S ecnochimica Acta Vol. 47B, No. 13, pp. 1435-1446, IS92 dinted in Great B&in.
05&?4-8547/92$5.00+ .cQ 0 1992 Pergamon Press Ltd
Comparisons between a Grimm-type glow discharge and a jet-assisted glow discharge source viewed axially and laterally P. R. BANKS and M. W. BLADES* Department
of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Zl (Received 30 March 1992; accepted 14 July 1992)
Abstract-Comparisons have been made between three types of glow discharge with regard to analytical performance. These glow discharge types include a Grimm-type glow discharge, a jet-assisted glow discharge viewed axially to the jet-assisted plasma plume and the same discharge viewed laterally 25 mm from the sample surface where its characteristics are sufficiently different to warrant reclassification. The axially viewed, jet-assisted glow discharge provides the highest detection power, but laterally viewing allows the potential for greater dynamic range, greater freedom from spectral interferences and a detection power comparable to the Grimm-type glow discharge. This makes lateral viewing of the jetiassisted plasma plume attractive for simultaneous multi-element applications.
1. INTR~DUC~~~N ALTHOUGHglow discharges can be used to directly analyze solid samples, the sputtering process is tempered by the re-deposition of sputtered material which reduces the sampling efficiency [l]. This problem has been addressed by the Analyte Corporation’s Atomsourcem [2]. Both the AtomsourceTM and discharges modelled on its design have been the subject of scientific scrutiny [3-171. The AtomsourceTM is familiar in that it uses the basic hollow anode of the Grimm glow discharge brought to within a mean free path of the cathode to form an obstructed discharge [18]. However, it also uses six jets, arranged in a hexagonal pattern around the inner circumference of the anode, that direct support gas at the sample surface. It has been demonstrated that the use of jet-assisted, support-gas flows significantly increases sample loss rates relative to a Grimm-type glow discharge [8, 171 largely through a reduction in re-deposition
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Despite the increase in sample loss rate, the intensities of some emission lines from sputtered atoms decrease with increasing jet-flow rate [8, 14, 171. This effect occurs only for emission lines with appreciable absorption coefficients. It was found that the jets produce an increase in absorption path length invoked by the entrainment of sputtered atoms along the optical axis of the source, which can lead to self-absorption
P41*
An obvious remedy to this problem is to rotate the optical axis 90” so that the absorption path length is reduced. It has been shown in the jet-assisted glow discharge source that the lateral width of the jet-assisted plasma plume is significantly less than the axial length [20]. However, in rotating the axis, emission from the negative glow is obstructed: only the jet-assisted plume can be analyzed and at axial distances no closer than 25 mm from the sample surface with the present source geometry. Therefore, one would expect that a reduction in signal intensity would result from laterally viewing which could lead to a decline in analytical performance relative to axial viewing. The purpose of this study is to compare lateral with axial viewing with regard to analytical performance and self-absorption.
* Author to whom all correspondence
should be addressed. 1435
P. R. BANKS and M. W. BLADES
1436
r
l-i=
t
Auxiliary Support Gaslnlet
Jet-assisted Poskive Potential
-
b)
P.._
Jet suppon
Jet-assisted Plume Positive Potential
Lateral Viewed Emission
Fig. 1. Schematic diagram showing a cross-section of the jet-assisted glow discharge source displaying (a) axially viewed emission and (b) laterally viewed emission from the jet-assisted plasma plume.
