Thermodynamics of pyrope - majorite, Mg3Al2Si3O12 ...

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Thermodynamics of pyrope - majorite, Mg3Al2Si3O12 -Mg4Si4O12, solid solution from atomistic model

calculationsVictor Vinograd, Bjoern Winkler, Andrew Putnis, Herbert Kroll, Victor

Milman, Julian Gale, Olga Fabrichnaya

To cite this version:Victor Vinograd, Bjoern Winkler, Andrew Putnis, Herbert Kroll, Victor Milman, et al.. Thermody-namics of pyrope - majorite, Mg3Al2Si3O12 - Mg4Si4O12, solid solution from atomistic model calcula-tions. Molecular Simulation, Taylor & Francis, 2006, 32 (02), pp.85-99. �10.1080/08927020500501599�.�hal-00514976�

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Thermodynamics of pyrope - majorite, Mg3Al2Si3O12 - Mg4Si4O12, solid solution from atomistic model calculations

Journal: Molecular Simulation/Journal of Experimental Nanoscience

Manuscript ID: GMOS-2005-0059.R1

Journal: Molecular Simulation

Date Submitted by the Author:

25-Nov-2005

Complete List of Authors: Vinograd, Victor; University of Frankfurt, Institute of Mineralogy Winkler, Bjoern; University of Frankfurt, Institute of Mineralogy Putnis, Andrew; University of Muenster, Institute of Mineralogy Kroll, Herbert; University of Muenster, Institute of Mineralogy Milman, Victor; Accelrys

Gale, Julian; Curtin University of Technology, Nanochemistry Research Institute Fabrichnaya, Olga; Max-Planck-Institute for Metal Research

Keywords: Monte Carlo simulations, pyrope-majorite s.s., activity-composition relations, cubic/tetragonal transition

http://mc.manuscriptcentral.com/tandf/jenmol

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Table 1. The empirical interatomic potentials used in the present study. The notation [4]

and [6] refers to the coordination number of the associated species.

Buckingham

Interaction A (eV) ρ (Å) C (eV* Å6)

Al(core)-O(shell) 1262.2081 0.286370 0.0

Mg(core)-O(shell)

1432.8544 0.277265 0.0

Si(core)-O(shell) 1073.4668 0.298398 0.0

O(shell)-O(shell) 598.8996 0.314947 26.89746

Spring

Interaction K (eV* Å-2 ) O(core)-O(shell) 56.5598

Three-body

Interaction kθ (eV*rad-2 ) θ ( degree)

O(shell)-Si[4]- O(shell) 0.77664 109.47

O(shell)-Si[6]- O(shell) 2.2955 90.0

O(shell)-Al[4]- O(shell) 1.2883 109.47

O(shell)-Al[6]- O(shell) 1.8807 90.0

Note: The charges on the oxygen core and shell are 0.746527 and -2.446527, respectively. Cutoff distance for the Buckingham potentials is 12 Å.

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Table 2. Structural data and elastic stiffness coefficients for majorite as calculated with

the SLEC and DFT GGA in comparison with available experimental data. Cell parameters

XRD SLEC DFT-GGA

a (Å) 11.501a; 11.481b; 11.519c

11.494 11.670

c (Å) 11.48a; 11.406b; 11.420c

11.392 11.561

a/c 1.0018 1.0089 1.0094 Volume (Å3) 1518.494 a 1505.040 1574.474

Atomic coordinates

XRDa SLEC

(Core)

DFT-GGA

x 0.1253 0.1256 0.1287 y 0.0112 0.0120 0.0135

Mg1 16f

z 0.2587 0.2623 0.2665 x 0.0000 0.0000 0.0000 y 0.2500 0.2500 0.2500

Mg2 8e

z 0.6258 0.6227 0.6235 x 0.0000 0.0000 0.0000 y 0.0000 0.0000 0.0000

Mg3 8c

z 0.5000 0.5000 0.5000 x 0.0000 0.0000 0.0000 y 0.2500 0.2500 0.2500

Si1 4a

z 0.3750 0.3750 0.3750 x 0.0000 0.0000 0.0000 y 0.2500 0.2500 0.2500

Si2 4b

z 0.8750 0.8750 0.8750 x 0.1249 0.1261 0.1254 y 0.0065 0.0116 0.0116

Si3 16f

z 0.7544 0.7560 0.7568 x 0.0000 0.0000 0.0000 y 0.0000 0.0000 0.0000

Si4 8d

z 0.0000 0.0000 0.0000 x 0.0282 0.0257 0.0268 y 0.0550 0.0603 0.0588

O1 16f

z 0.6633 0.6685 0.6691 x 0.0380 0.0429 0.0447 y 0.9529 0.9540 0.9565

O2 16f

z 0.8562 0.8610 0.8611 x 0.2195 0.2248 0.2234 y 0.1023 0.1079 0.1069

O3 16f

z 0.8021 0.8070 0.8053 x 0.2150 0.2145 0.2117 y 0.9106 0.9161 0.9165

O4 16f

z 0.7000 0.7013 0.7028 x 0.9412 0.9353 0.9372 y 0.1617 0.1629 0.1638

O5 16f

z 0.4680 0.4690 0.4678 x 0.8960 0.8980 0.8977 y 0.2080 0.2102 0.2153

O6 16f

z 0.7851 0.7821 0.7833

Elastic constants Observed (GPa) SLEC (GPa)

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C11 286.4d 295.10

C44 85.0d 85.23

C66 93.2d 93.66

C12 83.0d 112.7

C16 1.4d 14.72

Bulk modulus 159.8d; 166.0e 170.14

References: aAngel et al. [9]; bKato & Kumazawa [8]; cHeinemann et al. [11]; dPacalo & Weidner [33]; eSinogeikin & Bass [34];.

