The Elastic Properties of β -Mg 2 SiO 4 Containing 0.73 wt.% of H 2 O to 10 GPa and 600 K by Ultrasonic Interferometry with Synchrotron X-Radiation

: We measured the elastic velocities of a synthetic polycrystalline β -Mg 2 SiO 4 containing 0.73 wt.% H 2 O to 10 GPa and 600 K using ultrasonic interferometry combined with synchrotron X-radiation. Third-order Eulerian finite strain analysis of the high P and T data set yielded K so = 161.5(2) GPa, G o = 101.6(1) GPa, and ( ∂ K s / ∂ P ) T = 4.84(4), ( ∂ G / ∂ P ) T = 1.68(2) indistinguishable from K so = 161.1(3) GPa, G o = 101.4(1) GPa, and ( ∂ K s / ∂ P ) T = 4.93(4), ( ∂ G / ∂ P ) T = 1.73(2) from the linear fit. The hydration of the wadsleyite by 0.73 wt.% decreases K s and G moduli by 5.3% and 8.6%, respectively, but no measurable effect was noted for ( ∂ K s / ∂ P ) T and ( ∂ G / ∂ P ) T . The temperature derivatives of the K s and G moduli from the finite strain analysis ( ∂ K S / ∂ T ) P = − 0.013(2) GPaK − 1 , ( ∂ G/ ∂ T) P = − 0.015(0.4) GPaK − 1 , and the linear fit ( ∂ K S / ∂ T ) P = − 0.015(1) GPaK − 1 , ( ∂ G/ ∂ T) P = − 0.016(1) GPaK − 1 are in agreement, and both data sets indicating the |( ∂ G/ ∂ T) P | to be greater than |( ∂ K S / ∂ T ) P |. Calculations yield ∆ V p( α - β ) = 9.88% and ∆ V S( α - β ) = 8.70% for the hydrous β -Mg 2 SiO 4 and hydrous α -Mg 2 SiO 4 , implying 46–52% olivine volume content in the Earth's mantle to satisfy the seismic velocity contrast ∆ V s = ∆ V P = 4.6% at the 410 km depth.


Introduction
Wadsleyite [β-(Mg, Fe)2SiO4] is a high-pressure polymorph of olivine [α-(Mg, Fe)2SiO4] stable from 410 to 525 km in the Earth's transition zone. Wadsleyite can incorporate varying amounts of water up to 3.3% as hydroxyl (OH) groups in the structure depending on the pressure and temperature and phase conditions [1][2][3][4]. The hydration of wadsleyite takes place by two H atoms substituting for one octahedral Mg in the structure [1,4,[5][6][7][8][9][10][11] that affects the elasticity when compared to the anhydrous phase, and most notably at high pressure (P) and temperature (T) due to the variability of the H atomic radius.
Previous studies [9,[12][13][14] have suggested that the Earth's transition zone could be a potential water reservoir, given the abundance of wadsleyite and ringwoodite in the region. The seismic velocity jumps at the 410-km depth have been associated with changes in the elastic wave speeds resulting from the transformation of olivine to wadsleyite. Still, the depth of the phase transition is reduced by OH incorporation into the olivine and wadsleyite structures.
The elastic properties of hydrous wadsleyite, and particularly the pressure (P) and temperature (T) derivatives of the elastic moduli, are essential for providing tighter constraints on the olivine content of the Earth's mantle by comparing laboratory elasticity data with the seismic velocity jumps at the 410-km depth. The data are also essential for constraining the water content in the wadsleyite phase that constitutes about 60% of the mineral assemblage in the transition zone region.
Data are already available for anhydrous Mg, and Mg-Fe wadsleyite at high pressure and room temperature [15][16][17], high pressure and high temperature [18][19][20], and, high temperature and room pressure [21][22][23][24], but the elasticity data for the hydrous wadsleyite are sparse compared to the anhydrous wadsleyite. A few static compression studies [6,25] have yielded information on the isothermal bulk modulus (KT) and its derivative (KT') for OH-bearing wadsleyite. However, the static compression studies do not provide information on the shear modulus. Secondly, fitting such P-V-T data to an equation of state requires a significant trade-off between the bulk modulus and its derivative [26].
Brillouin scattering measurements [27] on single-crystal wadsleyite containing 0.37, 0.84, and 1.66 wt.% water at room pressure (P) and room temperature (T) show that both the bulk (Kso) and shear (Go) moduli decrease linearly with water content in the wadsleyite. Measurements of the elasticity of wadsleyite containing 0.84 wt.% H2O to 12 GPa at room T [28] yielded the P-wave and Swave velocities that are 2.7% and 3.6%, respectively, lower than the corresponding data for the anhydrous wadsleyite [17]. The study also concluded that the effect of OH substitution in wadsleyite on the pressure derivatives of the bulk and shear moduli is immeasurable. There is no consistent effect on the pressure derivatives of K and G due to the hydration of wadsleyite from comparing the Brillouin scattering data on single-crystal hydrous Fe-bearing wadsleyite containing 0.24 wt.% H2O (Fe = 0.112) measured at high P and room T [29], with similar measurements carried out for wadsleyite containing 1.93 wt.% H2O (Fe = 0.112) [30], and combining the Fe-bearing wadsleyite data with the static compression studies and the Brillouin scattering data on end-member wadsleyite with 0.84 wt.% H2O [28] still does not reveal a clear trend regarding the variation of K and G with OH content in wadsleyite.
