Correction of Mass Bias Drift in Inductively Coupled Plasma Mass Spectrometry Measurements of Zinc Isotope Ratios Using Gallium as an Isotope Ratio Internal Standard* RAIMUND ROEHL AND JOHN GOMEZ California Public Health Foundation 2151 Berkeley Way Berkeley CA 94704 USA LESLIE R. WOODHOUSE Department of Nutritional Sciences University of California at Berkeley Berkeley CA 94720 USA Gallium was used as an isotope ratio internal standard to correct the mass-bias drift of an inductively coupled plasma mass spectrometer employed for zinc isotope ratio determinations. The Zn isotope ratios were measured to establish dietary Zn absorption in nutritional studies using 67Zn aZn and "Zn as tracers and &Zn as the reference isotope. Natural abundance Zn standards were analysed before and after each group of five samples.All samples and standards were adjusted to a common Zn concentration and spiked with Ga to obtain a 69Ga intensity roughly equal to that of &Zn. The measured 7'Ga:69Ga ratios served to monitor and correct for changes of the instrumental mass bias. Adding the two Ga isotopes to the data acquisition programme increased the cycle time of the mass spectrometer by only 25% and had no discernible effect on the precision of individual Zn isotope ratio determinations. Three different methods for correcting the measured Zn isotope ratios for mass-bias drift were compared (i) simple division of the Zn isotope ratios by the Ga isotope ratios; (ii) correction based on a power law; and (iii) correction by a regressien method which uses the relationships between the temporal changes of the 71Ga 69Ga ratio and the three Zn isotope ratios for each analytical batch.The correction based on regression consistently gave the best results because it avoided undercorrection or overcorrection of mass-bias drift; even when there was little or no correlation between the drift of the Zn and Ga isotope ratios the range of variation of the corrected data was not increased. On the average the regression correction method reduced the drift of the Zn isotope ratios by a factor of 2.5. The study included the derivation of equations which can be used to predict the achievable improvement in precision for the simple division and regression correction methods directly from the measured data i.e. without actually applying the corrections.Keywords Isotope ratio internal standard; inductively coupled plasma mass spectrometry; mass bias drift correction; gallium; zinc Inductively coupled plasma mass spectrometry (ICP-MS) con- tinues to gain importance in biological and nutritional studies using isotope tracers. The attractive features of ICP-MS are its applicability to almost all elements of the Periodic Table and its excellent sensitivity high sample throughput and ability * Presented at the 1994 Winter Conference on Plasma Spectrochemistry San Diego CA USA January 10-15 1994. I Journal of I Analytical I Atomic Spectrometry to determine isotope ratios with a good degree of short-term precision. Over the past two years the present authors have collabor- ated on studies of human Zn metabolism using 67Zn 68Zn and 70Zn tracers with 66Zn as the reference isotope.'-3 The samples analysed included purified extracts of human blood plasma urine and faecal material.Because of the large number of specimens ICP-MS is better suited for the isotope ratio measurements than thermal ionization mass spectrometry (TIM S). One limitation of isotope ratio determinations by ICP-MS with quadrupole instruments is that the measured ratios are subject to a small amount of mass bias (discrimination). This means that the observed isotope ratios differ slightly from those generally accepted for a given element; the relative differences of observed and accepted values typically range from 0 to 5%. The mass bias observed in ICP-MS can change with time due to variations in instrument conditions.Although the mass bias itself and its drift can partially be monitored and corrected for by frequently analysing solutions containing Zn isotopes at natural i.e. known abundances this approach does not provide a correction for each individual sample. Correction schemes which are based solely on analyses of natural abundance standards usually require the assumption that the mass-bias drift is linear with respect to time between analyses of the standards. Since the correction calculations in this case also demand considerable operator involvement e.g. manual parsing of the data the linear interpolation method was not evaluated in this study. A more direct approach to the correction of mass-dependent drift has been suggested by Longerich et u L ~ who added TI as an internal standard element with a constant isotope ratio to samples that were analysed for Pb isotopes.Changes in the observed T1 isotope ratio were used to correct the readings for the Pb isotopes for each individual sample. The correction equations were based on a power law designed to account for the mass difference of the isotopes forming a given ratio. A detailed evaluation of that method has been published by Ketterer et al.' Amarasiriwardena