Geochemistry

W. M. White Appendix: Spectometry

APPENDIX: A.1 SAMPLE EXTRACTION AND PREPARATION Isotopic analysis can be performed on minerals, rocks, soil, tissue, gases, and liquids – just about any- thing. In view of the broad possibilities, we will only cover a few generalities here. In most instances, the element to be analyzed must be first isolated and purified before it can undergo isotopic analysis in a mass . Purification is necessary to avoid isobaric interferences and well as to enhance ionization efficiency, This can be done in a number of ways: • Dissolving the sample and then chemically purifying it (a process that often involves chroma- tography) • Reacting the sample with a reagent to produce a gas. For example, oxygen isotopic analysis of silicates involves reacting the sample with a fluorine compound such as bromine pentafluoride

to produce CO2 gas; in the case of carbonates, the process is similar but the reagent is phospho- ric acid. Purification is then achieved either cryogenically, or more commonly, using a gas chromatograph. For organic compounds, pyrolysis (thermal decomposition) may preceed such reactions. • Heating or fusing a sample in vacuum to release a gas. This is the usual method for noble gases. Individual gas components are separated cryogenically. • Noble gases contained in vesicles and inclusions can be released by crushing in vacuum. • If the sample is a liquid, the element of interest can be extracted chromatographically from the solution. With instruments in which the analyte is a gas or a liquid, a chromatograph can be used to select a spe- cific compound, for example the alkenones discussed in Chapter 10. This technique is termed com- pound-specific isotopic analysis. Micro-analytical techniques in which solids can be directly analyzed have become increasingly im- portant in the last several decades. The first of these is secondary ion mass spectrometry (SIMS), in which an ion beam (generally Cs or O) is fired at at a polished surface to produce secondary ions that are then swept into the mass spectrometer. An alternative approach is laser ablation in which a finely focused laser is fired at a polished surface ablating a small pit. The ‘debris’ consists of very fine particles that are swept up by a carrier gas, generally argon and carried into the mass spectrometer. A variation of this technique is laser fluorination, in which a laser is focused on a small spot of the sample in the presence of fluorine gas, producing a reaction that releases the element of interest (for example, oxygen or argon). A.2 THE MASS SPECTROMETER In most cases, isotopic abundances are measured by mass spectrometry. One exception is, as we have seen, short-lived radioactive , the abundances of which are determined by measuring their de- cay rate, and in fission track dating, where the abundance of 238U is measured, in effect, by inducing fis- sion. Another exception is spectroscopic measurement of isotope ratios in the Sun and stars. Frequen- cies of electromagnetic emissions of the lightest elements are sufficiently dependent on nuclear mass that emissions from different isotopes can be resolved. This approach is useful in astronomy not only because it is the only possibility, but also because isotope ratios stars show very large variations. This technique is not sufficiently precise for most geochemical problems. A mass spectrometer is simply a device that can separate atoms or molecules according to their mass. There are a number of different kinds of mass operating on different principles. Undoubtedly, the vast majority of mass spectrometers are used by chemists for qualitative or quantitative analysis of organic compounds. We will focus exclusively, however, on mass spectrometers used for isotope ratio determination. Most iso-

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tope ratio mass spectrometers are of a similar design, the magnetic-sector, or Nier mass spectrometer*, a schematic of which is shown in Figure A.1. It consists of three essential parts: an , a mass analyzer and a detector. There are, however, several variations on the design of the Nier mass spec- trometer. Some of these modifications relate to the specific task of the instrument; others are evolution- ary improvements. We will first consider the Nier mass spectrometer, and then briefly consider a few

