A simple procedure is developed for simulating elastic compositions of substances as may occur in exogeologic bodies. It is based on a mixing law using acoustic slownesses obtained by averaging elasticities of crystallite aggregate components. The method applies to materials having an arbitrary number of constituents; illustrative examples are given for two-component solid solutions in the olivine and pyroxene families. The averaging algorithm is also applied to perovskite, post-perovskite, and alumina elasticities. Linear temperature and pressure coefficients are estimated for the perovskites; higher-order temperature coefficients are computed for alumina. Debye temperatures are given for the various simulants. 

Olivine, Perovskite, Pyroxene, Alumina, Acoustics, Slowness, Voigt-Reuss-Hill

Stern B, DeFilippo V, Ballato A (2017) Planetary Seismology Simulants. J Mater Appl Sci 1(1): 1003

Aspera ad astra had, for millenia, been cited only metaphorically. 1 No longer [1-14]. The exertions of still-nascent space programs have already yielded an abundance of results revealing properties of planetary [15-29], cometary [30-35], and stellar bodies [4]. In the near and in the farther future, probes will continue to amass information permitting inferences to be drawn about the constitution and evolution of the cosmos. At present, most of the data have been harvested by means of remote sensing modalities from proximate spacecraft orbits and fly-bys. Valuable as these are, a much fuller picture is to be had by supplementing these with information gleaned from on-site samplings of material constituents, and their relations to acoustic and seismic properties. This approach has proven its value in geology by allowing models of the Earth’s directly inaccessible interior to be constructed from, inter alia, a knowledge of the elastic properties of minerals, and acoustic wave propagation therein [36-43].

Seismology provides a direct probe of solid environments. Acoustic wave speeds and transit times yield data of material extent, composition, and gradation. These data, in turn, permit formulation of specific models (simulants) for interpreting physical features of exogeologic bodies, such as elastic coefficients determining lithospheric bending; these features are not immediately accessible to remote sensing methods. Such models do not by themselves yield unique characterizations of planetary, meteoric, or cometary interiors, yet they are an invaluable adjunct to other measurement modalities.

In this paper, a simple homogenization procedure is applied to convert single crystal elastic constants to isotropic elastic stiffnesses of microcrystalline aggregates, such as those commonly occurring in planetary bodies. These coefficients, for each mineral, are then combined with those of all other constituents in an algorithm yielding effective elastic parameters that characterize both seismic wave propagation and quasistatic deformations in the mineral composite. 2 Using data from the literature, example simulants are provided for olivines: forsterite–fayalite, and pyroxenes: enstatite–ferrosilite. These are known constituents of solid planetary bodies.

Models for acoustic wave propagation in geologic-type media (e.g., inhomogeneous, stratified, porous, non-linear, etc.) are quite well developed [44-51], and will not be discussed, nor will be reviewed the various modes of acoustic propagation. Considered only are the two canonical, non-dispersive, plane body wave types that propagate in homogeneous, linear media: shear (S) and pressure (P). 3 Other wave types and parameters may be considered to be admixtures of these. As examples, the seismic parameter Φ = VP 2 - (4/3) . VS 2 is a combination of P and S wavespeeds [52,53], as is the the velocity of a wave localized near the free surface of an isotropic elastic half-space (Rayleigh wave) [54].

The main thrust of this paper is presentation of a simple model of acoustic properties of mineralogical compositions that is generally applicable to many of the material systems likely to be encountered in exogeology. The WSD model, described subsequently, makes direct use of the S and P wave speeds, and is a means of computing velocities in mineral species as function of mole fractions of the constituents.

In as much as realistic macroscopic mineralogical deposits are found to be composed largely of randomly oriented polycrystalline aggregates, the starting point of the model is reduction of the elasticities of each crystal constituent to equivalent isotropic values by means of a simple extension of the Voigt-Reuss-Hill (VRH) procedure [55-58]. The extension (VRHx) yields a single set of relaxed quantities for each material, rather than a range of values. The VRHx values are then “mixed,” for specified mole fractions, by a time-of-flight assignment applied separately to the S and P wave slownesses. The results provide simulants [19] of various mineral combinations for comparison with experimental seismic data, thereby enabling inferences of internal composition and structure to be made. It is understood that, whereas a knowledge of material composition may permit computation of wave velocities, the reverse is generally not possible. Useful inferences about composition may, however, be drawn from a knowledge of wavespeeds when combined with ancillary data, and a more complete theoretical treatment.

While the WSD mixing algorithm may be applied to an arbitrary number of components, for didactic purposes two-component, solid-solution, examples are provided in the section on olivine (forsterite-fayalite) and pyroxene (enstatite-ferrosilite) [59-67]. These families are prominent constituents of the Earth’s mantle; moreover, forsterite has been found in cometary dust of probable supernova ejecta origin [4]. The model does not depend on the constituents being members of a solid-solution family, only that the material assemblage be considered “isotropic” on the scale of the acoustic wavelengths used. General references to material properties, and methods of their determination are found in [68-86].

In view of their geological and planetary importance, VRHx values are included here for perovskite [87,88], post-perovskite [89-95], (both rhombic class mmm), and for α-alumina (trigonal class 3m) [96,97]. Least-squares fits are also provided for critical properties of α-alumina versus temperature. These enable rapid and accurate estimates of elasticities and velocities, etc., as function of temperature in a wide range.

 We anticipate that future applications of the WSD procedure will include other prominent mineral constituents such as:

(a) Garnets [98-101], a major component of Earth’s mantle. The two main families, Ca3 (Cr, A?, Fe)2 (SiO4 )3 , and (Mg, Mn, Fe)3 A?2 (SiO4 )3 , crystallize in cubic point group m3m.

(b) Periclase (MgO, m3m), another abundant planetary mantle constituent [102-104].

(c) Spinel (MgA?2 O4 , m3m), and m3m analogs, such as peridotite (Mg, Fe) (A?, Cr)2 O4 and pleonaste (Mg, Fe)A?2 O4 [105].

(d) Topaz (A?2 SiO4 F2 , rhombic mmm) and A?2 SiO5 analogs such as andalusite (mmm), sillimanite (mmm), and kyanite (triclinic 1) [106,107].

(e) Merrillite (Ca9 NaMg(PO4 )7 ), which occurs prominently in extraterrestrial rocks, and whitlockite (Ca9 Mg(PO4 )6 (PO3 OH). These are members of trigonal, acentric class 3m, rendering them both piezoelectric and pyroelectric [108-115].

Many mineral species of interest presently lack precision determinations of elastic constant values, not only at standard pressure and temperature, but particularly for those values existing at the ambient conditions where the minerals occur.

Crystal elasticities reduced to isotropy:

The Voigt-Reuss-Hill procedures [55-58] for space-averaging crystal elastic stiffnesses (Voigt) and compliances (Reuss) are not discussed in detail here.

In brief, stiffness (Clm) and compliance (Slm) = (Clm)-1 matrices are converted to isotropic matrix averages and via the relations (valid for the most general triclinic crystal):

Voigt:

<c11V>=[3. (c11 + c22 + c33) + 2. (c12 + c13 + c23) + 4. (c44 + c55 + c66)]/15 = [(c11 + c22 + c33) – (c12 + c13 + c23) + 3. (c44 + c55 + c66)]/15 = 2.<c44V>

Reuss:

<s11R> = [3. (s11 + s22 + s33) + 2. (s12 + s13 + s23) + (s44 + s55 + s66)]/15 = [4. (s11 + s22 + s33) – 4. (s12 + s13 + s23) + 3. (s44 + s55 + s66)]/15 = <s44R>/2

A self-consistent relaxation method (VRHx):

The Voigt and Reuss formulas produce upper and lower bounds on the elasticities of a given crystal of any anisotropy. Hill [57] found that an arithmetic average of the bounds agreed better with experiment in many cases. The VRH homogenization procedure has been augmented over the years by many variants; these improve on the VRH method, producing closer bounds, ability to incorporate measures of texture and inclusions, treatment of multiphase materials, etc. [116-145]. Any of these advanced methods may be applied in the WSD model discussed in the next section. For the purpose of illustrating the WSD model here, a simple homogenization scheme, referred to as the selfconsistent relaxation (VRHx) method [58] is applied to the Voigt and Reuss matrices cV and sR. It employs “interleaved averaging,” iteratively relaxing the Voigt-Reuss extremes, to arrive at matrices such that = -1, replacing bounds with consistent mean values.