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Table 3. The comparison between empirically and ab initio calculated differences in the

lattice energies of selected configurations.

Energy difference

Empirical

potentials

Ab initio

D-(A+B)/2 0.0508 0.0744

C-B 0.1054 0.1556

Notations: A – pyrope, d3Ia ; B – majorite, I41/a; C - Majorite, P4132; D - Majorite50, 3Ia . Values are in eV

per octahedral atom.

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Table 4. The final parameters of the fitted cluster expansion.

Pairwise energies

Mg-Al Al-Si Mg-Si Mg-Al Al-Si Mg-Si Distance (Å)

0 GPa 20 GPa 4.9511 -13.849 -12.332 -32.350 -21.125 -21.611 -54.338 5.7178 -17.769 -16.637 -46.718 -20.852 -18.233 -51.553 8.0883 -9.239 -8.113 -16.804 -11.700 -9.697 -24.627 9.4852 -6.203 -4.772 -8.486 -7.239 -4.259 -13.335 9.9072 -4.526 -3.923 -5.208 -7.152 -3.093 -8.592

11.4400 -2.321 -3.344 -5.148 -5.540 -6.009 -7.435 12.4676 -2.948 -0.165 0.664 -7.252 -3.375 -1.886 12.7916 -2.093 -1.855 -1.742 -4.913 -1.385 -2.286

The elastic term

Parameter 0 GPa 20 GPa A12 610.85 906.51 A21 640.00 899.48

Quaternary energies

Cluster type 0 GPa 20 GPa MgMgMgSi -3.8054 -4.0740 MgSiMgSi -12.7228 -14.4620 MgSiSiSi -3.3496 -4.3289 AlAlAlSi 2.7180 1.0271 AlAlSiSi 5.0292 3.2307 AlSiAlSi -2.0462 -3.1154 AlSiSiSi 1.6866 2.3576

AlAlAlMg -0.8564 -0.0710 AlAlMgMg 0.0363 2.2000 AlMgAlMg -6.4528 -6.8823 AlMgMgMg -1.7643 -2.1405

MgAlSiSi 2.1632 1.0320 MgSiAlSi -7.2655 -4.7004

AlSiMgMg -0.4698 0.0186 AlMgSiMg -5.8033 -7.8190 MgSiAlAl 0.0 0.0 MgAlSiAl 0.0 0.0 MgMgSiSi 0.0 0.0

MgMgMgMg 0.0 0.0 SiSiSiSi 0.0 0.0

AlAlAlAl 0.0 0.0 Note: the values are in kJ per mole of octahedral cations, or per 1.5 moles of Si[4] atoms.

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Table 5. Coefficients of the Redlich-Kister polynomial for the excess free energy in

the pyrope – majorite solid solution.

N A1n A2n*10-3

A3n*10-6

[kJ/mol] [kJ/mol/K] [kJ/mol/K2]

1 20.7761 2.91736 -2.3053

2 -11.1357 19.77231 -5.979

3 -17.4504 22.02424 -4.9062

4 14.7575 -24.45104 8.7821

5 40.7301 -36.20679 8.1251

6 -13.7634 19.75529 -6.5512

7 -17.2006 10.69285 -1.4385

N B1n B2n*10-3

B3n*10-6

[kJ/mol] [kJ/mol/K] [kJ/mol/K2]

1 -0.6976 2.1641 -0.3205

2 3.1557 -0.0637 0.1517

3 -0.2216 9.8304 -3.9812

4 -16.0881 20.3068 -5.5879

5 -24.0941 1.1400 4.6094

6 23.0833 -26.5699 7.2069

7 37.4186 -16.7439 0.0

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For Peer Review Only1.6 1.8 2 2.2 2.4 2.6

Interatomic distance (experiment)

1.6

1.8

2

2.2

2.4

2.6

Inte

rato

mic

dis

tan

ce (

calc

ula

ted

) Mg-O (DFT GGA)

Si-O (DFT GGA)

Si-O (SLEC)

Mg-O (SLEC)

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For Peer Review Only0 4 8 12 16 20 24 28 32

Static lattice energy (GULP) [kJ/mol]

0

4

8

12

16

20

24

28

32F

itte

d e

ner

gy

[k

J/m

ol]

a

Majorite, C2/c

Majorite, I 41/a

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For Peer Review Only0 5 10 15 20 25 30

Excess enthalpy (GULP) [kJ/mol]

0

4

8

12

16

20

24

28

32F

itte

d e

nth

alp

y [

kJ/

mo

l]b

Majorite, C 2/c

Majorite, I 41/a

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For Peer Review Only0 5 10 15 20 25 30

Excess enthalpy (GULP) [kJ/mol]

