Nuclear volume and mass dependent fractionation of cerium isotopes

Abstract

Equilibrium cerium isotope fractionations in cerium-bearing minerals and aqueous species are estimated using electronic structure calculations that include both nuclear volume and mass dependent effects. As with europium and uranium, the nuclear volume effect in redox reactions goes in the opposite direction from equilibrium mass-dependent fractionation for ¹⁴²Ce/¹⁴⁰Ce because of the larger nuclear charge radius of the ¹⁴²Ce nucleus. Mass-dependent effects dominate ¹³⁶Ce/¹⁴⁰Ce and ¹³⁸Ce/¹⁴⁰Ce fractionations because ¹³⁶Ce, ¹³⁸Ce, and ¹⁴⁰Ce share very similar nuclear charge radii. ¹⁴²Ce/¹⁴⁰Ce is predicted to be lower in most Ce(IV)-bearing species than in coexisting Ce(III)-bearing species, particularly at high temperatures. However, species with Ce:Zr substitution, such as zirconium-rich compositions along the stetindite-zircon (CeSiO₄-ZrSiO₄) solid solution series and a model Ce-subsituted baddeleyite (CeZr₃O₈), may show higher ¹⁴²Ce/¹⁴⁰Ce at ambient to low-T metamorphic temperatures because of higher effective force constants acting on the smaller, more snug substitution sites. Ce(III)P(V) charge-coupled substitution into zircon is likewise associated with high ¹⁴²Ce/¹⁴⁰Ce relative to other Ce(III) species. ¹³⁶Ce/¹⁴⁰Ce and ¹³⁸Ce/¹⁴⁰Ce fractionations will tend to favor more massive isotopes in Ce(IV)-bearing species, by 0.1–1.0‰ at 25°C and 0.01–0.1‰ at 727°C depending on the species present. The models predict ~0.3‰ higher ¹⁴²Ce/¹⁴⁰Ce in Ce(III)-bearing solution than coexisting Ce(IV)-solids at ambient temperatures, roughly agreeing with measurements. Zircon in equilibrium with typical silicate melts is predicted to be slightly enriched in ¹⁴⁰Ce relative to ¹⁴²Ce, ¹³⁸Ce, and ¹³⁶Ce. Supplementary calculations based on ¹⁴¹Pr-Mössbauer spectroscopy literature suggest somewhat (~1/3) smaller nuclear volume fractionation effects than the electronic structure models.

Introduction

This study uses electronic structure models of cerium compounds, with supplemental information provided by isomer shift data from the ¹⁴¹Pr-Mössbauer spectroscopy literature, to estimate equilibrium cerium-isotope fractionations for geochemically relevant species and simpler analogs. Cerium abundance variations and partitioning have been extensively studied to gain insight into both igneous and low-temperature processes (e.g., Hanson, 1980; Smythe and Brenan, 2015; Zhang and Shields, 2022). Cerium isotope fractionation in nature has also recently become a topic of study in geochemistry (e.g., Nakada et al., 2013a, 2017; Laycock et al., 2016; Hu et al., 2021; Pourkhorsandi et al., 2021; Bonnand et al., 2023; Bai et al., 2024).

A series of studies by Nakada and coworkers (Nakada et al., 2013a, 2013b, 2016, 2017) on fractionation associated with adsorption onto iron- and manganese-oxides and hydroxides are particularly important for motivating the present study. Initially, Nakada et al. (2013a) measured isotopic fractionations between aqueous Ce(III) and adsorbed Ce(III) on ferrihydrite, adsorbed Ce(IV) on δ-MnO₂ (birnessite), and directly precipitated CeO₂ (cerianite). They found higher ¹⁴²Ce/¹⁴⁰Ce ratios in the aqueous phase in all three experiments, with a particularly strong effect of ~0.4‰ for adsorption onto δ-MnO₂. All experiments showed linear relationships between the precipitated fraction and isotopic composition, suggestive of batch equilibrium fractionation. This behavior contrasts with Nd- and Sm-isotope fractionations in δ-MnO₂ adsorption experiments, in which more massive isotopes are preferentially adsorbed or exchanged into the metal oxide phase (Nakada et al., 2013b). Nakada et al. (2017) found an effect of pH and/or aqueous carbonate concentration on the ¹⁴²Ce/¹⁴⁰Ce depletion of adsorbed cerium on δ-MnO₂, and made electronic structure models to estimate mass dependent fractionation factors for aqueous Ce(III) species. More recently, Bonnand et al. (2023) infer a ¹⁴²Ce/¹⁴⁰Ce fractionation of –0.2 to –0.35‰ in aqueous Ce³⁺→Ce⁴⁺ oxidation experiments at room temperature. The weaker fractionation (–0.2‰) is based on a Rayleigh model fit of measured fractionation versus the proportion of cerium oxidized, and is the preferred interpretation by the authors. Notably, these fractionations are also reversed with respect to typical redox reactions at equilibrium. Nakada et al. (2016) and Bai et al. (2024) interpret ¹⁴²Ce/¹⁴⁰Ce measurements of natural iron- and manganese-oxide precipitates on the basis of these experiments and theoretical models. Bai et al. (2024), in particular, conclude that fractionation systematics in a natural weathering profile can be more complex than might be expected from any single fractionation mechanism.

Although the possibility of nuclear volume isotope fractionation has been discussed in the context of measurements (Nakada et al., 2013b, 2017), as well as an initial theoretical study (Schauble, 2023), a focused attempt to predict both the nuclear volume component and mass-dependent components of fractionation in Ce(IV) species relative to Ce(III) species is still lacking. In addition to being motivated by the pioneering measurements and reconnaissance theoretical modeling, a focused theoretical study of cerium isotope fractionation has promise because the ¹⁴²Ce isotope is unique among the stable cerium isotopes in having a larger nuclear charge radius, whereas ¹³⁶Ce, ¹³⁸Ce, and ¹⁴⁰Ce have nearly the same nuclear charge radii (e.g., Angeli and Marinova, 2013; Fricke and Heilig, 2004). This indicates that equilibrium nuclear volume fractionations could generate measurable, diagnostic mass-independent signatures in ¹⁴²Ce/¹⁴⁰Ce vs. other isotope pairs. The 145.1 keV ¹⁴¹Pr-Mössbauer system, adjacent to cerium in the periodic table, provides an independent means to estimate nuclear volume fractionation effects. Although the large charge radius of ¹⁴²Ce relative to other cerium isotopes underpins the potential significance of nuclear volume fractionation in this system, it should be noted that the deviation is only roughly half as large as between ¹⁵¹Eu and ¹⁵³Eu (δ<r²> ≈ ~0.3 fm² vs. ~0.6 fm²; Angeli and Marinova, 2013). This will tend to make nuclear volume isotope fractionation in cerium more subtle than in europium (Schauble, 2023), likely by a roughly similar 1:2 factor.

Methods

The total equilibrium cerium isotope fractionation factor α_1xx–140 (where 1xx indicates one of the other Ce isotopes) between two substances CeX and CeZ is defined as:

  

$${\alpha }_{1xx–140}=\frac{{\left({}^{1\mathit{xx}}\mathit{Ce}/{}^{140}\mathit{Ce}\right)}_{CeX}}{{\left({}^{1\mathit{xx}}\mathit{Ce}/{}^{140}\mathit{Ce}\right)}_{CeZ}}$$

In this study, α_1xx–140 is estimated by combining mass dependent and nuclear volume components:

  

$$\begin{array}{c}\hfill ln{\alpha }_{1xx–140}=ln{\alpha }_{1xx–140}^{NV}+ln{\alpha }_{1xx–140}^{\mathit{MD}}\end{array}$$(1)

(Schauble, 2007). This relationship is a simplification of the one proposed by Bigeleisen (1996), omitting the ‘BOELE’ term (a mass-dependent correction to the Born-Oppenheimer approximation), hyperfine, and anharmonic effects. The omitted terms may not be significant given the uncertainties in the mass dependent and nuclear volume components (e.g., Bigeleisen, 1996; Zhang and Liu, 2018). As in the previous paper on europium isotope fractionation (Schauble, 2023), here the general fractionation factor α is used rather than fractionation relative to atomic vapor (β) because the nuclear volume component of fractionation in atomic vapor is not calculated as easily as the mass-dependent component, so a different material (endmember CePO₄-monazite) is used as the standard of comparison instead.

Other substances studied include cerianite-Ce (CeO₂), stetindite (CeSiO₄—a zircon-like structure), Ce-bastnaesite (CeCO₃F), cerium sesquioxide (hexagonal A-type Ce₂O₃), cerium(III) iodide nonahydrate (Ce[H₂O]₉.I₃) representing aqueous cerium at low pH, and cerium(IV) sulfate tetrahydrate as an example of Ce(IV)-bearing salts and hydrates. Because rare-earth element budgets in rocks are often dominated by trace substitutions into other phases and REE mixing, Ce-substituted zircon (Ce(IV):ZrSiO₄ and Ce(III)P(V):ZrSiO₄), baddeleyite (Ce:ZrO₂), clinopyroxene (Ce:CaMgSi₂O₆, in which the Ce³⁺/Ca²⁺ charge mismatch is balanced by Al³⁺/Si⁴⁺ substitution), and apatite (Ce:Ca₅(PO₄)₃F, in which Ce³⁺/Ca²⁺ charge mismatch is balanced by Na⁺/Ca²⁺ substitution) are also studied. Y- and La-rich variants of bastnaesite are also modeled. In the zircon substitutions, cerium concentration effects are investigated by modeling single cerium atom substitutions into progressively larger unit cells and supercells (e.g., Wang et al., 2017, on calcium isotope fractionation in orthpyroxene). Birnessite (often abbreviated as δ-MnO₂, but actually a mixed-valence defect structure containing significant water, sodium, & etc. in natural samples) would be a valuable addition because of its significance in soil and seafloor rare earth element partitioning, but its chemistry is too complex to model adequately in this study.

