Nuclear field vs. nucleosynthetic effects as cause of isotopic anomalies in the early Solar System

Earth and Planetary Science Letters 247 (2006) 1 – 9 www.elsevier.com/locate/epsl

Nuclear field vs. nucleosynthetic effects as cause of isotopic anomalies in the early Solar System Toshiyuki Fujii a,b,⁎, Frédéric Moynier b , Francis Albarède b b

a Research Reactor Institute, Kyoto University, 2-1010 Asashiro Nishi, Kumatori, Sennan, Osaka 590-0494, Japan Laboratoire de Sciences de la Terre, UMR 5570 CNRS, Ecole Normale Supérieure de Lyon, 46, Allee d'Italie, 69364 Lyon Cedex 7, France

Received 24 February 2006; received in revised form 17 April 2006; accepted 20 April 2006 Available online 5 June 2006 Editor: R.W. Carlson

Abstract Even in the absence of nucleosynthesis, mass-independent isotope effect due to nuclear field can lead to a number of isotope anomalies found in refractory inclusions of chondrites. Isotope anomalies observed for Ba, Ca, Sr, Ti, and Cr in FUN inclusions may simply be explained by the nuclear field effects of these elements. A whole class of isotopic heterogeneities may therefore reflect evaporation/condensation processes in the solar nebula rather than nucleosynthetic effects. We speculate that the current estimates of the r-process component of the normal solar material may be affected by non-mass dependent fractionation effects. © 2006 Elsevier B.V. All rights reserved. Keywords: isotope anomalies; FUN; mass-independent; nuclear field; nucleosynthesis

With very few exceptions, notably oxygen, the bulk isotopic abundances of the elements in the Solar System are believed to reflect the mixing of different components produced by different nucleosynthetic processes [1–3]. Most of the isotopic variability is accounted for by four types of processes: mass-dependent thermodynamic fractionation, radioactive decay, spallation by cosmic rays, and incomplete mixing in the solar nebula. Isotopic anomalies are understood as the rather unusual deviations from the mean solar proportions remaining after correction for radiogenic ingrowth and massdependent isotopic fractionation [4–6]. The anomalous abundances observed, notably for Mg, Si, Ca, Ti, Cr, Sr, ⁎ Corresponding author. Research Reactor Institute, Kyoto University, 2-1010 Asashiro Nishi, Kumatori, Sennan, Osaka 590-0494, Japan. Tel.: +81 724 51 2469; fax: +81 724 51 2634. E-mail address: [email protected] (T. Fujii). 0012-821X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2006.04.034

Ba, Nd, and Sm (see [7] for a review), in the so-called FUN inclusions (referring to Fractionated and Unknown Nuclear effects), which are unusual calcium–aluminumrich inclusions (CAIs) found in the Allende carbonaceous chondrite, epitomize the occurrence of isotopic anomalies in early nebular condensates. Some isotopic anomalies cannot, however, be accounted for by nucleosynthetic processes [6,7] and we therefore explore the possibility that they may instead be due to mass-independent isotope fractionation induced by nuclear field shift. We will discuss the extreme conditions of the solar nebula in which these fractionation effects are the most likely to be operative. The standard theory [8,9] holds that mass-dependent fractionation arises as a quantum mechanical effect due to different zero-point vibrational energies for different isotopes. The theory of first-order mass-dependent fractionation has recently been revised to include the

2

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

so-called nuclear field shift effect [10,11]. The spatial distribution of protons in the nucleus, which impacts the charge distribution interacting with electronic shells, obeys symmetry requirements that in turn cause the nuclear charge radii to vary unevenly with the number of neutrons characterizing the different isotopes of a same element [12]. The ensuing shift of the nuclear field imparts a mass-independent character to the electric field around the nucleus of the different isotopomers [11] and therefore to mass fractionation among coexisting species. The variability of nuclear spins from one isotope to another has also been considered to be an extra cause of mass-independent isotope fractionation. This effect has been discussed with reference to both systems at equilibrium [11,13,14] and kinetic fractionation [15]. Bigeleisen [11] concluded for uranium that, at equilibrium, this effect can safely be neglected with respect to the nuclear field-shift effect. A number of experiments, mostly involving solvent extraction and liquid chromatography, confirmed the existence of mass-independent fractionation as predicted by the nuclear field shift effect (see [16]). We will demonstrate that, for several elements, the pattern of isotopic anomalies observed in FUN inclusions follows that of their nuclear charge radii.