Figure A.1. The magnetic sector or Nier mass spectrometer. This instrument uses a 60° magnetic sec- tor, but 90° magnetic sectors are also sometimes used. other kinds of mass spectrometers. A.2.1 The Ion Source As its name implies, the job of the ion source is to provide a stream of energetic ions to the mass ana- lyzer. Ions are most often produced by either , for solid-source mass spectrometers, electron bombardment, for gas-source mass spectrometers, or by inductively exciting a carrier gas into a plasma state in the case of inductively coupled plasma–mass spectrometers (ICP-MS). In thermal ionization ass spectrometry (TIMS), used for metals such as Sr, Nd, and Pb, a solution con- taining the element of interest is dried or electroplated onto a ‘filament’, a think ribbon of high- temperature metal, such as Re, Ta, or W, welded to supports. The ribbon is placed in the instrument and heated under vacuum by passing an electric current of several amperes through it, the sample evaporates, and some fraction of the atoms, depending on the element, ionize. The alkali metals ionize quite easily; the ionization efficiency for Cs, for example, approaches 100%. For some other elements, it can be 0.1% or less. In some cases, multiple filaments are used to achieve higher ionization efficiency. + - Ions produced in this way may be either the pure metal, for example Sr , or a radical such as OsO4 . Ionization efficiency can sometimes be increased by using a suitable substrate with a high work func- tion. The greater energy required to evaporate the atom results in a higher likelihood of its also being ionized. Tantalum oxide, for example, is a good substrate for analysis of Sr. Ionization efficiency can also be increased by altering the chemical form of the element of interest so that its evaporation tem- perature is increased (ionization is more likely at higher temperatures). For example, when a silica gel suspension is loaded along with Pb, the evaporation temperature of Pb is increased by several hundred degrees, and the ionization efficiency improved by orders of magnitude. Finally, the sample may be loaded in a particular chemical form in order to (1) form a positive rather than negative ion (or visa versa) and (2) provide a molecule of high mass, such as cesium metaborate where boron is the element of interest, to minimize mass fractionation, or to promote or inhibit the formation of oxides.

* It was developed by Alfred Nier of the University of Minnesota in the 1930's. Nier used his instrument to determine the isotopic abundances of many of the elements. In the course of doing so, however, he observed variations in the ratios of isotopes of a number of stable isotopes as well as Pb isotopes and hence was partly responsible for the fields of stable and radiogenic isotope geochemistry. He also was the first to use a mass spectrometer for geochronology, providing the first radiometric age of the solar system. In the 1980's he was still designing mass spectrometers, this time miniature ones which could fly on spacecraft on interplanetary voyages. These instruments provided meas- urements of the isotopic composition of atmospheric gases of Venus and Mars. Nier died in 1994.

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Electron bombardment is used when the analyte is a gas, such as CO2, H2, or N2. The gas is flow slowly into the evacuated source through a small orifice. In this case, a current is passed through a filament, normally Re, giving off electrons that strike the analyte molecules, and ionize them by knocking off one + + of the electrons. Except for noble gases, the ion is generally a polyatomic species such as H2 or CO2 . Most thermal ionization mass spectrometers employ a turret source in which a number of samples (typically 6 to 20) can be loaded. The turret is rotated to bring each sample into position for analysis. Gas source mass spectrometers often employ automated gas inlet systems, which allow for automated analysis of many samples. They also have so-called dual inlet systems that allow for rapid switching between a sample and a standard gas, insuring both are analyzed under similar conditions allowing correction for instrumental mass fractionation and other effects. A third type of ion source is an inductively coupled plasma (ICP), which is slowly replacing thermal ionization for many applications. An ICP operates by passing a carrier gas, generally Ar, through an induction coil in which an electric current alternates at radio frequencies. This excites the gas into ~7000°K plasma, in which the gas is completely ionized. The analyte is aspirated, generally as a so- lution, into the plasma and is ionized by the plasma. Alternatively, the sample can be introduced into the gas stream as an aerosol produced by laser ablation. The ions flow through an orifice into mass spectrometer. Initially, quadrupole mass spectrometers, which are commonly used for analysis of or- ganic compounds, were employed for these instruments. However, quadrupoles cannot achieve the same level of accuracy as magnetic sector instruments, although they are widely used for elemental analysis. Magnetic sector ICP-MS instruments came on the market a decade after quadrupole ICP-MS instruments and now achieve accuracies competitive with thermal ionization instruments. Combined with their generally higher ionization efficiency and hence higher sensitivity, they produce results that are superior to thermal instruments for several elements. As they continue to develop they may entirely replace thermal ionization instruments. More information on ICP-MS can be found in Vanhaecke and Degryse (2012). In secondary ion mass spectrometry (SIMS), the sample is introduced directly into the evacuate source chamber and ions produced by ion sputtering, in which an ion beam is fired at the sample surface to produce secondary ions. The advantage of this technique is that very small areas, with diameters of the order of 10’s of microns and less, can be analyzed. The amount of analyte ions produced, however, is small, so this approach is most useful when the element of interest is at high concentration (Pb and Hf in zircon, or oxygen in silicates, for example). Even in this case, however, precision is generally lower than with other techniques. Another method of ion production is resonance ionization. In this technique, a laser, tuned to a frequency appropriate for ionization of the element of interest, is fired at the sample. After the ions are produced, they are accelerated by an electrostatic potential, typically in the range of 5-20 kV for magnetic sector mass spectrometers. The ions move through a series of slits between charged plates. The charge on the plates also serves to collimate the ions into a beam. Generally the po- tential on the plates can be varied somewhat; in varying the potential on the plates, one attempts to maximize the beam intensity by 'steering' as many ions as possible through the slits. Thus the source produces a narrow beam of nearly mono-energetic ions (secondary ions are not mono-energetic and must be energy-filtered, which we will discuss below. A.2.2 The Mass Analyzer The function of the mass analyzer is two-fold. The main purpose is to separated the ions according to their mass (strictly speaking, according the their mass/charge ratio). But as is apparent in Figure A.1, the mass analyzer of a sector mass spectrometer also acts as a lens, focusing the ion beam on the detec- tor. A charged particle moving in a magnetic experiences a F = qv × B A.1

