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Physics of the Earth and Planetary Interiors 208-209 (2012) 1–10

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X-ray diffraction from stishovite under nonhydrostatic compression to 70 GPa: Strength and elasticity across the tetragonal ? orthorhombic transition Anil K. Singh a,⇑, Denis Andrault b, Pierre Bouvier c,d a

Materials Science Division, National Aerospace Laboratories, Council of Scientific and Industrial Research (CSIR), Bangalore, India Laboratoire Magmas et Volcans, Université B. Pascal, Clermont-Ferrand, France c European Synchrotron Research Facility, 6 Rue Jules Horowitz, F-38043 Grenoble Cedex, France d Laboratoire des Materiaux et du Genie Physique, CNRS, Grenoble Institute of Technology, 3 parvis Louis Neel, F-38016 Grenoble, France b

a r t i c l e

i n f o

Article history: Received 27 November 2011 Received in revised form 28 June 2012 Accepted 2 July 2012 Available online 10 July 2012 Edited by Kei Hirose Keywords: Stishovite Nonhydrostatic compression High pressure Phase transition Elastic moduli Compressive strength

a b s t r a c t The tetragonal phase of silica (stishovite) was synthesized at high pressure and temperature in a laserheated diamond anvil cell. Nonhydrostatic pressure condition was produced by pressurizing the sample without any pressure transmitting medium. The tetragonal?orthorhombic transition could be detected from the X-ray diffraction patterns at 40 GPa. In contrast, the orthorhombic phase has been reported to occur only above 60 GPa in an earlier experiments under hydrostatic pressure. However, the transition pressures derived from the square of the symmetry-breaking strain versus pressure data in the two cases differ only marginally, the values being 44(8) GPa and 49(2) GPa under nonhydrostatic and hydrostatic compressions, respectively. We combine the d-spacings measured under nonhydrostatic and hydrostatic compressions to derive a parameter Q(hkl) that contains the information on differential stress t (a measure of compressive strength) and single-crystal elasticity. The compressive strengths derived from the average value of Q(hkl) and line-width analysis agree well. It increases from 4 GPa at 20 GPa to 8 GPa at 40 GPa and decreases as the transition pressure is approached. In the orthorhombic phase, t increases with pressure monotonically. The mean crystallite size of the sample decreases from 5000 Å to 1000 Å as the pressure is increased from 20 GPa to 45 GPa and remains nearly unchanged between 45 GPa and 70 GPa. The single-crystal elastic moduli derived from the X-ray diffraction data indicate that ðC 11  C 12 Þ decreases rapidly as the transition pressure is approached. Line-width analysis of the diffraction lines suggests that near-hydrostatic pressure condition is achieved by laser annealing of the compressed sample. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Tetragonal phase of silica with octahedral coordination (stishovite) has been the subject of many studies because of its importance in several disciplines of science (Hemley et al., 1994; Ross et al., 1990). Stishovite has been found to occur naturally in meteorites (Goresy et al., 2000). A significantly higher SiO2 content in the mid-ocean ridge basalts compared to the mid-mantle suggests that SiO2 could be present as a free phase in the subducted slabs (Hirose et al., 2005; Guignot and Andrault, 2004). Stishovite is also observed in diamond as inclusions, confirming its importance in deep Earth processes (Joswig et al., 1999). The six-fold coordination of silicon results in extremely high strength, the highest among the oxides (Leger et al., 1996). Synthesis of this phase in the laboratory is achieved by subjecting silica to high pressure and temperature. Stishovite at ambient ⇑ Corresponding author. Tel.: +91 80 2508 6299; fax: +91 80 2527 0098. E-mail addresses: [email protected], [email protected] (A.K. Singh). 0031-9201/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.pepi.2012.07.003

temperature and high pressure exhibits a structural transition to orthorhombic phase with CaCl2 structure (Tsuchida et al., 1989). Several investigators have studied the equation of state (EoS) using diamond anvil cell (DAC) to pressurize the sample and X-ray diffraction to measure the volume change. Ross et al. (1990) reported EoS up to 15 GPa by single-crystal X-ray diffraction. The pressure was rendered hydrostatic by neon or ethanol–methanol pressure medium. Hemley et al. (2000) used hydrogen as the pressure medium and measured EoS to 70 GPa. Andrault et al. (1998) used laserannealing to produce near-hydrostatic pressures and proposed EoS to 120 GPa. Andrault et al. (2003) carried out another set of measurements with NaCl as a pressure transmitting medium combined with laser annealing to obtain improved hydrostaticity of pressure below 50 GPa and laser annealing without pressure medium above 50 GPa. These measurements used more reliable Pt EoS for the pressure estimation. The pressure-volume data proposed by different investigators are in reasonable agreement in the overlapping pressure range. We use in this analysis the data obtained by Andrault et al. (2003) as the reference.

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The spectroscopic studies (Kingma et al., 1995) suggest that the shear moduli ðC 11  C 12 Þ decreases with increasing pressure and vanishes at the transition pressure (50 GPa). This transition shows the characteristics of second order transition. Andrault et al. (1998) showed that the a axis of the tetragonal phase splits into the a and b axes of the orthorhombic phase above the transition pressure. These data show that the symmetry-breaking strains are large and the nonsymmetry-breaking strains small. Obviously, the pressure dependence of strength and elasticity of silica across the transition is of great interest. Only limited studies on direct measurement of strength and elastic moduli of stishovite are available (Cordier and Rubie, 2001; Li et al., 1996). Several investigators (Cohen, 1992;Lee and Gonze, 1995; Karki et al., 1997) carried out the first-principles calculation of elastic moduli over a wide pressure range and found that C 11  C 12 vanishes at the transition pressure. Carpenter et al. (2000, 2006) used the compressibility data of Andrault et al. (1998) and evaluated the constants in the Landau free energy expansion. This allowed the evaluation of the singlecrystal elastic moduli as functions of pressure in both the tetragonal as well as orthorhombic phases. Shieh et al. (2002) analyzed the high-pressure diffraction data on stishovite recorded using the radial diffraction geometry and derived the strength and elastic moduli using the lattice-strain equations (Singh et al., 1998a,b). These data (Shieh et al., 2002) also suggest that the shear modulus ðC 11  C 12 Þ decreases with pressure and tends to vanish at the transition pressure. We recorded X-ray diffraction patterns from stishovite under nonhydrostatic compression using conventional diffraction geometry wherein the primary X-ray beam passes parallel to the load axis of the DAC. The strength and elastic moduli cannot be derived using these data alone. However, by combining the d-spacings measured under nonhydrostatic and hydrostatic compressions, we could determine the strength of silica in both tetragonal and orthorhombic phases, and the single-crystal elastic moduli of the tetragonal phase. We also carried out line-width analysis to determine strength and crystallite size as a function of pressure.