This observation is corroborated by experiments involving the same elements with similar patterns of massindependent isotope fractionation. Differences in mean-squared nuclear charge radii [12,17], δ〈r2〉, are plotted for the isotopes of Ca, Sr, Ba, Ti, and Cr, which are all elements with an even number of protons (Fig. 1). As can be seen most clearly for Ti and Ba, the mean-squared radius of the charge distribution in the nucleus, 〈r2〉, of an odd atomic mass-number nuclide (with odd number neutrons) is smaller than the value expected from the adjacent isotopes with even atomic mass numbers (i.e., with even number neutrons) [12]. In addition, nuclei with a magic number of neutrons (N = 20, 28, 50, 82, or 126) consist entirely of closed shells of neutrons [12] and therefore these nuclei are particularly compact. For example, 40Ca (N = 20), 48Ca, 50Ti, and 52Cr (N = 28), and 88Sr (N = 50) have smaller values of 〈r2〉. This characteristic is most prominent for Ca and Cr. Striking isotopic anomalies are present in the alkaline-earth elements of FUN inclusions. The case for Ba in FUN inclusions, notably in the EK-1-4-1 and C-1 inclusions, which are the most representative samples of FUN inclusions from the Allende chondrite,

Fig. 1. Variation of the mean-squared radii with mass number for a) Ca, b) Ti, c) Cr, d) Sr, and e) Ba. Data are taken from [12,17], which also reports the error bars on the radii.

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

is particularly illuminating. While EK-1-4-1 is unique in its isotopic properties, a number of inclusions are closely related to C-1 in their isotopic compositions [18,19]. The mass-independent isotope effects are calculated in ε units (relative deviations of the isotopic ratios with respect to the reference values in parts per 10,000) using the following equation (for derivation see Appendix A).  emi ¼

m2 ðmi −m1 Þ 2 yhr im1 ;m2 yhr im1 ;mi − mi ðm2 −m1 Þ 2

 a

ð1Þ

in which mi stands for the atomic mass of a nuclide indexed with the variable i and a is an adjustable parameter depending on temperature T as 1/T and representing the overall extent of mass-independent fractionation. By substituting the ε values measured in the samples and the literature δ〈r2〉 and mass data for all isotope combinations into the equation above, the

3

parameter a can be determined by regression. The precision on a is mostly dependent on that of the ε values, but its magnitude depends on the process by which fractionation takes place, in particular on the temperature (see below), and on the proportion of anomalous and unfractionated element. As in the original literature [20], the measured Ba isotope compositions are first normalized to a reference value for 134Ba/138 Ba. The ε values plotted as a function of mass number show that the abundances of the other isotopes corrected for mass-fractionation in EK-1-4-1 and C-1 clearly are anomalous (Fig. 2 and Table 1), and the isotopic variation patterns closely follow the variation of nuclear charge radius. The C-1 inclusion may contain a smaller fraction of Ba that was exposed to mass-independent fractionation processes than inclusion EK-1-4-1. Both enrichment (EK-1-4-1) and depletion (C-1) of odd atomic-mass isotopes with respect to the even atomic-mass isotopes are observed. Furthermore,

Fig. 2. Isotopic anomalies of alkaline-earth elements (Ba, Ca, and Sr). ε values are in parts per 10,000. a) Ba(EK-1-4-1), b) Ba(C-1), c) Ca(EK-1-4-1), d) Ca(C-1), and e) Sr(USNM 1623-5). Open symbols are for literature data [19,20,24] (errors are 2σ uncertainties) and closed symbols for the data calculated using Eq. (1). The isotope pairs used for normalization are 134Ba/138Ba, 40Ca/44Ca, and 86Sr/88Sr.

4

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

Table 1 δ〈r2〉 and calculated ε values Ba (m1 = 138, m2 = 134) Mass number

δ〈r2〉m1,mi a) [fm2]

Ek-1-4-1(a = − 284.89 [fm− 2]) Calculated εmi

134 135 136 137 138

− 0.040 ± 0.007 − 0.068 ± 0.006 − 0.034 ± 0.004 − 0.055 ± 0.005 0

b)

4

[10 ]

0 10.9 ± 0.1 4.1 ± 0.0 12.9 ± 0.1 0

C-1 (a = 23.55 [fm− 2])

Measured εmi

b)

4

[10 ]

0 13.4 ± 1.0 −0.8 ± 0.6 12.3 ± 0.4 0

Calculated εmi c) [104]

Measured εmi c) [104]