Nuclear volume fractionation

The nuclear volume component α^NV of the field shift isotope fractionation is determined by the electronic energy difference (per cerium atom) between isotopically substituted species of two substances CeX and CeZ:

  

$$ln{\alpha }_{1xx–140}^{\mathit{NV}}=\frac{\left(E\left[{}^{1\mathit{xx}}\mathit{CeZ}\right]–E\left[{}^{140}\mathit{CeZ}\right]\right)–\left(E\left[{}^{1\mathit{xx}}\mathit{CeX}\right]–E\left[{}^{140}\mathit{CeX}\right]\right)}{RT}$$(2)

Here is the molar gas constant and T is the absolute temperature. In contrast to mass-dependent fractionation, where the vibrational energies of the more massive ¹⁴²Ce-isotopologues are lower (i.e., they are more stable thermodynamically) relative to less massive cerium isotopologues, the nuclear volume component of the field shift effect leads to higher energies in species with the larger (more voluminous) ¹⁴²Ce-nucleus.

In an earlier europium-focused study (Schauble, 2023) the nuclear volume effect on energy was estimated primarily through scaling tabulated ¹⁵¹Eu-Mössbauer isomer shifts (converted to energy units ΔE_isomer = (hν_γ/c)·ΔIS) by the ratio of the isotopic difference in ground-state nuclear mean-squared radii to the isomeric change in the mean-squared ¹⁵¹Eu radius:

  

$$(E\left[{}^{153}\mathit{EuZ}\right]–E\left[{}^{151}\mathit{EuZ}\right])–(E\left[{}^{153}\mathit{EuX}\right]–E\left[{}^{151}\mathit{EuX}\right])={\mathrm{\Delta }E}_{\mathit{isomer}}\frac{⟨r_{153}^2⟩–⟨r_{151}^2⟩}{⟨r_{151*}^2⟩–⟨r_{151}^2⟩}$$(3)

Here 151* represents the excited 7/2⁺ nuclear state of ¹⁵¹Eu at 21.5 keV used in Mössbauer spectroscopy, and ΔE_isomer the difference in the isomer shifts of the two substances in units of energy. In that study the same method was adapted to estimate nuclear volume effects in cerium by scaling measured ¹⁴¹Pr-Mössbauer isomer shifts, assuming that electron density changes in praseodymium-bearing species are identical to their cerium-bearing analogs, i.e.,

  

$$(E\left[{}^{1\mathit{xx}}\mathit{CeZ}\right]–E\left[{}^{140}\mathit{CeZ}\right])–(E\left[{}^{1\mathit{xx}}\mathit{CeX}\right]–E\left[{}^{140}\mathit{CeX}\right])={\mathrm{\Delta }E}_{\mathit{isomer}}\frac{⟨r_{1xx}^2⟩–⟨r_{140}^2⟩}{⟨r_{141*}^2⟩–⟨r_{141}^2⟩}$$(4)

where <r²_141*> and <r²₁₄₁> are tabulated mean-squared charge radii for ¹⁴¹Pr in the 145.4 keV excited state and ground state, respectively. Schauble (2023) estimate a value of 21 ± 2 for the mean square charge radius ratio, which will also be used in the present work. Although cerium and praseodymium are adjacent in the lanthanide series and share in common an empty or partially filled 4f electron shell in their chemically accessible oxidation states, this method is necessarily crude because the number of electrons, bonding, ionic radii, effective charges & etc. are not identical. In addition, published ¹⁴¹Pr-Mössbauer measurements are sparser and less precise than for ¹⁵¹Eu, limiting the scope of materials that can be studied via scaled isomer shifts. For these reasons, this scaling method is used mainly as a check on electronic structure calculations in the present study.

Here the primary method of estimating the nuclear volume component of equilibrium fractionation is first-principles electronic structure theory, combined with calibrated Density Functional Theory calculations using Projector Augmented Waves (DFT-PAW; Schauble, 2013). The method is based on calculating ground state relativistic electronic structures for a calibration set of molecules, that are then compared to Fermi contact densities |Ψ(0)²| (i.e., electron densities at the center of the nucleus) calculated at the DFT-PAW level for the same species. Relativistic all-electron models allow the direct determination of the nuclear volume effect of isotope substitution on the electronic energy, whereas DFT-PAW calculations are well-suited for minerals and other species that are best modeled with a periodic boundary condition. The calibration set of molecules are modeled at the Dirac Hartree-Fock level using an exact two component (X2C) Hamiltonian, as implemented in the 2019 version of the DIRAC software package (Gomes et al., 2019). In the case of open-shell atoms and molecules (i.e., Ce³⁺ vapor and Ce(III)-bearing species, which have a [Xe]4f¹ electronic configuration in the limit of ionic bonding, as well as Ce²⁺ vapor—[Xe]4f²), the energies of individual spin-orbit states can be resolved from the average-of-configuration open-shell Hartree-Fock solution via a subsequent configuration interaction calculation (Visser et al., 1992). In general, however, it is found that there is little difference between ¹⁴²Ce-¹⁴⁰Ce isotope effects in the resolved ground state relative to the 4f¹ configurational average in the Ce(III)-bearing calibration species (0.01 J/mol or less), in the vapor-phase ion Ce³⁺ (<0.01 J/mol), or between the resolved ³H₄ ground state of Ce²⁺ ([Xe]4f²) and the 4f² configurational average (0.05 J/mol). The calculations use all-electron double-zeta quality basis sets for Ce (Gomes et al., 2010) and the cc-pVDZ basis set family for other elements (Dunning, 1989; Woon and Dunning, 1993). Test calculations with larger (and presumably more accurate) basis sets indicate that models using the Dyall double-zeta + cc-pVDZ basis sets are sufficiently accurate, given other sources of error. Nuclear charge radii tabulated by Angeli and Marinova (2013) are used for the stable Ce isotopes, with a mean squared radius difference of 0.286 fm² between ¹⁴⁰Ce and ¹⁴²Ce (note that Angeli and Marinova, 2013 quote a slightly smaller difference, 0.281 fm², than their tabulated radii yield; the discrepancy is larger than would be consistent with a rounding error in the tabulation, but the radius difference is still well within the range found elsewhere, e.g., Fricke and Heilig, 2004; Nadjakov et al., 1994, revised 2015 at http://cdfe.sinp.msu.ru/cgi-bin/muh/radchartnucl.cgi?zmin=58&zmax=58&tdata=123456). DFT-PAW Fermi contact densities |Ψ(0)²| are calculated using the PBE gradient-corrected functional of Perdew et al. (1996) as implemented in the Abinit software package (https://www.abinit.org/; Gonze et al., 2016). A PAW dataset for with a small cutoff radius for oxygen is taken from the original ATOMPAW library (http://users.wfu.edu/natalie/papers/pwpaw/periodictable/oldperiodictable) in order to avoid overlapping core volumes in species with short bond lengths. The cerium PAW dataset is taken from the family developed by Topsakal and Wentzcovitch (2014). For other elements, PAW data sets from the JTH library are used (https://www.abinit.org/downloads/PAW2; this includes updates of the library described in Jollet et al., 2014). Plane wave basis sets are extended to a kinetic energy cutoff of 40 hartrees (1088 eV), with the fine grid cutoff set at 100 hartrees (2721 eV). A Hubbard U correction of 4.5 eV is applied to the 4f orbitals of cerium, this correction is within the ~2.5–6 eV range chosen by other recent studies of cerium oxide species (e.g., Fabris et al., 2005; Pourovskii et al., 2007; Da Silva et al., 2007; Adelstein et al., 2011; Amadon, 2012; Kanoun et al., 2012; Qi et al., 2013) and cerium nitride (Topsakal and Wentzcovitch, 2014). Crystal structures are optimized until residual forces acting on each atom are less than 1 × 10^–4 hartree/a₀ (8 × 10^–12 N) and cartesian components of the lattice stress tensor are less than 1 × 10^–6 hartree/a₀³ (3 × 10⁷ N/m² or 300 bar). Vacuum surrounding molecular species is approximated by placing each molecule in a cubic periodic box 24 a₀ (12.7 Å) on a side.

The calibration set includes a mix of real and hypothetical species representing the Ce(IV) and Ce(III) oxidation states: CeF₄, Ce(OH)₄, CeCl₆^2–, Ce(H₂O)₉⁴⁺, CeF₄^–, Ce(H₂O)₆³⁺, and Ce(H₂O)₉³⁺. The choice of species is a compromise between computational tractability, relevance to natural environments, and wide variation in bonding properties and coordination. The molecular structure for CeF₄ is taken from measurements by Giricheva et al. (1998); Ce(OH)₄ from the theoretical and matrix isolation study of Fang et al. (2016); CeF₄^– from electronic structure calculations of Vent-Schmidt et al. (2016); and CeCl₆^2–, Ce(H₂O)₉⁴⁺, Ce(H₂O)₆³⁺, and Ce(H₂O)₉³⁺ from in vacuo geometry optimizations at the PBE/def2-TZVP level in the present study using the GAMESS and ORCA software packages (Schmidt et al., 1993; Neese, 2012; for details on the def2-TZVP basis set family see Dolg et al., 1989; Gulde et al., 2012; and Weigend and Ahlrichs, 2005). The calibration calculations (Table 1; Fig. 1) indicate very close grouping of ¹⁴²Ce-¹⁴⁰Ce energy differences by oxidation state. Ce(IV) species all have higher electron densities and larger ¹⁴²Ce-¹⁴⁰Ce isotopic energy shifts than Ce(III) species—in this sense the results may not be very sensitive to the particular choice of these calibration molecules. The slope of the calibration line indicates that a shift of 1 e^–/a₀³ in the Fermi contact density corresponds to a nuclear volume energy effect of 0.045 ± 0.003 J/mol (1σ) between ¹⁴²Ce and ¹⁴⁰Ce, but only –0.005 ± 0.0003 J/mol between ¹³⁸Ce and ¹⁴⁰Ce and between ¹³⁶Ce and ¹⁴⁰Ce. The quoted uncertainty is based solely on the statistical quality of the linear regression, and does not incorporate uncertainties in tabulated nuclear charge radii or systematic errors in the electronic structure models. It is notable that this calibration implies a higher sensitivity of ¹⁴²Ce-¹⁴⁰Ce field shift energies to the DFT-PAW Fermi contact density than is implied by scaled ¹⁴¹Pr Mössbauer isomer shifts, which suggest approximately 0.030 ± 0.03 J/mol per e^–/a₀³. The difference lies mainly in the measured difference of ~0.5 mm/sec (0.02 J/mol) in the ¹⁴¹Pr isomer shift of the Pr(IV) fluorides CsPrF₅ and Cs₂PrF₆, relative to PrO₂ (Kapfhammer et al., 1971) in the three published tabulations of the difference in ^141*Pr-¹⁴¹Pr nuclear charge radii used in Schauble (2023) to calculate the scale factor (Kapfhammer et al., 1971; Groves et al., 1973; Kalvius and Shenoy, 1974). The present DFT-PAW models find no significant difference in electron densities at the cerium nucleus in CeO₂ vs. CsCeF₅ or Cs₂CeF₆, and only a small difference between PrO₂ and Cs₂PrF₆. Nor do the all-electron relativistic electronic structure models find large differences among the gas-phase Ce^IV species CeF₄, Ce(OH)₄, CeCl₆^2–, and Ce(H₂O)₉⁴⁺. In vacuo all-electron relativistic electronic structure models of field shifts in PrF₄, PrF₄^–, Pr(H₂O)₆³⁺, and Pr(H₂O)₈⁴⁺ are very similar to their Ce-analogues (on a J/mol/fm² basis), which confirms a key assumption of the ¹⁴¹Pr-Mössbauer method.