Temperature is an important variable, in addition to the pressure that affects geophysical properties, including the elastic wave velocities in the Earth's mantle. However, compared to the limited data currently available for the anhydrous wadsleyite [18,[20][21][22][23][24], there is as yet no study to investigate the effect of temperature on the elasticity of hydrous wadsleyite. Ultrasonic and Brillouin scattering techniques are well-established for investigating the elasticity of materials. The Brillouin scattering measures the elasticity of the single-crystal as a function of the crystallographic direction within the sample. However, the utilization of the technique to study the hydrous material at high T is challenging compared to the pressure study. The ultrasonic technique employs either dense synthetic polycrystalline or relatively large single-crystal specimens. However, the main difficulty is to make suitable high-acoustic polycrystalline specimens or relatively large single-crystals of the hydrous high-pressure phase containing structurally-bound water. In addition to the sample difficulty, the OH retention in the hydrous sample during the high P and T ultrasonic studies is an essential factor to consider in the studies.
We report the synthesis of a dense isotropic polycrystalline specimen of hydrous wadsleyite (β-Mg2SiO4) containing 0.73 wt.% H2O, and the ultrasonic elasticity data for the sample measured to 10 GPa and temperatures to 600 K using ultrasonic interferometry techniques combined with synchrotron X-ray diffraction. We compare our data with those of previous studies of Mg and Mg-Fe bearing hydrous wadsleyite, evaluate the effect on the elasticity due to OH incorporation in the wadsleyite, and discuss the impact of the temperature derivatives of the elasticity of wadsleyite on the olivine content of the Earth's transition zone.

Sample Synthesis and Characterization
The synthetic polycrystalline specimen of hydrous wadsleyite (β-Mg2SiO4) used in this study was hot-pressed in the 2000-ton uniaxial split-cylinder apparatus (USSA-2000) of the Kawai-type [31,32] at the Stony Brook High-Pressure laboratory, under conditions of 15.5 GPa and 1000 °C for 3 h within a 14/8 mm cell assembly.
The starting mixture consisted of pure forsterite (α-Mg2SiO4) and brucite [Mg(OH2)] calculated to yield 0.778 wt.% H2O, with additional silica (SiO2) to maintain chemical charge balance due to the breakdown of brucite. Before weighing, we dried the α-Mg2SiO4 and the SiO2 powders at 1000 °C for 24 h and the Mg(OH)2 at 350 °C for 24 hours. The mixture was ground several times to a fine-grained homogenous powder in an agate mortar under alcohol. The powder mixture was dried in an oven at 150 °C for 24 h and then densely packed in an Au75Pd25 capsule previously annealed at 900 °C. The capsule was crimped, cold-sealed in the air, and placed inside a NaCl sleeve inside a graphite resistance furnace. The NaCl insulates the capsule from the graphite furnace and also provides a pseudo-hydrostatic environment for the sample during heating at high pressure. A previous study [33] has demonstrated that NaCl loses most of its shear strength at about 300 °C, thus minimizing nonhydrostatic stress around the sample.
The hot-pressing procedure used in the study is similar to those described in detail by previous investigators [34][35][36]. We first increased the pressure to 1 GPa, and then the sample was preheated to 200 °C to relax the NaCl around the sample. Subsequently, we increased the pressure slowly to 15.5 GPa in 13 h, and then the temperature was increased to 1000 °C in 15 min. After maintaining the sample at the P and T conditions for 3 h, the temperature was rapidly decreased to 200 °C in 10 min and then maintained at the T condition throughout the depressurization that took 15 h.
The recovered sample was cylindrical, about 2.7 mm in diameter and 2.2 mm long. Before further characterization, we ground both ends of the sample roughly flat and collected X-ray diffraction spectra of the surfaces on a Rigaku Ultima IV diffractometer (Tube voltage-40 kV, Tube current-30 mA, Cu Kα λ = 1.54059 Å, 10-70° 2θ, 0.02° step size, 0.4 minute/step) in Bragg-Brentano geometry with a D-tex Ultra solid-state detector. The spectra confirmed the specimen to be single-phase wadsleyite (β-Mg2SiO4).
The bulk density of the sample was determined by Archimedes' method using distilled water in which we add a few drops of an organic solvent to reduce surface tension on the sample and immersed parts. We corrected the measured bulk density for the effect of air buoyancy on the sample weight in air and also the change in the density of the fluid with temperature. We obtained ρ = 3.435 (5) g/cc, which is in excellent agreement with the theoretical X-ray density (ρ = 3.436 g/cc) based on the single-crystal X-ray systematics developed [6] that report the variation of the density of wadsleyite with H2O content.
We carried out a scanning electron microscopy (SEM) examination of both a fractured and a polished surface of the wadsleyite specimen using an LEO-1550 FEG SEM with electron dispersive X-ray spectroscopy (EDAX) operating at the high tension of 20 kV. The samples were sputter-coated with gold for the analysis. We observed, as evident in the photomicrograph of the fractured surface of the sample in Figure 1a, that the synthetic polycrystalline wadsleyite sample is homogeneous and fine-grained. The average grain size of the sample is about 3-5 µm. Examination of the polished surface of the sample (Figure 1b) reveals the grains to be well-equilibrated and well-developed with sharp straight edges meeting at high angles. The water content of the polycrystalline hydrous wadsleyite (β-Mg2SiO4) sample was measured using the Fourier transform infrared spectroscopy (FTIR) technique at the Key Laboratory of High-Temperature and High-Pressure Study of the Earth's Interior, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang, China. Previous studies [37,38] have described in detail the set-up and experimental procedure of the FTIR technique.
