Precise Measurement of Magnesium Isotopes in Fe‐Mg Minerals Using a Multi‐collector SHRIMP Ion Microprobe

The distribution of Mg isotopes in minerals is becoming increasingly relevant in Earth science. Usually, they are determined by dissolving mineral concentrates and, after purifying Mg with ion exchange resins, analysing the resulting solutions by TIMS or, most often, MC‐ICP‐MS. When applied to individual minerals, these methods are slow and prone to contamination from impurities in the concentrates, inconveniences that may be avoided using spot analysis techniques such as LA‐MC‐ICP‐MS or SIMS, albeit at the price of a large instrumental mass fractionation (IMF) and isobaric interferences, most prominent in the former. Here, we studied the potential of the multi‐collector SHRIMP II ion microprobe for measuring Mg isotopes in Fe‐Mg silicates and oxides. We found that, when corrected for the divergence of the Mg ion paths within the sample chamber caused by the Earth's magnetic field, the SHRIMP's IMF overwhelmingly depends on the mineral species, and the effects of variable chemical composition are negligible. We propose that the IMF is caused by the force constant difference, ∆F, between "hard" and "soft" bonds linking the ions of the studied element to the mineral lattice. Given that ∆F is a constant for each mineral species, we calculated IMF‐correction factors for the most common Mg‐bearing minerals. The thus‐calculated correction factors permit the analysis in the same session, and with reasonable accuracy (within ~ 0.3‰ of the δ26Mg determined by SN‐MC‐ICP‐MS analyses of concentrates), of samples from different mineral species, facilitating the application of Mg isotopes to terrestrial studies.