Table 1.

Calibration data for the DFT-PAW method

Molecules	E[142Ce]-E[140Ce]	|Ψ(0)2|
	(J/mol)	(e–/a03)
CeF4	11820.98	502.29
Ce(OH)4	11820.99	502.18
CeCl62–	11821.02	497.52
Ce(H2O)94+	11821.03	502.07
CeF4–	11819.94	478.43
Ce(H2O)63+	11819.93	477.39
Ce(H2O)93+	11819.91	477.21
Ions
Ce2+	11818.83	432.84
Ce3+	11819.78	474.38
Ce4+	11820.95	533.24

Estimated energy differences between ¹⁴²Ce and ¹⁴⁰Ce isotopologues calculated with all-electron relativistic models are compared to Fermi contact densities (|Ψ(0)²|) estimated with density functional theory. Energy differences for Ce(II) and Ce(III) species include spin-orbit coupling for the resolved ground state.

Fig. 1.

Electron density at the cerium nucleus for Ce-bearing ions and molecules, from density functional theory using projector augmented waves (DFT-PAW) versus all-electron relativistic estimates of the field shift between ¹⁴²Ce and ¹⁴⁰Ce in the same species. The solid line is a best fit linear regression to molecules, and the dotted line is the corresponding fit to atomic ions.

An alternate calculation of the difference in ^141*Pr-¹⁴¹Pr mean-squared nuclear charge radii in Groves et al. (1973), using PrO₂ to represent the Pr(IV) end member rather than CsPrF₅ and Cs₂PrF₆, yields 0.008 fm², versus their main published value of 0.013 fm². This alternate result is fully compatible with the present DFT-PAW calibration. However, this alternate calculation does not resolve the question of whether the discrepancy arises from errors in Mössbauer measurements in Pr(IV)-bearing fluorides relative to PrO₂, from an unexpectedly large difference in electronic structure between PrO₂ and CeO₂, or from shortcomings in the present all-electron and DFT-PAW electronic structure models. Because of this discrepancy, it is expected that the ¹⁴¹Pr-Mössbauer scale factor method (using 21 ± 2) will predict ~1/3 smaller field shift fractionations than the DFT-PAW method.

The vapor phase atomic ions Ce²⁺, Ce³⁺, and Ce⁴⁺ were also modeled for comparison with Ce-bearing compounds. All-electron relativistic calculations indicate a similar ¹⁴²Ce-¹⁴⁰Ce energy shift [e.g., 1.2 J/mol for atomic Ce⁴⁺-Ce³⁺ vs. 1.1 J/mol for Ce(H₂O)₉⁴⁺-Ce(H₂O)₉³⁺], consistent with predominantly ionic REE-O and REE-F bonding, and/or limited bonding effects on the 4f and 6s orbital occupations and structures. Interestingly, calculated DFT-PAW Fermi contact densities show a much higher sensitivity for atomic ions than molecules, changing by 59 e^–/a₀³ between Ce⁴⁺ vs. Ce³⁺ but only 25 e^–/a₀³ between Ce(H₂O)₉⁴⁺ vs. Ce(H₂O)₉³⁺. This means that the DFT-PAW method would yield very different (and likely incorrect) estimates of nuclear field shift fractionations if calibrated only with atomic ions, rather than focusing on more realistic molecular species.

Based on the recent analyses and tabulations of Ce-isotope nuclear charge radii by Fricke and Heilig (2004), Cheal et al. (2003), and Angeli and Marinova (2013), uncertainties in charge radii of the cerium isotopes are likely to be smaller than the regression uncertainty, at least for ¹⁴²Ce-¹⁴⁰Ce and ¹⁴²Ce relative to the other isotopes. Charge radius differences between ¹³⁶Ce, ¹³⁸Ce, and ¹⁴⁰Ce are so small that the relative uncertainties as a fraction of the (small) estimated field shift component between these three isotopes could be substantial.

Mössbauer data

Measured ¹⁴¹Pr-Mössbauer isomer shifts used in the present work are tabulated in Table 2, and are drawn from the studies of Kapfhammer et al. (1971), Groves et al. (1973), and Moolenaar et al. (1996) (including only the CeF₃-source data from the latter study). In contrast to ¹⁵¹Eu, the ¹⁴¹Pr-Mössbauer system is much less well studied, and the literature sources have both limited overlap and relatively large measurement uncertainties. The Kapfhammer et al. (1971), Groves et al. (1973), and Moolenaar et al. (1996) datasets show good consistency, based on four phases (PrO₂, Pr₆O₁₁, Pr₂O₃, and PrF₃) measured in common. Cross-correlation of the isomer shifts for these phases (ignoring measurement uncertainties) yields R² > 0.98, and regression slopes of within 8% of unity (with Groves et al., 1973, showing slightly more apparent sensitivity than the other two). Because Moolenaar et al. (1996) used a CeF₃ source while the others used CeO₂, there is an offset of ~0.84 mm/sec between reported isomer shifts. The strong correlation suggests that the three datasets can be used interchangeably after correcting for this offset. Isomer shift data for the crystals PrO₂, Pr(OH)₃, Pr₂O₃ (in the cubic C-type structure), and metallic Pr are used here.

Table 2.

¹⁴¹Pr Mössbauer isomer shifts vs. CeO₂

	Isomer shift	Uncertainty	Other studies	Scaled 142Ce-140Ce isotope shift	103 ln αNV142–140 at 298.15 K (vs. CeO2)
	(mm/sec)			(J/mol)
Pr(IV) species
PrO2	–0.06	0.02	0.0a, 0.02b	–0.06	0.024
Pr(III) species
Pr(OH)3	–0.74	0.02	–0.89a	–0.73	0.293
Pr2O3	–0.80	0.03	–0.80a, –0.78b	–0.79	0.317
Pr(0) species
Pr-metal	–0.83	0.06		–0.82	0.329

All data are from Kapfhammer et al. (1971) unless otherwise specified.

^a Data from Groves et al. (1973)

^b Data from Moolenaar et al. (1996), corrected to the CeO₂ reference frame by subtracting 0.84 mm/sec.

Moolenaar et al. (1996) note a strong dependence of the isomer shift on the oxidation state of praseodymium, particularly among Pr(III) species, which range only from 0–0.2 mm/sec relative to praseodymium in a CeF₃ host crystal. Pr(IV) species appear to range more widely (0.8–1.5 mm/sec), though this range includes only one oxide (PrO₂), along with two geochemically exotic cesium praseodymium(IV) fluoride species, CsPrF₅ and Cs₂PrF₆. Unfortunately, isomer shifts for CsPrF₅ and Cs₂PrF₆ appear only to have been measured once (Kapfhammer et al., 1971), despite their importance for calibrating the charge radius difference between ¹⁴¹Pr and ^141*Pr.

A fourth ¹⁴¹Pr Mössbauer study by Bent et al. (1971) shows a similar general correlation between oxidation state and isomer shift, but appears to be inconsistent with the others for one or more substances, and data from that study is not used here. This inconsistency was not fully appreciated during preparation of an initial Ce-isotope investigation (Schauble, 2023), in which the Bent et al. (1971) isomer shift for PrPO₄-monazite was used to represent typical Ce(III) species. However, their reported isomer shift between PrO₂ and PrPO₄-monazite (0.72 ± 0.08 mm/sec, 0.017 ± 0.002 J/mol) is similar to the shift between PrO₂ and Pr₂O₃ and from Kapfhammer et al. (1971) (0.74 ± 0.04 mm/sec, 0.017 ± 0.001 J/mol), and so the choice makes little practical difference for comparisons between that work and the present study.

Mass dependent fractionation

The mass dependent component of fractionation, α^MD, is determined by the reduction in vibrational frequencies when a more massive isotope is substituted for a less massive one. In the limit of harmonic vibrations, a simplified formula for molecules (Urey, 1947) and its extension to phonons in crystals (Elcombe and Hulston, 1975) can be used to estimate this component. Because precise measurements of these vibrational frequency shifts aren’t available, and (unlike europium) Nuclear Resonance Inelastic Xray Scattering (NRIXS) measurements are not yet feasible for either cerium- or praseodymium-bearing species, it is necessary to rely on some type of model to approximate them.