(Hammersley, 1996). A four parameter pseudo-Voigt function (Sànchez-Bajo and Cumbrera, 1997) with a linear background term was fitted to each peak, and peak position and full width at half maximum (FWHM) determined. The overlapping peaks could be resolved by fitting a linear combination on two, three, or four pseudo-Voigt functions. To study the effect of temperature annealing on the magnitude of nonhydrostatic stresses, the sample in Run# SiO2_01 was compressed to 72 GPa and then annealed for 600 s by laser heating. The sample was subjected to four further annealing operations of 600 s duration each at increasing laser powers. The diffraction patterns were recorded after each annealing step. The average of unit cell dimensions computed from the 111, 200, and 220 lines from Pt (or Au) was used to compute the volume compression Vð0Þ=V, where Vð0Þ and V are the unit cell volumes at ambient and high pressure, respectively. The ambient pressure unit cell parameters of Pt and Au were taken as 3.9213 Å (JCPDS 040802) and 4.0786 Å (JCDPS 040784), respectively. Pressure was computed using measured volume compressions of Pt in the equation of state (EoS) proposed by Holmes et al. (1989). In case of Au pressure marker, the EoS proposed by Holzapfel (1991), Holzapfel (1996) with K 0 ¼ 166:5 GPa and K 00 ¼ 5:739 was used to compute pressures. The value of K 00 was fixed by using the procedure suggested by Singh (2007). It may be noted that the pressures thus computed and those computed from other EoSs for Au (Anderson et al., 1989;Takemura and Dewaele, 2008) are close. The difference is 0.4% at 10 GPa and increases to 2% at 70 GPa. These differences are within the uncertainties in the determination of pressures. It may be noted that the compression at ambient temperature introduces nonhydrostatic stress field in the sample. It was assumed that the marker pressure equaled the mean normal stress of the sample. The pressures at different locations in the cell were equal within the uncertainty of the pressure measurement. This indicated the absence of large pressure gradients across the compressed stishovite samples. 3. Lattice strain theory 3.1. General

2. Experimental details Stishovite samples were synthesized from quartz at pressures in the range 10–20 GPa and 2000 K in a laser-heated DAC mounted with 250 lm culet diamonds. Re-gaskets were pre-indented to a thickness of 40 lm, before a 80 lm hole was laser drilled at its center. For experiments in Run# SiO2_01, stishovite synthesis was carried out by heating the mixture of quartz and 1 wt.% of Pt powder at 16 GPa and 2000 K for 15 min. Pt was used as the pressure marker. In Run# SiO2_02, stishovite was synthesized by heating a mixture of quartz and Au (1 wt.%) at 10 GPa and 2000 K for 10 min. Au acted as pressure marker in these runs. The X-ray diffraction experiments were carried out at the ID27 beamline of the European Synchrotron Radiation Facility (ESRF) using angle-dispersive diffraction mode (Andrault and Fiquet, 2001; Mezouar et al., 2005; Schultz et al., 2005). A channel-cut and water-cooled monochromator was set to produce monochromatic X-ray beam (wavelength 0.3738 Å). A bent silicon-crystal mirror was used to focus the beam. The full width at half maximum of the X-ray beam was less than 5  5 lm on the sample. To ensure nonhydrostatic stress condition the stishovite samples were compressed without any pressure transmitting medium in steps from 16 to 72 GPa for Run# SiO2_01, and from 10 to 66 GPa for Run# SiO2_02. After each compression, diffraction patterns were recorded at five different locations in the sample, one being at the center of the hole containing the sample. The two-dimensional diffraction images were recorded on an image plate and converted to intensity-2h data using the Fit2d code

The nonhydrostatic stress state at center of the sample compressed in a diamond anvil cell (DAC) can be described with reference to a set of three orthogonal axes: x and y axes in the plane of the anvil face, and z-axis parallel to the load direction. The stress component along the z-axis is denoted by r3 . Since the stresses are cylindrically symmetry along the z-axis, the stress components r1 and r2 along x- and y-axis are equal. The off-diagonal terms in the stress tensor are taken to be zero. Singh et al. (1998a,b) showed that the d-spacings measured on a polycrystalline sample under nonhydrostatic compression is given for all crystal systems by the following expression

dm ðhklÞ ¼ dP ðhklÞ½1 þ ð1  3 cos2 WÞQðhklÞ:

ð1Þ

here dP ðhklÞ is the d-spacing under the mean normal stress rP ¼ ð2r1 þ r3 Þ=3; W is the angle between the diffraction vector and load axis of the DAC, and Q ðhklÞ is given by,

QðhklÞ ¼

t X ½G ðhklÞ1 : 6

ð2Þ

here, t ¼ ðr3  r1 Þ. The term GX ðhklÞ denotes the diffraction shear modulus for the set of planes ðhklÞ and is expressed by the relation:

o h i1 n GX ðhklÞ ¼ a½GXR ðhklÞ1 þ ð1  aÞ½GðVÞ1

ð3aÞ

On substituting for GX ðhklÞ from Eq. (3a) in Eq. (2) we get the familiar equation (Singh et al., 1998a,b),

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A.K. Singh et al. / Physics of the Earth and Planetary Interiors 208-209 (2012) 1–10

Q ðhklÞ ¼

o tn a½GXR ðhklÞ1 þ ð1  aÞGðVÞ1 6

ð3bÞ

The parameter a determines relative weights of the two types of shear moduli. GXR ðhklÞ is the diffraction shear modulus calculated under the assumption of stress continuity across the grain boundaries, the averaging being done only over the crystallites that contribute to the diffracted intensity at the point of observation. GðVÞ is the shear modulus of the aggregate under Voigt limit (Voigt, 1928). A detailed discussion of the various shear moduli used in the theory is given by Singh (2009). The value of a ranging between 0.5 and 1 has been used in the literature. It is often assumed that a ffi 1 is valid at high stresses. However, recent studies (Chen et al., 2006; Singh, 2009; Singh and Liermann, 2011) point out that a can take values greater than 1. As noted in the preceding section, compressed stishovite samples in this study are free from gradients and the complications arising thereof are not relevant to this study. The aggregate elastic properties for the tetragonal system are given by,

1 ½ðC 11  C 12 Þ þ ðC 11 þ C 33  2C 13 Þ þ 3ð2C 44 þ C 66 Þ ð4aÞ 15 1 KðVÞ ¼ ð2C 11 þ C 33 þ 2C 12 þ 4C 13 Þ ð4bÞ 9 1 ½GðRÞ1 ¼ ½4ðS11  S12 Þ þ 4ðS11 þ S33  2S13 Þ þ 3ð2S44 þ S66 Þð4cÞ 15 1 ½KðRÞ ¼ 2S11 þ S33 þ 2S12 þ 4S13 ð4dÞ

GðVÞ ¼

The diffraction shear modulus under Reuss limit is given by,

h

GXR ðhklÞ

i1

    2 4 2 2 2 2 2 ¼ S11 2  5l3 þ 3l3  6l1 l2  S12 1  l3  6l1 l2     2 4 2 4  S13 1  5l3 þ 6l3  S33 l3  3l3   2 4 2 2 þ 3S44 l3  l3 þ 3S66 l1 l2 2 2

2

½dm ðhklÞ  dP ðhklÞ ð1  3 cos2 WÞdP ðhklÞ

ð4eÞ 2

ð5Þ

Eq. (5) can be easily obtained by rearranging the terms in Eq. (1). This equation requires the d-spacings measured under both nonhydrostatic and hydrostatic pressures for the estimation of Q ðhklÞ. It may be noted that W ¼ 90  h for the conventional geometry. Eq. (5) can also be used if the d-spacings are measured at any W using radial diffraction pffiffiffi geometry. However, Eq. (5) is not valid for W ¼ cos1 ð1= 3Þ because, for this value of W; Q ðhklÞ term vanishes in Eq. (1) from which Eq. (5) is derived. 3.2. Estimation t Singh et al. (1998a,b) derived the following relation that has been used extensively in estimating the value of t

t ¼ 6GhQðhklÞif

ð7Þ

It is also possible to estimate t from the unit cell volumes V m and V P measured under nonhydrostatic and hydrostatic pressures, respectively. Singh and Balasingh (1994) derived the following relation that connects t with V m and V P :

t¼

2GðV m  V P Þ V P hð1  3 cos2 WÞi

ð8aÞ

The term hQ ðhklÞi can also be obtained by combining Eqs. (7) and (8a). This gives

hQ ðhklÞi ¼

ðV m  V P Þ 3V P hð1  3 cos2 WÞi

ð6Þ

Here, G is the shear modulus of isotropic aggregate and hQ ðhklÞi denotes the average of all measured Q ðhklÞ. The term f ffi 1. Thus,