We carried out the spectra measurements using a Fourier transform vacuum infrared spectroscopy (FTIR) spectrometer (Vertex-70V and Hyperion-1000 infrared microscope). We collected infrared spectra of the original polycrystalline sample from 350 to 8000 cm −1 wavenumbers, as well as the recovered specimen from the high pressure and high-temperature ultrasonic studies.
The specimens were doubly polished to a thickness of about 60 µm for the IR analysis. The IR absorption of the sample was measured using unpolarized radiation with a Mid-IR light source, a CaF2 beam splitter, and an MCT detector with a 100 µm × 100 µm aperture. We collected 512 scans for each spectrum in the analysis. In Figure 2, we have overlapped the infrared spectra acquired for a sliced piece of the original sample and the remaining sample recovered from the ultrasonic velocity measurements to demonstrate that there was no change in the water content during the high P and T ultrasonic studies; the data agree within the uncertainties (10%) of the FTIR measurements. The Paterson calibration [39] was adopted to precisely determine the water content from FT-IR absorption data using, where COH is the molar concentration of hydroxyl (ppm wt. of H2O or H/106 Si), Bi is the density factor (4.08 cm × 104 cm H/106 Si), ξ is the orientation factor (1/3), and K(v) is the absorption coefficient at wavenumber v in cm −1 . The integration was from 3000 to 3750 cm −1 . The analysis yielded identical water contents of 0.73 (7) wt.% for both specimens. The listed uncertainty of 10% is mainly due to errors in the sample thickness measurement, in addition to the current unavailability of a standard water content calibration for hydrous wadsleyite. For ultrasonic studies, both ends of a sliced cylindrical piece of the sample were ground and polished flat and parallel to within 1/4λ of visible light, using 9, 6, 3, 1, and ¼ µm diamond compounds in succession. The polished sample, having a length of 1.017 mm, was cored to a 2 mm diameter for the ultrasonic studies. For internal consistency and data comparison, we also prepared a sample of anhydrous β-Mg2SiO4 that we have measured using the same the high P and T technique described subsequently for the hydrous sample.
The synchrotron X-ray diffraction pattern for the hydrous wadsleyite (β-Mg2SiO4) containing 0.73 (7) wt.% of H2O used in this study is shown in Figure 3 and compared with one taken with the press open (ambient condition) at the end of the high P and T ultrasonic experiments.
The spectra are sharp and show no uncharacteristic peaks, indicating that there is no significant residual stress in both the original and the recovered specimens, and also that the wadsleyite phase did not change during the high P and T experiments. We conducted the ultrasonic measurements below 600 K to avoid the dehydration of the hydrous β-Mg2SiO4 and also to avoid a possible backtransformation of the sample to the low P and T olivine phase.

Elasticity Measurements at High Pressure and High Temperature
The high P and T ultrasonic velocity measurements of the hydrous wadsleyite sample were carried out by ultrasonic interferometry techniques. The pressure generating device was a 250-ton hydraulic DIA-type multi-anvil apparatus equipped with a DDIA module installed at the 6-BM-B beamline at the advanced photon light source (APS) of the Argonne National Laboratory and interphased with in-situ X-ray diffraction and X-radiographic techniques. The experimental setup and the directly integrated acoustic system combined with pressure experiments (DIASCOPE) for data acquisition are described in detail in a previous study [40]. The DIASCOPE is an adapted ultrasonic interferometry technique for fast acoustic travel time measurements. A previous study [41] provides a detailed description of the main features and pressure-generating mechanism of the DIA-type highpressure apparatus. The cell assembly and the acoustic piezoelectric transducer/cubic carbide anvil arrangement used in this study are the same as illustrated in Figure 1 of a previous study [42], and as described in other previous studies [43]. We utilized a dual-mode 10° Y-cut LiNbO3 transducer capable of generating and receiving P and S waves simultaneously. An alumina buffer rod of length 2.0 mm and 2.0 mm in diameter ground and polished flat on both ends is inserted between the specimen and the tungsten carbide (WC) anvil, allowing high-frequency acoustic signals  to propagate to and from the sample. A BN sleeve houses the sample supported at the far end by a NaCl disc; the NaCl also serves as an in-situ pressure marker. A 2 µm gold Au foil was inserted at the top and bottom of the specimen to improve the mechanical coupling at the interfaces between the specimen and cell components as the sample was pressurized and to enhance the transmission and reception of the acoustic signal. The two Au foils delineate the specimen in the X-radiographic images acquired for determining the specimen length at each P-T condition [44,45]. A W/Re 3%-W/Re 25% thermocouple located at the interface between the sample and the NaCl in-situ pressure standard monitors the sample temperature.
We have designed the pressure-temperature (P-T) path of the ultrasonic experiments used in this study and shown in Figure 4 to minimize deviatoric stress in the sample. We initially compressed the sample to about 2 GPa. We then heated it to 600 K, where we collected travel times of acoustic compressional (P) and shear (S) waves, X-ray diffraction of the sample, the NaCl in-situ pressure standard, and a sample image. Data were subsequently collected at 100 K intervals as we decreased the temperature down to ambient temperature along the isobar. We then increased the pressure moderately and repeated the procedure until six cycles of data collection were completed up to the peak pressure of 9.8 GPa, followed by four data collecting sequences on decompression. At each P and T point, we waited for about 3 min to equilibrate the pressure and temperature within the sample before acquiring the data. The P and T path in the current study differs from that of previous standard studies [16,19,20,[46][47][48] in which the sample is first compressed to the maximum pressure, followed by heating and subsequent data collection during cooling along an isobaric path. Temperatures in the traditional studies were relatively high to relax stress accumulated in the sample during compression to high pressure. However, given the limited temperature of 600 K in the current study, we minimized the stress in the sample throughout the experiment by heating the sample to the peak temperature at each pressure step of the operation before the data acquisition.