Instead of using vibrational frequencies, mass dependent fractionation can also be estimated directly from force constants (Bigeleisen and Mayer, 1947), following a similar procedure to that described in Widanagamage et al. (2014). In this method, Ce atoms in relaxed structures are displaced from their minimum-energy position in each of three orthogonal Cartesian directions, and the resulting energy vs. displacement relationship is used to numerically estimate the force constant (spring constant) acting on that atom. Mass dependent fractionation can then be calculated from the force constants using an adapted form of the expression given by Bigeleisen and Mayer (1947):

  

$$\begin{array}{c}\hfill {\beta }_{1xx–140}^{MD}\approx 1+\left(\frac{h^2}{96{\pi }^2{k}_B^2{T}^2}\right)\left(\frac{m^{1xx}Ce–m^{140}Ce}{m^{1xx}Ce\times m^{140}Ce}\right)A\end{array}$$(5)

Here β^MD_1xx–140 is the mass dependent fractionation factor between a substance and atomic vapor, N_A is the Avogadro constant, k_B is the Boltzmann constant, T is the absolute temperature, m^1xxCe and m¹⁴⁰Ce are isotopic masses, and A is the sum of force constants opposing displacement of a cerium atom in three perpendicular directions. β^MD_142–140 is then converted to fractionation relative to CePO₄ monazite using:

  

$$\begin{array}{c}{\alpha }_{1\mathit{xx}–140}^{\mathit{MD}}[X]={\beta }_{1\mathit{xx}–140}^{\mathit{MD}}[X]/{\beta }_{1\mathit{xx}–140}^{\mathit{MD}}[{\mathit{CePO}}_4]\hfill \end{array}$$(6)

Force constants and phonon frequencies for Ce-bearing species are estimated with DFT calculations, performed using the PBE functional (Perdew et al., 1996). The same Projector Augmented Wave (PAW) dataset from the REE library generated by Topsakal and Wentzcovitch (2014) is used for calculating mass-dependent fractionation in Ce(IV)-bearing species as was used for calculating Fermi contact densities. However, it proved difficult to achieve model convergence with Ce(III)-bearing species using this dataset, and an alternative PAW dataset from the PSLibrary collection (Ce.pbe-spdn-kjpaw_psl.1.0.0.UPF; Dal Corso, 2014) was used instead. The PSLibrary PAW dataset treats the 4f electron of Ce³⁺ as part of the non-reactive core, simplifying model parameterization and greatly improving DFT model convergence behavior. Cross-testing with a PSLibrary PAW dataset (Ce.pbe-spdfn-kjpaw_psl.1.0.0.UPF) that treats 4f electrons as valence (so it can be applied consistently to both Ce(III) and Ce(IV)-species) for hexagonal A-type Ce₂O₃, rhombohedral CeOF (chosen for this test as a tractably simple Ce(III) species), CeO₂-cerianite, and (Ce_0.5Zr_0.5)SiO₄-zircon indicates agreement in α^MD_142–140 within 0.09‰ at 298 K, and within 0.01‰ at 1000 K, between Ce(III)- and Ce(IV)-bearing phases. The PSLibrary 4f-valence PAW dataset requires such a high cutoff energy (~200 Rydberg, 2720 eV) that it is unfortunately not practical to use it to model mass-dependent fractionation in more complex materials.

A combination of PAW and ultrasoft pseudopotentials is used to represent other elements, often following recommendations of the Standard Solid State Pseudopotential project (https://www.materialscloud.org/discover/sssp). All mass dependent fractionations reported here are based on models with plane wave basis sets generated at a kinetic energy cutoff of 75 Rydberg (1020 eV). A cutoff of 600 Rydberg (8163 eV) was used for the charge density and potential. DFT force constant and phonon frequency calculations are made with the Quantum Espresso software package (Giannozzi et al., 2009; https://www.quantum-espresso.org/). Crystal structures are relaxed (optimized) to the nearest local energy minimum before calculating vibrational frequencies and/or force constants.

Correlation of calculated phonon mode frequencies with literature Raman, infrared, and inelastic neutron scattering measurements on CeO₂-cerianite (Clausen et al., 1987), A-type Ce₂O₃ (Sethi et al., 2019), and CePO₄-monazite (from the compilation of Lalla et al., 2021) indicates that the present models systematically underestimate frequencies by 7.5 ± 1% (1 s.e.). For CePO₄-monazite, only frequencies less than 500 cm^–1 are included in the correlation—higher-frequency modes are expected to be dominated by the internal vibrations of phosphate groups and have little direct impact on Ce-isotope fractionation despite carrying the lion’s share of the statistical weight in a linear regression. To correct the apparent systematic underestimation of vibrational frequencies, a scale factor of 1.075 is applied in the calculation of the mass-dependent component of fractionation from phonons, and an equivalent scale factor of 1.075² = 1.1556 is applied to directly calculated force constants. The frequency scale factor is roughly consistent with those previously determined for strontium- and calcium-bearing minerals (e.g., Widanagamage et al., 2014; Antonelli et al., 2019) for models constructed with very similar parameters. Comparing observed frequencies for molecular CeF₄ in noble-gas matrices (Ne and Ar; Bukhmarina et al., 1992 as tabulated in Giricheva et al., 1998) vs. gas-phase calculations using the PBE functional and the atom-centered def2-TZVP basis set family also suggests a frequency scale factor of ~5–7%.

In order to more accurately compare mass dependent fractionations estimated with the force constant and phonon methods, a simplified form of the approximation of Dauphas et al. (2017) is applied to force constant-based models. In this method, the curvature of ln β^MD vs. 1/T² is estimated via a polynomial regression of phonon-based fractionation factors over a temperature range from 250 K (–23°C) to 5000 K. Taken to 2^nd order in 1/T², this yields the relationship

  

$$\begin{array}{c}1000ln{\beta }_{142–140}^{\mathit{MD}}=\frac{A_1}{T^2}+B_{\mathit{fit}}\left(\frac{{A_1}^2}{T^4}\right)\hfill \\ B_{\mathit{fit}}=\mathrm{–0.0306}\pm 0.0014\hfill \end{array}$$(7)

This polynomial reproduces phonon-based mass dependent ¹⁴²Ce/¹⁴⁰Ce fractionations within 0.05‰ at temperatures above 263 K (–10°C), and 0.02‰ above 333 K (60°C), errors of smaller magnitude than other sources of uncertainty, such as the choice of PAW dataset. It seems reasonable to assume that the equation will approximate nonlinearity vs. 1/T² for the force-constant based estimates with similar accuracy.

The mass dependent components of ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce fractionations are determined by scaling estimated ¹⁴²Ce/¹⁴⁰Ce fractionations using the high temperature limit of the equilibrium mass scaling relationship (e.g., Young et al., 2002):

  

$$\begin{array}{c}\frac{ln{\beta }_{1\mathit{xx}–140}^{\mathit{MD}}}{ln{\beta }_{142–140}^{\mathit{MD}}}=\frac{1/m{}^{1\mathit{xx}}\mathit{Ce}–1/m{}^{140}\mathit{Ce}}{1/m{}^{142}\mathit{Ce}–1/m{}^{140}\mathit{Ce}}\hfill \end{array}$$(8)

For ln β^MD_138–140/ln β^MD_142–140 the ratio is –1.0268, and for ln β^MD_136–140/ln β^MD_142–140 the ratio is –2.0835.

Molecular and crystal structures modeled in this study are listed in Supplementary Material.

Results

Estimated nuclear volume and mass dependent components of equilibrium ¹⁴²Ce/¹⁴⁰Ce, ¹³⁸Ce/¹⁴⁰Ce, and ¹³⁶Ce/¹⁴⁰Ce fractionation are shown in Tables 3 and 4, and net fractionations are shown in Table 5. These are also plotted in Fig. 2. Fitted equations for calculating fractionation components as functions of temperature are given in Table 6.

Table 3.

Calculated DFT-PAW Fermi contact densities, field shifts, and field-shift fractionation factors for cerium-bearing species relative to CePO₄ monazite

		Field shifts (J/mol)			103 ln αNV at 298.15 K			103 ln αNV at 1000 K
Ce(IV) species	|Ψ(0)2| (e–/a03)	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce
CeSiO4 - stetindite	504.7	1.2716	–0.148	–0.139	–0.513	0.060	0.056	–0.153	0.018	0.017
Ce1/2Zr1/2SiO4 - zircon	507.2	1.3851	–0.161	–0.151	–0.559	0.065	0.061	–0.167	0.019	0.018
Ce1/4Zr3/4SiO4 - zircon	508.3	1.4353	–0.167	–0.157	–0.579	0.067	0.063	–0.173	0.020	0.019
Ce1/8Zr7/8SiO4 - zircon	508.5	1.4450	–0.168	–0.158	–0.583	0.068	0.064	–0.174	0.020	0.019
CeO2 - cerianite	502.4	1.1666	–0.135	–0.127	–0.471	0.055	0.051	–0.140	0.016	0.015
Ce1/4Zr3/4O2 - baddeleyite	505.3	1.2972	–0.151	–0.142	–0.523	0.061	0.057	–0.156	0.018	0.017
Ce(SO4)2(H2O)4	502.3	1.1614	–0.135	–0.127	–0.469	0.054	0.051	–0.140	0.016	0.015
Ce(III) species
CePO4 - monazite	476.7	0	0	0	0	0	0	0	0	0
Ce1/4Zr3/4P1/4Si3/4O4 - zircon	479.9	0.1455	–0.017	–0.016	–0.059	0.007	0.006	–0.017	0.002	0.002
Ce1/2Na1/2Ca4(PO4)3F - apatite	477.5	0.0329	–0.004	–0.004	–0.013	0.002	0.001	–0.004	0.000	0.000
Ce2O3 A-type (hexagonal)	478.1	0.0615	–0.007	–0.007	–0.025	0.003	0.003	–0.007	0.001	0.001
Ce1/4Ca3/4MgAl1/4Si7/4O6 - diopside	476.8	0.0004	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Ce(CO3)F - bastnaesite	476.7	–0.0029	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000
La5/6Ce1/6(CO3)F - bastnaesite	476.7	–0.0041	0.000	0.000	0.002	0.000	0.000	0.000	0.000	0.000
Y5/6Ce1/6(CO3)F - bastnaesite	476.8	0.0032	0.000	0.000	–0.001	0.000	0.000	0.000	0.000	0.000
Ce(H2O)9.I3	477.9	0.0516	–0.006	–0.006	–0.021	0.002	0.002	–0.006	0.001	0.001

Table 4.