After oxygen, magnesium is the most abundant element in the solid Earth; it has three isotopes, 24 Mg, 25 Mg and 26 Mg, with molar fractions of 0.78992, 0.10003 and 0.11005, whose relative abundances are expressed as delta values: where xx is either 25 or 26, and the reference value is from the DMS3 reference material ( 25 Mg/ 24 Mg = 0.126851; 26 Mg/ 24 Mg = 0.139796: Galy et al. 2003, Young andGaly 2004).
In the three-isotope diagram (δ 25 Mg vs. δ 26 Mg, Figure 1), materials fractionated from a uniform reservoir define a straight line: the theoretical slope of which ranges between ß = 0.500 and ß = 0.521, depending on the mass fractionation law chosen (Davis et al. 2015).All terrestrial and most meteoritic materials plot along a line with slope between ß = 0.511 and 0.521 corresponding to the predicted equilibrium and kinetic mass fractionation laws (Galy et al. 2001, 2003, Young and Galy 2004), which is duly called the terrestrial fractionation line (TFL).Some meteoritic materials, especially the CAIs, fall to the right because they contain radiogenic 26 Mg (Gray andCompston 1974, Lee andPapanastassiou 1974).The excess 26 Mg (Figure 1) is usually reported as Δ 26 Mg = δ 26 Mg measured -δ 25 Mg measured ß -1 and generally attributed to the decay of the high heat-producing and short-lived 26 Al isotope, incorporated only in minerals accreted early during the condensation of the Solar nebula.
The first bulk measurements of Mg isotopes were done with thermal ionisation mass spectrometry (TIMS) on dissolvedand ion exchange purifiedrock samples and mineral concentrates (op.cit.).The first spot analyses of minerals were done with secondary ion mass spectrometry (SIMS) using single-collector and relatively simple ion microprobes such as the Chicago AEI IM-20 (e.g., Hutcheon et al. 1978, Russell et al. 2000, Richter et al. 2002) or the Canberra SHRIMP I (Ireland et al. 1986).Both TIMS and SIMS produced a large instrumental mass fractionation (IMF) so that the resulting 25 Mg/ 24 Mg was normalised to a reference value of 0.12663, then believed to approximate the Earth's mean (Catanzaro et al. 1966), and the measured 26 Mg/ 24 Mg ratios were duly corrected according to the exponential law of mass fractionation: 26 Mg =24 Mg corrected ¼ 0:12663= 25 which yielded values of Δ 26 Mg that are precise enough for meteoritic studies.However, the lack of a suitable method for correcting the IMF and the subsequent necessity of normalisation prevented studying the Mg isotope fractionation occurring in geological processes, that is, to determine the position along the TFL (see Figure 1).Such studies were only possible with the development of multicollector (MC) ICP-MS because this technique, when operating in solution-nebulisation mode (SN-MC-ICP-MS) causes a uniform instrumental mass fractionation, which is easily correctable (Young and Galy 2004, and references therein).Determining Mg isotopes after dissolving the sample also permits separating the matrix components and potential interferents with ion-exchange resins before analyses.
SN-MC-ICP-MS copes well with bulk-rock analysis but not with individual minerals because it is time-consuming, lacks spatial resolution and is prone to contamination by impurities within the mineral concentrates.These disadvantages are suppressed by analysing spots of solid mineral grains with laser ablation (LA) MC-ICP-MS or SIMS (see revision in Chaussidon et al. 2017), albeit at the price of introducing new drawbacks, such as large and variable IMF and potential isobaric interferences.
Magnesium isotope measurement using large-radius ion microprobes suffer from little, if any, isobaric interferences, especially if cryogenic pumps evacuate the sample chamber until a vacuum is in the E -08 Torr range, but have a large IMF, the causes of which are still poorly understood (see references below).In principle, we can consider the IMF as consisting of two components: one related to the mineral lattice and the other related to the chemical composition of a given mineral.The former occurs when reference materials and unknowns belong to different mineral species.The latter occurs when reference materials and unknowns belong to the same mineral species but have different compositions.Ireland et al. (1986), using a single collector SHRIMP I ion microprobe, considered only the different-mineral effect and proposed IMF correction factors relative to kaersutite, which they used as a reference material.In contrast, authors working with multi-collector ion microprobes (CAMECA 1270and 1280, e.g., Kita et al. 2012, Luu et al. 2013, Ushikubo et al. 2013, Fukuda et al. 2020, Richter et al. 2021) have repeatedly reported IMF effects for the same mineral species that depend on the chemical composition, proposing corrections equations derived from reference materials of known Mg isotope ratios and variable chemical compositions.
Preliminary work with the Granada MC SHRIMP IIe has shown that the chemical composition-related IMF is negligible in the single-collection mode, confirming the result of Ireland et al. (1986), but it may be noticeable in the multi-collection mode.Multi-collection is inherently more precise than single collection and, therefore, highly desirable for precise measurement of Mg isotopes.For this reason, we undertook a study to understand the reasons for the instrumental mass fractionation of Mg isotopes in the MC SHRIMP and how it might be corrected for olivine, clinopyroxene, orthopyroxene, amphibole, spinel, garnet, cordierite and biotite, the results of which are reported here.We found that if the local effects of the Earth's magnetic field on the Mg isotopes ionic paths within the sample chamber are well corrected, the IMF overwhelmingly depends on the differences in the mineral nature.This observation permitted us to calculate IMF correction factors for the minerals mentioned above, enabling reasonably accurate determination of Mg isotopes in different minerals using a single reference material instead of employing a different reference material for each mineral species and then correcting the results according to its specific chemical composition.
After dissolution and chromatographic purification, the six gem-quality minerals and the Ronda peridotite minerals were analysed using MC-ICP-MS.The Adrar Sutuff granulite minerals were not analysed in this way.However, we included them in the study because their bulk composition differs substantially from the peridotitic minerals; they likely have identical Mg isotope ratios because of diffusion after being equilibrated at T ≥ 1000 °C (Molina et al. in review, Molina et al. 2018) and the Mg isotope bulk-rock composition was known.
To assess the influence of the bulk chemical composition, Fe mainly, on the instrumental mass fractionation of Mg isotopes, we made three synthetic glasses with identical Mg isotope ratios, similar MgO mass fraction ($ 5% m/m) and different FeO (0, 5 and 10% m/m), with the remaining elements varying concomitantly.The compositions of all studied materials, glasses and minerals are given in Table 1.

