135 (1997) 325-334
Optimization of multicollector isotope-ratio measurement of strontium and neodymium Kenneth R. Ludwig Berkeley Geochronology
Center, 2455 Ridge Road, Berkeley, CA 94709, USA
Received 30 June 1995; accepted 9 July 1996
Abstract For isotope-ratio analyses of Sr and Nd whose precisions are constrained by amplifier and ion-counting noise, significant improvements in efficiency/precision can be realized by appropriate choices of peak vs. background measurement times, peak-jumping modes, and ratios for fractionation normalization. For Nd, theoretical limits on internal precision predict that
the time-efficiency of dynamic analyses should be similar to that of static in most cases (slightly better in others), and that for Sr, dynamic analyses should be actually more efficient than static. Equations for the optimum mix of background times and peak-top times are given. Keywords:
1. Introduction I. 1. Static and dynamic modes of data collection For multi-collector thermal-ionization mass spectrometry of elements with at least one isotopic ratio involving no radiogenic isotopes, the analyst generally has a choice of two analytical protocols: static, wherein each isotope is assigned a collector and the position of the ion beam is invariant; or dynamic (Lenz and Wendt, 19761, wherein isotope-ratio data are acquired in two or more steps ‘, with the ion
’For this paper, a step is defined as a suite of collector measurements at single mass position: a cycle is defined as the number of sequential steps necessary to calculate an isotope ratio corrected for mass fractionation, isobaric interferences, and collector efficiencies. The number of steps per cycle is 1 for static analyses, and typically :! or 3 for dynamic analyses. 0009-2541/97/$17.00 Copyright PII SOOOS-2541(96)00120-9
beams magnetically switched so that different masses arrive in the same collectors in each step. The static method has the advantages of simplicity and the ability to deal with large mass-separations, but can only be as accurate as the amplifier gains and collector efficiencies are known, and as precise as they are constant. The dynamic method has the advantage that neither precise nor accurate amplifier gains/collector efficiencies are necessary, but involves a time penalty because of the settling times required between mass-switching steps. Dynamic analyses are more robust than static, in that fewer demands on hardware performance and calibration are made (Thirlwall, 1991a; Fletcher, 1986a). However, dynamic analyses are often avoided because of a large presumed dynamic timepenalty (e.g., Makishima and Nakamura, 1991). The calculations in this paper, however, show that this is not true, and that in some common instances dy-
0 1997 Elsevier Science B.V. All rights reserved.
namic analyses than static. 1.2. Noise-related ment
should be even more time-efficient
limits to isotope-ratio
The precision and accuracy of isotope-ratio data acquired by multi-collection are affected by a long list of factors. Some of the more common are: uncorrected isobaric interferences, poor peak-flat for one or more collectors, variable clipping of the ion beam in the z-direction, use of an inappropriate mass-fractionation law, poorly-calibrated amplifier gains or collector efficiencies (static only), invalid background correction, beam-independent amplifier noise, and ion-counting errors. With the exception of the amplifier/collector calibrations, however, only the last two factors - amplifier noise and ion-counting errors are unavoidably present, and thus provide fundamental limits on performance for even a perfectly functioning mass spectrometer. The simple mathematical nature of these two sources of noise permits their effect on internal (within-run) isotope-ratio precision * to be estimated with straightforward error-propagation equations, as has been done previously for single-collector isotope-ratio analyses (Ludwig, 1986) but with the additional generality that multicollector analyses are unaffected by ion-beam instability (excepting from ion-counting noise). For a well-maintained mass spectrometer of modern design, experience suggests that such noise will be the dominant source of isotope-ratio measurement error down to precisions somewhere in the neighborhood of lo-40 ppm, depending on both the mass spectrometer and sample preparation. Regardless of the exact precision at which amplifier and ion-counting noise are overwhelmed by other factors (that is, where external precision begins to significantly exceed internal), such noise will be the limiting factor in many types of analyses.
The “internal pm&ions” calculated in this paper are not necessarily comparable to the within-run precision calculated by existing mass-spectrometer control programs, as such programs may not propagate all sources of error (e.g., background error) in the calculated ratios.