Calculated equilibrium mass dependent fractionation factors for cerium-bearing species relative to CePO₄ monazite

		103 ln αMD at 298.15 K			103 ln αMD at 1000 K
Ce(IV) species	Method	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce
CeSiO4 - stetindite	Phonon	0.392	–0.402	–0.816	0.037	–0.038	–0.077
Ce1/2Zr1/2SiO4 - zircon	Phonon	0.771	–0.791	–1.605	0.073	–0.075	–0.153
Ce1/4Zr3/4SiO4 - zircon	Phonon	1.046	–1.074	–2.180	0.100	–0.102	–0.208
Ce1/8Zr7/8SiO4 - zircon	Force constant	0.892	–0.916	–1.859	0.086	–0.088	–0.179
CeO2 - cerianite	Phonon	0.304	–0.312	–0.633	0.027	–0.028	–0.056
Ce1/4Zr3/4O2 - baddeleyite	Phonon	0.660	–0.678	–1.376	0.062	–0.063	–0.128
Ce(SO4)2(H2O)4	Force constant	0.404	–0.415	–0.843	0.038	–0.039	–0.080
Ce(III) species
CePO4 - monazite	Phonon	0	0	0	0	0	0
Ce1/4Zr3/4P1/4Si3/4O4 - zircon	Force constant	0.853	–0.876	–1.778	0.082	–0.084	–0.171
Ce1/2Na1/2Ca4(PO4)3F - apatite	Force constant	0.153	–0.157	–0.318	0.014	–0.014	–0.029
Ce2O3 A-type (hexagonal)	Phonon	0.141	–0.145	–0.293	0.012	–0.013	–0.026
Ce1/4Ca3/4MgAl1/4Si7/4O6 - diopside	Force constant	0.180	–0.185	–0.375	0.017	–0.017	–0.035
Ce(CO3)F - bastnaesite	Phonon	0.007	–0.007	–0.014	0.001	–0.001	–0.001
La5/6Ce1/6(CO3)F - bastnaesite	Force constant	–0.063	0.065	0.132	–0.006	0.007	0.013
Y5/6Ce1/6(CO3)F - bastnaesite	Force constant	0.083	–0.085	–0.172	0.007	–0.008	–0.015
Ce(H2O)9.I3	Phonon	0.181	–0.186	–0.377	0.018	–0.018	–0.037

Table 5.

Net (total) equilibrium fractionation factors, relative to CePO₄-monazite

	103 ln α at 298.15 K			103 ln α at 1000 K
Ce(IV) species	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce	142Ce-140Ce	138Ce-140Ce	136Ce-140Ce
CeSiO4 - stetindite	–0.121	–0.343	–0.760	–0.116	–0.020	–0.060
Ce1/2Zr1/2SiO4 - zircon	0.212	–0.726	–1.544	–0.093	–0.056	–0.134
Ce1/4Zr3/4SiO4 - zircon	0.467	–1.007	–2.117	–0.073	–0.082	–0.189
Ce1/8Zr7/8SiO4 - zircon	0.309	–0.848	–1.795	–0.088	–0.068	–0.160
CeO2 - cerianite	–0.167	–0.257	–0.581	–0.113	–0.011	–0.041
Ce1/4Zr3/4O2 - baddeleyite	0.137	–0.617	–1.319	–0.094	–0.045	–0.111
Ce(SO4)2(H2O)4	–0.064	–0.361	–0.791	–0.101	–0.023	–0.064
Ce(III) species
CePO4 - monazite	0	0	0	0	0	0
Ce1/4Zr3/4P1/4Si3/4O4 - zircon	0.795	–0.869	–1.772	0.065	–0.082	–0.170
Ce1/2Na1/2Ca4(PO4)3F - apatite	0.140	–0.155	–0.317	0.010	–0.014	–0.029
Ce2O3 A-type (hexagonal)	0.116	–0.142	–0.291	0.005	–0.012	–0.025
Ce1/4Ca3/4MgAl1/4Si7/4O6 - diopside	0.180	–0.185	–0.375	0.017	–0.017	–0.035
Ce(CO3)F - bastnaesite	0.008	–0.007	–0.014	0.001	–0.001	–0.002
La5/6Ce1/6(CO3)F - bastnaesite	–0.062	0.065	0.132	–0.006	0.006	0.013
Y5/6Ce1/6(CO3)F - bastnaesite	0.081	–0.085	–0.172	0.007	–0.008	–0.015
Ce(H2O)9.I3	0.160	–0.184	–0.375	0.011	–0.017	–0.036

Fig. 2.

Equilibrium ¹⁴²Ce/¹⁴⁰Ce (a–c), ¹³⁸Ce/¹⁴⁰Ce (d–f), and ¹³⁶Ce/¹⁴⁰Ce (g–i) fractionations relative to CePO₄-monazite, as functions of the reciprocal of the absolute temperature. Left panels (a, d, and g) show the mass dependent component of fractionation, middle panels (b, e, and h) show the nuclear volume component of fractionation, and right panels (c, f, and i) show the total fractionation for each isotope pair. The gray shading indicates Ce(III)-species, and pale yellow shading indicates Ce(IV)-species. Ce-substituted zirconium minerals zircon and baddeleyite show distinct mass-dependent affinities for massive isotopes, with Ce(IV)-substituted species indicated with golden shading, and the Ce(III)-substituted zircon (Ce_1/4Zr_3/4P_1/4Si_3/4O₄) marked with a dashed red line.

Table 6.

Polynomial fits of fractionation factors relative to CePO₄ monazite

	142Ce/140Ce			138Ce/140Ce			136Ce/140Ce
Ce(IV) species	A1	A2	C	A1b	A2b	C	A1b	A2b	C
CeSiO4 - stetindite	37051	–2.20 × 108	–152.9	–38044	2.26 × 108	17.7	–77195	4.58 × 108	16.7
Ce1/2Zr1/2SiO4 - zircon	73658	–5.22 × 108	–166.6	–75632	5.36 × 108	19.3	–153467	1.09 × 109	18.2
Ce1/4Zr3/4SiO4 - zircon	100258	–7.36 × 108	–172.6	–102945	7.56 × 108	20.0	–208888	1.53 × 109	18.9
Ce1/8Zr7/8SiO4 - zircona	86844	–6.23 × 108	–173.8	–89171	6.40 × 108	20.2	–180939	1.30 × 109	19.0
CeO2 - cerianite	27079	1.82 × 107	–140.3	–27805	–1.87 × 107	16.3	–56420	–3.80 × 107	15.3
Ce1/4Zr3/4O2 - baddeleyite	61950	–3.05 × 108	–156.0	–63610	3.13 × 108	18.1	–129072	6.35 × 108	17.0
Ce(SO4)2(H2O)4a	38444	–1.74 × 108	–139.7	–39474	1.79 × 108	16.2	–80098	3.63 × 108	15.3
Ce(III) species
CePO4 - monazite	0	0	0	0	0	0	0	0	0
Ce1/4Zr3/4P1/4Si3/4O4 - zircon	82944	–5.82 × 108	–17.5	–85167	5.97 × 108	2.0	–172813	1.21 × 109	1.9
Ce1/2Na1/2Ca4(PO4)3F - apatitea	14144	–2.49 × 106	–4.0	–14523	2.56 × 106	0.5	–29468	5.19 × 106	0.4
Ce2O3 A-type (hexagonal)	12320	4.67 × 107	–7.4	–12650	–4.79 × 107	0.9	–25669	–9.73 × 107	0.8
Ce1/4Ca3/4MgAl1/4Si7/4O6 - diopsidea	16744	–1.91 × 107	0.0	–17192	1.96 × 107	0.0	–34886	3.98 × 107	0.0
Ce(CO3)F - bastnaesite	638	–1.76 × 107	0.3	–655	1.81 × 107	0.0	–1328	3.68 × 107	0.0
La5/6Ce1/6(CO3)F - bastnaesitea	–6386	1.14 × 108	0.5	6557	–1.17 × 108	–0.1	13306	–2.38 × 108	–0.1
Y5/6Ce1/6(CO3)F - bastnaesitea	7444	3.85 × 107	–0.4	–7643	–3.95 × 107	0.0	–15509	–8.01 × 107	0.0
Ce(H2O)9.I3	17698	–1.50 × 108	–6.2	–18172	1.54 × 108	0.7	–36874	3.63 × 108	0.7

Fits have the form 10³ ln α^Total = A₁/T² + A₂/T⁴ + C/T. The first two terms correspond to mass-dependent fractionation, and the third to the nuclear volume component. The fits are expected to reproduce estimated fractionation factors within 0.02‰/amu for temperatures ≥263 K (–10°C) and 0.01‰/amu for temperatures ≥298 K (25°C). The polynomial fit for 10³ ln β^MD_142–140 [CePO₄] = 89056/T² – 3.2341 × 10⁸/T⁴ over the same temperature interval.

^a A₂ terms determined via eq. 7.

^b A₁ & A₂ terms determined via eq. 8

For exchanges between different oxidation states of cerium, the mass-dependent component of equilibrium fractionation partitions more massive isotopes into the more oxidized species, i.e., high ¹⁴²Ce/¹⁴⁰Ce, low ¹³⁸Ce/¹⁴⁰Ce, and low ¹³⁶Ce/¹⁴⁰Ce in Ce(IV)-bearing phases. This is consistent with the systematics seen for mass dependent fractionation in other elements. Ce(IV) vs. Ce(III) fractionations are ~0.1–0.5‰/amu at 298 K (27°C). Because mass-dependent fractionation scales in approximate proportion to 1/T², it becomes much more subtle at elevated temperatures, shrinking to ~0.05‰/amu or less at 1000 K (727°C). Fractionations among Ce(III) species are generally much smaller, ≤0.1‰/amu at 298 K. This agrees well with previous predictions that non-redox mass-dependent REE-isotope fractionation is limited (Nakada et al., 2017; Hu et al., 2023; Schauble, 2023). There is also a fairly restricted range of predicted mass-dependent fractionation among Ce(IV)-bearing species. However, there is a noted preference for high mass isotopes in crystallographic sites where cerium substitutes for zirconium in zircon and baddeleyite (as estimated by models of ordered crystals with low Ce/Zr ratios), relative to pure Ce(IV) phases such as CeO₂-cerianite and stetindite. Interestingly, Ce(III)P(V) substituted zircon, Ce_1/4Zr_3/4P_1/4Si_3/4O₄, shows a similar relationship to other Ce(III) species, and in fact has mass-dependent fractionation properties closely resembling Ce(IV)-substituted zircon. This appears to be analogous to Ca-isotope fractionation in species where calcium substitutes into magnesium sites, for instance in Ca-poor pyroxene structures (e.g., Huang et al., 2010; Wang et al., 2017). For both cerium and calcium, substitution for a smaller cation is associated with affinity for the massive isotopes at equilibrium. In the present case the Zr⁴⁺ ion (0.98 Å crystal radius in VIII-fold coordination; Shannon, 1976) is smaller than Ce⁴⁺ (1.11 Å). For Ce-substituted zircons with Ce/Zr ratios of 1/3 and 1/7, and a reconnaissance model with a ratio of 1/15, there is some scatter but no clear systematic trend of increasing affinity for massive Ce-isotopes towards the lowest Ce/Zr ratio, suggesting that 1/3 and 1/7 models will be reasonably accurate analogues for typical natural Ce(IV)-substituted zircons and baddeleyites with much lower Ce/Zr. Likewise, mass dependent fractionation in the Ce(III)-substituted structure Ce_1/4Zr_3/4P_1/4Si_3/4O₄ does not appear to be strongly sensitive to concentration or the details of the substitution structure: a reconnaissance model with a 1/15 Ce/Zr ratio is very similar, and the model at a 1/3 ratio with Ce and P substituting on the closest Zr-Si distance in the structure—tabulated with the main results—predicts very similar mass dependent fractionation to a model with substitution on the next-nearest Zr-Si distance.