The VRHx method begins with the Voigt stiffness matrix: cV = cV (0) and the Reuss compliance matrix: sR = sR (0). Each is then inverted to give sV (0) = (cV (0))-1 and cR (0) = (sR (0))-1. From these, the arithmetic (Hill) averages are formed: cVR (1) = (cV (0) + cR (0))/2 and sRV (1) = (sR (0) + sV (0))/2.

These are again inverted to yield: sVR (1) = (cVR (1))-1 and cRV (1) = (sRV (1))-1, and are further averaged via the inversions: sVR (n) = (cVR (n))-1 and cRV (n) = (sRV (n))-1

and recursions: cVR (n+1) = (cVR (n) + cRV (n))/2 and sRV (n+1) = (sRV (n) + sVR (n))/2.

The numerical process is doubly convergent, and rapidly yields “best” isotropic matrices, and = -1. From these coefficients and mass density, the S and P wave speeds for substance k are found:

The WSD additivity model

The model proposed here has its genesis in the WinkelmannSchott work that was originally applied to glass mixtures [146]. The modern version is due to Dragic, who refined it, and successfully applied it to germanium doping of silica glass [147]. The Winkelmann-Schott-Dragic (WSD) model proceeds as follows: Given the mole fraction (fk (mol)), mass density (ρk ), and molecular mass (Mk ) of each constituent k, the corresponding volume fraction (fk (vol)), and weighted average mass density (ρ(mix)) are first determined. For the simple two-component case the relations are:

From the volume fraction of each isotropic constituent and its corresponding inverse wave speed [38,42], i.e., “slowness” (1/VSk, 1/VPk), the transit time for an S or P wave traversing its fraction of a unit distance is determined; the sum over all components yields the total slowness associated with the S or P wave in the composite. Continuing the two-component example, the relations are:

Using both the S and P wavespeeds so determined, one finds the isotropic elastic constants of the mixture. Key here is the use of slownesses in determining average elasticities. Slowness is proportional to transit-time, and has a direct physical meaning, whereas the meanings of velocity averaging, or other criteria [52,53], are less obvious.

The elasticities of the composite, (introducing the Lamé constants l, m [37-39] that characterize isotropic media), are then Isotropic stiffnesses:

Young’s modulus, = 

Bulk modulus, = 

Poisson’s ratio, = 

Isotropic compliances:

The WSD procedure is given here only in schematic outline; as with the VRHx method, it can be generalized in various ways to accommodate texture, etc.

Application: Olivines and pyroxenes

The olivine series [59-66] consists of solid solutions between the end members forsterite (Mg2 SiO4 ) and fayalite (Fe2 SiO4 ). These crystalize in centric, rhombic point group mmm. Table 1 presents the WSD model results for 0(10)100 mole % of fayalite in fosterite. The pyroxene silicate solid solution series [67] extends from enstatite (MgSiO3 ) to ferrosilite (FeSiO3 ); all members also have mmm symmetry. Table 2, for ferrosilite in enstatite, is the complement to Table 1. In Table 1 and Table 2, the experimental data appearing in [83] have been used. Columns labeled “” and “ΘDmix” in Tables 1 & 2, and subsequently in Table 7, are pertinent to the section on perovskite and post-perovskite, and are discussed there.

Reasonable agreement is seen in Table 3 (olivine) and Table 4 (pyroxene) between the WSD model predictions and experimental results. A more comprehensive comparison could not be made because of the paucity of experimental data; these results are from naturally occurring specimens containing admixtures of other substances [83]. In all cases the agreement is within < 5% (olivine) and < 8% (pyroxene). Given the vagaries of the natural samples, the agreement with the WSD model is encouraging, although an unqualified justification of the procedure cannot be adduced from these results.

Values reported in Tables 1-4 are for room temperature (RT), and zero applied pressure. The WSD procedure generalizes for conditions of spatially varying temperature and pressure. This requires a knowledge of the first- and higher-order temperature and pressure coefficients of the elasticities of the constituents. Many of the requisite data are not yet available, but an example is provided in Table 5 for naturally occurring forsterite and olivine, using the data in [83].

Application: Perovskite and post-perovskite

Because it is considered a significant component of telluric bodies, [16-29], the VRHx procedure is applied to perovskite (MgSiO3 ), with the results reported in Table 6. This table provides VRHx computations from the data of [87], and the simulations of [88,93,94], including those of the post-perovskite phase. Pressure- and temperature-coefficient estimations derived from [88,93,94] are also given.

Alumina

Another important geological and planetary constituent is alumina (A?2 O3 ) in its various phases. The careful and accurate measurements of stiffnesses of α-alumina, (point group symmetry 3m), in an extended temperature range [96] have been used to compute the VRHx elasticities, and associated quantities, as a function of temperature in the range 300 to 1800K. The results are expressed here as coefficients of power series expansions in temperature. The coefficients T(n) are derived from least-squares fits of values tabulated at 50K intervals in the cited range.

Each quantity y(T), appearing in Tables 8, 9, and 10, is computed as follows:

In Tables 9 and 10, T0 = 296K; superscript “x” refers to values obtained from the VRHx procedure. The units of the temperature coefficients T(n) are 10-3(n+1)/Kn ; the correlations of the curve-fits with the tabulated values in each instance yields a goodness-of-fit R2 > 99.9%.

Debye temperature

The Debye temperature (ΘD) [148] is used in considerations of specific heats of elastic solids. It provides a quantitative estimate demarcating classical (Dulong-Petit) and quantum regimes. For our purposes, it affords a simple indication that the VRHx results agree with a more accurate computation [149]. The Debye procedure globally averages the three plane acoustic wave speeds in a crystal over all propagation directions. In terms of the space-averaged elastic stiffness (cm) characterizing wave m, the requisite Debye expression is:

The VRHx algorithm also produces a global space averaging; the resultant equivalent elastic stiffness corresponding to cD is:

For the VRHx (and consequently the WSD) procedures to render accurate results it is necessary that cx ≈ cD. The Debye temperature of a single component, linearly elastic solid is given by:

where h is Planck’s constant, k is Boltzmann’s constant, q is the number of atoms in the molecule, ρ is the mass density, N is Avogadro’s number, and M is the molecular mass. V is the average acoustic wave speed: V = (C/ρ)1/2, where C is either cD, (Θ = ΘD), the true average elastic stiffness, or ,, (Θ = Θ Dmix), the VRHx approximate average elastic stiffness. For mixtures, appropriate average values of ρ, q, and M are used.

