Detection Limits
and Spectral Interferences in Atomic Emission Spectrometry
R e detection limit serves as both the foundation and the skeleton of the basic methodology of an analytical techniaue A
D
etection limits are important figures of merit in all branches of analytical chemistry. In routine analyses, they are intuitively interpreted as the lowest concentrationthat one can determine with a particular method. In research and instrument development, where one is primarily concerned with the experimental determinationof numbers that reflect the power of the method or the instrument, detection limits are used to promote methods or instruments and their manufacturers. The true content of detection limits is the set of relationshipsthat describes the dependence of the detection limits on the physical variables of the analytical systems they characterize. Because discussions of detection limits are scattered throughout the literature and sometimes
P.W.J.M. Boumans Philips Research Laboratories 0003-2700/94/0366-459A/$04.50/0 0 1994 American Chemical Society
are incoherent or ambiguous, it has been difficult to understand this content. However, it is my view that the messages detection limits convey now have been unified; the means exist for systems to communicateconcrete messages about strong and weak features that are hidden behind the numerical values of detection limits. Detection limits are usually determined experimentally with ideal samples that contain only analytes at very low concentrations. Such detection limits provide useful information about the intrinsic performance of instruments, particularly if the measurements are obtained according to procedures that include the acquisition of additional data. The results may provide insight into system diagnosis,possible improvements,optimal exploitation of new technologies,and unbiased communication of results that can be unambiguously interpreted beyond the strict environment in which they were collected.
A real sample comprises a matrix that may adversely affect the detection limit, the precision, and the accuracy of the measurement. In atomic emission spectrometry (AES),for example, the matrix may substantially worsen the detection limit, chiefly because of spectral interference. A detailed analysis has not only revealed the precise reasons for this prob lem but has also uncovered ways that it can be addressed. AES is a field in which the study of the fundamentalaspects of detection limits has received considerable attention. D e velopments in instrumentation,particularly the advent of steady plasma sources such as the inductively coupled plasma (ICP), have fostered in-depth investigations whose results can be extended and applied to all AES methods with either liquid or solid sampling. With proper adaptations, these results can also be a p plied to atomic mass and fluorescence spectroscopy,including X-ray fluores-
Analytical Chemistry, Vol. 66, No. 8, April 15, 1994 459 A
cence spectroscopy. Although detection limits will be discussed here in terms of ICP-AES (1,2),there is no limit on the scope of applications. Historical perspective
For the measurement of detection limits and the theoretical discussion of their dependence on system variables, either of two approaches may be followed (3): the SNR approach,which uses the signal-tonoise ratio (SNR), or the SBR-RSDB approach, which uses the signal-to-background ratio (SBR) and the relative standard deviation of the background (RSDB) . Each approach is correct, and each yields the same detection limits. Over the past 50 years, however, a major problem has been that protagonists of one approach did not understand the essentials of the other approach. Nevertheless, concepts from one were used with the other, which caused not only confusion but also horrible errors. Fortunately,it now a p pears that this problem has disappeared. Kaiser described the basis of the SBRRSDB approach (4),but its acceptance outside continental Europe was hampered, possibly because it was published in German but more likely because it was originally associated with photographic detection, which has lost much of its attraction since the early 1960s. In photographic detection, noise is manifested as fluctuations in optical density (“blackening”) and a difference between a line signal and the background is primarily observed as a difference in optical densities. Because a photographic emulsion has a logarithmic response, a density difference translates into a difference between the logarithms of two intensities (x), which is equivalent to the logarithm of an intensity ratio (log xA/xB).This is the origin of the ratio concept underlying SBR and RSDB. Further development of the theory expanded its scope beyond photographic detection. Using the source background as a natural reference level greatly facilitates the practical application of physical relationshipsas well as the transfer of data on detection limits and related quantities between equipment.Thus, one can avoid the cumbersome absolute intensity calibrations with a standard source, which would be required in the SNR approach to 460 A
Figure 1. Block diagram illustratesthe dependence of SBR, RSDB, and detection limit on the characteristics of the source, spectrometer, and detector when background is defined in terms of radiant energy only. The SBR value that dictates the detection limit is the SBR as measured, which depends on the SBR in the source, the physical width of the spectral line, and the bandwidth of the spectrometer. The relevant value of RSDB may be directly measured or calculated from an RSDB function in which the coefficients for the noise contributions have been previously measured under conditions representative of the pertinent equipment. (Adapted with permission from Reference 16.)
