# Seismic Wave Speeds Derived from Nuclear Resonant Inelastic X-ray Scattering for Comparison with Seismological Observations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### Alternative Expressions for the Debye Speed under Various Simplifying Assumptions

## 3. Results and Discussion

#### 3.1. Variations in the Debye Speed

#### 3.2. Extracting Seismic Wave Speeds

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NRIXS | Nuclear Resonant Inelastic X-ray Scattering |

## Appendix A. Seismic Wave Speeds

#### Appendix A.1. Hexagonal Symmetry

#### Appendix A.2. Cubic Symmetry

## Appendix B. Experimental Measurements of Cobalt

**Table A1.**The Debye ${v}_{D}$ and seismic wave speeds ${c}_{p}$ and ${c}_{s}$ of cobalt at pressures of zero to 40 GPa. The alloy’s density in $\mathrm{g}/{\mathrm{cm}}^{3}$ ($\rho $ is given in column 2 [45]), while the strength of anisotropy (${A}^{L}$; [22]), and the Debye speed calculated using Equation (7) and the elements of the elasticity of cobalt (${v}_{D}$; [44]) are given in columns 3 and 4. The material’s seismic wave speed (${c}_{p}$; column 5) is compared against longitudinal wave speeds obtained using the full non-linear equations of Equation (21) (${c}_{p}^{nl}$; column 6), and experimentally measured values determined using ultrasonics (${c}_{p}^{US}$; column 7; [46] ), inelastic X-ray scattering (${c}_{p}^{IXS}$; column 8; [45]), and impulsive stimulated light scattering (${c}_{p}^{ISLS}$; column 9; [47]). Columns 10 through 14 are the same as columns 5 through 9 except for shear/transverse wave speeds.

Pressure | $\mathit{\rho}$ | ${\mathit{A}}^{\mathit{L}}$ | ${\mathit{v}}_{\mathit{D}}$ | ${\mathit{c}}_{\mathit{p}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{n}\mathit{l}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{U}\mathit{S}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{I}\mathit{X}\mathit{S}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{I}\mathit{S}\mathit{L}\mathit{S}}$ | ${\mathit{c}}_{\mathit{s}}$ | ${\mathit{c}}_{\mathit{s}}^{\mathit{n}\mathit{l}}$ | ${\mathit{c}}_{\mathit{s}}^{\mathit{U}\mathit{S}}$ | ${\mathit{c}}_{\mathit{s}}^{\mathit{I}\mathit{X}\mathit{S}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{I}\mathit{S}\mathit{L}\mathit{S}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 8.8 | 0.09 | 3.44 | 5.7 | 5.7 | 5.8 | - | - | 3.1 | 3.1 | 3.1 | - | - |

11 | 9.3 | 0.06 | 3.67 | 6.2 | 6.2 | - | 6.3 | 6.1 | 3.3 | 3.3 | 3.3 | 3.1 | - |

40 | 10.3 | 0.03 | 4.18 | 7.3 | 7.3 | - | 7.3 | - | 3.7 | 3.7 | - | 3.9 | - |

## Appendix C. Mean Projected Wave Speed for Randomly Oriented Samples

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**Figure 2.**Percent variation of ${v}_{\widehat{\mathbf{k}}}$ as a function of the strength of anisotropy ${A}^{L}$ for the five hcp iron alloys (black circles) in Table 1. The dashed line represents a linear best fit to the data.

**Figure 3.**Distribution of the deviation of the Debye speed obtained using acoustic wave speeds estimated using the Voigt (red), Reuss (blue), and Voigt-Reuss-Hill (grey) averaged elasic moduli with respect to the true Debye speed. One million randomly generated tensors of materials with hexagonal symmetry and identical seismic wave speeds are used.

**Figure 4.**The dependence of the deviation of the Debye speed ${v}_{D}$ from that based upon the Voigt average, ${v}_{D}^{V}$, and the seismic speeds for the hcp iron alloys in Table 1 as a function of the strength of anisotropy ${A}_{L}$. The semi-transparent grey circles represent the difference between ${v}_{D}$ (Equation (7)) and ${v}_{D}^{V}$ (Equation (3)) for ${10}^{6}$ randomly generated transversely isotropic elastic tensors with the same seismic wave speeds as those of pure iron (Table 1). The red circles correspond to the difference between ${v}_{D}$ and ${v}_{D}^{V}$ for the five hcp iron alloys (Table 1) and the blue and purple circles denote the corresponding deviation in inferred acoustic longitudinal and transverse wave speeds, respectively (Table 3). The dashed lines represent linear best fits to the acoustic wave speed data.

