Anal. Chem. 1994,66, 2336-2341
Factors ContrdHng Precision and Accuracy in I soto p e - R a t b W W r b Mass Spectrometry Dawn A. Merrlttt and J. M. Hayes’ Biogeochemical Laboratories, Departments of Chemistty and of Geological Sciences, Geo@y BuiMing, Indiana University, Bloomlngton, Indiana 47405- 1403
The performance of systems in which picomole quantities of sample are mixed with a carrier gas and passed through an isotope-ratio mass spectrometer system was examined experimenfally and theoretidy. Two difierent mass spectrometers were used, both having electron-impact ion sources and Faraday cup collector system. One had an accelerating potential of 10kV and accepted 0.2 mL of He/min, producing, d e r thase conditions, a maximum efficiency of 1 C02 molecular ion collected per 700 molecules introduced. Comparable figures for the second inshvarent were 3 kV, 0.5 mL of He/min, and 14 OOO molecules/ion. Signal pathways were adjusted so that response times were
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signal-processing pathways but also by physical processes such as the growth and decay of space charges within the ion source and sorption and desorption of molecules at surfaces along the sample-transfer pathway. For example, the ion current recordings in Figure 2 (the voltage signals, not the calculated ratios) indicate only small deviations from a perfect square wave. Effects are exaggerated, however, when ratios are considered. At that point, any mismatch between time constants or any differential sorption-desorption phenomena wilt produce significant transients in computed ratios. Since perfect elimination of such phenomena is impossible, systems for the acquisition and processing of data must be tolerant of them. An idealized view of electronic effects is shown in Figure 3, which depicts calculated ion currents and ion current ratios rather than actuaI recordings. In this case, transients expected from mismatched time constants have been modeled by assumption of perfect RC behavior. Time constants assigned to the mass-44, -45, and -46 signals were 170, 180, and 300 ms, respectively, and the effects on square waves of realistic amplitude were determined by calculation of instantaneous signals and ratios as a function of time. The resulting wave forms are similar to those in Figure 2a but not identical, in spite of the fact that the assigned time constants match those measured prior to the recording of the data in Figure 2a. It follows that (as is likely) the actual signal pathways do not display ideal RC behavior and/or that nonelectronic effects are present. 2940 Analytical Chsmistry. Vd. 66, No. 14, Ju& 15, 1994
Resolution and minimization of eleotronic effects are demonstrated in the sequence of recordings shown in Figure 2. In part b of the figure, adjustment of the time constant of the mass-45 amplifier allowed nearly complete elimination of overshoot on the 45Rtrace. Because the time constant on the higher resistance mass-46 amplifier had already been minimized, elimination of transients on the &R trace would have required lengthening the mass-44 time constant and consequent degradation of the response time of the overall system. Accordingly, the presence of overshoot in theaRtecord, which is involved only indirectly in the calculation of carbon isotopic abundanm, was accepted. The third set of recordings (Figure 2c) indicates the effect of maximizing the gas dynamic conductance of the ion source and thus confirms the presence of nonelectronic effects. The return of the 45Rtrace to the background level occurs significantly more promptly when the “window”in the ion sourceis opened to thegreatest possible extent. The ”window” is an externally adjustable shutter provided by the manufacturer and is intended to allow optimizationof ion source sensitivity and samplechange-over times when the mass spectrometer