2. EXPERIMENTAL The jet-assisted glow discharge source used in this study has been described elsewhere [19, 201. Figure l(a) and (b) are schematics of the source which demonstrate axial (Fig. l(a)) and lateral viewing (Fig. l(b)). If there exists no support-gas flow through the jets, then the jetassisted source operates as a Grimm-type glow discharge. When laterally viewing, the side pumping port used when axially viewing, remains closed. Emission spectra were collected using the same system described previously [20]. Line shapes were obtained using the method also described earlier using a pure copper sample (>99%) [14]. Axially viewed calibration curves presented in this article contain data that have been published previously [14]; the laterally viewed calibration curve was obtained using the same monochromator and linear photodiode array [14]. Signal standard deviations using a pure copper sample (>99%) at different jet-tlow rates were obtained by the following procedure. The emission line was monitored for 10 s for any one jetflow rate and an average signal determined. This process was repeated five times to achieve five averaged signals for each jet-flow rate investigated. The standard deviation of the five averaged signals was then reported as the signal standard deviation. Limits of detection for copper using the Cu I 521.82 nm emission line for the three types of glow discharge were obtained using similar discharge conditions: constant voltage, constant current (600 V, 50 mA), and similar equipment settings. The monochromator and PMT used
Glow discharge comparisons viewed axially and laterally
1437
the same as that used for the emission spectra and signal standard deviations, but the data acquisition system was changed to RC Electronics (Santa Barbara, CA) Computerscope ISC-16 System. The ISC-16 offers 1Zbit resolution for A/D conversion which was adequate for acquiring background RSD. The background and background RSD were obtained by setting the monochromator to a wavelength of 531.0 nm and sampling the background signal for 6.4 s at a rate of 40 Hz using a current amplifier setting three orders of magnitude larger than that used to acquire the Cu I signal. An average background signal and its RSD were calculated from the resulting 256 points. The dark current contribution was found to be at least an order of magnitude less than the total background measurement and therefore the recorded background signals were attributed to the radiant background. were
3. GAS DISCHARGE COMPARISONS 3.1. Copper spectra Figure 2(a) is a copper spectrum obtained using no support-gas flow through the jets, axially viewed. When used in this mode, the jet-assisted glow discharge source emulates a Grimm glow discharge. From this point on in the manuscript, the zero-jetflow rate case will be referred to as a Grimm-type glow discharge. The spectrum displays-significant resonance line emission at 324.75 and 327.40 nm as well as prominent green lines at 510.55, 515.32 and 521.82 nm. The second order Cu II 224.70 nm emission line (due to a charge transfer reaction between the copper ground state atom and a metastable argon ion [21]) is also prominent in the spectrum at a wavelength of 449.40 nm. Also evident is Ar II emission (i.e. in the vicinity of 350 nm) indicative of energetic electrons that have achieved sufficient energy for support-gas ion excitation through acceleration in the cathode dark space [22]. Figure 2(b) is a copper spectrum (axially viewed) obtained under similar operational conditions to Fig. 2(a) except the jet-flow rate has been increased to 30 cl min-l. It is apparent that the relative intensities of some emission lines has changed and that the emission intensity scale for the spectrum has increased. Self-absorption of the resonance lines is apparent by comparing the relative intensities in Fig. 2(a) and (b): the resonance lines possess smaller intensities at the greater jet-flow rate. Other emission lines, however, demonstrate increases in intensity with the larger jet-flow rate: both the Cu II and the green lines show increases by a factor of about three. The increasing green line intensities explain the green hue of the jet-assisted plume when copper is used as a cathode. Furthermore, the Cu I 402.26 and 406.26 nm lines show increases in intensity by a factor of five. Conversely, Ar II and Ar I emission at 415.86 and 420.02 nm, respectively, appears unchanged in intensity, in agreement with previous findings [14] for the two cases, which suggests that the use of jets increases sampling rather than excitation efficiency. Figure 2(c) is a copper spectrum obtained under similar conditions to Fig. 2(b) except viewing is lateral: only the jet-assisted plume can be sampled using this optical axis. The first obvious difference between this spectrum and Fig. 2(a) and (b) is the intensity of the emission lines, which are typically an order of magnitude reduced. Also, the absence of any Ar II emission indicates that the ionizing plasma typically associated with the negative glow is no longer being sampled. Although the resonance lines have been restored as the most intense lines in the spectrum, their intensity relative to the Cu I 402.26 and 406.26 nm lines is much smaller than in Fig. 2(a): for example the ratio between Cu I 324.75 and Cu I 406.26 nm in Fig. 2(a) is about 8; in Fig. 2(c), this ratio is less than 3. At face value, this suggests that the self-absorption problem associated with using appreciable jet-flow rates has been reduced but not eliminated. The Cu II 224.70 and Cu I 510.55 nm emission lines also demonstrate an intensity reduction relative to other emission lines in the spectrum. However, the Cu II line emission intensity relative to other lines is restored by increasing the jet-flow rate; but the green line at 510.55 nm remains diminished relative to other lines regardless of what jet flow rate is used [20]. This behavior is also evident for the other two emission lines at 570.02 and 578.21 nm, which similarly decay to the low-lying
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(4
1, Wavelength
,.‘; 550
450
600
(nm)
6.
l+,,‘t 550 Wavelength
I.
