**Abbreviations**

The following abbreviations are used in this manuscript:

NRIXS Nuclear Resonant Inelastic X-ray Scattering

#### **Appendix A. Seismic Wave Speeds**

The acoustic wave speeds *cm* for a given propagation direction **q**ˆ may be determined from the eigenvalues and eigenvectors of the Christoffel Matrix (*Mik* ≡ <sup>Λ</sup>*ijklq*<sup>ˆ</sup>*jq*<sup>ˆ</sup>*l*, where <sup>Λ</sup>*ijkl* are the elements of the fourth-order elasticity tensor) through the Christoffel Equations (e.g., [34–37]),

$$\rho c\_{m}^{2}(\clubsuit) = M\_{ik}\mathfrak{p}\_{i}^{m}\mathfrak{p}\_{k}^{m} = \Lambda\_{i\slash k\choose{k}}\mathfrak{q}\_{j}\mathfrak{q}\_{l}\mathfrak{p}\_{i}^{m}\mathfrak{p}\_{k}^{m} = \mathfrak{p}\_{m}\mathfrak{q}\_{l}:\Lambda:\mathfrak{p}\_{m}\mathfrak{q}\_{l} \tag{A1}$$

where **p**ˆ *m* is the unit polarization direction with the superscript and subscript *m* denote the mode of the acoustic phonon. Equation (A1) is solved by using the characteristic equation

$$\det|M\_{ik} - \rho c^2 \delta\_{ik}| = 0,\tag{A2}$$

where *δij* is the Kronecker delta. This equation results in the three eigenvectors (**p**ˆ *m*) and eigenvalues (*cm*). For isotropic materials, the eigenvectors and eigenvalues correspond to a longitudinal wave if the polarization direction is parallel to the propagation direction (i.e., **p**ˆ 1 · **q**ˆ = 1) and two transverse waves if the polarization direction is perpendicular to the propagation direction (i.e., **p**ˆ 2 · **q**ˆ = **p**ˆ 3 · **q**ˆ = 0). For weakly anisotropic materials, the eigenvectors of *Mij* are not necessarily orthogonal to **q**ˆ.

In order to derive analytical expressions for the wave speeds of hexagonal and cubic symmetries with weak anisotropy, it is useful to consider the problem in the spherical coordinate system with the unit vectors

$$\begin{aligned} \dot{\mathbf{r}} &= \sin\theta\cos\phi\mathbf{\hat{x}} + \sin\theta\sin\phi\mathbf{\hat{y}} + \cos\theta\mathbf{\hat{z}},\\ \dot{\theta} &= \cos\theta\cos\phi\mathbf{\hat{x}} + \cos\theta\sin\phi\mathbf{\hat{y}} - \sin\theta\mathbf{\hat{z}}, \quad \text{and} \\ \dot{\phi} &= -\sin\phi\mathbf{\hat{x}} + \cos\phi\mathbf{\hat{y}}, \end{aligned} \tag{A3}$$

where **x**ˆ, **y**ˆ, and **z**ˆ are unit vectors in the Cartesian coordinate system, and *θ* and *φ* are polar and azimuth angles, respectively, in the spherical coordinate system. These angles are typically defined relative to an axis of crystal symmetry (Figure 1). The propagation unit vector can also be expressed as

$$\mathbf{\dot{q}} = \sin\theta\cos\phi\mathbf{\dot{x}} + \sin\theta\sin\phi\mathbf{\dot{y}} + \cos\theta\mathbf{\dot{z}}.\tag{A4}$$

ˆ

## *Appendix A.1. Hexagonal Symmetry*

For transversely isotropic materials such as hexagonal close-packed (hcp) iron, the elastic stiffness is described by five independent parameters *A*, *C*, *F*, *L*, and *N* [38] that are related to the elasticity tensor Λ as

$$\begin{aligned} \Lambda\_{1111} = \Lambda\_{2222} = \mathfrak{c}\_{11} = A & \Lambda\_{1133} = \Lambda\_{2233} = \mathfrak{c}\_{13} = F & \Lambda\_{1212} = \mathfrak{c}\_{66} = N \\ \Lambda\_{1313} = \Lambda\_{2323} = \mathfrak{c}\_{44} = L & \Lambda\_{1122} = \mathfrak{c}\_{12} = A - 2N & \Lambda\_{3333} = \mathfrak{c}\_{33} = \mathbb{C} \end{aligned} \tag{A5}$$

where the subscripts indicate the indices of the fourth-order tensor. Note that, in the limiting case of an isotropic material with zero anisotropy, *A* = *C* = *κ* + 43*<sup>μ</sup>*, *L* = *N* = *μ*, and *F* = *κ* − 23*<sup>μ</sup>*, where *κ* and *μ* are the elastic moduli, the incompressibility and rigidity, respectively.

