2.4 Isotope dilution
Isotope dilution is generally agreed to be the
supreme analytical method for very accurate concentration determinations. In
this technique, a sample containing an element of natural isotopic composition
is mixed with a ‘spike’ solution, which contains a known concentration of the
element, artificially enriched in one of its isotopes. When known quantities of
the two solutions are mixed, the resulting isotopic composition (measured by
mass spectrometry) can be used to calculate the concentration of the element in
the sample solution. The element in question must normally have two or more
naturally occurring isotopes, one of which can be enriched on a mass separator.
However, in some cases a long-lived artificial isotope is used.
2.4.1 Analysis
technique
Before
use, the isotopic composition of a ‘spike’ must be accurately determined by
mass spectrometry. This measurement cannot be normalized for fractionation,
because there is no ‘known’ ratio to use as a fractionation monitor. Therefore,
several long runs are generally made, from which the average midpoint of the
run is taken to be the actual spike composition. The concentration of the spike
is generally determined by isotope dilution against standard solutions (of
natural isotopic composition) whose concentrations are themselves calculated
gravimetrically. Metal oxides are generally weighed out, but if these are
hygroscopic (e.g. Nd2O3) then accurate weighing requires
part of a metal ingot.
A
simple example of isotope dilution analysis is the determination of Rb
concentration for the Rb)Sr dating method. Typical mass spectra of natural, spike, and mixed
solutions are shown in Fig. 2.21.
Fig. 2.21. Summation of spike (hatched) and natural
ion beams to generate aggregate mixed peaks, as illustrated by rubidium isotope
dilution.
In
the mixture, each isotope peak is the sum of spike (S) and natural (N)
material. Hence,
87Rb moles 87N +
moles 87S
)))
= R = ))))))))))))))) [2.6]
85Rb moles 85N +
moles 85S
But the number of moles of an isotope is equal
to the number of moles of the element as a whole, multiplied by the isotopic
abundance. If the total number of moles of natural and spike Rb are represented
by MN and MS, then:
MN @ %87N + MS
@ %87S
R = ))))))))))))))))) [2.7]
MN @ %85N + MS
@ %85S
where percentages indicate the isotopic
abundances in the spike and natural solutions. This equation is rearranged in
the following steps:
R ( MN @ %85N + MS
@ %85S ) = MN @ %87N + MS
@ %87S
R @ MN @ %85N + R @ MS @ %85S = MN @ %87N + MS
@ %87S
R @ MN @ %85N ! MN @ %87N = MS
@ %87S ! R @ MS @ %85S
MN
( R @ %85N ! %87N ) = MS
( %87S !
R @ %85S )
( %87S ! R @ %85S
)
MN =
MS @
))))))))))))))) [2.8]
( R @ %85N ! %87N )
If we insert figures for the isotopic abundance
of natural Rb, and the isotopic abundance of a typical spike, such that
27.83/72.17 is the natural 87Rb/85Rb ratio, and 99.4/0.6
is the spike 87Rb/85Rb ratio, then the number of moles of
the natural Rb is given by:
(99.4 ! R @ 0.6)
MN = )))))))))))
@ MS [2.9]
(R @ 72.17 ! 27.83)
where R
is the measured isotope ratio.
But number of moles = molarity H mass, so the molarity of the
natural sample is given by:
(99.4 !
R @ 0.6) wtS
MolarityN = ))))))))))))
@ ))) @
MolarityS [2.10]
(R @ 72.17 ! 27.83) wtN
Molarity is then multiplied by atomic weight to
yield concentration:
(99.40 !
R @ 0.6) wtS Conc.S
ConcN = At. wtN @ )))))))))))))
@ )) @
))) [2.11]
(R @ 72.17 ! 27.83) wtN At.wt.S
Because there are only two isotopes of Rb, no
internal correction for fractionation is possible in the measurement of 87Rb/85Rb.
However, in the isotope dilution analysis of Sr, fractionation correction based
on 88/86 measurement is possible, which allows a much more accurate 84/86
(spike Sr / natural Sr) measurement to be made.
