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# Modeling nuclear volume isotope effects in crystals

Edited by Mark H. Thiemens, University of California at San Diego, La Jolla, CA, and approved April 4, 2013 (received for review September 18, 2012)

## Abstract

Mass-independent isotope fractionations driven by differences in volumes and shapes of nuclei (the field shift effect) are known in several elements and are likely to be found in more. All-electron relativistic electronic structure calculations can predict this effect but at present are computationally intensive and limited to modeling small gas phase molecules and clusters. Density functional theory, using the projector augmented wave method (DFT-PAW), has advantages in greater speed and compatibility with a three-dimensional periodic boundary condition while preserving information about the effects of chemistry on electron densities within nuclei. These electron density variations determine the volume component of the field shift effect. In this study, DFT-PAW calculations are calibrated against all-electron, relativistic Dirac–Hartree–Fock, and coupled-cluster with single, double (triple) excitation methods for estimating nuclear volume isotope effects. DFT-PAW calculations accurately reproduce changes in electron densities within nuclei in typical molecules, when PAW datasets constructed with finite nuclei are used. Nuclear volume contributions to vapor–crystal isotope fractionation are calculated for elemental cadmium and mercury, showing good agreement with experiments. The nuclear-volume component of mercury and cadmium isotope fractionations between atomic vapor and montroydite (HgO), cinnabar (HgS), calomel (Hg_{2}Cl_{2}), monteponite (CdO), and the CdS polymorphs hawleyite and greenockite are calculated, indicating preferential incorporation of neutron-rich isotopes in more oxidized, ionically bonded phases. Finally, field shift energies are related to Mössbauer isomer shifts, and equilibrium mass-independent fractionations for several tin-bearing crystals are calculated from ^{119}Sn spectra. Isomer shift data should simplify calculations of mass-independent isotope fractionations in other elements with Mössbauer isotopes, such as platinum and uranium.

Among the various causes of mass-independent isotope fractionation discovered over the past 40 years, the nuclear field shift effect (1) is unique in that it is an equilibrium thermodynamic phenomenon (and may also impart an identifiable and distinct signature in disequilibrium conditions) (2, 3). It has also proven amenable to prediction by electronic structure methods, at least in simple molecular systems (e.g., refs. 4⇓⇓–7). The goal of the present study is to extend the scope of systems that can be modeled to include condensed phases and crystals, and to speed up calculations for complex materials while retaining enough accuracy to be useful.

Nuclear field shift effects result from the finite volume and sometimes nonspherical shapes of atomic nuclei, which are slightly different from one isotope to another. These differences in size and shape affect the coulomb potential felt by bound electrons, and thus the electronic structures of atoms and molecules. Nuclear field shifts have long been known in optical spectra of atoms and ions, but Bigeleisen (1) and Fujii and coworkers (8) were the first to recognize that the same phenomenon could lead to chemical fractionation of isotopes. They showed that uranium isotope separation via ion exchange columns (the Asahi process) tracked nuclear charge radii (or more specifically, the effective mean-square charge radius), rather than nuclear masses (1, 8, 9). Field shift effects are also expected to scale in proportion to the difference in electron densities inside the nucleus of the relevant atoms in the fractionating species (10). This electron density at the nucleus, indicated by |Ψ(0)|^{2}, is called the contact density:where *δE*_{FS} is the volume component of the isotopic field shift energy driving fractionation, |Ψ(0)|^{2}_{A,B} are the contact densities of the fractionating species, and Δ<*r*^{2}> is the difference in mean-squared charge radii of the isotopic nuclei. Note that nonspherical nuclei may show additional fractionation effects (e.g., ref. 11), which are beyond the scope of the present study. Bigeleisen showed that the magnitude of observed uranium isotope fractionation was consistent with optical measurements of nuclear field shifts in uranium vapor. Correlations between nuclear charge radii and isotope fractionation have been used to suggest that nuclear field shift effects contribute to isotope separations in other elements (e.g., refs. 12⇓–14). Knyazev and Myasoedov (15) used relativistic electronic structure theory calculations on atoms and atomic ions to show that nuclear field shift isotope fractionation will be more pronounced for nuclei with high atomic number. Schauble (4) and Abe et al. (5, 6) took the calculations a step farther, directly modeling the electronic structures of mercury-, thallium- and uranium-bearing molecules while explicitly varying nuclear charge radii.