Sample preparation and SHRIMP analysis
Handpicked mineral grains were mounted with highvacuum epoxy in a SHRIMP megamount.These were 35 mm in diameter and were screwed from their back in the mount holder to avoid metal and abrupt topographic changes facing the extraction plate.This configuration was designed by Ickert et al. (2008) to minimise the fractionation of oxygen isotopes during the extraction of secondary ions in grains located off-centre of the mount.
Once mounted and polished, mineral grains were studied by optical and SEM microscopy to detect cracks and inclusions, and then coated with a 10 nm thick gold layer to be analysed in the SHRIMP IIe/mc ion microprobe at the IBERSIMS laboratory of the CIC-University of Granada, Spain.The primary beam was composed of 16 O 16 O -, filtered with a Wien mass filter and set to a yield of ≈ 6 nA.The primary ion optics were configured with a 120 μm Kohler aperture to produce a $ 18 μm diameter spot on the megamount surface.The secondary beam was guided through the 100-μm source slit to the electrostatic analyser, maximising its signal of the post-ESA monitor.From here, the secondary beam was focused and guided with the post-ESA quadrupole to the collector through the magnet.The collector consisted of three Faraday cups arranged to detect each Mg isotope peak at the same magnetic field, yielding precisely matched peak shapes.Additionally, their 200 μm entrance slits were moved vertically so that the vertical scanning of the secondary beam with the post-ESA quadrupole produced matched patterns for the three cups.The ion current arriving at the Faraday cups was measured with e7600 electrometers designed around Thermo Finnigan DC amplifier modules.The electrometers were set with 10 11 Ω resistors, V/F input sensitivities up to 50 V and V/F gains of ×1 for 24 Mg + , and ×10 for 25 Mg + and 26 Mg + .The Earth's magnetic field (EMF), able to fractionate Mg isotopes in the secondary beam (see below), was neutralised with a current of -550 mA in the Helmholtz coils surrounding the SHRIMP sample chamber.This value depends on the local EMF and must be determined empirically.To this end, we first set to 0 mA the current applied to the Helmholtz coils, centring the 25 Mg + beam on the axial cup horizontally with the QT1 quadrupole, measuring the 25 Mg/ 24 Mg and 26 Mg/ 24 Mg ratios, moving the QT1 horizontal steering to the left by -50 bits, measuring the ratios again, moving QT1 horizontal steering to the right by 50 bits, re-measuring the ratios and comparing the three measurements.If these differed, we increased slightly the current applied to the Helmholtz coils, repeating the procedure until the three measurements matched.
Spots to be analysed were burned for about 2 min before measurements.After pre-sputtering, the secondary beam was guided horizontally and vertically to maximise the signal on the axial cup ( 25 Mg + ); the magnet position was fixed with a mass scan in the same cup, and the isotope ratios were measured in two sets of ten scans of 10 s each, with a total measuring time of 200 s per spot.In these conditions, the mass resolving power (M/ΔM) at 10% height was 2500 and isobaric interferences by double ions or hydrides were undetected.Small grains of the San Carlos olivine, measured every four unknowns (Hu et al. 2016), were used as the reference material.Data reduction was done with in-house software (available from F. Bea upon reasonable request) written in the STATA programming language.The software corrects the instrumental drift using kernel-weighted local polynomial smoothing of the isotope ratios of the measured reference material vs. the measurement time.It then applies the resulting function to the unknowns to compute their isotope ratios.In total, we performed 585 and 172 spot analyses of minerals and synthetic glasses, respectively (online supporting information Tables S1 and S2).

