Intercomparison of pressure standards (Au, Pt, Mo, MgO, NaCl …duffy.princeton.edu/sites/default/files/pdfs/2012/... · 2015-12-29 · where P is pressure, K 0 is the isothermal

Intercomparison of pressure standards (Au, Pt, Mo, MgO, NaCland Ne) to 2.5 Mbar

S. M. Dorfman,1,2 V. B. Prakapenka,3 Y. Meng,4 and T. S. Duffy1

Received 8 March 2012; revised 30 June 2012; accepted 12 July 2012; published 30 August 2012.

[1] We have performed a series of co-compression experiments on Au, Pt, Mo, MgO andNaCl to extend internally consistent pressure calibration and characterize shear strength ofpressure media to 2.5 Mbar. Measured unit cell volumes of calibrants show differencesbetween existing pressure scales of �10% above 2 Mbar. A new comprehensive pressurescale is proposed in good agreement with recent reduced shock isotherms for Au, Mo andPt and with internal agreement to 3% at Mbar pressures. Deviatoric stresses wereanalyzed for each material based on diffraction line broadening and line shifts due toanisotropic strain. The measured strength of Ne at 2.5 Mbar is 8 GPa. Diffraction linewidth analysis suggests that deviatoric stress conditions at 2.5 Mbar are similar for He andNe media.

Citation: Dorfman, S. M., V. B. Prakapenka, Y. Meng, and T. S. Duffy (2012), Intercomparison of pressure standards (Au, Pt,Mo, MgO, NaCl and Ne) to 2.5 Mbar, J. Geophys. Res., 117, B08210, doi:10.1029/2012JB009292.

1. Introduction

[2] Uncertainty in experimental stress conditions is animportant source of error in measurements of phase equilibriaand physical properties at deep Earth pressures. This error isevident in the range of reported pressures for key mantlephase transitions such as the perovskite–post-perovskiteboundary in MgSiO3 [Murakami et al., 2004; Hirose et al.,2006]. Accurate pressures are also necessary for measure-ments of the equation of state and compressibility of mantlephases. Differences of up to 15% have been observed amongequation of state measurements for materials such as MgSiO3

post-perovskite [e.g., Ono et al. 2006a] and Fe [Dewaeleet al., 2006; Sata et al., 2010] largely due to pressure scaledifferences at Mbar conditions.[3] An absolute pressure scale can be determined from

independent measurements of density/volume together withpressure or compressibility. Commonly used pressure scalesare often derived from shock compression experiments inwhich measurements of the shock and particle velocitiesconstrain pressure, volume and specific internal energythrough the application of Hugoniot equations [Nellis, 1997].Static pressure scales based on shock compression data

include those for gold [Jamieson et al., 1982; Yokoo et al.,2008], platinum [Jamieson et al., 1982; Holmes et al., 1989;Yokoo et al., 2008], magnesium oxide (MgO) [Jamiesonet al., 1982] and molybdenum [Hixson and Fritz, 1992].However, the correction for thermal effects on the shockHugoniot is uncertain, and the size of the necessary thermalcorrection grows as shock temperatures increase [Nellis, 2007].Alternatively, an absolute pressure scale can be derived froma combination of adiabatic elastic properties from Brillouinscattering or ultrasonic interferometry and volume fromX-ray diffraction [Zha et al., 2000]. Such measurements havebeen restricted to low pressures and require uncertain extra-polations to be applied at Mbar pressures. These primarymethods for pressure calibration are the basis for pressuremeasurements in diamond anvil cell (DAC) experiments.[4] An ideal internal pressure standard is inert and com-

pressible, with a simple crystal structure, strong X-ray dif-fraction pattern and no phase transitions over the experimentalpressure range. Commonly used standard materials includegold and platinum especially due to their additional roles aslaser absorbers in high-temperature experiments. Pressurecalibrants also used as insulating/pressure-transmitting mediainclude magnesium oxide, sodium chloride and neon. Anotherconsideration in calibrant choice is overlap of diffraction peaksof the standard with the sample. Au, Pt and MgO all havesimple face-centered cubic (FCC) structures, and their dif-fraction peak positions are similar at Mbar conditions.Molybdenum, as a body-centered cubic (BCC) metal, offerspotential to be a useful alternative to FCC standards.[5] Recent experiments have assessed the consistency of

primary equations of state for X-ray diffraction standardsby simultaneous X-ray diffraction measurements [Dewaeleet al., 2004; Dewaele and Loubeyre, 2007; Fei et al., 2007;Dewaele et al., 2008b; Hirose et al., 2008; Sata et al., 2010].Shock-based calibrations for metals have been observed to

1Department of Geosciences, Princeton University, Princeton, NewJersey, USA.

2Now at Earth and Planetary Science Laboratory, École PolytechniqueFédérale de Lausanne, Lausanne, Switzerland.

3Consortium for Advanced Radiation Sources, University of Chicago,Argonne, Illinois, USA.

4High-Pressure Collaborative Access Team, Carnegie Institution ofWashington, Argonne, Illinois, USA.

Corresponding author: S. M. Dorfman, Earth and Planetary ScienceLaboratory, École Polytechnique Fédérale de Lausanne, Station 3, CH-1015Lausanne, Switzerland. ([email protected])

©2012. American Geophysical Union. All Rights Reserved.0148-0227/12/2012JB009292

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, B08210, doi:10.1029/2012JB009292, 2012

B08210 1 of 15

deviate at Mbar pressures by as much as 10% [Dewaeleet al., 2004, 2008b]. Furthermore, most co-compressionexperiments have been limited to maximum pressures of100–150 GPa. As static experiments have now pushedmeasurements of both calibrants [Sata et al., 2010; Takemuraet al., 2010; Sakai et al., 2011] and planetary materials [e.g.,Tateno et al., 2010] to the multi-Mbar regime, additionalcross-calibration experiments are needed to test the reliabilityof these scales to higher pressures.[6] Theoretical calculations can provide complementary

constraints on material behavior at high pressures and tem-peratures [e.g.,Wentzcovitch et al., 2010]. Density functionaltheory (DFT) calculations of density and elastic propertieshave inherent approximations but are not limited by experi-mentally feasible conditions and do not suffer from uncer-tainties due to pressure calibration. These methods provideimportant tests of experiments and have been used to deriveindependent pressure standards [e.g., Tsuchiya, 2003].Hybrid equations of state for pressure calibration have beenproduced based on a combination of theoretical constraintsand experimental data [Dorogokupets and Oganov, 2006;Dorogokupets and Dewaele, 2007; Sun et al., 2008; Wuet al., 2008; Matsui et al., 2009; Matsui, 2009; Jin et al.,2010, 2011].[7] At high compressions, differences between the func-

tional forms used to parameterize experimental and theoret-ical equations of state also become important. The third-orderBirch-Murnaghan equation [Birch, 1947] derived from aseries expansion of free energy is the following:

P ¼ 3

2K0

V0

V

� �7=3

� V0

V

� �5=3" #

1þ 3

4½K′0 � 4� V0

V

� �2=3

� 1

" # !;

ð1Þ

where P is pressure, K0 is the isothermal bulk modulus atambient pressure, V0 is the zero-pressure volume, and K ′0 isthe pressure-derivative of the bulk modulus at ambientpressure. The equation of state originally derived by Staceyet al. [1981] based on the Rydberg interatomic potential[Rydberg, 1932] and popularized by Vinet et al. [1986] hasan exponential form:

P ¼ 3K01� fnf 2n

exp½hð1� fnÞ�; ð2Þ

where h is 3/2(K′0 � 1) and fV is (V/V0)1/3. As pressure

approaches infinity, volume approaches zero and so thisform has been argued to be more suitable for modelingsolids under extreme compression [Vinet et al.,1987; Hemleyet al., 1990]. At high pressures, the Birch-Murnaghan andVinet equations yield different sets of parameters V0, K0 andK ′0. Specifically, the Vinet equation yields a larger value forK ′0 for a given data set. This difference is important becausecomparing EOS parameters derived from fits of experimen-tal data to independent measurements by ultrasonic inter-ferometry and Brillouin spectroscopy is an important test ofreliability.[8] Additional uncertainty in stress conditions arises due

to the non-hydrostatic stresses produced in the DAC. Tominimize deviatoric stresses, DAC experiments employpressure-transmitting media. Solid noble gases Ar, Ne andHe are widely used as pressure media due to their chemicalinertness and low strength. Of these, He solidifies at thehighest pressure (11.5 GPa [Besson and Pinceaux, 1979])and is known to provide the most hydrostatic stress condi-tions up to at least 40–50 GPa [Mao et al., 1988; Takemuraand Dewaele, 2008; Klotz et al., 2009]. Only limited mea-surements of the strengths of Ne and He media have beenreported to higher pressures [Takemura, 2001; Takemuraet al., 2010].[9] In this work we compress combinations of Au, NaCl,

MgO, Mo and Pt in the diamond anvil cell in solid noble gaspressure media up to 2.5 Mbar. We use synchrotron dif-fraction to determine the unit cell volumes and estimate themagnitude of deviatoric stresses. A new self-consistentpressure scale is proposed for these materials based on therecently proposed MgO scale of Tange et al. [2009]. Thisscale is then compared to previous shock measurements andelasticity data.

