Inner core composition paradox revealed by sound velocities of Fe and Fe-Si alloy

Knowledge of the sound velocity of core materials is essential to explain the observed anomalously low shear wave velocity (VS) and high Poisson’s ratio (σ) in the solid inner core. To date, neither VS nor σ of Fe and Fe-Si alloy have been measured under core conditions. Here, we present VS and σ derived from direct measurements of the compressional wave velocity, bulk sound velocity, and density of Fe and Fe-8.6 wt%Si up to ~230 GPa and ~5400 K. The new data show that neither the effect of temperature nor incorporation of Si would be sufficient to explain the observed low VS and high σ of the inner core. A possible solution would add carbon (C) into the solid inner core that could further decrease VS and increase σ. However, the physical property-based Fe-Si-C core models seemingly conflict with the partitioning behavior of Si and C between liquid and solid Fe.

Introduction Line 31 : "Recent … models", I have the feeling that these models are not that recent … Maybe a more detailed comment of the recent literature around metal/silicate partitioning would be required here to discuss the Si or O budget in the core, as well as the loss of volatile elements. This should be also put in perspective in the conclusion, as the present paper state a relatively high C content in the core. Line 38-42: This part is a bit weird. First, the authors state that there are no data on Fe at core pressure. And the sentence after, they discuss the difference between seismological observables and pure Fe at core conditions. So there are data at core pressures! I would recommend to make a more extensive discussion of experimental (static or dynamic) and theoretical dataset, the latest allowing the comparison with observables.

Results
The fact that the Birch law is still valid is a relatively important message from the present article. However, there are experimental results using LH-DAC setup with a somehow different result, finding some temperature effect on the Birch law, in agreement with Brown data (Lin et al., 2005;Sakamaki et al., 2016). Similar discussion could be performed by comparison with recent data on hcp Fe-Si at high temperature (Sakairi et al., 2018). The authors should discuss the fact that static compression find a deviation with the Birch law. The decrease of the VB induced by melting is relatively clear on the present dataset. What is the temperature of melting found for the Fe-Si alloy? Is it in agreement with static compression dataset? Could the author make supplementary figures presenting the Hugoniot in a P-T graph, and comparing with previous static and dynamic phase diagram results? Such comparison would be helpful, but not central to the present manuscript. Discussion and conclusion Line 156 : The authors discuss a composition of 3.6wt%Si that would fit the density, but not the other seismological observations. I would suggest to present this on the different graphs of Figure  3, instead of the 8.6 wtSi, that is not in agreement with any of the observables. With this figure, it is not possible to see the fact that Si could not satisfy the four presented observables. Also, I am wondering what are the uncertainties on the observables? Could the author present them on the Figure 3? Without comparing with the strongest effect due to premelting, how the present effect of T on VS could be compared with numerical simulations results? This should be commented also… Line 163 : larger instaed of "lager" Line 188: ideal instead of "idea" Regarding the effect of carbon and silicon on the phase diagram and elastic properties of hcp Fe, the authors should compare their results with the present list of recent bibliography : (Edmund et al., 2019;Li et al., 2018;Miozzi et al., 2020;Pamato et al., 2020) These different papers discuss the compatibility of an FeSiC ternary system as an inner core. This study consists of measurements of density, compressional velocity, and bulk sound speed on iron and an iron-silicon alloy from shock experiments. These values are used to calculate shear velocity and Poisson's ratio and demonstrate that an iron-silicon alloy cannot match all the measured values of these properties in the inner core. The study goes on to propose that an ideal mixture of their measured iron-silicon alloy and an iron-carbon alloy can match all the inner core values, despite this mixture being geochemically unlikely (thus the "paradox"). Examinations of iron alloys at core relevant conditions are fundamental to mineral physics. The experimental data in this study appear to be novel, well collected, and well analyzed. The conclusion that neither temperature nor Si alloying can account for the measured inner core properties is quite interesting. However, the extension of the analysis to carbon-bearing alloys is less well supported. No velocity data were collected on iron-carbon alloys, so it is unclear why C (and Fe7C3 in particular) was chosen above all other light element alloys. On top of this, there is insufficient discussion of important calculations throughout the paper, and the properties of the alloy mixture should be more carefully treated.