Sample preparation and SN-ICP-MS analysis
Sample digestion: Mineral particles were finely crushed in an agate mortar, then dissolved and purified using procedures previously established (Ke et al. 2016, Li et al. 2016, Gao et al. 2019).Garnets and other refractory minerals were digested in high-pressure bombs with concentrated HNO 3 -HF (1:2 in v/v) in an oven at 190 °C for 48 h.The other samples were routinely dissolved in PFA beakers with concentrated HNO 3 -HF (1:3 in v/v) at 130 °C over 8 h.All the sample solutions were then dried and refluxed with concentrated HNO 3 -HCl (1:3 in v/v) and HNO 3 sequentially.Finally garnets and staurolite were dissolved in 0.5 mol l -1 HCl-95% acetone, and other samples in 1 mol l -1 HNO 3 , ready for purification.
The Mg cuts of garnets and staurolite were evaporated to dryness and dissolved in 100 μl 1 mol l -1 HNO 3 .All the samples containing $ 20 μg Mg in 100 μl 1 mol l -1 HNO 3 were loaded on a PFA column with 2.3 ml of Bio-Rad AG50W-X8 (200-400 mesh) resin.Matrix elements such as Na, K, Ti etc. were eluted in the first 23 ml 1 mol l -1 HNO 3 , and Mg was then collected by 15 ml 1 mol l -1 HNO 3 with the other matrices remaining hosted on the resin (e.g., Ca and Al).After one purification step, most of the matrix elements were efficiently removed.To ensure the purity of the collected Mg, the same column chemistry was performed twice.Recoveries of Mg are > 99.6% for step 1 and > 99.8% for step 2 (Li et al. 2016, Gao et al. 2019).The whole procedure blank was < 10 ng and rarely reached 20-30 ng, less than 0.2% of the sample Mg processed and thus considered negligible.
Isotope measurement: Isotope measurements were conducted on a Neptune Plus MC-ICP-MS in the Isotope Geochemistry Laboratory at China University of Geosciences, Beijing, following the 25 Mg-26 Mg double spike procedure (He et al. 2022).Sample solutions were introduced to the plasma in 0.15 mol l -1 HNO 3 using a Scott double pass quartz glass spray chamber and a PFA self-aspiration micro nebuliser. 24Mg, 25 Mg and 26 Mg were simultaneously collected on the L3, C and H3 Faraday cups under a lowmass resolution mode.
After measuring the concentration of each sample solution, bracketing GSB calibrator (a mono-elemental standard solution, China Iron and Steel Research Institute) and sample solutions were spiked with a 25 Mg-26 Mg double spike and prepared in 1 μg ml -1 , with q within the AE4% range of the optimised one (0.472 AE 0.02).The total ion intensity was ≥ 70 V.Each measurement consisted of a 3 s idle time and eighty cycles of 4.194 s integration time.Each analytical sequence for each sample was repeated four times.A set of under-spiked and over-spiked GSB standard solutions, with q ranging from 0.452, 0.462, 0.482 to 0.492, were analysed in each sequence to correct the inaccuracy arising from preparation of the critical mixtures.The Mg isotope ratios are reported as δ values relative to DSM-3:

Results and discussion
As mentioned, SIMS analysis causes a variably intense instrumental mass fractionation of Mg isotopes.Given that the secondary beam arising from different minerals is unlikely to behave differently during the flight from the source slit to the magnet and the detectors, the IMF should occur within the sample chamber, be it related to the sputtering process or the extraction and subsequent guidance of the secondary beam to the source slit, or both.
During sputtering, one of the main factors able to fractionate isotopes of the same element is the force constant difference between the bonds linking the isotopes of the studied element to the mineral lattice, ΔF = F hard -F soft .Coordination sites with "soft" bonds tend to be preferentially populated with the 'lightest' isotopes, whereas those with "hard" bonds do it with the heaviest ones (e.g., Albarède 2009, p. 53).On collision with primary ions, mineral species with different ΔF but the same 25 Mg/ 24 Mg and 26 Mg/ 24 Mg will release ions of the three isotopes at different rates, thus yielding different isotope ratios that will depend on how the force constant differences of reference material and unknowns compare.
Magnesium isotopes have also shown noticeable IMF when reference materials and minerals belonging to the same mineral species have different chemical compositions, such as often happens in solid solutions between Mg and Fe endmembers (e.g., Kita et al. 2012, Luu et al. 2013, Ushikubo et al. 2013, Fukuda et al. 2020, Richter et al. 2021, etc.).This effect might arise because (i) the concentration of Fe ions affects the distribution of Mg isotope ions between bonds of different force constants, (ii) the variable composition of the ion cloud affects the ionisation efficiency of each isotope differently, or (iii) the variable composition of the ion cloud affects the flight path of the Mg isotopes from the mineral surface to the source slit.As illustrated below, the last hypothesis seems more likely, especially if the effects of the Earth's magnetic field on the ion flight paths are not corrected.