0

5

10

15

20

25

30F

itte

d e

nth

alp

y [

kJ/

mo

l]

20 GPa

Majorite I 41/a

Majorite C 2/c

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For Peer Review Only0 0.2 0.4 0.6 0.8 1

Mole fraction of majorite

0

5

10

15

20

25

30E

nth

alp

y o

f d

iso

rder

[k

J/m

ol]

Pyr Maj

20 GPa

3673 K

1073 K

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For Peer Review Only0 1000 2000 3000 4000

Temperature, K

0

0.2

0.4

0.6

0.8

1L

RO

par

amet

er

Phillips et al., 1992

(NMR)

Angel et al., 1989

(XRD)

Majorite

20 GPa

0 GPa

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For Peer Review Only1000 2000 3000 4000

Temperature, K

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8P

rob

abil

ity

dif

fere

nce

, 3

Pij

kl(S

i) -

Pij

kl(V

a)MgMgSiSi

MgMgMgMg+SiSiSiSi

MgSiMgSi

MgMgMgSi+MgSiSiSi

0 GPa

MajoritePage 15 of 116


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0

2

4

6

8

10

Co

nfi

gu

rati

on

al e

ntr

op

y [

J/m

ol/

K]

20 GPaIdeal entropy

3673 K

1073 K

Pyr Maj

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-20

-15

-10

-5

0

Fre

e en

erg

y o

f m

ixin

g [

kJ/

mo

l]1073 K

3673 K

MajPyr

20 GPa

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0

0.2

0.4

0.6

0.8

1

Act

ivit

y

2673 K

1073 K

20 GPa

Pyr Maj

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1366.4

1366.6

1366.8

1367

1367.2U

nit

cel

l v

olu

me

[A

]3

o

2273 K, 20 GPa

Pyr Maj

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Thermodynamics of pyrope - majorite, Mg3Al2Si3O12 -

Mg4Si4O12, solid solution from atomistic model calculations

V. L. VINOGRADa,*

, B. WINKLERa, A. PUTNIS

b, H. KROLL

b, V. MILMAN

c,

J. D. GALEd and O. B. FABRICHNAYA

e

a University of Frankfurt, Institute of Mineralogy,

Senckenberanlage 30, 60054 Frankfurt a.M., Germany

b University of Münster, Institute of Mineralogy,

Correnssrasse 24, 48149, Münster, Germany

c Accelrys, 334 Cambridge Science Park,

Cambridge CB4 0WN, United Kingdom

d Nanochemistry Research Institute, Curtin University of Technology,

PO Box U1987, Perth 6845, Western Australia.

e Max-Planck-Institute for Metal Research,

Heisenbergstr 3, 70569 Stuttgart, Germany

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Abstract

Static lattice energy calculations, based on empirical pair potentials have been performed

for a large set of different structures with compositions between pyrope and majorite, and

with different states of order of octahedral cations. The energies have been cluster

expanded using pair and quaternary terms. The derived ordering constants have been

used to constrain Monte Carlo simulations of temperature-dependent properties in the

ranges of 1073 – 3673 K and 0 – 20 GPa. The free energies of mixing have been

calculated using the method of thermodynamic integration. At zero pressure the

cubic/tetragonal transition is predicted for pure majorite at 3300 K. The transition

temperature decreases with the increase of the pyrope mole fraction. A miscibility gap

associated with the transition starts to develop at about 2000 K and xmaj=0.8, and widens

with the decrease in temperature and the increase in pressure. Activity-composition

relations in the range of 0-20 GPa and 1073- 2673 K are described with the help of a

high-order Redlich-Kister polynomial.

Keywords: Pyrope-majorite s.s., Monte Carlo simulations, cubic/tetragonal transition,

activity-composition relations.

* Corresponding author. Tel. +49 69 798 22764

E-mail address: [email protected]

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1. Introduction

Recent progress in the understanding of the chemical composition and thermo-physical

properties of the Earth’s mantle has been achieved by comparing geophysical data on

seismic velocities with the corresponding properties of theoretically calculated

equilibrium mineral assemblages subjected to high pressures and temperatures, e.g. [1].

Pyrope- and majorite-rich garnet is thought to be an important phase of variable

composition in the transition zone of the Earth’s mantle (410-650 km). Theoretical

calculations show [2, 3] that the mole fraction of majorite, Mg4Si4O12, component in the

aluminosilicate garnet increases gradually with depth. This corresponds to an increase of

the site fractions of Mg and Si in the octahedral position of the garnet structure, which

occurs via the coupled substitution 2Al3+ = Mg2+ + Si4+. Any realistic model of mineral

transformations in the transition zone requires knowledge of the thermodynamic activities

of pyrope and majorite in the garnet solid solution. This study is concerned with atomistic

modelling of thermodynamic mixing effects associated with this substitution.