This method has been used in a number of studies to predict or rationalize mass-independent isotope fractionations (e.g., refs. 7, 16, and 17). However, the all-electron relativistic electronic structure calculations that form the basis for these studies are computationally intensive, requiring self-consistent solutions for four-component wave functions involving typically thousands of basis functions. As a result, these studies are limited to modeling systems with ∼20 or fewer atoms, even though some of the most interesting observations of mass-independent isotope effects are in aqueous solutions (3, 7, 9), liquid metals (2, 18, 19), and solids (e.g., refs. 20 and 21). Advances in computer performance, more efficient parallelization of relativistic electronic structure codes, and implicit treatment of long-range interactions will make studies of larger systems more feasible in the future, but the method is now practical mainly for relatively small molecules and molecular clusters. This highlights a need for new methods of modeling nuclear volume isotope fractionation in extended systems and condensed phases. This is particularly true for metals like liquid mercury, in which characteristic bond properties may emerge only for fairly large clusters (e.g., refs. 22 and 23). In addition, it would be useful to develop methods that could be adapted to dynamical systems, for instance through ab initio molecular dynamics, to investigate thermal and compositional effects, and activation energies involving reactive intermediates.

Density functional theory (24) using the projector augmented wave formalism (DFT-PAW) is a promising alternative for these calculations. DFT, in conjunction with plane wave basis sets, is well suited to modeling systems with periodic boundary conditions, including metals (e.g., ref. 25), and has been adapted for molecular dynamics simulations as well. Projector augmented waves (26, 27) enable efficient plane wave calculations in the space between atoms while also preserving information about the inner portion of atomic electronic structures. These favorable properties have previously been harnessed for a variety of modeling applications, including studies of Mössbauer isomer shifts (28) that influenced the design of the present work. Mössbauer isomer shifts and nuclear volume isotope effects have the same dependence on electron density at the nuclei of interest [varying in proportion to changes in Δ|Ψ(0)|^{2}], so the apparent success of DFT-PAW in Mössbauer studies is encouraging. The close relationship between Mössbauer shifts and nuclear field shift isotope fractionation will be explored further in the present study.

In this study, contact densities calculated with DFT-PAW are calibrated against relativistic all-electron calculations of nuclear volume effects on energies of isotope substitution. Calibrated DFT-PAW models are then used to estimate the nuclear volume component of equilibrium isotope fractionation in crystals. The calibration set includes tin-, cadmium-, and mercury-bearing species, sampling geochemically relevant bond partners, oxidation states, and coordination arrangements. Previous studies have shown that chemical changes in contact densities of molecules with exotic electronic structures are not always well characterized by the Dirac–Hartree–Fock (DHF) method (i.e., HgF_{4}) (29), so more sophisticated coupled-cluster with single, double (triple) excitation [CCSD(T)] calculations are performed on most of the same species. It is worth noting that DHF appears to work reasonably well with molecules more closely related to geochemically relevant species (e.g., ref. 7).

Recent observations of mass-independent fractionation have been attributed to field shift effects in all three elements (2, 18, 19, 30, 31); Cd- and Hg-isotope fractionation between liquid metal and the vapor phase are of particular interest (2, 18, 19). Tin is also the heaviest element in a recent DFT-PAW Mössbauer calibration study (28) and has been the focus of other calibration studies (e.g., refs. 32 and 33). Building on this earlier calibration work, Mössbauer isomer shifts are calibrated against both DFT-PAW and all-electron calculations, allowing estimation of equilibrium Sn-isotope fractionations in complex tin-bearing mineral species using isomer shift data from the literature.