ð8bÞ

pffiffiffiffiffi The Eqs. (8a) and (8b) are not valid if W ¼ cos1 ð1= 3Þ for the reasons discussed after Eq. (5). The maximum value of t that the sample material can support equals the compressive strength at a pressure rP . 3.3. Estimation of C ij In principle, all the 6 elastic moduli of stishovite can be estimated if Q ðhklÞ for six independent reflections are measured. The term t that appears in the expression for Q ðhklÞ can be determined from Eqs. (7), (8a) or (8b). However, the value of a cannot be determined independently and an assumed value has to be used. Shieh et al., 2002 assumed a ¼ 1 while interpreting the diffraction data on stishovite. If the compression data under hydrostatic pressure are used then additional relations become available that contain terms in Sij . One such relation is

d lnðc=aÞ=dP ¼ S11 þ S12  S33  S13

Here, l1 ¼ ch=M; l2 ¼ ck=M; l3 ¼ al=M, and M ¼ ½a l þ c2 ðh þ k Þ1=2 . The terms a and c are the unit cell parameters of the tetragonal cell and hkl are the indices of the reflections. The aggregate shear and bulk moduli are denoted by G and K, respectively. The terms V and R denote Voigt (Voigt, 1928) and Resus (Reuss, 1929) limits, respectively. The term Q ðhklÞ can be directly measured by radial diffraction geometry (Singh, 1994; Mao and Hemley, 1996; Merkel and Yagi, 2005). However, for the diffraction patterns taken with the conventional diffraction geometry wherein the incident X-ray beam is parallel to the load axis, Q ðhklÞ can be estimated from the following relation,

Q ðhklÞ ¼

t ffi 6GhQ ðhklÞi

3

ð9Þ

Another relation is provided by the compressibility. Since the measurements are made on polycrystalline samples, it is essential to consider the two limiting stress-strain states. The compressibility (1=KÞ of the aggregate under the Voigt and Reuss limits for the tetragonal system can be obtained from Eqs. (4b) and (4d), respectively. The measured K under hydrostatic pressure corresponds to:

K ¼ ½KðVÞ þ KðRÞ=2:

ð10Þ

Using the single-crystal elastic moduli of stishovite (Carpenter et al., 2000, 2006) it can be shown that KðVÞ is only 5% higher than KðRÞ at ambient pressure and the difference between the two decreases with increasing pressure. At the tetragonal?orthorhombic transition pressure, the difference is 0.5%. Since Q ðhklÞ is expressed in terms of Sij , it is convenient to use in this study K = K(R), with K given by Eq. (4d). 3.4. Line-width analysis The increase in the diffraction-line widths under nonhydrostatic compression has been used to estimate the micro-stains and the grain size of crystalline solids. The strain and size broadenings for the angle-dispersive diffraction data are related by the equation (Langford, 1971;Stokes and Wilson, 1944)

½ð2whkl cos hhkl 2 ¼

k2

þ D2

 2 4pmax 2 sin hhkl Ehkl

ð11Þ

here, 2whkl is the full width at half maximum on 2h-scale of the reflection hkl; D is the average crystallite size (apparent size), Ehkl is the single-crystal Young’s modulus in the direction ½hkl, and k is the wavelength. It is assumed in the derivation of Eq. (11) that all stresses between zero and the maximum stress pmax occur in the crystallites with equal probability (Langford, 1971;Stokes and Wilson, 1944). Singh (2004) and Singh et al. (2004) extended this model to interpret the X-ray diffraction line-widths data taken

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A.K. Singh et al. / Physics of the Earth and Planetary Interiors 208-209 (2012) 1–10

4.4

Unit cell parameters (Ao)

Intensity (arb. scale)

2000 14.9 GPa 51.8 GPa 1600

B

1200

C

A

D

800

6

8

10

12

14

16

18

(a)

4.2

b

4.0

a

this study

3.8

hydrostatic 2.6

c

0.20

20

2θ (degree)

(b - a) Ao

0.15

Fig. 1. Diffraction patterns at 14.9 and 51.8 GPa of the sample mixed with Pt. Shift of the lines towards higher 2h due to pressure, and increased line broadening due to nonhydrostatic compression are seen in the patterns at 51.8 GPa. The stishovite sample at 14.9 GPa is free from nonhydrostatic compression effects as this pressure is close to the synthesis pressure and high temperature during the synthesis relaxes the nonhydrostatic stresses. The peaks in the regions marked A, B, C, and D on the 51.8 GPa plot show splitting of the peaks due to the tetragonal?orthorhombic transition (see Fig. 2 for details).

0.10

0.05

(b)

0.00 1040

50

1200 1100

210

1000 880 900

(a)

(b)

12.0

12.5 14.0

800 15.0 1120

14.5

031 002

1080

301 1080

130

Pt (220)

The single-crystal Young’s modulus for the tetragonal system (Nye, 1960) required in Eq. (11) is given by

1 4 4 4 2 2 2 2 ¼ ðl1 þ l2 ÞS11 þ l3 S33 þ l1 l2 ð2S12 þ S66 Þ þ l3 ð1  l3 Þð2S13 þ S44 Þ Ehkl ð12Þ and for the orthorhombic system by

1 4 2 2 2 2 4 2 2 4 ¼ l1 S11 þ 2l1 l2 S12 þ 2l1 l3 S13 þ l2 S22 þ 2l2 l3 S23 þ l3 S33 Ehkl 2 2

2 2

2 2

For orthorhombic system (Singh et al., 1998a), l1 ¼ hdðhklÞ=a; l2 ¼ kdðhklÞ=b, and l3 ¼ ldðhklÞ=c.

1000

4. Results and discussions

(d)

(c)

ð13Þ

1040

310

1000

100

Fig. 3. (a) The lattice parameters a; b, and c as functions of pressure. (b) The difference ðb  aÞ (filled circles) increases gradually (dashed line) with increasing pressure up to 45 GPa and steeply (solid line) at higher pressures. The data under hydrostatic compression are shown by unfilled circles.

þ l2 l3 S44 þ l1 l3 S55 þ l1 l2 S66

1040

75

Pressure (GPa)

960

800 11.5 1120

25

211 121

Intensity (arb. unit)

0

1300

120

4.1. Features of the diffraction data

960

960 16.0

16.5

17.0

17.5

18.0

18.5

2θ (degree) Fig. 2. The expanded views of the regions marked a–d in the diffraction pattern at 51.8 GPa (Fig. 1). The splitting of some of the lines in the tetragonal phase suggests onset of the transition to the orthorhombic phase. The separation between (120) and (210) reflections of the orthorhombic phase is barely seen.

under nonhydrostatic compression. Several studies (e.g. Singh et al., 2004, 2006, 2007, 2008) show that 2pmax ffi t. A few studies have used energy-dispersive diffraction data to derive strength (e.g. Chen et al., 2002; He and Duffy, 2006).

