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					 Gigahertz Ultrasonic Interferometry at High P-T: New Tools for a Thermodynamic
                              Equation of State (EoS)

  S. D. Jacobsen1, H. A. Spetzler1, H. -J. Reichmann2, J. R. Smyth1, S. J. Mackwell3, R. J. Angel4, and W. A. Bassett5
          1
            CIRES and Department of Geological Sciences, University of Colorado, Boulder, CO 80309 U.S.A.
                                  2
                                    GeoForschungsZentrum, D-14474 Potsdam, Germany
                       3
                         Bayerisches Geoinstitut, Universität Bayreuth D-95440 Bayreuth, Germany
                    4
                      Department of Geological Sciences, Virginia Tech, Blacksburg, VA 24060 U.S.A.
                  5
                   Department of Geological Sciences, Cornell University, Ithaca N. Y. 14853 U.S.A.


       Abstract. A new method of generating shear waves with near-optical wavelength has been developed for
       GHz-ultrasonic interferometry (GUI). The new acoustic technique features a P-to-S conversion upon
       reflection inside an MgO buffer rod, and is first used to determine the full set of ambient P-T elastic
       constants (cij) for magnesiowüstite-(Mg,Fe)O. In other developments, P-wave travel times have been made
       in olivine to 250 ºC at ~2.5 GPa in a resistance-heated ultrasonic diamond anvil cell (DAC), demonstrating
       that acoustic coupling can be maintained at high temperature in a hydrostatic (alcohol) pressure medium.
       The new GUI tools bring us closer to obtaining a complete travel-time equation of state.
       Key words: Ultrasonics, Elasticity, Equations of State, Diamond-Anvil Cell, Magnesiowüstite.