exchange information between different equipment or laboratories. Flame emission spectrometrists in the 1950s and 1960s and plasma source emission spectrometrists in the 1970s and 198Os,particularly in North America, viewed the photographic plate as a curious antiquity. They did not see a need to use European idiosyncracies such as SBR and RSDB in their work, which was dominated by the straightforwardSNR concept (5).They did borrow the SBR concept for optimization purposes, but blending SBR and SNR turned out to be a ticklish endeavor. This confusing situation has been clarified by a detailed theoretical analysis of both appmaches and their interrelationship (3,5-7). Practical applications of the SBR-RSDB approach to assess classical ICP systems us@g photomultipliers (PMTs) as d ectors (6, 7-9) and advanced ICY systems based on array detectors (10-jZ) suggest that this approach is finding wider acceptance. This fact, coupled with the recent design and publication of software for data collection and
7
Analytical Chemistry, Vol. 66, No. 8, April 15, 1994
processing using the SBR-RSDB a p proach (I.?), reflects my belief that the approach offers distinct advantages. Principles
Detection limit and background-limited noise. Conventionally,a detection limit (cL) is experimentally defined as the analyte concentrationthat yields a net analyte signal (xA) equal to k times the standard deviation (oB)of the background (x,) c L =
OB
’A”0
where co is the concentrationyielding a net analyte signal x,. Equation 1is the formulation of the detection limit in terms of the SNR theory. The right-hand side of the equation is the quotient of the net signal at the detection limit (koB)and the sensitivity (xA/c0). The numerical value of the factor k is important in the context of the statistical interpretation of the detection limit as a practical analytical figure of merit (14). For the sake of uniformity, the use of k = 3 is generally recommended.The use of
k = 2@ is preferable in fundamental discussionsto emphasize the statistical interpretation and to ease connecting the detection limit with precision (3). The standard deviation oBis usually determined by measuring the variability of the background signal using an analytefree “blank” sample. In Equation 1it is assumed that near the detection limit the system is limited by background noise. The complications inherent in the treatment of the situation in which both background and analyte signal noise are covered (15)is of little interest here. This is because it may be disputed whether the conditions under which analyte signal noise perceptibly contributes to the total noise at the detection limit are acceptable as a basis of viable analytical methods. If such conditionsoccur, usually as a result of a lack of detector sensitivity, the relevant system is far from optimal. SBR-RSDB approach. Equation 1essentially contains the quotient of the background noise (oB)and the net analyte signal (x,); only the value of the quotient has meaning in comparisons of results obtained on different equipment. Numerical values of oB and x, in terms of units such as volts, amperes, or counts have meaning only if used in the context of the same instrument. Transfer of data between equipment requires the use of a standard source for intensity calibrations. This impractical approach is rarely used in analytical laboratories. A viable alternative is to normalize oB and x, to the background signalx,, d e fined as the readout signal obtained for the radiant background of the source (i.e., without possible contributionsfrom detector dark current or readout offsets). This is the essence of the SBR-RSDB approach, which is formalized by dividing the numerator and the denominatorof Equation 1by x,. The common form of this equation is CO
cL = k x 0.01 x RSDB x SBR
which is equal to c,/SBR, is often preferred. Clearly this equation should yield exactly the same detection limit as Equation l. The difference is not in the final results but in the ease with which the detection limit can be broken down into practically measurable functions. To avoid absolute measurements, the SBR-RSDB approach uses two relative quantities, SBR and RSDB. They can be unambiguously determined experimentally for any system and can be expressed as functions of the physical variables of the system. This approach permits a detailed assessment of the system and an unbiased comparison of different systems. Figure 1 (16) illustrates this concept schematically for the variables associated with the three components of an emission spectrometric system: source, spectrometer, and detector. This concept has been examined theoretically (3,6,7) and practically (8-11),and tools for further applications are available (17). The SBR function
Figure 1shows that the measured SBR ([SBR] ),,,, depends on both the source and the spectrometer. The term “source” refers to the combination of the excitation
source and the sample introduction device. Actually, only a part of the source is used for observations; this part is characterized by the “source SBR ([SBR]source), which is dictated by typical source parameters such as temperature, sample transport efficiency, type of discharge gas, and gas flow rates. Recognition that the source SBR is a characteristic independent of the spectrometer is paramount.The spectrometer converts the source SBR into the measured SBR on the basis of two variables: the physical width of the spectral line and the spectral bandwidth of the spectrometer. The result of a rigorous treatment of the pertinent relationshipsin terms of spectral radiances may be condensed into a simple equation in which the measured SBR is the product of the source SBR and fopt, which is a function of the physical width of the spectral line and the bandwidth of the spectrometer (3,17). This function has been determined semiempirically for the ICP using general physical principles, and its appropriateness has been independentlyconfirmed theoretically (9).The semiempirical function has been used to construct the curves in Figure 2 (18)for spectral lines having physical widths ranging from 1to 16 pm.
