Anal. Chem. 1991. 63. 164-169
164
Effect of the Source/Absorber Width Ratio on the Signal-to-Noise Ratio of Dispersive Absorption Spectrometry Thomas C. O'Haver D e p a r t m e n t of Chemistry a n d Biochemistry, Uniuersity of Maryland, College Park, Maryland 20742
is the absorption coefficient (cm-I), 1 is the path length (cm), and the integral extends over the frequency region for which radiant power exists a t the detector. The absorbance A is given by A = -log (1 - M). Let us assume that the spectral distribution of the incident spectral radiant power @(Jv) is described by a single Gaussian function:
Conventional slgnai-to-noise theory for absorption spectrometry is ordinarily divided into two main cases: the monochromatic or narrow-band case, in which spectral distribution of the light source is much narrower than the spectral distribution of absorption, and the polychromatic or wide-band case, in which the spectral distribution of the light source is much greater than the the spectral distribution of absorption, as is the case in low-resolution continuum-source atomic absorption. The narrow-band case is usually assumed to be optimum for analytical absorption measurement and is a requirement for adherence to Beer's law. This paper extends the conventional theory for the intermediate case in which the source and absorber spectral distributions are comparable. It is shown that in some cases improved detection limits can be obtained when the spectral width of the light source is comparable to the absorption line width.
(2) where a,,, is the maximum spectral radiant power a t the center of the line or band (U'/Hz) and G(u,vo,.lv,) is a Gaussian function (unitless) of unit height centered a t vo and with full width a t half-maximum Aus. For a dispersive spectrophotometer based on a continuum source and a monochromator, + ( u ) represents the spectral distribution of light exiting from the monochromator and Avs is its spectral band-pass. For line-source atomic absorption, AuS is a characteristic of the hollow cathode lamp and is primarily a function of its temperature and pressure. We assume that the absorption consists of a single absorption line or band centered on the source distribution. Two basic absorption distributions, Gaussian and Lorentzian, are evaluated. Gaussian absorption is a reasonable approximation for condensed-phase molecular UV-visible spectrometry, and Lorentzian absorption is a reasonable approximation for infrared spectrometry. In atomic absorption, absorption profiles are typically Voigt profiles, which are intermediate between Gaussian and Lorentzian, corresponding to the Doppler and collisional contributions to the total line shape, respectively:
INTRODUCTION The signal-to-noise ratio theory of analytical spectrometry has proven to be useful in understanding, designing, and optimizing spectrometric measurement systems ( I , 2). The signal-to-noise ratio expressions for absorption spectrometry developed by previous investigators have ordinarily been based on the assumption that the spectral distribution of the light source is much narrower than the spectral distribution of absorption (monochromatic or narrow-band light source). This is the usual case for analytical absorption measurement and is a requirement for adherence t o Beer's Law. Signalto-noise theories for atomic absorption (AA) ( I ) have in addition considered the wide-band case, in which the spectral width of the light source is much greater than the absorption line width, as in low-resolution continuum-source A.4. It is usually argued that the wide-band case is not optimum hecause many of the photons in the wider source spectral distribution will not be absorbed by the narrower absorption distribution. On the other hand, it can just as easily be argued that the narrow-band case is not optimum either, because in that case many of the absorbers in the wider absorption spectral distribution will not overlap with photons in the narrower source spectral distribution and therefore will not be observed. Clearly, qualitative arguments such as these are not conclusive. The purpose of this paper is to consider explicitly the effect o f t h e relative widths of the source and absorber spectral distributions on the signal-to-noiseratio of absorption measurement.
Gaussian absorption:
= S ( @ ( v ) ) d v / ~ ( @ " ( ' ~ ~=) d u
S(@o(v))(le - h ( ~ " ) d o / S ( ~ , ( i ~ ) )(1) dii -
where @ , ( v ) is the incident spectral radiant power (h'/Hz), @(Y) is the transmitted spectral radiant power (WIHz),k ( v ) 0003-2700/91~0363-0164$0250112
K
(3)
h ( u ) = -L(i~,v0,Av8) (4) Ava where k ( v ) is the absorption coefficient (cm~'),K is a coefficient that includes the absorber density or concentration ( K has units of Hz/cm), G and L are the unit Gaussian and Lorentzian functions respectively (unitless), and Aua is the full width at half-maximum of the absorption spectral profile (Hz). The integral of k ( v ) over the entire absorption band is assumed to be proportional to the total number of absorbers in the light path and independent of Ava. For simplicity, stray light is assumed to be zero, the absorbers are assumed to be nonluminous (Le. no thermal emission or fluorescence), the absorption width is assumed to be independent of the concentration of absorbers (110 resonance broadening, dimer or excimer formation. or other intermolecular interactions), the absorption coefficient is assumed to be independent of incident radiant power (no saturation), the spacial distribution of the absorbers across the light beam is assumed to be uniform, and the spectral response of the detector is assumed to be uniform across the source spectral profile. Signal-to-NoiseOptimization of the Source Width, Au,. Suppose that it is possible to vary the source width, Aus, of the light source while its peak spectral radiant power the peak absorption and spectral width of the absorbers, and all other factors in the experiment are kept constant. In a convrntional dispersive absorption measurement. this could Lorentzian absorption:
THEORY In the general formulation for absorption spectrometry ( I ) , the fraction CY of incident radiant power absorbed is given by (1
K
h ( v ) = -G(u,v,,Ai~~) Ava
L
1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
165
Table I. Dependence of Signal-to-Noise Ratio on Source Width, for Two Different Models of Source Intensity Dependence on Source Width dependence of source intensity on source spectral'width, Au, total, peak, J+(u)du model 1 model 2
constant aAu,
a Au, a Au:
source-width-dependent portion of the signal-to-noise ratio expression source photon detector noise noise noise a adAu, aAu, a Au: a aAu,
to have models for the variation of measurement noise with source spectral radiant power. We confine the analysis to low absorbances, which is the simplest case as well as the most interesting in terms of detection limit. The signal can then be defined as the difference between the measured incident and transmitted source intensities, that is, the numerator of eq 1,which is equal to aJa0(v)dv. Because the peak spectral radiant power +ma of the source is assumed to be constant, J+o(v)dv is proportional to the source width Av,. The noise can be defined as the standard deviation of J+O(v)dv. Three limiting noise models are considered. Case 1. Source Flicker and Background Flicker Noise. Noise
Figure 1. The effect of changing the source width, Av,, of the light source while the peak spectral radiant power, amXand the peak absorption and spectral width of the absorbers are kept constant. The light-gray bands on the left represent the spectral distribution of the incident source spectral radiant power, the dark bands on the left represent the transmission of the absorber (1 and the light-gray bands on the right represent the transmitted spectral radiant power %(v). The dotted lines on the transmitted spectral radiant power distributions show the incident spectral radiant power distribution for comparison. (a) The source width is much less than the absorption width. (b) The source and absorber widths are comparable. (c) The source width is much greater than the absorption width.
be accomplished by varying the spectrometer dispersion while the mechanical slit width is kept constant. This is illustrated in Figure 1. (The effect of varying the slit width is considered in the following section.) Since the number of absorbers and the absorption width and spectral distribution is held constant, there is one true value for the the peak absorption, which is assumed to be directly proportional to the number of absorbers. The absorbance is a good estimate of this true peak absorption only when Av,