is operated with a conventional dual inlet. Under the conditions employed here (notably the high throughput of He), maximum sensitivity was obtained when the window was opened to 25% of maximum. However, as shown by the tailing 45R trace in Figure 2b, the temporal response is degraded. When the window was fully closed, sensitivity was decreased by 5-10% and tailing was only slightly w m e than that shown in Figure 2b. With the window was opened to 50% of maximum, temporal response was greatly improved (Figure 2c) but sensitivity was alsoonly 50%of maximum. Further increases in source conductancedid not improve temporal performance but did result in further losses of sensitivity and were thus avoided. The 4SRsignal shown in Figure 2b is consistent with from the an expected effect, namely, the removal of 12C1602 ion source more rapidly than l3CI6O2 and 12C170160. Meaautrement of Isotopic Abuadances. In isotope-ratiomonitoring measurements, ion currents are integrated as samples of interest pass through the ion source. Integration intervals may be brief-only a few seconds. Sample pressures and resulting signal levels may vary widely. Observations of standards may be infrequent. Questions then arise concerning methods for selection and definition of integration intervals, attainable levels of precision and accuracy, and optimization of procedures. These can be addressed by examination of system performance as procedures are varied, but it is useful first to consider an absolute standard of performance, namely, the “shot-noise limit”. This refers to the precision that would be obtained if the ion beam itself were the only significant noise source,16 with all transducers, amplifiers, and other components of the signal pathway being completely free of noise. It represents a level of performance that can be improved only by redesign of the measurement so that more ions are observed or collected. Calculation of the shot-noise limit is based entirely on the statistics of ion collection. For the precision-controlling ratio in the present work, R = 4si/44i, the standard deviation will ~~
(16)Petcnon, D.W.; Hayes, J. M. In Contemprary Topica in Analytical and CIfnicot Chcmislry; Hercules, D. M., Hieftje, G. M., Ma.; Plenum hbliihing: New York, 1978; Vol. 3, pp 217-252.
sample and standard have not been equal, this expression becomes
be given by 0:
= ( a R / ~ 7 ~ ' i ) ~+~(eR/a44i)2a,2 ,,2
+ 1/"Asbnbd)
a: = 0.00446( 1/"Ammplc
where a45 and a" are the standard deviations of currents. Evaluating the derivatives, we can write a:
+
= R * [ ( u ~ ' / ~ '(~~) ~ ~ / " i ) ~ ]
For any ion current, i = Nqdt, where N is the number of ions collected in a time interval, t . Since, at the shot-noise limit, the variance in N is N , it follows that (ai/i)2is 1/N and that (aR/R)2
= (1/45N) + (1/"N)
(10)
Recalling that R = 45i/44i= 4sN/44N,this expression can be rewritten in terms only of "N: = (1/"N)[(1
+ R)/R1
(11)
The sensitivity of the mass spectrometer can be described in terms of the number of molecular ions received at the collector per molecule introduced to the ion source. If this factor is termed the efficiency and assigned the symbol E, we can write
"N
EmNA/(l -k R )
(12)
where m is the number of moles of sample gas introduced and N A is Avogadro's number. It follows that
(oR/R)' = (1 + R)2/EmNAR
(13)
Considering the definition of 6, we can write that
= (a6/aRsamplc)2uRsamplc
+ (as/aRsbndard)2u,,,ard
2
(14) and Rsbnhd are essentially where, in practice,values of Rmmplc equal, differing only in the third or fourth significant figure. The expression can thus be rewritten in terms of a single R value. If RampIe and R ~ ~ nhave ~ r dbeen determined with equal precision, the two terms on the right of eq 14 are equal, and evaluating the derivatives, we obtain an expression for the maximum attainable (i.e., shot-noise limited) performance: U?
