An analytical formulation of isotope fractionation due to self-shielding
Introduction
Photochemical processes can be a significant source of isotope fractionation in the terrestrial sedimentary rock record (e.g., Farquhar et al., 2000a), and have been proposed to play a key role in determining the distribution of light stable isotopes in the solar system (Thiemens and Heidenreich, 1983, Clayton, 2002). Gas-phase photochemistry is often a source of mass-independent isotope fractionation (MIF), which, in elements with 3 or more stable isotopes, leaves a readily identified isotopic signature, in comparison to the more usual mass-dependent fraction (MDF) processes. Two primary processes are responsible for photochemical MIF: a non-statistical and isotope-dependent population of rovibrational energy levels in a molecule, and ultraviolet shielding effects during photodissociation. As an archetypal example of the former, ozone formation exhibits a remarkable mass-independent isotope signature (Thiemens and Heidenreich, 1983, Gao and Marcus, 2001), and in the atmosphere is readily transferred to a wide array of oxygen-containing gaseous species (Lyons, 2001, Thiemens, 2006). Transfer of a positive O-MIF signature from O3 to stratospheric CO2 via O(1D) (Yung et al., 1991) yields a small negative O-MIF signature in tropospheric O2 by mass balance (Bender et al., 1994, Luz et al., 1999, Young et al., 2014). Subaerial oxidation of continental sulfides transfers a portion of the negative O-MIF in O2 to the resulting sulfates, allowing an important constraint to be placed on atmospheric CO2 and net primary productivity over parts of the Paleoproterozoic and Neoproterozoic (Bao et al., 2008, Crockford et al., 2018). Sulfur isotope MIF signatures extend even further back in time to Archean rocks, and serve as a critical proxy of O2 in the early atmosphere (Farquhar et al., 2000a, Farquhar et al., 2000b, Bekker et al., 2004). The chemical physics responsible for the O-MIF effect seen in O3 formation is likely to involve non-statistical effects in the formation of symmetric versus asymmetric isotopomers of O3 (Gao and Marcus, 2001), but a definitive experimental demonstration of this mechanism has yet to made. The mechanism of sulfur MIF observed in ancient rocks is also debated, but may involve sulfur allotropes addition reactions (Babikov, 2017, Harman et al., 2018).
Self-shielding isotope effects arise from the UV absorption properties of isotopologues of a given molecule. Usually, a statistical distribution of states exists in the molecule. Of greater importance for this case is the nature of the absorption spectrum. For a line-type absorption spectrum, in which transitions occur between specific rovibronic states, abundance-dependent saturation occurs in an optically-thick column of gas, leading to sometimes massive mass-independent isotope effects. (I will refer to this as a MIF isotope effect, but a significant mass-dependent component can also be present depending on the nature of the absorption spectra.) For oxygen isotopes, CO is the archetypal self-shielding molecule, with numerous astrochemical examples known (Bally and Langer, 1982, van Dishoeck and Black, 1988, Lyons and Young, 2005, Smith et al., 2009.) A key application of self-shielding is to understand the distribution of O isotopes in the solar system, and, in particular, why the Sun is 60‰ lighter in 17O and 18O than are the terrestrial planets (Clayton, 2002, Lyons and Young, 2005). N2, isoelectronic with CO, also undergoes strong self-shielding with isotopic implications for astrochemical environments (e.g., Heays et al., 2014) and planetary atmospheres (Liang et al., 2007).
Here, I derive analytical expressions for O and S isotope fractionation due to self-shielding for the case of idealized spectra. ‘Idealized’ means that perturbations to the isotopic cross sections due to interactions among excited electronic states will be neglected, as will line overlap for line-type absorption spectra. I have recently discussed the isotopic effects of line overlap for pressure broadened SO2 (Lyons et al., 2018), a process that is not easily treated analytically. Band overlap is accounted for in the latter portion of this work.
