The suitability of a hyperbolic calibration curve for continuum source atomic absorption spectrometry (CS-AAS) using electrothermal atomization and array ...
Characterization of hyperbolic calibration curves for continuum source atomic absorption spectrometry with array detection Desmond N. Wichems,† Robert E. Fields‡ and James M. Harnly* USDA, ARS, BHNRC, Food Composition Laboratory, Building 161, BARC-East, Beltsville, MD 20705, USA Received 17th June 1998, Accepted 11th September 1998
The suitability of a hyperbolic calibration curve for continuum source atomic absorption spectrometry (CS-AAS) using electrothermal atomization and array detection was critically examined for Ag (328.1 nm), Cd (228.8 nm) and Pb (283.3 nm). For each element, 20–25 calibration standards were used to cover six orders of magnitude of concentration. The same hyperbolic shape, with appropriate offsets of the X and Y axes, was found to be suitable for fitting each series of standards. The inflection point of the hyperbolic calibration curve, and hence the X and Y offsets, were dependent on the element and atomization temperature. These shifts arose from fundamental differences in the shape of the absorbance profile due to the ratio of Doppler and collisional broadening. As few as two standards (one above and one below the inflection point) were used to construct calibration curves covering six orders of magnitude of concentration. The use of four standards (two above and two below the inflection point) reduced the recalibration precision by a factor of two and, in general, introduced less uncertainty into the quantification of an unknown than the absorbance noise associated with the analytical measurement.
Introduction Continuum source atomic absorption spectrometry (CS-AAS ) with array detection can generate calibration curves with extended analytical ranges through the measurement of intensities in the wings of the absorption profile. This has been shown theoretically and with limited experimental data.1 In principle, there is no limit to the range of CS-AAS calibration curves. In practice, however, the range will be limited by the size of the array, the ability to clean up the furnace between atomizations and/or spectral interferences. Despite these limitations, it is anticipated that the routine, useful calibration range of CS-AAS will be five or six orders of magnitude. This calibration range greatly exceeds that of conventional, linesource AAS (using hollow cathode or electrodeless discharge lamps) and is equal to or greater than that of inductively coupled plasma optical emission spectrometry (as currently practised). With a continuum source and array detection, two fundamentally different approaches can be taken towards computing absorbances: wavelength integrated absorbance ( WIA) and wavelength specific absorbance ( WSA).1 The WIA approach sums the absorbances for all the pixels that cover the absorption profile and provides a single calibration curve. The WSA approach uses absorbances computed for selected pixels at the center and in the wings of the profile to construct a family of calibration curves. WSA is restricted to using an entrance slit of approximately the same width as each pixel, whereas WIA can be employed with any slit width.1 If the detector read noise is limiting, as is the case with a linear photodiode array (LPDA), the signal-to-noise ratio (SNR) will improve with the square root of the entrance slit width and WIA is the only useful approach. With a charge coupled device (CCD) array, the detector read noise will usually be insignificant, shot noise † Present address: 13210 Grenoble Dr., Rockville, MD 20853, USA. ‡ Present address: SpectruMedix, 2124 Old Gatesburg Road, State College, PA 16803-2200, USA.
will be dominant and the SNR will be independent of the entrance slit width. In this case, a narrow entrance slit width can be used to preserve the inherent resolution of the spectrometer. For the shot noise limited case, either WIA or WSA can be used; both approaches will provide similar SNRs. This study will focus on the WIA approach, which can be used with either an LPDA or CCD detector. The shape of the single calibration curve constructed with the WIA approach is theoretical predictable. Mitchell and Zymansky2 demonstrated that absorbance, integrated over the absorption profile, is linear with respect to concentration [a slope of 1.0 when log (absorbance) is plotted versus log (concentration)] at low concentrations and is proportional to the square root of the concentration (a slope of 0.5 on a log–log plot) at higher concentrations. A computer model1 was used to demonstrate that if the concentration axis is normalized by the intrinsic concentration (i.e., a concentration of 1.0 gives a time and wavelength integrated absorbance of 0.0044 pm s), then stray light determines the inflection point, the point at which the slope changes from 1.0 to 0.5 on the log–log plot. The appeal of the single calibration curve of the WIA approach is enhanced by the ability to fit all the curve with a well defined equation that matches the theoretical shape of the analytical data.1 Half of a hyperbola can be rotated and its coefficients adjusted ( Table 1) to provide linear plots with slopes of 1.0 and 0.5 joined by a region of varying smoothness. The smoothness of the curve at the inflection point is dependent Table 1 Hyberbola used for all calibration curves: Ax2+Bxy+ Cy2+Dx+Ey+F=0 A B C D E F
12.0 −36.2 24.2 0.0 0.0 Variable
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only on a single coefficient, F. It was shown that this hyperbola could be used to fit every calibration curve, theoretical or experimental.1 Differences in position due to sensitivity (without normalization) or stray light can be accommodated by shifting the X and Y axis coordinates of the hyperbola or the data set. Initial modeling in this laboratory suggested that, with correction of all computed absorbances for stray light, the X and Y axis coordinates for each element would be the same.1 The computer model, however, was relatively unsophisticated. It assumed only collisional broadening for the shape of the absorbance profile; Doppler broadening and hyperfine structure were ignored. Both of these factors will affect the shape of the absorbance profile and have an influence on the inflection point. Although the area of the absorbance profile may be constant, hyperfine structure and the ratio of collisional to Doppler broadening (the a parameter in the Voigt profile) will determine the height-to-width ratio. While hyperfine structure is constant for each element, the a parameter is temperature dependent. Hence it can be anticipated that the inflection point will vary between elements and as a function of temperature. In this study, experimental calibration curves were constructed for three elements using four standards per decade, ranging from a concentration close to the detection limit to a concentration where the profile width exceeded the width of the array. The position of the inflection point was compared between elements and as a function of temperature. In addition, the precision for repeat determinations of a calibration curve (recalibration precision) was evaluated using two, three and four standards.