## Methods

DFT-PAW calculations are made with the ABINIT software package (34, 35) using the Perdew–Wang functional in the local density approximation (PW) (36) and the Perdew, Burke, and Ernzerhof functional in the generalized gradient approximation (PBE) (37). PAW data sets are generated using AtomPAW (38); most use parameters available from online AtomPAW and ABINIT repositories (www.wfu.edu/∼natalie/papers/pwpaw/periodictable/oldperiodictable.html; www.abinit.org/downloads/PAW2). Tin, cadmium, and mercury datasets are constructed using a scalar relativistic treatment (39) and finite Gaussian nuclear charge distributions. Each nuclear charge radius is determined by an approximate polynomial fit of experimental radii vs. atomic number (38). There does not appear to be a compatible public PAW data set for mercury, so datasets for gold (40) have been adapted for this study. All electron DHF and CCSD(T) calculations are performed with the DIRAC software package (41, 42), using uncontracted relativistic Gaussian basis sets (43⇓–45) for heavy atoms (*Z* > 35) and correlation-consistent nonrelativistic basis sets for light atoms (46⇓–48). Relatively simple (and computationally faster) double-ζ basis sets (e.g., cc-pVDZ) are used unless otherwise specified. In DHF calculations, the Dirac–Coulomb Hamiltonian for each molecule is simplified by the inclusion of an interatomic correction for some small-component integrals (49). Calculations are further simplified (and accelerated) by setting a screening threshold for integral direct calculations of Fock matrices (50). Energies and contact densities in all-electron models are calculated using Gaussian nuclear charge distributions set to match the root-mean-squared charge radii tabulated in ref. 51 for tin and cadmium and ref. 52 for mercury, following the procedure of ref. 4. ^{124}Sn, ^{114}Cd, and ^{198}Hg are the reference isotopes. ^{116}Sn, ^{110}Cd, and ^{202}Hg radii are used to calculate isotopic shifts. Further details of the DFT-PAW and all-electron coupled-cluster calculations are available (*SI Text* and Table S1).

### DFT-PAW Calibration Results.

Calculated electron densities and energies for Sn-, Cd-, and Hg-bearing molecules are presented in Fig. 1 and Table S2.

### Tin and Cadmium.

DFT-PAW and DHF electron densities are strongly correlated (Fig. 1 *A–C*) and fall close to a 1:1 match in electron density variation. In fact, regression slopes are within 1 SE of unity (tin: PW, 0.96 ± 0.06; PBE, 0.97 ± 0.06; cadmium: PW, 1.04 ± 0.04; PBE, 1.00 ± 0.05). Total contact densities also match well (e.g., DHF, 188,242 e^{–}/a_{0}^{3}; DFT-PAW, ∼188,000 e^{–}/a_{0}^{3} for Sn^{4+}). Tests with point-nucleus DFT-PAW models show much more variable contact densities (∼1.6 × DHF) and much higher absolute contact densities (>400,000 e^{–}/a_{0}^{3}) (28). DHF model contact densities increase by less than 2% if a much larger “quadruple-ζ” basis set is used. DFT-PAW to DHF model correlations are only slightly affected by the exclusion of molecules for which CCSD(T) results are not available. Correlations tighten if Sn^{2+} and Sn^{4+} are removed from the calibration set, suggesting different systematic errors in DHF and DFT-PAW models of multiatom molecules, relative to ions. The finite-nucleus results agree well with a previous point-nucleus full-potential linear muffin-tin orbital study of Mössbauer isomer shifts, in which an average integrated density was calculated within a volume equivalent to the ^{119}Sn nucleus (32).

Energy differences resulting from isotope substitution calculated at the CCSD(T) level are even more strongly correlated with DFT-PAW contact densities (Fig. 1 *D–F*), except for the PBE functional and Cd-bearing species, where the correlation with DHF results is slightly higher—0.982 vs. 0.979. DFT-PAW models using the PW (local density approximation) and PBE (gradient-corrected) functionals have almost the same correlation coefficients (*R*^{2} = 0.98–0.99).