Due to practical challenges, few experimental data is available to constrain the

thermodynamics of mixing in the pyrope-majorite binary system. Akaogi et al. [4] have

determined heats of dissolution in molten 2PbO*B2O3 of several cubic garnets with

compositions in the range of 0 < xmaj <0.58. They have observed an approximately linear

variation of the heat of dissolution in this compositional range. The dissolution enthalpy

of the majorite end-member with the tetragonal structure has been measured by Yusa et

al. [5]. Using a linear extrapolation of the data of Akaogi et al. [4], Yusa et al. [5] have

estimated the enthalpy of the cubic/tetragonal transition in majorite to be 20.4 ± 5.0

kJ/mol per 12 oxygen atoms. The combination of the data of Akaogi et al. [4] and Yusa et


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al. [5] suggests a positive enthalpy of mixing at intermediate compositions with a value

of about 12 ± 3.0 kJ/mol at xmaj=0.6. Phase equilibrium studies cannot be used to tightly

constrain mixing properties due to large uncertainties in the pressure calibration and in

the equilibrium composition of the garnet phase. The reported maximal solubility of

majorite in garnet in an assemblage with wadsleyite and stishovite varies between

different studies by up to 20 mole %. [6, 7]. However, the lack of thermodynamic data is

partially compensated by an abundance of structural information. It is known that pure

majorite at low temperatures crystallizes in the tetragonal space group I 41/a [8]. Pyrope-

rich compositions have a cubic structure with space group d3Ia . The cubic/tetragonal

transition is caused by long-range ordering of Mg and Si in the octahedral site. Majorite

samples synthesized at 1800 °C, 17 GPa [9] and at 2000 °C, 17.7 GPa [10] show nearly

complete Mg/Si order. Heinemann et al. [11] have located the cubic/tetragonal transition

at xmaj=0.8 in samples synthesized at 2000 °C and 19 GPa. However, Heinemann et al.

[11] and Hatch and Ghose [12] noted that the true temperature of the order/disorder

transition cannot be judged based on the synthesis temperatures. The tetragonal majorites

are twinned, thereby suggesting that they have a higher symmetry precursor. Heinemann

et al. [11] and Hatch and Ghose [12] argued, therefore, that the cubic/tetragonal transition

occurs during the quench and thus the transition boundary lies somewhere below 1800

°C. This assumption has been questioned by Wang et al. [13] who have observed tweed

microstructure in majorite-rich samples synthesized at 2300 °C, just below the melting

temperature, which is about 2600 °C for pure majorite [14]. Wang et al. [13] concluded

that the transition occurs at about 2300 °C. The location of the transition boundary is a

matter of an ongoing debate, e.g. [15], which has important implications for models of

the rheological properties of the transition zone. If the stability field of tetragonal

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majorite-rich garnets is intersected by the mantle geotherm, it is most likely that the

elastic properties of majorite are altered due to the presence of mobile domain walls. For

example, a drastic change in the viscoelasticity due to the cubic/tetragonal transition has

recently been observed experimentally in Ca1-xSrxTiO3 perovskite [16]. Heinemann et al.

[11] have noted that the increase of the pyrope mole fraction should enhance disorder on

octahedral sites and move the transition to even lower temperatures, thus decreasing the

likelihood for an intersection of the I 41/a stability field by the geotherm. In this sense, the

pyrope-majorite system behaves then analogously to the plagioclase solid solution where

an increase in the albite mole fraction leads to a rapid decrease of the 1/1 IC transition

temperature [17]. Referring to the same analogy, one can expect that the mixing enthalpy

has a kink at the transition line and that a miscibility gap, associated with the boundary,

develops at some low temperature. Such a gap should be reflected in the thermodynamic

activities of pyrope and majorite. The commonly used ‘two sublattice’ model [4, 5, and

14] ignores this complication. In this model, the configurational entropy related to the

octahedral atoms is assumed to be ideal for the Mg-rich and Si-rich sublattices, i.e. Al is

allowed to mix randomly either with Mg in the Mg-rich sublattice or with Si in the Si-

rich sublattice. Being consistent with the complete long-range order in pure majorite, the

‘two sublattice’ model offers a reasonable approximation for the configurational entropy

of well ordered tetragonal samples. However, the application of this model to pyrope-rich

cubic garnets cannot be easily justified: In the cubic phase Mg, Si and Al should be

allowed to mix within the same sublattice. Since the composition of the majoritic garnet

widely varies in different Mg-Al-Si-O mineral assemblages [3, 4], it is desirable to

develop a model which is applicable to the whole compositional range and to both cubic

and tetragonal garnets. Based on empirical static lattice energy and quantum mechanical


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total energy calculations, as well as Monte Carlo modelling, this study aims to constrain

the cubic/tetragonal transition and mixing properties along the pyrope – majorite binary

in the pressure and temperature range relevant for the transition zone.

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2. Simulation methodology

The methodology we adopt in this study has been established in a series of recent

simulation studies [18-24]. It consists of the following steps:

– Development and testing of a set of empirical interatomic potentials.

– Static lattice energy calculations (SLEC) on a set of structures with randomly varied

cation configurations.

– Finding a simple equation that describes the energies of the simulated structures. This

procedure is known as the ‘cluster expansion’.

– Using the cluster expansion model to obtain temperature-dependent properties by

Monte Carlo simulation.

– Calculation of the free energies of mixing and ordering by thermodynamic integration

of the Monte Carlo results.

– Refitting the free energies to simple polynomial equations useful for phase equilibrium

calculations.