1. Introduction                                                measurements are carried out on (Mg0.423Fe0.541 0.036)O
      Knowledge of the elastic properties and crystal          to 3.6 GPa, resulting in determination of ∂c11/∂P. Also,
chemistry of minerals plays an important role in               P-wave travel-times have been measured in San Carlos
interpreting the composition and mineralogy of Earth's         olivine to 2.6 GPa, and 250 ºC without a bonding agent
interior from seismological observation. GHz-ultrasonic        between the sample and the anvil. Although the pressure
interferometry (GUI) is a relatively new tool among the        could not be measured at high temperature, this
arsenal of acoustic or optical methods that can be used to     experiment demonstrates that acoustic coupling between
measure elastic-wave travel times or velocities in solids      a sample and the diamond can be maintained at high
for determination of elastic moduli [1].                       temperature in a hydrostatic alcohol pressure medium.
      In recent years, GUI has successfully reduced the              We are in the final stages of adapting the new shear
sample-size requirements for single-crystal ultrasonics to     wave technology to a sturdy Al2O3 buffer rod shaped for
about 50 microns, increasing by manifold the variety of        use in the DAC. If both P- and S-wave travel times can
natural and synthetic phases that can be studied with this     be measured at high pressures and temperatures, the
method. In turn, small sample sizes allow mating of the        complete travel time equation of state (CT-EoS)
GHz technology to a diamond-anvil cell (DAC) [2]               described by Spetzler et al. [6, 7, 8] may be obtained. If
where P-wave measurements have been carried out on             the CT-EoS is measured in conjunction with an
MgO to a maximum hydrostatic pressure of 6 GPa [3]             independent EoS (such as the Birch-Murnaghan P-V
and to 207 ºC at 4.5 GPa in a solid KBr pressure medium        EoS), the variable pressure is determined by two
[4]. However, GUI has been essentially without shear           independent volume (length) measurements, resulting in
waves because very thin (~30µm) modified piezoelectric         an absolute pressure scale. The new GUI tools are aimed
shear transducers produced only relatively low quality         at this larger goal.
shear over a very narrow frequency band from about
590–605 MHz [5]. Shear is required for determination of        2. Experimental
the complete set of elastic constants (cij) and isotropic            Seismologists have long known about the
bulk (KS) and shear (G) moduli of minerals.                    conversion of elastic wave particle motion in solids
      Here we present a new method of generating shear         between compressional (P) and transverse (S) upon
waves with near-optical wavelength. The new acoustic           reflection described by the acoustic analogy of Snell’s
technique features a P-to-S conversion upon reflection         law. We borrowed this idea to generate GHz-frequency
inside a single-crystal MgO buffer rod, and is used first      shear waves by the P to S conversion on the internal facet
to determine the complete set of elastic constants for         of an oriented MgO prism. The input P-wave is generated
(Mg,Fe)O with compositions spanning the solid solution.        at a sputtered ZnO thin-film transducer. The incident P-
(Mg,Fe)O is a dense monoxide of the nominally B1 or            wave strikes the conversion facet of the MgO prism at an
rocksalt structure that is expected to coexist with silicate   angle of incidence such that the incident P-wave is
perovksite-(Mg,Fe,Al)SiO3 in Earth's lower mantle (660–        orthogonal to the reflected S-wave. This geometry takes
2900 km depth).           In the DAC, [100] P-wave             advantage of the orthogonal pure-mode [100] directions
in cubic MgO, as well as forcing the symmetric return of                                                     between periclase (MgO) and pre-reacted (Mg,Fe)O
the elastic wave (PSSP) for detection at the source                                                          powders. The periclase single crystals were packed
transducer. For MgO, the angle of incidence is 54º. The                                                      tightly in the (Mg,Fe)O powders inside an alumina
new bench-top shear wave generator is illustrated in                                                         crucible. Interdiffusion was carried out in a gas furnace
Figure 1a. Interferometry can be performed by bonding a                                                      operating at 1450ºC and 10-2 Pa oxygen fugacity (fO2) for
sample to the prism face which back-reflects the S-wave.                                                     approximately 200 hours. The furnace was cooled at a
A set of two tone bursts are introduced into the prism,                                                      rate of about 300 ºC/h. The original crystal plates were
spaced in time such that the second PSSP echo overlaps                                                       excavated from the sintered powders and re-polished on
with the first sample echo. The frequency is stepped, and                                                    the [100] faces. Microprobe analysis reveals that the Fe
an interference pattern is produced (Figure 1b) from                                                         distribution in the single crystals is homogeneous on the
which the travel-time as a function of frequency can be                                                      level of sampling, about every 10–20 µm. The first suite
determined (Figure 1c). In this way, we have produced                                                        of (Mg,Fe)O single-crystals have ΣFe/(ΣFe+Mg) ratios
useable transverse wave energy observed continuously                                                         of 0.058, 0.149, 0.270, 0.366, 0.561, and 0.749. A
from 0.6 to 2.1 GHz. This range is limited at high-                                                          second suite of samples having ΣFe/(ΣFe+Mg) ratios of
frequency only by the upper-limit of the RF generator.                                                       0.239, 0.527, and 0.783 were prepared in the same way.
                                                                                                             However, these samples were subsequently annealed at
                                       incident P-wave                                                       1300ºC and 10-5 Pa fO2 for 20 hours and quenched at
                                       from transducer                     normal                   (a)      about 30 ºC/s.
                                                     P                               sample echo
                                                                                                                   Compressional and shear-wave travel-times were
                                                                                                             measured in the [100] and [111] directions for all the
                                      co




                                                    θip            S-wave              PSSP echo
                                         nv




                                                            θrs
                                                                                                             (Mg,Fe)O samples. The thickness of each sample was
                                            ersi
                                                 on




                   MgO
                                                                        quasi-P                              determined using a digital micrometer with an accuracy
                                                    fac




                                                                        reflection
                                                                                                             of ± 1 µm. The orientation of the samples was checked
                                                       et




                   [100]
                                                                    single-crystal MgO buffer rod            to be within ± 1º by the X-ray precession method.
                                                                                                    sample
                                                                                                             Combined with sample densities calculated from the
                                                                                                             measured cell parameters and compositions, the
                                                                                                   (b)       velocities were used to calculate the variation of the
      Normalized Amplitude