0 − 0.9 ± 0.0 − 0.3 ± 0.0 − 1.1 ± 0.0 0

0 −1.8 ± 0.7 0.6 ± 0.6 −0.6 ± 0.4 0

Ca (m1 = 44, m2 = 40) Mass number

δ〈r2〉m1,mi a) [fm2] − 0.283 ± 0.006 − 0.068 ± 0.008 − 0.158 ± 0.007 0 − 0.288 ± 0.009

40 42 43 44 48

EK-1-4-1 (a = −242.10 [fm− 2])

C-1 (a = 55.54 [fm− 2])

Calculated εmi b) [104]

Measured εmi d) [104]

Calculated εmi b) [104]

Measured εmi d) [104]

0 − 16.2 ± 0.1 22.3 ± 0.2 0 126.8 ± 1.3

0 17.4 ± 2.9 7.8 ± 10.3 0 133.6 ± 7.3

0

0

3.7 ± 0.0 − 5.1 ± 0.0 0 − 29.1 ± 0.3

0.3 ± 2.4 −2.3 ± 8.7 0 −30.0 ± 4.4

Sr (m1 = 88, m2 = 86) USNM 1623-5 (a = −315.57 [fm− 2])

δ〈r2〉m1,mi a) [fm2]

Mass number

84 86 87 88

0.111 ± 0.015 0.047 ± 0.006 0.005 ± 0.003 0

Calculated εmi b) [104]

Measured εmi e) [104]

− 4.7 ± 0.1 0 5.8 ± 0.0 0

−4.6 ± 1.5 0 5.8 ± 0.5 0

Ti (m1 = 48, m2 = 46) Mass number

46 47 48 49 50

δ〈r2〉m1,mi f) [fm2] 0.110 ± 0.006 0.020 ± 0.025 0 −0.134 ± 0.025 −0.164 ± 0.008

EK-1-4-1 (a = − 361.79 [fm− 2])

C-1 (a = − 153.71 [fm− 2])

Calculated εmi b) [104]

Measured εmi g) [104]

Calculated εmi b) [104]

Measured εmi g) [104]

0 12.2 ± 0.3 0.0 29.8 ± 0.7 22.7 ± 0.2

0 12.7 ± 1.6 0 18.8 ± 2.2 36.9 ± 2.4

0 5.2 ± 0.1 0 12.7 ± 0.3 9.7 ± 0.1

0 5.2 ± 0.8 0 − 7.0 ± 1.0 − 51.2 ± 1.1

Cr (m1 = 52, m2 = 50) Mass number

50 52 53 54 a b c d e f g h

δ〈r2〉m1,mi a) [fm2] 0.073 ± 0.022 0 0.062 ± 0.018 0.159 ± 0.030

EK-1-4-1 (a = 206.98 [fm− 2])

C-1 (a = − 82.39 [fm− 2])

Calculated εmi b) [104]

Measured εmi h) [104]

Calculated εmi b) [104]

Measured εmi h) [104]

0 0 20.0 ± 0.4 46.9 ± 1.7

0 0 16.2 ± 0.7 48.5 ± 0.7

0 0 − 7.9 ± 0.2 −18.7 ± 0.7

0 0

[17]. Errors are originated from δ〈r2〉. No errors are set for a. [20]. Averaged values of measured values [24]. [19]. [12]. [35]. [36].