et aL6 reported the use of Ga to monitor changing ICP-MS response factors for Zn isotopes. However those workers did not correct the "Zn 68Zn ratios measured in their study based on the observed 71Ga 69Ga ratios 'since small variations in these two isotope ratios are not always correlated'.The Ga isotope ratios merely served to indicate the presence of excessive drift or noise and the need for instrument maintenance. The purpose of the present study was to demonstrate that Journal of Analytical Atomic Spectrometry January 1995 Vol. 10 15Ga can be used effectively as an isotope ratio internal standard (IRIS) to correct for mass-bias drift in Zn isotope ratio measurements by ICP-MS. Gallium is an almost ideal IRIS for Zn isotope ratio measurements because it is a rare element with a mass close to that of Zn. It has two isotopes of comparable abundance 69Ga and 71Ga neither of which suffers elemental isobaric interferences from singly charged ions (Fig. 1). One possible drawback of Ga is that its ionization potential of 6.00 eV is quite different from that of Zn (9.39 eV).This means that Ga and Zn may respond differently to changes in plasma conditions; however this should not affect the isotope ratios of the two elements. Potential interferences in the determination of the Zn isotopes from singly charged species or in the determination of either Zn or Ga from doubly charged ions of Bay La Ce and Pr can be avoided by appropriate sample clean-up procedures (cf. under Experiment a1 ). To implement the gallium IRIS method natural abundance Zn standards were analysed before and after each group of five samples. The Zn isotopes 67Zn 68Zn and 70Zn served as tracers in the nutritional studies and 66Zn was employed as the reference isotope. All samples and standards were adjusted to a common Zn concentration of 250 or 500 pg 1-' and spiked with Ga to obtain a 69Ga intensity roughly equal to that of 66Zn.Use of the more abundant 64Zn as the reference isotope would have limited the total Zn concentration to a lower level (to avoid detector shutdown for 64Zn) and resulted in lower count rates for the least abundant 70Zn isotope. Concentration levels of both elements were selected to maintain count rates with an ultrasonic nebulizer which yielded average isotope ratios with standard errors of the mean of <0.17% (corre- sponding to relative standard deviations (RSDs) of <0.5% for ten replicate measurements). Four isotope ratios 67Zn 66Zn 68Zn 66Zn 70Zn 66Zn and 71Ga 69Ga were measured for each sample or standard. Changes in the 71Ga 69Ga ratio were used to correct the observed Zn isotope ratios.Three different methods for correcting the measured Zn isotope ratios for mass-bias drift were evaluated and compared (i) simple division of the Zn isotope ratios by the Ga isotope ratios; (ii) correction based on the power law equations given by Longerich et al.;4 and (iii) correction by a regression method. For the latter method the results for the natural abundance Zn standards were examined and the relationships between the drift of the 71Ga 69Ga ratio and the three Zn isotope ratios were established through linear regression. This approach to correcting for mass-bias drift is analogous to the analyte-inter- nal reference correlated method (AIRCM) described by Lorber and Goldbart7 for ICP atomic emission measurements with internal standards.All three correction schemes evaluatd in this study are described in more detail under Theory. The performance of the three mass-bias drift correction methods was evaluated for several case studies with different drift behaviour. This evaluation focused on the results for the 70 r 1 60 8 50 40 Dm 30 C 3 a .2 20 2 10 - n u ~ ~~ 64 65 66 67 68 69 70 71 Atomic mass Fig. 1 Relative abundances of zinc and gallium isotopes natural abundance standards because they are the only samples for which the true Zn isotope ratios are known. It was found that the regression correction method consistently gave the best results in terms of reducing the variability of isotope ratios measured for the natural abundance standards. By using customized corrections for each batch of analyses it was possible to avoid undercorrection or overcorrection of mass- bias drift which can occur with fixed correction equations.The shortcomings of the drift correction by simple division and the superior performance of the correction based on regression could be explained by considering the mathematical and statistical principles underlying those two methods. EXPERIMENTAL All isotope ratio measurements were carried out with a Perkin- Elmer SCIEX Elan 500 ICP mass spectrometer equipped with a Cetac U-5OOOAT ultrasonic nebulizer (Cetac Technologies) and a Gibson 212B autosampler (Gilson Medical Electronics). Table 1 gives the instrumental conditions used in this study. The samples analysed were purified extracts of human urine. Zinc was extracted and purified using a modification of the method published by Friel et aL8 All acids used in the preparation of samples for ICP-MS were ultrapure double sub-boiling distilled in quartz.The acids and NH40H were