                                        2                                                                    elastic constants with composition for (Mg,Fe)O, also
                                                                                                             plotted in Figure 2:
                                        1
                                                                                                                 c11 = 287(2) – 85(9)x + 16(7)x2 GPa               (1)
                                        0
                                                                                                                 c12 = 95(3) + 99(23)x – 123(51)x2 + 51(31)x3 GPa (2)
                                                                                                                 c44 = 156(2) – 154(7)x + 44(5)x2 GPa              (3)
                                        -1


                                        -2
                                                                                                                                          300                                   (Mg,Fe)O
                                                                                                                 elastic constant (GPa)




                                      141.0                                                        (c)                                    250      c11
            S-wave travel-time (ns)




                                                                         m(1) = 117

                                      140.5
                                                                                                                                          200
                                      140.0
                                                    average time = 139.64(2) ns                                                                    c44
                                      139.5          m(1) = 116                                                                           150

                                      139.0
                                                                         m(1) = 115                                                       100
                                      138.5                                                                                                        c12
                                                                                                                                          50
                                              0.8                 0.9           1.0         1.1      1.2
                                                                   Frequency (GHz)                                                               0.0     0.2      0.4    0.6      0.8     1.0
                                                                                                                                                MgO                                     Fe0.95O
                                                                                                                                                               Σ Fe/(Σ Fe+Mg)
   Fig 1. (a) The P-to-S shear wave prism. (b)
   Interferometry data with a 374 µm-thick sample of
   (Mg,Fe)O. (c) Resulting travel-time data showing the                                                         Fig 2. Variation of elastic constants with
   consequence of choosing the wrong value of m, the                                                            composition for (Mg,Fe)O. The more stoichiometric
   number of wavelength in the round trip.                                                                      suite is represented by open symbols.

      The shear wave prism is first used to measure the                                                           The rate of change in the elastic constants with
elastic constants of (Mg,Fe)O. Two suits of (Mg,Fe)O                                                         composition is greatest between MgO and (Mg,Fe)O
single-crystals were prepared with varying degrees of                                                        with ~25 mol% FeO, such that adding Fe into periclase
non-stoichiometry by the interdiffusion of Fe and Mg                                                         has a greater effect on the elastic properties than adding
Mg to wüstite. The elastic anisotropy of (Mg,Fe)O has            ethanol pressure medium along with several rubies. In
rather unusual behavior, being essentially constant for the      this unusual case, a very good ultrasonic signal was
range 0–25 mol% FeO, but then decreases linearly with            obtained from the sample without any bonding agent,
Fe-content such that wüstite is elastically isotropic. The       such as glue. The open cell travel time of 22.72(2) ns was
elastic properties of (Mg,Fe)O having similar total-Fe,          obtained, giving an initial length of 94.8 µm, calculated
but varying Fe3+ contents are identical within uncertainty.      from the zero-pressure elastic constant c33 = 233.5 GPa
      A sample of (Mg,Fe)O with ΣFe/(ΣFe+Mg)= 0.56               obtained by Abramson et al. [11] for San Carlos olivine.
(13% ferric) was selected for high-pressure ultrasonics.         The pressure was raised, and [001] P-wave travel times
This sample was chosen in part because it is also the            were measured to a maximum pressure of 2.5 GPa,
subject of a single-crystal static compression study to 9        resulting in ∂c33/∂P = 5.6(2). The high-P San Carlos data
GPa so the measured K0T and K' [9] may be used in                are shown in Figure 4a.
calculation of the change in length with pressure. The
zero-pressure properties are: ρ0 = 4847(19) kg/m3; a0 =
4.2621(2) Å; K0T = 156(1) GPa; VP[100] = 7096(25) m/s,
                                                                                                 252
resulting in c11 = 244(2) GPa. The value of ∂KT/∂P =                                                         San Carlos olivine