2.7 ± 1.6 − 23.2 ± 1.8

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

this isotopic pattern reproduces the mass-independent fractionation effects consistently observed in chemical fractionation experiments on Ba [21–23]. Overall, the calculations reproduce the anomalous isotopic variations well, indicating that the nuclear field shift effect accounts adequately for the anomalous isotopic variations of Ba observed in FUN inclusions. The Ca isotopic anomalies [24] found in EK-1-4-1 and C-1 are shown in Fig. 2c–d (46Ca is not considered because of its very low abundance and thus large resulting uncertainty on the data). Again, the pattern of isotopic variations, especially for C-1, is similar to that of 〈r2〉. A large excess (EK-1-4-1) and a large deficit (C-1) are seen for 48Ca. In spite of a small discrepancy for 42 Ca, which possibly represents a true nucleosynthetic anomaly, the calculation reproduces, as for Ba, the isotopic anomalies well. Most experimental work on Ca isotope fractionation during chemical exchange [25] unfortunately only report results on 40Ca, 44Ca, and 48 Ca. In some systems, however, isotope separation factors of the 40Ca–44Ca and 40Ca–48Ca pairs show a breakdown of the mass-dependant law (∝ δm / mm′), which may also be assigned to the nuclear field shift effect. These experimental observations suggest that the anomalous isotopic variations of 48Ca found in FUN inclusions may result from mass-independent chemical isotope fractionation. The relationship between the nuclear field shift effect and mass-independent isotope fractionation was first predicted [26] for Sr and since confirmed for various chemical media [27]. As for Ba and Ca, a similar dependence of Sr isotope anomalies in FUN inclusions [28] on the nuclear field shift effect is expected. Because 87 Sr/86Sr corrected for radiogenic ingrowth from 87Rb over 4.5 Ga is similar to the basaltic achondrite best initial ratio [4], we considered that the value of ε87 is zero. It was pointed out [6], however, that the renormalization of Papanastassiou and Wasserburg's [28] data results in an excess of 87Sr. Hence, we decided not to discuss the Sr anomalies in EK-1-4-1 and C-1. However, in the refractory inclusion USNM 1623-5 from the Vigarano carbonaceous chondrite, which is thought to be isotopically similar to C-1 [19], the predicted anomaly is present. Surprisingly, once the 87Sr abundance in USNM 1623-5 has been corrected for radiogenic ingrowth from 87Rb over 4.56 Ga, the anomaly (ε87 = 5.8) is consistent with the effect predicted by the nuclear field shift theory (Fig. 2e). It nevertheless remains small with respect to the magnitude of radioactive effects. Even Mg seems isotopically anomalous [29] in EK1-4-1 and C-1. Once their Mg isotope compositions

5

are normalized to the terrestrial 25Mg/24Mg ratio, a deficit of 26Mg of about 3‰ is found in both inclusions [7]. The lack of correlation between the excesses of 26Mg and the Al/Mg ratios in the different phases of the inclusions of C-1 and EK-1-4-1 demonstrates that the observed isotopic anomalies cannot be attributed to the decay of 26Al, and thus the possibility of some unidentified nucleosynthetic effect on one or both of the other two isotopes of Mg (24Mg and 25Mg) has been considered [30]. Since Mg has only three isotopes, evidence of the nuclear field shift effect can only be based on one nuclide Nonetheless, the mean-squared charge radii [31] decrease in the order 24Mg > 26Mg > 25Mg. Remarkably, a mass-independent isotope fractionation effect has also been observed in chemical extraction experiments [32]. The isotopic anomalies of Mg may therefore also be attributable to the nuclear field shift effect. Mass-independent fractionation effects also have been observed in chemical exchange experiments on Ti and Cr that are successfully explained by nuclear field shift effects [16,33,34]. This work prompted us to review the isotopic evidence on FUN inclusions. The isotopic variations of Ti [35] and Cr [36] found in EK-14-1 are shown in Fig. 3a and c. There is a significant discrepancy at mass 50 between literature and predicted abundances, but the overall isotopic pattern of Ti is reproduced by the nuclear field shift theory. For Cr, the observed excesses of 53Cr and 54Cr are also well predicted by the δ〈r2〉 dependence. In contrast, the isotopic pattern of Ti and Cr observed in C-1 (Fig. 3b and d) are different from those of EK-1-4-1 and these patterns are poorly reproduced by the nuclear field shift effect. This suggests that the isotopic anomalies observed in C-1 are of either radioactive (53Cr) or nucleosynthetic origin. Isotopic anomalies of other transition metals, Fe [37] and Zn [38], have also been detected in FUN inclusions. The isotopic analysis of Fe and Zn by thermal ionization mass spectrometry (TIMS) is known to be arduous [39], which accounts for the rather large analytical errors. A mass-independent isotope effect caused by the silica-gel method has also recently been discussed [40]. Given the rapid changes in the performances of multiple-collector inductively coupled mass spectrometers (MC-ICP-MS), notably for the analysis of Fe [41] and Zn [42], the evaluation of the suitability of our model for these elements should be postponed until more data become available. The isotopic anomalies observed on Sm [43] in FUN inclusions cannot be even approximately reproduced by the nuclear field-shift theory. The mass-independent

6

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

Fig. 3. Isotopic anomalies of transition metal elements (Ti and Cr) for a) Ti in EK-1-4-1, b) Ti in C-1, c) Cr in EK-1-4-1, and d) Cr in C-1. The data have been normalized for mass-dependent isotope fractionation using a,b) 46Ti/48Ti [35] and c,d) 50Cr/52Cr [36], and errors are 2σ uncertainties. Open symbols are for literature data and closed symbols for the data calculated using Eq. (1). Since the pattern of isotopic variations of Ti observed in C-1 is markedly different from that of , we fitted our calculation result to the literature ε47 and calculated ε49 and ε50.