obtained from Seastar Chemicals and all other chemicals were from Sigma. Approximately 40 ml of thawed acidified urine was centrifuged at 5000 m s-' to remove any solids. The pH of the urine was adjusted to 5.3 f 0.2 with NaOH and 0.07 ml of 8.0moll-' ammonium acetate was added. The urine was applied to 2.0 ml Bio-Rad Chelex 100 columns (Bio-Rad Laboratories) which had been converted to the NH4+ form by adding 10 ml of 2.0 ml I-' NH40H and rinsing with 15 ml of water to remove the excess NH40H. The urinary macro- minerals (Cay Mg Na and K) were eluted off the Chelex 100 columns with 5 ml of 2.0 moll-' ammonium acetate and the trace elements were eluted with 10ml of 2,5moll-' HN03 and collected.The trace element fraction containing the Zn was evaporated to dryness made up in 2.5mol1-' HCl and applied to Bio-Rad AGl-X8 columns (0.8 ml) equilibrated in 2.5moll-' HCl. Zinc was eluted off the columns with 0.005moll-' HCl after washing the resin with ten column volumes each of 2.5mol1-' HCl and 0.5mol1-' HCl. The pure Zn fractions were evaporated to dryness in Teflon beakers on a hot-plate and diluted with 1% v/v HN03 to Zn concen- trations of typically 250 or 500 pg 1-' based on measurements by flame atomic absorption spectrometry (Smith-Hieftje 22 spectrometer Thermo Jarrell Ash). Prior to analysis by ICP-MS samples were spiked with the Ga IRIS to a final concentration of 0.02-0.046 mg 1-'. The Zn and Ga concentrations were selected such that 66Zn Table 1 Instrumentation and data acquisition parameters ICP-MS (Perkin-Elmer SCIEX Elan 500) Forward power/kW Plasma gas flow rate/l min-' Intermediate gas flow rate/l min-' Nebulizer gas flow rate/l min-l Sample flow rate/ml min-l Heater temperature/"C Condenser temperature/"(= Data acquisition Dwell time Ultrasonic nebulizer (Cetac U-5OOOAT) Sweeps per reading Readings per replicate Number of replicates Points per spectral peak 1.25 14 1.4 1.4 1 .o 140 5 35 ms for 70Zn 7 ms for 200 1 10 1 all other nuclides 16 Journal of Analytical Atomic Spectrometry January 1995 Vol.10and 69Ga yielded intensities in the 2 x 105-5 x lo5 counts s-’ range.This avoided potential signal losses due to gain suppres- sion at high count rates and ensured comparable precision (counting statistics) for all samples and standards. All ICP-MS measurements were made over a three month period rep- resenting a variety of conditions in terms of prior instrument use and detector response. A natural abundance Zn standard was analysed before and after each group of five samples. Individual sample analyses required 4 min (including sample wash-out time) and the total analysis time for sample batches ranged from 1.1 to 4.5 h. Adding the two Ga isotopes to the data acquisition programme increased the cycle time of the mass spectrometer by only 25% and had no discernible effect on the precision of individual Zn isotope ratio determinations. It should be noted that the dwell time for 70Zn was five times longer than for the other nuclides to obtain 70Zn 66Zn ratios with a repeatability comparable to that of the other isotope ratios (Table 1).The acquired isotope ratio data were saved in ASCII format report files and the regression and mass-bias correction calcu- lations were carried out off-line using the commercially avail- able spreadsheet programs Quattro Pro (Borland) or Excel (Microsoft). Quattro Pro was also used for the modelling calculations described under Theory. THEORY Internal reference (standardization) methods relate temporal variations of an analyte signal to those of a reference signal. The reference signal should be constant when noise drift or matrix interferences are absent. Any observed variations in the reference signal are considered to be due to some perturbation of the measurement system and are used to correct the analyte signal based on the assumed relationship between the two signals.The success of the correction depends largely on the assumptions about the nature of that relationship. It should be noted that when internal standardization is applied to isotope ratio measurements the two ‘signals’ are isotope ratios rather than intensities. Correction by Simple Division The simplest assumption about the relationship between the analyte and reference signals is that variations in the analyte signal are directly proportional to variations in the reference signal. In this case the correction is performed by dividing the analyte signals for all samples and standards by the correspond- ing reference signals.This approach which has been termed the line ratio internal reference method (LRIRM) by Lorber and Goldbart? is widely used in determinations of elemental concentrations by ICP atomic emission spectrometry and ICP-MS. The Myers-Tracy correction methodg is a well- known example of the LRIRM. In this paper the LRIRM is also referred to as ‘correction by simple division’. Before applying any corrections to the data collected for this study the results for the natural abundance Zn standards in each batch of analyses were normalized by setting the isotope ratios measured for the first standard equal to 1.000. Although