The classical Debye procedure averages each of the three elastic plane wave velocities over all space directions. As a practical computational matter, this necessitates covering a sphere with a grid of finite patches, computing the velocities at a convenient point within each patch, then summing the result over all patches to approximate the true values that would be obtained in the limit as the patch sizes diminish to zero. Depending on the grid size and shape, this has, in the past, resulted in either a loss of accuracy, or excessive computational effort. The essential difficulty is the impossibility of uniformly tessellating a sphere, familiar as the “dimples on a golf ball” situation. However, an accurate and efficient algorithm based on Fibonacci sequences [150] has been used to arrive quickly at accurate results for cD to compare with . Using F20 = 6,765 averaging points, Table 10 gives an example for α-A?2 O3 ; ΘD and Θx agree within 0.75%. This is a representative value; the agreement in all cases is within a few percent.

Table 1: Room temperature WSD model predictions as function of mole % of fayalite in forsterite

Mole	Vol	ρ	M	VS	VP	<c11>	<c12>	<c44>	<cmix>			<ν>	ΘDmix
%		Mg/m3	g/mole	km/s		GPa							K
fosterite	0	3.221	140.69	5.016	8.587	237.5	75.4	81.1	99.7	129.4	759.7	0.241	759.7
10	10.55	3.345	147	4.776	8.349	233.2	80.6	76.3	94.2	131.5	723.1	0.257	723.1
20	20.98	3.467	153.31	4.56	8.127	229	84.8	72.1	89.3	132.9	690.2	0.270	690.2
30	31.28	3.588	159.62	4.365	7.918	225	88.3	68.4	84.9	133.8	660.2	0.282	660.2
40	41.45	3.707	165.92	4.188	7.723	221.1	91	65	81	134.4	633	0.292	633
50	51.5	3.825	172.23	4.027	7.539	217.4	93.3	62	77.4	134.7	608.1	0.300	608.1
60	61.43	3.941	178.54	3.879	7.365	213.8	95.2	59.3	74.2	134.7	585.2	0.308	585.2
70	71.25	4.056	184.85	3.744	7.202	210.4	96.7	56.9	71.2	134.6	564.1	0.315	564.1
80	80.94	4.17	191.16	3.619	7.047	207.1	97.9	54.6	68.5	134.3	544.6	0.321	544.6
90	90.53	4.282	197.46	3.503	6.901	203.9	98.8	52.6	66	133.8	526.6	0.326	526.6
fayalite	100	4.393	203.77	3.396	6.762	200.8	99.5	50.7	63.7	133.3	509.8	0.331	509.8
Input values: [83]
Abbreviations: ρ: mass density; M: molecular mass; VS , VP: S and P wave speeds; : VRHx isotropic stiffnesses; : equivalent Debye stiffness; , , : VRHx isotropic compressibility, Young’s modulus, and Poisson’s ratio; ΘDmix: equivalent Debye temperature

Table 2: Room temperature WSD model predictions as function of mole % of ferrosilite in enstatite

Mole	Vol	ρ	M	VS	VP	<C11>	<C12>	<C44>	<Cmix>			<ν>	ΘDmix
%		Mg/m3	g/mole	km/s		GPa							K
enstatite	0	3.198	100.39	4.864	8.078	733.1	57.3	75.7	92.5	107.8	184	0.216	733.1
10	10.45	3.282	103.54	4.694	7.884	707.1	59.3	72.3	88.6	107.6	177.2	0.225	707.1
20	20.8	3.365	106.7	4.537	7.7	683	61	69.3	85.1	107.2	171	0.234	683
30	31.04	3.448	109.85	4.392	7.527	660.6	62.3	66.5	81.8	106.6	165.2	0.242	660.6
40	41.18	3.529	113	4.257	7.363	639.7	63.4	64	78.8	106	159.7	0.249	639.7
50	51.22	3.61	116.16	4.131	7.207	620.3	64.3	61.6	76	105.4	154.7	0.255	620.3
60	61.17	3.69	119.31	4.014	7.059	602.1	65	59.4	73.5	104.6	149.9	0.261	602.1
70	71.02	3.769	122.47	3.904	6.919	585	65.6	57.4	71.1	103.8	145.5	0.267	585
80	80.77	3.847	125.62	3.801	6.785	569	66	55.6	68.8	103	141.3	0.271	569
90	90.43	3.925	128.77	3.704	6.658	553	66.3	53.8	66.8	102.2	137.4	0.276	553
ferrosilite	100	4.002	131.93	3.613	6.536	539.7	66.5	52.2	64.8	101.3	133.7	0.28	539.7
Input values: [83]
Abbreviations: ρ: mass density; M: molecular mass; VS , VP : S and P wave speeds; ij> : VRHx isotropic stiffnesses;mix> : equivalent Debye stiffness; ,,, , : VRHx isotropic compressibility, Young’s modulus, and Poisson’s ratio; ΘDmix: equivalent Debye temperature

Table 3:Comparison of WSD model predictions with experimental values reported in [83]

Mole	ρ	VS	VP	<C11>	<C44>	<Cmix>
%	Mg/m3	km/s		GPa
7.0 a	3.308	4.845	8.419	234.4	77.7	95.88	130.9	194.5
7.0 b	3.311	4.883	8.418	234.6	79	97.2	129.4	196.8
7.5 a	3.314	4.833	8.407	234.2	77.4	95.5	131	194
7.5 b	3.299	4.892	8.385	232	78.9	97.1	126.7	196.1
8.1 a	3.321	4.819	8.393	234	77.1	95.2	131.1	193.5
8.1 b	3.316	4.871	8.382	233	78.7	96.8	128.1	195.9
9.0 a	3.332	4.799	8.372	233.6	76.7	94.7	131.3	192.7
9.0 b	3.325	4.832	8.371	233	77.6	95.7	129.53	194.1
Mole % fayalite in forsterite.
Abbreviations: ρ: mass density; VS , VP : S and P wave speeds; : VRHx isotropic stiffnesses; : equivalent Debye stiffness; ,,, : VRHx isotropic compress ibility, Young’s modulus, and Poisson’s ratio. a: WSD prediction; b: experimental value (natural specimens)

Table 4:Comparison of WSD model predictions with experimental values reported in [83]

Mole	ρ	VS	VP	<C11>	<C44>	<Cmix>
%	Mg/m3	km/s		GPa
06 a	3.249	4.761	7.96	205.8	73.6	90.2	107.7	179.9
06 b	3.272	4.753	7.834	200.8	73.9	90.3	102.3	178.7
15.2 a	3.325	4.611	7.787	201.6	70.7	86.7	107.4	173.9
15.2 b	3.335	4.749	7.845	205.2	75.2	91.9	105	182.1
20 a	3.365	4.537	7.7	199.5	69.3	85.1	107.2	171
20 b	3.354	4.72	7.781	203.1	74.7	91.2	103.5	180.7
Mole % ferrosilite in enstatite
Abbreviations: ρ: mass density; VS , VP : S and P wave speeds;  : VRHx isotropic stiffnesses; : equivalent Debye stiffness; ,,, , : VRHx isotropic compressibility, Young’s modulus, and Poisson’s ratio. a: WSD prediction; b: experimental value (natural specimens)

Table 5: First-order temperature (Tc(1)) and pressure (Pc(1) ) coefficients of elastic stiffnesses (cλµ) of naturally occurring forsterite and olivine; input data: [83]

forsterite	(Mg2SiO4)99.9(Mn2SiO4)0.01 ; ρ = 3,224 [kg/m3]
λµ →	11	22	33	44	55	66	23	31	12
Tc(1)	100.8	140.6	120.3	199.5	162.6	186.7	62.3	119.2	162.8
Pc(1)	25.8	32.8	27.9	32.5	20.4	29.3	55.7	70.3	73.1
olivine	(Mg2SiO4)92.72(Fe2SiO4)7.24(Mn2SiO4) 0.04 ; ρ = 3,311 [kg/m3]
λµ→	11	22	33	44	55	66	23	31	12
Tc(1)	105	144.2	121.7	198.1	166.6	198.6	67.5	131.3	158.1
Pc(1)	24.7	32.2	27.1	33.6	21	29.2	49.7	62.6	71.4
Abbreviations:  ρ: mass density First-order temperature coefficient of elastic stiffness: Tc(1) = (Δc/ΔT)/c in [ppm/K]  First-order pressure coefficient of elastic stiffness: Pc(1) = (Δc/ΔP)/c in [1/TPa]