isotopic variations of Sm in FUN inclusions appear to be genuine nucleosynthetic anomalies related to a mixture of s- and r-processes, which probably reflects the very large neutron cross-sections of lanthanides. For Nd [20], there is unfortunately no r-contribution free isotope pair available for normalization [7,20], which prevents us from using Eq. (1). Nevertheless, when the original Bigeleisen equation (Eq. (A1) in Appendix A) is used, most of the Nd isotopic anomalies found in EK-1-4-1 and C-1 can be reproduced, with the significant exception of 146 Nd for EK-1-4-1. This agreement, however, does not stand out as decisive evidence, because Eq. (A1) contains two free parameters, which gives the adjustment considerable leeway. A fair proportion of the isotopic anomalies observed in FUN inclusions of the Allende chondrite already 25 yrs ago are, therefore, consistent with Bigeleisen's theory [11] relating nuclear field shift effect to massindependent isotope fractionation and chemical exchange experiments. The effect of nuclear spin, from which further mass-independent fractionation is expected [11,13,14], remains to be added to the theory. Mass-dependent effects take place in a variety of multiphase nebular and planetary systems, notably between solids and ambient gases during condensation and vaporization, and between minerals and ambient fluids during metamorphism on planetary bodies. The

potential sites in which non-mass dependent fractionation is operative should essentially be the same. The apparent restriction of the non-mass dependent fractionation effects to refractory inclusions seems to at least point to very high-temperature processes, such as evaporation and condensation of silicate material in the inner nebular disk. Bigeleisen's theory [10] suggests that at high temperature, the effect of field shift (in 1/T) dominates mass-dependent effects (in 1/T 2 ). The direction of fractionation in a particular mineral, and therefore its relative excess/deficit of individual nuclides, depends on the particular phase change (e.g., vaporization or condensation) and of the extent of the reaction. These effects should be most visible in the relative stable isotope abundances of some elements. With the exception of a few elements, such as H, O, C, and S, the patterns of mass-dependent isotope fractionation in the Solar System remain essentially unknown. It can be seen from Eq. (1) that both mass-dependent and mass-independent fractionation are intimately related and therefore that the presence of visible massindependent effects may hint at strong mass-dependent fractionation effects showing through the standard steps of mass-fractionation corrections used, notably for data obtained by thermal-ionization mass spectrometry. For example, the recently described excesses of 54 Cr characteristic of different planetary objects [44] may

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

be a prime example of mass-independent fractionation and may signal non-nucleosynthetic isotopic heterogeneities of the Solar Nebula related to variable proportions of volatile and refractory components. The small and outward increasing excess of 53Cr in the Solar System (< 0.5 ε unit) is too systematic to be an artifact of the analytical procedure and may reflect the effect of a temperature gradient on the scale of the nebular disk. The potential overlap of mass-independent effects with radiogenic ingrowth suggests that a re-evaluation of some chronological results may also be in order, notably for 87Sr, 26Mg, and 53Cr. For instance, the time scales of planetary accretion inferred from 53Cr and 26Al are not fully consistent [45,46] and the possibility that some of the ages provided by these chronometers may be biased by mass-independent effects should be investigated. As a final note, we will take the speculation a little further. A first analysis of the Ba and Nd isotopic abundances in FUN inclusions led Clayton [4] to conclude that their r-process excesses did not fit the model proportions and thus suggested the existence of unusual nuclear anomalies. Consolmagno and Cameron [6] reinvestigated the issue and assumed instead that the r-process abundances resemble those observed in normal solar material. This may be interpreted in two ways. Either, Consolmagno and Cameron's original explanation is correct and their method gives the best estimates of the r-process abundances, or the excesses of r-process nuclides obtained by subtracting the s-process contribution to the solar abundances are pervasively polluted by nonmass dependent effects. Acknowledgment The authors wish to thank Malcolm McCulloch for his insightful suggestions at the early stages of this work, Dimitri Papanastassiou and an anonymous reviewer for constructive comments, and Janne Blichert-Toft for help in improving the English of this paper. This work was supported by the scientist exchange program of the Japan Society for the Promotion of Science. TF thanks Takafumi Hirata for his kind encouragements towards this collaborative research. Appendix A. Derivation of Eq. (1) In the theory of Bigeleisen [10,11], the isotope fractionation factor α is defined as,    2 hc 1 h ym lna ¼ B ðA1Þ fs  A þ kT 24 2pkT mm V