We assume that the polarization vectors are orthonormal, and use **p**ˆ 1 = **r**ˆ, **p**ˆ 2 = *θ*, and **p**ˆ 3 = *φ*ˆ. Substituting these expressions for the propagation and polarization vectors (Equation (A3)) and the transversely isotropic elastic tensor (Equation (A5)) into the Christoffel equations (Equation (A1)) gives the first-order perturbation solutions for the longitudinally polarized wave speed *c*1 and the two transversely polarized wave speeds *c*2, and *c*3 (e.g., [39–41]),

$$\begin{aligned} \rho c\_1^2 &= A - 2(A - F - 2L)\cos^2\theta + (A + \mathbb{C} - 2F - 4L)\cos^4\theta, \\ \rho c\_2^2 &= L + (A + \mathbb{C} - 2F - 4L)\cos^2\theta - (A + \mathbb{C} - 2F - 4L)\cos^4\theta, \quad \text{and} \\ \rho c\_3^2 &= N + (L - N)\cos^2\theta. \end{aligned} \tag{A6}$$

These wave speeds are given as functions of the polar angle *θ*, the angle between the symmetry axis and the propagation direction **q**ˆ (Figure 1). Because *c*1 gives speed for a wave propagating in the direction of polarization, this coresponds to seismic P-wave speed while *c*2 and *c*3 corresopnd to seismic S-wave speeds. Note that the wave speeds is independent of the azimuthal angle *φ*.

Using the the first order perturbation solutions for the wave speeds (Equation (A6)), the Debye speed (Equation (2)) is expressed as

$$\frac{1}{w\_D^3} \approx \frac{1}{6} \int\_0^\pi \sin \theta \left( \frac{1}{c\_1^3(\theta)} + \frac{1}{c\_2^3(\theta)} + \frac{1}{c\_3^3(\theta)} \right) d\theta. \tag{A7}$$

Note, that this expression with wave speeds defined in Equation (A6) is valid only for weakly anisotropic case where the polarization directions for the three acoustic waves can be assumed to be orthogonal. For strongly anisotropic materials, the polarization vectors and the wave speeds should be calculated numerically via the characteristic equation (Equation (A2)).

Taking the average over the unit sphere, the expression for the directionally dependent seismic wave speeds based upon Equation (A6) become

$$\begin{aligned} \rho \left< c\_1^2 \right> &= \frac{1}{15} (8A + 3C + 4F + 8L), \\ \rho \left< c\_2^2 \right> &= \frac{1}{15} (2A + 2C - 4F + 7L), \quad \text{and} \\ \rho \left< c\_3^2 \right> &= \frac{1}{3} (2N + L). \end{aligned} \tag{A8}$$

These expressions are compatible with the Voigt averaged <sup>Λ</sup>*ijkl*moduli which are given as

$$\begin{aligned} \kappa\_0^V &= \frac{1}{9}(4A + \mathcal{C} + 4F - 4N), \quad \text{and} \\ \mu\_0^V &= \frac{1}{15}(A + \mathcal{C} - 2F + 6L + 5N), \end{aligned} \tag{A9}$$

since *ρ c*22 + *c*23 /2 = *μV*0 and *ρ c*21 = *κV*0 + 4/3*μV*0 . However, the Reuss averaged <sup>Λ</sup>−<sup>1</sup> *ijkl*−<sup>1</sup> moduli are given as

$$\begin{aligned} \mu\_0^R &= \frac{(A-N)\mathbb{C} - F^2}{A - N + \mathbb{C} - 2F}, \quad \text{and} \\ \mu\_0^R &= \frac{15LN[(A-N)\mathbb{C} - F^2]}{6(L+N)[(A-N)\mathbb{C} - F^2] + LN(4A + \mathbb{C} + 4F - 4N)}, \end{aligned} \tag{A10}$$

which are not compatible with seismological observations. These expressions are consistent with those obtained by other authors such as Watt and Peselnick [42].