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where B is the strength, v is the particle velocity, and q is its charge (bold is used to de- note vector quantities). Note that force is applied perpendicular to the direction of motion (hence it is more properly termed a torque), and it is also perpendicular to the magnetic field vector. Since the force is always directed perpendicular to the direction of motion, the particle begins to move in a circular path. The motion is thus much like swinging a ball at the end of a string, and we can use equation for a centripetal force: v2 F = m A.2 r This can be equated with the magnetic force: v2 m = qv × B A.3 r The velocity of the particle can be determined from its energy, which is the accelerating potential, V, times the charge: 1 Vq = mv2 A.4 2 Solving A.4 for v2, and substituting in equation A.3 yields (in non-vector form): V 2Vq 2 = B A.5 r m Solving A.5 for the mass/charge ratio: m B2r2 = A.6 q 2V relates the mass/charge ratio, the accelerating potential, the magnetic field, and the radius of curvature of the instrument. If B is in gauss, r in cm, and V in volts, this equation becomes: m B2r2 = 4.825 × 10−5 A.7 q V with m in unified units and q in units of electronic charge. For a given set of conditions, a heavier particle will move along a curve having a longer radius than a lighter one. In other words, the lighter isotopes experience greater deflections in the mass analyzer. A typical radius is 27 cm for a TIMS instrument, which typically operating potential is 8 to 10 kV. In- struments with smaller radii are used in gas source mass spectrometers for light elements such as C, O, and Ar and larger radii are used in SIMS instruments where it is necessary to separate isobaric interfer- ences. Masses are selected for analysis by varying the magnetic field (note that in principle we could also vary the accelerating potential; however doing so has a second order effect on beam intensity, which is undesirable), generally in the range of a few thousand gauss for a TIMS instrument. It was shown in the 1950's that if the ions entered the magnetic field at an angle of 26.5° rather than at 90°, the effective radius of the mass analyzer doubles (i.e., a 27 cm instrument will have an effective ra- dius of 54 cm). This design trick, employed in all modern mass spectrometers, results in higher resolu- tion (better separation between the masses at the collector). An additional advantage of this ‘extended geometry’ is that the ion beam is focused in the ‘z’ direction (up-down) in addition to the x-y direction. This is an important effect because it allows the entire ion beam to enter the detector, which in turn al- lows the use of multiple detectors. In addition, modern mass spectrometers have further modifications

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to the magnet pole faces to produce a linear focal plane, which is helpful in the multiple collectors currently in use. Collisions of ions with ambient gas re- sult in velocity and energy changes and cause the beam to broaden. To minimize this, the analyzer is evacuated to 10-6 to 10-9 torr (=1mm Hg ≈ 10-3 atm). Some- what lower vacuums can be tolerated in the ion source. Where very high resolu- tion is required or the ion beam is not ini- tially mono-energetic, as in SIMS or ICP- MS instruments, an energy filter is em- ployed. This is simply an electrostatic field. The force is not pro- portional to velocity, as it the magnetic field. Instead, ions are deflected accord- ing to their energy. The radius of curva- ture is given by: 2V R = A.8 V2

where V2 is the electrostatic potential of Figure A.2. A double-focusing or Nier-Johnson mass spec- the energy filter and V is the energy of trometer with both magnetic and electrostatic sectors. Af- the ions (equal to the accelerating poten- ter Majer (1977). tial). Instruments employing both mass and energy filters are sometimes called double-focusing instruments. An example is shown in Figure 4.24. A.2.3 The Detector In general, ions are 'collected', or detected, at the focal plane of the mass spectrometer in one of two ways. The most common method, particularly for high precision mass spectrometers, is with a '', which is shown schematically in Figure A.3. As the name implies, this is a metal cup, gener- ally a few millimeters wide and several centime- ters deep (the depth is necessary to prevent ions and electrons from escaping). After passing through a narrow slit, ions strike the Faraday cup and are neutralized by electrons flowing to the cup from ground. The ion current into the cup is determined by measuring the voltage developed across a resistor as electrons flow to the cup to neutralize the ion current. The voltage is ampli- fied, converted to a digital signal, and sent to a computer that controls data acquisition. In most mass spectrometers, the resistor has a value of Figure A.3. Schematic drawing of a Faraday cup 1011 ohms. Since V = IΩ, an ion current of 10-11 A configured for positive ions. Electron flow would will produce a voltage of 1 V. Typical ion cur- reverse for negative ions. rents are on the order of 10-15 to 10-10 A. In the de-