2. Experiments

[10] High-pressure experiments were conducted usingdiamond anvil cells. Sample materials were commercialpowders of Pt, Au, NaCl, MgO and Mo with grain sizes of0.5–5 mm and purities greater than 99.9%. Samples wereground together and loaded into the DAC as compactedpowders or, in one case, a loose powder. Samples weretypically less than 5 mm thick and half the diameter of thesample chamber. Beveled anvils with inner culet diametersof 50–100 mm were used. Sample chambers were obtainedby drilling a hole of diameter 1/3–1/2 of the culet diameter ina �25-mm thick Re gasket. Ne or He gas was loaded as apressure-transmitting medium using the COMPRES/GSE-CARS high-pressure gas loading system [Rivers et al., 2008].A small ruby chip was used for pressure calibration duringgas loading. One sample of NaCl was loaded together with anAu foil and no medium present. Loading configurations foreach run are listed in Table 1.[11] Samples were compressed at room temperature. Syn-

chrotron X-ray diffraction was carried out at beamlines 16-ID-B (HPCAT) and 13-ID-D (GSECARS) of the AdvancedPhoton Source. The X-ray beam was focused with Kirkpa-trick-Baez mirrors to �3–6 mm. The tails of the beam wereblocked by an 8–20 mm pinhole, minimizing diffractionpeaks from the Re gasket. 2-D diffraction images wererecorded on a MarCCD detector. Images were calibrated

Table 1. Loading Configurations for Experimental Runs

Run SampleaPressureMedium

Culet(mm)

PressureRange(GPa) l (Å)

ANN11 1:8 Au:NaCl Ne 50 34–253 0.393545AN012 Au, NaCl none 75 1–164 0.415369MNN11 1:1 MgO:NaCl Ne 50 42–265 0.398585MNN13 1:1 MgO:NaCl Ne 50 99–255 0.398585MMH12 2:5 Mo:MgO He 50 43–92 0.3344MMH6* 2:5 Mo:MgO He 50 60–213 0.3344PMN3 2:5 Pt:MgO Ne 75 13–177 0.3344PMH6 1:8 Pt:MgO He 50 54–228 0.3344PNH7 1:8 Pt:NaCl He 75 42–192 0.3344PMNN7 1:6:6 Pt:NaCl:MgO Ne 100 5–98 0.3344

aAll samples were compacted powders except MMH6, which was loadedas loose powder. Sample ratios are by weight.

DORFMAN ET AL.: INTERCOMPARISON OF PRESSURE STANDARDS B08210B08210

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using a CeO2 standard and integrated with Fit2D software[Hammersley et al., 1996]. Diffraction peaks were fit toVoigt line shapes.

3. Results

[12] Representative diffraction patterns and the variation ofmeasured d-spacings with pressure are shown in Figures 1–4.The combinations of materials examined in our experiments(Table 1) provide redundancy and cross-checks. For mostruns, FCC (Au, MgO and Pt) and BCC (Mo and NaCl B2)materials were paired to minimize diffraction peak overlaps.Unit cell volumes (Tables S1–S6 of the auxiliary material)were determined by averaging lattice parameters from 3–5 diffraction lines for most materials.1 Exceptions were Ne,for which only (111) and (200) were observed, and MgO, forwhich (111) is much weaker than (200) and (220). Weak orpoorly resolved diffraction peaks were not used to determinepressures or deviatoric stresses.

3.1. Equations of State

[13] Equation of state fits to the volume data were madeusing both the third-order Birch-Murnaghan equation andthe Vinet equation. All data sets for Au, MgO, Mo, NaCl B2,Ne and Pt were simultaneously fit via non-linear leastsquares to determine equation of state parameters that min-imize the pressure differences among all the different mate-rials relative to the fixed MgO reference scale:

c2 ¼Xni¼0

DPið Þ2Pi

; ð3Þ

where for all n pairs of volume data, Pi is the pressure deter-mined by each calibrant, DPi is the difference between thesepressures and Pi is their mean. Equation of state parametersV0, K0 and K′0 are difficult to fit simultaneously due to trade-offs [e.g., Angel, 2000]—when all parameters are allowed tovary, uncertainty on K0 for each material is up to 20%—butfor most of these materials V0 and K0 are well-constrained byindependent measurements such as X-ray diffraction andultrasonic interferometry at ambient conditions. Values for V0

Figure 1. (a) Selected diffraction patterns for Au and NaCl in Ne up to 259 GPa (run ANN11). All peaksare labeled for Au, NaCl B2, Ne and Re gasket. Gold (200) and NaCl B2 (110) overlap above 200 GPa.(b) Pressure-dependence of d-spacings observed in the same run. These d-spacings were used to fit unitcell volumes except in cases of peak overlaps.

1Auxiliary materials are available in the HTML. doi:10.1029/2012JB009292.


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were fixed for all materials. Fits were performed with andwithout fixing K0 for Au, Mo and Pt. The resulting fitparameters using the Vinet equation are given in Table 2 andare compared to previously reported values in Tables 3–4.In the case of fits with fixed parameters, the uncertaintiesfrom the fitting are artificially low due to the constrainednature of the fit. We suggest that realistic uncertainties on K0

and K ′0 are �2%. Fits are shown together with volume datafrom this study and previously reported equations of state inFigures 5–9. Also shown are co-compression data reported inother studies on these materials [Sata et al., 2002; Ono et al.,2006b; Hirose et al., 2008; Sata et al., 2010].[14] Our fit results using the Vinet equation of state shown

in Figures 5–9 gave lower error and better agreement withambient elasticity measurements than the Birch-Murnaghanequation, as also reported in other previous studies to highcompressions [Vinet et al., 1987; Hemley et al., 1990; Onoet al., 2006b]. The strongest difference is seen for Ne, themost compressible material examined. Fitting errors inequation of state parameters for Ne are as much as an orderof magnitude lower when derived from the Vinet equationthan the Birch-Murnaghan equation. A Vinet equation fit alsoproduced K0 in better agreement with measured zero-pressure

values for Au, Mo and Pt than the Birch-Murnaghanequation.[15] MgO was selected as the reference for the new

equation of state determination and was fixed in our fit. Thehigh-pressure behavior of MgO may be better constrainedthan that of any other material: its simplicity and geophysi-cal importance have motivated multiple studies by shockcompression [Marsh, 1980; Duffy and Ahrens, 1995], staticcompression [Duffy et al., 1995; Dewaele et al., 2000;Speziale et al., 2001; Tange et al., 2009] high-pressureelasticity [Sinogeikin and Bass, 2000; Zha et al., 2000; Liet al., 2006; Kono et al., 2010] and theory [Dorogokupetsand Dewaele, 2007; Wu et al., 2008]. Data from a varietyof studies and methods were integrated by Speziale et al.[2001] to produce a single consistent P–V–T equation ofstate based on the Birch-Murnaghan equation. Morerecently, this unified equation of state approach was adoptedby Tange et al. [2009] combining 1-atm thermal expansion,1-atm high-temperature KS data, 300 K high-pressure data(based on absolute pressure scales), and shock compressiondata. The Tange et al. [2009] equation of state yields anambient pressure derivative of the bulk modulus, K′0 = 4.37when using the Vinet equation (Table 3). This value is

Figure 2. (a) Selected diffraction patterns for Pt and NaCl in He up to 188 GPa (run PNH7). All peaksare labeled for Pt, NaCl B2 and Re gasket. (b) Pressure-dependence of d-spacings observed in the samerun. These d-spacings were used to fit unit cell volumes except in cases of peak overlaps.