## *Appendix A.2. Cubic Symmetry*

For cubic materials such as body-centered cubic (bcc) iron, the elastic stiffness is described by three independent parameters *c*11, *c*12, and *c*44 that are related to the elasticity tensor Λ as

$$\begin{aligned} \Lambda\_{1111} &= \Lambda\_{2222} = \Lambda\_{3333} = \mathfrak{c}\_{11}, \\ \Lambda\_{1122} &= \Lambda\_{2233} = \Lambda\_{3311} = \mathfrak{c}\_{12}, \quad \text{and} \\ \Lambda\_{3131} &= \Lambda\_{2323} = \Lambda\_{3131} = \mathfrak{c}\_{44}. \end{aligned} \tag{A11}$$

Note that, in the limiting case of an isotropic material with zero anisotropy, *c*12 corresponds to the Láme parameter *λ* = *κ* − 23*<sup>μ</sup>*, *c*44 corresponds to the shear modulus *μ*, and *c*11 = *c*12 + 2*c*44. For ease of notation, we will use the parameters *λ*, *μ*, and *η* to describe the cubic material where *λ* = *c*12, *μ* = *c*44, and *η* = *c*11 − *c*12 − 2*c*44.

ˆ

In case of hexagonal symmetry, the unit vectors *θ* and *φ*ˆ provided the fastest and slowest polarization directions for the two shear waves. This is not the case for cubic material, and even though we can assume **p**ˆ 1 = **r**ˆ, the polarization for the shear waves would depend upon an additional angle *ξ* such that **p**ˆ 2 = cos *ξ* ˆ *θ* + sin *ξφ*ˆ and **p**ˆ 3 = cos(*ξ* + *π*/2)<sup>ˆ</sup>*<sup>θ</sup>* + sin(*ξ* + *π*/2)*φ*ˆ. The angle *ξ* depends upon *θ* and *φ* and is obtained through the expression

$$\cot 2\xi = \frac{\left(1 + \cos^2 \theta\right) \sin^2 2\phi - 4 \cos^2 \theta}{\sin 4\phi \cos \theta}. \tag{A12}$$

We substitute the expressions for the propagation and polarization vectors and the elasticity tensor into the Christoffel equations (Equation (A1)). The first order perturbation solutions for the longitudinally polarized wave speed *c*1 and the two transversely polarized wave speeds *c*2 and *c*3 are (e.g., [40,43])

$$\begin{cases} \rho c\_1^2 = \lambda + 2\mu + \eta(\cos^4\phi \sin^4\theta + \sin^4\phi \sin^4\theta + \cos^4\theta), \\ \rho c\_2^2 = \mu + \eta\chi(\theta, \phi, \xi), \\ \rho c\_3^2 = \mu + \eta\chi(\theta, \phi, \xi + \pi/2), \end{cases} \tag{A13}$$

where

$$\begin{split} \chi(\theta,\phi,\xi) &= \cos^2\phi\sin^2\theta \left(\cos\zeta\cos\phi\cos\theta - \sin\zeta\sin\phi\right)^2 \\ &+ \sin^2\phi\sin^2\theta \left(\cos\zeta\sin\phi\cos\theta + \sin\zeta\cos\phi\right)^2 + \cos^2\zeta\cos^2\theta\sin^2\theta. \end{split} \tag{A14}$$

Using these wave speeds lead to an expression for the Debye speed (Equation (2)),

$$\frac{1}{v\_D^3} \approx \frac{1}{12\pi} \int\_0^{2\pi} \int\_0^{\pi} \sin\theta \left( \frac{1}{c\_1^3(\theta,\phi)} + \frac{1}{c\_2^3(\theta,\phi)} + \frac{1}{c\_3^3(\theta,\phi)} \right) d\theta d\phi,\tag{A15}$$

and the directionally averaged wave speeds

$$\begin{aligned} \rho \left< c\_1^2 \right> &= \lambda + 2\mu + \frac{3}{5}\eta, \quad \text{and} \\ \rho \left( \left< c\_2^2 \right> + \left< c\_3^2 \right> \right) &= 2\mu + \frac{2}{5}\eta. \end{aligned} \tag{A16}$$