2.1. The empirical potentials

A set of empirical interatomic potentials has been developed in this study using the relax-

fitting procedure [25] as implemented in the GULP program [26]. The set involves two-

parameter Metal-Oxygen (M-O) Buckingham potentials, three-body O-M-O angle-

bending terms and the shell model for the oxygen polarisability [27-29]. Following

Vinograd et al. [30] we have multiplied formal cation and anion charges by the common


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factor 0.85, so that the charges of Mg, Al, Si and O have been reduced to the values of

1.7, 2.55, 3.4 and -1.7, respectively. Such a reduction leads to a much better

transferability of the potentials within mono- and complex oxides. The possible reason

for this improvement is that in dense structures the formal charges on cations cause too

strong a cation-cation repulsion. Vinograd et al. [30] noted that the cation-cation

distances tend to be too long in the formal-charge models. The reduction of the charges

by a small common factor removes this problem, but conveniently preserves the charge

balance. The Buckingham Si(core)-O(shell) and O(core) – O(shell) potentials have been

fitted to structural and elastic constants of stishovite, coesite and α-quartz. The Al – O

potential has been fitted to the data on corundum and the three Al2SiO5 polymorphs. The

Mg – O potential has been fitted to the data on periclase, spinel (MgAl2O4), forsterite,

wadsleyite, ringwoodite, ilmenite, perovskite, pyrope and cordierite. The three-body

terms have then been refitted to relevant subsets of the above-mentioned minerals. The

elastic constants have been adopted from the compilation of Bass [31]. The structural

constants have been taken from the online American Mineralogist Crystal Structures

Database [32]. The potential set is reproduced in Table 1. Table 2 compares the predicted

and observed structural and elastic parameters of majorite. The data for majorite have not

been included in the fit, and therefore can be used as a test of the transferability and

accuracy of the potentials.

‘[Insert table 1 about here]’


‘[Insert fig. 1 about here]’

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Deleted: Table 1. The empirical interatomic potentials used in the present study. The notation [4] and [6] refers to the coordination number of the

Deleted: predicted

Deleted: as calculated with the SLEC and DFT GGA in Deleted: the

Deleted: comparison with available experimental data.¶

... [17]

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2.2. Quantum mechanical calculations as a test for the accuracy of the potentials

While calculations based on empirical potentials are computationally very efficient, the

predictive power of this approach requires an independent test. Here we compare selected

SLEC results with parameter-free quantum mechanical calculations. For crystals, most

quantum mechanical calculations are currently based on the density functional theory

(DFT) [35-39]. While DFT itself is exact [35], practical calculations require an

approximation for the treatment of the exchange and correlation energies. Here we use

the generalized gradient approximation (GGA), in the form suggested by Perdew, Burke

and Ernzerhof [40]. Results based on GGA calculations are generally in better agreement

with experiment than those obtained with the local density approximation, LDA [41-43].

The study of structures with large unit cells requires a computationally efficient

approach. Here we use the computational scheme in which the charge density and

electronic wavefunctions are expanded in a basis set of plane waves. To avoid explicit

description of tightly bound core electrons, the approach employs ‘ultrasoft’

pseudopotentials [44,45] which mimic the screening of the Coulomb potential of the

nucleus by the core electrons.

Both the academic and commercial versions of CASTEP were used in the present

study for the DFT-GGA calculations. The cut-off energy for the plane wave expansion

was 380 eV. Calculations were performed for ordered tetragonal I 41/a majorite, using

the primitive cell and a k-point sampling of 4x4x4. In all calculations, all symmetry

independent structural parameters were varied simultaneously in the search for the

ground state geometry. The parameters of the ‘relaxed’ (optimized) structure are given in


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Table 2 and in Fig. 1 where they are compared to experimental values and to the results

of the calculations based on empirical potentials. We also computed the ground state

structures of cubic, P 41 32 majorite, d3Ia pyrope and the intermediate, Maj50, structure

with 3Ia symmetry in order to compare the differences of their total energies to those of

the static lattice empirical-potential based calculations. Table 3 shows the differences in

the lattice energy between the configurations of the same composition. The difference in

energies of the cubic P 41 32 and tetragonal I 41/a majorites constitutes 0.105 eV (10.2

kJ/mol) and 0.156 eV (15.0 kJ/mol) for the SLEC and ab initio calculations, respectively.

The energy of the intermediate structure 3Ia is larger than the energy of the equal

mixture of pyrope and tetragonal, I 41/a, majorites by 4.9 kJ/mol (SLEC) and by 7.2

kJ/mol (DFT GGA). These results show that the ab initio and SLEC results are consistent

with each other, but also point to the limitations of these methods. The differences in the

results could be due to limited transferability of the empirical potentials and due to the

“underbonding” usually observed in DFT GGA calculations. DFT GGA calculations

slightly overestimate lattice parameters, which is also seen in the present results for I 41/a

majorite (a = 11.67 Å, c = 11.56 Å). Fig. 1 shows that the distances of short Si-O and Mg-

O bonds, predicted ab initio, are in a very good agreement with the experiment, while the

length of Mg-O bonds with distances larger that 2.3 Å is exaggerated. In view of these

limitations, the 30% difference between the SLEC and DFT GGA could be considered

reasonable. The atomic coordinates and bond distances calculated with the SLEC agree

well with the data of Angel et al. [9]. However, the a/c ratios calculated with the SLEC

and DFT GGA disagree with the very small ratio reported by Angel et al. [9]. On the

contrary, our results compare well with the ratios reported by Kato & Kumazawa [8],

Heinemann et al. [11] and McCommon and Ross [46]. In summary, we conclude that the

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Deleted: the …inaccuracy of modeling interatomic interactions with…or

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Deleted: It is possible that this overestimation could alter the

Deleted: The a/c ratios and atomic coordinates calculated with Deleted: The atomic coordinates and bond distances predicted both by

... [49]

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new potentials predict reasonable structural parameters and reasonable energy differences

between the structures selected for the ab initio test. These potentials will be used below

to predict the energies of a larger set of structures different in the composition and the

state of order.


