In order to explore the effect of analyzing anisotropic single crystals and polycrystalline aggregates, elastic tensor elements of five iron alloys with hexagonal-close packed (hcp) [

17,

18,

19,

20] and one with a body-centered cubic (bcc) [

21] structure are considered (

Table 1). The five hcp iron materials are—a pure iron at

$6000{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{K}$ with a density of

$13.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$[

17], a Fe-Si alloy with 12.5 at% Si at 360 GPa and

$6900{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{K}$[

20], a Fe-Ni-Si alloy with 10 at% Ni and 21.25 at% Si with density of

$12.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ at 360 GPa and

$6500{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{K}$[

18], a Fe-Si-C alloy with 4.2 at% Si and 0.7 at% C with density of

$13.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ at 360 GPa and

$6500{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{K}$[

19], and a Fe-S-C alloy with 2.1 at% S and 0.7 at% C with density of

$13.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ at 360 GPa and

$6500{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{K}$ [

19]. The last material considered is a Fe-Si bcc alloy at 360 GPa and

$6000{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{K}$ with density of

$13.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ [

21]. The seismic wave speeds

${c}_{p}$ and

${c}_{s}$ are calculated from the elastic tensor elements (

Table 1) such that they are comparable with the seismically observed wave speeds [

14].

To investigate how various parameters are influenced by anisotropy, we introduce an absolute measure of anisotropy,

${A}^{L}$, that quantifies the log-Euclidean distance between the Voigt averaged elastic tensor

$\u2329{\mathsf{\Lambda}}_{ijkl}\u232a$ and the Reuss averaged elastic tensor

${\u2329{\mathsf{\Lambda}}_{ijkl}^{-1}\u232a}^{-1}$[

22,

23]. The log-Euclidean distance may be written in terms of the Voigt and Reuss averaged isotropic moduli

${\kappa}_{0}$ and

${\mu}_{0}$ as [

22]

where the superscripts

V and

R denote the Voigt and Reuss averages, respectively. The Voigt and Reuss averages coincide for the case of isotropic material [

24], and for this case,

${A}^{L}$ yields a value of zero.

#### 3.1. Variations in the Debye Speed

We first calculate the mean projected wave speed

${v}_{\widehat{\mathbf{k}}}$ for all possible incident photon wave vectors

$\widehat{\mathbf{k}}$ (Equation (

5)). This requires numerically calculating the wave speeds (eigenvalues) and polarization vectors (eigenvectors) as described in the

Appendix A, and then numerically integrating over the phonon propagation directions,

$\widehat{\mathbf{q}}$ (Equation (

5)). The minimum and maximum values represent the spread of possible mean projected wave speeds that may be measured for a given single crystal sample due to changes in orientation of the crystal with respect to the incoming X-ray beam (

Figure 1,

Table 2). For the transversely isotropic hcp materials considered in this study, the maximum projected mean wave speed corresponds to an incident X-ray beam aligned along the axis of symmetry, and the minimum corresponds to an incident X-ray beam in the plane perpendicular to the axis of symmetry. The range in

${v}_{\widehat{\mathbf{k}}}$ caused by this orientation effect is proportional to the strength of anisotropy, with a

$20\%$ increase per unit of anisotropy (

Figure 2). For the hcp pure iron,

${v}_{\widehat{\mathbf{k}}}$ varies by about

$40\%$, whereas for the less anisotropic hcp Fe-S-C alloy, it is only

$13\%$ (

Table 2). As expected from the symmetry in the scattering matrix for cubic materials [

11], the projected mean wave speed

${v}_{\widehat{\mathbf{k}}}$ is constant and is identical to the Debye speed for the bcc Fe-Si alloy (

Table 2).

In order to examine the Debye speed

${v}_{D}={\u2329{v}_{\widehat{\mathbf{k}}}^{-3}\u232a}^{-1/3}$ (Equation (

7)) of powdered samples, an additional integration over the wave vector

$\widehat{\mathbf{k}}$ is needed. Because of cubic symmetry,

${v}_{\widehat{\mathbf{k}}}={v}_{D}$ for the bcc Fe-Si alloy (

Table 1). The Fe-Si-Ni alloy which has the lowest seismic shear wave speed of

$3.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$ (

Table 1) results in the lowest

${v}_{D}$ value of

$3.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$, and the hcp Fe-S-C alloy which had the highest seismic shear wave speed

$4.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$ results in the highest

${v}_{D}$ value of

$4.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{s}$. The Debye speed

${v}_{D}$ in Equation (

7) can also be calculated approximately using the first-order propagation and polarization vectors (see

Appendix A) and obtaining the corresponding wave speeds which we denote by

${v}_{D}^{q}$. This procedure may either be done by numerical matrix multiplication, or for hexagonal and cubic symmetries, using the analytical expressions as described in the

Appendix A. Use of these analytical expressions for wave speeds and associated polarization vectors to calculated Debye speed is only valid for weakly anisotropic materials, however, the approximate values of

${v}_{D}^{q}$ are nearly identical to the exact values

${v}_{D}$ (

Table 2). Even for the case of the highly anisotropic materials, such as pure iron, the error is less than half a percent. On the other hand, Debye speeds based upon various isotropic averages,

${v}_{D}^{V}$,

${v}_{D}^{R}$, and

${v}_{D}^{VRH}$ (Equation (

10)) from

${v}_{D}$ show significant deviations, between 0.5 to 33%, including bcc Fe-Si alloy (

Table 2). The largest deviations occur with

${v}_{D}^{V}$, the averaging scheme that is most relevant for comparison with seismic wave speeds.