Correction of Earth's magnetic field and the effects of variable chemical composition
The Earth's magnetic field causes 24 Mg + , 25 Mg + and 26 Mg + to follow somewhat divergent horizontal trajectories from the sputtered spot to the source slit, which, being narrower than the secondary beam, truncates each isotope differently (Figure 2a).Consequently, minor variations in the secondary beam trajectory will cause intolerable variations in the measured isotope ratios.In the SHRIMP, the EMF effects can be amended using two horizontal Helmholtz coils surrounding the sample chamber, one above and the other below the secondary beam plane.When the magnetic field created by the current applied to the coils (see above, Sample preparation and SHRIMP analysis) compensates for the local EMF, the three Mg isotopes do not diverge (Figure 2b), thus being equally truncated by the source slit.In this situation, minor shifts in the secondary beam trajectory would have little or no effect on the measured isotope ratios.
These shifts may arise from small errors in the guidance lenses during the pre-analysis adjustments; if so, they will occur randomly without correspondence to the chemical composition.Minor but still perceptible variations in the trajectory of Mg ions might also arise from interaction with variable amount of Fe ions within the ionic cloud resulting from sputtering.To check this idea, we used the three synthetic glasses with the same Mg isotope composition but different Mg/Fe ratio (Table 1).We ran two measurement sessions, with and without EMF correction.Each session consisted of thirty spots of randomly interspersed fragments of each glass plus the San Carlos olivine as reference material measured every four unknowns.Measurement results are shown in Figure 3 and given in Table S2.
The measurements with no EMF correction show large dispersions and the replicates of each material plot along slightly curved and steeper-than-TFL alignments (Figure 3a).Each glass alignment cuts the TFL at different points, the coordinates of which increase with the glass iron content, thus indicating a compositional effect.In contrast, the EMFcorrected measurements plot perfectly aligned to the TFL yielding less dispersion, and the means of the three glasses cannot be distinguished statistically (p-value > 0.14, Table 2).These data suggest that the glasses' bulk compositions affected Mg isotope ratios only when the EMF was not properly neutralised.Given that there is no reason for not expanding this conclusion to minerals, we thus accept as a working hypothesis that the extensive IMF caused by the SHRIMP, either in single-collector or in multiple-collector when the EMF is adequately corrected, does not depend on the chemical composition but primarily relates to the different structures of reference    1 and 3).When the EMF is uncorrected, neither the glasses nor the San Carlos Olivine plot along the TFL, the replicates' variation increases greatly, and the three glasses intersect the TFL at coordinates increasing with the glass iron concentration.See text for details.materials and minerals.This idea finds support from the fact that olivines with Fo 90 (from a peridotite) and Fo 68 (from a mafic granulite) measured correcting the EMF yielded δ 26 Mg values that differ by 0.68‰ (Table 3), and this variation probably arises from the bulk composition differences of their respective host rocks, δ 26 Mg ≈ -0.24‰ in the San Carlos peridotite (mean of mineral compositions reported by Hu et al. 2016) and δ 26 Mg ≈ -0.77‰ in the mafic granulite (Table 3).
Figure 3b also shows that the EMF-corrected analyses still have some dispersion along the TFL, which is prominent in the glasses but still perceptible in the San Carlos olivine.In the glasses, this phenomenon probably reflects the presence of variable amounts of small crystallites (< 1 μm) within the analysed spot.In the minerals, however, the variation must stem from another reason.Excluding variations in the chemical composition because the dispersion happens in all minerals, even those remarkably homogeneous, the most plausible explanations are lattice anisotropy, isotopic heterogeneity or fluctuations in the sputter rate caused by variable primary beam intensity.In most cases, mineral structures are anisotropic; thus, it is logical that the intensity of the matrix effects related to the structure and, hence, the sputtering rate might vary with the crystallographic orientation of the analysed grains.Isotopic heterogeneities would cause the same effect; however, the fact that different specimens of the same mineral, even those equilibrated at 1000 °C, always yield similar dispersion favours the idea of structural anisotropy (constant) over isotopic heterogeneity (variable).Independently of the particular reason, this effect causes the error averaging analyses of the same mineral to always be higher than the error of every single analysis; accordingly, to get a reasonably representative isotope composition of each mineral requires the analysis of at least twenty replicates.