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sign of the collector, care must be exercised that ions or free electrons produced when the ion strikes the cup cannot escape from the cup. Most modern mass spectrometers now employ a number of Faraday cups arrayed along the focal plane so that several isotopes can be collected simul- taneously. The spacing of the Faraday cups varies from element to element. Positioning of cups can be done using stepping motors under computer control. In some cases, fixed cups are used and electric fields are used to deflect specify masses into the cups. When ion currents are weak, a ‘multiplier’ is used as the detector. These increase the signal-to-noise ra- tio, but at some cost in precision. In an electron mul- tiplier, illustrated in Figure A.4, the ion strikes a charged dynode. The collision produces a number of

free electrons, which then move down a potential Figure A.4. Schematic of the . gradient. Each one of the electrons strikes a second electrode, again producing a number of free electrons. This process continues through a series of 10 or so electrodes to produce a cascade or shower of electrons. The net effect is an amplification of the signal of typically 100. An alternative is a Daly detector (named for its inventor), illustrated in Figure A.5. Ions strike a charged elec- trode, producing electrons as in the

. These electrons Figure A.5. Schematic of an electron multiplier. then strike a fluorescent screen produc- ing light, which is later converted to electrical signal. The net effect is also an amplification of a factor of 100. Multipliers may be used in either “analog” or “pulse count” mode. In analog model, the ion beam current is simply amplified and measured. In pulse count mode, rather than measuring the ion beam current, individual ions are counted. When an ion strikes the detector, an electrical “pulse” is produced. In pulse count mode, these pulses are counted by specialized electronics. Pulse count mode is useful only at low beam intensities; at higher ion beam intensities, an analog detector, such as the faraday cup, provides superior results. A.3 ACCELERATOR MASS SPECTROMETRY Traditionally, cosmogenic nuclides have been measured by counting their decays. In the past few decades, the utility of these cosmogenic nuclides in geochemistry and geochronology has been greatly enhanced by the advent of accelerator mass spectrometers, providing both more precise results from old applications (e.g., 14C dating) and new applications (identifying subducted sediment with 10Be). Mass spectrometry is a much more efficient method of detecting atoms that counting their decays in most instances. For example, the 14C/12C ratio in the atmosphere is 10-12. One gram of this carbon pro- duces about 15 beta decays per minute. But this gram contains about 1010 atoms of 14C. Even at an effi- ciency of ion production and detection of only 1%, 70µg of carbon can produce an ion beam that will re- sult in detection of 36,000 atoms 14C per hour. It would take 65 years for the same amount of carbon to produce 36,000 beta decays. However, there are some severe limitations with conventional mass spec- trometry in measuring very small isotope ratios (down to 10-15). Two problems must be overcome: limi-

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Figure A.6. The accelerator mass spectrometer at the University of Rochester. F.C. 1, 2, & 3 are faraday cups, T of F detectors are time of flight detectors. After Litherland (1980). tations of resolution, and isobaric interferences. Conventional mass spectrometers have resolutions of only about a ppm at ∆m of 1 u, and a fraction of a ppm at ∆m of 2 u. What this means is that for every106 or 107 12C atoms that arrived at the 12C position in the detector, about 1 12C atom will arrive at the 14C position. If the 14C/12C ratio is 10-14, some 107 more 12C would be detected at the 14C position than 14C atoms! The techniques involved in accelerator mass spectrometry vary with the element of interest, but most applications share some common features that we will briefly consider. Figure 4.25 is an illustration of the University of Rochester accelerator mass spectrometer. We will consider its application in 14C analy- sis as an example. A beam of C– is produced by sputtering a graphite target with Cs+ ions. There are several advantages in producing, in the initial stage, negative ions, the most important of which in this case is the instability of the negative ion of the principal atomic isobar, 14N. The ions are accelerated to 20keV (a higher energy than conventional mass spectrometers) and separated with the first magnet, so that only ions with m/q of 14 enter the accelerator. The faraday cup FC 1 (before accelerator) is used to monitor the intensity of the 12C beam. In the accelerator, the ions are accelerated to about 8 MeV, and electrons removed (through high-energy collisions with Ar gas) to produce C4+ ions. The reason for producing multiply-charged ions is that there are no known stable molecular ions with charge greater than +2. Thus the production of multiply charged ions effectively separates 14C ions from molecular 12 isobaric ions such as CH2. The now positively charged ions are separated from residual ions through two more magnetic sectors, and an electrostatic one (which selects for ion energy E/q). The final detec- tor distinguishes 14C from residual 14N, 12C, and 14C by the rate at which they lose energy through inter- action with a gas (range, effectively).