41. 600
(nm)
Fig. 2 . Continued.
copper metastable levels (1.39 and 1.64 eV). This relative reduction in emission intensity suggests that these low energy states are not as efficiently excited in the jetassisted plume relative to the negative glow which supports the belief that excitation in the jet-assisted plume is dominated by metastable collision-produced electrons at energies between 7.32 and 7.68 eV [20] introduced into the plume by the metastable
1439
Glow discharge comparisons viewed axially and laterally
0.6
3
0.6
d
E
z
2 E
6
$ O.* 6 : f f
0.2
0.0 300
450 Wavelength
550
600
(nm)
Fig. 2. Copper spectra from 300 to 600 nm which have been corrected for the spectral response of the monochromator and PMT used (discharge power: 25 W; corrected meter pressure: 2.9 mbar). (a) Grimm-type glow discharge; (b) axially viewed jet-assisted glow discharge, jetflow rate: 30 cl min-*; (c) laterally viewed jet-assisted glow discharge, jet-flow rate: 30 cl min-I.
collision cycle. This cycle relies on the production of energetic electrons through the collision of two argon atoms existing in metastable states: after collision, one argon atom becomes ionized, the other relaxes to the ground state. The energy discrepancy is imparted to the resulting electron allowing it to produce atomic excitation. For this supposition to be true, the resonance lines should demonstrate a similar reduction in relative intensity when viewed laterally since all five emission lines are produced through spontaneous emission from the 3d1°4P states (either 2P3n or *Pr12). This may explain the relative intensity difference between the resonance lines and the Cu I 406.26 nm emission line evident in Fig. 2(a) and (c). In the jet-assisted plasma plume, the width of visible emission in the lateral direction is about an order of magnitude less relative to its length in the axial direction at a jet-flow rate of 30 cl min-l, so one would expect that the reduction in absorption path length in viewing laterally would substantially reduce, if not eliminate self-absorption. To determine whether self-absorption or the selective excitation of high energy states by the metastable collision cycle is responsible for the reduction in relative intensity originating from the 3d1°4p states and that from the 3d1°5d states of Cu I 402.26 and 406.26 nm in Fig. 2(a) and (c), line shapes of the Cu I 324.75 nm emission line can be examined for line shape broadening attributable to self-absorption [14]. 3.2. Evaluation of self-absorption Figure 3(a) shows the line shapes of the resonance line using a jet-flow rate of 30 cl min-l for both axial and lateral viewing. It is evident that the line shape for the laterally viewed case is significantly narrower than the axial case. This is also true for comparisons with a Grimm-type glow discharge with resonance line emission collected axially (Fig. 3(b)): the laterally viewed emission line displays a significantly narrower line shape, despite the use of a 30 cl min-’ jet-flow rate. It has been shown previously that this is not due to pressure broadening [7]. Therefore it appears that through
P. R. BANKSand M. W. BLADES
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Position
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5 200
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-5
0
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Fig. 3. Cu I 324.75 nm emission line shape comparisons (discharge power: 30 W). (a) (-C) Axially viewed jet-assisted glow discharge, jet-flow rate 30 cl min-‘, 3.3 mbar; (U) laterally viewed jet-assisted glow discharge, jet-flow rate 30 cl min-I, 3.3 mbar. (b) (-@-) Grimm-type glow discharge, 3.3 mbar; (U) laterally viewed jet-assisted glow discharge, jet-flow rate 30 cl min-‘, 3.3 mbar. (c) (+) Laterally viewed jet-assisted glow discharge, jet-flow rate 90 cl min-‘, 4.7 mbar; (-Cl-) laterally viewed jet-assisted glow discharge, jet-flow rate 30 cl min-I, 5.7 mbar.