7

where m′ and m indicate the masses of the light and heavy isotopes, respectively, and δm = m − m′. fs is the field shift, T the temperature, k and h the Boltzman and Planck constants, respectively, c the velocity of light, and A and B are adjustable constants. The field shift is proportional to the isotopic difference in nuclear charge radius, δ [12]. Hence, at constant temperature, Eq. (A1) can be simplified as, lna ¼ yhr2 im;m V  a þ

ym b mm V

ðA2Þ

in which a and b are new adjustable coefficients. We note that a is a function of 1/T and b a function of 1/T2. Since there is still no physical model for evaluating the scaling factor a, it must be estimated from the experimental values of ε's by employing Eq. (A2). The sign and magnitude of a reflect the combination of all the unknown processes, both natural and analytical, that led to the observed isotope composition of the sample. Let us consider a system with three or more isotopes with masses m1, m2, and mi (i = 3, 4, …, i), in which an isotope pair m1 and m2 is used for normalization. Let us note Rmi/m1 the abundance ratio of the isotope of mass mi to the isotope of mass m1. The isotope deviation of this ratio in a sample with respect to a standard can be written in the ε notation (reported in parts per 10,000) as, emi ¼

RCmi =m1 RN mi =m1

−1

ðA3Þ

in which N stands for a ‘normal’ reference, and C for a ratio corrected for instrumental fractionation (using the exponential law). The mass bias in the mass spectrometer is best represented by an exponential mass fractionation law with factor β [47], so that,  −b mi RCmi =m1 ¼ RM  ðA4Þ mi =m1 m1 where M stands for a raw measured ratio before instrumental fractionation. If the variations of mi isotopic abundances result from chemical isotope fractionation, the correlation therefore should be, RM mi =m1 RN mi =m1

−1 ¼ yhr2 im1 ;mi  a þ

mi −m1 b mi m1

ðA5Þ

From Eqs. (A3)–(A5), and lnα = ln(1 + ε) this relationship can be rewritten   b  m1 mi −m1 2 1 þ em i ¼ 1 þ yhr im1 ;mi  a þ b mi mi m1 ðA6Þ

8

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9

We then take the logarithm of this expression. Since ε ≪ 1, the first-order expansion ln (1 + ε) ≈ ε is now used throughout so that:   m1 mi −m1 emi ¼ bln b ðA7Þ þ yhr2 im1 ;mi  a þ mi mi m1 If |m1 − mi| ≪ m1, this expression further simplifies to: emi

mi −m1 mi −m1 ¼ −b þ yhr2 im1 ;mi  a þ b mi mi m1

m2 −m1 m2 −m1 þ yhr2 im1 ;m2  a þ b m2 m2 m1

[9]

[10]

[11]

ðA8Þ [12]

Since εm2 is normalized to be zero for the isotope pair, m1 and m2, we obtain 0 ¼ −b

[8]

[13]

ðA9Þ [14]

so that the resulting value of β is b¼

m2 1 yhr2 im1 ;m2  a þ b m1 m2 −m1

Substituting Eq. (A10) into Eq. (A8) gives   m2 ðmi −m1 Þ 2 emi ¼ yhr2 im1 ;mi − yhr im1 ;m2  a mi ðm2 −m1 Þ

ðA10Þ

[15]

[16]

ðA11Þ

which is identical to Eq. (1). As expected from the correction of the mass-dependent bias, the final equation does not contain the mass effect coefficient B, which is a function of 1/T2. By comparing εmi and δ〈r2〉m1,mi, Eq. (A11) can be used to test for a possible effect of the nuclear field shift.