                                                                   c33 elastic constant (GPa)
5.5(2) is also taken from the compression study. In the                                          248
open cell, a good P-wave ultrasonic signal was obtained
at zero pressure from 0.8 to 2.0 GHz, and the travel-time                                        244

tp = 14.60(2) ns was obtained from the interference data                                         240
From which the zero-pressure length is calculated: l0 =
51.80 ± 0.10 µm.                                                                                 236
      The cell was loaded with a 4:1 methanol:ethanol                                                                                                (a)
                                                                                                 232
pressure medium along with several rubies to act as a
pressure standard. The [100] P-wave travel time was                                              228
measured at nine different pressures up to 3.6 GPa. At                                                      0.0        0.5     1.0     1.5    2.0    2.5   3.0
each pressure, the sample length and density is obtained                                                                      Pressure (GPa)
from the P-V equation of state. The high-P travel-times,                                         22.7
sample length, and density are used to calculate the [100]
P-wave velocity and c11 elastic constant as a function of                                        22.6        San Carlos olivine
                                                                        [001] travel-time (ns)




pressure. For this composition, we obtain ∂VP[100]/∂P =                                                       P ~2.5 GPa
                                                                                                 22.5
85(5) (km s-1GPa-1) and ∂c11/∂P = 7.6(3). The variation
                                                                                                 22.4
of c11 with pressure for the iron-bearing phase is shown
in Figure 3 against MgO with ∂c11/∂P = 9.35(13)                                                  22.3
obtained in a similar experiment by Reichmann et al. [10]
                                                                                                 22.2


                                                                                                 22.1                                               (b)
            360
                                                                                                 22.0
                             MgO                                                                        0         50         100     150     200    250    300
            340        ∂c11/∂P = 9.35(13)                                                                                    Temperature (ºC)
            320
                                                                            Fig. 4. (a) Variation of c33 with pressure in [001]
c11 (GPa)




            300                (Mg0.423Fe0.541 0.036)O                      San Carlos olivine. (b) Variation of the P-wave
                                 ∂c11/∂P = 7.6(3)                           travel time with temperature at ~2.5 GPa.
            280
                                                                       At the highest pressure (~2.5 GPa), the temperature
            260
                                                                 was raised and P-wave travel times were measured at
            240
                                                                 100, 180, and 250 ºC, shown in Figure 4b. High
                                                                 temperatures are achieved by external heating of a Mo-
                  0     1      2       3       4         5   6   wire tightly wrapped around the carbide seats. The upper
                               Pressure (GPa)                    and lower anvils are heated independently by two
                                                                 temperature controllers reading the thermocouple in
            Fig. 3. Variation of the c11 elastic constant with   contact with each diamond. The temperature of one anvil
            pressure for MgO [10] and (Mg,Fe)O containing        was driven manually, while the other was controlled
            ~56 mol% FeO.                                        automatically to the set-point determined by the
                                                                 temperature of the other diamond. With this technique,
     In a similar DAC, a sample of [001] oriented San            the temperature of the anvils could be controlled together
Carlos olivine (Fo90) was loaded in a 4:1 methanol:              to within ~5 ºC. The cross-pressure cross-temperature
derivative of c33 is not reported here because the pressure     at temperatures up to 400 ºC. This range is readily
could not be measured at simultaneous high temperature,         obtainable in our existing ultrasonic/X-ray diamond cell
and, because the recorded temperature reflects the              and should be large enough to obtain meaningful P-T
temperature of the anvils rather than at the center of the      derivatives [6]. We are adapting the new shear wave
sample chamber. However, this pilot experiment                  generator to a more durable Al2O3 buffer rod shaped for
dutifully illustrates that acoustic coupling can be             use in the diamond cell, shown in Figure 5. Once both P-
maintained at high temperatures (≤ 250 ºC) in a liquid          and S-wave travel time measurements can be made at
pressure medium, motivating future single crystal               high P-T in the diamond anvil cell, the full potential of
ultrasonic measurements at simultaneous high P-T.               GHz-ultrasonic interferometry can be realized.