Table 6: VRHx algorithm applied to measured [87] and simulated [88,93,94] stiffness values at 120 GPa for perovskite and post-perovskite

perovskite MgSiO3 (M = 100.39 g/mole)
ρ	VS	VP	<C11>	<C12>	<C44>	<Cmix>	Ref.
Mg/m3	km/s		GPa
5.332 [94]	7.682	13.981	1042.3	413	314.7	391	[88]
5.332 [94]	7.45	13.63	990.6	398.8	295.9	368	[93]
5.332 [94]	7.675	14.186	1073	444.8	314.1	391.2	[93]
5.332 [94]	7.942	14.579	1133.3	460.7	336.3	418.5	[88]
5.332 [94]	7.635	14.115	1062.4	440.7	310.8	387.1	[94]
4.108 [87]	6.692	10.937	491.4	123.5	184	224.2	[87]
Tc(1) (10-6/K)→			-3.29	-3.08	-3.44		[94]
Pc(1) (1/TPa)→			4.16	5.78	3.07		[93]
post-perovskite MgSiO3 (M = 100.39 g/mole)
ρ	VS	VP	<C11>	<C12>	<C44>	<Cmix>	Ref.
Mg/m3	km/s		GPa
5.407 [94]	7.525	13.62	1003	390.7	306.2	380.1	[93]
5.407 [94]	7.835	14.279	1102.4	438.6	331.9	412.5	[93]
5.407 [94]	7.781	14.157	1083.7	428.9	327.4	406.8	[94]
Tc(1) (10-6/K)→			-5.67	-7.36	-4.56		[94]
Pc(1) (1/TPa)→			4.95	6.13	4.2		[93]
Abbreviations: First-order temperature coefficient of elastic stiffness: Tc(1) = (Δc/ΔT)/c First-order pressure coefficient of elastic stiffness: Pc(1) = (Δc/ΔP)/c ρ: mass density; M: molecular mass; VS , VP : S and P wave speeds;  : VRHx isotropic stiffnesses;  : equivalent Debye stiffness.

Table 7: VRHx algorithm applied to measured [87] and simulated [88,93,94] elastic values for perovskite and post-perovskite

perovskite MgSiO3 (M = 100.39 g/mole)
ρ	11>	12>	44>			<ν>	ΘDmix	T	P	Ref.
Mg/m3	1/TPa			GPa			K		GPa
5.332 [94]	1.238	-0.351	3.178	622.7	807.9	0.284		0	100	[88]
5.332 [94]	1.313	-0.377	3.38	596	761.7	0.287	1347	0	100	[93]
5.332 [94]	1.231	-0.361	3.184	654.2	812.2	0.293	1389	0	120	[93]
5.332 [94]	1.153	-0.333	2.973	684.9	867	0.289		0	120	[88]
5.332 [94]	1.244	-0.365	3.217	647.9	803.9	0.293	1380	3,000	120	[94]
4.108 [87]	2.264	-0.455	5.436	246.1	441.8	0.201	1091	300	0	[87]
post-perovskite MgSiO3 (M = 100.39 g/mole)
ρ	11>	12>	44>			<ν>	ΘDmix	T	P	Ref.
Mg/m3	1/TPa			GPa			K		GPa
5.407 [94]	1.275	-0.358	3.266	594.8	784	0.280	1325	0	100	[93]
5.407 [94]	1.173	-0.334	3.013	659.9	852.8	0.285	1383	0	120	[93]
5.407 [94]	1.19	-0.337	3.055	647.2	840.4	0.284	1371	3,000	120	[94]
Abbreviations: First-order temperature coefficient of elastic stiffness: Tc(1) = (Δc/ΔT)/c First-order pressure coefficient of elastic stiffness: Pc(1) = (Δc/ΔP)/c ρ: mass density; , ,<ν> , : VRHx isotropic compliances, compressibility, Young’s modulus, and Poisson’s ratio; ΘDmix: equivalent Debye temperature; T: temperature; P: pressure

Table 8: Higher-order temperature coefficients of elastic stiffnesses and compliances of α-A?2 O3

GPa	Cµ(RT)	T°	T¹	T²	T³	T(4)
C11	497.3	+1.198488	–64.460817	–50.110218	+28.337272	–5.501432
C33	500.9	+0.176642	–71.486857	–15.490214	–9.699813	+7.521084
C44	146.8	+0.717945	–165.017400	–28.311180	–15.398713	–1.804866
C12	163.8	+0.170563	–4.211675	–21.881483	–0.862603	+4.668163
C13	116	+1.530500	–86.598114	+17.459421	–72.519021	+37.063488
C14	-21.9	+9.442588	+238.788360	–252.815870	+166.215030	–47.879821
C66	166.75	+1.703358	–94.052449	–63.974887	+42.678919	–10.496273
1/TPa	Sµ (RT)	T°	T¹	T²	T³	T(4)
S11	2.3524297	-1.128903	+85.615147	+67.357057	-30.874116	8.297397
S33	2.1730056	-0.018508	+67.276682	+26.211000	+2.906003	-3.215423
S44	6.9481222	-0.350667	+179.500980	+53.505900	+5.694479	-2.355433
S12	-0.7059941	-2.053974	+182.558470	+92.611856	-23.083560	+5.874893
S13	-0.3812867	+0.811997	+24.693309	+107.382030	-124.548690	+49.903673
S14	0.4562635	+7.396371	+515.499870	-57.316463	+152.578980	-46.226595
S66	6.1168476	-1.342452	+107.993250	+73.186583	-29.075665	+7.738180
Input data: [96]
Abbreviations:  Nth-order temperature coefficient of elastic stiffness: Tc(n) = (Δc/ΔT)/(n! c)  Nth-order temperature coefficient of elastic compliance: Ts(n) = (Δs/ΔT)/(n! s)

Table 9: Higher-order temperature coefficients of elastic stiffnesses, compliances and mass density of α-A?2 O3 reduced to isotropy by the VRHx procedure

GPa	C×(RT)	T°	T¹	T²	T³	T(4)
C11	471.210551	0.865804	-89.332734	-32.859794	9.143476	1.555329
C12	145.306373	0.918558	-18.785851	-9.431711	-29.923438	18.090020
C44	162.952089	0.842297	-120.786410	-43.305726	26.562106	-5.816879
Y×	402.716269	0.842664	-110.330780	-41.182737	21.843169	-3.811046
K×	253.941099	0.885934	-62.421180	-23.922837	-5.759213	7.862764
ν×	0.235689	0.152056	51.411132	31.223218	-37.961271	15.609018
1/TPa	S×(RT)	T°	T¹	T²	T³	T(4)
S11	2.483138	-0.824258	109.806000	54.521691	-12.786087	2.196537
S12	-0.585249	-0.691949	161.489570	91.890974	-49.250966	18.943261
S44	6.1367731	-0.799026	119.663740	61.649617	-19.741494	5.390812
Mg/m3	ρ(RT)	T°	T¹	T²	T³	T(4)
ρ	3.982	0.053034	-17.093205	-15.079869	10.240228	-2.549397
Input data: [96]
Abbreviations:  Nth-order temperature coefficient of elastic stiffness: Tc(n) = (Δc/ΔT)/(n! c)  Nth-order temperature coefficient of elastic compliance: Ts(n) = (Δs/ΔT)/(n! s)