this normalization was not necessary for the correc- tion calculations it made it easier to compare the relative drift of the Ga and Zn isotope ratios with time (cf. Figs.5-7). The LRIRM was implemented by applying eqn. ( 1 ) to the normalized isotope ratios for the natural abundance Zn standards MZn (..zn). where the left superscript N indicates normalized ratios M stands for the isotope masses 67,68 or 70 and the subscripts c and m denote corrected and measured ratios respectively. As mentioned in the introduction the performance of the LRIRM was tested by applying it only to the results for the natural abundance standards because they are the only samples for which the true Zn isotope ratios are known. For a perfect internal reference method the drift-corrected normalized Zn isotope ratios for those standards should all be 1.000.The effectiveness of any internal reference method can be discussed in terms of a precision improvement factor p which is the ratio between the RSDs of the uncorrected and corrected signals for the analyte (indicated by the subscript A) RSDA (uncorrected) = RSDA (corrected) It should be noted that the RSDs in eqn. (2) include both short-term variations from one sample to another (noise) and long-term variations within an analytical batch (drift). They do not refer to the repeatability of individual measurements (analysis repeats) that are averaged to produce the results for a single sample. In the context of this paper precision improve- ment means that the variability of results for identical standards within a batch is reduced.An internal reference method which accomplishes that actually improves the accuracy of the results for individual samples. For the LRIRM the achievable improvement in precision can be predicted from the coefficient of correlation r between the analyte and internal reference signals and a noise pro- portionality factor n which is the ratio between the RSD of the analyte signal and that of the reference signal RSDR. RSDA RSDR n = - (3) Using this definition for n which differs somewhat from that given by Myers and Tracy,g the precision improvement factor for the LRIRM is predicted by eqn. (4). n J2TC-G PLRIRM = (4) Myers and Tracy defined the noise proportionality factor (for which they used the symbol b) the same way as given in eqn. (3) only for cases with completely proportional deviations in the two signals.For cases without complete correlation they defined it as the ‘slope of the linear regression representing the least squares best fit of the experimental set of point^'.^ With this latter definition the noise proportionality factor becomes zero when the two signals are totally uncorrelated. Close inspection of eqn. (4) reveals that in order to achieve any improvement in the precision of the analyte signal ( p > l) two conditions must be met (i) r must be positive and close to 1.00; and ( i i ) n must also be close to 1.00. There are many practical situations when those conditions are not met and when the LRIRM actually results in an overcorrection or increase in noise (p < 1). The precision improvement factors found for three analytical batches examined in this study are listed in Table 2 in the Discussion section.For purposes of illustration it is convenient to plot the logarithm of the precision improvement factor p so that combinations of n and r resulting in improved precision yield positive values whereas a degradation of precision is indicated by negative values. A log(p) of 1.0 means that precision is improved by a factor of ten whereas a log(p) of - 1.0 indicates that precision is degraded by a factor of ten. Fig. 2 dramatically illustrates the very limited range of r and n values for which the LRIRM improves precision. For all combinations of r and n represented by the shaded area in that figure the LRIRM results in negative log( p ) values. Journal of Analytical Atomic Spectrometry January 1995 Vol.10 171.5 5 Fig.2 Three-dimensional surface plot of the logarithm of the pre- cision improvement factor p as a function of the correlation coefficient r and the noise proportionality factor n for correction by simple division (LRIRM) Correction Using Power Law The application of an internal reference method to isotope ratio measurements by ICP-MS was first reported by Longerich et aL4 who assumed that the relationships between the analyte signals (three different Pb isotope ratios) and the reference signal (the 205Tl '03Tl ratio) could be described by equations based on a power law. Those equations were designed to account for the mass differences between the isotopes forming a given ratio. Power law correction equations are widely used in TIMS to correct for mass fractionation effects in the ion source." Russell et a1.l' and Wasserburg et a1." compared a variety of fractionation 'laws' including those based on linear power exponential and Rayleigh distillation law equations.Both groups of authors concluded that the fractionation process in thermal ionization sources is still not well understood and that there is no universally best correction method. At this time