In this section we discuss some interesting features exhibited by the diffraction data recorded under nonhydrostatic compression. Andrault et al. (2003) laser-annealed the compressed samples of stishovite before conducting high-pressure X-ray diffraction experiments. The laser-annealing is expected to relax the nonhydrostatic stresses. A quantitative estimate of the nonhydrostatic stress component t in laser-annealed samples presented in Section 4.6.2 suggests that the stress state in such experiments is, indeed, near-hydrostatic. We use the data of Andrault et al. (2003) in this study as the reference data under hydrostatic compression. The diffraction patterns of a mixture of stishovite and Pt at 14.9 and 51.8 GPa are shown in Fig. 1. The expanded views of the selected regions of the pattern at 51.8 GPa are shown in Fig. 2. The (120),

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A.K. Singh et al. / Physics of the Earth and Planetary Interiors 208-209 (2012) 1–10

48

Unit cell volume ( A3 )

X 10

-4

40

-- this study -- hydrostatic

(e1 - e2)

2

30

20

10

this study hydrostatic

46

fit - tetra. cell vol. 44

fit - ortho. cell vol.

42 40

(a)

X 10

-3

(a) 0 16

38 0

(e1+e2) this study

25

50

75

e3 this study

12

this study (1) this study (2) hydrostatic

-PΔV (kJ/mol.)

15

Strain

100

20

8 4

(b) (e1+e2) hydro.

0

10

5

e3 hydro.

(b)

0

-4 40

60

80

100

120

140 50

Pressure

125

-3

Fig. 5. (a) A comparison of the unit cell volume versus pressure in this study with those under hydrostatic pressure. Because of nonhydrostatic pressure effect the cell volume at a given pressure in this study is larger than the corresponding volume under hydrostatic pressure. (b) The strain energy at a given pressure in this study is larger than that under hydrostatic pressure. (1) data with a0 computed using a second order Birch equation and (2) computed using a linear extrapolation.

16

25

X 10

(121), (130), and (031) reflections of the tetragonal phase were found to split into pairs (120, 210), (121, 211), (130, 310), and (031, 301) of the orthorhombic phase. At pressures below 50 GPa these pairs of reflections were not well resolved. To examine the possibility of the orthorhombic phase occurring at pressures below 50 GPa, we derived the cell parameters using both tetragonal and orthorhombic systems. Even at pressures as low as 30 GPa, the orthorhombic cell fitted better than the tetragonal cell. At 40 GPa, the orthorhombic cell gave standard errors in a and c parameters that were nearly one-fifth the corresponding values for the tetragonal cell. It is interesting to compare the start pressure of the tetragonal?orthorhombic transition observed in various studies. The start pressure of a transition is defined as the lowest pressure at which a transition is found to occur experimentally (Singh, 1983). The orthorhombic phase was found to occur at 40 GPa in another study, wherein the pressure does not appear to be hydrostatic (Kingma et al., 1996). However, in the experiments under hydrostatic pressures, Andrault et al. (2003) observed the orthorhombic phase only above 65 GPa whereas Hemley et al. (2000) found this transition above 58 GPa. The start pressure of the tetragonal?orthorhombic transition in silica appears to be sensitive to the presence of nonhydrostatic stresses. From the molecular and lattice dynamics studies, Dubrovinsky and Belonoshko (1996) found that presence of a differential stress of only 1.5–2.5 GPa brings down the transition pressure to 40 GPa from 80 GPa under hydrostatic pressure. The transition pressures found in different experiments indicate similar trend but are less sensitive to the presence of differential stresses than suggested by the theory. The low transition pressure in the experiments under nonhydrostatic pressure bears a close similarity with the transition

100

Pressure (GPa)

120 210 121 211

220 110

14 12

Q (hkl)

Fig. 4. A comparison of the spontaneous strains measured in this study under nonhydrostatic pressure with those under hydrostatic pressure (Andrault et al., 2003). (a) The square of the symmetry breaking strain versus pressure data. (b) Nonsymmetry-breaking strains ðe1 þ e2 Þ and e3 versus pressure data.

75

20

-3

20

X 10

0

15 10 8

10

6

5

4 2

(a) 25

50

(b) 75 0

0 25

50

75

Pressure (GPa) Fig. 6. (a) A comparison of Q ðhklÞ as function of pressure for reflections 110 and 220; (b) for pairs (120, 210) and (121, 211).

pressures in the Al-bearing silica. Lakshtanov et al. (2007) found 24 GPa for the transition pressure in hydrous Al(5%)-bearing silica. Bolfan-Casanovaa et al. (2009) found the transition pressure to be 23 GPa in dry Al(4%)-bearing silica. It has been suggested that the low transition pressure of Al-bearing stishovite is caused by chemical pressure that destabilizes the tetragonal phase. Taking 46.51 Å3 and 47.12 Å3 as the unit cell volumes of the pure and Al(4%)-bearing stishovite, the volume strain in the latter is 0.013, which corresponds to a chemical pressure of 4 GPa. This should

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Here, a; b, and c are the lattice parameters of the orthorhombic cell, and the subscript 0 denotes the lattice parameters of the tetragonal cell extrapolated into the stability field of the orthorhombic phase. The symmetry-breaking and nonsymmetry-breaking strains are defined as

12 from Eq. (5)

X 10

-3

6

from Eq.(8b) 8

4

(a) 0 25 from line-width analysis from Eq. (7) from Eq. (8a) Shieh et al (2002)

t (GPa)

20 15 10 5 0

(b) 0

20

40

60

80

Pressure (GPa) Fig. 7. (a) The hQ ðhklÞi versus pressure plot. (b) The t versus pressure data. A large decrease in t is seen at the transition pressure. The solid lines indicate a cubic polynomial fit to the present t  P data. The change of curvature seen at 65 GPa is an artifact due to the choice of the fitted function.

result in an increase and not a decrease in the transition pressure relative to the transition pressure of pure silica. Therefore, the concept of chemical pressure does not seem to explain the low transition pressure of Al-bearing silica. The local strains at the Al3+-sites resulting from the lattice distortion may be an important factor in destabilizing the tetragonal phase. The lattice parameters as a function of pressure are shown in Fig. 3a. At any given pressure above 18 GPa, the parameters a and b in the present experiments are larger than those under hydrostatic compression. This is a typical effect produced by the nonhydrostatic stresses. A few data points below 18 GPa do not show this deviation. It may be noted that these pressures are close to the pressure at which stishovite was synthesized. The high temperature used in the synthesis had relaxed the nonhydrostatic stresses. Therefore, the pressures below 20 GPa are near-hydrostatic. We note from Fig. 3a that the c-parameter is not affected much by the presence of nonhydrostatic stresses. The evolution of ðb  aÞ with increasing pressure is shown in Fig. 3b. The data exhibit two distinct regions. Below 45 GPa, ðb  aÞ increases gradually and rapidly above this pressure. The hydrostatic pressure data, however, show an abrupt increase at the transition pressure. 4.2. Spontaneous strains We follow the notation used by Carpenter et al. (2000, 2006), and define the spontaneous strains e1 ; e2 , and e3 as follows