The materials in

Table 1 only represent a small subset of the possible variations in the elasticity tensor. In order to explore the deviations introduced by the approximations more thoroughly, we generate one million random transversely isotropic elastic tensors whose elements are sampled from a uniform distribution subject to the restriction that the resulting compressive wave speeds are greater than the shear wave speeds. The Debye speed

${v}_{D}$ is compared against

${v}_{D}^{V}$,

${v}_{D}^{R}$, and

${v}_{D}^{VRH}$ (

Figure 3). The Debye speeds calculated from the seismically compatible average,

${v}_{D}^{V}$, are all greater than

${v}_{D}$ while the Debye speeds calculated using the Reuss average are all less than

${v}_{D}$, which is consistent with the Voigt and Reuss averages representing upper and lower bounds, respectively, for the elastic moduli. The Debye speeds calculated using the Voigt-Reuss-Hill averages are roughly distributed about zero, and thus are a “best case scenario” that is most consistent with its true value (

Figure 3). In fact, Anderson [

9] showed that the isotropic values calculated with the Voigt-Reuss-Hill average may be used to accurately estimate a material’s Debye speed, however, one must keep in mind that the acoustic wave speeds based upon Voigt-Reuss-Hill averages are incompatible with observed seismic wave speed [

14]. Finally, we shows that the seismic wave speeds of Cobalt at pressures of zero to 40 GPa are accurately calculated using the Voigt average and consistent with independent experimental measurements (

Appendix B).

In order to explore the deviation between the Debye speeds estimated using the seismic wave speeds

${v}_{D}^{V}$ and the true Debye speed

${v}_{D}$ as a function of the strength of anisotropy

${A}^{L}$, the difference,

${v}_{D}-{v}_{D}^{V}$ are examined (

Figure 4). The seismic wave speeds for all one million tensors are forced to have identical seismic wave speeds (set to be those of pure iron in

Table 1), and yet the difference, despite the scatter in the data, increases with increasing anisotropy roughly at a rate of about 17% per unit of anisotropy. The distribution of the deviations also has some structure (

Figure 4). For a given strength of anisotropy, there is an upper and lower limit on the deviation between

${v}_{D}$ and

${v}_{D}^{V}$. These limits, as well as the range in between, result from the partitioning of anisotropy into various elastic tensor elements. Because the elasticity tensors are randomly generated with the only condition being the constant seismic wave speeds, there are tensors for which anisotropy is purely in the bulk modulus but not in shear modulus, and vice versa. The shallowly dipping upper limit corresponds to cases where anisotropy is all in the bulk modulus, only affecting the longitudinal acoustic waves, and the steeper lower limit corresponds to anisotropy that is purely in the shear modulus. The scatter of points between the two limits represent various levels of anisotropy partitioning.

#### 3.2. Extracting Seismic Wave Speeds

Section 2 and the previous subsection showed that the Debye speed extracted from the partial density of states (Equation (

6)), is not equal to the Debye speed estimated from that material’s seismically observed wave speeds (Equation (

3)). This result stems partly from the fact that the average of the inverse cube of a function (Equation (

12)) is not generally equivalent to the inverse cube of the average (Equation (

13)) and because seismic waves “see” the material’s isotropic properties in a specific manner. Therefore, the Debye speed estimated from seismic wave speeds is an upper bound for the material’s Debye speed

${v}_{D}$. This distinction is critical for geophysical applications of NRIXS, which typically seek to compare the experimentally observed Debye speed

${v}_{\widehat{\mathbf{k}}}$ or

${v}_{D}$ with seismically observed acoustic wave speeds

${c}_{p}$ and

${c}_{s}$. In this section, we focus on the issues associated with extracting seismic wave speeds based upon

${v}_{D}$.

A common procedure used to extract seismic wave speeds from NRIXS experiments is to use independent estimates of the the bulk modulus

$\kappa $ and density

$\rho $ from an equation of state based upon X-ray diffraction (e.g., [

25,

26,

27,

28,

29,

30]). For isotropic material, manipulating the expressions for the acoustic wave speeds (Equation (

9) and noting that

$\kappa ={\kappa}_{0}^{V}$ and

$\mu ={\mu}_{0}^{V}$ in order to be compatible with the quantities provide by an Earth model such as PREM) combined with Equation (

3) results in three equations for the three unknowns

$\mu $,

${c}_{p}$, and

${c}_{s}$ in terms of the known quantities

${v}_{D}$,

$\kappa $, and

$\mu $ such that

these equations are non-linearly dependent on

${c}_{p}$ and

${c}_{s}$, and are often solved by first linearizing the equations as is done in Equation (

15) of Sturhahn & Jackson [

3].