there does not appear to be a theoretical basis for the assumption that the power law accurately describes the mass bias observed in quadrupole ICP-MS instruments. However Longerich et ~ 1 . ~ and Ketterer et al.' found empiri- cally that the power law correction works well for Pb isotope ratio measurements with T1 as the internal standard.In the present study the power law correction was applied to the normalized Zn isotope ratios for the natural abundance Zn standards using eqns. (5)-(7). N f 67Zn\ 67Zn \66Zn) (%)c= 71Ga [ ( 6*Ga),lii2 70Zn ( %)c 70Zn (s) The superscripts and subscripts are the same as for eqn. (1). No attempt was made to model the potential improvement in precision that can be achieved with the power law correction method because its performance is largely dependent on how well the power law predicts the actual behaviour of each of the analyte isotope ratios. It should be noted however that for the 68Zn 66Zn ratio the results of the power law correction and the LRIRM are identical because 68Zn and 66Zn have the same mass difference as 71Ga and 69Ga.Correction Using Regression In this internal reference method the mathematical relation- ships used for the correction are derived from regression analysis of experimental data for the analyte and reference signals rather than being based on a priori assumptions. Lorber and Goldbart7 called this approach the analyte-internal reference correlated method (AIRCM). The terms 'regression correction method' and AIRCM are used interchangeably in this paper. In a very generalized implementation of the AIRCM the functional relationships between the analyte and the reference signals may be found by least squares fitting a power series of the form y = bo + blx + b2x2 + ... + b,x" to a set of experimen- tal data where y and x represent the analyte and reference signals respectively.However for small signal fluctuations the quadratic and higher terms can be neglectedY7 resulting in linear correction equations. Since this could be verified in the present study using multivariate regression analysis the relationships between the changes of the 71Ga 69Ga ratio and those of the three Zn isotope ratios were determined by linear regression yielding equations of the form where superscripts and subscripts are the same as for eqn. (1) and bo and bl are the intercepts and slopes of the regression lines. This procedure was performed separately for each analyt- ical batch using the normalized results for the natural abun- dance Zn standards. The normalized Zn isotope ratios for all standards within a batch were then corrected for mass-bias drift using eqn.(9) N/MZn\ where b is the slope of the regression curve relating the relative drift of the Zn and Ga isotope ratios for the natural abundance zinc standards and the superscripts and subscripts are the same as for eqn. (1). The intercepts of the regression lines boy were not used in the correction calculations based on eqn. (9). This made it easier to visually compare the relative drift of the Zn isotope ratios with and without the applied correction (cf. Figs. 5-7 below). It had no effect on the precision improvement achieved by the regression correction method. In cases in which intercepts were significantly different from zero the normalized and corrected isotope ratios for the natural abundance standards within a batch were systemati- cally shifted by a mean value of bo from their ideal values of 1.000.The regression line intercepts were taken into account in the calculations correcting the results for actual samples for the mass bias itself. Mass-bias drift correction of the sample 18 Journal of Analytical Atomic Spectrometry January 1995 Vol. 10Zn isotope ratios was accomplished by applying eqn. (10). This equation is identical to eqn. (9) except that the Zn isotope ratios are not normalized. The mass bias itself was corrected using eqn. (1 l) (11) where the subscript f indicates the final isotope ratios the subscript c denotes the drift-corrected ratios the subscript n indicates the literature values for the natural abundance ratios and the term with the left superscript S is the mean of the experimentally determined isotope ratios for the natural abun- dance standards (after drift correction).It should be noted that the latter term is (1 + b,) times as large as the isotope ratio measured for the first standard in a batch. The improvement in precision that can be achieved with the AIRCM depends only on the coefficient of correlation between the reference and analyte signals and not on the noise pro- portionality factor n. The variance of the analyte signal before correction is aAZ and the regression accounts for r2 of this variance. Therefore the variance remaining after the correction is (1 - r)crA2 and the RSD of the corrected signal is RSDA (corrected) = RSDA (uncorrected) x 41 - r2 (12) Inserting eqn. (12) into eqn. (2) yields the predicted precision improvement factor for the regression correction 1 PAIRCM = - JS Eqn.