Data Acquisition and Analysis
We determine the P and S wave travel times by the pulse-echo-overlap (PEO) method [40] with a standard deviation of 0.2 ns (0.2%) and 0.5 ns (0.1%) for the P and S waves, respectively. A correction 1.437 ns and 0.272 ns was applied to all the P and S wave travel times, respectively, for the effect of the 2 µm Au bond between the sample and the buffer rod, using a previously established procedure [49].
The two-way corrected P and S wave travel times are tabulated in Table 1 and plotted as a function of pressure in Figure 5a,b, showing the travel times of both the P and S waves to decrease linearly and steadily with pressure along the isotherms. The sample chamber for ultrasonic measurements in the cubic anvil DIA-type apparatus is composed of an alumina buffer rod and other cylindrical parts. These components impose axial deviatoric stress in the cell assembly that could cause vertical stress, σ1 to be different from the lateral stress σ2 = σ3, as observed in previous studies [33,41]. Previous ultrasonic experiments in the multianvil press with synchrotron radiation [18][19][20][46][47][48] only measured vertical stress σ1. However, recent replacement of the lateral WC anvil with a sintered diamond has afforded measurement of the lateral stress σ2 = σ3, in addition to the vertical stress. The pressure listed in Table 1 is the weighted average of the three components given by P = (2σ1 + σ3)/3, yielding a more precise pressure than previously possible. The axial stress was generally higher than the lateral stress by up to 0.4 GPa by heating the sample to 600 K. However, the difference diminished linearly with decreasing temperature, and ultimately reversed signs around ambient temperature where the vertical stress became moderately higher than the lateral stress. The analysis shows that while the temperature derivatives of the elastic moduli are sensitive to the stress differences between σ1 and σ2 = σ3, the pressure derivatives of the elastic moduli are only moderately affected by the difference.
The sample volume in Table 1 is also a weighted average of data from the two detectors given by V = (2V1 + V3)/3, where V1 and V2 = V3 is the sample volumes measured by the axial and lateral detectors, respectively.
The sample length was determined from the unit cell volume data using the relation l = lo (V/Vo) 1/3 , where l and V are the length and weighted unit cell volume, respectively; subscript zero represents ambient values. We observed considerable scatter in the lengths determined from the X-radiographic method, which we ascribed to the relatively small temperature interval (100 K) between measurements. We found changes in successive specimen lengths to be close to the precision (0.2-0.4%) of the ranges determined from the pixels data. In Table 1, we tabulated the specimen pressures calculated using the equation of state of NaCl [50] with a precision of 0.8%. We compute the listed sample densities in Table 1 from the unit-cell parameters from the X-ray data using the relationship ρ = ρo(Vo/V), where ρ and V are density and weighted unit cell volume, respectively; subscript zero represents ambient values.
The travel time data were combined with the lengths to calculate the P and S velocities at high P and T. Uncertainty in the travel time, when combined with uncertainty in specimen length (0.1%), yields uncertainty in the velocity of about 1%. The P and S wave velocities are given in Table 1 and used to calculate the longitudinal L = ρVP = (Ks + 4G/3), shear (G = ρVS 2 ), and bulk (Ks = L − 4G/3) moduli, also listed in Table 1.
We have plotted the P and S wave velocities from this study as a function of pressure ( Figure  6a,b), and the elastic moduli Ks and G are also plotted as a function of pressure in Figure 7a,b, respectively. The acoustic velocities, as well as the moduli data, are observed to increase systematically and linearly over the entire pressure range for each isotherm. For consistency in extrapolating to the high pressures of the transition zone, we fit the shear (G) and compressional (Mp = K + 4G/3) moduli, where K is the bulk modulus to functions of Eulerian strain {ε (ε = [1 − (ρ/ρo) 2/3 ]/2} of the third order: The coefficients of the polynomials are related to the bulk (Ks) and shear (Go) moduli and their pressure derivatives at zero pressure as follows: where Ks, and Go, are the elastic moduli and Ks', and Go' and their pressure derivatives, respectively. Previous ultrasonic studies have shown that the data collected at the end of the cooling cycles at 300 K (cooled data) are generally under pseudo-hydrostatic conditions due to cell relaxation resulting from heating. By fitting the 300 K data from the cooling cycles to Equations (2) and (3), we obtain Ks = 161.1(4) GPa, Go = 101.6(2) GPa, Ks' = 4.89 (7), Go' = 1.69(4) and from linear regression of the cooled data, we obtain Ks = 161.6(9) GPa, Go = 101.7(2) GPa, Ks' = 5.03(1), Go' = 1.79(4), where numbers in parenthesis represent the fitting uncertainty. Following the previous procedure [47], we have treated each modulus as a linear function of pressure and temperature by fitting the P-T-KS and P-T-G data sets separately to the linear Equation of the form:

M (P,T) = Mo + (∂M/∂P)T(P-Po) + (∂M/∂T)P(T-To)
Where Mo is the ambient pressure and temperature Ks or Go, and (∂M/∂T)P and (∂M/∂P)T are the temperature and pressure derivatives, respectively, of the elastic modulus M. In fitting, we exclude all data acquired before the sample was heated after each pressure increase because the data are likely to have residual stress caused by the cold compression. Fitting the entire P-T-Ks and P-T-G data sets separately to Equation (3)

Ambient Elastic Bulk (Ks) and Shear (Go) Moduli
In Table 2, we present the adiabatic bulk (Ks) and shear (Go) moduli and their pressure derivatives (∂Ks/∂P)T, and (∂G/∂P)T, respectively, for the hydrous wadsleyite (β-Mg2SiO4) containing 0.73 (7) wt.% H2O and compare the data with those from previous studies of a hydrous β-Mg2SiO4, as well as Mg-and Fe-bearing anhydrous wadsleyite. Data for the anhydrous β-Mg2SiO4 polycrystalline sample measured in this study by the identical procedure are also listed in the table to validate the interferometry method in the DIA-type apparatus used for the measurements and to serve as an internally consistent data set for direct comparison with the hydrous wadsleyite data.