Theory The width of an absorbance profile in a graphite furnace is determined primarily by Doppler and collisional (with the inert fill gas, Ar) broadening. The most accurate means of computing linewidths is the Voigt profile,3 which convolutes the Gaussian (Doppler) and Lorentzian (collisional ) broadening:
P
e−y2 a2+(l −y)2 r where y is an integration variable and k(x)=
a p
(1)
Dl c √ln 2 (2) Dl D 2(l−l ) √ln 2 0 (3) l= r Dl D where l is the wavelength center and Dl and Dl are the 0 D C Doppler and collisional widths, respectively. The value a is known as the a parameter or the damping constant. Both the Doppler and collisional widths are temperature dependent:4 a=
S C A
Dl =K l D D 0
1 Dl =K l s 2P C C 0 C f T
T m
1 1 + m m
that although the areas or integrated absorbances are equal, the shapes are not. This factor influences the shape of the calibration curves as shown in Fig. 2, where log (absorbance) is plotted versus log (normalized concentration) for four different values of the a parameter. Each plot has a slope of 1.0 at low concentration and 0.5 at high concentrations. As the a parameter changes from 1.66 to 0.42, however, the inflection point shifts to lower values of concentration. The data shown in Fig. 2 show the curve shapes that would be observed for a static analyte concentration, e.g., the curve shape that would be observed for flame AAS. At low concentrations, the absorption profiles [in the intensity domain, Fig. 1(B)] maintain a constant width and the computed absorbance at all points increases linearly with increasing concentration. At higher concentrations, the minimum intensity at the line center approaches a limit set by stray light. As concentration continues to increase, the transmitted intensity at the line center reaches a minimum determined by the stray light. In the model data presented in Fig. 2, the stray light was 5%. At this point, integrated absorbance will continue to grow owing to absorbance in the wings of the profile. Although the absorbance at each wavelength is increasing linearly, the width of the profile is increasing as the square root of concentration. Consequently, the integrated
(4)
BD
D
(5) f where K and K are the Doppler and collisional coefficients, D C respectively, T is the absolute temperature, m is the mass of the analyte, m is the mass of the fill gas, P is the pressure of f f the fill gas and s 2 is the collisional cross-section for the C analyte and fill gas. Fig. 1(A) shows Lorentzian, Gaussian and Voigt absorbance profiles with the same area. The Voigt profile in Fig. 1(A) was modeled with equal values for Dl and Dl . It can be seen D C 1278
Fig. 1 (A) Absorbance profiles and (B) absorption profiles for three different peak shapes with equal area: (a) Gaussian (Doppler broadening only), (b) Voigt (Doppler and collisional widths equal ) and (c) Lorentzian (collisional broadening only).
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Fig. 2 Modeled calibration curves illustrating the effect of the a parameter on the inflection point; a=(a) 1.66, (b) 1.25, (c) 0.83 and (d) 0.42.