+
= 2 X IO6 (aR/R)' = 2 X 106(1 R)'/EmNAR (15)
where b6 is the standard deviation characteristic of the distribution of observations of 6, and R is the ion current ratio characteristic of the gas being analyzed (approximately 0.01 1 for 45/44 in the mass spectrum of C02). In this expression, the product EmNA is simply the number of mass-44 ions collected. That quantity is also given by "itlq,, where "i is the mass-44 ion current, t is the integration time, and qc is the electronic charge. The value of "i is in turn given by "V/Rf, where"Vis the mass-44signal and Rfisthe feedbackresistance of the mass-44 electrometer. Substituting qc = 1.6 X 10-19 C, Rf = 3 X lo8 n, and R = 0,011, a simplified expression for the shot-noise-limited precision as a function of the integrated signal (A, Vas) is obtained: a: = O.O0892/"A
(16)
For instances in which the integrated ion currents for the
(17)
Definition of Integration Intervals. Addition of the sample to the stream of gas flowing through the ion source produces a sharp rise in ion current, and as outlined in the Experimental Section, the data system can be programmed to define the beginning of a sample zone in terms of an operator-selected slope threshold.'' The default used in the present work was 1.O mV/s (equivalent to 30 fmol*C*s-2).The end of a sample zone, or "peak", was defined in terms of the decreasing negative slope as the ion current returned to baseline values, the operator-selected threshold in this case being -1 .O mV/s. On average, integrated ion current ratios determined within these automatically selected limits differed by less than 0.01% from those resulting from manually selected limits placed arbitrarily before the beginning and after the end of sample zones. The precisionof 6 values from automaticallydefined peaks is always superior to that obtained with manually selected limits of integrati~n,'~ except where manual techniques allow removal of artifacts, as with transients at the borders of square peaks. Stability. In conventional, dual-inlet procedures, each integration of sample ion currents is immediately preceded and/or followed by an equivalent observation of ion currents derived from an isotopic standard. In isotope-ratio-monitoring procedures, observations of samples and standards may be separated by a considerable interval. In order to examine limitations imposed by instabilities in components and operating conditions, experiments were performed in which ion current ratios from a single gas sample were observed repeatedly. An early observation in the series was arbitrarily taken as "the standard", and all subsequent observations were treated as unknown samples. Any sample vs standard differences which then appeared indicated the Occurrence of drifts that, in real measurements, would degrade the accuracy of computed isotope ratios unless multiple standards were distributed throughout the run in order to allow recognition of and correction for such drifts. Results from two sets of observations using the 10-kV instrument are summarized in Figure 4. In the first case (Figure 4a), gas samples were repeatedly introduced in the absence of carrier gas. The background pressure in the ion source was 1O-* Torr and the pressure during sample introduction was 2 X lo-' Torr. The sample introduction rate for these measurements was 0.45 nmol of C02/s. The resulting signal level (mass 44) was 4.8 V, and integration of ion currents for 15 s was thus expected to produce-at the shot-noise limit-a population of 6 values with a mean of zero and a standard deviation of 0.01 1%. In fact, as shown by the minimal variations in the lines tracking the variations in the apparent 6 value, the drift rate was only 0.00038960/min and the scatter of 6 values around the central trend corresponded to a standard deviation of 0.017960. The two runs indicated are typical and were separated in time by 6 months. For many applications, the indicated stability is great enough that observation of a single standard could provide calibration for isotopic analyses over a period of more than 1 h. As shown in Figure 4b, drift rates up to 10-fold higher were encountered when a stream of helium carrier gas (0.2 mL/min, indicated AmlyticalChemkby, Vol. 66,No. 14, Ju& 15, 1994
2541
0 0.3
.
'
30
60
I
I
90 ,
i
120 I
i
30
0
60
90
120
,
a
& d ! -2
-
................. ..,' ........
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- a
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60
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30
source pressure 3 X lV Torr) was admitted to the ion source. The four lines shown summarize results of four typical runs and correspond to drift rates as high as O.OOSL/min. The cause of these drifts is not easy to identify. Their origin is clearly not electronic (if electronic, they ought also to have affected the runs depicted in Figure 4a), but the "obviousn idea that some varying background component is present in the He carrier gas must be set aside when it is recalled that the zero baseline for integration of ion currents is automatically reset at the beginning of each integration interval (see Experimental Section). It follows that the drift is associated with some variation in properties or performance of the ion source or analyzer system. Conceivably, a slightly unstable mass discrimination is associated with the presence of the He plasma in the source. The 3-kV instrumentwas lessstable, both with and without carrier gas in theion source (Figure 5). Because nonstandard, prototype amplifiers are employed on that system, this low performance indicates nothing about characteristics of instruments that may be in use elsewhere. There are, however, two broader elements of significance. First, thedrift in the presence of carrier gas is not markedly worse than that in the 10-kV instrument. The different ion source design in the 3-kV instrument is therefore not associated with any significantly different level of unstable mass discrimination if, indeed, that is the cause of the drifts seen in Figure 4b. Second, however severe these instabilities may appear, they have not hindered 2342
A m W I C h e m l ~ W y ,Vol. 66,No. 14, Ju& 15, 1994
'
.