e1 ¼ ða  a0 Þ=a0

ð14Þ

e2 ¼ ðb  a0 Þ=a0

ð15Þ

e3 ¼ ðc  c0 Þ=c0

ð16Þ

V s ¼ e1 þ e2 þ e3

ð17Þ

esb : ¼ ðe1  e2 Þ ¼ ða  bÞ=a0

ð18Þ

ensb : ¼ ðe1 þ e2 Þ ¼ ða þ b  2a0 Þ=a0

ð19Þ

The values of a0 can be obtained by extrapolating the measured aparameter of the tetragonal cell as a function of pressure with the help of a suitable function. Andrault et al. (2003) reported that the Birch-Murnaghan equation with K ¼ 250:9 GPa and K 00 ¼ 5:48 fitted well to the a3 versus pressure data for the tetragonal phase. The present unit cell data contain nonhydrostatic-pressure effects that cause dilatation of the cell parameters. Since this effect increases with increasing pressure, the intrinsic curvature in the cell parameter versus pressure data tends to become less pronounced. Further, the scatter in the present cell parameter versus pressure data is much larger than in the case of hydrostatic pressure data. These factors make the extrapolation of the data difficult. A Birchlike equation did not fit the a3 versus pressure data any better than a linear equation: a0 ¼ 4:1439ð45Þ  0:00253ð16ÞP. The data below 20 GPa were left out during this fit as these did not contain the effect of nonhydrostatic stresses. We used this linear relation to compute a0 in stability field of the orthorhombic phase. The square of symmetry-breaking strain is largely determined by ða  bÞ and the uncertainty in the determination of a0 has only marginal effect. The e2sb: versus pressure data above 45 GPa fall on a straight line (Fig. 4a). The slope of the line is nearly 30% larger than that for the hydrostatic-pressure data (Andrault et al., 2003). The transition pressure as determined from the intercept of the line on the pressure axis is 44(8) GPa from the present data as compared to 49(2) GPa from the hydrostatic-pressure data. This difference is not significant if the combined error of the two estimates is considered. It must be pointed out that the start pressure of the transition in the present study (nonhydrostatic compression) is much lower than the transition pressure obtained from the e2sb: -P plot. This trend is reversed under hydrostatic compression. The start pressure of the transition is 66 GPa but the transition pressure derived from the e2sb: -P plot is 49 GPa (Andrault et al., 2003). The ðe1 þ e2 Þ and e3 versus pressure data are shown in Fig. 4b. The strains e3 were computed using c0 obtained by linear extrapolation [c0 = 2.6641(15) – 0.00163(9)P] of the c versus pressure data of the tetragonal phase. The large scatter in both nonsymmetry breaking strains arises from large scatter in lattice parameter versus pressure data. These strains are sensitive to the uncertainties in a0 and c0 . The strain ðe1 þ e2 Þ appears to increase with pressure. This trend is similar to that observed under hydrostatic compression. The strain e3 measured under hydrostatic compression also increases with pressure. The present results suggest that the average e3 in the 18–45 GPa range is 0.0001(3). Ideally, this value should be zero. The data in the 50–70 GPa range give an average value of 0.013(4). But for the large scatter, e3 also shows the trend that is seen in the hydrostatic compression data. Eq. (17) suggests that for small strains ðe1 þ e2 þ e3 Þ equals the volume strain associated with the transition. Since both ðe1 þ e2 Þ and e3 are positive, a positive V S is obtained from Eq. (17). Consideration of the unit cell volumes suggests that the orthorhombic phase is more compressible than the tetragonal phase resulting in a negative volume strain of the transition (see the next section). Carpenter (2006) argued that this discrepancy arises because of our inability to obtain a0 and c0 with sufficient precision and suggested that small ðe1 þ e2 Þ and a negative e3 would bring the V S computed from Eq. (17) in agreement with V S obtained from the volume measurement.

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4.3. Strain energy of the transition The volume strain associated with the transition increases with increasing pressure. At any pressure above the transition pressure, the orthorhombic phase shows slightly lower cell volume than V 0 , the volume obtained from the extrapolation of the EoS of the tetragonal phase (Fig. 5a). The data under hydrostatic pressure also exhibit a similar trend (Andrault et al., 2003). Because of the presence of nonhydrostatic stresses the unit cell volume in the present experiments at any given pressure above 20 GPa is larger than that obtained from hydrostatic-pressure data. The effect of the nonhydrostatic compression on the measured unit cell volume increases with increasing pressure. This tends to suppress the intrinsic curvature in the pressure-volume data. Two procedures were used to compute V 0 . In the first procedure, the Birch equation with fixed K 00 ¼ 4 and adjustable zero pressure cell volume Vð0Þ was fitted to the PV data in the range 20–45 GPa. It gave K 0 ¼ 546ð22Þ GPa and Vð0Þ ¼ 45:65ð8Þ. In the second procedure, a linear relation was used [V 0 ¼ 45:46ð7Þ  0:0658ð25ÞP] to compute V 0 . The two sets of estimates of V 0 differ by less than 0.1% at pressures below 45 GPa. However, V 0 computed from the Birch equation is consistently larger than the corresponding values obtained from the linear relation above 45 GPa The difference between the two sets is 0.15% at 50 GPa and increases to 0.8% at 70 GPa. For this reason, the strain energy P DV computed using V 0 from the Birch equation is larger than that computed using V 0 from the linear relation. Fig. 5b shows the strain energies computed from both sets of V 0 . The present estimates of the strain energy are larger than those derived from the hydrostatic-pressure data. We refrain, at this stage, from offering any further explanation for the large PDV under nonhydrostatic compression and present the data merely as a record of the experimental observation. 4.4. Determination of Q ðhklÞ and t The Q ðhklÞ-values were computed from Eq. (5). The dm ðhklÞ used were measured in this study. The standard errors in dm ðhklÞ estimated from the standard errors in 2h were 105  104 Å and appeared to be gross underestimates. In another approach, we used the standard errors in a; b, and c to computed the standard errors in dm ðhklÞ. These errors ranged between 104 and 103 , and were used to estimate the errors in Q ðhklÞ. The dP ðhklÞ were computed from the data of Andrault et al. (2003) by the following procedure. For the tetragonal phase, a third-order Birch-Murnaghan equation was used to calculated aP at the relevant pressures using K 0 ¼ 250:9 GPa and K 00 ¼ 5:48, and cP using a linear relation: cP ¼ 2:6654  0:001581P (Andrault et al., 2003). These aP and cP values were used to compute dP ðhklÞ. For the orthorhombic phase, quadratic equations were fitted to the aP ; bP , and cP versus P data. The following relations were obtained: aP ¼ 8:95  106 P 2  4:91039  103 P þ 4:1929; bP ¼ 1:589  105 P 2  3:6245 103 Pþ 4:1616, and cP ¼ 2:52  106 P2  1:667  103 P þ 2:6636. These equations were used to compute aP ; bP , and cP at any desired pressure, which in turn were used to compute dP ðhklÞ. The data for (110) and (220) reflections are shown in Fig. 6a. Ideally, the Q ðhklÞ-values should be identical for these reflections but differ slightly due to uncertainties in dm ðhklÞ and dP ðhklÞ. Fig. 6b shows the data for the pairs (120, 210) and (121, 211) in the orthorhombic phase. The hQ ðhklÞi in the tetragonal phase increases with increasing pressure and shows a small dip at the transition pressure (Fig. 7a). In the orthorhombic phase, hQ ðhklÞi begins to increase again. At the transition pressure, the present show a distinct drop in the hQ ðhklÞi  P plot. The data presented by Shieh et al. (2002) do not show this feature. The strength t computed from Eq. 7 are shown in Fig. 7b. The strength is 4 GPa at 20 GPa and increases to 8 GPa at 40 GPa and decreases as the transition

pressure is approached. The strength shows a monotonic increase in the orthorhombic phase. The strength data of Shieh et al. (2002) show a similar trend except that the present estimates of strength are higher. Unlike the drop in t near the transition pressure, which is largely caused by the shear modulus, the anomaly in the hQ ðhklÞi  P data at the transition pressure is purely an experimental observation and is of primary significance as it carries the signature of the transition.

