Predicted elastic properties of the hydrous D Phase at mantle pressures:

Implications for the seismic anisotropy of subducted slabs near 670 km

discontinuity and in the lower mantle.

David Mainprice*a, Yvon Le Page b and John Rodgers c

aGéosciences Montpellier UMR CNRS 5243, Université Montpellier II, 34095

Montpellier, France bICPET, National Research Council of Canada, Canada.

cToth Information Systems Inc., Ottawa, Canada.

* corresponding author tel: +33-4671432832; fax: +33-467143603

email: David.Mainprice@gm.univ-montp2.fr

Submitted to Earth and Planetary Sciences Letters 22nd February 2007

Predicted elastic properties of the hydrous D Phase at mantle pressures:

Implications for the seismic anisotropy of subducted slabs near 670 km

discontinuity and in the lower mantle.

David Mainprice*a, Yvon Le Page b and John Rodgers c

aGéosciences Montpellier UMR CNRS 5243, Université Montpellier II, 34095

Montpellier, France bICPET, National Research Council of Canada, Canada.

cToth Information Systems Inc., Ottawa, Canada.

* corresponding author tel: +33-4671432832; fax: +33-467143603

email: David.Mainprice@gm.univ-montp2.fr

Abstract

The dense hydrous magnesium silicate called the D phase is the most likely

candidate to recycle hydrogen into the lower mantle in subduction zones. As

seismology represents the preferred method to detect this mineral in subduction

zones, the single crystal elastic tensor has been calculated using first principle

methods. A triple cell (P

!

3_

m1) was used to account for an ordered ideal

composition of the D phase to lower mantle pressures. The elastic tensor of the

mineral at ambient conditions is predicted to be C11 = 387.7, C33=287.7,

C44=100.4, C12=108.0, C13=51.1, C14=-14.6 in GPa with bulk modulus at zero

pressure Ko=163 GPa and a statistical error of about 4.0 GPa for all constants.

The pressure derivatives of the elastic constants have been determined to

pressure of 80 GPa. Our tensor correctly predicts the published experimental

measurements of linear compressibility along the a- and c-axes, the logarithmic

pressure derivative of the c/a axial ratio and bulk modulus at zero pressure. It

contradicts the experimentally measured elastic tensor reported by Liu et al.

Solid State Comm. 132 (2004) 517-520, which fails reproduce the experimental

compressibility data.

Using the elastic constants and density predicted to 85 GPa the seismic

anisotropy of P- and S-waves has been calculated as a function of pressure. The

single crystal has high P-wave velocities and shear wave splitting in the basal

plane below 20 GPa. At high pressures, the maximum P-wave velocities have

three fold symmetry normal to a-axes in the basal plane, where as the maximum

shear wave splitting is parallel to a-axes with twofold symmetry. The D phase

has high Vp and Vs anisotropy at ambient pressure, 17.6% for Vp and 19.9% for

Vs. The Vp anisotropy decreases with increasing pressure to about 30 GPa,

while it remains constant at 8% from 30 to 85 GPa. The Vs anisotropy decreases

to 17% at 20 GPa and then increases to 22% at 85 GPa. The isotropic velocity

ratios of S to P (Rs/p), bulk sound to S (Rφ/s) and density to S (Rρ/s) are

calculated as a function of pressure. The Rs/p shows very little variation with

pressure with low value of 0.6, whereas both Rφ/s and Rρ/s increase with

pressure to peak values at about 60 GPa of 2.8 and 2.0 respectively.

The presence of the D phase in subducted horizontal ‘stagnant’ plates in the

circum Pacific could contribute to a number of seismic observations, such as the

observed velocity heterogeneity at depths between 500 and 1000 km and the fast

S-waves travelling horizontally with horizontal polarization in this depth range.

Further the lower density of the D phase compared with anhydrous minerals

could influence buoyancy of the hydrated ‘stagnant’ slabs.

Keywords: D phase, ab initio ; elastic properties ; subduction ; seismic

anisotropy; lower mantle

1. Introduction

Subduction zones are regions where plates composed of a thin veneer of

sediments, hydrated oceanic crust and upper mantle are transformed by

metamorphic processes into an exotic patchwork of high-pressure minerals at

great depths [1]. Many of the minerals postulated to exist at great depth in

subduction zones have only been synthesised in the laboratory at extreme

conditions of pressure and temperature, with no natural occurrences having been

reported [2]. One group of minerals that are expected to be important

components of the hydrated mantle, which is taking part in the recycling of

hydrogen at depth, are the dense hydrous magnesium silicates (DHMS) [2-4].

The first DHMS were discovered in the pioneering high pressure experimental

petrology work of Ringwood and Major [5], who synthesised the minerals

known as the alphabet phases, A, B and C. Later other alphabet phases were

recognised, of which the D (Liu, 1987)[6], E (Kanzaki, 1991)[7] and

superhydrous B (Gasparik, 1993)[8] are now well established. Other phases

initially called F (Kranaki, 1991)[7] and G (Ohtani et al., 1997; Kudoh et

al.1997)[9-10] are now considered (see Frost [3]) to be the same as the D phase

originally described by Liu [6](1987). Liu synthesised the D phase in a laser

heated diamond anvil cell at pressures greater than 22 GPa from serpentine

starting material. He was the first to recognise the potential of the mineral as a

water storage site in the mantle. The D phase is the DHMS which is stable at the

highest pressures, and hence the most likely candidate to host hydrogen into the

lower mantle at pressures of 25 GPa (Frost and Fei, 1998) to 44 GPa (Shieh et

al. 1998)[11-12] and temperatures up to 1400°C. In addition the D phase can

contain between 10 to 18 weight percent water (Yang et al., 1997)[13], making

it one of the most water rich minerals stable at high pressure, which can coexist

with ultramafic mantle assemblages. The ultimate depth of the D phase is

limited by the reaction D phase + ferripericlase = Mg-perovskite + water (Frost,

2006) [2], the D phase may be present to depths of 1200-1500 km [14].