As with the transversely isotropic case, these speeds are consistent with the Voigt averaged moduli given as

$$
\kappa\_0^V = \lambda + \frac{\eta + 2\mu}{3} \quad \text{and} \quad \mu\_0^V = \mu + \frac{\eta}{5}, \tag{A17}
$$

since *ρ c*22 + *c*23 /2 = *μV*0 and *ρ c*21 = *κV*0 + 4/3*μV*0 . On the other hand, the Reuss averaged moduli are given as

$$
\kappa\_0^R = \kappa\_0^V \quad \text{and} \quad \mu\_0^R = \frac{5\mu \left(\eta + 2\mu\right)}{4\mu + 3\left(\eta + 2\mu\right)}, \tag{A18}
$$

and they are not compatible with seismic wave speeds.

#### **Appendix B. Experimental Measurements of Cobalt**

All examples in the main text of this paper are based on theoretical calculations of hcp and bcc iron alloys. Here, we briefly report and analyze the results obtain using the elements of the elasticity tensor of cobalt obtained from [44] which have been determined using measurements from inelastic X-ray scattering experiments. Note that, because of weak anisotropy represented by strength of anisotropy (*AL*) between 0.03 and 0.09 (Table A1), the wave speeds calculated from the elements of the elasticity tensor following the seismically relevant averaging scheme [14] are consistent with their experimentally measured values (e.g., [45]). At this levels of anisotropy, the NRIXS derived Debye speeds and the expressions in Equation (21) provide accurate measurements of the material's seismic wave speeds.

**Table A1.** The Debye *vD* and seismic wave speeds *cp* and *cs* of cobalt at pressures of zero to 40 GPa. The alloy's density in g/cm<sup>3</sup> (*ρ* is given in column 2 [45]), while the strength of anisotropy (*AL*; [22]), and the Debye speed calculated using Equation (7) and the elements of the elasticity of cobalt (*vD*; [44]) are given in columns 3 and 4. The material's seismic wave speed (*cp*; column 5) is compared against longitudinal wave speeds obtained using the full non-linear equations of Equation (21) (*cnl p* ; column 6), and experimentally measured values determined using ultrasonics (*cUS p* ; column 7; [46] ), inelastic X-ray scattering (*cIXS p* ; column 8; [45]), and impulsive stimulated light scattering (*cISLS p* ; column 9; [47]). Columns 10 through 14 are the same as columns 5 through 9 except for shear/transverse wave speeds.


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

The partial density of state measured by NRIXS experiments are proportional to *v*<sup>−</sup><sup>3</sup> **k**ˆ (Equation (5)). However, materials composed of randomly oriented anisotropic crystal with bulk or macroscopically isotropic properties (e.g., powdered samples) are well represented by averaging over the incident wave vector direction **k** ˆ . In this section we will show that for all materials,

$$
\left\langle v\_{\hat{\mathbf{k}}}^{-3} \right\rangle = v\_D^{-3}.\tag{A19}
$$

The eigenvalues (*λm*) of the Christoffel Matrix **M** are equal to the squared seismic the wave speeds (*λm* = *<sup>c</sup>*2*m*; Equation (A1)), such that Equation (5) can be written as

$$\sum\_{m=1}^{3} \int \frac{d\Omega\_{\mathbf{\hat{q}}}}{4\pi} \frac{\left|\mathbf{\hat{k}} \cdot \mathbf{\hat{p}}\_{m}\right|^{2}}{c\_{m}^{3}} = \int \frac{d\Omega\_{\mathbf{\hat{q}}}}{4\pi} \sum\_{m=1}^{3} \frac{\left|\mathbf{\hat{k}} \cdot \mathbf{\hat{p}}\_{m}\right|^{2}}{\lambda\_{m}^{3/2}}.\tag{A20}$$