Glow discharge comparisons viewed axially and laterally
1441
laterally viewing the jet-assisted plume, self-absorption can be significantly reduced relative to even a Grimm glow discharge which has previously been shown to suffer from self-absorption under certaitrconditions [23]. Furthermore, Fig. 3(c) demonstrates that an increase in jet-flow rate from 30 to 90 cl min-l produces no measurable difference in line shape for the resonance line, provided the jet-assisted plume is viewed laterally. This indicates that there is little lateral spread of sputtered atoms invoked by increasing the jet-flow rate. Therefore, the absorption path length remains unchanged producing no increase in self-absorption. The underlying premise for this argument is that the physical linewidth substantially contributes to the effective linewidth and that line broadening is attributable to selfabsorption. If the physical linewidth does not significantly contribute, then the effective profile represents the instrumental broadening and any linewidth changes produced by self-absorption broadening would not be measurable. In a previous paper, the spectral bandwidth and the reciprocal linear dispersion for the Cu I 324.75 nm resonance line using the Leco Plasmarray were quoted to be 9.7 pm and 2.37 pm per pixel, respectively [14]. Using these values, effective and physical linewidths for the profiles in Fig. 3 can be computed. For example, using the profile for the Grimm-type glow discharge case in Fig. 3(b), the effective linewidth is 15.5 pm and the physical linewidth, using the reported bandwidth and quadratic subtraction, is 12.1 nm. So it appears that the physical linewidth does contribute significantly to the effective profile; but is the broadening attributable to self-absorption? The Cu I resonance line possesses hyperfine structure such that it is a doublet with an apparent Doppler width of 4.7 pm at 1100 K
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A semi-quantitative evaluation of self-absorption broadening can be obtained by examining theoretical spectral profiles of the absorption factor as described by ALKEMADE et al. [25]. The extent to which a resonance line suffers from self-absorption broadening depends on the value of nofZ/b, where no is the ground state population, f is the oscillator strength of the transition and 1 is the absorption path length. To evaluate nofl/b, we first need an estimate of b:
where au,, is obtainable from the Doppler width already mentioned. Assuming T = 1100 K, 6vr, = 1.3 x lOlo Hz and therefore: b = 6.0 x lOlo Hz. FERREIRAand HUMAN have reported no and I values for a Grimm-type glow discharge [26]: no ranged from about l-3 x 1013 cme3 and 1 = 0.8 cm operating at approximately the same conditions used for Fig. 3. The value of no was limited to measurements that extended over only 3 mm from the cathode surface, so it is difficult to obtain a useful value of no over the full path length of 8 mm. We have assumed that the possible average value of no over this path length will range from about 0.5 to 2 x 1013 cmm3. Therefore, nofZ/b will range from lo-40 cm-*, v = 0.16 for the copper atom resonance line used [27]). Therefore, it seems that the copper atom resonance emission line, at this particular value of nofZ/b may be already significantly self-absorbed. From ALKEMADE et al. [25], their figure gives a range of values of nofZlb corresponding to half-widths at half height of between 1.6 and 2.3 corrected wavelength differences. To evaluate the actual broadening, the Doppler width of the resonance line must be known. If the Dopplerbroadened line width is 4.7 pm at a temperature of 1100 K due to its being a doublet, then:
x_h, = (1.6-2.3)(4.7 2Vln2
pm) = 4 5-6 5 pm ’
’
’
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P. R. BANKSand M. W. BLADES
Therefore, the broadening over the whole profile would range from about 9 to 13 pm. The experimentally determined physical linewidth of 12.1 pm is within this range. Furthermore, it is noted that for n&l/b values greater than 40 cm+ s, relatively small increases produce significant line broadening according to Alkemade’s figure. This suggests that the broadening observed viewing axially with appreciable jet-flow rates is real and attributable to self-absorption. It is apparent that the use of the jets extends the luminous portion of the discharge by about an order of magnitude, so one would expect a corresponding increase in 1. This coupled with an increase in sample loss rate by a factor of three could lead to an increase in nofl/b by an order of magnitude (i.e. = 400) relative to the Grimm-type case and a severely broadened emission line through self-absorption. With viewing laterally, I should be of the same magnitude as the Grimm-type case, but since no should be significantly lower due to the analysis volume being removed from the sample surface, one would expect a narrower line shape. It is difficult to be more quantitative than this as no data presently exists for no and 1 for jet-assisted glow discharges. However, the results from the Grimm-type discharge demonstrate the plausibility for the explanation of line broadening through selfabsorption. It appears that by viewing the jet-assisted plume laterally, self-absorption has been reduced relative to using no support gas flow through the jets. These findings support the premise of selective excitation of high energy states by the metastable collision cycle. Low-lying levels, like the 3d1°4p states of copper, are underpopulated relative to higher lying states, like the 3d1°5d states, provided the energy of the upper state does not exceed the upper energy limit for metastable collision-produced electrons. 