The exploration of the structure, composition and dynamics of subduction

zones requires accurate interpretation of geophysical observables, such as the

lateral and vertical variation of seismic velocities. To achieve this goal, the

mineral physics community has to provide elastic properties of candidate

minerals for subduction zones at the appropriate physical conditions of the

Earth’s interior. Iwamori (2004) [15] finds that the D phase is 46.6 and 54.7

modal percent of water saturated Lherzolite and Harzburgite respectively, at

pressures between 12 and 28 GPa, hence the D phase is twice as abundant as

any other mineral in rocks of these compositions. Iwamori’s results suggest that

the D phase will be approximately 50 percent of a subducted peridotite between

350 and 800 km depth in a subduction zone. Similarly Ohtani et al. 2004 [14]

reported 52.6 to 57.0 percent D phase for hydrous peridotite composition in the

pressure range 18 to 35 GPa. Combining both studies gives a depth range from

350 to 950 km over which the D phase is the dominant mineral along a cold slab

geotherm [16]. Such large volume fractions of the D phase are likely to have an

important impact on seismic properties in subduction zones. In this paper we

will explore the elastic properties of the D phase to lower mantle pressures using

first principles atomic scale modelling. Where available, we will compare model

predictions with experimental data. We will use the elastic single crystal

constants to calculate the upper bounds of the contribution of the D phase

seismic anisotropy of hydrated slabs at lower mantle pressures.

2. Computation details

2.1 Preliminary structure modeling

The initial structure model was taken from the refinement [13] of a disordered

material with space group P

!

3_

1m and cell formula Mg1.11 Si1.89 H2.22 O6. The

crystal structure of phase D is based on a hexagonal closest-packed array of

oxygen atoms. All cations except hydrogen are in octahedrally coordinated sites.

The phase D structure is composed of alternating layers of SiO6 and MgO6

octahedra stacked along c-axis. The layers of SiO6 octrahedra have a structure

similar to Brucite, except that one in three silicon sites are vacant. The

octrahedra of MgO6 occur above and below every vacant silicon site. The

hydogens are situated in the MgO6 layers bonded to the oxygens of the SiO6

octrahedra (see Figure 1). As quantum simulations cannot handle mixed

occupancy or H disorder, we created a corresponding ordered and stoichiometric

model in the triple cell (or supercell) with edges a-b, a+2b, c (Table 1b, Figure

1), resulting in space group P

!

3_

m1. CuKα1 X-ray powder patterns calculated for

the two models (a) and (b) from Table 1 were indistinguishable to the eye even

for their very weak reflections, confirming the equivalence of the two structure

models.

2.2 Ab initio computations

The elastic calculation and interpretation scheme was that described in [17]. It

was implemented here with two strain magnitudes of 0.5 and 0.75%. Input data

files for VASP [18-20] were generated by Materials Toolkit [21] and used the

following execution parameters: GGA PAW potentials [22]; electronic

convergence at 1x10-7 eV; convergence for forces 1x10-4 eV/Å; Davidson-

blocked iterative optimization of the wave functions in combination with

reciprocal-space projectors [23]; reciprocal space integration with a Monkhorst-

Pack scheme [24]; and a Methfessel-Paxton smearing scheme of order 1 and

width 0.2 eV for energy corrections [25]. Spin polarization corrections were not

used. The k-mesh grid used for all optimizations was 3x3x5. A maximum of 30

iterations for preliminary optimization of the undistorted structure and 20

iterations for elastic calculations ensured proper convergence of atom relaxation,

calculated energy and stress. Elastic calculations required about 4 cpu x days

each on 2.8-gigahertz Intel Xeon PCs running parallel VASP.4.6.3 under Linux.

Starting from the ordered structural model in Table 1, we first optimized its

atom coordinates, retaining its zero-pressure experimental cell data. The

corresponding calculated pressure was -5.45 GPa. This value constitutes the

offset between experimental and calculated pressures. Such an offset is expected

and is due to slight imperfections in the representation of the core electron

density for calculated ab initio potentials. This offset is used later in the

interpretation of results as a uniform correction to all calculated pressures.

Re-starting from the optimized coordinates in the experimental zero-pressure

cell, we then performed cell-and-coordinates optimizations at calculated

pressures from -10 to +80 GPa in 5 GPa steps, thus covering the approximate

range of true pressures from -5 to +85 GPa. The additional data points at -5 and

+85 GPa allow derivation of sensible polynomial expressions that are valid in

the known stability range extending from 0 to 80 GPa. Convergence turned out

to be slow, probably due to in part to the relatively high number of atoms per

primitive cell and in part to small drifts from constrained atom coordinates.

Three rounds of optimizations with respectively 30, 20 and 20 iterations were

required to reach convergence of cell data and constrained atom coordinates.

Results from those structure optimizations are listed in Table 2.

We then used the Materials Toolkit [21] implementation of the procedure

described in [17] to generate appropriate cell distortions of the above optimized

models for space group P-3m1. This gave least-squares numerical values for the

independent elastic tensor coefficients (Table 3). No least-squares standard

uncertainty on the individual tensor coefficients exceeded 4 GPa. It should be

stressed that those values are expressed in the IRE reference system [26] for the

P

!

3_

m1 and P

!

3_

1m space groups with an orthogonal frame with X1 axis parallel

to a-axis, X3 axis parallel to the c-axis and the mutually perpendicular X2 axis

completing the right-handed system. Equivalent isotropic values for the bulk

modulus, the shear modulus and Young's modulus are of course independent

from the reference system used. They are also shown in Table 3 together with

the density as well as the P-wave and S-wave velocities. Table 4 lists

coefficients for third-degree polynomial least-squares fit to all those quantities.

Those polynomial expressions allow straightforward computation of smoothed

values for the quantity, as well as its first and second derivatives at any pressure

between 0 and 80 GPa.

3. Comparison with experimental data

Over the last 10 years first principle calculations have become increasingly

important in the exploration of the elastic properties of minerals at high

pressure, see for example the review by Karki et al. 2001[27]. However, it is of

utmost importance to ensure that model calculations reproduce experimental

data. The initial requirement to make accurate elastic constant calculations is the

predicted unit cell parameters must be well optimised. The change in the cell

parameters with pressure is show in Figure 2 for the experimental results of

Frost and Fei (1999) [28] and compared with our ab inito predictions using

VASP. The agreement is very good in the experimental pressure range to 30

GPa and predictions to higher pressures accord well with values at lower

pressures. Our fitting of the data of Frost and Fei (1999)[28] to a third order

Birch-Murnaghan finite strain equation of state [29] gives bulk modulus at zero

pressure Ko=166±6 GPa, very similar to their quoted value of Ko=166±3 GPa

and our calculated value of Ko=163±4 GPa. Tuschiya et al. (2005)[30] report a

first principles calculation of Ko=147.3 and 185.8 GPa for their hydrogen off-

centered and hydrogen centered structure models respectively.