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consistent with more recent studies involving inversion ofMgO data sets independent of an empirical pressure scale[Kennett and Jackson, 2009] and combined elasticity-volume data at high pressures [Kono et al., 2010]. However,the Tange et al. [2009] scale is stiffer than most other MgOscales listed in Table 3. The uncertainty in the fit of thisequation of state to the primary data was measured to be0.8 GPa. We confirm below that the Tange et al. [2009] MgOscale is consistent with shock isotherms for Mo, Au and Pt(Figures 5–6).[16] The equation of state fit for Mo in this study is

dependent only on the MgO scale (Table 1). Measuredvolumes are in excellent agreement with the shock equationof state of Mo [Hixson and Fritz, 1992] when MgO pres-sures are calibrated by Tange et al. [2009] (Figure 5). The fitis also in good agreement with the co-compression study byDewaele et al. [2008b] and the X-ray and neutron diffractionstudy by Zhao et al. [2000]. An extrapolation of ultrasonicelasticity measurements by Liu et al. [2009] gives muchlower compressibility than other studies. Our work is thefirst static quasi-hydrostatic experiment to constrain the

equation of state of Mo relative to other commonly usedcalibrants to multi-Mbar conditions.[17] Gold has many of the attributes of an ideal pressure

standard, and has been widely studied as a result (Table 4).However, pressure scales for this material deviate by morethan 10% at Mbar conditions. Multiple ultrasonic studiesconsistently report K0 values close to 167 GPa, but K ′0measurements range from 5.0–6.5 [Daniels and Smith, 1958;Hiki and Granato, 1966; Golding et al., 1967; Biswas et al.,1994; Song et al., 2007]. Lower K ′0 was observed by shockcompression [Jamieson et al., 1982] and X-ray diffraction[Takemura, 2001; Shim et al., 2002], though these studieswere later found to give systematically low volumes for Aurelative to other data [Takemura and Dewaele, 2008]. Yokooet al. [2008] reported new shock compression data up to580 GPa and found a stiffer shock wave equation of statethan earlier shock experiments. Reduction of the shock dataincluded the electronic thermal pressure effect [Yokoo et al.,2009], and the reduced isotherm is stiffer than the earliershock isotherm of Jamieson et al. [1982]. These new shock

Figure 3. (a) Selected diffraction patterns for Mo and MgO in He up to 207 GPa (run MMH6). All peaksare labeled for Mo, MgO and Re gasket. (b) Pressure-dependence of d-spacings observed in the same run.These d-spacings were used to fit unit cell volumes except in cases of peak overlaps.


5 of 15

compression data and recent cross-calibration studies [Feiet al., 2007; Takemura and Dewaele, 2008] appear to con-verge on K′0 values of 5.9–6.0 when using the Vinet equationof state. Our equation of state for Au (Figure 6) is in excellent

agreement with the shock-reduced equation of state by Yokooet al. [2009] and with recent cross-calibration studies [Feiet al., 2007; Takemura and Dewaele, 2008]. Data fromprevious studies by Ono et al. [2006b]; Hirose et al.

Figure 4. (a) Selected diffraction patterns for Pt and MgO in He up to 226 GPa (run PMH6). All peaksare labeled for Pt, MgO and Re gasket. (b) Pressure-dependence of d-spacings observed in the same run.These d-spacings were used to fit unit cell volumes except in cases of peak overlaps.

Table 2. Vinet Equation of State Parameters From Simultaneous Fit to All Standards Using MgO [Tange et al., 2009] as a Reference

Aua MgOb Moc NaCl B2d Nee Ptc

K0 Fixed for MetalsV0 (Å

3) 67.85f 74.698f 31.17f 41.35f 88.967f 60.38f

K0 (GPa) 167f 160.6f 261f 24.2 � 0.3 1.04 � 0.02 277f

K ′0 5.88 � 0.02 4.37f 4.19 � 0.02 5.76 � 0.03 8.48 � 0.04 5.43 � 0.02

K0 Free for MetalsV0 (Å

3) 67.85f 74.698f 31.17f 41.35f 88.967f 60.38f

K0 (GPa) 167 � 3 160.6f 271 � 3 26.4 � 0.3 1.02 � 0.02 280 � 2K ′0 5.84 � 0.02 4.37f 3.89 � 0.09 5.49 � 0.03 8.50 � 0.04 5.29 � 0.07

aK0 of Au based on ultrasonic interferometry measurements [Daniels and Smith, 1958; Hiki and Granato, 1966; Golding et al., 1967; Biswas et al.,1981].

bEquation of state for MgO by Tange et al. [2009] fixed as reference.cK0 for Mo and Pt from ultrasonic measurements given in Every and McCurdy [2011].dV0 for NaCl B2 taken from Bukowinski and Aidun [1985].eV0 for Ne taken from Batchelder et al. [1967].fParameters fixed during fit.


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[2008]; Sata et al. [2010] have systematically slightlyhigher Au volumes relative to NaCl B2 volume thanobserved in our study near Mbar pressures (Figure 6).[18] Platinum is less compressible than gold but also

commonly used as a pressure calibrant and laser absorberand so its equation of state has also received much attention.Calibrations based on reduced shock data by Holmes et al.[1989] and Yokoo et al. [2009] exhibit lower compressibil-ity than recent X-ray diffraction studies [Dewaele et al.,2004; Fei et al., 2007]. The equation of state fit for Pt inthis study (Figure 7) is in good agreement with the reducedshock equation of state by Yokoo et al. [2009], but thevolume data show some systematic deviations from theshock data below 1 Mbar. Within runs, the data scatter iscomparable to previous studies by Sata et al. [2002, 2010]but is larger than errors reported by Dewaele et al. [2004].Measured volumes agree with the annealed data by Sata et al.[2002], but our data fall below volumes measured for Pt bySata et al. [2010] by 1%. More work is needed to investigatethese uncertainties and better resolve the equation of stateof Pt.[19] Since NaCl B2 and solid Ne are not stable at ambient

conditions, their zero-pressure equation of state parameterscannot be measured directly. In these fits, a theoreticalvalue for V0 was chosen for NaCl B2 [Bukowinski andAidun, 1985] and a cryostatic experimental value for Ne[Batchelder et al., 1967] as in Fei et al. [2007]. K0 and K′0for these phases differ by up to 20–40% between Birch-Murnaghan and Vinet fits. Our volume data for NaCl B2 andMgO agree very well with previous studies [Sata et al.,2002; Hirose et al., 2008; Sata et al., 2010], but the MgOcalibration by Tange et al. [2009] requires the NaCl B2

phase to be less compressible than previous equations ofstate [Ono et al., 2006b; Fei et al., 2007; Sakai et al., 2011](Figure 8). The previous equations of state of NaCl B2 phaseby Fei et al. [2007] and Sakai et al. [2011] are in very goodagreement, but for our data both of these NaCl scales givehigher pressures than the MgO scale by Tange et al. [2009]and Au scale by Fei et al. [2007] by �15 GPa at 2.5 Mbar.For Ne, our data exhibit some scatter but our scale is in goodagreement with Fei et al. [2007] (Figure 9). Results for Nerelative to MgO and NaCl B2 scales agree within observedvolume variation.[20] Pressure differences given by our comprehensive

pressure scale, DPi, for materials co-compressed with MgOand NaCl B2 respectively are presented in Figure 10.Although pressure scales determined in this study are inagreement with the scales by Fei et al. [2007] and Dewaeleet al. [2008b] at lower pressures, they deviate with increas-ing pressure to as much as a 10% difference at 2.5 Mbar.This difference is observed principally in the compressibilityof NaCl B2, which we have found to be overestimated byprevious studies relative to every other calibrant. Our com-prehensive fit brings these calibrants into agreement above2 Mbar within the scatter of the data. For metals Au, Pt, andMo, our equation of state parameters (Tables 3 and 4) are inbest agreement with the most recent reduced shock waveisotherms [Hixson and Fritz, 1992; Yokoo et al., 2009].

3.2. Pressure Error and Differential Stress

[21] Error in equation of state measurements increases atMbar pressures due both to measurement precision and tonon-hydrostatic stresses. The uncertainty in volume due todiffraction resolution and peak fitting results in increasing

Table 3. Vinet and Birch-Murnaghan (B-M) Equation of State Parameters for MgO, NaCl B2 and Ne from Selected Previous Work.