.-
E) io3-
Continuous background
(2)
where RSDB is the relative standard deviation of the background (expressed as a percentage) and SBR is associated with concentration cw Because a numerical value of SBR has a defined meaning only if the value of co is also stated, the use of the background equivalent concentration,
Figure 2. Line sensitivity (left) and signal-to=backgroundratio (rightj as functions of spectral bandwidth for lines with physical widths ranging from 1 to 16 pm. For comparison, the diagrams include a curve that represents the behavior of continuous background. The curves are based on calculations using a semiempirical model that involves a slight discontinuity. (Adapted with permission from Reference 18.)
Analytical Chemistry, Vol. 66, No. 8, April 15, 1994 461 A
The left-hand diagram illustrates the behavior of the line sensitivities and the continuous background as functions of spectral bandwidth. Accordingly, lines that have the same intensity at 1-pm bandwidth may differ by 1order of magnitude in intensity at 20-pm bandwidth. Logically, the broader the lines the more closely they resemble a continuum and the less their increase in intensity “lags” behind that of the continuum. The corollary is that an SBR decreases with increasing bandwidth (the narrower the line, the sharper the decrease), as illustrated in the right-hand diagram. Calculations involving the function&, permit the conversion of SBRs, and thus detection limits, between spectrometers or the reduction of the values of measured SBRs to the corresponding values of source SBRs. This approach can remove bias in assessments of the detection capabilities of different sources without the need to make observationson the same spectrometer.The practical application of the approach requires the availability of data on physical linewidths, knowledge of the bandwidth of the relevant spectrometers and, for convenience, the availability of a computer program to perform the calculations. For the ICP, the physical widths of - 350 prominent lines of 65 elements have been measured with a high-resolution spectrometer (19). A computer program (in IBM-PC format) (17) is associated with a database of measured physical linewidths (19) or, for spectral lines broadened by hyperfine structure, as the result of mod~cationsof the original data made during the development of the program and in interaction with it. It enables the user to perform calculations and to view simulated line profiles, with SBR+1 as an ordinate, on the screen. By default the calculations are performed for linewidths valid in an ICP with a Doppler temperature of 6300 K, but the temperature used by the program may be varied between 100 and 10,OOOK. New data may also be included in the database. Figure 3, which is based on this software, is equivalent to Figure 2 and primarily covers the behavior of three real spectral lines (curveswith points). The behavior is peculiar because the Cu and Ce lines are doublets and the In line is a
Cu 324754 pm Ce395254pm In 230606pm
-1 I
\ I
0
5
1
30 Bandv..- _.
,r
,
Figure 3. SBR as a function of spectral bandwidth for two doublets (Cu and Ce) and a triplet (In). The curves with points are for the actual structures; those without points represent the behavior if the strongest component is treated as an isolated, simple line. The data underlying the curves were obtained with a spectrum simulation program using measured, physically resolved spectra as the primary data. The various curves have been normalized. (Adapted with permission from Reference 17.)