3.3. Relative sensitivity and precision Figure 4(a) displays the variation in the Cu I 521.82 nm emission signal viewed axially and the signal standard deviation as the jet-flow rate is changed. It is apparent that the signal increases with increasing jet-flow rate, as does its standard deviation. However, the increase in signal standard deviation is not as significant as the gain in signal amplitude up to 20 cl min-l, so that an optimum precision, expressed as a %RSD, is available using a jet-flow rate of 20 cl min-’ when viewing axially (Fig. 4(b)). This constitutes a gain in precision by about a factor of two relative to a Grimmtype glow discharge, which is represented at the y-axis (0 cl min-’ jet-flow rate). This improvement is rather questionable, however, since the error associated with the evaluation of the precision nearly spans this difference. Figure S(a) repeats the experiment shown in Fig. 4(a), except using lateral viewing. It is apparent that the sensitivity, expressed on a relative scale from the currentamplified PMT output, is about an order of magnitude smaller when viewing laterally relative to axially, provided the same jet-flow rate is used. This magnitude is reduced when viewing laterally, however, if larger jet-flow rates are used. Flow rates greater than 30 cl min-l can not be used when viewing axially with the present design due to deposit of sputtered material on the optical window. The signal standard deviation is also substantially reduced when viewing laterally, so the resultant %RSD is approximately the same as for a Grimm-type glow discharge (Fig. 5(b)). In addition, the signal standard deviation closely follows the signal, and therefore little change in %RSD is seen with increasing jet-flow rate. For all types of glow discharge, the %RSD of the signal is impressive: at any flow rate the precision is approximately 0.3%. In addition, the signal standard deviation appears to follow the signal as the jet-flow rate is manipulated, indicating that the dominant noise source in the plasma is flicker noise. 3.4. Calibration curves It has already been established that the use of resonance lines when axially viewing may be precluded if the transition possesses a large absorption coefficient due to the self-absorption of the emission line [14]. Figure 6 contains three nickel calibration curves using the Ni I 352.45 nm emission line. The axially viewed jet-assisted glow discharge, using a jet-flow rate of 30 cl min- l, displays roll-over indicative of self-
Glow discharge comparisons viewed axially and laterally
1443
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26
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;
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0
20
10
30
Jet Flow Rate (cl min“)’
0.04
,
,
r
0
10
20
30
Jet Flow Rate (cl min.‘)
Fig. 4. Sensitivity and precision for axial viewing (discharge power: 25 W, corrected meter pressure: 3 mbar). (a) (-O-) Cu I 521.82 nm signal intensity and (-CL) signal standard deviation. (b) %RSD in the Cu I 521.82 nm emission signal.
absorption, whereas the other two, a Grimm-type glow discharge and a laterally viewed jet-assisted glow discharge, using a jet-flow rate of 50 cl min-l, display linear curves over the concentration range studied. In addition, this study in Section 3.2 has shown that the degree of self-absorption is measurably less for lateral viewing of a jet-assisted glow discharge relative to the Grimm-type glow discharge. Therefore, one may expect that the dynamic range for glow discharge atomic emission analyses may be extended by using a laterally viewed jet-assisted glow discharge. 3.5. Limits of detection BOUMANSet al. have recently demonstrated the use of a new method for calculating limits of detection that permit relative comparisons of different optical spectrometers [28]. In this approach, the Limit of Detection (LOD) is given by the equation: LOD = O.O3(RSDB)m (SBR) where RSDB is the relative standard deviation in the background, m is the concentration of analyte used in the determination, and SBR is the signal to background ratio. The background should be obtained using a blank sample at the same wavelength as the signal. However, this presents a problem when using solid-sampling glow discharges: it is difficult to obtain a conducting, solid blank. For this reason, it has been decided to use a point on the spectrum in close proximity to the selected copper emission line at 521.82 nm that displays no discrete emission at the same current amplifier gain setting. This may preclude comparison with other glow discharge spectrometers, but it does allow comparison between axial and lateral viewing for the jet-assisted glow discharge source. Table 1 contains the parameters used in calculating the LOD using a sample of commercial grade copper (taken to be 99% Cu). The detection limit
BANKS
P. R.
1444
M. W. BLADES
and
(4
30
40
50
60
Jet Flow Rate(cl
70
60
90
min.‘)
(b)
o.3 ~
go.2 K 9 0
0.1 I
o.ol______ 30
40
50
60
Jet Flow
70
60
90
Rate (cl mine’)
Fig. 5. Sensitivity and precision for lateral viewing (discharge power: 25 W, corrected meter pressure was set by the flow of support gas through the jets: 2.9-5.2 mbar for 30-90 cl mitt-’ respectively). (a) (-O-) Cu I 521.82 nm signal intensity and (-CL) signal standard deviation. (b) %RSD in the Cu I 521.82 nm emission signal.