60
"
'
90
l
i
120
t, min
t, min Fbun 4. Results of repeatedanalyses of a single sample using the l 0 k V instrument. Each llne represents a contlnuow series of obscwvatlons;except as noted, condltkns were not varied and the Hnes differ due only to random fluctuatlona in drift. I f the charactedstics of the ion source and analyzer were perfectly constant, Le., H drift were absent, aU observations would scatter around the zero Une with no long-term trends. (a) Samples (0.45 miof CO&, 15 s,shot-ndseilmlted u = 0.011460,intervals between observations 185 or 285 s) admitted in the absence of c a w gas, P= 2 X lo-' Torr. (b) same a8 (a) except for presence of carrier gas, 0.2 mL of He/mh, indicated P= 3 X 10-6 Torr.
"
........
Flgure 5. Resutts of repeated analyses of a single sample using the 3-kV instrument, as for Figwe 4 except 0.86 nmoi of CO&, 15 8 , shot-noise ilmlted u = 0.027460, intervalbetween observetkns = 185 8. (a) P= 2 X le7Torr. (b) He carrier gas,0.5 mL/mln, kdlceted P= 3 x 10-6 Torr. Tabla 1. P"of Imtoplo Morrrurmmrb VI Frequmcy of IntroducHon of Standard8
flow rate (mL of He/ t,b MS' min) h 1 1
2 2
0 0.2 0 0.5
tf
max time reprebetween sentative standardizations, h I l U X drift Ub08rate, (snl)d (obsd)e uob I us I %/min 460 % 3ud 0.1%
4.5 56 0.00050 0.011 9.0 112 h0.0045 0.011 7.0 123 h0.085 0.026 5.5 86 hO.015 0.027
0.017 >4.5 >4.5 0.021 1.5 x.3 0.29 0.3 0.090 1-1.5 0.3-0.5
*
* Mass spectrometer: (1) 10-kV MAT 252; (2) 3-kV Delta-S. Total time represented by all observations of stability. Number of discrete measurementsof6 representedby all observationsof stability. d Standard deviation that would be observed if system operated at shot-noise limit (eq 16). e Standard deviationobserved for measurementsof 6 calibrated Since this procedure recognizesand ap lies as described by Ricci et corrections for drift, this value reflects the scatter of points arounathe trends shown in Figures 5 and 6.
the very successfuluse of this instrument in numerous isotoperatio-monitoring GCMS analyses (e.g., refs 10 and 11). Drift characteristics and the observations on which they are based on summarized in Table 1. The significance of the tabulated "representativemaximum drift rates" can be inferred by comparing the data shown in Figures 4 and 5 with the numerical entries in Table 1; the tabulated drift rates provide a quantitative indication of the differences between the instruments and operating conditions noted. The practical effects of these instabilities depend on the nature of the drifts (e.g., if drifts were perfectly linear, their effects could be neutralized by use of only two standards, one at the beginning of an analytical run and one at the end). The columns headed "maximum time between standardizations" are based on longterm experience. For example, the accuracy of applied
~
Tabla 2. SensttMty and Pr.drkn of I8otop+Rallo-Monltorlng AMly8.S sample flow rate 1 fF (mL of (moleStd Qsi nal cule/ signal (sn1)d MSa pcakb ion) n0 (Ves) (%)
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 C
I%{
&zI\
0 0 0.2 0.2 0.2 0.2 0.2 0.4 0 0.5 0.5 0.5 0.5 0.5 0.5
1.8 4.7 2.0 4.0 0.060 0.17 0.29 0.29 8.6 8.6 0.2 1.0 5.0 10 35
SW,9 SW, 1 1 SW, 10 SW, 10 G, 14 G, 22 G,31 G, 20 SW, 15 SW, 15 G, 12 G, 17 G, 23 G,29 G,42
(8.s)
19 46 20 40 2.2 6.9 14 3.6 13 11 0.5 2.1 9.5 21 40
1.01
1
,
,
I
1
(2)
2800 2900 2900 2800 790 710 610 2300 19000 24000 12000 14000
0.025 0.017 0.016 0.016 0.061 0.047 0.042 0.082 0.28 0.058 0.16 0.079 15000 0.039 14000 0.031 25000 0.013
5 5 5 5 5 4 5 4 6 6 4 4 4 4 3
19 46 20 40 48 25 40 42 13 11
23 15 16 16 18
0.022 0.014 0.021 0.015 0.046 0.029 0.021 0.037 0.018 0.019 0.095 0.049 0.027 0.022 0.019
-1.01'
I
I
10
1000
100
C,pmol ............ ........