Broadly stated, my objective is to illustrate the origin of self-shielding fractionation effects, and to explore the range of possible MIF from self-shielding, particularly for oxygen isotopes in CO, and for sulfur isotopes with application to Archean S-MIF signatures. More specifically, my objective is to provide analytical expressions that will be of use to geochemists who are attempting to assess the importance of self-shielding isotope effects in photochemical experiments or in Earth and planetary environments. But underlying the approach I take here is the issue of how much spectroscopic detail is needed to account for photo-induced isotope signatures. The molecular spectroscopy of molecules, even for diatomics, is extremely complex when numerous electronic excited states, and the rotational and vibrational levels within them, are involved. However, photodissociation-derived isotope effects are quantified by photodissociation rate coefficients, which are quantities that are integrated over the relevant wavelength region for dissociation. It is therefore reasonable to expect that perfectly complete spectroscopic data for each isotopologue of a given molecule are not necessary to quantitatively capture the photodissociation-derived isotope effects. This work explores the veracity of that expectation by deriving analytic expressions for isotopic photodissociation rate coefficients based on simplified absorption cross sections, and comparing to previously published laboratory photodissociation experiments. As is shown an analytic formulation of isotopic dissociation rate coefficients can capture much of the behavior observed in experiments.
Section snippets
Photodissociation rate coefficients
Photochemical reactions are classified as unimolecular reactions, and in simplest form can be written asIn general, the products A and B are in excited states, but this does not influence the self-shielding isotope effects described here. However, excited state products often have greater subsequent reactivity versus ground state products, and it is therefore generally necessary to know the isotopic variation in the product branching ratios for a given reaction (e.g., Jiang et al., 2019
Types of gas-phase absorption spectra
Most large isotope effects in photochemical reactions arise from photodissociation, usually requiring UV photons. Photon absorption by a molecule in an electronic and (usually) vibrational ground state creates an excited electronic state in the molecule, with a wide array of possible vibrational and rotational states, depending on the exact photon energy and wavefunction overlap. If the excited electronic state is unbound, the molecule directly dissociates into constituent atoms, in a process
Self-shielding by non-overlapping line-type spectra
Simplified but representative cross sections are assumed (Fig. 3) as an approximation to actual line-type spectra (Fig. 2a). Making this approximation allows an analytical evaluation of Eqs. (6), (7). This approximation is most relevant to bands in a molecule such as CO, with very high peak-to-valley ratios. Similar approximations are applicable to N2, O2, and other diatomic molecules. The absorption cross sections for Fig. 3 are given by
Line-type spectra for symmetric molecules
Symmetric molecules in which an isotope of a non-central atom is substituted, e.g., of the form X2 or XYX in which X is substituted, undergo a reduction in symmetry. For linear or diatomic molecules with a point group the reduction in symmetry increases the number of rotational lines by about a factor of 2. For other molecules entirely new bands are formed, e.g., the 4.6 micron band in CH3D which is not present in CH4. Here, I consider the case of symmetric diatomic or linear triatomic
Self-shielding by overlapping box-type spectra
Spectral overlap is a common feature of nearly all isotopic spectra. Rovibronic lines in spectra of diatomic molecules are often spaced fairly widely apart and the lines are sufficiently narrow that line overlap does not occur frequently for isotopologues. This is true for most dissociating bands of CO and N2, but not all diatomics have narrow linewidths as evidenced by S2 (Stark et al., 2018). Triatomics and larger molecules usually exhibit less line-type structure in their absorption spectra,
Results
I have presented a derivation of accurate analytic expressions for isotopic photodissociation rate coefficients for idealized non-overlapping line-type and overlapping box-type cross sections. The idealized cross sections were chosen to illustrate self-shielding-derived isotope fractionation in molecules with absorption spectra dominated by narrow, mostly non-overlapping, lines such as occur in electronic transitions in CO and N2, and in molecules with a significant component of broad,
Discussion
The analysis presented here can be improved and expanded in several ways. For the overlapping box-type spectra, the overlapping rectangles could be replaced by overlapping triangular bands. For SO2 triangles would yield a better fit to the pseudo-continuum than do the rectangular boxes, and will still yield a closed-form solution. Another possibility is the addition of narrow lines to the top of a box-like band. This would be a representation of both the pseudo-continuum and the line-type
Conclusions
The complexity of electronic absorption spectra for diatomic and polyatomic molecules for predissociating transitions nearly always requires a numerical treatment of isotope fractionation during photodissociation. Using simplified cross sections, I have presented an analytic formulation of isotope fractionation due to self-shielding by the primary isotopologue of a molecule undergoing dissociation. The formulation considers two basic features common in cross sections: line-type cross sections,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was funded by grants from the NASA Emerging Worlds program (grant #80NSSC18K0592) and the NASA Exobiology program (grant #80NSSC19K0475).
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