Using calibrated electron densities from DFT, nuclear volume fractionation is predicted to favor high ¹⁴²Ce/¹⁴⁰Ce in Ce(III)-bearing species relative to Ce(IV)-bearing species, by 0.4–0.6‰ at 298 K and 0.1–0.2‰ at 1000 K, depending on the substances compared. This is less extreme than the ¹⁵³Eu/¹⁵¹Eu fractionation between Eu²⁺- and Eu³⁺-bearing species, mainly because of the much larger difference in nuclear charge radii between europium isotopes. The nearly constant charge radii for ¹³⁶Ce, ¹³⁸Ce, and ¹⁴⁰Ce lead to minimal nuclear volume fractionations among these isotopes, with both ¹³⁶Ce/¹⁴⁰Ce and ¹³⁸Ce/¹⁴⁰Ce typically ~0.06‰ higher in Ce(IV)-bearing species at 298 K, and ~0.02‰ higher at 1000 K.

The nuclear volume component of fractionation seems to be only weakly dependent on factors other than oxidation state, with only ~0.1‰ variation in ¹⁴²Ce/¹⁴⁰Ce between different Ce(IV) species and ~0.03‰ variation between different Ce(III) species at 298 K. Ce(IV) and Ce(III) for Zr(IV) substitution in zircon and baddeleyite is associated with slightly lower predicted ¹⁴²Ce/¹⁴⁰Ce relative to other species with the same oxidation state. There is very little difference between zircon models with Ce/Zr of 1/3 and 1/7.

Compared to the calibrated DFT estimates, nuclear volume fractionations determined from scaled Mössbauer isomer shifts are qualitatively similar but weaker, with ¹⁴²Ce/¹⁴⁰Ce predicted to be ~0.3‰ lower in CeO₂ relative to Ce₂O₃ and Ce(OH)₃ at 298 K. The calibrated DFT-PAW method predicts a difference of 0.5‰ for the first two crystals. This muting of the predicted nuclear volume component of fractionation is expected, based on the observations made in the methods section above. The ultimate cause of the discrepancy is not clear, however. Because ¹⁴¹Pr-Mössbauer data are only available for a limited range of geochemically relevant species, the results and discussion below will focus on calibrated DFT-PAW results. Based on ¹⁴¹Pr-Mössbauer results, it appears that pure Ce-metal will be similar to Ce(III) species in its nuclear volume fractionation behavior.

In general, total ¹³⁶Ce/¹⁴⁰Ce and ¹³⁸Ce/¹⁴⁰Ce fractionations are predicted to be dominated by the mass dependent component at ambient, metamorphic, and even magmatic temperatures (especially ¹³⁶Ce/¹⁴⁰Ce). The net fractionation for both ratios thus follows the rule of thumb that high oxidation states are enriched in heavy (i.e., more massive) isotopes at equilibrium. In contrast, nuclear volume plays a significant role in ¹⁴²Ce/¹⁴⁰Ce fractionation at all temperatures and dominates at igneous conditions. ¹⁴²Ce/¹⁴⁰Ce will be lower in Ce-substituted zircon and baddeleyite, relative to Ce(III)-phases, at temperatures above ~600 K (327°C), and it will be lower in the other modeled Ce(IV) species at most geochemically relevant temperatures. The estimated redox fractionation between CeO₂-cerianite and Ce(H₂O)₉³⁺ (based on Ce(H₂O)₉I₃) is –0.3‰ at 298 K (25°C). However, the near-cancellation of the mass-dependent and nuclear volume components suggests that the uncertainty of this estimate (a small difference between opposing, independently determined components) could be large in relative terms.

The main sources of uncertainty in estimated ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce fractionations at low temperatures (<500 K) likely come from errors in calculated vibrational frequencies and isotopic frequency shifts. Variation of up to ~0.05‰/amu at 298 K in the mass dependent component of Ce(IV) vs. Ce(III) fractionation when alternate PAW datasets are used to model the electronic structure of cerium (see Methods section above) represent a rough ballpark estimate on such uncertainty. A further test can be made by comparing models of CeO₂-cerianite, stetindite, CePO₄-monazite, and Ce₂O₃ made with the PBEsol functional (Perdew et al., 2008) to the present PBE-based results. Here no frequency scale factor is applied to the PBEsol-based vibrational frequencies, and the mean offset in 1000 ln β^MD_1xx–140 is –0.02‰/amu at 298.15 K relative to the PBE-based results. However, this appears to mainly reflect slight systematic underestimation of vibrational frequencies by the PBEsol functional, and the estimated fractionations between Ce(IV)-Ce(III) pairs are all within ~0.01–0.02‰/amu of their PBE-based counterpart models at 298 K. This is consistent with the ±1% (1 s.e.) statistical uncertainty in the PBE frequency scale factor, which propagates to errors of ca. 0.01–0.02‰/amu in the mass dependent component of fractionation at 298 K, and 0.001–0.002‰/amu at 1000 K. However, the frequency scale factor is calibrated using a limited data set of measured vibrational frequencies, and actual errors may be somewhat larger than the ±1% figure suggests.

Uncertainty in the nuclear volume component likely contributes significantly to uncertainty in overall fractionations only for ¹⁴²Ce/¹⁴⁰Ce fractionation, because the magnitude of this component is very small for ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce. For ¹⁴²Ce/¹⁴⁰Ce, plausible variability in calibrated electron densities of ~1–2 e^–/a₀³ is likely to be comparable to observed differences between models constrained at measured crystal structures versus fully relaxed structures. ±2 e^–/a₀³ propagates to ±0.04‰ in an estimated ¹⁴²Ce/¹⁴⁰Ce nuclear volume fractionation at 298 K (but less than ±0.01‰ for ¹³⁸Ce/¹⁴⁰Ce or ¹³⁶Ce/¹⁴⁰Ce), and ±0.01‰ at 1000 K. Propagated errors from the nominal statistical uncertainty in the slope of the DFT calibration regression are similar in magnitude. However, this uncertainty does not address much larger potential discrepancies in Mössbauer-based estimates of the nuclear volume component, or in DFT vs. all electron models calibrations using the atomic ions Ce²⁺, Ce³⁺, and Ce⁴⁺. If the Mössbauer-based method is correct, nuclear volume ¹⁴²Ce/¹⁴⁰Ce fractionation for Ce(III) vs. Ce(IV) species will typically be ~0.2‰ smaller at 298 K, and ~0.05‰ smaller at 1000 K, than estimated with the preferred DFT calibration method.

Discussion

The present results indicate that equilibrium cerium isotope fractionation is more complex than ¹⁵³Eu/¹⁵¹Eu fractionation, which is expected to be dominated by the nuclear volume effect (Schauble, 2023). Cerium isotope fractionation also appear to be more complex than light element isotope fractionations, which are typically mass dependent. Instead, the predominant fractionating property depends on the isotope pair, temperature, and phases of interest. In Ce(IV)-Ce(III) redox pairs, ¹³⁸Ce/¹⁴⁰Ce and especially ¹³⁶Ce/¹⁴⁰Ce are predicted to be mainly mass-dependently fractionated at typical crustal temperatures, with nuclear volume taking on an increasing role at high-grade metamorphic and igneous temperatures. However, even at ambient temperatures it may be possible to detect deviations from normal mass dependence with sufficiently precise measurements of these three isotopes. For ¹⁴²Ce/¹⁴⁰Ce, in contrast to the other isotope ratios, nuclear volume effects are much more pronounced, determining the sign of room-temperature fractionation in Ce(IV)-Ce(III) equilibria for several studied phases (notably excepting Ce(IV)-substituted baddeleyite and zircon). Even for these latter phases, strong deviations from normal mass dependence ought to be detectable in a three-isotope measurement (¹⁴²Ce/¹⁴⁰Ce + ¹³⁸Ce/¹⁴⁰Ce or ¹⁴²Ce/¹⁴⁰Ce + ¹³⁶Ce/¹⁴⁰Ce) made at a precision comparable to recent ¹⁴²Ce/¹⁴⁰Ce studies (Fig. 3). At most metamorphic and all igneous temperatures, the nuclear volume effect dominates for all studied redox pairs. The results indicate that measurable ¹⁴²Ce/¹⁴⁰Ce isotope fractionation is likely to be found in any geochemical environments where Ce(IV) and Ce(III) coexist—even at igneous temperatures. The range of elements with significant predicted and/or observed field shift fractionations now extends at least as low as atomic number 58, in addition to examples at higher atomic numbers 63 (Eu) (Schauble, 2023) and 80–92 (Hg, Tl, Pb, and U) (e.g., Nomura et al., 1996; Bigeleisen, 1996; Abe et al., 2008; Schauble, 2007; Yang and Liu, 2015; however see Fujii et al., 2009 for a review of effects observed in lower-Z elements).

Fig. 3.

Equilibrium ¹³⁸Ce/¹⁴⁰Ce fractionation vs. ¹⁴²Ce/¹⁴⁰Ce fractionation, showing departures from normal mass-dependent fractionation relationships. Shading is similar to Fig. 2, with Ce(IV) species in pale yellow and Ce(IV)-substituted zirconium species in gold. Ce(III)-substituted zircon is marked with a dashed red line. The high-temperature limit of the equilibrium mass dependent scaling relationship (Eq. 8) is shown with a blue dash-dotted line. The effect of temperature on Ce(IV)-Ce(III) fractionation is indicated with dotted contour lines.