Geology 135 (1997) 325-334
The equations in Appendices B-E (and thus the various diagrams in this paper) are dependent on the following assumptions. (1) Mass fractionation follows a power law: ‘Rij =TRij( 1 + (y)mf-m’ where R,, = ratio of isotope i to isotope j; P indicates fractionated according to a power-law; T indicates the true ratio; CY is the fractionation per unit mass difference; and mi, mj, and mk are the masses of isotopes i, j, and k, respectively (Russell et al., 1978).0ther laws - particularly exponential may be more accurate (Russell et al., 1978; Wasserburg et al., 1981; Fletcher, 1986a,b; Thirlwall, 1991a,b); however, essentially all laws of interest for Sr and Nd analyses can be thought of as small corrections to a power law (e.g., Thirlwall, 1991a, p.93) with negligible effect on the error propagation equations. (2) With the exception of 14*Nd/ 144Nd-normalized Nd, enough moveable collectors are available to measure all relevant peaks, including those for isobaric-interference monitoring, during each measurement step. (3) Amplifier noise is constant and independent of the ion beam. (4) The analyzed Sr and Nd are unspiked and have typical “common” Sr and Nd isotope ratios CE7Sr/ *‘Sr = 0.71, ‘43Nd/ ‘44Nd = 0.5 1). (5) For static analyses, a fixed amount of time is allocated to calibration of amplifier gains (collector efficiencies are assumed to be constant and precisely known a topic of extensive implications, but beyond the concerns of this paper). Whether this time is used for calibrations before each analysis or pooled in some way and spread over several analyses is unimportant. (6) The times required for filament warm-up, ion-beam focussing, and peak-centering (except for additional peak-centers required by dynamic analyses) are ignored, as they are the same for static and dynamic analyses. Also, some reasonable and typical values must be assigned for instrumental characteristics and operator
K.R. Ludwig/ Chemical Geology 135 (1997) 325-334
choices. The values used for this paper are:(l) Amplifier noise is 35 FV/S (la>, typical for modem amplifiers with a lO”-0 resistor. (2) For static analyses, calibration of the collector gains requires 5 min per analysis. (3) For dynamic analyses, peak-jump settling requires a delay of 2 s. Integration time per cycle (not including settling times) is 14 s for a 2-step dynamic cycle and 18 s for 3-step cycle. Thus when settling times are included, the 2-step and 3-step cycles will take 18 and 24 s, respectively, with average integration times per step of 7 and 6 s, respectively. When background times are taken into account, these parameters result in dynamic time-penalties of 15-24% (increasing the setthng time to 3 s would increase the dynamic time penalties to 24-32%). An additional 4-5% time-penalty is incurred by dynamic analyses because of extra peak-centering tasks, modelled here as taking an extra 6 s per peak-center per dynamic step, and performed once for every 12 cycles of data acquisition. (4) For Nd, the only reference isotope considered for radiogenic and normalizing ratios is ‘44Nd. The following are free (variable) parameters in the equations: (1) Size of the ion beam. (2) Relative time (compared to peak-top measurement) spent measuring collector backgrounds. (3) Relative tim.es spent on the different mass positions for the different steps of each measurement cycle (dynamic only). Note that the effect of different reasonable choices regarding amplifier gain-calibration times, peak-jump settling times, and dynamic peak-centering times are not particularly large, and can be easily modelled using the precision equations in Appendices B-E as a starting point.
2. Optimizing background times The precision of the collector background measurements, which must be subtracted from the raw readings to get net ion-beam intensities, obviously affects the precision of the final isotopic ratios. The importance of the background precisions, however, depends in detail on a large number of variables, including ion-beam size, isotopic ratios, and the
Millivolts ‘%r Fig. 1. Optimum analyses.
manner in which the different collector measurements are used to calculate the isotope ratios. For static Sr analyses, the optimum background/peak-top time-ratio (Eq. (B-121, Ap“s:“$ B; Fig. 1) decreasesspm N 0.f for 400 mV _ 0.3 for 6-10 V Sr. The time-effiiaency penalty for very short backgrounds is significant (Fig. 2): for example, an analysis at 3 V of 88Sr with 10 s of background and 15 sets of 8-s peak-top measurements would result in _ 40% worse precision (of the corrected 87Sr/ 86Sr ratio) per minute of measurement time, or N 80% more measurement
Background time/Peaktop time Fig. 2. Internal error in *‘Sr/@‘Sr as a function of the background/peaktop time-ratio. Total (peak-top plus background) measurement time was 30 min. considering neither collector gain-calibration time (static analyses) nor peak-jump settling time (dynamic analyses).
Geology 135 (1997) 325-334
Millivolts ‘“Nd Fig. 3. Optimum background/peaktop
time-ratio for 14’Nd/ ‘44Nd analyses with different normalizing
time to achieve a given precision. For dynamic Sr, much less background time (relative to total peak-top time) is needed; the optimum ratio ranges from mV 88Sr to 0.1-0.2 at 3-10 V 0.3-0.4 at -400 88Sr. The sensitivity of dynamic-analysis precision to non-optimal backgrounds times is also less than static.
isotopes (relative to ‘44Nd).