**Figure 5.**Comparison of ${c}_{s}$ against estimates of ${c}_{s}$ derived from the Debye speed. The solid grey line represents a one-to-one line. The dashed lines represent $\pm 5\%$ of the shear speed. Points to the left of the grey line underestimate ${c}_{s}$ where as points to the right overestimate ${c}_{s}$.

**Table 1.**The elastic constants and seismic wave speeds of iron alloys (column 1; [17,18,19,20,21]). The alloy’s density in $\mathrm{g}/{\mathrm{cm}}^{3}$ ($\rho $; column 2), seismic compressional wave speed in km/s (${c}_{p}$; column 3), shear wave speed in km/s (${c}_{s}$; column 4), the strength of anisotropy (${A}^{L}$; column 6; [22]), and the elastic tensor elements in GPa (columns 6 through 11) are summarized.

Material | $\mathit{\rho}$ | ${\mathit{c}}_{\mathit{p}}$ | ${\mathit{c}}_{\mathit{s}}$ | ${\mathit{A}}^{\mathit{L}}$ | ${\mathit{c}}_{11}$ | ${\mathit{c}}_{33}$ | ${\mathit{c}}_{13}$ | ${\mathit{c}}_{44}$ | ${\mathit{c}}_{66}$ | ${\mathit{c}}_{12}$ |
---|---|---|---|---|---|---|---|---|---|---|

hcp Fe | 13.0 | 11.8 | 3.9 | 1.5 | 2150 | 1685 | 990 | 140 | 60 | 2030 |

hcp Fe-Si | 13.1 | 11.3 | 3.9 | 0.3 | 1674 | 1855 | 1120 | 176 | 137 | 1400 |

hcp Fe-Si-Ni | 12.5 | 11.8 | 3.3 | 1.4 | 1816 | 1964 | 1224 | 80 | 49 | 1718 |

hcp Fe-Si-C | 13.1 | 11.6 | 3.7 | 0.6 | 1712 | 2066 | 1263 | 164 | 91 | 1530 |

hcp Fe-S-C | 13.1 | 11.8 | 4.2 | 0.3 | 1831 | 2091 | 1214 | 183 | 173 | 1485 |

bcc Fe-Si | 13.6 | 11.5 | 4.2 | 1.7 | 1562 | 1562 | 1448 | 366 | 366 | 1448 |

**Table 2.**Exact and approximate Debye speeds calculated from the elastic tensor elements of hcp and bcc iron alloys (column 1). Percent variation (column 2) of the difference of the extreme values of ${v}_{\widehat{\mathbf{k}}}$ (column 3) with respect to the average of the extreme values due to variations in $\widehat{\mathbf{k}}$ are given. The true Debye speed ${v}_{D}$ (column 4), and the approximate Debye speed obtained using the analytical expressions for ${c}_{m}$ given in the Appendix A (column 5) are compared. The last three columns give the values obtained using Equation (3) where the elastic moduli $\kappa $ and $\mu $ are estimated using the Voigt, Reuss, and Voigt-Reuss-Hill averages, respectively. The speeds are all in units of km/s.

Material | $\pm \%$ | $\left[min{\mathit{v}}_{\widehat{\mathit{k}}},max{\mathit{v}}_{\widehat{\mathit{k}}}\right]$ | ${\mathit{v}}_{\mathit{D}}$ | ${\mathit{v}}_{\mathit{D}}^{\mathit{q}}$ | ${\mathit{v}}_{\mathit{D}}^{\mathit{V}}$ | ${\mathit{v}}_{\mathit{D}}^{\mathit{R}}$ | ${\mathit{v}}_{\mathit{D}}^{\mathit{V}\mathit{R}\mathit{H}}$ |
---|---|---|---|---|---|---|---|