Our Ks and Go data from the four analytical techniques are consistent and in excellent agreement with the mutual uncertainty of the data. We have selected our finite strain results for comparison with previous studies. Our finite strain Ks = 161(5) GPa and Go = 101.6(2) GPa for the hydrated sample are 5.2% and 8.7%, respectively, lower than the corresponding values for anhydrous β-Mg2SiO4 (Table  2) from this study. We observe a comparable reduction in the Ks and Go by comparing the values for the hydrous β-Mg2SiO4 from this study with results from other previous studies for anhydrous wadsleyite [16,18,21,51]. An earlier investigation of the elasticity of a single crystal of wadsleyite (β-Mg2SiO4) containing 0.37-1.66 wt.% H2O by Brillouin spectroscopy [27] observed both Ks and Go to decrease linearly with increasing water content in wadsleyite. The linear relationships given by Equations (4) and (5) of the study [27], correlating the bulk and shear moduli respectively, with water content in wadsleyite predict Ks and Go of 161.4 GPa and 106.0 GPa, respectively, for hydrous β-Mg2SiO4 containing 0.73 wt.% H2O as measured in this study. Whereas our Ks = 161.4 GPa is identical with the predicted value, the Go = 101.4 GPa from this study is 4% smaller than predicted by Equation (5) of the Brillouin scattering study [27]. Our data show a stronger elasticity effect on G compared to Ks caused by the hydration in contrast to Brillouin scattering measurements on single-crystal wadsleyite [27,28] that report a comparable decrease in Ks and G of 7.6% and 7.0%, respectively, by dissolving one weight percent water in wadsleyite.
First principle calculations [52] report that one weight percent of water dissolved in wadsleyite reduces the Ks and G by ~5.6% and ~6%, respectively, which though also comparable, are out of the range of the Brillouin scattering data. The variability in the current data on the effect of wadsleyite hydration on the elasticity is mainly due to uncertainty in the water content measurements, generally around 10% [27,28]. Currently, standard water-content calibrations are available only for the olivine and ringwoodite phases, but not the wadsleyite phase. Resolving the magnitude and the relative contributions of the water effect on Ks and G requires the establishment of a conventional water content calibration for wadsleyite, as wells as systematic measurements of OH-bearing wadsleyite samples.

Pressure Derivatives of Elastic Moduli
As shown in Table 2, the pressure derivative of the adiabatic bulk modulus, Ks' = 4.84(2) for hydrous β-Mg2SiO4 with 0.73 wt.% H20 is identical to Ks' = 4.81(2) for our anhydrous β-Mg2SiO4 and also in agreement with previously reported Ks' = 4.56 (23) [19], within the mutual data uncertainties. The wadsleyite specimens used in this study have been hot-pressed by similar techniques and measured at high pressure using similar P and T paths and ultrasonic procedures.
The Ks' value for the hydrous wadsleyite is moderately higher than previous ultrasonic data for β-Mg2SiO4 (Ks' = 4.5(1) [16]). The Brillouin Scattering data (Ks' = 4.3(2)) for single-crystal β-Mg2SiO4 [17] measured to 15 GPa at room T. Our Ks' is also higher than the aggregate modulus Ks' = 4.5 from the first principle calculations [52] for β-(Mg0.875Fe0.125)2SiO4) wadsleyite. The Ks' = 4.1(1) for single-crystal wadsleyite containing 0.84 wt.% H2O measured to 12 GPa at room T by Brillouin spectroscopy [28], and Ks' = 4.13 (8) reported for Fe-bearing single-crystal wadsleyite from Brillouin scattering studies [29] are in excellent agreement, but both are lower than the other published data. Figure 8 is a plot of the elastic K and G moduli collected at the end of each cooling cycle (300 K) for the hydrous and anhydrous β-Mg2SiO4 as a function pressure. The line is the linear regression of the compression data. The overlap among the compression and decompression data yields a robust pressure dependence of the elastic K and G moduli for each composition. The similarity of the Ks' and Go' for the hydrous and the anhydrous β-Mg2SiO4 phases shown in Figure 8 suggests that the pressure derivatives of the elastic moduli are insensitive to the OH content of the wadsleyite, at least up to 0.73 wt.% of the study. For comparison, we also plotted the Brillouin scattering single-crystal measurements carried out to 12 GPa and room T for containing 0.84 wt.% H2O [28]. We note that whereas the Brillouin scattering shear modulus data further validates the insensitivity to the OH content in wadsleyite, the slope of the bulk modulus with pressure (Ks') is slightly lower than from the current study, as shown in Table 2.  Figure 8. Elastic bulk (K) and shear (G) moduli at 300 K for hydrous (triangle) and anhydrous (diamond) β-Mg2SiO4 as a function of pressure. Open and filled symbols are data acquired on compression and decompression, respectively. Lines are linear fit to the compression data. Stars are Brillouin scattering single-crystal data [27]. Indicated are 5.3% and 8.6% decreases in Ks and Go, respectively, due to hydration of wadsleyite by 0.73%.