References [1] F. Kaeppeler, R. Gallino, M. Busso, G. Picchio, C.M. Raiteri, Sprocess nucleosynthesis—classical approach and asymptotic giant branch models for low-mass stars, Astrophys. J. 354 (1990) 630–643. [2] E.M. Burbidge, G.R. Burbidge, W.A. Fowler, F. Hoyle, Synthesis of the elements in stars, Rev. Mod. Phys. 29 (1957) 547–650. [3] G. Wallerstein, I. Iben Jr., P. Parker, A.M. Boesgaard, G.M. Hale, A.E. Champagne, C.A. Barnes, F. Kaepeler, V.V. Smith, R.D. Hoffman, F.X. Timmes, C. Sneden, R.N. Boyd, B.S. Meyer, D.L. Lambert, Synthesis of the elements in stars: forty years of progress, Rev. Mod. Phys. 69 (1997) 995–1084. [4] D.D. Clayton, On strontium isotopic anomalies and odd-A pprocess abundances, Astrophys. J. 224 (1978) L93–L95. [5] D.D. Clayton, An interpretation of special and general isotopic anomalies in r-process nuclei, Astrophys. J. 224 (1978) 1007–1012. [6] G.J. Consolmagno, A.G.W. Cameron, The origin of the Fun anomalies and the high-temperature inclusions in the Allende meteorite, Moon Planets 23 (1980) 3–25. [7] J.L. Birck, An overview of isotopic anomalies in extraterrestrial materials and their nucleosynthetic heritage, Geochemistry of

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25] [26]

Non-Traditional Stable Isotopes, Rev. Mineral. Geochem. 55 (2004) 25–64. H.C. Urey, The thermodynamic properties of isotopic substances, J. Chem. Soc. (1947) 562–581. J. Bigeleisen, M.G. Mayer, Calculation of equilibrium constants for isotopic exchange reactions, J. Chem. Phys. 15 (1947) 261–267. J. Bigeleisen, Temperature dependence of the isotope chemistry of the heavy elements, Proc. Natl. Acad. Sci. U. S. A. 93 (1996) 9393–9396. J. Bigeleisen, Nuclear size and shape effects in chemical reactions. Isotope chemistry of the heavy elements, J. Am. Chem. Soc. 118 (1996) 3676–3680. W.H. King, Isotope Shifts in Atomic Spectra, Plenum Press, New York, 1984. T. Fujii, T. Hirata, Y. Shibahara, K. Nishizawa, Mass-dependent and mass-independent isotope effects of zinc in chemical exchange reactions using liquid chromatography with a cryptand stationary phase, Phys. Chem. Chem. Phys. 3 (2001) 3125–3129. D.A. Knyazev, G.K. Semin, A.V. Bochkarev, Nuclear quadrupole contribution to the equilibrium isotope effect, Polyhedron 18 (1999) 2579–2582. N.J. Turro, Influence of nuclear-spin on chemical-reactions— magnetic isotope and magnetic-field effects (a review), Proc. Natl. Acad. Sci. U. S. A. 80 (1983) 609–621. T. Fujii, D. Suzuki, K. Gunji, K. Watanabe, H. Moriyama, K. Nishizawa, Nuclear field shift effect in the isotope exchange reaction of chromium(III) using a crown ether, J. Phys. Chem., A 106 (2002) 6911–6914. P. Aufmuth, K. Heilig, A. Steudel, Changes in mean-square nuclear-charge radii from optical isotope shifts, At. Data Nucl. Data Tables 37 (1987) 455–490. D.A. Papanastassiou, C.A. Brigham, The identification of meteorite inclusions with isotope anomalies, Astrophys. J. 338 (1989) L37–L40. R.D. Loss, G.W. Lugmair, A.M. Davis, G.J. MacPherson, Isotopically distinct reservoirs in the solar nebula: Isotope anomalies in Vigarano meteorite inclusions, Astrophys. J. 436 (1994) L193–L196. M.T. McCulloch, G.J. Wasserburg, Barium and neodymium isotopic anomalies in the Allende meteorite, Astrophys. J. 220 (1978) L15–L19. T. Fujii, E. Hamanishi, Y. Shibahara, J. Inagawa, K. Watanabe, K. Nishizawa, Temperature dependence of isotope effects of barium using a crown ether, J. Nucl. Sci. Technol. 39 (2002) 447–450. K. Nishizawa, K. Nakamura, T. Yamamoto, T. Masuda, Separation of strontium and barium isotopes using a crownether—different behaviors of odd-mass and even mass isotopes, Solvent Extr. Ion Exch. 12 (1994) 1073–1084. A. Kondoh, T. Oi, M. Hosoe, Fractionation of barium isotopes in cation-exchange chromatography, Sep. Sci. Technol. 31 (1996) 39–48. T. Lee, D.A. Papanastassiou, G.J. Wasserburg, Calcium isotopic anomalies in the Allende meteorite, Astrophys. J. 220 (1978) L21–L25. K.G. Heumann, Isotopic-separation in systems with crown ethers and cryptands, Top. Curr. Chem. 127 (1985) 77–132. K. Nishizawa, T. Satoyama, T. Miki, T. Yamamoto, M. Hosoe, Strontium isotope effect in liquid–liquid extraction of strontium chloride using a crown ether, J. Nucl. Sci. Technol. 32 (1995) 1230–1235.