The experimental determination of the single crystal elastic constants of the

D phase has been reported by Liu et al. 2004 [31]. Several small crystals of the

D phase with sizes up to about 80 µm in diameter were synthesized at 19 GPa

and 1000°C. Samples composed of several small single crystals were studied by

Brillouin spectroscopy at ambient conditions. Although spectra were recorded

from several individual crystals, for a given crystallographic plane, most of the

data were incomplete. However, their data obtained from one particular

crystallographic plane yield satisfactory results and to increase reliability the

spectrum was recorded several times on the same plane. The elastic constants

from Liu et al.’s study using the P

!

3_

1m space group are C11=284.4±3.0,

C33=339.4±9.1, C44=120.7±1.9, C12=89.4±4.2, C13=126.6±3.1 and C14=-

4.7±1.4 in GPa. For comparison our calculated elastic constants at 0 GPa in

Table 4 in the P

!

3_

m1 space group the C11 = 387.7, C33=287.7, C44=100.4,

C12=108.0, C13=51.1, C14=-14.6 in GPa with a statistical error of about 4.0

GPa. We can see that for Liu et al. data C11< C33, whereas our calculated Cij

have C11> C33. This very fundamental difference is not caused the space group

setting. Both space groups have the same point group symmetry

!

3_

m, and use

the same Cartesian reference axes for the elastic tensor given above. This is

evidenced by the fact that both studies report the same 6 independent Cij (e.g.

Nye [32]). The C11 is the stiffness parallel to the a-axis, whereas the C33 is the

stiffness parallel to c-axis. The different relative magnitudes of C11 and C33

between the experimental elastic constants and the theory calculations is then a

major discrepancy with no counterpart to our knowledge in oxides or silicates.

Three parameters related to the compressibility of the crystal can be used to

test the validity of the elastic tensors against experimental data. Firstly, the

linear compressibility (β) along the a-and c-axes of the crystal at a given

pressure can be calculated from the compliance tensor Sij from the formula

given by Nye [32] as,

β = (S11+S12+S13) - (S11+S12-S13-S33)cos2θ

for trigonal crystals where θ is the angle between the c-axis and the direction

under consideration. Secondly, the logarithmic pressure derivative of the axial

ratio c/a can be calculated at given pressure by the formula originally given by

Franck and Wanner (1970) [33] as

dln(c/a)/dp = -(C11+C12-C33-C13)/(C33(C11+C12)-2C132) = S11+S12-S13-S33

The analysis is valid for any crystal with uniaxial symmetry of linear

compressibility about the unique c-axis, which are tetragonal, trigonal and

hexagonal symmetry crystals of all classes. Linear compressibility being strain

per unit change in pressure, it then is a second rank tensor, representable by an

ellipsoid similar to thermal expansion which is strain per unit change in

temperature. Note that 4 of the 6 independent compliance tensor Sij values are

used in the calculation of both β and dln(c/a)/dp. These parameters are therefore

a stringent test of the validity of the elastic tensor. The sign of C14 determines

the azimuth of elastic anisotropy about the c-axis and the magnitude of C14

determines the degree of departure from a hexagonal symmetry, which implies a

zero value for C14. Figure 3 illustrates the change of linear compressibility

along the a-and c-axes as a function of pressure. The experimental data of Frost

and Fei (1999) [28] show that the c-axis is more compressible than the a-axis at

low pressure and that both axes have approximately the same compressibility at

the highest experimental pressure of 30 GPa. The first principle calculations

show exactly the same trend and that both axes have approximately the same

compressibility from 40 to 85 GPa. In contrast the linear compressibility

calculated using the elastic constants of Liu et al. have the a-axis more

compressible than the c-axis in contradiction with the experimental results.

Figure 4 shows the logarithmic pressure derivative of the c/a axial ratio

calculated in two different ways; firstly from the unit cell axes from the

experiments of Frost and Fei (1999) [28] and the VASP models, and secondly

from the Cij stiffnesses measured by Liu et al. [31] and the VASP models. The

derivatives calculated from the unit cell axes agree very well between

experiment and theory up to 20 GPa. Above 20 GPa the agreement is less

apparent, but there is some scatter in the experimental c/a ratios above 20 GPa

(Figure 2) so the derivatives may be less reliable. The derivatives calculated

from the Cij stiffnesses show that the VASP results agree well with the

experimental data, and of course with VASP results from the unit cell axes.

However, the derivative calculated from the Cij stiffnesses of Liu et al. [31]

produces a positive derivative at ambient pressure, whereas the experimental

data of Frost and Fei (1999) [28] and the VASP results indicate a clearly

negative derivative at pressures of less than 20 GPa. Unfortunately in the Liu et

al. paper, an inadvertent change in the sign of their calculated derivative to

negative, led them to report that their results were compatible with the data of

Frost and Fei (1999) [28]. Thirdly, we calculate the Voigt-Reuss-Hill average

bulk modulus at zero pressure from the single crystal elastic constants, the

elastic tensor of Liu et al. [31] gives Ko=175.3±14.8 GPa. Note the rather large

error bar given in their paper, whereas our calculated tensor gives Ko=163±4

GPa, which is much closer to the experimental result of Frost and Fei (1999)

[28] of Ko=166±3 GPa.

In this section we have compared our calculated structural and elastic data

for the D phase with experimental data to 30 GPa. The agreement between

theory and experiment is very good suggesting we can have confidence in our

predictions to higher pressures. However, we have unexpectedly revealed that

there is a major inconsistently between the experimentally measured single

crystal elastic constants reported by Liu et al. [31] and experimentally measured

compressibility as well as our own calculated elastic constants.

4. Elastic and seismic properties as a function of pressure

Hydrostatic pressure is now known to have a major influence on the elastic

properties of minerals at the high pressures found in the Earth’s mantle (e.g.