The form of the Debye speed (Equation (

3)) used in Equation (

21) is only applicable to isotropic materials, and significant errors may result when applied to anisotropic materials. The difference between the seismically relevant Debye speed

${v}_{D}^{V}$ (Equation (

3)) and the observed Debye speed

${v}_{D}$ (Equation (

7)) maps directly into the the estimated acoustic wave speeds, especially into shear wave speed, since the Debye speed is heavily weighted by the shear wave speed.

In order to estimate the magnitude of the errors associated with these approximations, acoustic wave speeds are estimated using the expressions in Equation (

21) and the theoretical values of

${v}_{D}$,

${\kappa}_{0}^{V}$, and

$\rho $ using the five hcp iron alloys in

Table 1. Since the full elastic tensors for the materials are available, both the bulk modulus

${\kappa}_{0}^{V}$ and the Debye speed

${v}_{D}$ are calculated exactly. The inferred seismic wave speeds (

Table 3) using the linear solution of Sturhahn & Jackson [

3] show that the linear approximation results in errors in the compressional wave speed on the order of

$10\%$ and errors in the shear wave speed of 10–30%. Therefore, for iron alloys relevant for the Earth’s inner core, linearization of Equation (

21) results in significant errors due to the terms neglected when linearizing the set of equations. Solving the non-linear system of equations (Equation (

21)) captures the compressional wave speed to within several percent, however, significant errors of

$25\%$ in shear wave speed remain (

Figure 4). The errors associated with these solutions are smaller than those of the linearized solutions, however, they are substantial, especially for the shear wave speed. The source of the error is the assumption that

${v}_{D}\approx {v}_{D}^{V}$, that is, relating

${v}_{D}$ directly to the seismic speeds even though the material is not isotropic (the third equation in Equation (

21)).

Previous estimates of the error due to anisotropy in the seismic speeds obtained from the Debye speed are of the order of a few percent [

32], while the errors in this study are found to be considerably larger (

Table 3). Bosak et al. [

32] significantly underestimates the error due to their choice of averaging scheme. They use an averaging scheme [

33] which yields acoustic wave speeds that are similar to those obtained by the Voigt-Ruess-Hill average. However, when comparing with seismic wave speeds from a model such as PREM [

13], the wave speeds should be calculated using the Voigt average [

14].

Finally, Anderson [

10] argued that the ratio

${c}_{s}/{v}_{D}$ is roughly a constant value of

$0.9\pm 0.001$, and thus for isotropic materials, the Debye speed may be related to the shear wave speed as [

31]

Surprisingly, this simple estimate of

${c}_{s}$ (

Table 3) out-performs all other estimates of

${c}_{s}$. Motivated by this result, we perform a simple regression against the strength of anisotropy and find that

where

${c}_{s}$ and

${v}_{D}$ are given in km/s. This relationship is able to provide a good fit to

${c}_{s}$ with less than one percent error (

Figure 5). Unfortunately, the strength of anisotropy is not typically known, and in that case, the regression

can predict the shear wave speed to within ∼

$5\%$ (

Figure 5).

The problem of estimating seismic wave speeds can be turned around to estimate Debye speeds based upon seismically constrained wave speeds. Consider, for illustrative purposes, the seismically observed isotropic wave speeds of the inner core from the PREM model (

${c}_{p}=11.1$ km/s and

${c}_{s}=3.6$ km/s; [

13]). These isotropic values can be used to calculate Debye speed of

${v}_{D}=4.10$ km/s if the inner core is isotropic (Equation (

3)). Alternatively, using the five elastic constants describing transversely isotropic inner core from seismological observations (

${c}_{11}=1577$ GPa,

${c}_{33}=1647$ GPa,

${c}_{13}=1259$ GPa,

${c}_{44}=168$ GPa,

${c}_{66}=151$ GPa; [

14]), the Debye speed is

${v}_{D}=4.06$ km/s (Equation (

2)). This value of the Debye speeds is most similar to that of the bcc iron alloy (4.04 km/s), and least similar to that of the hcp Fe-Si-Ni alloy (2.90 km/s;

Table 2). However, comparison of the seismic wave speeds (

Table 1) suggests the opposite result, with the hcp Fe-Si-Ni alloy (

${c}_{p}=11.8$ km/s and

${c}_{s}=3.3$ km/s) being more consistent with the seismically observed values than those of the bcc Fe-Si alloy (

${c}_{p}=11.5$ km/s and

${c}_{s}=4.2$ km/s). This is due to the fact that the higher strength of anisotropy of the bcc Fe-Si crystal lowers its Debye speed, hence the comparison of the Debye speed for anisotropic material ideally should be done with the strength of the anisotropy of the material.