Comparison with previous measurements and predictions

Previous work on cerium isotope fractionation has mainly concentrated on low-temperature aqueous-mineral systems, such as the adsorption of Ce(III) from solution onto Fe- and Mn-oxides (e.g., Nakada et al., 2013a, 2017). These studies have often found that aqueous Ce(III) has higher ¹⁴²Ce/¹⁴⁰Ce than coexisting cerium adsorbed, associated with, or otherwise incorporated into δ-MnO₂—most likely as a Ce(IV)-species—behavior that contrasts with other rare earth elements (Nakada et al., 2013b), and is contrary to the typical expectation that mass dependent fractionation will concentrate massive isotopes in species with higher oxidation states. Similar ¹⁴²Ce/¹⁴⁰Ce depletion is also observed in CeO₂ or amorphous Ce(OH)₄ directly precipitated from Ce(III) solution (Nakada et al., 2013a; Bonnand et al., 2023). Cerium in ferrihydrite, likely in the +3 oxidation state, may not show consistent fractionation with respect to coexisting aqueous solutions (Nakada et al., 2013a). The present results broadly agree with prior measurements and experiments in estimating a ¹⁴²Ce/¹⁴⁰Ce fractionation of approximately –0.3‰ between CeO₂ and Ce(H₂O)₉I₃ (as an analogue for aqueous Ce(III), particularly in low-pH solution). The other non-zirconium Ce(IV) phases studied here are also expected to have lower ¹⁴²Ce/¹⁴⁰Ce than Ce(H₂O)₉I₃ at ambient temperatures. It should be noted, however, that more complex redox/isotope fractionation systematics in natural systems are implied by field studies, e.g., Bai et al. (2024) on fractionation during weathering and soil formation.

In a combined experimental and theoretical study, Nakada et al. (2017) proposed that changing aqueous speciation of Ce(III) with varying pH exerted some control on fractionations during adsorption to δ-MnO₂. In particular, their theoretical models of mass-dependent fractionation appear to show that Ce(III)-CO₃^2– complexes will have lower ¹⁴²Ce/¹⁴⁰Ce than aquo-complexes, and that this decrease is progressive as the CO₃^2–/Ce ratio of the complex increases (~0.04‰ per CO₃^2– at 298 K). Experiments at pH increasing from 7–11 likewise showed a progressive decrease in the solution-mineral fractionation, by about 0.2‰. In agreement with previous work, the solution-mineral fractionation factor was found to be positive at all pH conditions studied. Although the present study mainly focuses on crystalline species and was not able to include models of Fe(III) oxide/oxyhydroxide or MnO₂ phases, it is worth noting that the Ce(H₂O)₉I₃ crystal used in the present study as an analog for aqueous Ce(III) yields a somewhat higher 10³ ln β^MD than the Nakada et al. (2017) theoretical results suggest (1.15‰ vs. 0.94‰ at 298 K).

Reconnaissance estimations of fractionation factors for Ce(III)(H₂O)₉³⁺ using a more “apples-to-apples” method comparison consistently show somewhat higher 10³ ln β^MD than reported by Nakada et al. (2017) for Ce(III)(H₂O)₉³⁺ and Ce(III)CO₃(H₂O)₇⁺. Test calculations on Ce(III)(H₂O)₉³⁺ yield 10³ ln β^MD = 1.11‰ at 298 K for a frequency-scaled PBE/def2-TZVP model—quite close to the model result for crystalline Ce(H₂O)₉I₃—and 1.03‰ for both an unscaled B3LYP/def2-TZVP model and one in which the def2-TZVP basis set is replaced with the SDD basis set Nakada et al. (2017) used for cerium (the effective core potential components of these two basis sets are identical). A B3LYP/def2-TZVP model of the Ce(III)CO₃(H₂O)₇⁺ carbonato complex also yields 10³ ln β^MD = 1.03‰ at 298 K, consistent with no fractionation relative to the aqua ion. This is slightly inconsistent with the prediction in Nakada et al. (2017) that the carbonato complex will have ~0.04‰ lower ¹⁴²Ce/¹⁴⁰Ce. It should be pointed out that the conductor-like polarizable continuum treatment of implicit solvation is used for the present models is not quite the same as the IEF-PCM implicit solvation model of Nakada et al. (2017), so the original method is not reproduced exactly. However, it would be puzzling if this minor discrepancy yielded a large deviation in the final result. In vacuo models without implicit solvation are slightly closer to the Nakada et al. (2017) result: 10³ ln β^MD = 0.98‰ for unsolvated Ce(H₂O)₉³⁺ via B3LYP/def2-TZVP. Some (and possibly all) of the offset between the present PBE and B3LYP results can be attributed to vibrational frequency scaling—published general purpose vibrational frequency scale factors for these two functionals typically differ by 2–4% (e.g., Laury et al., 2012; Kesharwani et al., 2014; National Institute of Standards and Technology, 2022), whereas in the present comparison PBE results are scaled by 7.5% and B3LYP are unscaled. This discrepancy is large enough to explain a ~0.1‰ PBE vs. B3LYP offset in 10³ ln β^MD for ¹⁴²Ce/¹⁴⁰Ce in Ce(H₂O)₉³⁺.

Reasons for analyzing multiple cerium isotopes

The existence of distinct isotope pairs that are dominantly controlled by mass dependent fractionation at equilibrium (¹³⁸Ce/¹⁴⁰Ce, ¹³⁶Ce/¹⁴⁰Ce) alongside the field-shift sensitive ¹⁴²Ce/¹⁴⁰Ce is unique among isotope systems studied to date. Most previously studied systems show dominantly mass-dependent fractionation, a category that encompasses the intensively investigated H, C, N, O & S stable isotopes as well as more recently developed “non-traditional” isotope systems such as Mg, Ca, Fe, Mo & etc. (see, however, e.g., Fujii et al., 2009 for a review of possible laboratory field shift fractionations found across the periodic table). The range and structures of stable nuclei for the few elements that display significant field shift/nuclear volume fractionation in nature is such that field shift effects are always superimposed on mass dependent fractionation. A possible exception is the mercury isotope system, in which charge radii are nearly invariant for the ¹⁹⁸Hg/¹⁹⁹Hg isotope pair, and for the ²⁰⁰Hg/²⁰¹Hg isotope pair. However, this apparent exception only highlights how distinctive cerium is, because the most interesting natural mercury-isotope fractionations appear to involve a poorly understood, possibly magnetically controlled photochemical kinetic effect (e.g., Tsui et al., 2020).

The fortuitous separation of mass dependent fractionation in ¹³⁸Ce/¹⁴⁰Ce or ¹³⁶Ce/¹⁴⁰Ce from mixed mass dependent and field shift fractionation effects ¹⁴²Ce/¹⁴⁰Ce in may provide a useful means to discriminate equilibrium from kinetic isotope fractionation signatures, so long as non-radiogenic ¹³⁸Ce/¹⁴⁰Ce or ¹³⁶Ce/¹⁴⁰Ce variations can be measured with sufficient precision as supplements to the ¹⁴²Ce/¹⁴⁰Ce measurements currently being performed. At the simplest level, one might expect diffusive or transport-controlled fractionations to yield approximately linear mass dependence across ¹⁴²Ce/¹⁴⁰Ce as well as the other isotope pairs. In contrast, any equilibrium process, particularly if it involves oxidation or reduction of cerium, will generate a substantial divergence (i.e., lowering ¹⁴²Ce/¹⁴⁰Ce in the Ce(IV)-bearing species). Disequilibrium fractionations in which reaction rates depend on activation energies or partitioning among multiple species (including column chemistry) may show greater or lesser dependence on nuclear volume, depending on the details of species stability, the transition state, reaction coordinate, and transport limitations. As an example, the relatively well-studied ¹⁴²Ce/¹⁴⁰Ce fractionation of roughly –0.2‰ to –0.4‰ between CeO₂ (or Ce(IV) associated with δ-MnO₂) vs. aqueous Ce(III) (e.g., Nakada et al., 2013a; Bonnand et al., 2023) only reflects equilibrium isotope partitioning if ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce are both also higher in the Ce(III) phase than in the Ce(IV) phase. If ¹³⁸Ce/¹⁴⁰Ce or ¹³⁶Ce/¹⁴⁰Ce are lower in Ce(III), that strongly suggests a disequilibrium fractionation mechanism.

On the other hand, failure to account for the non-mass dependent component of ¹⁴²Ce/¹⁴⁰Ce fractionation during the processing of mass spectrometry data, or during isotopically spiked isotope exchange experiments, may introduce artifacts that reduce analytical precision or mislead experimental interpretations. From either perspective (as a feature, or as a bug), efforts to measure more than one cerium isotope ratio appear worthy of pursuit in future work.

Zircon/melt fractionation

One potential area of interest for future cerium isotope studies is the mineral zircon, which is a unique geochemical archive. This includes potential information about conditions on the Hadean Earth (e.g., Trail et al., 2011). Zircon is observed to strongly concentrate cerium relative to other rare earth elements, mainly because of a strong affinity for Ce(IV) (e.g., Smythe and Brenan, 2016). This affinity for Ce(IV) suggests that systematic isotope fractionation relative to melts is likely. To investigate this potential more closely, a simplified model for zircon-melt fractionation of cerium isotopes is developed here. The model is based on previous studies of Ce(IV)/Ce(III) speciation in silicate melts and coexisting zircon, as well as the present results for isotopic fractionation among Ce-substituted zircon and other crystals. It is beyond the scope of this work to attempt a detailed and fully realistic model of trace cerium in silicate melts and igneous crystals, but hopefully even a simplified framework will be helpful for designing and interpreting measurements.