Method Reference V0 (Å3) K0 (GPa) K ′0 K″0 Form

MgOShock Jamieson et al. [1982] 74.7a 160a 4.4 Vinetb

Brillouin Sinogeikin and Bass [1999] 163.2 4.0 �0.04 B-MUltrasonics Li et al. [2006] 74.70a 161.3 4.24 B-M

Kono et al. [2010] 74.71a 160.9 4.35 B-MDorogokupets and Dewaele [2007] 160.3 4.18 Vinet

Theory Wu et al. [2008] 75.19 160 4.23 �0.0281 B-MFei [1999] 74.78 160 4.15 B-M

X-ray Dewaele et al. [2000] 74.71a 161a 4.01 VinetSpeziale et al. [2001] 74.71a 160.2 3.99 B-M

Others Kennett and Jackson [2009] 74.686a 160.34 4.34 B-MTange et al. [2009] 74.698 160.6 4.37 Vinet

NaCl B2Theory Ueda et al. [2008] 41.11 28.45 5.16 Vinet

Fei et al. [2007] 41.a 26.9 5.3 VinetX-ray Ono [2010] 39.28 39.26 4.72 Vinet

Sakai et al. [2011] 37.73a 41.4 5.0 VinetThis study 41.35a 24.2 � 0.3 5.76 � 0.04 Vinet

Others Dorogokupets and Dewaele [2007] 29.72 5.14 Vinet

NeTheory Tsuchiya and Kawamura [2002a] 60.87 6.53 7.5 Vinet

Finger et al. [1981] 88.967a 0.744 7.071 0.051 B-MHemley et al. [1989] 88.967a 1.097 9.23 B-M

X-ray Fei et al. [2007] 88.967a 1.2 8.2 VinetDewaele et al. [2008a] 88.936a 1.07 8.4 VinetThis study 88.967a 1.04 � 0.02 8.48 � 0.04 Vinet

aParameter fixed during fit.bPreviously published data fit to equation of state parameters for this study.


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uncertainty in pressure at small volumes. In our experimentsthis uncertainty reaches �3% at 2.5 Mbar, as observed fromwithin-run data scatter (Figure 10). Additional systematicerror in equation of state measurements between materialsand between experimental runs can be due to differences insample loading conditions and/or material properties. Thiserror is assessed for our measurements by evaluating thehydrostaticity of the stress state.[22] Hydrostatic pressure in the diamond anvil cell can

only be achieved within a fluid medium. Because no liquidsare stable above 12 GPa at room temperature [Besson andPinceaux, 1979], purely hydrostatic stress is not possible athigher pressure. Low differential stress, or quasi-hydrostaticcompression, can be maintained by using soft solid mediasuch as noble gases. In the diamond anvil cell, the stress stateis approximately uniaxial, with the maximum stress along thecompression axis and the minimum stress in the radialdirection. The difference between maximum and minimumstresses, or differential stress, t, supported by a solid mediumwill be no greater than its yield strength. The yield strengthof the medium also limits the radial pressure gradient acrossthe sample [Sung et al., 1977]. Powder samples also exhibit adistribution of stress differences between crystallites, ormicrostresses, resulting from grain contacts. These non-hydrostatic stresses produce anisotropic lattice strain and

systematic errors in volume determination. Differential stressand microstress can be measured from their effects on dif-fraction line shifts [Singh, 1993] and widths [Weidner et al.,1994; Takemura, 2001]. These measurements can be used todetermine yield strengths [Duffy, 2007] and give comple-mentary constraints on the hydrostaticity of sample condi-tions at high compression.3.2.1. Peak Widths[23] Figure 11 shows diffraction peak widths (FWHM)

measured for Pt and Au loaded in He and Ne media. Thewidth of diffraction peaks depends on instrumental resolu-tion, grain size and microstresses. Instrumental resolutionmay vary especially for experiments conducted at differentfacilities due to X-ray beam divergence and detector positionand resolution. Typical instrumental resolution for this workis indicated by the range of peak widths observed for theCeO2 standard (Figure 11). With increasing pressure, dif-fraction peaks broaden due to differential stress [Takemura,2001; Weidner et al., 1994]. Grain size reduction by shear-ing can also result in peak broadening. In addition, the stressdistribution is greater when the sample supports a significantstress gradient relative to the area sampled by the X-raybeam.[24] Takemura and Dewaele [2008] reported diffraction

peaks widths in Au samples compressed in a Hemedium. The

Table 4. Vinet and Birch-Murnaghan (B-M) Equation of State Parameters for Au, Pt and Mo From Previous Workand This Study

Method Reference K0 (GPa) K′0 Form

AuDaniels and Smith [1958]Hiki and Granato [1966]

Ultrasonics Golding et al. [1967] 166.3–167.2 5.2–6.6Biswas et al. [1994]Song et al. [2007]

Shock Jamieson et al. [1982] 175.3 5.05 Vinetb

Yokoo et al. [2009] 167.5 5.94 VinetFei et al. [2007] 167 6.0 Vinet

X-ray Takemura and Dewaele [2008] 167 5.9 VinetThis study 167a 5.88 � 0.02 Vinet

Theory Tsuchiya [2003] 166.7 6.12Greeff and Graf [2004] 167.9 5.64Dorogokupets and Dewaele [2007] 167 5.9 Vinet

Others Shim et al. [2002] 167 5.0 B-MHeinz and Jeanloz [1984] 167 5.5 B-M

PtHolmes et al. [1989] 266 5.81 Vinet

Shock Yokoo et al. [2009] 276.4 5.48 VinetFei et al. [2007] 277 5.08 Vinet

X-ray Zha et al. [2008] 273.5 4.7 B-MThis study 277a 5.43 � 0.02 Vinet

Theory Matsui et al. [2009] 273 5.2 VinetOno et al. [2011] 290.8 5.11 B-M

Other Dorogokupets and Dewaele [2007] 277.3 5.12 Vinet

MoUltrasonics Katahara et al. [1979]

Liu et al. 2009 261–263 4.4–4.7Shock Hixson and Fritz [1992] 260 4.19 VinetX-ray Dewaele et al. [2008b] 261a 4.06 Vinet

This study 261a 4.19 � 0.02 VinetDewaele et al. [2008b]

Theory Liu et al. [2009] 255–295 4–4.5Zeng et al. [2010]

aParameter fixed during fit.bPreviously published data fit to equation of state parameters for this study.


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measured peak widths varied from run to run (Figure 11).This may reflect differences in experimental factors such asdeformation of the gasket, distribution of the pressuremedium, grain size reduction, and details of grain interactionswhich can be different even in runs conducted with nearlyidentical experimental protocols. We obtain peak widths inAu with a Ne medium up to 245 GPa (run ANN11). Ourmeasured peak widths for the Au (111) line display a trendconsistent with the lower range of measurements in a Hemedium, while the (200) line shows widths closer to thehigher range (Figure 11). The minimum peak widthsobserved by Takemura and Dewaele [2008] were within therange of our CeO2 standard, indicating these experimentslikely have comparable instrumental resolution. This indi-cates that the differential stresses in Ne may be similar tothose in He even up to Mbar pressures.[25] Also shown in Figure 11 are peak widths for Pt

compressed together with MgO or NaCl in He and Nemedia. Pt peaks widths are greater than Au peak widths,probably reflecting the smaller grain size of the Pt sample(0.5–2 mm). Larger peak widths for run PNH7 especially at50–125 GPa suggest this run was less hydrostatic than otherruns. At high pressures (>50 GPa), peak widths for a givenreflection measured in He (run PMH6) or Ne (run PMN3)

are very similar, again suggesting that the stress differencessupported by Ne and He are similar to Mbar conditions.3.2.2. Diffraction Line Shifts[26] Differential stress results in variation in the lattice

parameter as a function of crystallographic orientation (hkl)[Singh, 1993]. In the axial diffraction geometry in the DAC,interplanar spacings are measured near the minimum straindirection. Anisotropic lattice strain results from elasticanisotropy. The elastic anisotropy of a cubic crystal can beexpressed by the elastic anisotropy factor, S = 1/(C11 � C12)� 1/(2C44), where the Cij are elastic stiffnesses. For an iso-tropic material, S = 0, whereas a positive value of S impliesthat the Young’s modulus is greater in the [111] directionthan the [100] direction. The effect of deviatoric stress on thelattice parameter for a given hkl is given by Singh [1993] andSingh and Takemura [2001]:

aðhklÞ ¼ M0 þM1½3ð1� 3sin2qÞGðhklÞ�; ð4Þ

where

M0 ¼ aP 1þ at3ð1� 3 sin2qÞ S11 � S12 � 1� a�1

2GV

� �� ; ð5Þ

Figure 5. Volume data for Mo fit to the Vinet equation ofstate (green) with reference to MgO [Tange et al., 2009] andprevious measurements of the equation of state of Mo[Hixson and Fritz, 1992; Liu et al., 2009; Dewaele et al.,2008b].

Figure 6. Volume data for Au from this study and previouswork [Ono et al., 2006b; Hirose et al., 2008; Sata et al.,2010] fit to the Vinet equation of state (gold) with referenceto NaCl B2 scale from this work. Equation of state curves forAu from previous work are plotted in green [Jamieson et al.,1982], red [Yokoo et al., 2009] and blue [Fei et al., 2007].