 Table 10: Higher-order temperature coefficients of acoustic longitudinal and shear velocities and Debye temperatures of α-A?2 O3 reduced to isotropy by the VRHx procedure

km/s	,  (RT)	T°	T¹	T²	T³	T(4)
	10.8781959	0.409514	-36.177269	-9.759152	-2.308254	2.633480
	6.397044	0.388270	-51.670599	-16.983074	7.264243	-1.462067
K	ΘD,Θx	T°	T¹	T²	T³	T(4)
ΘD	1026.4758	0.349702	-59.713768	-18.890236	8.128135	-1.394143
Θx	1034.1824	0.408064	-55.945675	-21.109861	10.016589	-2.001376
Input data: [96]; ΘD computed using procedure in [150]
Abbreviations: ,  S and P wave speeds; K: isotropic compressibility; ΘD and Θx  : equivalent Debye temperatures

 

A simple computational procedure (VRHx) relaxing single crystal elastic constants in a self-consistent manner has been used to obtain equivalent isotropic elastic stiffnesses of microcrystalline aggregates, such as those that commonly occur in exogeologic bodies. These coefficients, for each mineral, are then combined with those of all other constituents in a “mixing” algorithm to yield effective elastic parameters characterizing both seismic wave propagation and quasi-static deformations in the mineral composite.

Data from the literature are used to compute simulants for olivines: forsterite- fayalite, and for the pyroxenes: enstatiteferrosilite, as examples of the WSD mixing procedure. The VRHx algorithm is further applied to other geological constituents: perovskite, post-perovskite, and alumina. Variations with pressure and temperature are also considered.

The thrusts of this paper are introduction of a simple homogenization procedure converting crystal elasticity data to isotropic values, and use thereof in a physically realistic mixing formula that provides a rapid estimate of acoustic properties.

It is suitable for use with much more sophisticated and elegant programs, such as BurnMan [A1-A3] and Perplex [A4- A8], that are available for producing detailed and accurate phase diagrams, phase equilibria, and extensive thermodynamic data of mineral mixtures.

Elastic nonlinearities, whose treatment in its modern form began with Francis Murnaghan [A9], is also incorporated in the advanced programs. Tables 5 to 10 of the present paper contain some pressure and temperature derivatives of elasticities of selected minerals. These are illustrative of what may be gleaned from the present procedure, and can be used with the advanced algorithms [A1, A4].

[A1] https://burnman.org

[A2] Cottaar S, Heister T, Rose I, Unterborn C. BurnMan - a lower mantle mineral physics toolkit. Geochemistry, Geophysics, Geosystems. 2014; 15(4): 1164-1179.

[A3] Myhill R, Cottaar S, Heister T, Rose I, Unterborn CT, Dannberg J, Martin-Short R. BurnMan: Towards a multidisciplinary toolkit for reproducible deep Earth science. In AGU Fall Meeting Abstracts 2016 Feb.

[A4] http://www.perplex.ethz.ch

[A5] Connolly JAD. Multivariable phase diagrams: an algorithm based on generalized thermodynamics. Am J Sci 1990; 290: 666-718.

[A6] Connolly JAD, Petrini K. An automated strategy for calculation of phase diagram sections and retrieval of rock properties as a function of physical conditions. J Metamorphic Geology. 2002; 20(7): 697-708.

[A7] Connolly JAD. The geodynamic equation of state: what and how. Geochem, Geophys, Geosysts. 2009; 10(10): Q10014.

[A8] Helffrich G, Connolly JAD. Physical contradictions and remedies using simple polythermal equations of state. Am Mineralogist. 2009; 94(11-12): 1616-1619.

[A9] Murnaghan FD. Finite deformations of an elastic solid. Am J Math. 1937; 59(2): 235-260.

The authors declare no conflicts of interest.

1. Cocconi G, Morrison P. Searching for interstellar communications. Nature. 1959; 184(4690): 844-846.

2. Dyson FJ. Search for artificial stellar sources of infrared radiation. Science. 1960; 131(3414): 1667-1668.

3. Dyson FJ. 21st Century spacecraft. Sci Am. 1995; 273(3): 114-116A.

4. Messenger SR, Keller LP, Lauretta DS. Supernova olivine from cometary dust. Science. 2005; 309(5735): 737-741.

5. Krimigis SM, Decker RB. The Voyagers’ odyssey. Am Sci. 2015; 103(4): 284-291.

6. Anglada-Escudé G, Amado PJ, Barnes J, Berdiñas ZM, Butler RP, Coleman GAL, et al. A terrestrial planet candidate in a temperate orbit around Proxima Centauri. Nature. 2016; 536(7617): 437-440.

7. Witze A. Nearby star hosts planet. Nature. 2016; 536(7617): 381- 382.

8. Lubin P. A roadmap to interstellar flight. J Br Interplanet Soc. 2016; 69(2-3): 40-72.

9. Hatzes AP. Earth-like planet around Sun’s neighbour. Nature. 2016; 536(7617): 408-409.

10. Popkin G. First trip to the stars. Nature. 2017; 542(7639): 20-22.

11. Heng K. A new window on alien atmospheres. Am Sci. 2017; 105(2): 86-89.

12. Farihi J, Parsons SG, Gänsicke BT. A circumbinary debris disk in a polluted white dwarf system. Nature Astronomy Lett. 2017; 1: art 0032. doi: 10.1038/s41550-016-0032.

13. Livio M, Silk J. Where are they? Phys Today. 2017; 70(3): 50-57.

14. Cockell CS. The laws of life. Phys Today. 2017; 70(3): 42-48.

15. Pieters CM, Head JW, Patterson W, Pratt S, Garvin J, Barsukov VL, et al. The color of the surface of Venus. Science. 1986; 234(4782): 1379- 1384.

16. Papike J, Taylor L, Simon S. In: Heiken GH, Vaniman DT, French BM, editors. Lunar sourcebook, a user’s guide to the Moon. Cambridge: University Press. 1991; 121-181.

17. Gertsch RE. In McKay MF, McKay DS, Duke MB, editors. Space resources, NASA SP-509. 1992; 3(1): 111-120.

18. Gudkova TV, Zharkov VN. Mars: Interior structure and excitation of free oscillations. Phys Earth Planet Inter. 2004; 142(1): 1-22. 19. Zheng Y, Wang S, Ouyang Z, Zou Y, Liu J, Li C, Li X, Feng J. CAS-1 Lunar soil simulant. Adv Space Res. 2009; 43(3): 448-454.

20. Nittler LR, Starr RD, Weider SZ, McCoy TJ, Boynton WV, Ebel DS, et al. The major-element composition of Mercury’s surface from MESSENGER X-ray spectrometry. Science. 2011; 333(6051): 1847- 1850.

21. Benešová N, ?ížková H. Geoid and topography of Venus in various thermal convection models. Studia Geophysica et Geodaetica 2012; 56(2): 621–639.

22. Stevenson D, Cutts J, Mimoun D, Arrowsmith S, Banerdt B, Blom P, et al. Probing the interior structure of Venus. Report of the Venus Seismology Study Team, Keck Institute for Space Studies (Caltech). 2015: 1-85.

23. Peplowski PN, Klima RL, Lawrence DJ, Ernst CM, Denevi BW, Frank EA, et al. Remote sensing evidence for an ancient carbon-bearing crust on Mercury. Nature Geosci Lett. 2016; doi: 10.1038/NGEO2669

24. Genge MJ, Larson J, Van Ginneken M, Suttle MD. An urban collection of modern-day large micrometeorites: Evidence for variations in the extraterrestrial dust flux through the Quaternary. Geology. 2016 Dec 5: G38352-1.