Analyses of minerals and calculation of IMF correction factors
The minerals studied include olivine, clinopyroxene, orthopyroxene, amphibole, spinel, garnet, cordierite and biotite; in this set, MgO ranges from 3 to 50% m/m, mol.MgO/(FeO+MgO) from 0.13 to 0.92, and δ 26 Mg (SN-MC-ICP-MS) from -1.88 to 0.28 (Table 1).Raw SHRIMP data (drift and EMF corrected) show a more considerable δ 26 Mg variation, from -10.1 to 5.7 (Table 3), with spinel and cordierite having the lowest and the highest values, respectively (Figure 4).Amphibole and biotite have Mg isotope ratios comparable to olivine; pyroxenes and garnet are moderately enriched in heaviest isotopes.In the threeisotope diagram (Figure 4), the raw data means of all analysed minerals fit well (R = 0.9998) on a straight line identical to the TFL.The slopecalculated by robust regressionis ß = 0.517, thus matching the Rayleigh fractionation line (Davis et al. 2015) and intermediate between the kinetic and equilibrium fractionation lines calculated by Young and Galy (2004).Table 3. Raw mean SHRIMP measurement results of the studied minerals (the full dataset is given in online supporting information Table S1).Note the large range of values, much higher than the SN-MC-ICP-MS data shown in From the raw data, we calculated IMF correction factors using two different approaches (Table 4): (i) based on the San Carlos olivine: 25 F ol = ( 25 Mg/ 24 Mg mineral) /( 25 Mg/ 24 Mg SC olivine ), (ii) based on SHRIMP to SN-MC-ICP-MS ratios: 25 F tm = ( 25 Mg/ 24 Mg SHRIMP )/( 25 Mg/ 24 Mg SN-MC-ICP-MS) , where subscripts "ol" and "tm" stand for "olivine" and "two methods", respectively.We used the San Carlos olivine as the SHRIMP calibration material but any other well-known mineral can be used equally.
The two methods yielded similar values: plotting 25 F tm against 25 F ol for each analysed specimen yields a regression line with slope =1 and correlation coefficient of 0.993 (Figure 5).This excellent correspondence is most remarkable because the 25 F ol coefficients result from comparing the San Carlos olivine (#Mg = 0.90) with minerals with very different composition (#Mg from 0.92 to 0.13), whereas the 25 F tm coefficients arise from two different measurements on the same mineral concentrate.The coefficients are also robust because different specimens of a given mineral yield the same values independently of the specimen composition.These observations confirm that if the EMF is adequately corrected, the matrix effects overwhelmingly depend on the mineral structure.
Although the two types of coefficients are similar, we preferred the 25 F tm ones (Table 4) because being based on two independent methods seems less prone to errors.The matrix-correction factor for 26 Mn/ 24 Mg are calculated as 26 F tm = 1 + ( 25 F tm -1)/0.517.We tested the IMF correction factors' efficacy in (i) the Ronda peridotite minerals, also measured by SN-MC-ICP-MS but not included in the F tm calculations to avoid tautologies, and (ii) the minerals of the high-T mafic granulite, not measured in that way, but probably isotopically homogeneous because of diffusion (Table 5).The SHRIMP IMF-corrected and the SN-MC-ICP-MS values of Ronda minerals differ by less than 0.3‰, except for the slightly serpentinised olivine, which differs by 0.52‰ probably because the ablated spot contained olivine partially transformed to serpentine, the structural effects of which differ from olivine.The minerals of the mafic granulite, which uncorrected vary from more than 2‰, yielded corrected δ 26 Mg in the range -0.64‰ to -0.97‰, i.e., within 0.3‰ and enclosing the SN-MC-ICP-MS bulk rock analysis, δ 26 Mg = -0.77‰,thus suggesting the F tm -based correction worked.
Table 6 summarises the recommended matrix correction coefficients.They must be considered provisional and amenable to improvement as more data become available, especially those based on a single specimen (garnet, biotite, cordierite).

Conclusions
• The multi-collector SHRIMP microprobe can quickly and precisely measure Mg isotopes in geologically fractionated materials despite the large differences in instrumental mass fractionation (IMF) between minerals inherent to the system.
• The IMF is mostly related to the force constant difference between the bonds linking the isotopes of the studied element to the mineral lattice, that is, the mineral structure and not to the chemical composition, provided the local Earth magnetic field (EMF) is adequately compensated.
• Uncompensated EMF causes the 24 Mg + , 25 Mg + and 26 Mg + trajectories within the sample chamber to diverge so that the source slit truncates them differently.Under these conditions, minor changes in the secondary beam guidance result in huge variations in the measured isotope ratios.The guidance fluctuations may arise from aleatory errors in the secondary beam-positioning lens or because of the variable composition of the ionic cloud resulting from sputtering.
• In the SHRIMP, the EMF was compensated by applying an empirically determined current to two horizontal Helmholtz coils, one above and the other below the plane of the secondary beam.The compensation prevented the ionic paths from diverging, thus cancelling the effects of the secondary beam guidance on the measured isotope ratios.With the EMF adequately compensated, each mineral species causes constant IMF effects, which can be easily corrected by applying suitable fractionation coefficients.
• The standard error measuring Mg isotope ratios in a single spot was typically around 0.007%, whereas the pointto-point error on approximately twenty analyses of a very homogeneous mineral such as San Carlos olivine was always larger, about 0.013%.We attributed this effect to the anisotropy of the mineral lattice.
• Mineral mass-fraction factors can be calculated in two ways: from mineral-to-reference material ratios measured in the same session (here, we used the San Carlos olivine as reference material) or based on the SHRIMP to SN-MC-ICP-MS ratios of the same minerals.These permit correction of the instrumental mass-fractionation caused by the SHRIMP within a ≈ 0.3‰ error range, which may be considered the method's accuracy.4 0 • The correction factors permit analysing different mineral species of variable composition with a single mineral as a reference material, thus simplifying the routine measurement of Mg isotopes in minerals and expanding their applicability to Earth Sciences.