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M1 ¼ � aPaSt3

; ð6Þ

GðhklÞ ¼ h2k2 þ k2l2 þ l2h2

h2 þ k2 þ l2ð Þ2 ; ð7Þ

and aP is the lattice parameter under hydrostatic pressure P.a is a measure of continuity of strain and can take valuesbetween 0.5 and 1 or between 1 and 2 [Singh, 2009]. GV isthe shear modulus under isostrain conditions. If we assumeM0 ≈ aP and a = 1, the product St can be derived directlyfrom the slope and intercept of a(hkl) with 3(1 � 3 sin2 q)G (hkl):

St ≈�3M1

M0: ð8Þ

[27] The accuracy of determination of the differentialstress, t, is limited by, among other factors, how well thedependence of S on pressure is known. S can be determined

from high-pressure elastic constants but experimental mea-surements are limited to low pressures and extrapolation toMbar conditions is uncertain. Instead, for most materials, weused theoretical calculations of Cijs from density functionaltheory. However, for Au, due to inconsistency betweentheoretical elastic constants [Tsuchiya and Kawamura,2002a; Greeff and Graf, 2004], we use pressure derivatives∂Cij/∂P from ultrasonic interferometry by Golding et al.[1967] as in Takemura and Dewaele [2008]. Extrapola-tions were performed using finite strain theory [Birch,1978]. Our samples include materials with both positiveand negative elastic anisotropy factor, S (Figure S1 of theauxiliary material). The magnitude of S decreases withpressure and several materials (NaCl, MgO, and Mo) areclose to isotropic at Mbar conditions. Due to their higherelastic anisotropy, Au, Ne, and Pt are more sensitive to non-hydrostatic stress and may give more reliable estimates for t.Conversely, the volumes of these materials are more likelyto show systematic deviation from the equation of state dueto lattice strain.

Figure 7. Volume data for Pt from this study and previouswork [Sata et al., 2002, 2010] fit to the Vinet equation ofstate (black) with reference to MgO [Tange et al., 2009].Equation of state curves for Pt from previous work are plot-ted in blue [Holmes et al., 1989], green [Fei et al., 2007] andred [Yokoo et al., 2009].

Figure 8. Volume data for NaCl B2 (blue filled dots) andfit (blue line) with reference to MgO [Tange et al., 2009]with other equations of state curves for the NaCl B2 phase[Ono et al., 2006b; Fei et al., 2007; Sakai et al., 2011]. Pre-vious measurements from co-compression experiments areshown in open circles [Sata et al., 2002; Hirose et al.,2008; Sata et al., 2010].


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[28] Examples of measured lattice parameters as a func-tion of 3(1 � 3 sin2 q) G (hkl) are shown for Pt in Figure S2of the auxiliary material (PNH7). Differential stress mea-sured in Pt was similar between runs with Ne and Hemedium (Figure 12), consistent with peak width measure-ments (Figure 11). Over 10–224 GPa, the observed t rangedfrom 1–4 GPa (with one outlier at 5 GPa). Above 100 GPa tis �2% of the pressure. This is higher than t measured forlaser-annealed samples in Dorfman et al. [2010], but lowerthan unannealed Pt in a NaCl medium [Dorfman et al., 2010]and previous measurements of the strength of Pt under non-hydrostatic conditions [Kavner and Duffy, 2003; Singh et al.,2008]. At Mbar pressures, samples loaded in Ne and He wereboth more hydrostatic than those in NaCl or no medium, butHe was not significantly more hydrostatic than Ne.[29] Differential stress was also directly evaluated for the

Ne medium from the (111) and (200) lines (Figure S3 of theauxiliary material). Our observations are consistent betweenruns and in excellent agreement with Takemura et al. [2010],given the same function for S(P) [Tsuchiya and Kawamura,2002b]: t is �3% of the pressure, and reaches values of 6–

10 GPa at 2.5 Mbar. While the strength of Ar has beensuggested to increase nonlinearly with pressure (Figure S3of the auxiliary material), the strength of Ne appears toincrease linearly to Mbar pressures. However, t measured inthe Ne medium is higher than t measured in either Pt or Ausamples. If Ne is stronger than either of these metals underMbar conditions, t in Pt and Au should be equal to their yieldstrengths. Higher t measurements in non-hydrostatic Ptsamples suggest this is not the case. This apparent contra-diction can be resolved by higher S in Ne, which wouldallow Ne to show more anisotropic strain at lower t. Tofurther evaluate the strength of Ne, more work is needed totest the reliability of theoretical predictions of the elasticanisotropy of these materials.[30] In summary, peak widths and relative line shifts were

analyzed in the most anisotropic materials in this study, Au,Ne and Pt. These measurements consistently show compa-rable quasi-hydrostatic stress conditions between Ne and Hemedia. The differential stress supported by the Ne and Hemedia is 2–3% of the pressure up to 2.5 Mbar. Equation ofstate measurements based on these data can be expected tohave error due to deviatoric stresses on the order of 2–3%.Further reduction of error in the comprehensive equation ofstate will require reduction of deviatoric stresses to <1%.

4. Summary

[31] Materials commonly used as pressure calibrants wereco-compressed in quasi-hydrostatic Ne and He media topressures up to 265 GPa. X-ray diffraction was used toanalyze the stress state in these materials. Although X-raydiffraction data cannot be used to create an absolute pressurescale, the simultaneous fit of these correlated data can beused to improve the relative agreement among pressurescales. A new comprehensive pressure scale for Au, MgO,Mo, NaCl B2, Ne and Pt is proposed with reference to theMgO scale by Tange et al. [2009] and internal agreement to3% at pressures up to 2.5 Mbar. Our Au, Mo and Pt scalesare in best agreement with reduced shock isotherms [Hixsonand Fritz, 1992; Yokoo et al., 2009], while the scale for Ne isin good agreement with the previous equation of state by Feiet al. [2007]. These experiments were conducted at 300 Konly. Future work should also extend these scales to hightemperatures for application to geophysics.[32] Diffraction line width and differential line shift anal-

yses suggest that deviatoric stress conditions at 2.5 Mbar arenot significantly different in He versus Ne media. The mea-sured strength of Ne at 2.5 Mbar is 8 GPa, in good agreementwith Takemura et al. [2010] but lower than stresses measuredin Pt or Au samples. More work is needed to determinereliable elastic anisotropy of materials for measurement ofdeviatoric stress. Errors in pressure calibration may be furtherreduced by annealing deviatoric stress.

[33] Acknowledgments. This work was supported by the NSF andthe Carnegie-DOE alliance center. High-pressure gas loading was supportedby the Consortium for Materials Properties Research in Earth Sciences(COMPRES) and GeoSoilEnviroCARS. Experiments were performed atGeoSoilEnviroCARS (Sector 13) and High Pressure Collaborative AccessTeam (HPCAT, Sector 16) at the APS. GeoSoilEnviroCARS is supportedby the NSF, DOE, and the State of Illinois. HPCAT is supported byDOE-BES, DOE-NNSA, NSF, and the W.M. Keck Foundation. Use ofthe APS was supported by the U.S. Department of Energy, Office of BasicEnergy Sciences.

Figure 9. Volume data for Ne from this study fit to theVinet equation of state (red). Pressures were calculated fromMgO (pink) and NaCl B2 (red) equations of state in Tangeet al. [2009] and this work. Also plotted are Birch-Murnaghan(green) and Vinet (black) equation of state for Ne by Feiet al. [2007]. Due to the high compressibility of Ne, the dif-ference between previously measured [Fei et al., 2007]Birch-Murnaghan and Vinet fits at 2.5 Mbar is significant.


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Figure 10. Differences between pressures determined from unit cell volumes of Au, MgO, Mo, NaCl B2,Ne and Pt simultaneously observed in this work (filled symbols) and previous studies [Sata et al., 2002;Ono et al., 2006b; Hirose et al., 2008; Sata et al., 2010] (open symbols). Pressures were calculated for allmaterials from equations of state from Tange et al. [2009], Fei et al. [2007], Dewaele et al. [2008b] andthis work. For all experimental runs with MgO, differences were calculated between Tange et al. [2009]MgO and pressures obtained from (a) previously published scales [Fei et al., 2007; Dewaele et al., 2008b]and (b) Dorfman scale. Similarly for experimental runs with NaCl, the difference between pressuresobtained from an NaCl B2 scale and pressures calculated for Au, Ne, Pt and MgO are shown for (c) pre-vious studies [Fei et al., 2007; Tange et al., 2009] (d) this work/Tange et al. [2009].


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Figure 11. Diffraction peak full width at half maximum normalized to diffraction angle (D(2q)/(2q))forPt (runs PMH6 and PMN3 in black, PNH7 and PMNN7 in gray) and Au (gold, run ANN11) loaded in He(filled symbols) and Ne (open symbols). Au in He are data from previous work by Takemura and Dewaele[2008]. A representative range of peak widths observed for CeO2 standard peaks (collected before runPMH6) is shown in blue.