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A.4 ANALYTICAL STRATEGIES Isotopic variations in nature are very often quite small. For example, variations in Nd (neodymium) isotope ratios are measured in parts in 10,000. There are exceptions, of course. He and Os isotope ratios, as well as Ar isotope ratios, can vary by orders of magnitude, as can Pb in exceptional circumstances (minerals rich in uranium). These small variations necessitate great efforts in precise measurements. In this section we will briefly discuss some of the methods employed in isotope geochemistry to reduce analytical error. One technique used universally to reduce analytical errors is to make a large number of measurements. Thus a value for the 3He/4He ratio reported in a paper will actually be the mean of perhaps 100 indi- vidual ratios measured during a 'run' or analysis. Any short-term drift or noise in the instrument and its electronics, as well as in the ion beam intensity, will tend to average out. The use of multiple collectors and simultaneous measurement of several isotopes essentially eliminates errors resulting from fluctua- tions in ion beam intensity. This, however, introduces other errors related to the relative gains of the amplifiers, which then require correction. A final way to minimize errors is to measure a large signal. The inherent uncertainty in measuring x number of counts is x . Thus the inherent uncertainty in mea- suring 100 atoms is 10%, but the uncertainty in measuring 1,000,000 atoms is only 0.1%. These 'count- ing statistics' are the ultimate limit in analytical precision, but they come into play only for very small sample sizes. A4.2 Correcting Mass Fractionation One of the most important sources of error in solid source mass spectrometry results from the ten- dency of the lighter isotopes of an element to evaporate more readily than the heavier isotopes (Chapter 8). This means that the ion beam will be richer in light isotopes than the sample remaining on the fila- ment. As the analysis proceeds, the solid will become increasingly depleted in light isotopes and the ratio of a light isotope to a heavy one will continually decrease. This effect can produce variations up to a percent or so per mass unit (though it is generally much less). This would be fatal for Nd, for exam- ple, where natural isotopic variations are much less than a percent, if there were no way to correct for this effect. Fortunately, a correction can be made. The trick is to measure the ratio of two isotopes that are not radiogenic; that is a ratio that should not vary in nature. For Sr, for example, we measure the ratio of 86Sr/88Sr. By convention, we assume that the value of this ratio is equal to 0.11940. Any devia- tion from the value is assumed to result from mass fractionation in the mass spectrometer. The simplest assumption about mass fractionation is that it is linearly dependent on the difference in mass of the iso- topes we are measuring. In other words, the fractionation between 87Sr and 86Sr should be half that be- tween 88Sr and 86Sr. So if we know how much the 86Sr/88Sr has fractionated from the 'true' ratio, we can calculate the amount of fractionation between 87Sr and 86Sr. Formally, we can write the linear mass frac- tionation law as: ⎡ RN ⎤ uv − 1 ⎢ RM ⎥ α(u,v) = ⎣ uv ⎦ A.9 Δmuv where α is the fractionation factor between two isotopes u and v, ∆m is the mass difference between u and v (e.g., 2 for 86 and 88), RN is the 'true' or 'normal' u/v isotope ratio (e.g., 0.11940 for 86/88), and RM is the measured ratio. The correction to the ratio of two other isotopes (e.g., 87Sr/86Sr) is then calculated as: C M Rij = Rij (1+α(i, j)Δmij ) A.10 where RC is the corrected ratio and RM is the measured ratio of i to j and

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α(u,v) α(i, j) = A.11 1−α(u,v)ΔMvj 86 If we choose isotopes v and j to be the same (e.g., to both be Sr), then ∆mvj = 0 and α(i,j) = α(u,v). (A convention that is unfortunate in terms of the above equations, however, is that we speak of the 86/88 ratio, when we should speak of the 88/86 ratio (= 8.37521)). Using the 88/86 ratio, the 'normalization' equation for Sr becomes:

87 c 87 M ⎡ ⎧ ⎫ ⎤ ⎛ Sr⎞ ⎛ Sr⎞ ⎢ ⎪ 8.37521 ⎪ ⎥ ⎜ 86 ⎟ = ⎜ 86 ⎟ 1+ ⎨ M −1⎬ 2 A.12 ⎝ Sr⎠ ⎝ Sr⎠ ⎢ 88 86 ⎥ ⎣ ⎩⎪ ( Sr / Sr) ⎭⎪ ⎦ A more accurate description of mass fractionation is the power law. The fractionation factor is: N 1/Δmuv ⎡ Ruv ⎤ α = ⎢ M ⎥ − 1 A.13 ⎣ Ruv ⎦

C M Δmi, j The corrected ratio is computed as: Ri, j = Ri, j [1+ α ] A.14

C M ⎡ 1 2 ⎤ or: Ri, j = Ri, j ⎢1 + αΔmi, j + Δmi, j (Δmi, j −1)α +…⎥ A.15 ⎣ 2 ⎦ Since α is a small number, higher order terms may be dropped. Finally, it is claimed that the power law correction is not accurate and that the actual fractionation is described by an exponential law, from which the fractionation factor may be computed as: N M ln[Ruv /Ruv ] and the correction is: α = A.16 m j ln(mu /mv )

αm j ⎛ m ⎞ ⎡ m2 m2 ⎤ RC = RM ⎜ i ⎟ = RM ⎢1 + αΔm −α i, j + α 2 i, j +…⎥ A.17 i, j i, j ⎜ m ⎟ i, j i, j 2m 2 ⎝ j ⎠ ⎣ j ⎦ The exponential law appears to provide the most accurate correction for mass fractionation. However, all of the above laws are empirical rather than theoretical. The processes of evaporation and ionization are complex, and there is yet no definitive theoretical treatment of mass fractionation during this proc- ess. For stable isotopes where the objective to discover the extent of natural fractionation, as well as for Pb, which has only one non-radiogenic isotope, this approach cannot obviously cannot be used. As we noted, gas source mass spectrometers used for light stable isotopes are designed to minimize instru- mental fractionation and furthermore they can quickly switch from sample to standard and back. A correction for any instrumental mass fractionation can be made based on the difference between the measured and ‘true’ isotopic composition of the standard. This approach is not practical for either TIMS or ICP-MS instruments, but the are several other approaches possible. First, standards can be analyzed intermittently and a correction applied to samples based on the observed mass fractionation in the standards; this was the traditional approach used for Pb isotopes. Second, for light elements such as Li or B, one can analyze a polyatomic molecule containing a much heavier element, increasing atomic mass and hence decreasing fractionation. Two other alternatives are available. In the first of these, for elements with 3 or more isotopes, a ‘spike’ consisting of two or more isotopes of the element in propor- tions very different from the natural ones can be added to the analyte. Knowing the precise composi- tion of the spike, the amount of spike of spike added, and that the ‘natural’ isotopic composition must be related to that of standard by mass fractionation, the true isotopic composition of the analyte can be

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calculated (e.g., Krabbenhöft et al., 2010). This approach is used quite successfully for Pb, but requires separate analysis of spiked and unspiked fractions (e.g., Galer, 1999). Finally, for ICP-MS analyses, the ‘spike’ can consist of isotopes of an element with masses similar to that of the element of interest. For example, Maréchal et al. (1999) used Cu (with isotopes 63Cu and 65Cu) to correct for isotopic fractiona- tion during analysis of Zn (isotopes 64Zn, 66Zn, 67Zn, 68Zn, and 70Zn). This approach must be used with some caution, however, since different elements in the same solution can fractionate to different extents (White et al., 2000). A4.1 Deconvolution of Results If the analyte is a polyatomic species, measured abundances of isotopologues must be deconvoluted to obtain the isotopic composition of the element or elements of interest. For example, carbon and oxy-

