The maximum plasma temperature along a line-of-sight is given by the equation [Bartels 1950]:

T = ([E.sub.u] - [E.sub.l])/[K.sub.B]/In(2h[c.sub.0.sup.5]/[[lambda].sub.0.sup.5]) + In(MY) - In([L.sub.max]) (2)

where [E.sub.u] and [E.sub.l] are the upper and lower energy of the levels involved in the transition, [k.sub.B] is the Boltzmann constant, h the Planck constant, [c.sub.0] the velocity of light in vacuum, [[lambda].sub.0] the center wavelength of the transition, [L.sub.max] the radiance of the self-reversal maximum. All quantities must be given in SI units. The Bartels functions M and Y are defined by the following equations in the most simple approach of the Bartels method:

M = [square root of ([E.sub.l]/[E.sub.u])]

Y = 0.736 + 0.264[p.sup.2]

p = 6/[pi]arctan ([M.sup.2]/ [square root of (1+[2M.sup.2]) (3)

where p is called inhomogeneity parameter assuming a parabolic temperature profile. This approach is of limited applicability. In more general approaches of Bartels method M would be a function of the spatial plasma temperature distribution and the plasma composition. In this case equations would have to be solved iteratively, like it was done here. For the lines evaluated here it was found that iteration yields almost the same temperatures as the most simple approach of Bartels method (Differences are less than 1 percent). But this is not a common behavior and usually it is necessary to apply the more general approaches of Bartels method as discussed in [Franke and Schneidenbach 2007, Schneidenbach and Franke 2008].

Without going into too much detail the Bartels functions can be given by these equations in their general form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Y = [2(1-p).sup.-1/2][(1 - [square root of (p)])/[(1 + [square root of (p)])].sup.1/(2[square root of (p)])] (4)

where [n.sub.u,l] are the population densities of the upper and the lower level, respectively, and [g.sub.u,l] are the corresponding statistical weights. [T.sub.m] represents the maximum temperature along the line-of-sight coordinate x at a given side-on position. Cylinder symmetry of the problem is assumed. The analytic approximation of Y in (4) has a higher accuracy of about 0.4 percent compared to that originally given by Bartels in (3). In M enters the so-called far line wing approximation, which requires a well developed self-reversal, allowing to separate the profile function into a product of a frequency dependent and a position dependent term. [PSI](x) represents that position dependent term which equals the profile width in case of a pure Lorentz line profile. Further approximations applied throughout this work are the assumption of a Boltzmann distribution of the excited level population and the ideal gas law. In this case all Bartels functions still depend on the side-on position and an iterative scheme has to be applied to yield plasma temperatures. It has to be noticed that ionization as well as radial cataphoresis and axial segregation violate the applicability of the ideal gas law, limiting the accuracy of the Bartels method. This is of particular interest, because a significant ionization and spatial demixing of additives in metal-halide lamps is expected [Dakin 1989, Hashiguchi and others 2002]. However, it reveals that the method is not sensitive to moderate deviations from spatially uniform partial pressures of additives [Funk and others 1970], as long as the main contribution to the Bartels functions arise by the integrals from regions around the discharge axis [Franke and Schneidenbach 2007]. The reasonable agreement of plasma temperatures obtained from self-reversed mercury lines and scandium lines can be motivated by this fact. A more detailed, quantitative analysis of this topic requires plasma composition and radiative transfer calculations which are beyond the scope of this paper.

For discussion of uncertainties of temperature evaluation the most simple approach is still followed. In this case equations (2) and (3) lead to a very simple and direct representation of the plasma temperature as a function of the self-reversal radiance for the Hg I 435.8 nm line:

T = 3.3005 x [10.sup.4]/36.562 + In(0.7062) - In([L.sub.max]). (5)

An equivalent equation can be given for the Sc I 473.8 nm line:

T = 3.0369 x [10.sup.4]/36.1463 + In(0.4777) - In([L.sub.max]). (6)

Error propagation now can tell us that even if the radiance of the self-reversal maximum is known with an uncertainty of 20 percent the plasma temperature can be given with an accuracy better than 3 percent. Furthermore it can bee seen from the equations above that only well-known atomic constants are needed for temperature evaluation. There is no need to know the radiators partial pressure or transition probabilities, which are often given with significant uncertainties.

In Table 2 results of temperature evaluation with Hg and Sc lines are shown. Axis temperatures determined with Hg and Sc lines are around the mean value of 5210 K.

TABLE 2. Central temperatures obtained from self-reversed lines of mercury and scandium. Line [[lambda].sub.0] (nm) [T.sub.0] (K) Hg I, 435.8 5240 Hg I, 546.1 5174 Sc I, 473.8 5213 Sc I, 474.1 5158 Sc I, 474.4 5267

Due to the fact that self-reversal vanishes around radial positions r/R=0.3 radial temperatures can be given up to this point and have to be extrapolated to an expected wall temperature of around 1000 K.