General comments • Grammar and typos
This manuscript, while not by any means difficult to read, contains an unusual number of errors in grammar and word choice. The text should be carefully proofread to fix these mistakes and ensure there are not any similar typos in the numbers or figures. • Quantity of supplemental information There is a lot of data in the supplement, and it is repeatedly referenced in the main text. The supplementary information currently contains significant portions of the results/analysis, and some of it should be moved unless there is a journal-imposed restriction. In particular, Figures S6, S7, and S8 would be of interest to most readers (S7 and S8 can probably be combined).
• Insufficient description of Birch's Law Birch's Law is invoked throughout the paper but is never explicitly defined or given a citation. Readers should be told what Birch's Law is, how you determined that your data are sufficiently linear to justify using it, and the significance of its applicability. • Insufficient justification of ideal mixing The analysis part of this study relies on the ideal mixing approximation holding for solid Fe alloys at inner core conditions. Is this likely to be true? Are there applicable non-ideal mixing models?
• Introduction of C is confusing The central "paradox" referenced in the title of this study is the difficulty in matching inner core physical properties with an Fe-Si alloy (which is predicted by geochemistry). This study says as much based on their experimental data, but then goes on to say that a C-bearing alloy (which was not part of their experimental dataset) could match all the geophysical parameters. Several things are unclear to me about this analysis: o Why was C selected? Could the parameters be matched with H, S, O alloys? o The values needed for Vp and Vs of Fe7C3 are not found in Table S3. Please be more explicit about how the calculation was performed, which values were used from references 16 and 33, and their uncertainties. o Why choose an alloy (Fe7C3) with no high temperature velocity data available? To what degree does the imposed (dVS/dT)P introduce uncertainty? Reference 33, for example, uses a wide range of (dVS/dT)P rather than the fixed value used here. o How was the "best fit" Fe-Si-C composition found? The error bars in Fig. 3 seem large enough that many compositions could have fit equally well.
Line comments • 11: Change "experimental" to "experiments" preexisting data on Fe and Fe-Si. Please clarify.
• 45: You should clarify which quantities were measured versus calculated. • 78: I assume "natural isotopic abundance" implies that you corrected for 57Fe enrichment in NRIXS data? If so, say that explicitly, perhaps in the relevant figure captions.
• 81: Final sentence of paragraph should be rewritten. • 102: These pressures are not, in fact, consistent with the cited study, which finds melting from 225-260 GPa. • 105: Why was this Si content chosen? • 153: A 6000 K ICB was an assumption even in the referenced study. Perhaps briefly justify this temperature choice. • 154: The "M" in PREM already stands for "model". Also, there have been recent revaluations of inner core properties (e.g., 10.1126/science.aav2296 and references therein), is PREM still the best point of comparison for this study? • 158: Perhaps similarity would be "not so surprising" but Fe and Fe-8.6Si appear identical. There are several studies which show at least a small effect of Si content on Vp (e.g., 10.1029/2003GL018405). • 166: How much higher than that of Fe? • 170: How is "near melting" different than "pre-melting"? Is there any melting outside of the shaded "melting region"? • 185: Mention that this is ~8 wt% C. • 198: This sentence just repeats Line 190. • 201: The core C content is controversial, maybe this should be reframed as the required 4% inner core C requiring a certain bulk core or bulk Earth abundance. • 202: You should cite a study of solid-liquid alloy C partitioning and comment on whether it is reasonable for the inner core to have all the C.
• 208: Be more specific about the types of data that could help explore this problem. • Figure 1: Please clarify how the boundaries of the "melt region" were determined and the associated pressures. I recommend using a different color for the Nguyen data and adjusting the "melting region" bar so that it does not cover the axes and tick marks. The revised version of this study is a substantial improvement, and the authors have done a particularly good job at clarifying the motivation behind their alloy mixing approach. There are just a few points that should be clarified before this study is ready for publication. Also, there are some lingering grammatical mistakes; I recommend letting a native English speaker proofread the final manuscript.
L40 and elsewhere: Should be "Birch's law" not "the Birch's law" L283: The authors should state that the best fit composition was chosen without considering the error bars of the endmembers, as explained in the reviewer response document. L249: I am still unsure of what values were used to calculate the Fe7C3 properties. Table S3 has not changed in the revision and does not seem like it contains some parameters (like shear measurements. All of the starting parameters and the actual equation used to calculate the derived (Table S4) values should be explicitly identified in the manuscript or supplement. L432: What is this "assumed value"? L435: Equations 10 and 11 should be explained. The reader shouldn't have to infer what the subscripts, dels, and deltas mean. Figure 5: There should be a sentence in the main text mentioning the Ni effect and justifying not including it in the analysis, as explained in the reviewer response document. Figure 6: The red circle and its error bars are hard to see and should be larger.