4.5. Single-crystal elastic moduli Eq. (3b) with Q ðhklÞ for reflections (110), (111), (120), and (121) provided four independent equations for the determination of C ij . In place of Q ð110Þ we used the average of Q ð110Þ and Q ð220Þ. The other observed reflections, (101), (130) and (031) could not be considered for the determination of C ij . The (101) reflection overlapped with (111) reflection of Pt (or Au), and (130) and (031) reflections showed excessive line broadening. These factors made reliable determination of Q ðhklÞ for the three reflections difficult. Two additional relations were provided by Eqs. (9) and (10). Following Shieh et al. (2002), we set a ¼ 1 in Eq. (3b) and extracted all the six elastic moduli shown in Fig. 8. As expected, ðC 11  C 12 Þ is found to decrease with increasing pressure and extrapolation of the data indicates that this difference vanishes at 46 GPa. Though the trends are similar, the present ðC 11  C 12 Þ-values are 50–70% larger than those computed from the data of Carpenter et al. (2000, 2006), referred to as CHM hereinafter. The modulus C 13 increases with increasing pressure and agrees reasonably well with the data of CHM at lower pressures but the pressure derivative is too large resulting in large deviations at higher pressures. The present data on C 33 starts with a good agreement with the data of CHM at 20 GPa but show a negative pressure derivative. The present C 66 -data show low scatter but the pressure derivative is negative. Because of the three data points marked by crosses, the present data on C 44 appears to exhibit very large scatter. If these three points are omitted then the present data show better agreement with the data of CHM. It must be emphasized that the use of just six equations to determine six elastic moduli does not offer the advantage of least-squares fit especially when Q ðhklÞ contain errors of measurements. Shieh et al. (2002) used only two reflections (110 and 121) together with two relations provided by Eqs. (9) and (10), and reduced the number of unknowns to four (C 11 ; C 12 ; C 13 , and C 33 Þ by feeding the values of C 44 and C 66 . When we followed similar procedure with our data, the C 11 ; C 12 ; C 13 , and C 33 showed

2000

1200 X

X

900 C33

600

1500 C44

X

300

1000 500

C66

0

0 C11

600

500

C13

400 400

300

C12

200

20

30

40

50

20

30

40

50

200

Pressure (GPa) Fig. 8. Elastic moduli of stishovite as a function of pressure. The fits through the present data and those of Carpenter et al. (2000, 2006), are shown by solid and dash-dot lines, respectively. As expected, ðC 11  C 12 Þ decreases with increasing pressure and tends to zero at transition pressure. See text for details.

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improved agreement with data of CHM. This improvement in the values of C 13 and C 33 is clearly due to the fact that the number of parameters is reduced by using the values of C 44 and C 66 . This suggests the possibility of determining C ij with much improved precision if Q ðhklÞ for larger number of reflections become available.

4.6. Diffraction line-width analysis

0.2

41

35

52

52

41

(a)

(b)

20 15 10

4000 5 2000

120 111 110

0 -5 1

2

3

4

5

1

6

2

3

4

5

6

Run number

40.5

0.1

(a) 0.0 0

20

40

60

Pressure (GPa) 40

6000

(b)

(c)

5000

30 4000 20 72.9 GPa 59.7 GPa

10

45.5 GPa

D (A)

X 10 -6

6000

Unit cell volume V (A3)

FWHM (degree)

0.3

[2w hklcosθ hkl ]2

Aprox. laser power (W) 35

D (A)

4.6.1. Estimation of D and t The full widths at half-maximum (FWHM) for a few reflections are shown at different pressures in Fig. 9a. The width-pressure plot starts with a large slope, which shows a marked decrease above 45 GPa. The diffraction line-width analysis was carried out using Eq. (11) to derive t and the crystallite (apparent) size D. The Ehkl values at different pressures were computed using the single-crystal elasticity data (Carpenter et al., 2000, 2006). The ð2whkl cos hhkl Þ2 versus ðsin hhkl =Ehkl Þ2 plots for a few pressure runs are shown in Fig.9b. The strength tðffi 2pmax Þ and crystallite size at different pressures were derived form the slope and intercept, respectively. Fig. 7b shows that the strength data derived from the width analysis are in reasonable agreement with those data obtained by the line shift method. Under nonhydrostatic compression, the crystallite size decreases from 4700 Å at 10 GPa to 1000 Å at 45 GPa and becomes nearly pressure-independent in the orthorhombic phase (Fig. 9c). It may be noted that the nonhydrostatic compression of the tetragonal phase t increases and D decreases with increasing pressure. Both the factors contribute to the line-width increase. However, the crystallite size remains practically unchanged in the orthorhombic phase and it is only the increase in t with pressure that contributes to the line width. This explains the decrease of slope of the width-pressure plot above 45 GPa (Fig. 9a).

4.6.2. Effect of laser annealing In this section we discuss the effect of successive cycles of laser annealing on the parameters D, t, and V. The sample with Pt pressure-marker compressed to 72.9 GPa was chosen for this study. The laser powers used for annealing were 35 W, 39 W, 49 W, 45 W, 52 W during the first through fifth annealing cycles, each cycle of 600 s duration. The heating was not homogeneous during annealing. After each annealing cycle the cell was allowed to cool to room temperature before recording the diffraction pattern. The average crystallite size determined from the line-width analysis after each annealing cycle showed an increase. After the fourth annealing cycle, the grain size was close to the initial value (Fig. 10a). The parameter t decreased with each annealing cycle and after the fifth cycle it dropped to 2 (4) GPa, a value comparable with the error in t (Fig. 10b). Since t is as a measure of nonhydrostatic stress component, vanishing t implies a stress state approaching hydrostatic pressure. The effect of laser annealing on the P  V point is shown in Fig. 10c. The sample pressure drops from 72.9 (20) GPa to 65.4(8) GPa and the cell volume from 40.28(6) Å3 to 39.91(3) Å3 after the first annealing cycle. This brings the P  V point close to the EoS under hydrostatic pressure (Andrault et al., 2003). Subsequent annealing produces small fluctuations in the pressure and cell volume but the P  Vpoint after each annealing cycle stays close to the EoS under hydrostatic pressure. The average pressure and cell volume for the last four heating cycles are 63.2(8) GPa and 40.10(4) Å3. This value compares well with the cell volume 40.17(5) Å3 obtained at the same pressure

t (GPa)

8

3000 2000

10

[sinθhkl / Ehkl ]

20 2

X 10 -8

run # 2 run # 3 run # 4 run # 5 run # 6 run # 1

39.5

hydrostat

65

70

75

80

Pressure (GPa)

1000 0

0

40.0

39.0 60

32.8 GPa

0

(c)

0

20

40

60

80

Pressure (GPa)

Fig. 9. (a) The full width at half-maximum (FWHM) for a few reflections at different pressures. The slope of the width versus pressure plot decreases above 45 GPa. (b) The ð2whkl cos hhkl Þ2 versus ðsin hhkl =Ehkl Þ2 plots for a few pressure runs. The line widths are in radian. The scatter in the plot increases with increasing pressure. (c) Variation of average crystallite size with pressure. A comparison of the t versus P data with those obtained by the line shift methods is shown in Fig. 7.

Fig. 10. Effect of successive cycles (10 min duration each) of laser annealing of the compressed silica sample. Run number (1) as compressed sample at 72.9 GPa; (2) laser power 35 W; (3) laser power 39 W; (4) laser power 49 W; (5) laser power 45 W; (6) laser power 52 W. (a) The crystallite size after each laser-annealing cycle increased and reached close to the initial value at the end of the fourth annealing cycle (run number 5). (b) The nonhydrostatic stress component t decreased to 2 GPa at the end of fifth heating cycle (run number 6). (c) After the first annealing (run number 2) the pressure drops from 72.9(20) GPa to 65.4(8) GPa and volume from 40.28(6) Å3 to 39.91(3) Å3 and the P–V point comes closer to the EoS under hydrostatic pressure (Andrault et al., 2003). During further annealing cycles, the pressure fluctuates between 64.3(4) GPa and 62.5(10) GPa and the corresponding volumes between 40.04(3) Å3 and 40.10(3) Å3.