triplet. The multiplet structure is well resolved at a very low bandwidth, but the components merge into each other as the bandwidth increases. The SBR behavior, orginally that of a single narrow line, changes into that of a broad line. If each line consisted of only the strongest component, its SBR would behave as depicted by the corresponding curves without points. Bandwidth correction of SBRs and detection limits. The crucial question is,
462 A Analytical Chemistry, Vol. 66, No. 8, April 15, 1994
“How large are the effects of bandwidth correction under practical conditions of spectrochemicalanalysis?”The data in Table 1, computed with the simulation program and the associated database (I7), give a general idea of the magnitude. The computed increase in SBR for changes in bandwidth is shown from a higher to a lower value, for example, from 25 to 3 pm (symbolized as 25/3 in the table heading). The results are based on spectral lines with physical widths in a range between 1and 20 pm; the high Values are associated with hyperfine structure composites. This range is common for sources such as ICPs that have a Dop pler temperature on the order of 6000 K. It is clear from these data that a change in bandwidth affects spectral lines of varying physical width differently. The maximum effect ranges from a factor of 8 to a factor of 1.1;a factor of 2-3 is the most common. Bandwidth corrections in comparisonsof detection limits do not have a dramatic effect, but because they are systematic,they may remove bias from the assessments. This type of data is useful in a typical research situation in which the primary interest may be in assessing the capabilities of sources separate from the spectrometers. In contrast, if the capabilities of a particular, complete (commercial) instrument are being assessed, it is the unmodified results that are important. Bandwidth determination. In principle, the bandwidth of a spectrometer may be estimated as the spectral slit (i.e., the product of the slit width and the reciprocal linear dispersion [5,20]).However, as the spectral slit decreases below 15 pm, aberrations increasinglyenhance the practical bandwidth above the value of the spectral slit, which makes an experimen-
tal determinationmandatory. This may be done conveniently with one or more ICP lines that have small physical widths (21). The RSDB function
The first approach to formulating the RSDB function expressed it in terms of the coefficients for source flicker noise, shot noise, and detector noise, and the background readout signal. For many years, the use of the equation in this form was justified because in all practical situations, detector noise was negligible. Recently, when systems with non-negligible detector noise were included (7,8), the equation was refined as RSDB
An additional feature of this approach is that the magnitudes of the terms in the RSDB equation reveal the relative importance of the various noise sources. This allows detailed analysis of the system; comparisonswith other systems; and predictions of the effects of changing properties of the system, categorized according to source (background radiance), spectrometer (throughput, bandwidth), and detector (sensitivity, dark current), as illustrated in Figure 1.
=
The background signal (2,) is expressed as the readout signal per unit time, and the equation also contains the detector’s dark current signal (2,) , expressed in the same readout units (per unit time) as 2,. The various versions of the RSDB hnction contain a set of coefficients whose values can be determined experimentally and subsequently can be substituted as parameters characteristic of the system. In turn, this implies that values of RSDB can be determined with the equation by inserting the numerical values of easily and precisely measurable variables into the relevant RSDB equation. The advantage of this approach is that one obtains far more consistent RSDB values than would be found with the common procedure of direct, repeated measurements of RSDB. The use of an RSDB function, based on straightforwardphysical principles, that incorporates experimentally determined numerical values of coefficients represents the application of a type of smoothing technique. On the other hand, use of an RSDB equation with particular coefficient values requires that the system remain constant.The system should be checked regularly by a direct RSDB determination based on a small set of measurements. Comparison of the value obtained from the check with the value derived from the equation will immediately show whether the system still behaves properly (i.e., whereby a tolerance of a factor of 2 may be acceptable).
-a C
d e
Figure 4. RSDB assessments of two ICP spectrometer systems. (a) System A and (b) System 6. The points are the experimental results from which the coefficients in Equation 3 have been determined. Curves a and c are the corresponding fitted curves, respectively; other curves represent extrapolations. Curves a, b, and c represent three different integration times (z): 5.36, 10, and 1 s, respectively; Curve d is for an extrapolation to dark current zero and z = 5.36 s. If the systems could be made free of shot noise and thus become flicker-noise dominated, Curve f would apply with aB= 0.8% (system A) and a, = 1.5% (system 6). Curve g is the corresponding extrapolation to a, = 0.4% and a, = 0.75%. Curve e shows what would happen to the actual systems if the flicker noise could be reduced to the latter values and if z were 5.36 s for both systems. (Adapted with permission from Reference 8.)