gen are commonly analyzed as CO2, of which there are 12 isotopologues (excluding those containing 14C). For example, ions detected in the collector set for m/z of 46 can be 12C18O16O, 13C17O16O, or 12C17O17O. Of course, isotopologues of 17O are not very abundant, and those containing two 17O atoms will be rare indeed, so mass 46 will consist mainly of 12C18O16O, but a correction must nevertheless be made. We do not a priori know the isotopic composition of either carbon or oxygen, but by solving a series of simul- taneous equations, we can nevertheless deduce both if we assume their isotopic compositions are re- lated to standards by mass dependent fractionation laws above. Another example is that of Nd, which is often analyzed as NdO+. 134Nd16O+ has a mass of 160, but so does 143Nd17O+ and 142Nd18O+. This situa- tion if further complicated by mass fractionation that occurs during analysis. Nevertheless, if one as- sumes that all Nd and O non-radiogenic isotopes are in their natural proportions, calculation of the 143Nd/144Nd is possible by solving a series of simultaneous equations. These calculations are often built into the software (or can be added) of the computers that control the instruments, so they are largely invisible to the user. A.4.3 Isotope Dilution Analysis Determining precise ages generally requires that we not only determine isotope ratios precisely, but that we also determine parent-daughter ratios precisely. In essence, this means we need to determine the concentrations of the parent and daughter elements. The mass spectrometer is designed for isotope ratio analysis, but with a technique called isotope dilution, we can use it for determining concentrations of elements. It is easiest to explain the technique by taking a concrete example. We can start with the simplest possible example, that of a mono-isotopic element such as Cs. All natural Cs is 133Cs. However, 135Cs is a U fission product with a half-life of 2.3 million years; on the scale of laboratory time, we can consider it to be stable. We take a known amount of rock containing an unknown amount of Cs in it and add a known amount of pure 135Cs 'spike' to it. If we separate the Cs from the rock and analyze it in the mass spectrometer, the ratio of 135Cs to 133Cs we observe will be directly proportional to the ratio of the amount of 135Cs we add to the rock to the amount of 133Cs in the rock to begin with. In other words: 135 135 133 moles Cs added Cs / Cs = 133 A.18 moles Csinrock Since we know how much 135Cs we added, we can easily calculate the moles of 133Cs in the rock. We can calculate the Cs concentration in ppm from this simply by multiplying by the atomic weight and divid- ing by the weight of rock we used in our analysis. This is the isotope dilution technique. The general case is slightly more complex because neither the natural element nor the ‘spike’ will be mono-isotopic. Let's take a second concrete example. Suppose we want to know the concentration of Rb is a rock. Rb has two isotopes, 85Rb and 87Rb. We make the assumption that the proportion of these two isotopes will not vary in nature. 85Rb constitutes 72.165% and 87Rb 27.835% of all Rb; the 87Rb/85Rb ratio is thus 0.3857. Now suppose we have available a 'spike' containing 99% 87Rb and 1% 85Rb. Adding a

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known amount of spike to the rock, and analyzing the isotopic composition, the measured 87Rb/85Rb will be equal to: ⎛ 87 Rb⎞ [ 87 Rb] + [ 87 Rb] N S A.19 ⎜ 85 ⎟ = 85 85 ⎝ Rb⎠ [ Rb] + [ Rb] M N S 87 85 87 87 87 where ( Rb/ Rb)M is the measured ratio and [ Rb]N is the moles of 'natural' Rb in the rock, [ Rb]S is the moles of 87Rb in the spike we added, etc. We can rearrange equation A.19 as: [ 85Rb] R 85Rb / 87 Rb − 1 87 s { M ( )S } [ Rb]N = A.20 1− R 85Rb / 87 Rb M ( )N 87 85 85 87 85 87 where RM is ( Rb/ Rb)M and ( Rb/ Rb)N is the natural isotope ratio and ( Rb/ Rb)S is the ratio in the 87 spike. Note that [ Rb]S would more conveniently be expressed as the (molar) concentration times the weight of spike added. To find the concentration of 87Rb, we would simply divide by the weight of the sample. To form the 87Rb/86Sr ratio, we would carry out an isotope dilution analysis for Sr, and calcu- late the amount of 86Sr using an equation similar to A.20. If our spike is a solution containing both Rb and Sr (which is the usual case), when we divide the equation for 87Sr by the equation from 86Sr, the terms for the weight of spike and weight of sample cancel; that is, weighing error is eliminated. If we are primarily interested in the concentration of an element such as Rb, rather than the 87Rb/86Sr 87 87 ratio, we can rewrite equation A.19 in an alternative form if we let [ Rb]N = N Rb where N is the total number of atoms ( or moles) of Rb in the rock and 87Rb is now the relative abundance of 87Rb in 'natural' 87 87 Rb, i.e., 0.27835. Similarly, [ Rb]S = S Rb where S is the total number of moles of 'spike' Rb added to the sample. Equation A.19 then becomes: ⎛ 87 Rb⎞ N 87 Rb + S 87 Rb N S A.20 ⎜ 85 ⎟ = 85 85 ⎝ Rb⎠ N Rb + S Rb M N S which we can rearrange as: ⎡ 87 Rb − 85Rb R ⎤ N = S ⎢ S S M ⎥ A.21 85Rb R − 87 Rb ⎣⎢ N M N ⎦⎥ To convert to moles to weight in grams, we multiply the right side by the atomic weight of natural Rb. If our spike concentration is in grams, we would divide by the atomic weight of Rb in the spike: ⎡ 87 Rb − 85Rb R ⎤ AW N = S ⎢ S S M ⎥ N A.22 85Rb R − 87 Rb AW ⎣⎢ N M N ⎦⎥ S Once again, it is convenient to express S, the amount spike Rb added, as the concentration times the weight. And to obtain the concentration of Rb in the rock rather than the weight, we would divide by the weight of the sample: ⎡ 87 Rb − 85Rb R ⎤ AW spike wt × spikeconc Rbconc = ⎢ S S M ⎥ N A.23 85Rb R − 87 Rb AW sample wt ⎣ N M N ⎦ S For elements consisting of more than two isotopes, it is possible to include corrections for mass frac- tionation in the isotope dilution calculation (e.g., Hofmann, 1971).