Isotope
dilution is potentially a very high-precision method. However, error
magnification may occur if the proportions of sample to spike which are mixed
are far from unity (Fig. 2.22). Consequently, it is generally believed that the
analysed peaks in an isotope dilution mixture should have an abundance ratio of
close to unity. In actual fact, the ideal composition of the mixture is
half-way between that of the natural and spike compositions. However, the best
precision normally required in an isotope dilution analysis is ca. 1 per mil
(0.1 %), which is two orders of magnitude worse than the precision normally achieved
in Sr or Nd isotope ratio measurements. Hence, significantly non-ideal spike)natural mixtures can be tolerated in
normal circumstances.
Fig. 2.22. Estimates of error magnification in
isotope dilution analysis as a function of the total number of moles of
sample/spike. Cases are shown for different percentages of spike isotope
enrichment, mixed with a natural sample with a 50)50 isotopic abundance ratio. After
DeBievre and Debus (1965).
The
only other sources of error in isotope dilution analysis are incomplete
homogenisation between sample and spike solution, and weighing errors. The
first of these can be overcome by centrifuging the sample solution to check for
any undissolved material, and repeating the dissolution steps as necessary
until complete solution is achieved. Given sufficient care, including the use
of non-hygroscopic standard material and correct balance calibration, spike
solutions can be calibrated to 0.1% accuracy (Wasserburg et al., 1981). The use of mixed spikes (e.g. Sm)Nd, Rb)Sr) then eliminates further weighing
errors in the analysis of these ratios in sample material. The result is that
isotope dilution accuracy can exceed 1% with ease, and 0.1% if necessary. This
compares very favourably with all other methods of concentration determination.
2.4.2 Double spiking
The selection of an arbitrary non-radiogenic
ratio for fractionation normalisation (e.g. 88Sr/86Sr =
8.37521) results in no loss of information for terrestrial samples. However,
for meteorites, use of such a procedure means that the true isotopes
responsible for certain anomalous isotope ratios cannot be uniquely identified.
An example is provided by inclusions EK 1-4-1 and C1 of the Allende chondritic
meteorite (Papanastassiou et al.,
1978), for which several mixing models of nucleosynthetic components are
possible (e.g. Clayton, 1978).
The
double-spike isotope dilution technique can be used to overcome this problem by
correcting for fractionation in the mass spectrometer source, thus allowing
comparison between all of the isotope ratios in a suite of samples, including
those used for fractionation normalisation. The theory of double spiking was
first investigated in detail by Dodson (1963). The calculations may be made
iteratively (e.g. Compston and Oversby, 1969) or algebraically (e.g. Gale,
1970). However, it is not usually possible to calculate ‘absolute’ values of
isotopic abundance because there is not normally any absolute standard to
calibrate the double spike.
Double
spiking has been used to study processes of Sr isotope fractionation in the
solar nebula, as sampled by inclusions and chondrules in the Allende meteorite,
relative to the composition of all terrestrial samples (Patchett, 1980). 88Sr/86Sr
versus 84Sr/86Sr
variations were found to fit a mass fractionation line very well (Fig. 2.23).
Ca)Al inclusions
were enriched in the lighter isotopes of Sr, which is surprising, since these
inclusions are thought to be relatively early condensates from the nebula,
which should therefore be enriched in heavy isotopes. The simplest explanation
for the effect is that the inclusions condensed from an isotopically light
region of the nebula.
Fig. 2.23. Plot of 88Sr/86Sr
against 84Sr/86Sr for samples from Allende chondrules.
Instrumental mass fractionation has been corrected using a double-spike
algorithm. After Patchett (1980).
Several
workers have investigated the use of double Pb spikes to allow within-run mass
fractionation correction of Pb isotope ratios. Most of these studies utilised
double stable isotope spikes such as 207Pb)204Pb (Compston and Oversby, 1969; Hamelin et al., 1985), which necessitate two
separate mass spectrometer runs, one with spike and one without. This is
because the spike interferes with the determination of natural 207Pb
and 204Pb. In contrast, the use of a 202Pb/205Pb
double spike allows both concentration determination and a correction for
analytical mass fractionation to be made on a single Pb mass spectrometer run
(Todt et al., 1996). These isotopes
are difficult to manufacture, but the procedure for 205Pb was
described by Parrish and Krogh (1987). The double spiking technique has not
been widely used for Pb, because between-run fractionation has been
successfully controlled by the silica gel loading technique. In addition, the
double-spike method may cause error magnification if the spike/sample ratio is
not close to optimal, or if the mass spectrometer run is not of high precision
(Fig. 2.24).