Karki et al. 2001 [27]). Our calculated single crystal elastic constants for the D

phase as functions of pressure are shown in Figure 5a. All moduli of the elastic

constants increase as a function of pressure. The increase of the modulus of C14

is considerable, enhancing the trigonal character (three fold symmetry about the

c-axis) of the elastic anisotropy, as opposed to the cylindrical symmetry of

elasticity that hexagonal crystallographic symmetry would produce. At about 40

GPa several pairs of elastic constants, C11 and C33, and C13 and C66 are

coincidentally nearly equal. At 40 GPa Tsuchiya et al. (2005) [30] observed in

their first principle calculations a change in pressure-induced hydrogen bonding

from asymmetric to symmetric OH bonding and suggested that important elastic

changes would occur at this pressure. The Voigt-Reuss-Hill average isotropic

bulk and shear moduli increase with pressure (Figure 5b), the bulk modulus

increasing more quickly than the shear modulus. We did not observe the

reported increase in the bulk modulus by 20 percent at 40 GPa reported by

Tsuchiya et al. (2005) [30] corresponding to the change from their hydrogen off-

centered structure (Ko=147.3) to the hydrogen centered structure (Ko=185.8

GPa). We instead observe a continuous variation of the bulk modulus with

pressure to 85 GPa, which could be taken for a change in slope in a study over a

smaller range of pressure. However, we did notice other changes in Cij at 40

GPa and our results also agree with their predictions that the compressibilities of

both a-and c-axes become almost equivalent above 40 GPa (Figure 3).

To understand the evolution of the seismic anisotropy of the D phase crystal

as a function of pressure we have calculated the seismic velocities for a given

propagation direction from the Cij and the density at each pressure using the

Christoffel equation (e.g. Mainprice 2007 [34]). The percentage anisotropy (A)

is defined here as A=200 (Vmaximum - Vminimum) / (Vmaximum +

Vminimum), where the maximum and minimum are found by exploring a

hemisphere of all possible propagation directions. Note that for P-wave

velocities the anisotropy is defined by the maximum and minimum velocities in

two different propagation directions, for example the maximum A is given by

the maximum and minimum Vp in a hemisphere. In contrast, in an anisotropic

medium there are two orthogonally polarized S-waves with in general different

velocities for each propagation direction, hence A can be defined for each

direction. Contoured upper hemisphere stereograms of P-wave velocity (Vp),

percentage shear wave splitting (AVs) defined by the A parameter above as well

as polarization of the fastest S-wave (Vs1) are shown in Figure 6. For Vp we

note that the fastest P-waves travel in the basal plane in almost every azimuth at

low pressure. In contrast at pressures greater than 20 GPa the fastest P-wave

velocities are along specific directions normal to a-axes in the basal plane. The

change in the distribution of P-wave velocities corresponds to the increase in

absolute magnitude of the negative C14 constant with pressure (Figure 5,Table

3) with increasingly clear three fold symmetry of the Vp distribution about the

c-axis (Figure 6). Similar symmetry effects with increasing pressure also occur

for S-waves with AVs anisotropy parameter being uniformly high in the basal

plane at low pressure changing to pronounced maxima parallel to a-axes at high

pressure. The centrosymmetric nature of elastic properties combined with the 3-

fold axis along X3 causes the apparent six fold symmetry of AVs maxima in the

basal plane. The out-of-basal plane minima display the expected three fold

symmetry. The polarization of the fastest S-wave in Figure 6 reveals that the

polarization is parallel to the basal plane at low pressure for high AVs values for

propagation directions in the basal plane. In contrast at high pressure high AVs

values parallel to the a-axes in the basal plane have inclined polarizations to the

basal plane, and the sense of inclination between adjacent a-axes is different,

hence there is really three fold symmetry not the apparent six fold symmetry

given by the AVs plot alone. As expected, Figure 6 clearly obeys Neumann’s

principle for the D phase at the various pressures.

Apart from the fine detail of elastic changes that occur in the D phase with

pressure we have plotted the variation of P- and S-wave anisotropy with

pressure (Figure 7). The seismic anisotropy of the D phase at low pressure is

high at 17.6 and 19.9 percent for P- and S-waves respectively, but their

evolution with pressure is very different. The P-wave anisotropy slowly

decreases to achieve a constant value of about 8 percent from 30 to 85 GPa. The

S-wave anisotropy decreases to a minimum value of 17 percent at 20 GPa and

then steadily increases to 22 percent.

Finally the isotropic velocities and their ratios can be calculated using the

isotropic bulk and shear moduli (Table 3, Figure 5) combined with the density

as a function of pressure using the classical relationships for P-wave (Vp), S-

wave (Vs) and bulk sound (Vφ) velocities,

Vp=√(K+4/3G)/ρ Vs=√G/ρ Vφ =√K/ρ = √(Vp2- 4/3Vs2)

where G, shear and K, bulk moduli and ρ is the density. The isotropic P-wave,

bulk sound and S-wave velocities incease with pressure (Figure 8), but the shear

wave velocity increases more slowly, resulting in the bulk sound velocity being

closer to the S-wave at low pressure and closer to the P-wave at high pressure.

Ratios of the three velocities and density have been used in various

seismological studies to characterise variations due to compositional, thermal or

pressure related response of seismic waves [35]. Karato and Karki (2001) [36]

gave a comprehensive overview of the use of these parameters and here we will

follow their notation,

Rs/p = dlnVs/dlnVp Rφ/s=dlnVφ/dlnVs Rρ/s = dlnρ/dlnVs

In this study we are only varying pressure, hence all ratios are calculated at

constant temperature, hence in practice the ratios are calculated as

Rs/p = (Vp/Vs)[(dVs/dP)/(dVp/dP)]

Rφ/s = (Vs/Vφ)[(dVφ/dP)/(dVs/dP)]

Rρ/s = (Vs/ρ)[(dρ/dP)/(dVs/dP)]

No correction has been applied for possible anharmonicity effects due to

temperature, as this would involve further approximations that are not justified

at the present time. From Figure 8 it can be seen that the ratio most directly

evaluated in seismology (Rs/p) using P-wave and S-wave velocities is more or

less constant over the entire range of pressure with rather low value at about 0.6.

The bulk sound velocity is not directly measured in seismology, but its value

can be estimated from Vp and Vs as given above, hence the Rφ/s ratio is an

accessible parameter for the Earth’s mantle. The value of Rφ/s increases with

pressure from 1.9 to a maximum of 2.8 at 55 GPa and then decreasing to 1.9 at

85 GPa. The evolution of Rρ/s is very similar to Rφ/s, increasing with pressure

from 0.8 to a maximum of 2.0 at 60 GPa and decreasing to 1.3 at 85 GPa. The

calculated velocity ratios indicate the combination of Vφ/Vs is the most

promising indicator of the D phase in subducted slabs in the deep mantle with

high values even when corrections for anharmonicity effects are not made.