Assessment of spectrometers. Figure 4 shows the use of RSDB in assessing several ICP spectrometers (8).The examples presented here refer to a 1-m (Spectrometer A) and a 0.4-m (SpectrometerB) monochromatorwith - 8- and 12-pm bandwidths, respectively. The spectrometers were different, as were the ICPs and the nebulizers, which were also assessed; both systems were equipped with photomultipliers as detectors. The points in the diagrams are experimental data obtained at 10 different (analytical) wavelengths ranging from 193.8 to 324.8 nm. A smoothing technique used with SpectrometerA provided an effective integration time of 5.36 s in contrast to Spectrometer B, in which the actual and effective integration time was 1s. This difference in integration time is reflected in the curves fitted through the experimental points: Curve a in Figure 4a and Curve c in Figure 4b. These fits provided for the numerical values of the coefficients a, and p in Equation 3; the third coefficient, zD, was obtained by a direct measurement. Once the coefficients are known, Equation 3 can be used to calculate RSDB for a variety of conditions,in particular other integration times, dark currents, and shotand flicker-noise coefficients. Thus, it allows extrapolationsto desirable or idealized conditions. The figures show several curves, most of which are based on extrapolations. From these curves, one can conclude that SpectrometerA is shot-noiselimited because Curve a lies well above Curve f, which represents the flicker noise limit (aB= 0.8%)for this system. The smoothing technique, which leads to an effective integration time of 5.36 s, is essential to keep RSDB at a reasonable level. If it were not used (Curve c with T = 1s), the system’s performance would be poor. Dark current reduction or ICP flicker noise reduction (actually nebulizer stability improvement) would affect RSDB only marginally. This can be deduced from the fact that the relevant curves (b, d, and e) are close to Curve a. For an appreciable improvement of RSDB at low wavelengths, both photon and dark current shot noise should be reduced. This can be achieved by increasing the throughput of the spectrometer and/or the sensitivity of
Analytical Chemistry, Vol. 66, No. 8, April 75, 7994 463 A
the PMT and decreasing the PMT dark current. For Spectrometer B, it is clear that the flicker noise limit (Curve f) is higher (a, = 1.5%),but in the absence of a smoothing technique, performance with the actual integration time of 1s is shotnoise limited (Curve c) . However, if smoothing were applied (7= 5.36 s), the system would be close to being flickernoise dominated. Consequently, improving the ICP and nebulizer stability (a, = 0.75%)would greatly affect the overall performance (Curve e), making it better than that of Spectrometer A. The focal length of Spectrometer B is smaller than that of A, which fosters a higher throughput and lower shot-noise level. However, it also entails a poorer spectral bandwidth, which partly offsets the gain in RSDB as a result of increased SBR. The detailed analysis of the spectrometers in terms of SBR, RSDB, and detection limit (8) shows that there is little statistical mystery and much straightforward physics in the behavior of SBRs, RSDBs, and detection limits, not only in research laboratories but also on commercial instruments. Even small variations among sets of detection limits obtained with different equipment can be explained in detail using a rational approach. The use of the SBR-RSDB approach made it possible to reveal the separate roles of the dark currents of the PMTs (detector shot noise), the sensitivity of these PMTs in conjunction with the background radiant flux at the detector (photon shot noise), the effective integration time (detector and photon shot noise), the nebulizers and ICPs (source flicker noise), and the spectral bandwidths of the spectrometers in connection with the physical widths of the lines (SBR and photon shot noise). The consistent application of the SBRRSDB approach during the development and evaluation of a new type of ICP array spectrometer (10, 11,22)not only demonstrated the applicability of this approach to array detectors but also fostered innovation. RSD of net line signal: precision
Very often it is not the detection limit but the precision of an analytical method, expressed in terms of the RSD or the confi-
dence interval of the concentration, that is crucial. Precision depends primarily on the RSD of the measured net line signal (RSDN) if the statistical error in the calibration is neglected. RSDN, in turn, is dictated by the fluctuations in the measured background signal and the measured gross line signal. The latter includes the fluctuations in the emitted net line signal (analyte flicker noise). These dependencies imply that precision and detection limit are functionally interrelated, at least to a certain extent (i.e., at the detection limit, RSDN is always 50%if, in Equation 1,k = Z f i and is - 50%if k = 3). Also, if the limit of determination (c,) is defined as the concentration associated with a 10%RSD, then C, = 5c,. At higher concentrations the connection between detection limit and precision is even further disrupted. Figure 5 illustrates the dependence of RSDN on the ratio of the concentration present to cLfor several values of parameter a,, which is the flicker noise coefficient associated with the emitted net line signal. The curves represent the function (3, 12), RSDN
=
which is valid for the situation in which a, = a, and SBR