### Mercury.

Contact densities and isotopic energy shifts calculated with DHF, CCSD(T), and DFT-PAW models are also well correlated for mercury-bearing species. The main exception is HgF_{4}, which Knecht et al. (29) studied and discussed in detail as an example of an electronic structure that is not reproduced well by either DFT or DHF methods. In the present study and in that by Knecht et al., DFT models get the relative change in contact density from HgF_{2} to HgF_{4} backward, whereas the CCSD(T) results are correct in sense and magnitude; DHF models get the sense correct but overestimate the magnitude. As a geochemical entity, HgF_{4} is highly exotic, however, and the excellent correlations seen in a variety of Hg^{0}, Hg^{+}, and Hg^{2+} species suggest that DFT-PAW is likely to be reliable for the forms of mercury typically found in natural environments, with the caveat that models of strange species may be inaccurate. The correlation of DFT-PAW contact densities with CCSD(T) energy shifts is notably better than the correlation between DHF energy shifts and Mulliken *s*-orbital occupancies, natural population analysis *s*-orbital occupancies, or atomic polar tensor charges, which were previously used to constrain liquid–vapor fractionation of mercury isotopes (19).

DHF calculations with the enhanced triple-ζ basis set used by Knecht et al. (29) suggest that the double-ζ basis set used in this study slightly underestimates contact densities (and their sensitivity to the chemical environment) by roughly 3% relative to a set constructed with additional, sharp basis functions to describe wave functions near the Hg nucleus. CCSD(T) model isotope substitution energies are only slightly sensitive (<1% difference for ^{198}Hg and ^{202}Hg in Hg^{0} vs. Hg^{2+}, ∼4% for HgF_{2} vs. Hg^{0}) to the choice of basis set. This basis set effect warrants further study, but appears to be smaller than other likely sources of error in DFT-PAW calculations of nuclear field shift fractionation.

The excellent correlation of CCSD(T) energies with DFT-PAW contact densities for all three elements suggests that it is possible to use calibrated DFT-PAW model contact densities to estimate isotope fractionations. Using linear regressions constrained to pass through atomic Sn^{2+}, Cd^{0}, and Hg^{0}, respectively, a 1 e^{–}/a_{0}^{3} change in contact density in a PW-PAW model is calculated to be equivalent to a 0.044 J/mol energy shift for ^{124}Sn/^{116}Sn substitution, a 0.30 J/mol energy shift for ^{114}Cd/^{110}Cd substitution, and a 0.034 J/mol energy shift for ^{202}Hg/^{198}Hg substitution. For PBE-PAW models, the corresponding sensitivities are 0.044 J/mol per e^{–}/a_{0}^{3}, 0.029 J/mol per e^{–}/a_{0}^{3}, and 0.035 J/mol per e^{–}/a_{0}^{3}. Regression slope uncertainties are ∼0.001 J/mol per e^{–}/a_{0}^{3} (1 SE). Scatter in the correlations suggest likely 1-σ errors of ±0.1 J/mol for ^{124}Sn/^{116}Sn and ^{114}Cd/^{110}Cd substitution, and ±0.4 J/mol for ^{202}Hg/^{198}Hg substitution, for either functional. To the extent that systematic errors in DFT-PAW models of solids are similar to those in the molecular calibration set, these energy uncertainties suggest likely errors of ±0.03‰ for the nuclear-volume component of tin and cadmium isotope fractionations at 25 °C, and ±0.1‰ for mercury; uncertainties will scale with Δ<*r*^{2}> for other isotope pairs, and as the reciprocal of the absolute temperature.