10
15
Nickel Concentration
20
25
(percent in sample)
Fig. 6. Nickel I 352.45 nm calibration curves (discharge power: 25 W): (-C) Grimm-type glow discharge, 9 mbar; (-CL) axially viewed jet-assisted glow discharge, jet-how rate 30 cl min-‘, 9 mbar; (-A-) laterally viewed jet-assisted glow discharge, jet-flow rate 50 cl min-*, 4.2 mbar.
reported here using a Grimm-type glow discharge is of the same order of magnitude of those reported elsewhere [29] for different elements providing an adequate benchmark to make our comparisons. When axially viewing, the LOD is reduced by a factor of two using a jet-flow rate of 30 cl min-l relative to a Grimm-type glow discharge. This is due largely to an increase in sampling efficiency, but an increase in background compromises the result
Glow discharge comparisons viewed axially and laterally Table 1. Copper limits of detection
1445
for a Grimm-type and both an axial viewed and lateral viewed jetassisted glow discharge
Figure of merit Limit of detection (ppm) Signal (a.u.) Background (a.u.) %RSD background SBR
Grimm-type
Axial jetassisted (30 cl min-I)
8 2302 1.44 0.443 1599
4 5310 1.82 0.443 2918
Lateral jetassisted (30 cl min-I) 50 47.8 0.123 0.641 389
Lateral jetassisted (90 cl min-I) 7 550 0.179 0.709 3073
somewhat. In comparing a Grimm-type glow discharge to a laterally viewed glow discharge using a jet-flow rate of 90 cl min-‘, no real difference is seen in the LOD. This equality is due to the balance between (1) an increased sampling efficiency coupled with a large reduction in background signal and (2) poorer transport efficiency to the viewing zone at an axial distance of 25 mm from the sample surface coupled with a larger RSDB for the lateral case. Using a jet-flow rate of 30 cl min-’ in conjunction with lateral viewing, transport efficiency is further reduced and any improvement in RSDB. or reduction in background is insufficient to produce an LOD comparable to the 90 cl min-’ case. If the pressure is reduced such that the jet flow is allowed to set the source pressure while maintaining the same power level (as in Fig. 5), transport efficiency using a jet-flow rate of 30 cl min-l is improved so that the signal intensity is increased when laterally viewing, but this is achieved at the cost of an increase in RSDB to the extent that the LOD is only marginally improved.
4. CONCLUSIONS Table 2 compares the analytical figures of merit determined in this study for each of the three gas discharges investigated: a Grimm-type and both an axially viewed and laterally viewed jet-assisted glow discharge. Choice for a particular glow discharge appears to depend upon application. If limit of detection is a motivating factor, then an axially viewed jet-assisted glow discharge appears to be the best choice. However, if multi-element analysis is required, then the laterally viewed jet-assisted glow discharge may be the best choice since it should provide the largest linear dynamic range and a relative freedom from spectral interferences due to an inherently simpler spectrum without much sacrifice in detection power. It should be noted, however, that the Grimm-type glow discharge provides excellent all-round performance. Table 2. Comparison of figures of merit for a Grimm-type and both an axial viewed and lateral viewed jet-assisted glow discharge
Figure of merit Limit of detection Sensitivity Dynamic range Precision Background Spectrum simplicity * Best performance. + Next best performance. - Worst performance.
Grimmtype + + + + +
Axial jetassisted (30 cl min-I) *I+ * _ */+ -
Lateral jetassisted (90 cl min-I) + * + * *
P. R. BANKSand M. W. BLADES
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Acknowledgements-Acknowledgement is made to the Natural Sciences and Engineering Research Council (NSERC) of Canada for partial support of this research and one of the authors (PRB).
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