0.5
0
............... ........ %Pw-
.......... .....
a As in Table 1. SW, square wave; G, Gaussian; width in seconds. Calculated using q 3. Calculated using q 17.
t -1.0
measurements has routinely been monitored by repeated analyses of mixtures containing a homologous series of n-alkanes with known isotopic compositions. Only three or four isotopic standards have been employed in chromatographic runs covering approximately 100 min, yet root-meansquare errors in 6 values assigned to the n-alkanes have seldom exceeded 0.2960 and are commonly as low as 0.05-0.10960. Sensitivity andPrecision. When standards are introduced frequentlyenough to neutralize effects of drift, the performance of this technique will be limited by instrument sensitivity and sample size. Results of trial analyses are summarized in Table 2, in which efficiencies of ionization and transmission are listed and observed standard deviations are compared to shotnoise limits. For the 3-kV system, approximately 15 000 molecules were required for each molecular ion collected under irm conditions. For Gaussian peaks containing from 200 to 35 000 pmol of C02,the observed standard deviation exceeded the shot-noise limit by a factor of 1.7 or less. For the 10-kV instrument under irm conditions, optimal tuning yielded sensitivity 20X higher than that observed with the 3-kV system. The observed efficiency of ionization and transmission is equal to that specified by the manufacturer for operation of the instrumentunder high-vacuum conditions. If the same tuning was used in the absence of carrier gas, efficiency decreased 4-fold. The observed precision under irm conditions fell short of the shot-noise limit by an average factor of 1.8. The failure of both instruments to achieve theoretical maximum levels of performance must indicate that not all noise sources have been eliminated, but the closeness of the approach to theoretical maxima is notable. For the 10-kV instrument, the observed standard deviation was still well below 0.1% even for samples as small as 60 pmol of C02. Dynamic Range. Samples with widely varying sizes were prepared using the exponential dilution system described in the Experimental Section. Standards isotopically identical to the samples were admitted at 20-min intervals using the variable-volume inlets of the mass spectrometers. Any
'
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I
...... '
0.1
3
.....
1 1
O
1
10
C,nmol Figure 6. Results of repeated analyses of samples of varylng slze prepared by exponential dliutlon. For a system dlsplaylng a complete absence of systematic errors related to sample size, points would scatter around the zero lines within boundaries Ike those indicated. Here, in both graphs, the envelopes have been placed at 15u (shotnoise limits calculated using eq 17). For performance atthe shot-noise . l 0 k V Instrument, limit, 99% of points would lie within 1 2 . 5 ~ (a) u A = 25~Vas. The observation that the f5u envelope etncbsea results for samples larger than 50 pmol indicates that the system operated within a factor of 2 of theoretlcai maxhnum pertormance In that sample range. (b) 3-kV instrument, = 23 Vas. Note differing horizontal axes.