(13) is symmetrical with respect to r i.e. the regression correction works equally well for positive and negative corre- lations between the analyte and reference signals. This is illustrated in Fig. 3. More importantly eqn. (13) and Fig. 3 show that the regression correction cannot degrade the pre- 1 .o Fig.3 Three-dimensional surface plot of the logarithm of the pre- cision improvement factor p as a function of the correlation coefficient r and the noise proportionality factor n for the regression correction method (AIRCM) cision of the analyte signal because even when r=0 the precision improvement factor is still 1.00. This somewhat surprising result can be understood by inspecting eqn.(9) above. Regression analysis of two completely uncorrelated signals yields a regression line with a slope b l of zero. In this case the corrected and uncorrected signals are identical and PAIRCM equals 1.00. A comparison of Figs. 2 and 3 clearly demonstrates the much broader useful application range of the AIRCM relative to the LRIRM. The PAIRCM values calcu- lated for three batches of samples are given in Table 2 in the Discussion section. Model Calculations In addition to applying different correction methods to the experimental data we also carried out model calculations to test the validity of the equations presented in this section over a broader range of conditions in terms of noise and drift than was possible with the experimental data available at this time.Eqns. (4) and ( 13) which predict the precision improvements expected for the LRIRM and AIRCM were tested by creating a variety of artificial data sets with variable amounts of Gaussian noise and systematic drift. Each data set contained 1000 numbers. After selecting one data set to represent the analyte and a different one to represent the reference the correlation coefficient noise proportionality factor and slope of the regression line were calculated. Then the 'analyte' data set was subjected to the two correction methods. A comparison of the actually observed and theoretically predicted precision improvement factors p is shown in Fig. 4. There was excellent agreement between the actual performance of both methods and theory (r > 0.9999) supporting the general validity of the derived equations.RESULTS A total of eleven case studies were evaluated for this work each one representing the results for a single batch of samples. Three of those case studies were selected for more detailed discussion in this paper because they represent the range of analytical results. For the other eight case studies the drift 2.0 1.5 1 .o - W 6 0.5 d - Q 0 - -0.5 -1.0 *AA A A I 0 0.5 1 .o 1.5 2.0 -1.0 -0.5 log( p predicted) Fig. 4 Comparison of observed and predicted logarithms of precision improvement factors p based on model calculations using the A LRIRM and 0 AIRCM Journal of Analytical Atomic Spectrometry January 1995 Vol. 10 19behaviour was intermediate between that found in the three selected cases. In eight of the eleven cases there was a significant and systematic drift of the mass bias for the Zn and Ga isotope ratios.The largest drift observed on a single day was close to 4% compared with the results for the first natural abundance standard in a batch. There was no apparent relationship between the extent of drift of the isotope ratios and the drift of the raw intensities for the individual isotopes. In two cases the raw intensity for the 66Zn reference isotope changed by more than 30% over a 3-4 h period while the isotope ratios varied less than & 1.5%. It is noteworthy that the mass bias most often increased with time during an analytical session. This may have been due to temperature effects or to gradual pressure changes in the quadrupole vacuum chamber.The degree to which the individual Zn isotope ratios followed the 71Ga 69Ga ratio was quite variable both in terms of the magnitude of the correlation coefficients and the slopes of the regression lines. Correlation coefficients for the three Zn isotope ratios and 11 case studies varied between 0.3282 and 0.9913. They were typically lowest for cases with very little drift. The slopes of the regression lines ranged from - 1.74 to 1.93; about 18% of all calculated values were negative. For each of the three selected case studies four figures are presented in this section; all of them are based on results for the natural abundance Zn standards [Figs. 5-7 (a)-(d)]. The first figure [(a)] shows the temporal changes of the Ga and Zn isotope ratios relative to the values measured for the first standard analysed in a given batch.The other three figures [(b)-(d)] show the results obtained with the simple division correction (LRIRM) the power law correction and the regression correction (AIRCM) respectively. 