Figure 12. Differential stress measured in Pt in Ne and He in this work. Also shown are previous mea-surements by this author of t in Pt in a NaCl medium with and without laser annealing. Previous measure-ments of the strength of Pt by other authors used radial diffraction [Kavner and Duffy, 2003] and peakwidths [Singh et al., 2008].


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ReferencesAngel, R. J. (2000), Equations of state, inHigh-Pressure, High-TemperatureCrystal Chemistry, Rev. Mineral. Geochem., vol. 41, edited by R. M.Hazen and R. T. Downs, pp. 35–60, Mineral. Soc. of Am., Washington,D. C.

Batchelder, D. N., D. L. Losee, and R. O. Simmons (1967), Measurementsof lattice constant, thermal expansion, and isothermal compressibilityof neon single crystals, Phys. Rev., 162(3), 767–775, doi:10.1103/PhysRev.162.767.

Besson, J. M., and J. P. Pinceaux (1979), Melting of helium at room temper-ature and high pressure, Science, 206(4422), 1073–1075, doi:10.1126/science.206.4422.1073.

Birch, F. (1947), Finite elastic strain of cubic crystals, Phys. Rev., 71(11),809–824, doi:10.1103/PhysRev.71.809.

Birch, F. (1978), Finite strain isotherm and velocities for single-crystaland polycrystalline NaCl at high pressures and 300�K, J. Geophys. Res.,83(B3), 1257–1268, doi:197810.1029/JB083iB03p01257.

Biswas, S. N., P. V. Klooster, and N. J. Trappeniers (1981), Effect of pres-sure on the elastic constants of noble metals from �196 to +25�C and upto 2500 bar: II. Silver and gold, Physica B + C, 103(2–3), 235–246,doi:10.1016/0378-4363(81)90127-3.

Biswas, S. N., M. J. P. Muringer, and C. A. T. Seldam (1994), Pressuredependence of shear modulus C44 of platinum from 77 to 298 K and upto 253 MPa, Phys. Status Solidi A, 141(2), 361–365, doi:10.1002/pssa.2211410214.

Bukowinski, M. S. T., and J. Aidun (1985), First principles versus sphericalion models of the B1 and B2 phases of NaCl, J. Geophys. Res., 90(B2),1794–1800, doi:10.1029/JB090iB02p01794.

Daniels, W. B., and C. S. Smith (1958), Pressure derivatives of the elasticconstants of copper, silver, and gold to 10,000 bars, Phys. Rev., 111(3),713–721, doi:10.1103/PhysRev.111.713.

Dewaele, A., and P. Loubeyre (2007), Pressurizing conditions in helium-pressure-transmitting medium, High Pressure Res., 27(4), 419–429,doi:10.1080/08957950701659627.

Dewaele, A., G. Fiquet, D. Andrault, and D. Hausermann (2000), P-V-Tequation of state of periclase from synchrotron radiation measurements,J. Geophys. Res., 105(B2), 2869–2877, doi:200010.1029/1999JB900364.

Dewaele, A., P. Loubeyre, and M. Mezouar (2004), Equations of state of sixmetals above 94 GPa, Phys. Rev. B, 70(9), 094112, doi:10.1103/PhysRevB.70.094112.

Dewaele, A., P. Loubeyre, F. Occelli, M. Mezouar, P. I. Dorogokupets, andM. Torrent (2006), Quasihydrostatic equation of state of iron above2 Mbar, Phys. Rev. Lett., 97(21), 215504, doi:10.1103/PhysRevLett.97.215504.

Dewaele, A., F. Datchi, P. Loubeyre, and M. Mezouar (2008a), High pres-sure–high temperature equations of state of neon and diamond, Phys. Rev.B, 77(9), 094106, doi:10.1103/PhysRevB.77.094106.

Dewaele, A., M. Torrent, P. Loubeyre, and M. Mezouar (2008b), Compres-sion curves of transition metals in the Mbar range: Experiments andprojector augmented-wave calculations, Phys. Rev. B, 78(10), 104102,doi:10.1103/PhysRevB.78.104102.

Dorfman, S. M., F. Jiang, Z. Mao, A. Kubo, Y. Meng, V. B. Prakapenka,and T. S. Duffy (2010), Phase transitions and equations of state of alka-line earth fluorides CaF2, SrF2, and BaF2 to Mbar pressures, Phys. Rev.B, 81(17), 174121, doi:10.1103/PhysRevB.81.174121.

Dorogokupets, P. I., and A. Dewaele (2007), Equations of state of MgO,Au, Pt, NaCl-B1, and NaCl-B2: Internally consistent high-temperaturepressure scales, High Pressure Res., 27(4), 431–446, doi:10.1080/08957950701659700.

Dorogokupets, P. I., and A. R. Oganov (2006), Equations of state of Al, Au,Cu, Pt, Ta, and W and revised ruby pressure scale, Dokl. Earth Sci.,410(7), 1091–1095, doi:10.1134/S1028334X06070208.

Duffy, T. S. (2007), Strength of materials under static loading in the dia-mond anvil cell, in Shock Compression of Condensed Matter- 2007:Proceedings of the Conference of the American Physical Society Topi-cal Group on Shock Compression of Condensed Matter, vol. 955,pp. 639–644, Am. Inst. of Phys., Melville, N. Y., doi:10.1063/1.2833175.

Duffy, T. S., and T. J. Ahrens (1995), Compressional sound velocity, equa-tion of state, and constitutive response of shock-compressed magnesiumoxide, J. Geophys. Res., 100(B1), 529–529, doi:10.1029/94JB02065.

Duffy, T. S., R. J. Hemley, and H.-K. Mao (1995), Equation of state andshear strength at multimegabar pressures: Magnesium oxide to 227 GPa,Phys. Rev. Lett., 74(8), 1371–1374, doi:10.1103/PhysRevLett.74.1371.

Every, A. G., and A. K. McCurdy (2011), Second and Higher OrderElastic Constants, Landolt-Börnstein: Numerical Data and FunctionalRelationships in Sciences and Technology - New Series, vol. 29a,pp. 11–17, Springer, Berlin.

Fei, Y. (1999), Effects of temperature and composition on the bulk modulusof (Mg, Fe)O, Am. Mineral., 84(3), 272–276.

Fei, Y., A. Ricolleau, M. Frank, K. Mibe, G. Shen, and V. B. Prakapenka(2007), Toward an internally consistent pressure scale, Proc. Natl. Acad.Sci. U. S. A., 104(22), 9182–9186, doi:10.1073/pnas.0609013104.

Finger, L. W., R. M. Hazen, G. Zou, H.-K. Mao, and P. M. Bell (1981),Structure and compression of crystalline argon and neon at high pressureand room temperature, Appl. Phys. Lett., 39(11), 892–894, doi:10.1063/1.92597.

Golding, B., S. C. Moss, and B. L. Averbach (1967), Composition and pres-sure dependence of the elastic constants of gold-nickel alloys, Phys. Rev.,158(3), 637–646, doi:10.1103/PhysRev.158.637.

Greeff, C. W., and M. J. Graf (2004), Lattice dynamics and the high-pressure equation of state of Au, Phys. Rev. B, 69(5), 054107,doi:10.1103/PhysRevB.69.054107.

Hammersley, A. P., S. O. Svensson, M. Hanfland, A. N. Fitch, and D.Hausermann (1996), Two-dimensional detector software: From real detec-tor to idealised image or two-theta scan, High Pressure Res., 14(4),235–248, doi:10.1080/08957959608201408.

Heinz, D. L., and R. Jeanloz (1984), The equation of state of the gold cal-ibration standard, J. Appl. Phys., 55(4), 885–893, doi:10.1063/1.333139.

Hemley, R. J., C.-S. Zha, A. P. Jephcoat, H.-K. Mao, L. W. Finger, andD. E. Cox (1989), X-ray diffraction and equation of state of solid neonto 110 GPa, Phys. Rev. B, 39(16), 11,820–11,827, doi:10.1103/PhysRevB.39.11820.

Hemley, R. J., H.-K. Mao, L. W. Finger, A. P. Jephcoat, R. M. Hazen, andC.-S. Zha (1990), Equation of state of solid hydrogen and deuteriumfrom single-crystal X-ray diffraction to 26.5 GPa, Phys. Rev. B, 42(10),6458–6470, doi:10.1103/PhysRevB.42.6458.

Hiki, Y., and A. V. Granato (1966), Anharmonicity in noble metals: Higherorder elastic constants, Phys. Rev., 144(2), 411–419, doi:10.1103/PhysRev.144.411.

Hirose, K., R. Sinmyo, N. Sata, and Y. Ohishi (2006), Determination ofpost-perovskite phase transition boundary in MgSiO3 using Au andMgO pressure standards, Geophys. Res. Lett., 33, L01310, doi:10.1029/2005GL024468.