25. Watters TR, Daud K, Banks ME, Selvans MM, Chapman CR, Ernst CM. Recent tectonic activity on Mercury revealed by small thrust fault scarps. Nature Geoscience. 2016; 9(10): 743-747.

26. Wilson RM. Circuitry made robust enough for Venus. Phys Today. 2017; 70(3): 19-21.

27. Vasavada AR. Our changing view of Mars. Phys Today. 2017; 70(3): 34-41.

28. Fischer-Gödde M, Kleine T. Ruthenium isotopic evidence for an inner Solar System origin of the late veneer. Nature. 2017; 541(7638): 525- 527.

29. Yingst RA, Berger J, Cohen BA, Hynek B, Schmidt ME. Determining best practices in reconnoitering sites for habilability potential on Mars using a semi-autonomous rover: A GeoHeuristic operational strategies test. Acta Astronautica. 2017; 132: 268-281.

30. Biele J, Ulamec S, Maibaum M, Roll R, Witte L, Jurado E, et al. The landing(s) of Philae and inferences about comet surface mechanical properties. Science. 2015; 349(6247): aaa9816.

31. Spohn T, Knollenberg J, Ball AJ, Banaszkiewicz M, Benkhoff J, Grott M, et al. Thermal and mechanical properties of the near-surface layers of comet 67P/Churyumov-Gerasimenko. Science. 2015; 349(6247): aaa0464.

32. Kofman W, Herique A, Barbin Y, Barriot JP, Ciarletti V, Clifford S, et al. Properties of the 67P/Churyumov-Gerasimenko interior revealed by CONSERT radar. Science. 2015; 349(6247): aaa0639.

33. Jutzi M, Asphaug E. The shape and structure of cometary nuclei as a result of low-velocity accretion. Science. 2015; 348(6241): 1355- 1358.

34. Thomas N, Sierks H, Barbieri C, Lamy PL, Rodrigo R, Rickman H, et al. The morphological diversity of comet 67P/Churyumov-Gerasimenko. Science. 2015; 347(6220): aaa0440.

35. Filacchione G, De Sanctis MC, Capaccioni F, Raponi A, Tosi F, Ciarniello M, et al. Exposed water ice on the nucleus of comet 67P/Churyumov– Gerasimenko. Nature. 2016; 529(7586): 368-372.

36. Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. J Acoust Soc Am. 1956; 28(2): 168-191.

37. Ewing M, Jardetzky W, Press F. Elastic waves in layered media. New York: McGraw-Hill. 1957.

38. Musgrave MJP. Crystal acoustics. San Francisco: Holden-Day. 1970.

39. Brekhovskikh LM. Waves in layered media. 2nd ed. New York: Academic Press. 1980. ISBN: 978-012-130560-4. Translation of Volny v sloistykh sredakh, 2nd ed. Moscow: Akademya Nauk. 1973.

40. Davis JL. Wave propagation in solids and fluids. New York: SpringerVerlag. 1988. ISBN: 978-1-4612-7950-1.

41. Norris AN, Sinha BK. The speed of a wave along a fluid/solid interface in the presence of anisotropy and prestress. J Acoust Soc Am. 1995; 98(2): 1147-1154.

42. Shuvalov AL, Every AG. Transverse curvature of the acoustic slowness surface in crystal symmetry planes and associated phonon focusing cusps. J Acoust Soc Am. 2000; 108(5): 2107-2113.

43. Sinha BK, Plona TJ. Wave propagation in rocks with elastic-plastic deformations. Geophys 2001; 66(3): 772-785.

44. Lay T, Wallace TC. Modern global seismology, vol 58. San Diego: Academic Press. 1995. ISBN: 978-0127-32870-6.

45. Anderson OL. Equations of state of solids for geophysics and ceramic science. Oxford, UK: Oxford University Press. 1995. ISBN: 0-19- 505606-X.

46. Cerveny V. Seismic ray theory. Cambridge, UK: Cambridge University Press. 2001. ISBN: 978-0521-36671-7.

47. Li C, van der Hilst RD, Engdahl ER, Burdick S. A new global model for P wave speed variations in Earth’s mantle. Geochem Geophys Geosyst. 2008; 9(5): art Q05018.

48. Aki K, Richards PG. Quantitative seismology. 2nd ed. Sausilito, CA: University Science Books. 2009. ISBN: 978-1891-38963-4.

49. Rudolph ML, Leki? V, Lithgow-Bertelloni C. Viscosity jump in Earth’s mid-mantle. Science. 2015; 350(6266): 1349-1352.

50. Ballmer MD, Schmerr NC, Nakagawa T, Ritsema J. Compositional mantle layering revealed by slab stagnation at ~ 1000-km depth. Science Adv. 2015; 1(11): e1500815.

51. Thio V, Cobden L, Trampert J. Seismic signature of a hydrous mantle transition zone. Phys Earth Planet Ints. 2016; 250: 46-63.

52. Chung DH, Wang H, Simmons G. On the calculation of the seismic parameter Φ at high pressure and high temperature. J Geophys Res. 1970; 75(26): 5113-5120.

53. Thomsen L. Elasticity of polycrystals and rocks. J Geophys Res. 1972; 77(2): 315-327.

54. White RM. Surface elastic waves. Proc IEEE. 1970; 58(8): 1238-1276.

55. Voigt W. Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Ann Phys (Leipzig). 1889; 274(12): 573-587.

56. Reuß A. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeits angew Math Mech. 1929; 9(1): 49-58.

57. Hill R. The elastic behaviour of a crystalline aggregate. Proc Phys Soc (London). 1952; A65(5): 349-354.

58. Ballato A. Voigt–Reuss–Hill moduli for ferroelectric aggregates. Intl Symp Appl Ferroelectrics (ISAF) Proc. 2006 Jul 30-Aug 03. Sunset Beach, NC. New York: IEEE; 208-211.

59. Suzuki I. Thermal expansion of perclase and olivine, and their anharmonic properties. J Phys Earth. 1975; 23(2), 145-159.

60. Hazen RM. Effects of temperature and pressure on the crystal structure of forsterite. Am Mineral. 1976; 61(11-12): 1280-1293.

61. Sumino Y, Nishizawa O, Goto T, Ohno I, Ozima M. Temperature variation of elastic constants of single-crystal forsterite between –190o and 400o C. J Phys Earth. 1977; 25(4): 377-392.

62. Sumino Y. The elastic constants of Mn2 SiO4 , Fe2 SiO4 and Co2 SiO4 , and the elastic properties of olivine group minerals at high temperature. J Phys Earth. 1979; 27(3): 209-238.

63. Suzuki I, Seya K, Takei H, Sumino Y. Thermal expansion of fayalite, Fe2 SiO4 . Phys Chem Minerals. 1981; 7(2): 60-63.

64. Suzuki I, Anderson OL, Sumino Y. Elastic properties of a single-crystal forsterite Mg2 SiO4 , up to 1,200 K. Phys Chem Minerals. 1983; 10(1): 38-46.

65. Isaak DG, Anderson OL, Goto T, Suzuki I. Elasticity of single?crystal forsterite measured to 1700 K. J Geophys Res: Solid Earth. 1989; 94(B5): 5895-5906.

66. Isaak DG, Graham EK, Bass JD, Wang H. The elastic properties of single-crystal fayalite as determined by dynamical measurement techniques. In Liebermann RC, Sondergeld CH, editors. Experimental techniques in mineral and rock physics, Vol. 141. Basel: Birkhäuser. 1994; 393-414. Reprint of Pure & Appl. Geophys. (PAGEOPH). 1993; 141(2-4): 393-414.