5. Discussion

Experimental petrology has shown that D phase is stable to at least 44 GPa

and temperatures of the order of 1400°C equivalent of about 1250 km depth

[11,12,14,15]. Upper mantle peridotite compositions subjected to transition zone

and lower mantle pressures in the presence of water are transformed to

aggregates containing approximately 50 percent D phase at temperatures that are

several hundred degrees less than the typical mantle geotherm [14,15]. It is

therefore expected that the D phase will be a volumetrically important

component in subducted hydrated plates just above and below the 670 km

seismic discontinuity in subduction zones. Seismic tomography is our main

source of information on the distribution of subducted plates below the Earth’s

surface [37-41]. Many tomography studies have shown that slabs in subductions

zones undergoing roll-back are often horizontal (frequently termed ‘stagnant’)

in the 670 km depth region, particularly around the circum Pacific region [37-

41]. The heterogeneity of seismic velocity, as measured by the root mean square

(RMS) velocity in seismic models, is more important in subducted slabs in the

depth range from 500 to 1000 km than rest of the mantle [38,41]. Some regional

studies have revealed that horizontally travelling S-waves with horizontal

polarization around 670 km have significantly faster velocities than vertically

polarized waves [42]. Can the presence of the D phase in subducted plates help

explain some of these observations?

Starting from the experimentally determined structure we have used first

principle methods to evaluate the compressibility and elasticity of the D phase to

lower mantle pressures. Using our elastic tensors for single crystal D phase as a

function of pressure we have shown that significant shear wave splitting (ca 18

percent) occurs in the basal plane (Figure 6) and the high degree of S-wave

anisotropy is preserved at high pressure. Although to our knowledge there are

no experimental deformation studies of the D phase, it seems likely that its

pronounced anisotropic layered structure (Figure 1) will favour slip on the basal

plane. The most probable preferred orientation of the D phase in a horizontally

sheared slab will be horizontally oriented basal planes which would produce

shear wave splitting for horizontal propagation of S-waves, with fastest wave

horizontally polarized, as is observed.

The heterogeneity in seismic tomography models in the 500 to 1000 km

depth range [38,41] could indeed be caused by the exotic patchwork of high-

pressure minerals predicted to exist in the hydrated subducted plates

[1,2,4,14,15]. The D phase is the densest of the hydrogen containing minerals,

but its density about 75 percent of a lower mantle anhydrous assemblage

(Smyth, 2006)[43]. The density contrast between the D phase and anhydrous

minerals could contribute to the observed high RMS velocities in subduction

zones. The lower density of the D phase could also influence buoyancy of the

hydrated slab and its eventual sinking into the deeper mantle.

Seismic velocities are often reported as differences or perturbations from a

reference velocity model (e.g. Kennett 2006)[44]. Seismic tomography of

subduction zones has been reported in terms of isotropic P-wave, S-wave and

bulk-sound velocities. Using velocity ratios we have predicted that combination

of S-wave and bulk sound velocities would be the most sensitive to the presence

of the D phase. Tomographic images of the Isu-Bonian subduction zone

modelled using S-wave and bulk sound velocities have been published by

Kennett and Gorbatov 2004[39]. The images show a strong difference between

the structure imaged using the S-wave and bulk sound velocities. A much

stronger positive perturbation (ca 0.6 percent faster velocites) is observed for S-

waves from ak135 reference model than for bulk sound velocities (ca 0.8

percent faster velocities) for the horizontal slab, which would have given an

Rφ/s of about 1.3. However, it is not straight forward to compare between the

perturbation due to pressure in our calculations at about 20 GPa (Rφ/s=2) for

100 percent D phase and the perturbation in the tomographic model of the cold

slab compared to an average (warmer) mantle reference model (ak135).

6. Conclusions

We have undertaken first principles calculation of the compressibility and

elastic properties of the dense hydrous magnesium silicate called the D phase.

We have introduced a novel triple cell with an ideal ordered composition in

order to achieve calculations at reasonable computational cost on this disordered

complex silicate. Our calculations have reproduced with good fidelity the

experimental results on the compression of the D phase confirming the

pertinence of our triple cell approach. When comparing our calculated elastic

tensor for the single crystal D phase with the experimentally measured tensor

reported by Liu et al. 2004 we discovered an important discrepancy.

Comparison of the three parameters calculated from the elastic tensors, the

linear compressibility along the a- and c-axes, the logarithmic pressure

derivative of the axial ratio c/a and Voigt-Reuss-Hill average bulk modulus at

zero pressure confirmed that our calculated tensor agreed closely with

independent experimental results, whereas the tensor of Liu et al. was in clear

contradiction.

Using our calculated elastic tensors and density to 85 GPa we have

calculated the single crystal seismic anisotropy and isotropic velocities of the D

phase to mantle pressures. We show that the D phase has significant seismic

anisotropy at ambient conditions, but this anisotropy is greatly reduced for Vp

and rather preserved or increased for Vs at lower mantle pressures. The presence

of 50 percent by volume in hydrated slabs suggests that D phase may be partly

responsible for strong shear wave splitting of horizontally propagating shear

waves in subduction zones.

Acknowledgements

DM would like to thank Don Issaks and Brian Kennett for helpful discussions

about velocity ratios. DM thanks Ross Angle for a copy of his program EOSfit6

used for the third order Birch-Murnaghan finite strain equation of state. DM

thanks Paul Jouanna for introducing him to atomic scale modelling.

References

[1] S.Poli, M.W.Schmidt, Petrology of subducted slabs. Annu. Rev. Earth Planet. Sci.

30 (2002) 207–235.

[2] D.J. Frost, The Stability of Hydrous Mantle Phases. Reviews in Mineralogy &

Geochemistry 62 (2006) 243-271.

[3] D.J. Frost, The stability of dense hydrous magnesium silicates in Earth’s transition

zone and lower mantle. In: Fei, Y., Bertka, C.M., B.O. Mysen (Eds.), Mantle Petrology:

Field Observations and High Pressure Experimentation: A tribute to Boyd, F.R., The

Geochem. Soc. Special Public, 6 (1999) 283–296.