Hirose, K., N. Sata, T. Komabayashi, and Y. Ohishi (2008), Simultaneousvolume measurements of Au and MgO to 140 GPa and thermal equationof state of Au based on the MgO pressure scale, Phys. Earth Planet.Inter., 167(3–4), 149–154, doi:10.1016/j.pepi.2008.03.002.

Hixson, R. S., and J. N. Fritz (1992), Shock compression of tungsten andmolybdenum, J. Appl. Phys., 71(4), 1721–1728, doi:10.1063/1.351203.

Holmes, N. C., J. A. Moriarty, G. R. Gathers, and W. J. Nellis (1989), Theequation of state of platinum to 660 GPa (6.6 Mbar), J. Appl. Phys.,66(7), 2962–2967, doi:10.1063/1.344177.

Jamieson, J. C., J. N. Fritz, and M. H. Manghnani (1982), Pressure mea-surement at high temperature in X-ray diffraction studies: Gold as aprimary standard, in High-Pressure Research in Geophysics, editedby S. Akimoto and M. H. Manghnani, pp. 27–48, Cent. for Acad. Publ.,Tokyo.

Jin, K., X. Li, Q. Wu, H. Geng, L. Cai, X. Zhou, and F. Jing (2010), Thepressure–volume–temperature equation of state of MgO derived fromshock Hugoniot data and its application as a pressure scale, J. Appl.Phys., 107(11), 113518, doi:10.1063/1.3406140.

Jin, K., Q. Wu, H. Geng, X. Li, L. Cai, and X. Zhou (2011), Pressure–volume–temperature equations of state of Au and Pt up to 300 GPaand 3000 K: Internally consistent pressure scales, High Pressure Res.,31(4), 560–580, doi:10.1080/08957959.2011.611469.

Katahara, K. W., M. H. Manghnani, and E. S. Fisher (1979), Pressure deri-vatives of the elastic moduli of BCC Ti-V-Cr, Nb-Mo and Ta-W alloys,J. Phys. F Metal Phys., 9(5), 773–790, doi:10.1088/0305-4608/9/5/006.

Kavner, A., and T. S. Duffy (2003), Elasticity and rheology of platinumunder high pressure and nonhydrostatic stress, Phys. Rev. B, 68(14),144101, doi:10.1103/PhysRevB.68.144101.

Kennett, B., and I. Jackson (2009), Optimal equations of state for mantleminerals from simultaneous non-linear inversion of multiple datasets,Phys. Earth Planet. Inter., 176(1–2), 98–108, doi:10.1016/j.pepi.2009.04.005.

Klotz, S., J. Chervin, P. Munsch, and G. L. Marchand (2009), Hydrostaticlimits of 11 pressure transmitting media, J. Phys. D Appl. Phys., 42(7),075413, doi:10.1088/0022-3727/42/7/075413.

Kono, Y., T. Irifune, Y. Higo, T. Inoue, and A. Barnhoorn (2010), PVTrelation of MgO derived by simultaneous elastic wave velocity and in situX-ray measurements: A new pressure scale for the mantle transitionregion, Phys. Earth Planet. Inter., 183(1–2), 196–211, doi:10.1016/j.pepi.2010.03.010.

Li, B., K. Woody, and J. Kung (2006), Elasticity of MgO to 11 GPa with anindependent absolute pressure scale: Implications for pressure calibration,J. Geophys. Res., 111, B11206, doi:10.1029/2005JB004251.


14 of 15

Liu, W., Q. Liu, M. L. Whitaker, Y. Zhao, and B. Li (2009), Experimentaland theoretical studies on the elasticity of molybdenum to 12 GPa,J. Appl. Phys., 106(4), 043506, doi:10.1063/1.3197135.

Mao, H.-K., R. J. Hemley, Y. Wu, A. P. Jephcoat, L. W. Finger, C.-S. Zha,and W. A. Bassett (1988), High-pressure phase diagram and equation ofstate of solid helium from single-crystal X-ray diffraction to 23.3 GPa,Phys. Rev. Lett., 60(25), 2649, doi:10.1103/PhysRevLett.60.2649.

Marsh, S. P. (Ed.) (1980), LASL Shock Hugoniot Data, Univ. Calif. Press,Berkeley.

Matsui, M. (2009), Temperature-pressure-volume equation of state of theB1 phase of sodium chloride, Phys. Earth Planet. Inter., 174(1–4),93–97, doi:10.1016/j.pepi.2008.05.013.

Matsui, M., E. Ito, T. Katsura, D. Yamazaki, T. Yoshino, A. Yokoyama,and K. Funakoshi (2009), The temperature-pressure-volume equation ofstate of platinum, J. Appl. Phys., 105(1), 013505, doi:10.1063/1.3054331.

Murakami, M., K. Hirose, K. Kawamura, N. Sata, and Y. Ohishi(2004), Post-perovskite phase transition in MgSiO3, Science, 304(5672),855–858, doi:10.1126/science.1095932.

Nellis, W. J. (1997), Dynamic high pressure effects in solids, in Encyclope-dia of Applied Physics, vol. 18, pp. 541–554, John Wiley, Hoboken, N. J.

Nellis, W. J. (2007), Adiabat-reduced isotherms at 100 GPa pressures, HighPressure Res., 27(4), 393–407, doi:10.1080/08957950701659734.

Ono, S. (2010), The equation of state of B2-type NaCl, J. Phys. Conf. Ser.,215, 012196, doi:10.1088/1742-6596/215/1/012196.

Ono, S., T. Kikegawa, and Y. Ohishi (2006a), Equation of state of CaIrO3-type MgSiO3 up to 144 GPa, Am. Mineral., 91(2–3), 475–478,doi:10.2138/am.2006.2118.

Ono, S., T. Kikegawa, and Y. Ohishi (2006b), Structural property of CsCl-type sodium chloride under pressure, Solid State Commun., 137(10),517–521, doi:10.1016/j.ssc.2006.01.022.

Ono, S., J. P. Brodholt, and G. David Price (2011), Elastic, thermal andstructural properties of platinum, J. Phys. Chem. Solids, 72(3), 169–175,doi:16/j.jpcs.2010.12.004.

Rivers, M., V. Prakapenka, A. Kubo, C. Pullins, C. M. Holl, and S. D.Jacobsen (2008), The COMPRES/GSECARS gas-loading system fordiamond anvil cells at the Advanced Photon Source, High PressureRes., 28(3), 273–292, doi:10.1080/08957950802333593.

Rydberg, R. (1932), Graphische Darstellung einiger bandenspektroskopischerErgebnisse, Z. Phys., 73(5–6), 376–385, doi:10.1007/BF01341146.

Sakai, T., E. Ohtani, N. Hirao, and Y. Ohishi (2011), Equation of state ofthe NaCl-B2 phase up to 304 GPa, J. Appl. Phys., 109(8), 084912,doi:10.1063/1.3573393.

Sata, N., G. Shen, M. L. Rivers, and S. R. Sutton (2002), Pressure-volumeequation of state of the high-pressure B2 phase of NaCl, Phys. Rev. B,65(10), 104114, doi:10.1103/PhysRevB.65.104114.

Sata, N., K. Hirose, G. Shen, Y. Nakajima, Y. Ohishi, and N. Hirao (2010),Compression of FeSi, Fe3C, Fe095O, and FeS under the core pressuresand implication for light element in the Earth’s core, J. Geophys. Res.,115, B09204, doi:10.1029/2009JB006975.

Shim, S.-H., T. S. Duffy, and K. Takemura (2002), Equation of state of goldand its application to the phase boundaries near 660 km depth in Earth’smantle, Earth Planet. Sci. Lett., 203(2), 729–739, doi:10.1016/S0012-821X(02)00917-2.

Singh, A. K. (1993), The lattice strains in a specimen (cubic system)compressed nonhydrostatically in an opposed anvil device, J. Appl. Phys.,73(9), 4278–4286, doi:10.1063/1.352809.

Singh, A. K. (2009), Analysis of nonhydrostatic high-pressure diffractiondata (cubic system): Assessment of various assumptions in the theory,J. Appl. Phys., 106(4), 043514, doi:10.1063/1.3197213.

Singh, A. K., and K. Takemura (2001), Measurement and analysis ofnonhydrostatic lattice strain component in niobium to 145 GPa undervarious fluid pressure-transmitting media, J. Appl. Phys., 90(7), 3269,doi:10.1063/1.1397283.