deviation of observed values of 6 from 0% represents effects of noise or systematic error. Results are graphically summarized in parts a (10-kV instrument) and b (3-kV instrument) of Figure 6. For the 10-kV instrument, there is no evidence for any systematic trend to positive or negative deviations at high or low sample sizes and thus, no evidence for any consequential nonlinearity in overall performance. For samples larger than 50 pmol of C02,virtually all observations fall within the indicated f 5 u envelope (where u is the shot-noise limit calculated by use of eq 17). Since 99% of all observations would be expected to fall within f 2 . 5 ~if shot noise were the only noise source, these observations are consistent with those above, demonstrating that actual performance of the system falls short of the theoretical maximum by roughly a factor of 2 in the sample range indicated. For smaller samples, decreased precision is observed, possibly because the handling and processing of such small quantities of gas is itself a significant noise source. For the 3-kV instrument (Figure 6b, note change in range of sample size axis), values of 6 tend slightly to drift to positive values at small sample sizes. Expressed in terms of the mass44signal maximum, this nonlinearity amounts to0.03-0.04%/ V. Relative to the shot-noise limits, the precision indicated by the data in Figure 6b, in which the envelope is placed at Ana!~tIca/Chemfstry,Vol. 66,No. 14, Ju& 15, 1994
2543
a
10
P
r5
0
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2000
loo00
t, sec
b 15.
10.
P 6’ 5-
0-
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I
eo00
1
KK)
Flgure 7. Representative ion current recordings (mlz 44) for sample peeks Introduced to the l o k V I n m n t together wlth varying amounts of “ba&gm”‘ C02. In (a) peaks of COPadmilted using the 1-13. samphg valve w m eq“d on c “ ! levels of backgroundC& &tnl&d from the variable-volume inlet of the mass spectrometer. At intervals, the flow of “background” C O 2 was intmupW and a wirbr peak d crtendard Conwas admitted from the alternate reservoif of the variable-volwne inlet (we Figure 1). I n (b) the exponentla1dlMh flask, InMelIy filled wlth pure He, was placed In the carrler gas supply line for the chromatographic cokmn and then purged wlth 284 ppm C02in Hb. ;cw a result, the concentratlon of C02 in the carrier gas asymptotically approached 284 ppm, a trend that roughly duplkated the riw in C Q lrveQ “dated with Increasing column bleed rates. Peaks of C02 were again injected by use of the sampUng valve, and the flow from the open @It was m d intermittently to aibw htroductlon of standards against zero background.
k5u, is marginally lower than that obtained in the experiments summarized in Table 2. A useful summary can be cast in terms of sample sizes required to obtain standard deviations of 0.1 and 0.5960, assuming that both instruments fall short of theoretical maximum performance by a factor of approximately 2. For the 10-kV instrument, with a 60 V*sstandard and E = 1/700 ions per molecule, 45 pmol of C02 will be required to obtain U6 = 0.1%. For the 3-kV system, with a 45 V*sstandard and E = 1/ 14 000 ions per molecule, the sample requirements for standard deviations of 0.1 and 0.5% would be 900 and 35 pmol of COz, respectively. Evaluation of Procedures for Background Correction. Particularly in gas chromatographic applications of isotoperatio-monitoring techniques, where peaks of C02 derived from the combustion of chromatographicallyseparated substances are likely to be superposed on substantial fluxes of C02 derived from the combustion of column bleed, corrections for contributions from background ion currents are critical if accurate results are to beobtained. Moreover, any imprecision in the background corrections will propagate and increase uncertainties in calculated values of 6. To test the procedures 2944
Ana&tk+aIChembtry,Vd. 88, No. 14, Ju& 15, 1994