Four components are relevant in the zircon-melt fractionation system: Ce(IV) substituted into the zircon structure, Ce(IV) in silicate melt, Ce(III) in zircon, and Ce(III) in silicate melt. For present purposes, these components are assumed to be represented by, respectively, Ce_1/4Zr_3/4SiO₄ zircon, CeSiO₄ stetindite, Ce_1/4Zr_3/4P_1/4Si_3/4O₄ zircon, and CePO₄ monazite. These choices are motivated by the idea that lattice strain effects on Ce-isotope partitioning appear to be important in zircon (and so these components are represented by Ce-substituted structures), whereas the coordination environment in silicate melt is likely to be more flexible in conforming to the preferred site size and coordination of cerium ions (perhaps best represented by stoichiometric Ce-bearing crystals). Alternative choices for these analogues could easily be explored using the results tabulated above.

The effective fractionation between zircon and melt is determined from the abundance-weighted average of the components in each phase:

  

$$ln\alpha =[X_{\mathit{zr}}^{\mathit{Ce}\left(\mathit{IV}\right)}{ln\alpha }_{\mathit{Ce}\left(\mathit{IV}\right)\mathit{zr}}+(1–X_{\mathit{zr}}^{\mathit{Ce}\left(\mathit{IV}\right)}){ln\alpha }_{\mathit{Ce}\left(\mathit{III}\right)\mathit{zr}}]–[X_{\mathit{melt}}^{\mathit{Ce}\left(\mathit{IV}\right)}{ln\alpha }_{{\mathit{CeSiO}}_4}+(1–X_{\mathit{melt}}^{\mathit{Ce}\left(\mathit{IV}\right)}){ln\alpha }_{{\mathit{CePO}}_4}]$$(9)

Where X^Ce(IV) represents the fractional abundance Ce(IV) in zircon or melt, as a fraction of total cerium in that phase (i.e., X_zr^Ce(IV) + X_zr^Ce(III) = X_melt^Ce(IV) + X_melt^Ce(III) = 1). Because CePO₄-monazite is the reference for reporting fractionations in this study, ln α_CePO4 = 0 by definition, and the equation can be simplified to:

  

$$ln\alpha =[X_{\mathit{zr}}^{\mathit{Ce}\left(\mathit{IV}\right)}{ln\alpha }_{\mathit{Ce}\left(\mathit{IV}\right)\mathit{zr}}+(1–X_{\mathit{zr}}^{\mathit{Ce}\left(\mathit{IV}\right)}){ln\alpha }_{\mathit{Ce}\left(\mathit{III}\right)\mathit{zr}}]–X_{\mathit{melt}}^{\mathit{Ce}\left(\mathit{IV}\right)}{ln\alpha }_{{\mathit{CeSiO}}_4}$$(10)

The fractional abundances of the oxidation states in melt are calculated as follows (Smythe and Brenan, 2015):

  

$$ln\left[\frac{X_{\mathit{Ce}\left(\mathit{IV}\right)}}{X_{\mathit{Ce}\left(\mathit{III}\right)}}\right]=\frac{1}{4}ln{\mathit{fO}}_2+\frac{13136}{T}–2.064\left(N\mathit{BO}/T\right)–8.878X_{H_{2}O}–8.955$$

Here ƒO₂ is the oxygen fugacity, T is absolute temperature, NBO/T is the ratio of non-bridging oxygens to tetrahedral cations in the melt structure, and X_H2O is the mole fraction of water. Typically NBO/T is low in high-silica, low liquidus temperature natural melts, and higher in higher temperature, less silicic melts. To approximate this trend, the NBO/T vs. temperature correlation observed by Loader et al. (2022) is adopted in the present study for illustrative purposes:

  

$$lnN\mathit{BO}/T=\frac{\mathrm{–13328}}{T}+9.1629$$

Cerium speciation in zircon is then determined by the oxidation-state specific zircon-melt distribution coefficients (Smythe and Brenan, 2016; Claiborne et al., 2018):

  

$$D_{\mathit{Ce}\left(\mathit{IV}\right)}^{\mathit{zr}}=e^{10245/T–4.3215}$$

  

$$D_{\mathit{Ce}\left(\mathit{III}\right)}^{\mathit{zr}}={10}^{\mathrm{–3}}$$

Calculated ¹⁴²Ce/¹⁴⁰Ce, ¹³⁸Ce/¹⁴⁰Ce, and ¹³⁶Ce/¹⁴⁰Ce fractionations between zircon and melt are shown in Fig. 4. The first panel shows the dependence of fractionation on oxygen fugacity at conditions that roughly correspond to the Bishop Tuff melt (Smythe and Brenan, 2016). It can be seen that the predicted zircon-melt fractionation is only sparingly sensitive to fO₂ over a range from the Iron-Wüstite buffer (IW) through the Fayalite-Magnetite-Quartz buffer (FMQ) to the Magnetite-Hematite buffer (MH) (the positions of the buffers are calculated using the tabulation of Frost, 1991). Within this wide range, Ce(III) is predominant in the melt while Ce(IV) is predominant in zircon, so that the fractionation is determined mainly by those two components. ¹⁴²Ce/¹⁴⁰Ce is lower in zircon because of the Ce(IV) vs. Ce(III) field shift effect, whereas ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce are lower in zircon because of a mass dependent fractionation enhanced by snug Ce(IV)-Zr(IV) and Ce(III)P(V)-Zr(IV)Si(IV) substitutions. The net result is that ¹⁴⁰Ce is enriched in zircon relative to ¹⁴²Ce, ¹³⁸Ce, and ¹³⁶Ce. At lower fO₂, the Ce(III) component becomes significant in zircon, gradually reducing the impact of the Ce(IV)-Ce(III) field shift effect, and the zircon becomes more uniformly enriched in massive isotopes. At high fO₂, Ce(IV) becomes significant in the melt, also shifting the system towards a mass-dependent fractionation pattern. Figure 4b generalizes this pattern to higher and lower temperatures, showing that ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce fractionations become notably more muted as temperatures increase, but ¹⁴²Ce/¹⁴⁰Ce is almost invariant because of the opposite directionality and different temperature dependences of the mass dependent and mass independent components. Note that in the vicinity of the FMQ fO₂ buffer, the overall shapes of the fractionation relationships have little sensitivity to melt polymerization and look roughly similar even if NBO/T is held constant. The temperature insensitivity of ¹⁴²Ce/¹⁴⁰Ce fractionation is emphasized in Fig. 4c, which shows a temperature cross-section at the FMQ buffer. Taken together, the results suggest that fO₂ variations will have limited impact on the instantaneous equilibrium zircon-melt ¹⁴²Ce/¹⁴⁰Ce, except when oxygen fugacity is either extremely high or extremely low. However, the bulk partitioning of cerium is sensitive to fO₂, so it is possible that complicating effects such as mass balance and Rayleigh-like distillation may correlate observed zircon-melt fractionations more strongly with oxygen fugacity in natural samples.

Fig. 4.

Equilibrium cerium isotope fractionations between model analogues for zircon and silicate melts. ¹⁴²Ce/¹⁴⁰Ce is shown with solid lines, ¹³⁸Ce/¹⁴⁰Ce with dot-dashed lines, and ¹³⁶Ce/¹⁴⁰Ce with dashed lines. (a) fractionations as a function of oxygen fugacity (log fO₂) for a melt composition and temperature roughly similar to the Bishop Tuff (1000 K, NBO/T near zero, H₂O oxide mole fraction = 0.2). The positions of the fO₂ buffers Iron-Wüstite (IW), Fayalite-Magnetite-Quartz (FMQ) and Magnetite-Hematite (MH) are calculated assuming 0.2 GPa pressure. The Bishop Tuff melt was likely near to or slightly above the FMQ buffer (e.g., Smythe and Brenan, 2016). (b) fractionation vs. the deviation of log fO₂ from the 0.2 GPa FMQ buffer at 900 K (black), 1100K (medium gray), and 1300 K (light gray), assuming correlated NBO/T and temperature. The oxide mole fraction of H₂O is assumed to be 0.05 at all temperatures. (c) fractionation vs. temperature at the 0.2 GPa FMQ buffer, assuming correlated NBO/T and temperature.

It should be emphasized that these model predictions are rough and tentative—as yet there is no empirical or experimental confirmation of field shift Ce-isotope fractionation in igneous rocks, nor is there as yet independent evidence for the proposed Ce(IV)-Zr(IV) and Ce(III)P(V)-Zr(IV)Si(IV) substitution effect on mass-dependent fractionation. The disagreement between Mössbauer- and DFT-based estimates of the nuclear field shift is also not resolved. Instead, the major goal of the present work is to motivate and guide future measurement studies to test these results.

Conclusions

Theoretical estimates of equilibrium cerium isotope fractionations predict contrasting behavior for ¹⁴²Ce/¹⁴⁰Ce, which shows subequal, often opposed mass-dependent and nuclear field shift effects, relative to isotope pairs without ¹⁴²Ce that are dominated by mass dependent fractionation. ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce will typically be lower (i.e., ¹⁴⁰Ce more enriched) in Ce(IV)-bearing species than coexisting Ce(III)-bearing species. ¹⁴²Ce/¹⁴⁰Ce will also be lower in Ce(IV)-bearing species than coexisting Ce(III)-bearing species at high temperatures, but may fractionate in either direction at ambient temperatures, depending on the balance of field shift and mass dependent components. For both oxidation states of cerium, substitution for the smaller Zr(IV) ion is associated with an increased preference for massive cerium isotopes relative to otherwise analogous phases. The predicted ¹⁴²Ce/¹⁴⁰Ce fractionation of –0.3‰ between CeO₂-cerianite and Ce(H₂O)₉I₃ at ~25°C is in qualitative agreement with recent experimental studies on adsorption/crystallization of Ce(IV) from Ce(III)-dominated aqueous solutions. When combined with experimental studies of cerium redox chemistry in silicate melts and partitioning into zircon, the present results predict a nearly temperature-invariant depletion of 0.05–0.1‰ in ¹⁴²Ce/¹⁴⁰Ce in zircon over a range naturally relevant oxygen fugacities, along with strongly temperature-dependent depletions of 0.02–0.2‰ in ¹³⁸Ce/¹⁴⁰Ce and ¹³⁶Ce/¹⁴⁰Ce.

Acknowledgments

The author is grateful for helpful discussions with Beth Ann Bell and Alex Sedlak during the preparation of the manuscript, and for thoughtful suggestions from two anonymous reviewers. This research is supported by US National Science Foundation grants EAR1530306 and EAR1524811, as well as funding from the UCLA Division of Physical Sciences.

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