For static Nd analyses, regardless of which isotope is used together with 144Nd to normalize for fractionation, optimum background/peak-top timeratios (Fig. 3) decrease from N 0.6 for small (200 mV> 144Nd ion-beams to u 0.2 for large (- 5 V> ion-beams. As for Sr, the sensitivity of the 143Nd/ 144Nd precision (Fig. 4) is moderately high, such that background times as short as 10 s per 120 s of peak-top measurement would increase the measurement time for an analysis by _ 75% (at 400 mV ‘44Nd). For dynamic Nd analyses (Fig. 41, optimum ratios of background time to total peak-top times (and sensitivity of precision to these ratios) are again smaller than for static: N 0.2 for < 0.2-V beams to < 0.1 for > 5-V beams.
3. Optimizing step times of dynamic analyses
Background timelpeaktop time Fig. 4. Internal error in I43Nd/ ‘44Nd as a function of the background/peaktop time-ratio, for different normalizing isotopes measure(relative to ‘44Nd). Total (peak-top plus background) ment time was 30 min, considering neither collector gain-calibration time (static analyses) nor peak-jump settling time (dynamic analyses).
For 2-step dynamic Sr and Nd (Appendices C-E, Eqs. C-2-C-7 and E-2-E-41, the sensitivity of the precision of the analysis to the step-2/step-1 timeratio is low, with an optimum value close to 1. For 3-step dynamic Sr or 3-step 142/144-normalized Nd, precision sensitivity to either the step-2/step-1 or step-3/step-1 time-ratios is also low, but setting the step-a/step-l time-ratio to 2 (instead of both at
1) results ficiency.
in a very slight (-
Geology 135 (1997) 325-334
9%) gain in time-ef-
of dynamic and static effkiencies
The curves in Figs. 5-8, derived using optimized background and dynamic-step times from the equations in Appendices, B-E show that: (1) For Sr (Figs. 5 and 6), 3-step dynamic analyses should be slightly more time-efficient 3 than 2-step dynamic, and significantly more precise than static. The minimum dynamic time-advantage, compared to static, of _’ 18% (2-step) or N 30% (3-step) occurs at 2-4 V *‘Sr, increasing to > 50% (3-step) for < 0.4 V or > 7 V of @Sr. (2) For Nd 4 (Fyigs. 7 and S), normalization to 146/144 is the least precise, least efficient method (compared to 142/144, 148/144, or 150/144 normalization). For 14.6/ 144normalized analyses, dynamic and static modes should yield similar precisions, though static is somewhat more efficient than dynamic for ion bleams of typical size (0.4-5 V 144Nd). Static 148/144 and 150/144 analyses should be slightly more precise and efficient than 146/ 144normalized analyses (provided the fractionation law is accurate). (3) For Nd, normalization to 142/144 (provided the Ce correction on 14*Nd is small) should yield the most precise data, as stated by Thirlwall (1991b). Quantitatively, the error-propagation equations indicate that time-advantage for 3-step dynamic analyses over either static or dynamic 146/144-normalized analyses will be more than a factor of 2. Two-step dynamic 142/144-~normalized analyses should be slightly more time-efficient than static (though at l-2 v ‘44Nd the difference is negligible), and only slightly less efficient than 3-step dynamic. There is an apparent paradox in the greater efficiency of dynamic analyses for Sr or 142/144-normalized Nd, which exists despite the dynamic time-
Fig. 5. Time required for a s’Sr/ “Sr analysis with an internal 2a precision of i-20 ppm, assuming optimum background and step times, with gain-calibration times, peak-jump settling times, and instrumental parameters as specified in text.
penalties of peak-jump settling and loss of continuous measurements of all isotopes. Examination of the error propagation equations shows that much of the increased precision per actual measurement second (that is, disregarding both static gain-calibration and dynamic settling times) of dynamic runs arises from the pooling of collector-background data for the different dynamic steps, permitting both a reduction in the background/peak-top time-ratio, and in some cases a further demagnification of background errors when the same background occurs in both the nu-
3 For a given precision analysis-time target. 4 Only static analys,es 150/144-normalized Nd.
or more precise evaluated
for a given
Fig. 6. relative ground settling
Analysis time (to f 20 ppm internal error in *‘Sr/ *‘Sr) to a 3-step dynamic analysis, assuming optimum backand step times, with gain-calibration times, peak-jump times, and instrumental parameters as specified in text.