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REFERENCES AND SUGGESTIONS FOR FURTHER READING Dickin, A. 1995. Radiogenic Isotope Geochemistry. Cambridge: Cambridge University Press. Elmore, D. and F. M. Phillips. 1987. Accelerator mass spectrometry measurements of long-lived radioi- sotopes. Science. 236: 543-550. Faure, G. 1986. Principles of Isotope Geology. New York: Wiley & Sons. Galer, S. J. G. 1999. Optimal double and triple spiking for high precision lead isotopic measurement. Chemical Geology, 157, 255-274. Hofmann, A. 1971. Fractionation corrections for mixed-isotope spikes of Sr, K, Pb. Earth and Planetary Science Letters, 10, 397-402. Krabbenhöft, A., Eisenhauer, A., Böhm, F., Vollstaedt, H., Fietzke, J., Liebetrau, V., Augustin, N., Peucker-Ehrenbrink, B., Müller, M. N., Horn, C., Hansen, B. T., Nolte, N. & Wallmann, K. 2010. Con- straining the marine strontium budget with natural strontium isotope fractionations (87Sr/86Sr∗, δ88/86Sr) of carbonates, hydrothermal solutions and river waters. Geochimica et Cosmochimica Acta, 74, 4097-4109, doi: 10.1016/j.gca.2010.04.009. Litherland, A. E. 1980. Ultrasensitive mass spectrometry with accelerators. Ann. Rev. Nucl. Part. Sci. 30: 437-473. Majer, J. R. 1977. The mass spectrometer. London: Wykeham Publications. Marèchal, C. N., Tèlouk, P. & Albarède, F. 1999. Precise analysis of copper and zinc isotopic composi- tions by plasma-source mass spectrometry. Chemical Geology, 156, 251-273, doi: 10.1016/s0009- 2541(98)00191-0. Vanhaecke, F. & Degryse, P. (eds.) 2012. Isotopic Analysis: Fundamentals and Applications Using ICP-MS, Weinheim: Wiley-VCH. Wasserburg, G. J., S. B. Jacobsen, D. J. DePaolo, M. T. McCulloch and T. Wen. 1981. Precise determina- tion of Sm/Nd ratios, Sm and Nd isotopic abundances in standard solutions. Geochimica et Cosmo- chimica Acta 45: 2311-2324. White, W. M., Albarede, F. & Telouk, P. 2000. High-precision analysis of Pb isotope ratios by multi- collector ICP-MS. Chemical Geology, 167, 257-270. Wunderlich, R. K., I. D. Hutcheon, G. J. Wasserburg and G. A. Blake. 1992. Laser-induced isotopic se- lectivity in the resonance ionization of Os. International Journal of Mass Specectrometry and Ion Processes, 115: 123-155. PROBLEMS 1. How long will a 143Nd+ ion spend traversing a 2 m mass spectrometer flight tube if it has been accel- erated by a potential of 10 kV?

2. What magnetic field (in gauss) is required to steer a 87Sr+ atom through a 90° sector, 30 cm radius mass spectrometer when the accelerating potential is set to 10 kV?

3. If the magnetic field in the mass spectrometer described above is set to intensity you have calculated, what would be the spacing in mm between a faraday cup collecting 87Sr+ and one collecting 86Sr+?

4. Show that the equation: 87 86 87'' 87' Sr/ Sr true = 86' 88'' ! 8.37521 corrects for both mass fractionation and differences in amplifier gain between two faraday cups. 87" is the measured signal intensity of 87Sr in cup 2, 87' is the measured signal intensity in cup 1.

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5. Suppose you measure a 87Sr/86Sr ratio of 0.70326 with a mass spectrometer. To monitor mass frac- tionation, you also measured 86Sr/88Sr and found a value of 0.12010. Assume the true value of 86Sr/88Sr is 0.11940. a. What is the true 87Sr/86Sr ratio according to the linear mass fractionation law? b. What is the true 87Sr/86Sr ratio according to the power mass fractionation law?

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