2.3. Supercell calculations

The SLEC calculations have been performed in the athermal limit with GULP in a 2x2x2

supercell (128 octahedral sites) on set of specially constructed structures (configurations)

with different compositions and ordering states. In the first set of calculations, the

pressure was fixed at 0 GPa. We have started with the ordered octahedral arrangement of

Mg and Si consistent with the I 41/a structure determined by Angel et al. [9] and

calculated its relaxed energy. The relaxed coordinates have been used as starting values

for a new structure in which one randomly chosen pair of Mg and Si was swapped. The

swapping has been repeated 60 times. During this exercise, the static energy increased

sharply during the initial steps. Then the rate of increase slowed down indicating the

approach to a set of random distributions. The 60 generated structures with variable

degrees of disorder have been added to the data base. The same procedure has been

repeated for structures with the compositions xmaj=0.75, xmaj=0.50 and xmaj=0.25, which

have been prepared from I 41/a majorite by replacing appropriate numbers of Mg and Si

with Al atoms. The whole set of configurations was used again in a further set of force

field calculations at 20 GPa.


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2.4. Cluster expansion

The aim of the cluster expansion is to find a simple equation, which fits the energies (or

enthalpies in the case of a nonzero pressure) of all simulated configurations and,

hopefully, predicts the energy of any other possible configuration. One popular form for

such an expansion is known as the Js formalism, e.g. [19,47],

0AB2/1 E+JPzE

n

n(n)

(n)

i ∑≈ (1)

where n(n)

(n)JP,z and

ABare the coordination numbers, frequencies of AB-type pairs and

ordering constants for pairs of the order n. Jn corresponds to the energy (or enthalpy) of

the exchange reaction

AA+BB=2AB (2)

between atoms A ∈ {Al, Mg, Si} and B ∈ {Al, Mg, Si} located at the n-th distance.

When this equation is applied to the excess energies, E0 is either assumed to be equal to

zero, e.g. [23], or fitted together with the Js, e.g. [24]. In order to determine the set of the

Js for each of the 240 structures prepared in the way described above we have calculated

the frequencies of occurrence of pairs of dissimilar atoms at 8 distances, as specified in

Table 4. Since there are 3 types of dissimilar pairs, Mg-Si, Mg-Al, and Al-Mg, each

configuration i is characterized by 24 frequencies, Pn (the factor 1/2z(n) is included in the

frequency value), and by a value of the relaxed excess energy Ei, where the excess value

is defined relative to the weighted sum of the relaxed energies of pyrope and I 41/a

majorite. The whole set of configurations has been thus characterized by the 240x24


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frequency matrix P and with the 240-element vector E. With each dissimilar pair we

associate a constant Jn . The vector J is traditionally found by solving the matrix equation

J=P-1

E using a least squares method. We adopt here a slightly different technique.

Vinograd et al. [24] in the study of pyrope-grossular system have observed that when the

fit is applied separately to configurations of the same composition, the E0 value varies

with the composition non-linearly. It was shown that this variation can be well

approximated with a Margules two-parameter equation

)( 21121210 Ax+Axxx=E 2 (3)

where xi are the mole fractions of the end-members. Physically, E0 describes the energy,

which is required to prepare a virtual crystal from the two end-members. In the virtual

crystal the exchangeable atoms, Mg, Al, and Si are “homogenized” forming a virtual

atom with the average size and charge. The state of the virtual crystal is the reference

state of the Js expansion. The Js describe the energetic advantage of separating pairs of

virtual atoms into the pairs of dissimilar atoms and thus include bond relaxation effects

and the Coulomb interaction. In the case of the pyrope-majorite solid solution the mixing

of the cations occurs not only in the solution, but also at the end-member (majorite)

composition. Therefore, E0 does not vanish at x2=1, but includes the energy of

homogenization of Mg2+ and Si4+. To take into account the non-zero value of E0 for

majorite, we modify Equation 3 as follows

)( 21121210 Ay+Ayyy=E 2 (4)

where y2= x2/2=Xmaj/2 and y1=1- y2. Since E0 is a configuration-independent term, its


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value must not be mapped onto the Js expansion. Therefore, the J vector is found from

J=P-1

E', where the vector E' is obtained from the vector E by subtracting the value of E0

from each element. Within this procedure we calculate the vector E'', which represents

the approximation to E' predicted with the Js expansion. Finally, we search for the values

of A12 and A21, which result in the best fit. The set of the Jn and Aij parameters is then

used to calculate energies of configurations within a Monte Carlo algorithm. However,

experience shows that the Js calculated in the described way often predict wrong ground

states. In order to avoid this problem we have designed a feedback algorithm (Fig.2) in

which the Js self-educate to predict the correct ground state. The algorithm involves a