### Application to Liquid–Vapor Fractionation of Cadmium and Mercury.

Previous electronic structure studies of liquid mercury have likened its behavior to crystalline analogs (e.g., ref. 53). Following a similar approach, DFT-PAW models of crystalline cadmium and mercury were constructed. Hypothetical simple-cubic, body-centered cubic, and face-centered cubic structures of mercury were also modeled, to examine the sensitivity of the contact density to bond environments and nearest-neighbor density in the crystal, following ref. 53. To the extent that liquid metals have similar electronic structures, these should give a rough approximation of their isotopic properties.

The results are shown in Table 1. As was noted with molecules, the choice of functional has only a minor effect on isotope substitution energies, 0.04 J/mol in the case of crystalline cadmium and ≤0.08 J/mol for mercury—less than the expected scatter in the calibration. The different hypothetical crystal structures of mercury also fall within a rather limited range, varying by up to 0.5 J/mol with the PW functional and 0.6 J/mol with the PBE functional. This relatively small variability is consistent with band structure calculations (53), which found roughly the same 6*s*-valence orbital occupations (76–82%; equivalent to 1.5–1.6 e^{–}/atom) for the simple-cubic, body-centered cubic, and face-centered cubic crystal structures. The 6*s*-related contact densities for these structures in the present study are 82–84% as large as for Hg^{0} vapor, which scales to 1.6–1.7 e^{–}/atom if possible changes in the shapes of the 6*s*-electron density distributions are ignored. In the case of cadmium, the calculated contact density of the crystal is consistent with 62–66% occupation of valence 5*s*-electronic bands (i.e., ∼1.3 e^{–}/atom). The higher valence *s*-orbital occupancy of solid mercury, relative to its group-12 neighbor cadmium, is expected because of relativistic stabilization of *s*-electrons in high atomic number elements.

The calculated ^{202}Hg/^{198}Hg fractionation between Hg^{0} vapor and crystalline-Hg ranges from −0.8‰ to −0.6‰ at 25 °C depending on the choice of structure. Combined with the equilibrium mass-dependent fractionation (−0.2 ± 0.1‰) estimated from lattice-dynamics models (19), the net theoretical fractionation of −1.1‰ to −0.7‰ closely matches recent liquid–vapor equilibration experiments [−1.5‰ to −1.0‰ (19); −0.9 ± 0.2‰ (18)]. Odd-isotope mass-independent signatures (Δ^{199}Hg and Δ^{201}Hg) are also in good agreement with experiment, if the compilation of nuclear charge radii of ref. 52 is used in the models. The somewhat poorer match with the compilation of ref. 51 is consistent with previous measurements of Δ^{199}Hg/Δ^{201}Hg (7, 19). There has not been an analogous equilibration experiment performed on cadmium, but the pattern of deviations from an expected “mass-dependent” fractionation mechanism (Fig. 2) crudely resembles the residuals found in a liquid-cadmium evaporation study (2), after a generalized power law correction was applied to measured isotope ratios in the most fractionated sample. The calculated deviation from mass-dependent fractionation in cadmium is more subtle than that measured among Cd(II) species in solvent extraction experiments (30), ∼0.013‰ vs. ∼0.16‰ in ^{111}Cd. The maximum deviation found between any species studied here (atomic Cd^{0} vs. ionic Cd^{2+}) is 0.06‰ at 25 °C.

### Applications to Crystals and Minerals.

Applying the same technique, equilibrium nuclear-volume driven ^{202}Hg/^{198}Hg fractionations in montroydite (HgO), cinnabar (HgS), and calomel (Hg_{2}Cl_{2}) relative to mercury vapor are also estimated. Fractionation increases with increasing oxidation state (e.g., Hg^{2+} in montroydite and cinnabar vs. Hg^{+} in calomel), and with the electronegativity of the bond partner (O vs. S). These trends are consistent with previous studies of molecules (4, 7). Interestingly, calomel is predicted to have slightly higher ^{202}Hg/^{198}Hg (0.1–0.2‰ at 25 °C) than gas-phase Hg_{2}Cl_{2}, even though it has the same basic molecular structure (55). This may reflect sharing of 6*s*-electrons from the mercury with chlorine atoms in neighboring molecules in the crystal structure to make incipient bonds. Nuclear-volume fractionations in cadmium-bearing minerals (CdO-monteponite and CdS-greenockite) are much smaller than their Hg-bearing analogs (<0.5‰ for ^{114}Cd/^{110}Cd at 25 °C) and may be dominated by mass-dependent effects at most relevant temperatures.