employed here, experiments have been performed in which additional, “background” C O 2 was deliberately added to the gas stream carrying the sample. The accuracy of the results and effects on precision were then examined. Ion current traces representative of data collected using the 10-kV instrument are shown in Figure 7. In the first experiment, sample peaks had Gaussian profiles and background levels of C02 were increased incrementally (Figure 7a). Corresponding deviations of observed S values from correct results are summarized graphically in Figure 8a, which includes data not only from the run shown in Figure 7a but also from additional runs in which samplesize and the isotopic difference between “sample” and “backgroundn C02 were varied. In neither of these cases or in similar experiments in which an exponential dilution system was used to produce a smoothly rising background (Figures 7b and 8b) was any evidence observed indicating the presence of systematicerrors related to the background corrections. During two tests of background corrections using the 3-kV instrument, sample C02 was introduced in the carrier gas stream (Gaussian profiles). Ion current traces were similar to those shown in Figure 7b. As shown by data in Figure 8c,
0.4
I
1
a T
-0.4
sample pmol ofCOz
T
I
0
2
4
I
6 I
60 60 60 60 60 60 60 60 60
60
I 0 0.4
c=
I
I
2
3
-I
170 170 170 170 170 170 170 170 170 170
bkgd‘ (V) 0.096 0.096 0.214 0.331 0.480 0.480 0.697 0.878 1.119 1.119 0.099 0.202 0.291 0.387 0.489 0.595 0.712 0.815 0.906 1.238
sample background area ample
?b*
(s)
n
1.0 0.25 1.0 1.0 1.0 0.25 1.0 1.0
4 4 5 4 4 4 4 4 3 3 4 3 5 4 4
1.0 0.25 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
s
4 4 4 3
4.2 4.2 1.9 1.1 0.76 0.76 0.49 0.39 0.28 0.28 4.6 2.2 1.6 1.2 0.93 0.76 0.64 0.56 0.50 0.37
11 11 4.9 3.2 2.2 2.2 1.5 1.2 0.9 0.9 29 14 9.8 7.4 5.8 4.8 4.0 3.5 3.1 2.3
Ad
08
(%I
(%)
obsd
eq27
0.02 0.14 0.00 -0.07 0.00 -0.01 -0.01 0.07 0.00 0.02 0.02 -0.01 0.03 0.00 0.02 0.00 0.01 0.04 0.00 0.21
0.089 0.098 0.061 0.078 0.11 0.13 0.13 0.16 0.20 0.22 0.047 0.11 0.073 0.052 0.10 0.086 0.10 0.16 0.17 0.15
0.049 0.049 0.054 0.059
0.064 0.068 0.070 0.076 0.082 0.095 0.028 0.03 1 0.033 0.034 0.036 0.038
0.040 0.042
0.044 0.048
Background signal measured at m / z 44 (V). Interval of time for measurement of background signals. e Ampl, amplitude corresponding to peak maximum for mlz 44 peak trace. A = 6~ - &.
-0.4
1 0
1 1
2
Mass-44 Background, nA Flgure 8. (a) Summary of isotopic resuits obtained in emments itke those depicted in Figure 7a. For each peak, the difference between the obswved and expected b value is plottedas a fun& of background slgnal levd. Error bars are f2u (shot-ndee nmit) as calculated uslng eq 27. Lengths of error bars vary because quantities of COP(both sample and standard) varied. I n addition, isotopic ditferences between sample and “background” C02 varled. The open circles represent 13.8 Vas samples (290 pmol) depleted in by 34% relative to the background; standards, 40 V.S. The open triangles represent 4.8 Vas samples (120 pmoi) with an isotopic composrtlon equal to that of the background; standards, 31 V.S. The open squares represent 2.2 Ves samples (60 pmd) enriched in by 19% relative to the background; standards, 48 V.S. (b) Results of experiments like that shown in Figure 7b. Error bars, f2u. Both series of points represent 5.0 V.s samplee by 19% relative to the (120 pmoi). Open circles: C02 enriched in background; standards 52 V.S. Open triangles: CO2 depleted In by 16% relative to the background: standards 38 V.S. (c) Resof background.correctedanalyses of standard +alkanes combusted on nm,S with the resulting C02 belng led to the ion source of the 3 k V instrument. In thls case, the error bars have been lengthenedto f4u in order to regularly interceptthe A = 0 line, indicatingth8tlhe process of combustion is itself a significant source of noise. open circles: 7-24 V.s samples (4-12 nmol of COP); standards, 11.6 V-s. Open triangles: 4-16 V-s samples (2-8 nmol of C02); standards, 9.9 V.S. Samples were depleted in by 0-10s relative to the background.