hcp Fe | $40\%$ | $\left[3.07,\phantom{\rule{3.33333pt}{0ex}}4.60\right]$ | 3.35 | 3.35 | 4.45 | 3.18 | 3.87 |

hcp Fe-Si | $16\%$ | $\left[4.04,\phantom{\rule{3.33333pt}{0ex}}4.76\right]$ | 4.23 | 4.22 | 4.47 | 4.16 | 4.32 |

hcp Fe-Si-Ni | $37\%$ | $\left[2.66,\phantom{\rule{3.33333pt}{0ex}}3.85\right]$ | 2.89 | 2.96 | 3.78 | 2.77 | 3.32 |

hcp Fe-Si-C | $28\%$ | $\left[3.53,\phantom{\rule{3.33333pt}{0ex}}4.66\right]$ | 3.79 | 3.83 | 4.21 | 3.67 | 3.95 |

hcp Fe-S-C | $13\%$ | $\left[4.38,\phantom{\rule{3.33333pt}{0ex}}4.97\right]$ | 4.54 | 4.56 | 4.77 | 4.47 | 4.62 |

bcc Fe-Si | $0\%$ | $\left[4.04,4.04\right]$ | 4.04 | 4.04 | 4.79 | 3.33 | 4.13 |

**Table 3.**Comparison of seismic wave speeds obtained from ${v}_{D}$ for iron alloys at inner core conditions (Table 1). The material’s true compressional wave speed (${c}_{p}$; column 2) is compared against those obtained using linearized form of Equation (21) (${c}_{p}^{l}$; column 3) and solving the full non-linear equations (${c}_{p}^{nl}$; column 4). Columns 5 through 7 are the same as columns 2 through 4 except for shear waves. The last column gives the shear wave speed estimated from the Debye speed using the relationship of Anderson et al. [31] (${c}_{s}^{A}$; Equation (22)).

Material | ${\mathit{c}}_{\mathit{p}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{l}}$ | ${\mathit{c}}_{\mathit{p}}^{\mathit{n}\mathit{l}}$ | ${\mathit{c}}_{\mathit{s}}$ | ${\mathit{c}}_{\mathit{s}}^{\mathit{l}}$ | ${\mathit{c}}_{\mathit{s}}^{\mathit{n}\mathit{l}}$ | ${\mathit{c}}_{\mathit{s}}^{\mathit{A}}$ |
---|---|---|---|---|---|---|---|

hcp Fe | 11.8 | 10.7 [–10%] | 11.5 [–3%] | 3.9 | 2.8 [–28%] | 3.0 [–23%] | 3.1 [–20%] |

hcp Fe-Si | 11.2 | 10.2 [–10%] | 11.2 [–1%] | 3.9 | 3.6 [–8%] | 3.7 [–5%] | 3.9 [–2%] |

hcp Fe-Si-Ni | 11.8 | 10.8 [–8%] | 11.5 [–2%] | 3.3 | 2.4 [–29%] | 2.6 [–22%] | 2.7 [–20%] |

hcp Fe-Si-C | 11.6 | 10.6 [–9%] | 11.4 [–1%] | 3.7 | 3.2 [–13%] | 3.4 [–9%] | 3.5 [–6%] |

hcp Fe-S-C | 11.8 | 10.6 [–10%] | 11.7 [–1%] | 4.2 | 3.9 [–7%] | 4.0 [–5%] | 4.1 [–1%] |

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**MDPI and ACS Style**

Delbridge, B.; Ishii, M. Seismic Wave Speeds Derived from Nuclear Resonant Inelastic X-ray Scattering for Comparison with Seismological Observations. *Minerals* **2020**, *10*, 331.
https://doi.org/10.3390/min10040331

**AMA Style**

Delbridge B, Ishii M. Seismic Wave Speeds Derived from Nuclear Resonant Inelastic X-ray Scattering for Comparison with Seismological Observations. *Minerals*. 2020; 10(4):331.
https://doi.org/10.3390/min10040331

**Chicago/Turabian Style**

Delbridge, Brent, and Miaki Ishii. 2020. "Seismic Wave Speeds Derived from Nuclear Resonant Inelastic X-ray Scattering for Comparison with Seismological Observations" *Minerals* 10, no. 4: 331.
https://doi.org/10.3390/min10040331