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from the equation of state of the orthorhombic phase proposed by Andrault et al. (2003). 4.7. Effect of t on the transition The nonhydrostatic stress component t in the tetragonal phase is large and pressure dependent (Fig. 7b). The presence of large t is expected to introduce features in the diffraction data that are absent when the compression is hydrostatic. Carpenter (2007) developed a rigorous formalism to examine the effect of externally imposed stresses, pi , on the transitions in perovskites. The notation for stress used by Carpenter (2007) is changed in this discussion to avoid any confusion with ri used in the present work. The stress state pi can be expressed in terms of ri for any given orientation of the crystallite. The presence of pi introduces a term ðp1  p2 Þðe1  e2 Þ in the Landau free energy expansion. Since ðe1  e2 Þ is proportional to the order parameter, Q, it is equivalent to adding a field term ðp1  p2 ÞQ (Carpenter, 2007). The field terms break the symmetry and give a tail in the order parameter extending into the stability field of the low symmetry phase. The term ðp1  p2 Þ can become large for certain orientations of the crystallites. These crystallites would transform at pressures (mean normal stress) lower than the transition pressure, P c , under hydrostatic compression. The crystallites which are oriented such that ðp1  p2 Þ ¼ 0 would be expected to transform at P c . This appears to explain qualitatively the features of the data in Fig. 3. A closer comparison of the theoretical predictions with the experimental results would require extension of Carpenter’s approach to the polycrystalline sample as applicable to X-ray diffraction. The intensity of a reflection ðhklÞ recorded using the conventional geometry, which is used in this study, arises only from a select group of crystallites that have the diffracting plane normal inclined at ðp=2  hÞ to the load axis of the DAC (Singh, 1993). As h depends on ðhklÞ, each reflection arises from a distinctly different group of crystallites. A rigorous approach would require evaluating the average effect on each reflection. Qualitatively, such groups of crystallites experience different magnitudes of ðp1  p2 Þ. Thus, each reflection is affected to different extent by the presence of t. This would explain a significantly larger scatter in the lattice parameters that is observed under nonhydrostatic compression as compared to the scatter under hydrostatic compression. 5. Conclusion The tetragonal?orthorhombic transition in silica under nonhydrostatic compression is observed at much lower pressures than it occurs under hydrostatic compression. However, the transition pressures derived from the square of the symmetry-breaking strain versus pressure data are not significantly different in the two cases. We use the offset between the d-spacing measured under nonhydrostatic and hydrostatic compression to derive a parameter Q ðhklÞ, which is central to the determination of the strength and elasticity of a solid by X-ray diffraction. This parameter has been measured thus far using radial diffraction geometry that requires specially designed experimental setup. The present diffraction data suggest that the compressive strength of stishovite increases with pressure and begins to decrease at 40 GPa. After reaching a minimum at the tetragonal?orthorhombic transition, the strength of the orthorhombic phase shows a monotonic increase with pressure. The single-crystal elastic moduli of stishovite derived in this study suggest that the shear modulus ðC 11  C 12 Þ decreases with increasing pressure and vanishes at the transition pressure. These results are in agreement with those obtained in an earlier study (Shieh et al., 2002). The strength-pressure data derived from the line-width analysis agree well with those derived from the

9

Q ðhklÞ-data. The grain size of stishovite under nonhydrostatic compression decreases from 5000 Å at ambient pressure to 1000 Å at 45 GPa and remains practically unchanged in the orthorhombic phase. The line-width analyses of diffraction profiles indicate that the sample pressure is rendered near-hydrostatic by laser annealing the nonhydrostatically compressed samples.

Acknowledgement The authors thank the reviewers for their comments on the paper. Michael Carpenter’s suggestions have resulted in a more comprehensive discussion of the results.

References Anderson, O.L., Issak, D.G., Yamamoto, S., 1989. Anharmonicity and the equation of state for gold. Journal of Applied Physics 65, 1534–1543. Andrault, D., Fiquet, G., 2001. Synchrotron radiation and laser-heating in a diamond anvil cell. Review of Scientific Instruments 72 (2), 1283–1288. Andrault, D., Angel, R.J., Mosenfelder, J.L., Le Bihan, T., 2003. Equation of state of the stishovite to lower mantle pressures. American Mineralogist 88, 301–307. Andrault, D., Fiquet, G., Guyot, F., Hanfland, M., 1998. Pressure-induced landau-type transition in stishovite. Science 23, 720–724. Bolfan-Casanovaa, N., Andrault, D., Amiguet, E., Guignot, N., 2009. Equation of state and post-stishovite transformation of Al-bearing silica up to 100 GPa and 3000 K. Physics of the Earth and Planetary Interiors 174, 70–77. Carpenter, M.A., 2006. Elastic properties of minerals and the influence of phase transitions. American Mineralogist 91, 229–246. Carpenter, M.A., 2007. Elastic anomalies accompanying phase transitions in (Ca,Sr)TiO3 perovskites: Part I. Landau theory and a calibration for SrTiO3. American Mineralogist 92, 309–327. Carpenter, M.A., Hemley, R.J., Mao, H.K., 2000. High-pressure elasticity of stishovite and the P4(2)/mnm $ Pnnm phase transition. Journal of Geophysical Research 105 (B5), 10807–10816. Chen, J. et al., 2006. Do Reuss and Voigt bounds really bound in high-pressure rheology experiments? Journal of Physics: Condensed Matter 18, S1049–S1059. Chen, J., Weidner, D.J., Vaughan, M.T., 2002. The strength of Mg0.9Fe0.1SiO3 perovskite at high pressure and temperature. Nature (London) 419, 824–826. Cohen, R.E., 1992. First-principles predictions of elasticity and phase transitions in high pressure SiO2 and geophysical implications. In: Syono, Y., Manghnani, M.H. (Eds.), High pressure Research: Application to Earth and Planetary Sciences. American Geophysical Union, pp. 425–431. Cordier, P., Rubie, D.C., 2001. Plastic deformation under extreme pressure using a multi-anvil apparatus. Materials Science and Engineering A, 38–43. Dubrovinsky, L.S., Belonoshko, A.B., 1996. Pressure-induced phase transition and structural changes under deviatoric stress of stishovite to CaC12-like structure. Geochimica et Cosmochimica Acta 60, 3657–3663. Goresy, A.E., Dubrovinsky, L., Sharp, T.G., Saxena, S.K., Chen, M., 2000. A Monoclinic post-stishovite polymorph of silica in the Shergotty meteorite. Science 288, 1632–1634. Guignot, N., Andrault, D., 2004. Equations of state of Na–K–Al host phases and implications for MORB density in the lower mantle. Physics of the Earth and Planetary Interiors, 107–128. Hammersley, J., 1996. Fit2d user manual, ESRF, Grenoble, France. He, D., Duffy, T.S., 2006. X-ray diffraction study of the static strength of tungsten to 69 GPa. Physical Review B 73, 134106. Hemley, R.J., Prewitt, C.T., Kingma, K.J., 1994. High-pressure behavior of silica. In: Heaney, P.J., Prewitt, C.T., Gibbs, G.V. (Eds.), Silica: Physical Behavior, Geochemistry and Materials Applications. Mineralogical Society of America, Washington, DC, pp. 41–81. Hemley, R.J., Shu, J.F., Carpenter, M.A., Hu, J., Mao, H.K., Kingma, K.J., 2000. Solid/ order parameter coupling in the ferroelastic transition in dense SiO2. Solid State Communications 114, 527–532. Hirose, K., Takafuji, N., Sata, N., Ohishi, Y., 2005. Phase transition and density of subducted MORB crust in the lower mantle. Earth and Planetary Science Letters 237, 239–251. Holmes, N.C., Moriarty, J.A., Gathers, G.A., Nellis, W.J., 1989. The equation of state of platinum to 660 GPa (66 Mbar). Journal of Applied Physics 66, 2962–2967. Holzapfel, W.B., 1996. Physics of solids under strong compression. Reports on Progress in Physics 59, 29–90. Holzapfel, W.B., 1991. Equations of state for ideal and real solids under strong compression. Europhysics Letters 16, 67–72. Joswig, W., Stachel, T., Harris, J., Baur, W., Brey, G., 1999. New Ca-silicate inclusions in diamonds – tracers from the lower mantle. Earth and Planetary Science Letters 173, 1–6. Karki, B.B., Stixrude, L., Crain, J., 1997. Geophysical Research Letters 24, 3269–3272. Kingma, K.J., Cohen, R.E., Hemley, R.J., Mao, H.K., 1995. Transformation of stishovite to a denser phase at lower mantle pressures. Nature 374, 243–245. Kingma, K.J., Mao, H.-k., Hemley, R.J., 1996. Synchrotron X-ray diffraction of SiO2 to multimegabar pressures. High Pressure Research 14, 363–374.