Figure 5 shows the measured temperature profile of the Hg I 435.8 nm line with a parabolic extrapolation to the wall, whereas Fig. 6 depicts a more detailed view of all measured temperature profiles near the arc center. It can be stated that temperatures determined with Bartels method from self-reversed Sc lines agree well within experimental uncertainties compared to temperatures obtained from self-reversed Hg lines.

This indicates that all these Sc lines are suitable for temperature determination in HID lamps with Bartels method. This is of particular interest, because the lower levels of the considered Sc lines are around 1.4 eV, whereas Bartels required in his theory a lower level energy of [E.sub.l] [much greater than] 1 eV. This condition is sufficiently fulfilled and proven for self-reversed Hg lines (546.1 nm & 435.8 nm), where the lower levels have energies higher than 4.8 eV, but not for the discussed Sc lines. But even for the Tl I line at 535.0 nm with a lower energy level of 0.966 eV the Bartels method was already found to work well [Funk and others 1970]. Hence it can be concluded that the Sc lines at 473.8, 474.1 and 474.4 nm are indeed well suited for temperature determination with Bartels method.

The most challenging restriction is to provide a sufficient spectral resolution for proper evaluation of self-reversal maxima which should be better than 0.01 nm. A more detailed analysis of Bartels method applied to scandium lines requires elaborate plasma composition and radiative transfer calculations [Franke and Schneidenbach 2007, Schneidenbach and Franke 2008] which are beyond the scope of this work.

Once the plasma temperature is known, partial pressures of the radiating species can be obtained from optically thin lines, applying Abel inversion to their side-on radiance. For LTE conditions the emission coefficient [epsilon] can be given as a function of the plasma temperature T by:

[epsilon](T) = hc/4[pi] x [A.sub.ul][g.sub.u]/[[lambda].sub.0] x p/Q(T)[[kappa].sub.B]T x exp x (-[E.sub.u]/[[kappa].sub.B]T) (7)

with the transition probability [A.sub.ul], the statistical weight of the excited (emitting) state [g.sub.u] and its energy [E.sub.u], the emission wavelength [[lambda].sub.0], the radiators partial pressure p, the Boltzmann constant [k.sub.B] and the partition function Q(T) for the given species.

Table 3 summarizes the partial pressures obtained for a mean central arc temperature of 5210 K, where error propagation yields:

TABLE 3. Partial pressure of mercury, sodium and scandium calculated with atomic data from Tab. 1. Line [[lambda].sub.0] (nm) Pressure Hg I, 576.9 6.28 bar Hg I, 579.1 6.58 bar Sc I, 364.7 0.44 mbar Sc I, 404.4 0.57 mbar Sc I, 407.5 0.64 mbar Sc I, 435.2 0.47 mbar Sc I, 546.8 0.64 mbar Na I, 514.9 0.31 mbar Na I, 615.4 0.29 mbar Na I, 616.1 0.31 mbar

[delta]p/p = [sigma]T/T[absolute value of [R.sub.u]/[[kappa].sub.B]T - 1] + [delta][epsilon]/[epsilon]. (8)

Therefore, the relative uncertainty in partial pressure determination scales with upper level energy [E.sub.u]. Hence, assuming 10 percent uncertainty in the emission coefficient and 3 percent uncertainty in plasma temperature (as discussed above), the uncertainty in partial pressure ranges from 20 percent to 66 percent for upper level energies from 2 eV to 9 eV. Keeping this in mind, different partial pressures obtained from single lines agree within experimental uncertainties.

The plasma temperatures of around 5210 K found here reasonably agree with values given by [Dakin 1989] who obtained a plasma temperature of about 5000K in the middle of the discharge. The Hg partial pressure given in this work is around 6.4 bar whereas [Dakin 1989] obtained a Hg partial pressure of about 4.7 bar which is within the experimental uncertainty. However, the partial pressures of additives (Na, Sc) given here with mean values of around 0.55 mbar for scandium and 0.30 mbar for sodium significantly deviate from the values given by [Dakin 1989] but they are in the same order of magnitude. This is acceptable because partial pressures of additives are sensitive to cold spot temperatures which may differ comparing our measurements with [Dakin 1989].

4 SUMMARY

Temperature determination utilizing self-absorbed spectral lines is well established for mercury high-pressure discharge lamps. If the self-reversed Hg lines are not superimposed by spectral lines of additives they are well suited for temperature measurements in multicomponent HID lamps, too. In this work it is demonstrated that also self-reversed Sc lines can be applied for plasma temperature determination. The three scandium lines analyzed here were found to be suitable for evaluation with the Bartels method, hence extending diagnostic options. The experimental uncertainty is found to be better than 3 percent for plasma temperatures. The most challenging requirement for the proposed method is a sufficient spectral resolution to resolve the self-reversed line structure.

[c] 2013 The Illuminating Engineering Society of North America

doi: 10.1582/LEUKOS.2013.09.03.002

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Jacob Zalach (1) and Steffen Franke PhD

(1.) INP Greifswald, Felix-Hausdorff-Str. 2, 17489 Greifswald, Germany *Corresponding author: Jacob Zalach, Email: zalach@inp-greifswald.de

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Article Type: | Report |

Date: | Jan 1, 2013 |

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