Fig. 2.24. Error propagation in Pb double-spike
analysis. (a) Effect of differing proportions of spike to natural Pb; (b)
effect of within-run precision. After Hamelin et al. (1985).
More
recent work on double spike TIMS Pb analysis has been done by by Powell et al. (1998), Galer (1999) and Thirlwall
(2000). However, the advent of plasma source mass spectrometry has offered an
alternative approach for fractionation correction in the analysis of common Pb
(see below).
References
Arden, J. W. and Gale N. H. (1974). New
electrochemical technique for the separation of lead at trace levels from
natural silicates. Anal. Chem. 46,
2)9.
Aston, F. W. (1919). A positive ray
spectrograph. Phil. Mag. (Series 6), 38, 707––14.
Aston, F. W. (1927). The constitution of
ordinary lead. Nature 120,
224.
Barovich, K. M., Beard, B. L., Cappel, J. B.,
Johnson, C. M., Kyser, T. K. and Morgan, B. E. (1995). A chemical method for
hafnium separation from high-Ti whole-rock and zircon samples. Chem. Geol.
(Isot. Geosci. Sect.) 121,
303–8.
Blichert-Toft, J., Chauvel, C. and Albarede, F.
(1997). Separation of Hf and Lu for high-precision isotope analysis by magnetic
sector-multiple collector ICP-MS. Contrib. Mineral. Petrol. 127, 248–60.
Brooks, C., Wendt,
Brooks, C., Hart, S. R. and Wendt,
Cameron, A. E., Smith, D. H. and Walker, R. L.
(1969). Mass spectrometry of nanogram-size samples of lead. Anal. Chem. 41, 525)6.
Cassidy, R. M. and Chauvel, C. (1989). Modern
liquid chromatographic techniques for the separation of Nd and Sr for isotopic
analyses. Chem. Geol. 74, 189)200.
Catanzaro, E. J. and Kulp, J. L. (1964). Discordant zircons from the Little Butte (
Chen, J. H. and Wasserburg, G. J. (1981).
Isotopic determination of uranium in picomole and sub-picomole quantities. Anal.
Chem. 53, 2060)7.
Christensen, J. N., Halliday, A. N., Godfrey,
L. V., Hein, J. R. and Rea, D. K. (1997). Climate and ocean dynamics and the
lead isotopic records in Pacific ferromanganese crusts. Science 277, 913–8.
Clayton,
Collerson, K. D., Kamber, B. S. and Schoenberg, R. (2002). Applications of accurate,
high-precision Pb isotope ratio measurement by multi-collector ICP-MS. Chem.
Geol. 188, 65–83.
Compston, W. and Oversby, V. M. (1969). Lead
isotopic analysis using a double spike. J. Geophys. Res. 74,
4338)48.
Cotte, M. (1938). Recherches sur l’optique
electronique. Ann. Physique 10,
333)405.
Crock, J. G., Lichte, F. E. and Wildeman, T. R.
(1984). The group separation of the rare-earth elements and yttrium from
geologic materials by cation-exchange chromatography. Chem. Geol. 45, 149)63.
Croudace,
Crumpler, T. B. and Yoe, J. H. (1940). Chemical
Computations and Errors, Wiley, pp. 189)90.
Daly, N. R. (1960). Scintillation type mass
spectrometer ion detector. Rev. Sci. Instrum. 31, 264)7.
David, K., Birch, J. L., Telouk, P. and
Allegre, C. J. (1999). Application of isotope dilution for precise measurement
of Zr/Hf and 176Hf/177Hf ratios by mass spectrometry
(ID–TIMS/ID–MC–ICP–MS). Chem. Geol. 157, 1–12.
Davis, D. W. (1982). Optimum linear regression
and error estimation applied to U)Pb data.
Dawson, P. H. (1976). (Ed.), Quadrupole Mass
Spectrometry and its Applications. Elsevier, 349 p.
DeBievre, P. J. and Debus, G. H. (1965).
Precision mass spectrometric isotope dilution analysis. Nucl. Instrum. Meth.
32, 224)8.
Dempster, A. J. (1918). A new method of positive ray
analysis. Phys. Rev.
11, 316-24.