The Go'= 1.55(1) from the current study of anhydrous β-Mg2SiO4 is in excellent agreement within the mutual data uncertainties with previous anhydrous studies Go' = 1.4(2) [17], Go' = 1.5(1) by [18] and in reasonable agreement with Go'= 1.4(1) for hydrous wadsleyite containing 0.84 wt.% H2O [28]. Our Go' is also in close agreement with Go' = 1.6 for β-(Mg0.875Fe0.125)2SiO4) from first-principle calculations [52]. However, as shown in Table 2, the Go'= 1.68(4) for the hydrous β-Mg2SiO4 containing 0.73 wt.% H2O from this study is marginally higher than the ultrasonic data for anhydrous wadsleyite [16,18,19] and the Brillouin scattering data for anhydrous wadsleyite [17] but in excellent agreement only with Go' = 1.64(4) from the single-crystal Brillouin scattering measurement of Fe/(Mg + Fe) = 0.112(2) wadsleyite containing 0.2492 wt.% H2O [29].   The investigators of the single-crystal wadsleyite elasticity containing 0.84 wt.% H2O at 12 GPa and room T [28] concluded that Ks' and Go' for the hydrous wadsleyite phase is indistinguishable from those of the anhydrous phase, citing the overlap of the Ks' and Go' from their study with the anhydrous data [17]. The Ks' values for the hydrous (0.73 wt.% H2O) and the anhydrous wadsleyite samples (Table 2; also see Figure 8) are identical, and there is also a close agreement between the Go' for the two compositions. We, therefore, conclude that the pressure derivatives of K and G are independent of the H2O content of wadsleyite (at least up to the 0.73 wt.% of this study). The agreement of the Ks' and Go' from this study with Mg and Mg-Fe wadsleyite data from previous studies further strengthens our assumption.

Temperature Derivatives of Elastic Moduli
We have tabulated values for (∂Ks/∂T)P and (∂G/∂T)P for the hydrous wadsleyite with 0.73 (7) wt.% water and the anhydrous β-Mg2SiO4 from this study in Table 2 and compared the results with data from previous studies on the anhydrous Mg and Mg-Fe wadsleyite. As observed in the table, the (∂Ks/∂T)P and (∂G/∂T)P results for the hydrous β-Mg2SiO4 obtained from fitting the entire data set linearly (see Equation (3)) and the third-order finite strain method to all the data is consistent and in good agreement within the mutual uncertainties of the analysis techniques. For consistency, we select the finite strain results for extrapolation to mantle conditions and comparison with previous data.
As observed in Table 2, our (∂Ks/∂T)P and (∂G/∂T)P data sets from the linear fitting and thirdorder finite strain analysis are consistent, and each shows |(∂G/∂T)P| to be moderately higher than |(∂KS/∂T)P|. We are currently analyzing elasticity data for wadsleyite containing 0.26 wt.%, 0.53 wt.%, and 1.00 wt.% H2O, respectively, and the preliminary results seem to corroborate the current observation of the |(∂G/∂T)P| being higher than |(∂Ks/∂T)P| for the 0.26 wt.% and 0.53 wt.% wadsleyite samples. Only the (∂Ks/∂T)P and (∂G/∂T)P for the 1.00 wt.% H2O wadsleyite shows |(∂Ks/∂T)P| > |(∂G/∂T)P|, as also observed for the anhydrous β-Mg2SiO4 in this study, as well as many previous studies of Mg-and Mg-Fe wadsleyite [21][22][23][24]. The (∂Ks/∂T)P and (∂G/∂T)P of −0.164(5) GPa/K and −0.130(3) GPa/K, respectively, from the ultrasonic measurements of San Carlos olivine to 8 GPa and 1073 K [19] exhibit similar patterns to those of most mantle phases where |(∂Ks/∂T)P| is greater than |(∂G/∂T)P|. Apart from our current data for the hydrous wadsleyite, only two previous ultrasonic high P and T studies on anhydrous β-Mg2SiO4 [18,19] and Mg(0.87)Fe(0.13) wadsleyite [20] also observe |(∂G/∂T)P| to be greater |(∂KS/∂T)P|. It is noteworthy that the anhydrous β-Mg2SiO4 data [18,19] are not two separate studies. Whereas the earlier study [18] used the linear function to fit the data, the latter [19] reanalyzed the data using the finite strain method. A previous study [27] has remarked on the difficulty of synthesizing a nominally anhydrous wadsleyite without a hydroxyl; despite the care taken, the sample still contained at least 50 ppm wt.% H20. The temperature derivative results for the β-Mg2SiO4 [18,19] and the Mg(0.87)Fe(0.13) wadsleyite [20] could reflect minor H2O in the samples if the starting materials were completely dry before the hot-pressing. A detailed examination of the β-Mg2SiO4 and [18,19] and Mg(0.87)Fe(0.13) wadsleyite [20] samples would have been useful in explaining the (∂Ks/∂T)P and (∂G/∂T)P observations of the studies. Nonetheless, based on the current (∂Ks/∂T)P and (∂G/∂T)P data for the hydrous wadsleyite (0.73 wt.% H2O) and the preliminary data for wadsleyite containing 0.26 wt.% and 0.53 wt.%, we infer that the observation showing the bulk modulus ((∂Ks/∂T)P) being affected more than shear modulus ((∂G/∂T)P) by temperature could be an intrinsic structural property of the wadsleyite hydration for relatively small OH content up to 0.73 wt.% H2O and if so, could have important implications for evaluating the orthosilicate content of the Earth's mantle.