[4] T.Kawamoto, Hydrous Phases and Water Transport in the Subducting Slab. Reviews

in Mineralogy & Geochemistry 62 (2006) 243-271.

[5] A.E.Ringwood, A.Major, High-pressure reconnaissance investigations in the system

Mg2SiO4-MgO-H2O. Earth Planet. Sci. Lett. 2 (1967) 130-133.

[6] L.-g. Liu, Effects of H2O on the phase behaviour of the forsterite–enstatite system at

high pressures and temperatures and implications for the earth, Phys. Earth Planet.

Inter. 49 (1987) 142–167.

[7] M. Kanzaki, Stability of hydrous magnesium silicates in the mantle transition zone,

Phys. Earth Planet. Inter. 66 (1991) 307– 312.

[8] T. Gasparik, The role of volatiles in the transition zone, J. Geophys. Res. 98 (1993)

4287– 4299.

[9] E. Ohtani, H. Mizobata, Y. Kudoh, T. Nagase, A new hydrous silicate, a water

reservoir, in the upper part of the lower mantle, Geophys. Res. Lett. 24 (1997) 1047–

1050.

[10] Y. Kudoh, T.Nagase, H.Mizobata, E.Ohtani, S.Sasaki, M.Tanaka, Structure and

crystal chemistry of phase G, a new hydrous magnesium silicate synthesized at 22 GPa

and 1050°C.

Geophys. Res. Lett. 24 (1997) 1051–1054.

[11] D.J. Frost, Y. Fei, Stability of phase D at high pressure and high temperature, J.

Geophys. Res. 103 (1998) 7463-7474.

[12] S.R. Shieh, H.-k. Mao, R. J. Hemley & L.C. Ming, Decomposition of phase D in

the lower mantle and the fate of dense hydrous silicates in subducting slabs. Earth

Planet. Sci. Lett. 159 (1998) 13–23.

[13] H. Yang, C.T. Prewitt, D.J. Frost, Crystal structure of the dense hydrous

magnesium silicate, phase D, Am. Mineral. 82 (1997) 651-654.

[14] E. Ohtani, K. Litasov, T. Hosoya, T. Kubo, T. Kondo, Water transport into the

deep mantle and formation of a hydrous transition zone, Physics of the Earth and

Planetary Interiors 143–144 (2004) 255–269.

[15] H.Iwamori, Phase relations of peridotites under H2O-saturated conditions and

ability of subducting plates for transportation of H2O, Earth Planet. Sci. Lett. 227

(2004) 57–71.

[16] S.M. Peacock, Fluid processes in subduction zone. Science 248 (1990) 329–337.

[17] Y Le Page & P.W. Saxe, Symmetry-general least-squares extraction of elastic data

for strained materials from ab initio calculations of stress. Phys. Rev. B 65 (2002)

104104.

[18] G. Kresse (1993). PhD thesis, Technische Universität Wien, Austria.

[19] G. Kresse. & J. Hafner, Ab initio molecular dynamics for open-shell transition

metals. Phys. Rev. B 48 (1993) 13115-13118.

[20] G. Kresse & J. Hafner, Ab initio molecular-dynamics simulation of the liquid-

metal–amorphous-semiconductor transition in germanium. Phys. Rev. B 49 (1994)

14251-14269.

[21] Y. Le Page & J.R.Rodgers, Quantum software interfaced with crystal structure

databases: Tools, results and perspectives. J. Appl. Cryst. 38 (2005) 697-705.

[22] G. Kresse & J. Joubert, From ultrasoft pseudopotentials to the projector

augmented-wave method. Phys. Rev. B 59 (1999) 1758-1775.

[23] E.R. Davidson, Methods in Computational Molecular Physics (1983) edited by

G.H.F. Diercksen & S. Wilson, Vol. 113, NATO Advanced Study Institute, Series C, p.

95. New York,: Plenum Press.

[24] H.J. Monkhorst & J.D. Pack, Special points for Brillouin-zone integrations. Phys.

Rev. B 13 (1976) 5188-5192.

[25] M. Methfessel & A.T. Paxton, High-precision sampling for Brillouin-zone

integration in metals. Phys. Rev. B 40 (1989) 3616-3621.

[26] J.G. Brainerd et al., Standards on piezoelectric crystals. Proc. IRE 37 (1949) 1378-

1395.

[27] B.B. Karki, L. Stixrude and R. Wentzcovitch, High-pressure elastic properties of

major materials of Earth’s mantle from from first principles. Rev. Geophys. 39 (2001)

507-534.

[28] D.J. Frost, Y. Fei, Static compression of the hydrous magnesium silicate phase D

to 30 GPa at room temperature, Phys. Chem. Minerals 26 (1999) 415-418.

[29] R.J. Angel, Equations of state. In Hazen, R.M., Downs, R.T. (Eds.), High

pressure, high-temperature crystal chemistry. Reviews in Mineralogy and Geochemistry,

41 (2000) 35-60.

[30] J. Tsuchiya, T. Tsuchiya & S. Tsuneyuki, First-principles study of hydrogen bond

symmetrization of phase D under high pressure. Amer. Mineral. 90 (2005) 44-49.

[31] L.-g. Liu, K. Okamoto, Y.-j. Yang, C.-c. Chen, C.-C. Lin, Elasticity of single-

crystal phase D (a dense hydrous magnesium silicate) by Brillouin spectroscopy. Solid

State Communications 132 (2004) 517–520.

[32] J.F. Nye, Physical properties of crystals – their representation by tensors and

matrices. Oxford University Press, Oxford, UK (1957).

[33] J.P. Franck and R. Wanner, Elastic constants of hcp He4. Physical Review Letters,

25 (1970) 345–348.

[34] D. Mainprice, Seismic anisotropy of the deep Earth from a mineral and rock

physics perspective. In ‘Treatise in Geophysics’ Editor-in-Chief G.Schubert, Volume 2

‘Mineral Physics’ Editor G.D.Price, Elsevier. In press.

[35] D.G. Isaak, O.L. Anderson and R.E. Cohen, The relationship between shear and

compressional velocities at high pressures : reconciliation of seismic tomography and

mineral physics, Geophys. Res. Lett. 19 (1992) 741-744.

[36] S.-I. Karato, B.B. Karki, Origin of lateral variation of seismic wave velocities and

density in the deep mantle. J. Geophys. Res. 106 (2001) 21771–21783.