DePaolo, D. J. and Wasserburg, G. J. (1976). Nd
isotopic variations and petrogenetic models. Geophys. Res. Lett. 3, 249)52.
Dickin, A. P., Jones, N. W., Thirlwall, M. and
Thompson, R. N. (1987). A Ce/Nd isotope study of crustal contamination
processes affecting Palaeocene magmas in Skye, NW Scotland. Contrib.
Mineral. Petrol. 96, 455)64.
Dodson, M. H. (1978). A linear method for
second-degree interpolation in cyclical data collection. J. Phys. E (Sci.
Instrum.) 11, 296.
Dodson, M. H. (1963). A theoretical study of
the use of internal standards for precise isotopic analysis by the surface
ionisation technique: Part I - General first-order algebraic solutions. J.
Sci. Instrum. 40, 289)95.
Dosso, L. and Murthy, V. R. (1980). A Nd isotopic
study of the
Eberhardt, A., Delwiche, R. and Geiss, J.
(1964). Isotopic effects in single filament thermal ion sources. Z. Natur.
19a, 736)40.
Edwards, R. L., Chen, J. H. and Wasserburg, G.
J. (1987). 238U)234U)230Th)232Th systematics and the precise measurement of
time over the past 500,000 years. Earth Planet. Sci. Lett. 81, 175)92.
Eugster, O., Tera, F., Burnett, D. S. and
Wasserburg, G. J. (1970). The isotopic composition of gadolinium and neutron
capture effects in some meteorites. J. Geophys. Res. 75, 2753)68.
Faul, H. (1966). Ages of Rocks, Planets, and
Stars. McGraw-Hill, 109 p.
Gale, N. H. (1970). A solution in closed form
for lead isotopic analysis using a double spike. Chem. Geol. 6, 305)10.
Galer, S. J. G. (1999). Optimal double and
triple spiking for high precision lead isotopic measurement. Chem. Geol.
157, 255)74.
Habfast, K. (1983). Fractionation in the
thermal ionization source. Int. J. Mass Spectrom. Ion Phys. 51, 165)89.
Halliday, A. N., Lee, D.-C., Christensen, J.
N., Rehkamper, M., Yi, W., Luo, X., Hall, C. M., Ballentine, C. J., Pettke, T.
and
Hamelin, B., Manhes, G., Albarede, F. and
Allegre, C. J. (1985). Precise lead isotope measurements by the double spike
technique: a reconsideration. Geochim. Cosmochim. Acta 49, 173)82.
Hooker, P., O’Nions, R. K. and Pankhurst, R. J.
(1975). Determination of rare-earth elements in U.S.G.S. standard rocks by
mixed-solvent ion exchange and mass spectrometric isotope dilution. Chem.
Geol. 16, 189)96.
Horn,
Houk, R. S. (1986). Mass spectrometry of
inductively coupled plasmas. Anal. Chem. 58, 97A–105A.
Houk, R. S., Fassel, V. A., Flesch, G. D.,
Svec, H. J., Gray, A. L. and Taylor, C. E. (1980). Inductively coupled argon
plasma for mass spectrometric determination of trace elements. Anal. Chem.
52, 2283–9.
Ingram, M. G. and Chupka, W. A. (1953). Surface
ionisation source using multiple filaments. Rev. Sci. Instrum. 24, 518)20.
Kalsbeek, F. and Hansen, M. (1989). Statistical
analysis of Rb)Sr isotope data by the ‘bootstrap’ method. Chem. Geol. (Isot.
Geosci. Section) 73, 289)97.
Krogh, T. E. (1973). A low contamination method
for hydrothermal decomposition of zircon and extraction of U and Pb for
isotopic age determination. Geochim. Cosmochim. Acta 37, 485)94.
Kuritani, T. and Nakamura, E. (2002). Precise
isotope analysis of nanogram-level Pb for natural rock samples without use of
double spikes. Chem. Geol. 186,
31–43.
Kurz, E. A. (1979). Channel electron
multipliers. Amer. Lab. 11
(3), 67)74.
Lee D.-C. and Halliday, A. N. (1995). Precise
determinations of the isotopic compositions and atomic weights of molybdenum,
tellurium, tin and tungsten using ICP source magnetic sector multiple collector
mass spectrometry. Int. J. Mass Spec. Ion Process. 146/147, 35–46.