### Relationship with Mössbauer Isomer Shifts.

Mössbauer spectroscopy, based on resonant absorption of gamma rays in certain isotopes, is a highly sensitive probe of the electronic environments of nuclei. The Mössbauer isomer shift, in particular, is of interest here because it shows the same dependence on electron density at the nucleus as nuclear volume isotope fractionation (e.g., refs. 28, 56):

where *δ*(*v*)_{isomer} is the isomer shift in units of velocity, *Z* is the atomic number of the nucleus, *e* is the charge of an electron, |Ψ(0)|^{2}_{A,B} are the contact densities at the nucleus in substances *A* and *B* (for Mössbauer studies these would be the source and absorber materials), Δ<*r*^{2}> is the change in root-mean-squared (rms) nuclear charge radius between the initial and final states of the Mössbauer isotope, *c* is the speed of light, and *E*_{γ} is the energy of the Mössbauer gamma ray. For a given Mössbauer isotope (or a given isotope pair, in the case of isotopic field shifts), it is expected that all of the terms appearing before the electron densities in Eq. **2** are constant (or very nearly so). To first order, the ratio of the difference in isomer shifts to the energy driving nuclear field shift fractionation will therefore be a constant multiple of the ratio of the change in the rms charge radius for the Mössbauer transition to the change in the rms charge radius of the field-shift isotopes:

If the relative differences in charge radii for the Mössbauer isomers and field-shift isotopes are known, this ratio can be calculated, allowing direct conversion of isomer shifts to predicted isotope fractionation factors. If the Mössbauer isomer charge radii are not known with sufficient accuracy, the isotope shift/isomer shift ratio can still be estimated by comparing calibrated DFT-PAW isotope substitution energies with measured isomer shifts. Once known, the ratio can then be used to convert isomer shifts to isotope substitution energies and thereby determine nuclear-field shift isotope fractionation factors.

As an example, calculated DFT-PAW energy shifts are compared with ^{119}Sn-Mössbauer isomer shifts for several tin-bearing crystals in Tables 2 and 3 and Fig. 3. An excellent correlation between DFT-PAW contact densities (modeled assuming point-charge nuclei) and isomer shifts has previously been reported (28); other electronic structure methods do similarly well (e.g., refs. 32 and 33). Note that the correlation lines do not pass through the origin because Sn^{2+} is used as the reference for DFT-PAW contact densities, whereas SnO_{2}-cassiterite is the reference absorber for isomer shifts. The slope of the correlation lines indicates a conversion factor of 0.43 ± 0.01 J/mol per mm/s for PW-PAW models, and 0.44 ± 0.01 J/mol per mm/s for PBE-PAW models. These imply Δ<*r*^{2}> values of 0.0082 ± 0.0001 fm^{2} and 0.0080 ± 0.0002 fm^{2} for the ^{119}Sn isomer pair, respectively, which are consistent with previous first-principles calibration studies (0.007–0.009 fm^{2}) (32, 33). The mean-squared radius difference found in ref. 28, 0.0048 fm^{2}, is smaller mainly because of the use of point-nucleus PAW datasets. Using the calculated sensitivities in Table 2, energy shifts and ^{124}Sn/^{116}Sn fractionation factors have been estimated for a suite of tin sulfide crystals. Many of these have complex structures with transition elements that would complicate first-principles modeling.