The integrand can be written in matrix form such that

$$\sum\_{m=1}^{3} \frac{\left|\hat{\mathbf{k}} \cdot \hat{\mathbf{p}}\_m\right|^2}{\lambda\_m^{3/2}} = \left(\hat{\mathbf{k}} \cdot \mathbf{P}\right) \cdot \mathbf{D}^{-3/2} \cdot \left(\hat{\mathbf{k}} \cdot \mathbf{P}\right),\tag{A21}$$

where **P** is a matrix whose columns are the normalized eigenvectors **p**ˆ *m* of **M**, and **D** is a diagonal matrix whose diagonal elements are the eigenvalues (*λm*) of **M**. Using **P***<sup>T</sup>* = **P**−<sup>1</sup> and **D**−3/2 = **P***<sup>T</sup>* · **M**−3/2 · **P**, the right-hand side of Equation (A21) can be written in terms of the Christoffel Matrix such that

$$\mathbf{P}\left(\hat{\mathbf{k}}\cdot\mathbf{P}\right)\cdot\mathbf{D}^{-3/2}\cdot\left(\hat{\mathbf{k}}\cdot\mathbf{P}\right) = \left(\hat{\mathbf{k}}\cdot\mathbf{P}\right)\cdot\mathbf{P}^T\cdot\mathbf{M}^{-3/2}\cdot\mathbf{P}\cdot\left(\mathbf{P}^T\cdot\hat{\mathbf{k}}\right) \\ = \hat{\mathbf{k}}\cdot\mathbf{M}^{-3/2}\cdot\hat{\mathbf{k}}.\tag{A22}$$

For ease of notation, let us temporarily denote the elements of **M**−3/2 by *Mij*, and represent the incident wave vector by the expression for the unit vector in Equation (A4). The above expression can then be explicitly written out as

$$\begin{split} \hat{\mathbf{k}} \cdot \mathbf{M}^{-3/2} \cdot \hat{\mathbf{k}} &= (M\_{11} + M\_{22}) \sin^2 \phi \sin^2 \theta + (M\_{12} + M\_{21}) \sin \phi \sin^2 \theta \cos \phi \\ &+ (M\_{23} + M\_{32}) \sin \phi \sin \theta \cos \theta + (M\_{13} + M\_{31}) \sin \theta \cos \phi \cos \phi \\ &+ M\_{33} \cos^2 \theta. \end{split} \tag{A23}$$

For randomly oriented sample, the above expression needs to be integrated over all incident wave vector directions, that is, integration over *<sup>d</sup>*Ω**k**<sup>ˆ</sup> , and only the first and last terms of the right-hand side Equation (A23) remain. Thus,

$$
\left\langle \hat{\mathbf{k}} \cdot \mathbf{M}^{-3/2} \cdot \hat{\mathbf{k}} \right\rangle = \frac{1}{3} \left( M\_{11} + M\_{22} + M\_{33} \right) \\
= \frac{1}{3} \text{Tr} \left( \mathbf{M}^{-3/2} \right),
\tag{A24}
$$

where Tr denotes the trace. Using the invariance of the trace, Tr **<sup>M</sup>**−3/2can be directly related to the material's wave speeds as

$$\operatorname{Tr}\left(\mathbf{M}^{-3/2}\right) = \operatorname{Tr}\left(\mathbf{D}^{-3/2}\right) = \sum\_{m=1}^{3} \lambda^{-3/2} = \sum\_{m=1}^{3} \frac{1}{c\_m^3}.\tag{A25}$$

Thus, the mean projected wave speed (Equation (5)), averaged over all incident wave vector directions, is equivalent to the material's Debye speed since

$$\begin{split} \left< \frac{1}{\nu\_{\hat{\mathbf{k}}}^{3}} \right> &= \int \frac{d\Omega\_{\hat{\mathbf{q}}}}{4\pi} \left< \sum\_{m=1}^{3} \frac{\left| \hat{\mathbf{k}} \cdot \hat{\mathbf{p}}\_{m} \left( \hat{\mathbf{q}} \right) \right|^{2}}{c\_{m}^{3} (\hat{\mathbf{q}})} \right> = \int \frac{d\Omega\_{\hat{\mathbf{q}}}}{4\pi} \left< \hat{\mathbf{k}} \cdot \mathbf{M}^{-3/2} \cdot \hat{\mathbf{k}} \right> = \frac{1}{3} \int \frac{d\Omega\_{\hat{\mathbf{q}}}}{4\pi} \operatorname{Tr} \left( \mathbf{M}^{-3/2} \right) \\ &= \frac{1}{3} \int \frac{d\Omega\_{\hat{\mathbf{q}}}}{4\pi} \sum\_{m=1}^{3} \frac{1}{c\_{m}^{3}} = \frac{1}{v\_{D}^{3}} \end{split} \tag{A26}$$