absorbance increases with the square root of concentration at higher values of concentration. The smaller the a parameter, the narrower and taller is the absorbance profile (Fig. 1). In the intensity domain, this means that the smaller the a parameter, the narrower and deeper is the absorption profile and the quicker the intensity at the line center reaches the stray light limit. Consequently, as the a parameter decreases, the inflection point will occur at lower values of concentration. The data in Fig. 2 were obtained by assuming that the spectral width of each pixel was equal to the half-width of the absorption profile at the focal plane. This relatively large spatial integration interval (with respect to the profile width) affects the shape of the calibration curve. Earlier modelers,5 who assumed a much smaller integration interval, found a significant overshoot, i.e., a region where the curve shape fails to behave as two intersecting straight lines. This is illustrated in Fig. 3(A), where the integration interval is one tenth of the profile half-width and the a value is 0.082. Fig. 3(B) shows the curve shape obtained with the same value for the a parameter and with an integration interval equal to the profile half-width (the same as in Fig. 2). It can be seen that the overshoot is dramatically reduced in the latter case. The overshoot will also be reduced by increasing values of the a parameter and integrating absorbance with respect to time. As the a parameter increases the Lorentzian wing structure becomes more important and the overshoot becomes less noticable; overshoot is barely visible at an a parameter value of 0.25. The absorbance signals for the graphite furnace are usually integrated with respect to time. Each calibration point is composed of the sum of absorbances in the linear region (slope 1) or, at higher concentrations, the sum of absorbances in the linear and non-linear region (slope 0.5). This integration process serves to dilute the contribution of the overshoot region. As a result of all three factors (spectral resolution, value of the a parameter for most elements and time integration), no visible overshoot was expected for any of the experimentally obtained calibration curves.
Experimental Equipment All experimental data were acquired using a prototype CS-AAS instrument which has been described previously.6 This instrument consists of a 300 W xenon arc lamp and power supply (Cermax LX300UV and PS300-1, respectively; ILC Technology, Sunnyvale, CA, USA), an electronic shutter and controller ( Uniblitz LS67 and T132, respectively; Vincent Associates, Rochester, NY, USA), an e´chelle spectrometer (Spectrospan V; ARL, Valencia, CA, USA), a 256 pixel LPDA and driver/amplifier board (Models S3904-256Q and C2325-22S, respectively; Hamamatsu, Norwalk, CT, USA) and a 486 PC (Comtech, Springfield, VA, USA). The graphite furnace was an HGA-500 and controller (Perkin-Elmer, Norwalk, CT, USA). The volume of solution injected was 20 ml. The furnace conditions used in this study are given in Table 2. Software All data acquisition and most of the processing were performed using programs developed under LabVIEW (National Instruments, Austin, TX, USA). Some data processing was also performed using a spreadsheet program ( Excel, Microsoft, Redmond, WA, USA). Table 2 Furnace atomization conditions Temperature/°C Time/s Step
Ramp
Hold
Ag (328.1 nm)
Cd (228.8 nm)
Pb (283.3 nm)
Dry Char Pre-Atomize Atomize Clean-up
15 5 1 0 1
15 5 9 5 4
250 650 20 1600 2500
250 350 20 1600 2500
250 350 20 1900 2500
Fig. 3 Modeled calibration curves with integration interval equal to (A) 1.0 and (B) 10: (a) Voigt profile, Dl =10 and Dl =10 (a=0.82); C D (b) Voigt profile, Dl =10 and Dl =1.0 (a=0.082); and (c) Lorentzian profile, Dl =1.0. C D D
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Calculations Wavelength and time integrated absorbance. Wavelength integrated absorbance, A , is computed as the sum of absorbances l for all the pixels covering the absorption profile:7 A =Dl ∑A (6) l p p where A is the abundance of pixel p and Dl is the width of p p the pixels in picometers. The wavelength and time integrated absorbance, A , is calculated by summing the wavelength l,t integrated absorbance over the desired time interval:8 A =Dt∑A l,t l where Dt is the time between computed absorbances.
(7)
Intrinsic mass. Time and wavelength integrated absorbance is normalized by the width per pixel (in picometers) and the time between absorbance computations (in seconds). Intrinsic mass is defined as the mass necessary to provide an integrated absorbance of 0.0044 pm s.8 The normalization factors used in this study were 0.016 s (1/60 Hz) for time integration and 3.38, 2.23, and 2.85 pm for wavelength integration for Ag (328.1 nm), Cd (228.8 nm) and Pb (283.3 nm), respectively. Hyperbolic curve. As previously discussed, half of a hyperbolic function was rotated and appropriate coefficients selected to provide a slope of 1.0 at the low concentration end and a slope of 0.5 at the high concentration end.1 The equation and coefficients for this hyperbola are given in Table 1. The F coefficient controls the degree of curvature between the two linear regions (F=0 provides two intersecting straight lines). The hyperbola is fitted to calibration data by offsetting the X and Y axis coordinates of the data or the hyperbola. F was varied to provide the best fit of the hyperbola to the calibration data. The goodness of fit was computed by summing the square of the vertical distance between individual data points and the hyperbola in the linear domain. The best fit was determined as the offsets providing the lowest sum of squares. Stray light. Stray light is defined as radiation from outside the selected spectral band pass which reaches the detector, i.e., has a negligible absorptivity. The total intensity, I , reaching T a pixel is equal to the sum of the useful intensity in the spectral band pass, I , and the stray light, I . The percentage of stray 0 sl light is calculated as I /I ×100% and will exceed 100% when sl 0 I