Li, W. X., Lundberg, J., Dickin, A. P., Ford,
D. C., Schwarcz, H. P., McNutt, R. H. and Williams, D. (1989). High-precision
mass spectrometric uranium-series dating of cave deposits and implications for
paleoclimate studies. Nature 339,
534)6.
Luais, B., Telouk, P. and Albarede, F. (1997).
Precise and accurate neodymium isotopic measurements by plasma-source mass
spectrometry. Geochim. Cosmochim. Acta 61, 4847)54.
Ludwig, K. R. (1980). Calculation of
uncertainties of U)Pb isotope data. Earth Planet. Sci. Lett. 46, 212)20.
Ludwig, K. R. (1997). Isoplot. Program and documentation, version 2.95. Revised edition
of U.S. Geol. Surv. Open-File report 91-445 (kludwig@bgc.org).
Lugmair, G. W., Scheinin, N. B. and Marti, K.
(1975). Search for extinct 146Sm, 1. The isotopic abundance of 142Nd
in the Juvinas meteorite. Earth Planet. Sci. Lett. 27, 79)84.
Luo, X., Rehkamper, M., Lee, D.-C. And
Halliday, A. N. (1997). High precision 230Th/232Th and 234U/238U
measurements using energy-filtered ICP magnetic sector multiple collector mass
spectrometry. Int. J. Mass Spec. Ion Process. 171, 105–17.
McIntyre, G. A., Brooks, A. C. Compston, W and
Turek, A. (1966). The statistical assessment of Rb)Sr isochrons. J. Geophys. Res.
71, 5459)68.
Manton, W.
Noble, S. R., Lightfoot, P. C. and Scharer, U.
(1989). A new method for single-filament isotopic analysis of Nd using in situ reduction. Chem. Geol. (Isot.
Geosci. Section) 79, 15)19.
Nier, A. O. (1940). A mass spectrometer for
routine isotope abundance measurements. Rev. Sci. Instrum. 11, 212)16.
O’Nions, R. K., Hamilton, P. J. and Evensen, N.
M. (1977). Variations in 143Nd/144Nd and 87Sr/86Sr
ratios in oceanic basalts. Earth Planet. Sci. Lett. 34, 13)22.
O’Nions, R. K., Carter, S. R., Evensen, N. M.
and
Papanastassiou, D. A., Huneke, J. C., Esat, T.
M. and Wasserburg, G. J. (1978). Pandora’s box of the nuclides. In: Lunar
Planet. Sci. IX, Lunar Planet.
Sci. Inst.,
Parrish, R. R. (1987). An improved
micro-capsule for zircon dissolution in U)Pb geochronology. Chem. Geol.
(Isot. Geosci. Section) 66,
99)102.
Parrish, R. R. and Krogh, T. E. (1987).
Synthesis and purification of 205Pb for U)Pb geochronology. Chem. Geol.
(Isot. Geosci. Section) 66,
103)10.
Patchett, P. J. (1980). Sr isotopic fractionation
in Ca)Al inclusions
from the Allende meteorite. Nature 283,
438)41.
Patchett, P. J. and Tatsumoto, M. (1980). A
routine high-precision method for Lu)Hf isotope geochemistry and chronology. Contrib.
Mineral. Petrol. 75, 263)7.
Pietruszka, A. J., Carlson, R. W. and Hauri, E.
H. (2002). Precise and accurate measurement of 226Ra–230Th–238U
disequilibria in volcanic rocks using plasma ionization multicollector mass
spectrometry. Chem. Geol. 188,
171–91.
Potts, P. J. (1987). Handbook of Silicate Rock
Analysis. Blackie, 622 p.
Powell, R., Hergt, J. and Woodhead, J. (2002).
Improving isochron calculations with robust statistics and the bootstrap. Chem.
Geol. 185, 191–204.
Powell, R., Woodhead, J. and Hergt, J. (1998).
Uncertainties on lead isotope analyses: deconvolution in the double-spike
method. Chem. Geol. 148, 95–104.
Richard, P.,
Roddick, J. C., Loveridge, W. D. and Parrish,
R. R. (1987). Precise U/Pb dating of zircon at the sub-nanogram Pb level. Chem.
Geol. (Isot. Geosci. Section) 66,
111)21.
