An analytical formulation of isotope fractionation due to self-shielding

Abstract

Isotope fractionation due to photochemical self-shielding is believed to be responsible for the enrichment of inner solar system planetary materials in the rare isotopes of carbon, nitrogen, and oxygen relative to the Sun. Self-shielding may also contribute to sulfur isotope mass-independent fractionation in modern atmospheric sulfates, although its role in the early Earth atmosphere has not yet been convincingly established. Here, I present an analytical formulation of isotopic photodissociation rate coefficients that describe self-shielding isotope signatures for 3 and 4-isotope systems broadly representative of O and S isotopes. The analytic equations are derived for idealized molecular spectra, making an analytic formulation tractable. The idealized spectra characterize key features of actual isotopologue spectra, particularly for CO and SO₂, but are applicable to many small molecules and their isotopologues. The analytic expressions are convenient for evaluating the magnitude of isotope effects without having to pursue involved numerical solutions. More importantly, the analytic expressions illustrate the origin of particular isotope signatures, such as the previously unexplained large mass-dependent fractionation associated with photodissociation of optically-thick SO₂. The formulation presented here elucidates the origin of some of these important isotopic fractionation processes.

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.