Constraints on grain size and stable iron phases in the uppermost Inner Core from multiple scattering modeling of seismic velocity and attenuation

We propose to model the uppermost inner core as an aggregate of randomly oriented anisotropic ``patches’’. A patch is defined as an assemblage of a possibly large number of crystals with identically oriented crystallographic axes. This simple model accounts for the observed velocity isotropy of short period body waves, and offers a reasonable physical interpretation for the scatterers detected at the top of the inner core. From rigorous multiple scattering modeling of seismic wave propagation through the aggregate, we obtain strong constraints on both the size and the elastic constants of iron patches. We perform a systematic search for iron models compatible with measured seismic velocities and attenuations. An iron model is characterized by its symmetry (cubic or hexagonal), elastic constants, and patch size. Independent of the crystal symmetry, we infer a most likely size of patch of the order of 400 m. Recent {\it bcc} iron models from the literature are in very good agreement with the most probable elastic constants of cubic crystals found in our inversion. Our study (1) suggests that the presence of melt may not be required to explain the low shear wavespeeds in the inner core and (2) supports the recent experimental results on the stability of cubic iron in the inner core, at least in its upper part.

💡 Research Summary

This paper addresses the long‑standing problem of reconciling the seismic observations of the Earth’s uppermost inner core—namely, isotropic P‑wave velocities, unusually low S‑wave speeds, and strong attenuation—with mineral‑physics constraints on iron at core conditions. The authors propose a physically simple yet mathematically rigorous model: the uppermost inner core is treated as an aggregate of randomly oriented anisotropic “patches”. Each patch consists of many iron crystals that share a common crystallographic orientation, while the orientation varies randomly from patch to patch. Because the patches are randomly oriented, the aggregate is macroscopically isotropic, but microscopically it is heterogeneous due to the intrinsic elastic anisotropy of the crystals.

The elastic stiffness tensor C(x) is decomposed into a spatial average C₀ (the Voigt average) and a fluctuating part δC(x) with zero mean. The spatial correlation of the fluctuations is described by a tensorial covariance Ξ multiplied by a scalar correlation function η(r)=exp(−r/a). The parameter a defines the correlation length; the effective patch (or grain) size is d=2a. This exponential form assumes convex, equiaxed patches, which is a reasonable approximation for polycrystalline iron.

Using the Dyson equation for the ensemble‑averaged Green’s function, the authors derive the effective wave numbers for coherent P‑ and S‑waves. In the First‑Order‑Smoothing Approximation (valid to second order in heterogeneity) the mass operators σ_P and σ_S are expressed in terms of the covariance tensor, the correlation function, and the wave vector. At low frequencies (λ≫d) the Born approximation is applied, allowing σ to be evaluated at the unperturbed wave number k₀=ω/V₀. The real part of σ modifies the phase velocity, while the imaginary part yields the attenuation α. The effective velocities V_e and quality factors Q are then given by:

1/V_e = 1/V₀ + (1/2) ω² V₀⁻¹ Re{σ(ω/V₀)}
α = (1/2) ω V₀⁻¹ Im{σ(ω/V₀)}
Q⁻¹ = ω α / V₀

These formulas capture both velocity dispersion and scattering‑induced attenuation in a single framework.

To connect the theory with real Earth data, the authors compile six iron elasticity datasets from the literature, covering three hexagonal‑close‑packed (hcp) and three body‑centered‑cubic (bcc) models. The datasets include both experimental measurements (e.g., X‑ray diffraction) and first‑principles calculations, and they span a wide range of anisotropy parameters. For each iron model the authors compute V_e and Q as functions of patch size d (100–800 m) and frequency (0.5–2 Hz), then compare the results with seismic constraints for the uppermost 200 km of the inner core: P‑wave speed ≈ 11 km s⁻¹, S‑wave speed ≈ 3.5–3.7 km s⁻¹, and P‑wave quality factor Q_P between 100 and 400.

The systematic inversion yields several robust conclusions:

1. Patch size – Across all iron models the best fit to the seismic data is obtained for a patch size of roughly 400 m (correlation length a≈200 m). This size is consistent with the scale of heterogeneities inferred from scattering analyses of PKIKP waves.

2. Hexagonal models – The hcp iron datasets generally predict S‑wave Voigt speeds of 4.0–4.4 km s⁻¹, substantially higher than the seismic value. Although strong anisotropy can increase scattering attenuation, the resulting effective S‑wave speeds remain too high, and the required anisotropy parameters are often inconsistent with laboratory measurements.

3. Cubic models – Recent bcc iron models (e.g., Vochčadlo 2007; Belonoshko et al. 2007) produce Voigt velocities and anisotropy parameters that, when combined with a 400 m patch size, match both the observed P‑wave velocity and the low Q_P values. The calculated attenuation is of the same order as the seismic estimates, indicating that scattering alone can account for the observed energy loss without invoking melt.

4. Implications for melt – Because the scattering attenuation derived from the bcc‑based patch model reproduces the observed Q_P, the authors argue that a pervasive liquid phase is not required to explain the low shear‑wave speeds. Instead, the reduction in effective S‑wave speed arises from the multiple scattering of waves across the anisotropic patches.

5. Stability of cubic iron – The agreement between the seismic inversion and the bcc elasticity datasets lends independent geophysical support to recent mineral‑physics proposals that body‑centered‑cubic iron may be the stable phase in the upper inner core, possibly stabilized by light elements such as Ni or Si.

Overall, the paper demonstrates that a relatively simple statistical description of the inner‑core microstructure, combined with rigorous multiple‑scattering theory, can bridge the gap between mineral‑physics predictions and seismic observations. The methodology provides a quantitative tool for extracting grain‑size information from seismic attenuation and velocity dispersion, and it opens the way for future studies that could incorporate depth‑dependent patch sizes, preferred orientation, or the presence of minor melt fractions.

In conclusion, the uppermost inner core is best described as an isotropic aggregate of ~400 m‑scale, randomly oriented anisotropic iron patches, most plausibly composed of a body‑centered‑cubic phase. Scattering from these patches alone explains the observed low shear‑wave speeds and strong attenuation, reducing the need to invoke a liquid component and reinforcing the case for cubic iron stability under core conditions.