Russell, W. A., Papanastassiou, D. A. and
Tombrello, T. A. (1978). Ca isotope fractionation on the Earth and other solar
system materials. Geochim. Cosmochim. Acta 42, 1075)90.
Shen, C.-C., Edwards, R. L., Cheng, H., Dorale,
J. A., Thomas, R. B., Moran, S. B., Weinstein, S. E. and Edmonds, H. N. (2002).
Uranium and thorium isotopic and concentration measurements by magnetic sector
inductively coupled plasma mass spectrometry. Chem. Geol. 185, 165–78.
Smythe, W. R. and Mattauch, J. (1932). A new
mass spectrometer. Phys. Rev. 40,
429-
Tanaka, T. and Masuda, A. (1982). The La)Ce geochronometer: a new dating method,
Nature 300, 515)18.
Thirlwall, M. F. (1982). A triple-filament
method for rapid and precise analysis of rare-earth elements by isotope
dilution. Chem. Geol. 35, 155)66.
Thirlwall, M. F. (1991a). High-precision
multicollector isotopic analysis of low levels of Nd as oxide. Chem. Geol.
(Isot. Geosci. Section) 94,
13)22.
Thirlwall, M. F. (1991b). Long-term
reproducibility of multicollector Sr and Nd isotope ratio analysis. Chem.
Geol. (Isot. Geosci. Section) 94,
85)104.
Thirlwall, M. F. (2000). Inter-laboratory and
other errors in Pb isotope analyses investigated using a 207Pb–204Pb double spike. Chem. Geol. 163, 299)322.
Thirlwall, M. F. (2002). Multicollector ICP-MS
analysis of Pb isotopes using a 207Pb–204Pb double spike
demonstrates up to 400 ppm/amu systematic errors in Tl-normalization. Chem.
Geol. 184, 255–79.
Thirlwall, M. F. and Walder, A. J. (1995). In
situ hafnium isotope ratio analysis of zircon by inductively coupled plasma
multiple collector mass spectrometry. Chem. Geol. 122, 241–7.
Thompson J.J. (1913). Rays of Positive
Electricity and their Application to Chemical Analysis, Longmans, Green and
Co. Ltd.
Titterington, D. M. and Halliday, A. N. (1979).
On the fitting of parallel isochrons and the method of maximum likelihood. Chem.
Geol. 26, 183)95.
Todt, W., Cliff, R. A., Hanser, A. and Hofmann,
A. W. (1996). Evaluation of a 202Pb – 205Pb double spike for
high-precision lead isotopic analysis. In: Basu, A. and Hart, S. R. (Eds.)
Earth Processes: Reading the Isotopic Code. Geophys. Monograph 95, American Geophysical
Vance, D. and Thirlwall, M. (2002). An
assessment of mass discrimination in MC-ICPMS using Nd isotopes. Chem. Geol.
185, 227–40.
Walder, A. J., Abell,
Walder, A. J. and Furuta, N. (1993). High
precision lead isotope ratio measurement by inductively coupled plasma multiple
collector mass spectrometry. Anal. Sci. 9, 675-80.
Walder, A. J., Platzner, I. And Freedman, P. A.
(1993b). Isotope ratio measurement of lead, neodymium and neodymium–samarium
mixtures, hafnium and hafnium–lutetium mixtures with a double focussing
multiple collector inductively coupled plasma mass spectrometer. J. Anal.
Atomic. Spectrom. 8, 19-23.
Wasserburg, G. J., Jacobsen, S. B., DePaolo, D.
J. McCulloch, M. T. and Wen, T. (1981). Precise determination of Sm/Nd ratios,
Sm and Nd isotopic abundances in standard solutions. Geochim. Cosmochim.
Acta 45, 2311)23.
Wendt,
White, W. M., Albarede, F. and Telouk, P.
(2000). High-precision analysis of Pb isotope ratios by multi-collector ICP-MS.
Chem. Geol. 167, 257)270.
York, D. (1966). Least-squares fitting of a
straight line.
York, D. (1967). The best isochron. Earth
Planet. Sci. Lett. 2, 479)82.
York, D. (1969). Least-squares fitting of a
straight line with correlated errors. Earth Planet. Sci. Lett. 5, 320)4.