Singh, A. K., H. Liermann, Y. Akahama, S. K. Saxena, and E. Menendez-Proupin (2008), Strength of polycrystalline coarse-grained platinum to330 GPa and of nanocrystalline platinum to 70 GPa from high-pressureX-ray diffraction data, J. Appl. Phys., 103(6), 063524, doi:10.1063/1.2891424.

Sinogeikin, S. V., and J. D. Bass (1999), Single-crystal elasticity of MgO athigh pressure, Phys. Rev. B, 59(22), R14141, doi:10.1103/PhysRevB.59.R14141.

Sinogeikin, S. V., and J. D. Bass (2000), Single-crystal elasticity of pyropeand MgO to 20 GPa by Brillouin scattering in the diamond cell, Phys.Earth Planet. Inter., 120(1–2), 43–62, doi:10.1016/S0031-9201(00)00143-6.

Song, M., A. Yoneda, and E. Ito (2007), Ultrasonic measurements of single-crystal gold under hydrostatic pressures up to 8 GPa in a Kawai-typemulti-anvil apparatus, Chin. Sci. Bull., 52(12), 1600–1606, doi:10.1007/s11434-007-0240-y.

Speziale, S., C.-S. Zha, T. S. Duffy, R. J. Hemley, and H.-K. Mao (2001),Quasi-hydrostatic compression of magnesium oxide to 52 GPa: Implica-tions for the pressure-volume-temperature equation of state, J. Geophys.Res., 106(B1), 515–528, doi:10.1029/2000JB900318.

Stacey, F. D., B. J. Brennan, and R. D. Irvine (1981), Finite strain theoriesand comparisons with seismological data, Geophys. Surv., 4(3), 189–232,doi:10.1007/BF01449185.

Sun, T., K. Umemoto, Z. Wu, J. Zheng, and R. M. Wentzcovitch (2008),Lattice dynamics and thermal equation of state of platinum, Phys. Rev.B, 78(2), 024304, doi:10.1103/PhysRevB.78.024304.

Sung, C., C. Goetze, and H.-K. Mao (1977), Pressure distribution in thediamond anvil press and the shear strenght of fayalite, Rev. Sci. Instrum.,48(11), 1386–1391, doi:10.1063/1.1134902.

Takemura, K. (2001), Evaluation of the hydrostaticity of a helium-pressuremedium with powder X-ray diffraction techniques, J. Appl. Phys., 89(1),662–668, doi:10.1063/1.1328410.

Takemura, K., and A. Dewaele (2008), Isothermal equation of state for goldwith a He-pressure medium, Phys. Rev. B, 78(10), 104119, doi:10.1103/PhysRevB.78.104119.

Takemura, K., T. Watanuki, K. Ohwada, A. Machida, A. Ohmura, andK. Aoki (2010), Powder X-ray diffraction study of Ne up to 240 GPa,J. Phys. Conf. Ser., 215, 012017, doi:10.1088/1742-6596/215/1/012017.

Tange, Y., Y. Nishihara, and T. Tsuchiya (2009), Unified analyses forP-V-T equation of state of MgO: A solution for pressure-scale pro-blems in high P-T experiments, J. Geophys. Res., 114, B03208,doi:10.1029/2008JB005813.

Tateno, S., K. Hirose, Y. Ohishi, and Y. Tatsumi (2010), The structure ofiron in Earth’s inner core, Science, 330(6002), 359–361, doi:10.1126/science.1194662.

Tsuchiya, T. (2003), First-principles prediction of the P-V-T equation ofstate of gold and the 660-km discontinuity in Earth’s mantle, J. Geophys.Res., 108(B10), 2462, doi:10.1029/2003JB002446.

Tsuchiya, T., and K. Kawamura (2002a), Ab initio study of pressure effecton elastic properties of crystalline Au, J. Chem. Phys., 116(5), 2121,doi:10.1063/1.1429643.

Tsuchiya, T., and K. Kawamura (2002b), First-principles study of sys-tematics of high-pressure elasticity in rare gas solids, Ne, Ar, Kr, andXe, J. Chem. Phys., 117(12), 5859, doi:10.1063/1.1502241.

Ueda, Y., M. Matsui, A. Yokoyama, Y. Tange, and K. Funakoshi (2008),Temperature-pressure-volume equation of state of the B2 phase of sodiumchloride, J. Appl. Phys., 103(11), 113513, doi:10.1063/1.3054331.

Vinet, P., J. Ferrante, J. R. Smith, and J. H. Rose (1986), A universal equa-tion of state for solids, J. Phys. C Solid State Phys., 19(20), L467–L473,doi:10.1088/0022-3719/19/20/001.

Vinet, P., J. Ferrante, J. H. Rose, and J. R. Smith (1987), Compressibilityof solids, J. Geophys. Res., 92(B9), 9319–9325, doi:10.1029/JB092iB09p09319.

Weidner, D. J., Y. Wang, Y. Meng, and M. T. Vaughan (1994), Deviatoricstress measurements at high pressure and temperature, AIP Conf. Proc.,309(1), 1025–1028, doi:10.1063/1.46284.

Wentzcovitch, R. M., Y. G. Yu, and Z. Wu (2010), Thermodynamic proper-ties and phase relations in mantle minerals investigated by first principlesquasiharmonic theory, Rev. Mineral. Geochem., 71, 59–98, doi:10.2138/rmg.2010.71.4.

Wu, Z., R. M. Wentzcovitch, K. Umemoto, B. Li, K. Hirose, and J. C.Zheng (2008), Pressure-volume-temperature relations in MgO: An ultra-high pressure-temperature scale for planetary sciences applications,J. Geophys. Res., 113, B06204, doi:10.1029/2007JB005275.

Yokoo, M., N. Kawai, K. G. Nakamura, and K. Kondo (2008), Hugoniotmeasurement of gold at high pressures of up to 580 GPa, Appl. Phys.Lett., 92(5), 051901, doi:10.1063/1.2840189.

Yokoo, M., N. Kawai, K. G. Nakamura, K. Kondo, Y. Tange, and T.Tsuchiya (2009), Ultrahigh-pressure scales for gold and platinum atpressures up to 550 GPa, Phys. Rev. B, 80(10), 104114, doi:10.1103/PhysRevB.80.104114.

Zeng, Z.-Y., C.-E. Hu, L.-C. Cai, X.-R. Chen, and F.-Q. Jing (2010), Latticedynamics and thermodynamics of molybdenum from first-principles cal-culations, J. Phys. Chem. B, 114(1), 298–310, doi:10.1021/jp9073637.

Zha, C.-S., H.-K. Mao, and R. J. Hemley (2000), Elasticity of MgO anda primary pressure scale to 55 GPa, Proc. Natl. Acad. Sci. U. S. A.,97(25), 13,494–13,499, doi:10.1073/pnas.240466697.

Zha, C.-S., K. Mibe, W. A. Bassett, O. Tschauner, H.-K. Mao, and R. J.Hemley (2008), P-V-T equation of state of platinum to 80 GPa and1900 K from internal resistive heating/X-ray diffraction measurements,J. Appl. Phys., 103(5), 054908, doi:10.1063/1.2844358.

Zhao, Y., A. C. Lawson, J. Zhang, B. I. Bennett, and R. B. V. Dreele(2000), Thermoelastic equation of state of molybdenum, Phys. Rev. B,62(13), 8766–8776, doi:10.1103/PhysRevB.62.8766.


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Correction to “Intercomparison of pressure standards(Au, Pt, Mo, MgO, NaCl and Ne) to 2.5 Mbar”

S. M. Dorfman, V. B. Prakapenka, Y. Meng, and T. S. Duffy

Received 27 September 2012; published 14 November 2012.

Citation: Dorfman, S. M., V. B. Prakapenka, Y. Meng, and T. S. Duffy (2012), Correction to “Intercomparison of pressurestandards (Au, Pt, Mo, MgO, NaCl and Ne) to 2.5 Mbar,” J. Geophys. Res., 117, B11204, doi:10.1029/2012JB009800.

[1] In the paper “Intercomparison of pressure standards(Au, Pt, Mo, MgO, NaCl and Ne) to 2.5 Mbar” by S. M.Dorfman et al. (Journal of Geophysical Research, 117,B08210, doi:10.1029/2012JB009292, 2012), the caption

of Figure S3 in the auxiliary material was incorrect. Thecorrect caption, which appears correctly in the readme file, is“Differential stress measured in Ne and Ar compared tomeasurements of Pt and Au samples.”

©2012. American Geophysical Union. All Rights Reserved.0148-0227/12/2012JB009800

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, B11204, doi:10.1029/2012JB009800, 2012

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Intercomparison of pressure standards (Au, Pt, Mo, MgO, NaCl …duffy.princeton.edu/sites/default/files/pdfs/2012/... · 2015-12-29 · where P is pressure, K 0 is the isothermal