Figure 1 .
Figure 1.The three Mg isotopes diagram.The terrestrial fractionation line (TFL) has a theoretical slope between 0.500 and 0.521, depending on the fractionation model chosen; it is straight except if the mass fractionation law chosen is linear (Davis et al. 2015).Δ 26 Mg = δ 26 Mg measured -ß -1 δ 25 Mg measured represents the excess of radiogenic 26 Mg attributed to the decay 26 Al isotope, characteristic of Al-rich meteoritic materials accreted soon after the Solar nucleosynthesis (see text for details).Terrestrial materials mostly plot along the TFL; unfortunately, instrumental mass fractionation also causes variations along the TFL, which, if uncorrected, mask the effects of geological processes.The star represents Bulk Earth's composition.

Figure 2 .
Figure 2. (a) Trajectories of 24 Mg + , 25 Mg + and 26 Mg + from the sample surface to the source slit.The Earth's magnetic field (EMF) can cause divergent trajectories, so the source slit truncates each isotope beam differently.Therefore, small variations in the secondary beam horizontal guidance will cause intolerable errors in the isotope ratio measurements.(b) If the Helmholtz coils are activated, the EMF can be compensated, eliminating the divergence and improving the results.The current applied to the Helmholtz coils depends on the local magnetic field.

Figure 3 .
Figure 3.The three synthetic glasses measured with the EMF corrected and uncorrected.When the EMF is corrected, the three glasses plot along the terrestrial fractionation line (TFL) and yield statistically indistinguishable δ 25 Mg and δ 26 Mg means despite the large compositional difference (see Tables1 and 3).When the EMF is uncorrected, neither

Figure 4 .
Figure 4.The SHRIMP results on the three-isotope diagram plot along the TFL.The mean δ 25 Mg and δ 26 Mg yielded a correlation coefficient R = 0.9998 with a slope ß = 0.517, characteristic of the Rayleigh mass fractionation law.The crosses represent 2s.Note the wide dispersion along the TFL, δ 26 Mg from -9.7 to 5.9, much higher than the SA-MC-ICP-MS analyses of the same minerals, δ 26 Mg from 1.88 to 0.26, Tables1 and 2. Cordierite and spinel occupy the heavier and lighter isotope-enriched extremes, respectively.The large variation along the TFL is caused by structural differences between the reference material (San Carlos olivine) and the other minerals.See text.

Figure 5 .
Figure 5. Plot of the olivine-based against the two methods-based IMF correction factors.The two kinds of independently calculated correction factors show excellent correspondence, with a correlation coefficient of R = 0.993, thus suggesting the robustness of the correction.

Table 2 .
Two-sample t-test with unequal variances comparing the mean ∂ 25 Mg of the glasses with no FeO (KM1), 5% FeO (KM2) and 10% m/m FeO (KM3) measured with the EFM corrected; we did not find a statistically significant difference in the means of ∂ 25 Mg.The same occurred with ∂ 26 Mg Pr(|T| > |t|) = 0.6744 Pr(T > t) = 0.6628

Table 4 .
Matrix-correction factors.25Mg/ 24Mg* and 25 Mg/ 24 Mg** represent raw SHRIMP data and SN-MC-ICP-MS measurements, respectively.Factors 25 F ol resulted from dividing each data in the SHRIMP column by the San Carlos olivine, 0.126836.Factors 25 F tm resulted from dividing the SHRIMP data by the corresponding ICP-MS data of each mineral

Table 6 .
Recommended matrix correction coefficients.The last column contains the valuesIreland et al.  (1982)determined with the SHRIMP-I, recalculated to be olivine-based instead of kaersutite-based.The authors did not specify whether the pyroxene was ortho-or clinopyroxene.Note the good correspondence with our 25 F tm factors