We assumed orthorhombic symmetry in the X-ray analysis of all the wadsleyite specimens of this study. However, some previous studies [53,54] observed a monoclinic distortion of hydrous wadsleyite. The monoclinic distortion is associated with cation vacancy stacking disorder of distinct modules [53] or poly-synthetic twinning [54]. Another study [7] has found only samples with water content greater than 0.5 wt.% to display the monoclinic symmetry. Further studies of the hydrous wadsleyite symmetry and cause would enhance understanding of the temperature derivatives elastic moduli phenomena of wadsleyite observed in this study.
Examination of all current T-derivatives data shows that the effect of Fe substitution or OH incorporation in wadsleyite on (∂G/∂T)P is immeasurable. Previous resonance ultrasound spectroscopy measurements of anhydrous β-Mg2SiO4 [21] concluded that there is no measurable difference in (∂KS/∂T)P and (∂G/∂T)P due to Fe substitution in the anhydrous wadsleyite up to XFe = 0.09 from observation of agreement of reported (∂KS/∂T)P and (∂G/∂T)P for Mg and Mg-Fe wadsleyite [18][19][20][21]. By including our hydrous wadsleyite data, we find the conclusion for (∂G/∂T)P to be strengthened, but less robust for the current (∂KS/∂T)P measurements.
The worldwide 410-km seismic velocity discontinuities in the Earth's upper mantle are generally attributed to the phase transition of the olivine (α) to the wadsleyite (β) phase. Previous studies have estimated the olivine volume of the mantle by comparing the seismic P and S wave velocity contrasts (∆Vp; ∆Vs) across the 410-km discontinuity caused by the phase change, with the corresponding laboratory velocity contrast between the olivine (α) and wadsleyite (β) phases [15,[18][19][20]55].
We have applied our new elasticity data for the hydrous wadsleyite (0.73 wt.% H2O) and adapted the analytical approach of previous studies [18][19][20], to determine the P and S wave velocity contrasts between the α and β phases at the 410-km P and T conditions. We use the third-order finite strain approach of previous studies [18][19][20] to calculate the P and S wave velocities of the hydrous wadsleyite from this study, the anhydrous α-Mg(0.9Fe0.1)2SiO4 [19] and other mantle phases at the 410km discontinuity. We applied the iron-partitioning data between the α-and the β-phases [56] and performed the calculations to 14 GPa along 1673 K-foot adiabat. We calculated the elasticity of the pyrolite lithology using the Voight-Reuss-Hill (VRH) averaging scheme. In addition to the anhydrous α-Mg(0.9Fe0.1)2SiO4, we also calculate the P and S wave velocities at 14 GPa using the Brillouin scattering elasticity data for hydrous olivine containing 0.8-0.9 wt.% H2O measured at room P and T [57] and room T and 14 GPa [58]. For the calculations, we follow the same approach to correct for the effect of Fe on the elastic bulk (K) and shear (G) moduli and densities of the α and β phases, as reported in the previous ultrasonic analysis [18][19][20] used to evaluate the olivine content of the mantle.
The range of the velocity contracts across the 410-km discontinuity from body wave studies is 3.8% to 4.9% for P waves and 4% to 5% for S waves [63]. We obtain a higher olivine volume content (50%) for the P wave that is close to the S-wave value (52%) by adopting the seismic velocity contrasts (∆VP = 4.9%; ∆VS = 4.6%) used by Liu and co-workers [20]. A comparison of the derived olivine volume content from the anhydrous α-Mg(0.9Fe0.1)2SiO4 [19] and the hydrous olivine [57,58] underscores the significance of our new data on the T-derivatives of the elastic moduli for the hydrous wadsleyite, as well as the importance of extending the measurements to the hydrous olivine and hydrous ringwoodite phases of (Mg, Fe)2SiO4. Figure 9 is an adaptation of Figure 3 from the previous ultrasonic measurements of the elastic moduli of β-(Mg)2SiO4 carried in a liquid pressure piston-cylinder apparatus to 3 GPa at room temperature [15]. The figure displays the trade-off of |(∂G/∂T)P| and |(∂KS/∂T)P| for the wadsleyite phase compatible with the orthosilicate content of an iso-chemical mantle model, calculated along a 1575 K-foot adiabat; the pressure dependence of the elastic moduli K's = 4.8 (2) and Go' = 1.7(1) used in the calculations are identical, within the mutual uncertainties of the data with values for the hydrous β-(Mg)2SiO4, with 0.73 wt.% H2O and the anhydrous β-(Mg)2SiO4 measured in this study. The study assumes that the seismic velocity contrasts ∆Vp = ∆Vs = 4.6% at 410-km in the Earth's mantle, as reported for regional seismic Earth model studies [59][60][61][62]. The calculations predicted that the values of (∂KS/∂T)P and (∂G/∂T)P of the orthosilicate content for an iso-chemical model mantle would be those with |(∂G/∂T)P| being greater than |(∂KS/∂T)P|. We note that the predicted orthosilicate content is much less sensitive to temperature change than to uncertainties in the seismic velocity contrasts, as shown in Figure 9.  