3. Application to Equations of State                                   DAC
      In this report we establish two important                                                    c-axis
                                                                                                                          conversion facet
                                                                                         single-crystal sapphire
developments in GUI. One, shear waves with near-
optical wavelength have been made with existing P-wave                                         S-wave
                                                                                                                                   P-wave
transducers by P-to-S conversion. Two, acoustic coupling
                                                                                                                   P-wave transducer
(with P-waves) has been maintained between a sample
and a diamond anvil at high temperatures (up to 250 ºC)
without a bonding agent in a hydrostatic (alcohol)
pressure medium. Possibly, the most powerful                       Figure 5. The new conversion buffer-rod will
application of obtaining high precision P- and S-wave              deliver shear-waves to the sample at high-pressure
travel times at high P-T will be determination of the              through the anvil.
complete travel-time equation of state (CT-EoS) [6].
      The CT-EoS is formulated as follows. First, P and         Acknowledgements
S travel times (tP and tS) must be made as a function of
pressure and temperature. In addition, knowledge of ρ0          References
and l0 (the initial density and sample length), along with      [1]H. A. Spetzler, G. Chen, S. Whitehead, I. C. Getting,
the thermal expansivity (α) and the heat capacity (CP) as          Pageoph., 141, 341 (1993).
a functions of temperature at zero-pressure are required.       [2]W. A. Bassett, H. -J. Reichmann, R. J. Angel, H.
First, tP and tS at P = 0 and any T are used to calculate the      Spetzler, J. R. Smyth, Am. Mineral., 85, 283 (2000).
VP and VS as a function of temperature (at P = 0) using l0      [3]H. -J. Reichmann, R. J. Angel, H. A. Spetzler, W. A.
and α(T). The velocities can then be used to calculate the         Bassett, Am. Mineral., 83, 1357 (1998).
adiabatic moduli KS and G at any T (and P = 0) using            [4]A. H. Shen, H. -J. Reichmann, G. Chen, R. J. Angel,
ρ(T) calculated from α(T). Since the heat capacity is              W. A. Bassett, H. Spetzler, Prop. Earth Planet. Mat.,
known as a function of temperature, the Grüneisen                  Am. Geophys.Un. Mon. 101, 71 (1998).
relation can be used to convert the adiabatic bulk              [5]G. Chen, R. Miletich, K. Mueller, H. A. Spetzler,
modulus to the isothermal one (KT), which, in turn can be          Phys. Earth Planet. Int., 99, 273 (1997).
used to calculate the density and sample length for a           [6]H. Spetzler, A. Yoneda, Pageoph., 141, 379 (1993).
small pressure step. At the first pressure, the same            [7]A. Yoneda, H. Spetzler, I. Getting, Proc. AIRAPT,
process is used to obtain the parameters as a function of          Am. Inst. Phys., 1609 (1994).
temperature. This requires knowledge of ∂CP/∂P and              [8]I. C. Getting, H. A. Spetzler, Proc. AIRAPT, Am. Inst.
∂α/∂P at constant temperature, which can be determined             Phys., 1581 (1994).
using the parameters already known [6]. In this way, the        [9]J. R. Smyth, S. D. Jacobsen, R. M. Hazen, Rev. Min.
CT-EoS delivers the thermodynamic variables; ρ, α, CP,             Geochem., 41, Min. Soc. Am., 157 (2000).
VP, VS, KS, KT, and G at any P-T within a reasonable            [10]H. -J. Reichmann, R. J. Angel, H. Spetzler, W. A.
range of the experiment.                                           Bassett, Am. Mineral., 83, 1357 (1998).
      If an independent EoS (such as the Birch                  [11]E. H. Abramson, J. M. Brown, L. J. Slutsky, and J.
Murnaghan P-V EoS) can be measured simultaneously,                 Zaug, J. Geophys. Res., 102, 12,253 (1997).
the two equations may be rewritten together allowing            [12]C. -S. Zha, H. -K. Mao, R. J. Hemley, Proc. Natl.
back-calculation or isolation of the pressure variable,            Acad. Sci., 97, 13,499 (2000).
thus obtaining an absolute pressure scale not dependent
on the properties any particular material. This idea was
recently explored using the alternative combination of
Brillouin Scattering and X-ray diffraction to 55 GPa (at
ambient temperature) by Zha et al [12].
      We are working towards focusing the new tools of
GUI towards obtaining the CT-EoS in the modest but
hydrostatic pressure range up to 10 GPa (in alcohol) and

				
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