67. Bass JD, Weidner DJ. Elasticity of single-crystal orthoferrosilite. J Geophys Res: Solid Earth. 1984; 89(B6): 4359-4371.

68. Anderson OL, Schreiber E, Liebermann RC, Soga N. Some elastic constant data on minerals relevant to geophysics. Revs Geophysics. 1968; 6(4): 491-524.

69. Ohno I. Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorhombic crystals. J Phys Earth. 1976; 24(4): 355-379.

70. Kranzberg M, Smith CS. Part I Materials in History and Society. Matls Sci Engrg. 1979; 37(1): 1-39.

71. Sumino Y, Anderson OL. Elastic constants of minerals. In: Carmichael RS, editor. CRC Handbook of Physical Properties of Rocks. Boca Raton, FL: CRC Press. 1984; 2: 39–138.

72. Nye JF. Physical properties of crystals. Clarendon, UK: Oxford University Press. 1985. ISBN: 978-0198-51165-6.

73. Ohno I, Yamamoto S, Anderson OL, Noda J. Determination of elastic constants of trigonal crystals by the rectangular parallelepiped resonance method. J Phys Chem Solids. 1986; 47(12): 1103-1108.

74. Mochizuki E. Application of group theory to free oscillations of an anisotropic rectangular parallelepiped. J Phys Earth. 1987; 35(2): 159-170.

75. Anderson OL, Isaak D, Oda H. High?temperature elastic constant data on minerals relevant to geophysics. Revs Geophysics. 1992; 30(1): 57-90.

76. Hunt CP, Moskowitz BM, Banerjee SK. Magnetic properties of rocks and minerals. In: Ahrens TJ, editor. A handbook of physical constants: Rock physics and phase relations. AGU Reference Shelf. Washington, DC: Amer Geophys Union. 1995; 189–204.

77. Anderson OL, Bass JD. Mineralogy and composition of the upper mantle. Geophys Res Lett. 1984; 11(7): 637-640.

78. Anderson OL. Theory of the Earth. Oxford: Blackwell Scientific. 1989. ISBN 0-86542-335-0

79. Anderson OL. New theory of the Earth. 2nd ed. Cambridge, UK: Cambridge University Press. 2007. ISBN: 978-521-84959-3.

80. Buffett BA, Garnero EJ, Jeanloz R. Sediments at the top of Earth’s core. Science. 2000; 290(5495): 1338–1342.

81. Isaak DG. In: Levy M, Bass H, Stern R, editors. Handbook of elastic properties of solids, liquids, and gases, Vol. III. Orlando, FL: Academic Press. 2001; 325-376.

82. Carlson RL. In: Levy M, Bass H, Stern R, editors. Handbook of elastic properties of solids, liquids, and gases, Vol. III. Orlando, FL: Academic Press. 2001; 377-461.

83. Bass JD. In Ahrens TJ, editor. Mineral physics and crystallography: A handbook of physical constants. AGU Reference Shelf Series 2. Washington, DC: Am Geophys Union. 1995; 45-63.

84. Avdeeva A, Moorkamp M, Avdeev D, Jegen M, Miensopust M. Three dimensional inversion of magnetotelluric impedance tensor data and full distortion matrix. Geophys J Int. 2015; 202(1): 464-481.

85. Carlson RW. Earth’s building blocks. Nature. 2017; 541(7636): 468- 470.

86. Dauphas N. The isotopic nature of the Earth’s accreting material through time. Nature. 2017; 541(7638): 521-524.

87. Yeganeh-Haeri A. Synthesis and re-investigation of the elastic properties of single-crystal magnesium silicate perovskite. Phys Earth Planet Inter. 1994; 87(1): 111-121.

88. Wentzcovitch RM, Ross NL, Price GD. Ab initio study of MgSiO3 and CaSiO3 perovskites at lower-mantle pressures. Phys Earth Planet Inter. 1995; 90(1-2): 101-112.

89. Monnereau M, Yuen DA. Topology of the postperovskite phase transition and mantle dynamics. PNAS. 2007; 104(22): 9156–9161. 90. Duffy TS. Mineralogy at the extremes. Nature 2008; 451(7176): 269- 270.

91. Duffy TS. Some recent advances in understanding the mineralogy of Earth's deep mantle. Philos Trans R Soc A. 2008; 366(1883): 4273– 4293.

92. Yoneda A, Fukui H, Xu F, Nakatsuka A, Yoshiasa A, Seto Y, et al. Elastic anisotropy of experimental analogues of perovskite and post perovskite help to interpret D′′ diversity. Nature communs. 2014; 5(3453): art 4453.

93. Iitaka T, Hirose K, Kawamura K, Murakami M. The elasticity of the MgSiO3 post-perovskite phase in the Earth’s lowermost mantle. Nature. 2004; 430(6998): 442-445.

94. Oganov AR, Ono S. Theoretical and experimental evidence for a post-perovskite phase of MgSiO3 in Earth’s D” layer. Nature. 2004; 430(6998): 445-448.

95. Carrez P, Ferr D, Cordier P. Peierls-Nabarro model for dislocations in MgSiO3 post-perovskite calculated at 120 GPa from first principles. Philos Mag. 2007; 87(22): 3229–3247.

96. Goto T, Anderson OL, Ohno I, Yamamoto S. Elastic constants of corundum up to 1825 K. J Geophys Res. 1989; 94(B6): 7588-7602.

97. Wittstruck RH, Emanetoglu N, Lu Y, Laffey S, Ballato A. Properties of transducers and substrates for high frequency resonators and sensors. J Acoust Soc Am. 2005; 118(3): 1414-1423.

98. Goto T, Ohno I, Sumino Y. The determination of the elastic constants of natural almandine-pyrope garnet by rectangular parallelepiped resonance method. J Phys Earth. 1976; 24(2): 149-156.

99. Sumino Y, Nishizawa O. Temperature variations of elastic constants of pyrope-almandine garnets. J Phys Earth. 1978; 26(3): 239-252.

100. Babuška V, Fiala J, Kumazawa M, Ohno I, Sumino Y. Elastic properties of garnet solid-solution series. Phys Earth planet Interiors. 1978; 16(2): 157-176.

101. Li L, Weidner DJ, Brodholt J, Alfe D, Price GD. Ab initio molecular dynamic simulation on the elasticity of Mg3 A?2 Si3 O12 pyrope. J Earth Sci. 2011; 22(2): 169-175.

102. Sumino Y, Ohno I, Goto T, Kumazawa M. Measurement of elastic constants and internal frictions of single-crystal MgO by rectangular parallelepiped resonance. J Phys Earth. 1976; 24(3): 263-273.

103. Sumino Y, Anderson OL, Suzuki I. Temperature coefficients of elastic constants of single crystal MgO between 80 and 1,300 K. Phys Chem Minerals. 1983; 9(1): 38-47.

104. Zhu Q, Oganov AR, Lyakhov AO. Novel stable compounds in the Mg-O system under high pressure. Physical Chemistry Chemical Physics. 2013; 15(20): 7696-7700.

105. Suzuki I, Ohno I, Anderson OL. Harmonic and anharmonic properties of spinel MgA?2 O4 . Am Mineral. 2000; 85(2-3): 304-311.

106. Northrup PA, Leinenweber K, Parise JB. The location of H in the high pressure synthetic  A?2 SiO4 (OH)2 topaz analogue. Am Mineral. 1994; 79(3-4): 401-404.

107. Winkler B, Hytha M, Warren MC, Milman V, Gale JD, Schreuer J. Calculation of the elastic constants of the A?2 SiO5 polymorphs andalusite, sillimanite and kyanite. Z. Kristallogr. 2001; 216(2): 67–70.