T. Fujii et al. / Earth and Planetary Science Letters 247 (2006) 1–9 [27] Y. Shibahara, K. Nishizawa, Y. Yasaka, T. Fujii, Effect of counter anion in isotope exchange reaction of strontium, Solvent Extr. Ion Exch. 21 (2003) 435–448. [28] D.A. Papanastassiou, G.J. Wasserburg, Strontium isotopic anomalies in the Allende meteorite, Geophys. Res. Lett. 5 (1978) 595–598. [29] T. Lee, D.A. Papanastassiou, Mg isotopic anomalies in the Allende meteorite and correlation with O and Sr effects, Geophys. Res. Lett. 1 (1974) 225–228. [30] G.J. Wasserburg, T. Lee, D.A. Papanastassiou, Correlated O and Mg isotopic anomalies in Allende inclusions: II. Magnesium, Geophys. Res. Lett. 4 (1977) 299–302. [31] I. Angeli, A consistent set of nuclear rms charge radii: properties of the radius surface R (N, Z), At. Data Nucl. Data Tables 87 (2004) 185–206. [32] K. Nishizawa, T. Nishida, T. Miki, T. Yamamoto, M. Hosoe, Extractive behavior of magnesium chloride and isotopic enrichment of magnesium by a crown ether, Sep. Sci. Technol. 31 (1996) 643–654. [33] T. Fujii, J. Inagawa, K. Nishizawa, Influences of nuclear mass, size, shape and spin on chemical isotope effect of titanium, Ber. Bunsenges. Phys. Chem. 102 (1998) 1880–1885. [34] T. Fujii, D. Suzuki, K. Watanabe, H. Yamana, Application of the total evaporation technique to chromium isotope ratio measurement by thermal ionization mass spectrometry, Talanta 69 (2006) 32. [35] F.R. Niederer, D.A. Papanastassiou, G.J. Wasserburg, Titanium isotope anomalies in Allende inclusions, Meteoritics 15 (1980) 339. [36] D.A. Papanastassiou, Chromium isotopic anomalies in the Allende meteorite, Astrophys. J. 308 (1986) L27–L30. [37] J. Völkening, D.A. Papanastassiou, Iron isotope anomalies, Astrophys. J. 347 (1989) L43–L46.

9

[38] J. Völkening, D.A. Papanastassiou, Zinc isotope anomalies, Astrophys. J. 358 (1990) L29–L32. [39] I.T. Platzner, K. Habfast, A.J. Walder, A. Goetz, Modern Isotope Ratio Mass Spectrometry, John Wiley & Sons, Chichester, 1997. [40] G. Manhès, C. Göpel, Heavy stable isotope measurements with thermal ionization mass spectrometry: non mass-dependent fractionation effects between even and uneven isotopes, Geophys. Res. Abstr. 5 (2003) 10936. [41] X.K. Zhu, Y. Guo, R.K. O'Nions, E.D. Young, R.D. Ash, Iron isotope homogeneity in the early Solar nebula, Nature 412 (2001) 311–313. [42] J.M. Luck, D.B. Othman, F. Albarède, Zn and Cu isotopic variations in chondrites and iron meteorites: Early solar nebula reservoirs and parent-body processes, Geochim. Cosmochim. Acta 69 (2005) 5351–5363. [43] G.W. Lugmair, K. Marti, N.B. Scheinin, Incomplete mixing of products from r-, p-, and s-process nucleosynthesis: Sm–Nd systematics in Allende inclusions, Lunar Planet. Sci. IX (1978) 672–674. [44] A. Trinquier, J.L. Birck, C.J. Allègre, 54Cr anomalies in the solar system: their extent and origin, Lunar Planet. Sci. XXXVI (2005) 1259. [45] J.L. Birck, C.J. Allegre, Manganese–chromium isotope systematics and the development of the early solar system, Nature 331 (1988) 579–584. [46] G.W. Lugmair, A. Shukolyukov, Early solar system timescales according to 53Mn–53Cr systematics, Geochim. Cosmochim. Acta 62 (1998) 2863–2886. [47] C.N. Maréchal, P. Telouk, F. Albarède, Precise analysis of copper and zinc isotopic compositions by plasma-source mass spectrometry, Chem. Geol. 156 (1999) 251.