[37] R.D. van der Hilst, S. Widiyantoro, E.R. Engdahl, Evidence for deep mantle

circulation from global tomography, Nature 386 (1997) 578-584.

[38] Y. Fukao, S. Widiyantoro, M. Obayashi, Stagnant slabs in the upper and lower

mantle transition region, Rev. Geophys. 39 (2001) 291-323.

[39] B.L.N. Kennett & A. Gorbatov, Seismic heterogeneity in the mantle - strong shear

wave signature of slabs from joint tomography. Phys. Earth Planet. Inter. 146 (2004)

87–100.

[40] J.Ritsema, H.-J. van Heijst and J.H. Woodhouse Global transition zone

tomography. J. Geophys. Res. 109 (2004) B02302, doi:10.1029/2003JB002610

[41] D. Zhao, Global tomographic images of mantle plumes and subducting slabs:

insight into deep Earth dynamics. Phys. Earth Planet. Inter. 146 (2004) 3-34.

[42] J.Wookey, J.-M. Kendall J.-M. and G. Barruol, Mid-mantle deformation inferred

from seismic anisotropy. Nature 415 (2002) 777-780.

[43] J.R. Smyth, Hydogen in High Pressure Silicate and Oxide Mineral Structures.

Reviews in Mineralogy & Geochemistry 62 (2006) 85-115.

[44] B.L.N. Kennett, On seismological reference models and the perceived nature of

heterogeneity. Phys. Earth Planet. Int. 159 (2006) 129-139.

Table 1 Models adapted from Yang et al. (1997) [13]. The first model is made stoichiometric within the experimental cell and contains disordered H atoms. The second model has triple cell volume and is also ordered on H atoms Yang et al. (1997) adjusted to be stoichiometric: Space group: P-31m, a= 4.7453 c= 4.3450 Å Formula: MgSi2O6H2, Z=1 Atom Wyckoff site x y z Occupancy symbol symmetry Si 2d 3.2 2/3 1/3 1/2 1 Mg 1a -3.m 0 0 0 1 O 6k ..m 0.63270 0 0.27160 1 H 6i ..m 0.53600 0 0.09100 1/3

Same structure referred to new axes [ a-b, a+2b, c ] and ordered for H: Space group: P-3m1, a= 8.2191 c= 4.3450 Å Formula: MgSi2O6H2, Z=3 Atom Wyckoff site x y z Occupancy symbol symmetry Si 6h .2. 0.33333 0 1/2 1 Mg1 1a -3m. 0 0 0 1 Mg2 2d 3m. 1/3 2/3 0.00000 1 O1 6i .m. 0.12243 2x 0.27160 1 O2 6i .m. 2y 0.54423 0.27160 1 O3 6i .m. 2y 0.21090 0.27160 1 H 6i .m. 2y 0.17867 0.09100 1

Table 2 Optimized structure results as a function of pressure. Calc. Actual a c x(Si) z(Mg2) x(O1) z(O1) y(O2) z(O2) y(O3) z(O3) y(H) z(H) press. press. (Å) ×10**5 (GPA) -10 -4.55 8.3039 4.3713 33953 03596 12640 27887 54933 29580 20837 24405 17292 03570 -5 0.45 8.2332 4.3088 33965 02921 12569 27567 54953 29249 20844 24359 17238 02997 0 5.45 8.1698 4.2566 33972 02397 12499 27310 54967 28953 20858 24346 17186 02520 5 10.45 8.1127 4.2130 33979 01982 12438 27108 54978 28695 20873 24354 17140 02123 10 15.45 8.0610 4.1743 33986 01632 12383 26943 54989 28463 20886 24369 17096 01767 15 20.45 8.0113 4.1423 33991 01352 12334 26810 54999 28269 20902 24386 17058 01477 20 25.45 7.9658 4.1130 33997 01115 12290 26703 55006 28083 20920 24409 17026 01231 25 30.45 7.9237 4.0864 34003 00912 12251 26613 55014 27916 20937 24429 16998 01014 30 35.45 7.8799 4.0594 34008 00719 12212 26534 55021 27746 20955 24453 16970 00806 35 40.45 7.8434 4.0376 34011 00581 12179 26475 55025 27604 20974 24473 16952 00661 40 45.45 7.8084 4.0173 34013 00463 12149 26426 55030 27474 20992 24492 16938 00540 45 50.45 7.7710 3.9968 34016 00347 12118 26381 55035 27343 21011 24512 16923 00422 50 55.45 7.7402 3.9791 34017 00262 12093 26344 55037 27228 21028 24529 16914 00336 55 60.45 7.7104 3.9623 34018 00186 12069 26311 55039 27119 21046 24545 16906 00261 60 65.45 7.6816 3.9469 34018 00121 12047 26282 55042 27019 21062 24559 16900 00198 65 70.45 7.6541 3.9317 34018 00060 12026 26256 55043 26922 21078 24572 16894 00139 70 75.45 7.6276 3.9172 34019 00007 12005 26232 55045 26830 21095 24585 16890 00089 75 80.45 7.6020 3.9036 34018 -00038 11986 26212 55047 26745 21110 24595 16887 00044 80 85.45 7.5770 3.8911 34018 -00080 11969 26188 55048 26662 21124 24608 16884 00006