108. Jolliff BL, Hughes JM, Freeman JJ, Zeigler RA. Crystal chemistry of lunar merrillite and comparison to other meteoritic and planetary suites of whitlockite and merrillite. Am Miner. 2006; 91(10): 1583- 1595.

109. Hughes JM, Jolliff BL, Rakovan J. The crystal chemistry of whitlockite and merrillite and the dehydrogenation of whitlockite to merrillite. Am Miner. 2008; 93(8-9): 1300-1305.

110. Adcock CT, Hausrath EM, Forster PM, Tschauner O, Sefein KJ. Synthesis and characterization of the Mars-relevant phosphate minerals Fe- and Mg-whitlockite and merrillite and a possible mechanism that maintains charge balance during whitlockite to merrillite transformation. Am Miner. 2014; 99(7): 1221-1232.

111. McCubbin FM, Shearer CK, Burger PV, Hauri EH, Wang J, Elardo SM, et al. Volatile abundances of coexisting merrillite and apatite in the martian meteorite Shergotty: Implications for merrillite in hydrous magmas. Am Miner. 2014; 99(7): 1347-1354.

112. Shearer CK, Burger PV, Papike JJ, McCubbin FM, Bell AS. Crystal chemistry of merrillite from Martian meteorites: Mineralogical recorders of magmatic processes and planetary differentiation. Meteor Planet Sci. 2015; 50(4): 649-673.

113. Xie X, Yang H, Gu X, Downs RT. Chemical composition and crystal structure of merrillite from the Suizhou meteorite. Am Miner. 2015; 100(11-12): 2753-2756.

114. Britvin SN, Krivovichev SV, Armbruster T. Ferromerrillite, Ca9 NaFe2+(PO4 )7 , a new mineral from the Martian meteorites, and some insights into merrillite-tuite transformation in shergottites. Eur J Miner. 2016; 28(1): 125-136.

115. Adcock CT, Tschauner O, Hausrath EM, Udry A, Luo SN, Cai Y, et al. Shock-transformation of whitlockite to merrillite and the implications for meteoritic phosphate. Nature Communs. 2017; 8: art 14667.

116. Kröner E. Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Zeits Physik. 1958; 151(4): 508-518.

117. Lotgering FK. Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures-I. J Inorg Nuc Chem 1959; 9(2): 113-123.

118. Lotgering FK. Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures-II. J Inorg Nuc Chem. 1960; 16(1-2): 100-108.

119. Hashin Z, Shtrikman S. On some variational principles in anisotropic and nonhomogeneous elasticity. J Mech Phys Solids. 1962; 10(4): 335-342.

120. Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behaviour of polycrystals. J Mech Phys Solids. 1962; 10(4): 343-352.

121. Gazis D, Tadjbakhsh I, Toupin R. The elastic tensor of given symmetry nearest to an anisotropic elastic tensor. Acta Cryst. 1963; 16(9): 917- 922.

122. Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids. 1963; 11(2): 127-140.

123. Hashin Z. On the elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J Mech Phys Solids. 1965; 13(3): 119-134.

124. Peselnick L, Meister R. Variational method of determining effective moduli of polycrystals: (A) Hexagonal symmetry, (B) trigonal symmetry. J Appl Phys. 1965; 36(9): 2879-2884.

125. Meister R, Peselnick L. Variational method of determining effective moduli of polycrystals with tetragonal symmetry. J Appl Phys. 1966; 37(11): 4121-4125.

126. Rietveld HM. A profile refinement method for nuclear and magnetic structures. J Appl Crystallog. 1969; 2(2): 65-71.

127. Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 1973; 21(5): 571-574.

128. Willis JR. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J Mech Phys Solids. 1977; 25(3): 185-202.

129. Kröner E. Self-consistent scheme and graded disorder in polycrystal elasticity. J Phys F: Metal Phys. 1978; 8(11): 2261-2267.

130. Claassen RS, Chynoweth AG. Part II Materials science and engineering as a multidiscipline. Matls Sci Engrg. 1979; 37(1): 41-102.

131. Watt JP. Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with orthorhombic symmetry. J Appl Phys. 1979; 50(10): 6290-6295.

132. Watt JP. Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with monoclinic symmetry. J Appl Phys. 1980; 51(3): 1520-1524.

133. Watt JP, Peselnick L. Clarification of the Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. J Appl Phys. 1980; 51(3): 1525-1531.

134. Hashin Z. Analysis of composite materials – A survey. J Appl Mech. 1983; 50(3): 481-505.

135. Dollase WA. Correction of intensities for preferred orientation in powder diffractometry: Application of the March model. J Appl Crystallog. 1986; 19(4): 267-272.

136. Weng GJ. The theoretical connection between Mori-Tanaka’s theory and the Hashin-Shtrikman-Walpole bounds. Int J Engrg Sci. 1990; 28(11): 1111-1120.

137. Hirsekorn S. Elastic properties of polycrystals: A review. Textures & Microstruct. 1990; 12(1-3): 1-14.

138. Bunge HJ. Partial texture analysis. Textures & Microstruct. 1990; 12(1-3): 47-63.

139. Zuo L, Humbert M, Esling C. Elastic properties of polycrystals in the Voigt-Reuss-Hill approximation. J Appl Crystallogr. 1992; 25(6): 751-755.

140. Horn JA, Zhang SC, Selvaraj U, Messing GL, Trolier-Mckinstry S. Templated grain growth of textured bismuth titanate. J Am Ceram Soc. 1999; 82(4): 921-926.

141. Seabaugh MM, Vaudin MD, Cline JP, Messing GL. Comparison of texture analysis techniques for highly oriented α-A?2 O3 . J Am Ceram Soc. 2000; 83(8): 2049-2054.

142. Wenk HR, Van Houtte P. Texture and anisotropy. Rep Prog Phys. 2004; 67(5): 1367-1428.

143. Furushima R, Tanaka S, Kato Z, Uematsu K. Orientation distribution – Lotgering factor relationship in a polycrystalline material – as an example of bismuth titanate prepared by a magnetic field. J Ceram Soc Jpn. 2010; 118(10): 921-926.

144. Man CS, Huang M. A simple explicit formula for the Voigt-Reuss-Hill average of elastic polycrystals with arbitrary crystal and texture symmetries. J Elasticity. 2011; 105(1): 29-48.

145. Köhler B, Endler I, Herzog T, Heuer H, Kopycinska-Müller M, Krueger P, et al. Piezoelectric properties of CVD deposited A?N layers as active material for ultrasonic transducers. Proc 11th European Conf Non-Destructive Testing (ECNDT 2014), 2014 Oct 6-10. Prague, Czech Republic. 3195-3196.

146. Winkelmann A, Schott O. Über die Elastizität und über die Zug- und Druckfestigkeit verschiedener neuer Gläser in ihrer Abhängigkeit von der chemischen Zusammensetzung. Ann Phys (Leipzig). 1894; 287(4): 697-729.

147. Dragic PD. Simplified model for effect of Ge doping on silica fibre acoustic properties. Electron Lett. 2009; 45(5): 256-257.

148. Debye P. Zur Theorie der Anomalen Dispersion im Gebiet der Langwellen Elektrischen Strahlung. Ber d Deut Phys Ges. 1913; 15(16): 777-793.

149. Anderson OL. A simplified method for calculating the Debye temperature from elastic constants. J Phys Chem Solids. 1963; 24(7): 909-917.

150. Hannay J, Nye JF. Fibonacci numerical integration on a sphere. J Phys A: Math Gen. 2004; 37(48): 11591-11601.