Monte Carlo annealing step, which is applied to the small 2x2x2 supercell. The idea of

using the small cell is that the Monte Carlo simulated configuration can be used as an

input for a new SLEC run. We start with a configuration, which is assumed to be a

candidate for the ground state and gradually increase its Monte Carlo temperature up to a

certain value and then gradually decrease it until the configuration freezes in the lowest

energy state. This new configuration is energy-minimized, and its energy and the

frequency numbers are added to the set of previously simulated structures. The new least

squares solution is then obtained and the Monte Carlo annealing is repeated with the new

values of the Js, A12 and A21 parameters. Typically, a few iterations are needed for the Js

to converge to values consistent with the correct ground state. This procedure relies on

the assumption that the empirical model (GULP) predicts the correct ground state.


‘[Insert fig. 3,a,b about here]’



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¶Figure 1. The feed-back algorithm which improves the accuracy of the cluster expansion in predicting ground state configurations.¶¶Table 4. The final parameters of the fitted cluster expansion

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However, this did not happen in the majorite case: the Js converged, but the low-energy

Monte Carlo configuration fluctuated between two structures with identical pair

expansions, but very different GULP energies. One of these configurations corresponded

to the correct I 41/a ground state, while the other one to an unknown triclinic, C 2/c,

structure with a much higher lattice energy. Since these structures had identical pair

expansions, the Js fit was not able to predict the energies of both of them accurately, but

resulted in the same compromise value. The C 2/c configuration was subjected to the

randomization procedure and 60 new randomized configurations were added to the

original set of configurations. Figure 3 shows the Js fit to the whole set of configurations.

The low accuracy of the fit suggests that the pair expansion misses some important

physics. One can easily recognize that I 41/a and C 2/c structures are identical only in

terms of the octahedral distribution. In fact, the C 2/c structure can be obtained from the I

41/a structure with a parallel shift of all octahedral atoms while leaving all other atoms

fixed (Fig. 4). One can also observe that this shift destroys the tetragonal symmetry of the

whole lattice: the four-fold axes in the I 41/a phase pass through the centers of the

tetrahedral clusters of octahedral atoms, where all octahedral atoms are the same, and

through the Si[4] sites, that center these clusters. In C 2/c, the tetragonal symmetry is lost

because the “all-same” tetrahedral clusters are swapped with the clusters that do not

possess four-fold symmetry. Since the C 2/c structure has a higher lattice energy than I

41/a, one can assume that the “all-same” arrangement with the Si[4] in the center has a

lower energy than the all-same arrangement around an empty site. In fact, the octahedral

atoms in garnet form a perfect BCC lattice, which can be built by close packing of

tetrahedral clusters. There are 6 such tetrahedra per one octahedral site. In garnets only ¼


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of these tetrahedra are filled with Si[4] atoms, the other ¾ of the clusters are empty. This

means that the octahedral distribution in garnets is made of two different types of

tetrahedral clusters, filled and empty. This observation immediately suggests a new form

of the cluster expansion, which is able to reflect the energetic difference of these cluster

types:

( )∑∑ +−+≈lk,j,i,

ijklijklijklijkl

n

n(n)

(n)

i EQPPaJPzE 0SiVa

AB32/1 (5)

The first term in this equation describes the interactions within the pairs, while the second

term reflects the tendency of a tetrahedral group to locate itself either around Si[4] or an

empty site. The Pijkl and aijkl denote the probability (frequency) and the multiplicity of the

tetrahedral group ijkl, where i,j,k,l ∈ {Al, Si, Mg}. The factor of 3 is needed to equalize

the numbers of the filled and vacant sites. With each ijkl group we associate an energetic

Qijkl constant. There are 21 symmetrically and chemically distinct groups. However, we

have observed that when all 21 Qijkl constants are allowed to vary in the fit, their

magnitudes become unreasonably large and some values correlate strongly with each

other. We then arbitrarily fixed six of the constants at a value of zero and varied only the

remaining 15 constants. The reduction to the 15-term expansion resulted in reasonable

(small) values of the Q constants (Table 4) and didn't lead to any noticeable decrease in

the fit accuracy compared to the 21-term expansion. Figure 5 illustrates the fit accuracy

of the 15-term expansion. Apparently, the accuracy of J-Q expansion is sufficient for the

energies of I 41/a and C 2/c structures to be correctly distinguished. The J-Q expansion

thus provides the possibility to greatly increase the speed of energy calculation without a

significant loss in the accuracy. This in turn makes it feasible to efficiently simulate a

Boltzmann probability distribution of the octahedral atoms with the Monte Carlo method.

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2.5. Monte Carlo simulations

Monte Carlo simulations have been performed using a 4x4x4 supercell with periodic

boundary conditions (1024 octahedral sites) with our own code. The swapping of sites

has been performed according to the Metropolis algorithm. The energy differences

between the subsequent steps have been calculated using equation (5). The temperature

dependent properties have been calculated on a grid of 32 different compositions across

the pyrope – majorite binary in the interval of 1073 to 3673 K with a step of 200 K. Each

point in the T-X space was annealed for 500,000 Monte Carlo steps and additional

500,000 steps were used for the calculation of the averages. The whole procedure was

repeated twice, with the J-Q sets corresponding to 0 and to 20 GPa. The results of both

calculation runs are qualitatively similar and therefore we plot only the 20-GPa set.