Figure 3 [15]. The heavy solid line represents calculations at a potential temperature of 1300 °C with a seismic velocity contrast for both P and S waves of 4.6% at the 410 km discontinuity. Vertical bars show the effect of changing the temperature by ±100 °C. The outlying solid lines represent the effect of a ±0.5% change in ∆VP or ∆VS, as indicated in the insert. The lines labeled orthosilicate content (percent) correspond to (δ∆VP) = ±0.5% at constant ∆VS (lines with short dashes with slope ~ 0) and to (δ∆VS) = ±0.5% at constant ∆VP (lines with long dashes with slope ~−3/4). Also shown are measured values of these parameters; star, from this study, for hydrous β-(Mg)2SiO4, (0.73 wt.% H2O); 1. the linear fit results to data (Equation (3) In Figure 9, we plot the data for |(∂G/∂T)P| and |(∂KS/∂T)P| obtained from the finite strain analysis (1) and the linear fit to the entire data set (2) from this study (star). For comparison, we also plot our new data for the anhydrous β-(Mg)2SiO4, (3) as well as high T data for anhydrous Mg-and Mg-Fe-bearing wadsleyite from previous studies [18][19][20][21][22][23][24]. The triangle and the diamonds are the Mg end-member and the Fe-bearing wadsleyite data, respectively. Our two new data sets predict about 45-50% orthosilicate content for the mantle, in excellent agreement with 46-52% from comparing the seismic velocity contrasts (∆Vp = ∆Vs = 4.6%) with the corresponding contrasts between the α and the β phases (∆Vp(α-β) = 9.88%; ∆VS(α-β) = 8.70%) for hydrated α-(Mg)2SiO4 and hydrated β-(Mg)2SiO4 at the 410-km depth obtained from the finite strain analysis. All current high-temperature data on anhydrous Mg and Mg-Fe wadsleyite, including our new β-(Mg)2SiO4, data (3) lead to velocity contrasts with ratio ∆Vp/∆Vs significantly different from unity for which there is no compelling seismological evidence [15].
The two ultrasonic data for anhydrous β-(Mg)2SiO4 [18,19], and anhydrous Mg(0.87)Fe(0.13) wadsleyite [20] also fall within the predicted range of preferred orthosilicate content of the Earth's mantle. However, as noted above, given that there is no information on the sample characterization used in the study, and given that our anhydrous β-(Mg)2SiO4 data from this study also lie out of the range, we presume that the wadsleyite sample used in the study might have contained some structural water. Figure 9 shows that the inferred |(∂G/∂T)P| and |(∂KS/∂T)P| values for the wadsleyite phase consistent with the isochemical model mantle are those for which |(∂G/∂T)P| is higher than |(∂KS/∂T)P|. It also indicates that as the temperature derivatives increase along the curve, the inferred olivine volume content of the model mantle also increases from ~40% to 70%. The inferred olivine content is only minimally affected by changing the temperature at the zero-pressure of the appropriate adiabat but strongly influenced by perturbating the seismic relative velocity contrast (∆VP/VP ≠ ∆VS/VS). For a given K and G of the olivine phase, the decrease in K and G of the wadsleyite caused by hydration, as observed in this study, leads to increases in the inferred olivine content of the mantle, while temperature derivatives decrease the content. As noted above, preliminary data for wadsleyite containing 0.26 wt.% and 0.53 wt.% H2O, show |(∂G/∂T)P| to be higher than |(∂Ks/∂T)P|. Additionally, we also observe that the (∂G/∂T)P and (∂KS/∂T)P values move towards higher olivine content with decreasing water OH content in the wadsleyite for constant pressure derivatives of the elastic moduli. Completion of the data analysis on the current OH-wadsleyite samples and any necessary additional elasticity measurements of hydrous wadsleyite could reveal the trend of the mantle olivine content variation with the elastic properties of wadsleyite.

Conclusions
We measured the compressional (P) and shear (S) wave velocities for a synthetic polycrystalline wadsleyite containing 0.73 wt.% H2O to 10 GPa and 600 K by ultrasonic interferometry techniques combined with synchrotron X-radiation. The adiabatic bulk (Ks) and shear (G) moduli and their pressure and temperature derivatives are determined using the third-order finite strain Equation Comparison of the body wave seismic velocity contrasts (∆Vs = ∆VP = 4.6%) with the corresponding contrasts between the hydrous β-Mg2SiO4 phase from this study and the hydrous α-Mg2SiO4 (∆Vp(α-β) = 9.88% and ∆VS(α-β) = 8.70%) calculated for the 410-km depth by third-order finite strain technique yields 46-52% volume content in the Earth's mantle. However, in comparison, the contrast for the hydrous β-Mg2SiO4 and anhydrous α-Mg(0.9Fe0.1)2SiO4 [19] yields ∆Vp(α-β) = 13.0% and ∆VS(α-β) = 14.70%, corresponding to a significantly lower olivine volume content (39.4-36%) for the mantle. The combined effect of OH and Fe on the high P and T elasticity of the wadsleyite and olivine phases need to be understood to improve the assessment of the olivine volume content of the Earth's mantle.