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).
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