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Lakshtanov, D.L. et al., 2007. The post-stishovite phase transition in hydrous alumina-bearing SiO2 in the lower mantle of the earth. National Academy of Sciences of the USA 104, 13588–13590. Langford, J.I., 1971. X-ray powder diffraction studies of Vitromet samples. Journal of Applied Crystallography 4, 164–168. Lee, C., Gonze, X., 1995. The pressure-induced ferroelastic phase transition of SiO2 stishovite. Journal of Physics: Condensed Matter 7, 3693–3698. Leger, J.M., Haines, J., Schmidt, M., Petitet, J.P., Pereira, A.S., da Jornada, J.A.H., 1996. Nature, 383–401. Li, B., Rigden, S.M., Liebermann, R.C., 1996. Elasticity of stishovite at high pressure. Physics of the Earth and Planetary Interiors 96, 113. Mao, H.-k., Hemley, R.J., 1996. Energy dispersive X-ray diffraction of micro-crystals at ultrahigh pressures. High Pressure Research 14, 257–267. Merkel, S., Yagi, T., 2005. X-ray transparent gasket diamond anvil cell high pressure experiments. Review of Scientific Instruments 76, 046109. Mezouar, M. et al., 2005. Development of a new state-of-the-art beamline optimized for monochromatic single-crystal and powder X-ray diffraction under extreme conditions at the ESRF. Journal of Synchrotron Radiation 12, 559–664. Nye, J.F., 1960. Properties of Crystals: Their representation by Tensor and Matrices. Clarendon Press, Oxford, p.145. Reuss, A., 1929. Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeitschrift Angewandte Mathematical and Mechanics 3, 49–58. Ross, N.L., Shu, J.F., Hazen, R.M., Gasparik, T., 1990. High pressure crystal chemistry of stishovite. American Mineralogist 75, 739–747. Sànchez-Bajo, F., Cumbrera, F.L., 1997. The use of the pseudo-Voigt function in the variance method of X-ray line-broadening analysis. Journal of Applied Crystallography 30, 427–430. Schultz, E. et al., 2005. Double-sided laser heating system for in situ high pressurehigh temperature monochromatic X-ray diffraction at the ESRF. High Pressure Research 25 (1), 71–83. Shieh, S.R., Duffy, T.S., Li, B., 2002. Strength and Elasticity of SiO2 across the Stishovite–CaCl2-type Structural Phase Boundary. Physical Review Letters 89, 255507. Singh, A.K., 1983. The kinetics of pressure-induced polymorphic transformations. Bulletin of Materials Science 5, 219–230. Singh, A.K., 1993. The lattice strains in a specimen (cubic system) compressed nonhydrostatically in an opposed anvil device. Journal of Applied Physics. 73, 4278-4286. Erratum: Journal of Applied Physics 74, 5920. Singh, A.K., 1994. The effect of stress anisotropy on the lattice strains measured with an X-ray diffraction opposed anvil setup. In: Schmidt, S.C., Shaner, J.W.,

Samara, G.A., Ross, M. (Eds.), High Pressure Science and Technology – 1993, AIP Conf. Proc. 309, 1629–1632. Singh, A.K., 2004. X-ray diffraction from solids under nonhydrostatic compression ? some recent studies. Journal of Physics and Chemistry of Solids 65, 1589–1596. Singh, A.K., 2007. Pressure–volume data for epsilon iron and platinum to earth-core pressures and above from one-parameter equation of state: Implication on platinum pressure scale. Physics of the Earth and Planetary Interiors 164, 75–82. Singh, A.K., 2009. Analysis of nonhydrostatic high-pressure diffraction data (cubic system): Assessment of various assumptions in the theory. Journal of Applied Physics 106, 043514. Singh, A.K., Balasingh, C., 1994. The lattice strain in a specimen (hexagonal system) compressed nonhydrostatically in an opposed anvil high pressure setup. Journal of Applied Physics 75, 4956–4962. Singh, A.K., Balasingh, C., Mao, H.K., Hemley, R.J., Shu, J., 1998a. Analysis of lattice strains measured under nonhydrostatic pressure. Journal of Applied Physics 83, 7567–7575. Singh, A.K., Jain, A., Liermann, H.P., Saxena, S.K., 2006. Strength of iron under pressure up to 55 GPa from X-ray diffraction line-width analysis. Journal of Physics and Chemistry of Solids 67, 2197–2202. Singh, A.K., Liermann, H.-P., 2011. Strength and elasticity of niobium under high pressure. Journal of Applied Physics 109, 113539. Singh, A.K., Liermann, H.-P., Akahama, Y., Kawamura, H., 2007. Aluminum as a pressure-transmitting medium cum pressure standard for X-ray diffraction experiments to 200 GPa with diamond anvil cells. Journal Applied Physics 101, 123526. Singh, A.K., Liermann, H.-P., Akahama, Y., Saxena, S.K., Men’endez-Proupin, E., 2008. Strength of polycrystalline coarse-grained platinum to 330 GPa and of nanocrystalline platinum to 70 GPa from high-pressure X-ray diffraction data. Journal Applied Physics 103, 063524. Singh, A.K., Liermann, H.P., Saxena, S.K., 2004. Strength of magnesium oxide under high pressure: evidence for the grain-size dependence. Solid State Communications 132, 795–798. Singh, A.K., Mao, H.K., Shu, J., Hemley, R.J., 1998b. Estimation of single-crystal elastic moduli from polycrystalline X-ray diffraction at high pressure: Application to FeO and iron. Physics Review Letters 80, 2157–2160. Stokes, A.R., Wilson, A.J.C., 1944. The diffraction of X-rays by distorted crystal aggregates – I. Proceedings of the Physical Society, London 56, 174–181. Takemura, K., Dewaele, A., 2008. Isothermal equation of state for gold with a Hepressure medium. Physics Review B 78, 104119. Tsuchida, Y., Yagi, T., 1989. A new, post-stishovite high-pressure polymorph of silica. Nature 340, 217–220. Voigt, W., 1928. Lehrbuch der Krystallphysik, pp.962. B. G. Teubner, Liepzig.