Although nuclear-volume isotope shifts calculated for Sn-bearing compounds are generally quite small (0.5‰ in ^{124}Sn/^{116}Sn for SnO_{2} cassiterite vs. SnO romarchite at 25 °C, 0.14‰ at 727 °C), and are likely dominated by mass-dependent fractionation at low and moderate temperatures (5.4‰ for SnO_{2} vs. SnO at 25 °C, 0.50‰ at 727 °C; (63)], they indicate that nuclear volume effects are a potentially measurable contributor to net isotopic fractionation of medium atomic-number elements in minerals. It should be noted that effects calculated in this study are smaller than the ∼6‰ field shift fractionation proposed earlier for dissolved tin species, based on solvent-extraction experiments (31).

The present calculations point to the potential utility of Mössbauer/DFT-PAW/Dirac fractionation calibrations in solids containing heavier elements such as platinum, mercury, thallium, and uranium, where nuclear volume effects can be major drivers of isotope fractionation at equilibrium. Transition, lanthanide, and actinide elements are particularly intriguing targets because their electronically complex naturally occurring compounds may be difficult to model. Another potential advantage is that, although uncertainties in mean-squared radius differences in isomers and isotopes will affect the slope of the isomer/isotope shift ratio, this type of uncertainty will generate a systematic error affecting all species proportionately. So the relative order and rough magnitude of interspecies mass-independent fractionations will be preserved even when fractionations are small.

## Conclusions

Calibrated DFT-PAW calculations provide a useful, computationally tractable method for estimating mass-independent isotope fractionations driven by changes in the sizes and shapes of nuclei. This method is applicable to condensed phases, including liquid mercury, that would be difficult to adequately model using molecular cluster methods. The close relationship between Mössbauer isomer shifts and equilibrium mass-independent fractionation makes it possible to harness the extensive Mössbauer spectroscopy literature to predict fractionations, including mineral phases with complex electronic structures that would not be suitable for DFT-PAW or all-electron methods. Previously observed trends in nuclear volume fractionation with oxidation state and ionic bonding are confirmed, as is the tendency of heavy elements (Hg) to fractionate more strongly than midrange elements (Sn and Cd). Even for these latter elements, however, mass-independent fractionation is likely to be a significant (if secondary) component of overall fractionation, especially at elevated temperatures.

## Acknowledgments

I thank M. Thiemens, and the PNAS editors and reviewers for encouragement and helpful suggestions to improve the manuscript. This study also benefitted from discussions with James Rustad, Ronald Cohen, Laurence Yeung, Abby Kavner, Bridget Bergquist, Sanghamitra Ghosh, and others. This work was funded by National Science Foundation Grants EAR-1047668 and EAR-0711411.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: schauble{at}ucla.edu.

Author contributions: E.A.S. designed research, performed research, analyzed data, and wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1216216110/-/DCSupplemental.

## References

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Fujii Y,
- et al.

^{235}U in U(IV)-U(VI) chemical exchange. Z Naturforsch A 44(5):395–398. - ↵King WH (1984)
*Isotope Shifts in Atomic Spectra (Physics of Atoms and Molecules)*. (Plenum, New York). - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Fujii T,
- Moynier F,
- Agranier A,
- Ponzevera E,
- Abe M

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵Saue T, et al. (2010) DIRAC, a relativistic ab initio electronic structure program, release DIRAC10. Available at http://dirac.chem.vu.nl. Accessed March 25, 2011.
- ↵Bast R, et al. (2011) DIRAC, a relativistic ab initio electronic structure program, release DIRAC11. Available at http://dirac.chem.vu.nl. Accessed May 25, 2012.
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Saue T,
- Fægri K,
- Helgaker T,
- Gropen O

- ↵
- ↵Fricke G, Heilig K (2004)
*Group I: Elementary Particles, Nuclei and Atoms.*Nuclear Charge Radii (Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology, New Series) (Springer, Berlin), Vol 20. - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Bird SRA,
- Donaldson JD,
- Silver J

- ↵
- ↵
- ↵
- ↵

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