Geology 135 (lY97) 325-334
calculation, as isotopic impurity of the spike perturbs the fractionation-normalizing ratio to a variable degree. But as Thirlwall (1991a) and Wendt and Haase (1996) have noted, dynamic analyses of spiked samples with full correction for collector efficiencies, spike isotopes, and power-law fractionation can still be performed. Should full correction for exponentiallaw fractionation be required (e.g., for precision/accuracy of better than +2-3%0 for sample/spike, or better than +20-30 ppm for 87Sr/ 86Sr or 143Nd/ 144Nd), the equations in Wendt and Haase (1996) can be further “polished” in a roughly similar manner to that used for unspiked double-collector data (Thirlwall, 1991a).
Fig. 7. Time required for a 14’Nd/ ‘44Nd analysis with an internal 2a precision of + 10 ppm, for different normalizing isotopes (relative to ‘44Nd). Assumes optimum background and step times, with gain-calibration times, peak-jump settling times, and instrtmental parameters as specified in text.
of the isotope-ratio
Acknowledgements Comments and suggestions by M. Thirlwall, I. Fletcher, and L. Neymark have substantially improved this paper.
Appendix A. Notation for Appendixes B-E 5. Dynamic analyses of spiked samples Addition complicates
of a spike isotope such as “‘Nd or s4Sr the equations for dynamic isotope-ratio
a, b, and c are measured, background-corrected peaks, in millivolts, for the radiogenic isotope (e.g., 87Sr), reference isotope (e.g., *‘Sr), and normalizing
E ._ l.E
5% I a $ ‘3= s d
Millivolts l”Nd Fig. 8. Analysis time (to + 10 ppm internal error in ‘43Nd/ ‘44Nd) relative to a 14’Nd/ ‘44Nd-normalized, 3-step dynamic, for different times, peak-jump settling normalizing isotopes (relative to ‘44Nd). Assumes optimum background and step times, with gain-calibration times, and instrumental parameters as specified in text.
K.R. Ludwig/ Chemical Geology 135 (1997) 325-334
i refers to a meaisotope (e.g., @Sr). respectively. sured interference-monitor isotope (e.g., 85Rb for tI=b/c T=true
r=b/a M = measured
p = t,/t, p = t,/t,
4 = ratio of interfering isotope to monitor isotope for the subscripted isotope, e.g., & = 87Rb/ 85Rb for I N
I, t,, t,, t, marmbv and m, - mb)/(m,
87Sr), R is the ratio u/b,
corrected for power-law
mass-fractionation. r2 = t/t, r3 = t,/t,
Sr, &, = 144Sm/ ‘47Sm for Nd, for 142-normalized Nd
& = 14*Ce/ 14’Ce
= ions per second per millivolt (N 62,000 for amplifiers with lo”-0 resistors) = amplifier dark-noise, lo, in mV/s (typically 0.03-0.04 for amplifiers with lo”-fi resistors) = time spent measuring backgrounds (a single background measurement is assumed to apply to all of the measured peaks) = time spent measuring peak-tops (static) = time for dynamic step-l, step-2, step-3 measurements, respectively, not including settling time are the atomic masses of the radiogenic, reference, and normalizing isotope, respectively, and
Appendix B. Static analyses for unspiked Sr, Nd, etc.
ion-counting errors for the three background-corrected peaks can be expressed as:
The equation for static data-collection, corrected for power-law mass-fractionation (Russell et al., 1978) can be written as:
(B-1) where the net peaks a, b, and c are calculated from
a=d-Z,-fq5&-Z,,) b = b’- Z, - c$,( i:, - Zi,) c = c’- Z, - +C( i’, - Zi,)
where d, b’, and C’are raw (uncorrected) measurements of the radiogenic isotope, reference isotope, and normalizing isotope, respectively; Z refers to the zero (background) measurement for the collector used for the subscripted isotope; i’ refers to the raw measurement of the interference-monitor isotope for the subscripted isotope (e.g., “Rb for 87Sr). The measurement error (in millivolts) from beam-independent amplifier noise ( = dark noise) and
P-8) assuming that the ion-counting noise has a Poisson distribution. Using Eqs. B-2-B-8, differentiation and rearrangement of Eq. (B-1) (and assuming that the monitored isobaric-interference peaks are small) then gives:
as the relative variance per total second of measurement time (i.e. peak-top plus background ignoring
Chemical Geolog! 135 (1997) 325-334