Table 3 Elastic tensor coefficients, elastic moduli, seismic velocities and density as a function of pressure Calc. True C11 C12 C13 C14 C33 C44 Bulk Shear Young P-wave S-wave Density press. press. (GPa) (GPa) (GPa) (GPa) (km/s) (km/s) (kg/m3) (GPa) (GPa) -9.76 -4.31 360.07 93.69 34.16 -11.15 246.69 86.34 139.33 111.24 263.57 9.19 5.71 3406.21 -5.07 0.38 389.99 109.68 52.81 -14.56 286.46 102.23 163.02 122.93 294.72 9.64 5.91 3515.21 -0.29 5.16 419.05 124.65 71.27 -18.57 338.46 115.42 187.97 133.70 324.23 10.07 6.08 3613.84 4.58 10.03 454.19 142.21 87.58 -21.38 380.81 128.28 211.97 144.81 353.84 10.46 6.25 3702.79 9.49 14.94 478.41 156.53 107.86 -24.29 435.78 141.45 236.60 154.39 380.42 10.81 6.37 3785.17 14.39 19.84 511.24 173.61 126.81 -26.72 470.31 155.57 260.04 164.71 407.97 11.14 6.53 3861.84 19.33 24.78 535.82 190.99 140.11 -29.24 505.15 160.94 279.29 169.86 423.67 11.34 6.57 3934.00 24.26 29.71 567.84 207.93 157.32 -29.67 539.63 168.94 301.71 177.77 445.75 11.60 6.66 4001.71 29.79 35.24 599.89 226.46 180.05 -33.27 584.83 185.66 328.31 188.66 474.98 11.93 6.80 4073.26 34.73 40.18 616.08 239.98 191.37 -35.16 612.05 188.89 343.06 191.43 484.21 12.03 6.80 4133.45 39.67 45.12 638.87 250.93 207.15 -37.69 645.54 194.79 361.42 197.11 500.34 12.20 6.86 4191.73 45.14 50.59 673.63 269.48 225.84 -38.99 673.51 206.52 384.63 206.24 524.89 12.45 6.96 4253.81 50.03 55.48 697.63 289.15 241.08 -41.19 702.51 208.26 404.34 208.72 534.21 12.59 6.96 4306.79 54.98 60.43 724.24 301.25 262.94 -42.82 734.45 225.00 426.30 218.91 560.73 12.84 7.09 4358.62 59.97 65.42 744.37 318.72 275.67 -45.64 755.75 225.86 442.66 220.23 566.68 12.92 7.07 4408.52 64.83 70.28 757.69 330.11 286.30 -46.13 773.32 227.37 454.84 221.69 572.10 12.97 7.05 4457.40 69.79 75.24 783.30 344.69 303.65 -47.38 806.12 237.97 475.16 229.27 592.48 13.16 7.13 4505.04 74.74 80.19 808.89 362.01 323.10 -49.32 837.25 248.63 496.82 235.84 610.84 13.35 7.20 4551.20 79.69 84.14 832.80 374.63 339.44 -50.88 862.85 255.88 515.05 241.62 626.81 13.49 7.25 4595.94

Table 4

Least-squares coefficients for third-degree polynomial expressions for various quantities. Those expressions allow calculation of smoothed values for the quantities as well as their first and second derivatives at any pressure between 0 and 80 GPa Y M0 M1 M2 M3 C11 (GPa) 387.7198 6.652357 -0.02623824 0.0001118997 C12 107.9675 3.456600 -0.00582115 0.0000270297 C13 51.1311 3.972499 -0.01512064 0.0000988316 C14 -14.6127 -0.689479 0.00529162 -0.0000264251 C33 287.6640 10.535530 -0.07624298 0.0003765922 C44 100.4233 3.184812 -0.03149970 0.0001832366 Bulk (GPa) 161.6448 5.357267 -0.02553026 0.0001357652 Shear 122.0325 2.488181 -0.02306420 0.0001212530 Young 292.4668 6.691189 -0.05802386 0.0003002348 P-wave (km/s) 9.614626 0.092484 -0.00098190 5.104530E-06 S-wave 5.898212 0.393821 -0.00051370 2.797300E-06

Y = M0 + M1*P + M2*P2 + M3*P3

Figure 1 View down c-axis (left-hand figure) and a-axis (right) of Yang, Prewitt and Frost

(1997) (YPF) structure results for phase D in space group P

!

3_

1m. The YPF cell unit shown by a black dotted outline with labels o at the origin, a, b and c cell unit lengths. The c vector points toward the viewer in the left-hand figure and the a vector in the right-hand figure. The blue shaded octahedral represent the SiO6 octrahedra with the blue Si at the center and red oxygens at the corners. The magnesium atoms (yellow) are also at the center of an octahedral oxygen cage. The hydrogen atoms (white) is the only atom that is not octrahedrally co-ordinated, and they are clearly seen to have trigonal three-fold distribution around the c-axis (left-hand figure). The view down the a-axis (right) illustrates the layered structure of the D phase. The solid black lines represent the edges of the triple unit cell (supercell) with as=a-b,

bs=a+2b, cs=c with symmetry P

!

3_

m1 used in the present study, as well as corresponding labels as and bs. The red lines at 120 degrees represent the two-fold axes a1, a2 and a3 in the basal plane.

Figure 2 Hydrostatic compression of D phase as a function of pressure. Comparison between the experimental data of Frost & Fei (1999) to 30 GPa at ambient temperature with theory predictions using VASP to 85 GPa. Open symbol are experimental data and filled symbols are theory calculations.

Figure 3 Linear compressibility of the a- and c-axes as a function of pressure. The open triangles and circle symbols are calculated from the unit cell axes from the experimental data of Frost and Fei (1999) and filled symbols from VASP calculations based the elastic constants at each pressure. The open circle with dot and open triangle point calculated from the elastic stiffnesses tensors of Liu et al.(2004).

Figure 4 Logarithmic pressure derivative of the axial ratio c/a as a function of pressure. The square symbols are the derivative calculated from the unit cell axes from the experimental data of Frost and Fei (1999) and VASP calculations based on cell optimisation at each pressure. The filled diamond and circles are the derivatives calculated from the elastic stiffnesses tensors of Liu et al.(2004) and from the theory calculations respectively.

Figure 5 Evolution of the elastic properties as a function of pressure.

a) Elastic constants Cij in GPa. Notice that two pairs of constants, C11 and C33, and C13 and C66 are nearly equal at 40 GPa.

b) Isotropic elastic moduli, K and G are bulk and shear elastic moduli respectively.

Figure 6 Evolution of P-wave velocity (Vp) distribution, anisotropy of S-waves (AVs) and fastest S-wave (Vs1) polarization as function of pressure. The short white lines on the right-hand figures are the fastest S-wave (s1) polarizations, the slowest S-wave (s2) polarisation are orthogonal to the short white lines, but not marked for clarity. Lambert azimuthal equal-area upper hemisphere projections with elastic tensor orthogonal axes X1 = a-axis (north), X2 = m-axis (east) and X3 = c-axis (centre). (see text for discussion).

Figure 7 Evolution of seismic anisotropy of P-waves and S-waves as a function of pressure, see text for definitions.

Figure 8 Evolution of isotropic seismic velocities (Vp P-wave,Vs S-wave, Vφ bulk sound) and velocity ratios as a function of pressure. See text for definitions of the ratios Rs/p, Rφ/s and Rρ/s.