1.02 Case 1 In this analytical batch all isotope ratios systematically increased with time [Fig. 5(a)]. The Ga IRIS followed the 68Zn 66Zn ratio most closely while the 67Zn 66Zn ratio changed at a slower rate (about 0.5 x ) and the 70Zn 66Zn ratio drifted more rapidly [about 2 x see Fig. 5 ( a ) ] . Consequently the simple division method undercorrected the 70Zn 66Zn ratio and overcorrected the 67Zn:66Zn ratio [Fig. 5(b)]. The power law and regression corrections were almost equally successful in removing drift for the natural abundance Zn standards [Fig. 5(c) and ( d ) ] .Of the 11 data sets examined for this study this was the only one for which the power law correction performed that well. It should be noted that with a limited range of mass-bias drift of a few percent the power law correction method gives very similar results to the regression method with fixed slopes of the regression lines of 0.5 1 and 2 respectively. This is due to the fact that the functions y=x1I2 y = x and y = x 2 have first derivatives with the approximate values of 0.5 1.0 and 2.0 when x is close to 1.00. (a) - Case 2 The results from this analytical batch demonstrate a failure of the power law correction. The Ga IRIS closely followed the 67Zn 66Zn and 70Zn 66Zn ratios. In comparison the 68Zn 66Zn ratio drifted more slowly [Fig.6 ( a ) ] . Since this behaviour is very different from that assumed for the power law correction [eqns. (5)-( 7 ) ] the 67Zn 66Zn ratios were undercorrected and the 68Zn 66Zn and 70Zn 66Zn ratios were overcorrected by this method [Fig. 6 ( c ) ] . Even the simple division correction method produced better results for the 67Zn 66Zn and 70Zn:66Zn ratios [Fig. 6 ( b ) ] . However by far the best results 1 l . O 1 ~ .oo !E 0.98 0.99 i I I I J 0 1 2 3 4 0.98 ' t 0 1 2 3 4 'Tirnelh Fig. 5 Results of case study 1. (a) Relative mass bias drift for the 67Zr1 66Zn (V) 68Zn 66Zn (+ ) "Zn 66Zn (A) and 71Ga 69Ga (m) ratios with time. Results for the Zn isotope ratios after correction using (b) simple division (c) the power law equations and ( d ) the regression correction method 20 Journal of Analytical Atomic Spectrometry January 1995 Vol.10al - a t ( C) 1.01 1 .oo 0.99 0.98 I 0 1 2 3 4 0 1 2 3 4 Time/h Fig. 6 Results of case study 2. (a) Relative mass bias drift for the 67Zn 66Zn (V) 68Zn 66Zn (+) "Zn 66Zn (A) and 71Ga 69Ga (m) ratios with time. Results for the Zn isotope ratios after (b) correction using simple division (c) the power law equations and (d) the regression correction method 1.00 I 0.99 - 0.98 ' I I I I I 0 1 2 3 4 5 0.99 I 1 . 1 I I 0 1 2 3 4 5 Time/h Fig. 7 Results of case study 3. (a) Relative mass bias drift for the 67Zn 66Zn (V) 68Zn &Zn (+) 70Zn 66Zn (A) and 7'Ga 69Ga (m) ratios with time. Results for the Zn isotope ratios after (b) correction using simple division (c) the power law equations and (d) the regression correction method Journal of Analytical Atomic Spectrometry January 1995 VoZ.10 21were achieved with the regression correction method which removed all systematic drift from the Zn isotope ratio data [Fig. 6(d)]. Case 3 This batch of samples required a total analysis time of 4.5 h and included nine natural abundance Zn standards. The original data for the standards showed only random variations within the error limits of the measurements; there was no systematic drift of the isotope ratios with time [Fig. 7(a)]. Owing to the absence of a strong correlation between the Ga IRIS and Zn isotope ratios (r < 0.523) the simple division and power law correction methods actually increased the overall spread of the data [Fig.7(b) and (c)]. The regression correction on the other hand had no adverse effect on the variation range of the Zn isotope ratio results [Fig. 7 ( d ) ] as was expected from theory. DISCUSSION Table2 is a compilation of the coefficients of correlation between the normalized Ga and Zn isotope ratios as well as the regression line slopes noise proportionality factors and precision improvement factors with their logarithms for the three case studies and three correction methods. The precision improvement factors listed were calculated from the experimen- tal data before and after correction and not from the predictive eqns. (4) and ( 1 3 ) given under Theory. For case studies 1 and 2 (Figs. 5 and 6) all correlation coefficients were in the range 0.813-0.991.The slopes of the regression lines for the 67Zn:66Zn ratio were 0.433 and 0.968 while those for the 68Zn:66Zn ratio were 0.901 and 0.535 and those for the 70Zn 66Zn ratio were 1.94 and 1.1 1 respectively. Figs. 5(b) and 6(b) confirm the theoretical prediction that the simple division correction method (LRIRM) only performs well when the correlation coefficient r is close to +1.00 and the slope of the regression line is also close to + 1.00. As Table 2 shows the regression line slope is approximately equal to the noise proportionality factor n for high values of r. The power law correction method is similarly restrictive in that it only performs well when r is close to $1.00 and the slopes of the regression lines are close to 0.500 for the 67Zn:66Zn ratio close to 1.00 for the 68Zn:66Zn ratio and close to 2.00 for the 70Zn:66Zn ratio.As pointed out earlier this type of relationship was rarely observed. For all 11 case studies the regression line slopes for the 67Zn 66Zn 68Zn 66Zn and 70Zn 66Zn ratios averaged 0.312 & 0.840 0.541 0.432 and 1.021 & 0.727 respectively. The IRIS/regression method always showed the best per- formance among the three correction methods tested. The RSD of the drift-corrected