ratio) and yields:
Differentiation to find the minimum and thus the optimum static p, gives:
of Eq. (B-9),
(k, + k2/,r2)
The triple-collector algorithm (e.g., VG Isotech, 1994) for “Sr/ 86Sr, wherein 3 adjacent collectors receive masses 85, 86, 87 in step 1, 86, 87, 88 in step 2, and 87, 88, 89 in step 3, and mass 85 is also monitored in steps 1, 2 and 3 by two other, lowermass collectors, can be expressed as: 41
Appendix C. Dynamic analyses for unspiked Sr The double-collector equation 87Sr/ 86Sr can be expressed as:
which upon differentiation l+T?+r3+p
where the subscripts 1 and 2 refer to the two steps of integration wherein two adjacent collectors receive masses 86 and 87 in step 1, then 87 and 88 in step 2 (mass 85 is monitored for Rb by two different lower-mass collectors for each step). Differentiation and rearrangement of Eq. (C-l) using Eqs. B-2-B-8 yields: 1+7+p 4b2
as the expression for relative variance per total second (again, including only peak-top plus background measurement time, ignoring, for dynamic, settling time between steps), where k, = [(I + ,‘)r’+
1]N2 + (Y+
(C-9) where (C-10)
k,=[(1+~2)r2+1]N2+(r+l)~ k, = [4(1 + 42)r2
+ 1 + 02]N2
1 + 0): (C-l 1)
(C-12) 1 - 13)’ (C-13)
(r2)opt = J_
Differentiation to find the minimum of Eq. (C-2), and thus the optimum TV (the step-2/step-1 time-
(l+Tz+P)b 1 + VT2 + b/P
Appendix D. Dynamic analyses for unspiked Nd normalized to 146Nd / 144Nd Using a triple-collector algorithm wherein three adjacent collectors receive masses: 144, 145, 146 in step 1; 143, 144, 145 in step 2; and 142, 143, 144 in step 3 (mass 147 is monitored in three different higher-mass collectors for each step), the equation for dynamic lG3Nd/ ‘44Nd normalized to ‘46Nd/ 144Nd can be expressed as: R=/S
which upon differentiation Eqs. B-2-B-8, yields UR’ 2=
Geology 135 (1997) 325-334
step 1, then 143 and 144 in step 2, with 14’Ce monitored in another low-mass collector in step 1 and 147Sm monitored in another high-mass collector in step 2, the double-collector equation can be expressed as: R=
Differentiation and rearrangement gives an error equation of the same form as Eq. (C-2), but with: k,=[r2+(l+c$$3*]N2+(r+8);
(D-2) per total second (peak-top surement time, where
Eqs. C-6 and C-7 define the optimum T* and p. For triple-collector 142/144-normalized Nd, wherein 3 adjacent collectors receive masses: 141, 142, 143 in step 1; 142, 143, 144 in step 2; and 143, 144, 145 in step 3 (with 14’Ce monitored in a low-mass collector in step 1, and ‘47Sm in two different high-mass collectors in steps 2 and 3), the corrected 143/144 can be expressed as: 4/
Differentiation of Eq. (D-2) for optimum 72 (the step-2/step- 1 time-ratio), r3 (the step-3/step- 1 time-ratio), and p yields: (1 Pop
Appendix E. Dynamic analyses for unspiked Nd normalized to 142Nd/ 144Nd Using a doublecollector adjacent collectors receive
algorithm wherein two masses 142 and 143 in
which yields an expression for relative variance similar to that of triple-collector Sr (Eqs. C-9, and C-14C- 16), except that k,=[r’+(l++j)B2]Ar2+(r+8); k, = [4r2 + 1 + (1 + #)02]N2
(E-6) + (4r+
1 + 8); (E-7)
(E-8) 1 -2r)2
References Fletcher, I.R., 1986a. Gain free data collection for (almost) all isotope ratios of Sm and Nd: 7-, 6- and 5-collector dynamic multicollector procedures. W. Aust. Inst. Technol., School Phys. Geosci., Rep. No. SPG 438/1986/AP 132, 17 pp. Fletcher, I.R., 1986b. Mass fractionation of Sm in the VG354 solid-source mass spectrometer - a preliminary review. W. Amt. Inst. Technol., School Phys. Geosci., Rep. No. SPG 427/1986/AP 128, 14 pp. Lenz, H. and Wendt, I., 1976. Use of a double collector for high-precision isotope ratio measurements in geochronology. Adv. Mass Spectrom., 7A: 565-568 (Proc. 7th Int. Conf. on Mass Spectrometry, Florence, Aug. 30-Sept. 3, 19761. Ludwig, K.R., 1986. Constraints on time-efficient data-taking strategies for single-collector, isotope ratio mass-spectrometers. U.S. Geol. Surv. Bull., 1622T: 219-221. Makishima, A. and Nakamura, E., 1991. Calibration of Faraday
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