there is no significant change in 6 values as a function of the magnitude or isotopic composition of background C02 relative to the sample. Effects on Precision. Table 3 summarizesresults of trials in which 60-and 170-pmol samples were repeatedly analyzed in the presence of varying backgrounds. The 10-kV system was used, and a small samplesizewas chosen toallow sample/ background ratios to reach low levels representative of the worst cases encountered in applied analyses. To further duplicate conditions encountered in practical work, times devoted to observation of background levels were minimized, being held to 1 s, maximum, or as little as 250 ms. The latter
condition corresponds to situations in which background levels are based on a single integration interval. This minimizes the precision of background measurement and thus maximizes chances for propagationof random errors. It might, however, still provide the best estimate of background levels when observations on either side of a minimum point clearly trend to higher values and thus indicate the presence of nonbackground components. The first entry in Table 3 represents observations made with no background C02 added to the sample stream. The “background signal” is, nevertheless, correctly represented as 96 mV, this nonzero level representing the offset current described in the ExperimentalSection. If the presence of this offset is ignored and the shot-noise limit for analysis of the 60-pmol sample in the absence of any background computed (eq 17), the result is ua = 0.046% As shown in Table 3, the observed standard deviation was-as expected based on the results summarized in the preceding section-about 2-fold worse, 0.089 or O.OS8%, depending on the time spent observing the background. Subsequent entries in the table show, remarkably, that, as background levels were increased so that sample/background ratios dropped as low at 0.28, observed standard deviations increased by only a factor of about 2. Twoquestionsarise: (i) Is noise associated with the correction for the offset currents responsible for the failure to achieve maximum performance? (ii) Can the relationship between precision and background levels be adequately modeled? When an ion current comprised of sample and background components is integrated so that a total number of ions, Nx, is observed, we can write N,=N+N, (18) where N represents the sample-derived component and Nb represents the sum of all background components (nonsample COz, ion currents from other residual gases in the ion source, and pseudo ion currents such as the electronic offsets applied Ana&?kal-,
Vol. 66, No. 14, Ju& 15, 1994
2348
to the summing points of the electrometers). The arithmetic of the required correction is trivial, N = NE - N b r but consideration of the propagation of errors is not. Ifvariations in NEand N b were independent, it would be correct to take a conventional approach, writing gN2
+ (dhr/dNb)2cN:
= (@N/dNz)2aN,2
(19)
where UNE and UNb represent, respectively, the standard deviations of NE and N b . However, NEand N b are strongly covariant. N b contributes directly to NE,and any increase in N b tends very strongly to produce an increase in NE. We have therefore approached this problem in terms of the analysis of variance, starting from the expression 2,
QNT
- 'N
2
+ 'Nb
2
(20)
At the shot-noise limit, each of these variances is q u a l to = NE,CTN~= N,and m2= N b . its related variable, thus m2 Equation 20 is thus equivalent to eq 18 and it might be concluded that application of the background correction imposes no penalty on the precision with which N can be measured. Indeed, this would be true ifthe measurement of the background were based on observation of N b ions, that is, a number of background ions equal to the number underlying the sample-derived signal. For a sample-related peak with width, w (i.e., some period of time), this would require that the background be observed in a sample-free interval also having width w. In practice, however (see Experimental Section), the estimate of N b is usually based on observation of the background during a small time interval immediately p r d i n g the peak, and it is advantageous to minimize that interval in order to allow more time for observation of samplerelated signals. Moreover, in a typical chromatogram, it can be difficult tofind sample-free intervals, and minimization of the time required for observation of background signals is desirable for that reason. If the background level is observed for some time interval rt,, we can write Nb
= nbW/fb
(21)
where n b is the number of ions collected in the interval lb. The corrected signal level is then given by
N = NT - nbW/tb
(22)
and we recognize that, even though the total variance in the integrated signal must still be given by NE,the variance associated with the background correction is less than N b by a factor tb/W. By difference, the variance associated with the corrected signal level is thus
- anb2 = N + (1 - tb/w)Nb (23) where the additional term, (1 - tb/W)Nb, represents the uN2 uN;
contribution of the background correction to the uncertainty in N. Resultant uncertainties in R can be derived from an expression analogous to eq 10: (24) = ('4SN/"N2 +