data for an entire batch was typically of a magnitude similar to the standard error of the mean of the individual isotope ratio determinations of 0.17%. This is illustrated by Figs. 5(d) 6 ( d ) and 7(d). The residual RSDs for all Zn isotope ratios in the three selected cases ranged from 0.09 to 0.32% with a mean value of 0.18%.The highest reduction in drift was obtained in case study 2 in which the precision of the 70Zn:66Zn ratio was improved by a factor of 7.6 (Table 2). Considering all 11 case studies the regression correction method reduced the variability of the three Zn isotope ratios for the natural abundance standards by an average factor of 2.5. The method performed equally well for all three Zn isotope ratios. By establishing the relationships between the analyte and reference isotope ratios for each analytical batch the regression correction method is very flexible. A particularly attractive feature of this method is that it can be applied to all situations without fear of degrading precision. Case study 3 illustrates that the regression correction does not increase the spread of the data even when the variations of analyte and reference isotope ratios are only weakly correlated.When there is no correlation at all the results remain unchanged. Negative correlations between analyte and reference isotope ratios are equally effective in reducing mass bias drift as positive ones. The degrees to which the IRIS/regression method can improve precision for a given analytical batch can be predicted directly from the coefficient of correlation between the analyte and reference isotope ratios using eqn. (13). This allows the analyst to decide quickly whether carrying out the correction calculations is worthwhile. It also means that any improve- ments in precision are statistically significant even if they are small because the improvements are inherent in the deposition (ie.correlation) of the experimental data. Although the correc- tion must be carried out off-line after a batch of analyses is completed the computations are relatively simple and can be performed easily with modern spreadsheet software. The regression method presented in this paper can also be applied to other types of ICP-MS measurements using internal standardization such as the conventional determination of elemental concentrations. In the area of isotope ratio measure- ments by ICP-MS the correction of mass-bias drift is particu- larly important in situations in which small errors in the experimentally measured isotope ratios are amplified by the calculations used to determine a derived quantity.This is the case in the computation of relative enrichments of isotope tracers3 and in the determination of elemental concentrations by isotope dilution mass spectrometry.12 Financial support for this work from the California Department of Health Services Hazardous Materials Laboratory (contract 91- 12843) is gratefully acknowledged. Table 2 Coefficients of correlation (r) between the normalized "Ga 69Ga ratio and the normalized Zn isotope ratios regression line slopes (bl) noise proportionality factors (n) and the precision improvement factors and their logarithms for the three case studies and three correction methods r bl n PLRIRM PAIRCM P ~ o w e r law LOg(PLRIRM) - Log(PPower law) LOg(PAIRCM) 0.8128 0.4333 0.5349 0.828 1.683 1.718 0.082 0.226 0.235 0.9437 0.9012 0.9544 2.844 2.844 3.027 0.460 0.460 0.48 1 0.9629 1.9361 1.9917 1.871 3.639 3.673 0.272 0.561 0.565 0.9885 0.9678 0.9787 6.486 1.991 6.622 0.812 0.229 0.821 0.9863 0.5348 0.5459 1.153 1.153 6.081 0.062 0.062 0.784 0.99 13 1.1117 1.1189 6.067 1.233 7.586 0.783 0.09 1 0.880 0.3282 0.1725 0.5237 0.542 0.883 1.059 - 0.266 - 0.054 0.025 0.5231 0.1827 0.3490 0.401 0.401 1.175 - 0.397 - 0.397 0.070 0.4600 0.3743 0.8117 0.849 0.561 1.125 .0.07 1 .0.251 0.051 22 Journal of Analytical Atomic Spectrometry January 1995 Vol. 10REFERENCES Morgan P. N. Woodhouse L. R. Serfass R. E. Roehl R. and King J. C. FASEB J. 1993 7 A279. Morgan P. N. Woodhouse L. R. Abrams S. A. Serfass R. E. Roehl R. and King J. C. FASEB J. 1994 8 A918. Woodhouse L. R. Morgan P. N. King J. C. Roehl R. and Gomez J. ICP lnf. Newsl. 1994 20 25. Longerich H. P. Fryer B. J. and Strong D. F. Spectrochim. Acta Part B 1987 42 39. Ketterer M. E. Peters M. J. and Tisdale P. J. J. Anal. At. Spectrom. 1991 6 439. Amarasiriwardena C. J. Krusheva A. Foner H. Argentine M. D. and Barnes R. M. J. Anal. At. Spectrom. 1992 7 915. Lorber A. and Goldbart Z. Anal. Chem. 1984 56 37. 8 Friel J. K. Naake V. L. Miller L. V. Fennessey P. V. and Hambidge K. M. Am. J. Clin. Nutr. 1992 55 473. 9 Myers S. A. and Tracy D. H. Spectrochim. Acta Part B 1983 38 1227. 10 Wasserburg G. J. Jacobsen S. B. DePaolo D. J. McCulloch M. T. and Wen T. Geochim. Cosmochim. Acta 1981 45 2311. 11 Russell W. A. Papanastassiou D. A. and Tombrello T. A. Geochim. Cosmochim. Acta 1978 42 1075. 12 Chiba K. Inamoto I. and Saeki M. J. Anal. At. Spectrom. 1992 7 115. Paper 4/01519C Received March 14 1994 Accepted September 12 1994 Journal of Analytical Atomic Spectrometry January 1995 Vol. 10 23