Figure 6 shows the isotherms of the excess configurational enthalpy together with the

original set of energies calculated via the use of explicit interatomic potentials. It is clear

that even at 3673 K the octahedral distribution significantly deviates from a random one.

The breaks in the isotherms reflect the cubic/tetragonal transition. The long-range order

(LRO) parameter variation across the transition was specially investigated at the majorite

composition (Fig. 7). The LRO parameter has been defined as follows


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AβAα

AβAαod

P+P

PP=Q

−, (6)

where PAα and PAβ are the probabilities of finding A atom (e.g. Mg) in two dissimilar

octahedral sites. These two sites become structurally distinct in the tetragonal majorite

[9]. However, these probabilities (frequencies) cannot be directly determined from the

site occupancies derived from the Monte Carlo simulations because the LRO fluctuates

between the three equally possible orientations. Therefore, we have calculated the LRO

parameter indirectly from the probabilities of AA pairs. The pair probabilities are much

less affected by the spontaneous changes in the LRO orientation. Statistical theories of

LRO suggest, e.g. [49], that at short distances the pair probabilities are functions of both

short-range order (SRO) and LRO parameters. However, the SRO correlations rapidly

vanish with distance and by measuring the AA probability at the maximum separation in

the Monte Carlo supercell one can be fairly sure that the SRO contribution is

insignificant. (This distance was equal to 22.89 Å in our simulations.) Therefore

4/)1()1)(1( 2ododod

2AAβAαAA QQQP=PP=P −=−+ (7)

From Equation 7 one can easily recalculate Qod (Fig. 7). One observes that the predicted

degree of LRO at 2000 °C is in good agreement with the experimental result of Phillips et

al. [10] subject to the assumption that the measured quantity corresponds to the synthesis

temperature. LRO completely disappears at about 3300 K at 0 GPa and at 3450 K at 20

GPa. The transition can be also visualized by monitoring the probabilities of the two

types of tetrahedral clusters of octahedral sites; those with the Si[4] in the middle and the

vacant ones. The difference of these probabilities exhibits a striking change at the


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transition temperature (Fig.8).



2.6. Thermodynamic integration

It has been shown [18,19,50] that the configurational free energy can be calculated from

Monte Carlo averaged excess energies using the method of λ-integration:

λdEF=F

λ

λ∫+0

0 (8)

In this equation F0 corresponds to the free energy of mixing of the solid solution with

zero ordering energy, which can be calculated theoretically:

)lnlnln( AlAlSiSiMgMg0 xxxxxxTRF ++= (9)

The integral describes the contribution to the free energy from the energy (or the enthalpy

in the case of a nonzero pressure), E , when it changes from the state of complete

disorder to the state of equilibrium order. To calculate λ

E for a state with an

intermediate degree of order defined by certain value of λ, 0< λ <1, one scales the Js and

Qs

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Deleted: (Myers, 1999; Dove, 2001)…λ−

Deleted: Note that the E0 term, i.e. the energy of the disordered state, contributes only to the F0 term, but is excluded from the integral. … change…due to the ordering…When

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ijkl

λ

ijkln

λ

n QQJJ λλ == , (10)

and simulates Boltzmann’s distribution constrained with λ

nJ and λ

ijklQ . Effectively, the

scaling means that the probabilities of microstates become more random than in the non-

scaled case. λ

E is then calculated using equation 5 with nominal (not scaled) values of

Js and Qs by taking a simple average over a sufficient number of the equilibrated

configurations. The aim of the scaling is to make the distribution more random without

decreasing the strength of the interactions. In our simulations, λ �was gradually increased

from 0 to 1 with a step of 0.025. The integral was calculated using Simpson’s method.

The configurational entropy was calculated with the equation

TEFS /)( −= (11)

where E is the average excess energy (or enthalpy) calculated with λ =1. Figure 9

shows the configurational entropy in the interval of 1073 – 3673 K. This function is

severely affected by SRO and LRO. Evidently, the cationic distribution tends to the one-

sublattice regime at low concentrations of majorite and at high temperatures. The two-

sublattice regime is observed only at very low temperatures and at very majorite-rich

compositions. The behaviour of the entropy at intermediate compositions cannot easily be

predicted from simple model assumptions. Figure 10 shows the free energy of mixing at

20 GPa in the same interval of temperature. The excess values are calculated with respect

to the mixture of pyrope and fully ordered majorite. One observes that the excess free

energy of mixing of majorite deviates significantly from zero only at temperatures above

2673 K. This is the effect of disorder in majorite. One can also see that the miscibility

gap starts to develop along the cubic-tetragonal boundary at about 2500 K and xmaj=0.8.


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At zero pressure this temperature is about 500 K lower. Curvature analysis of the free

energies, combined with the order-disorder analysis, permits the drawing of the T-X

phase diagram (Fig. 11)





2.7. Activity-composition relations

The free energy calculations suggest that the effect of disorder in majorite becomes

significant only above 2673 K, i.e. at temperatures significantly exceeding the

hypothetical mantle geotherm [3]. Therefore, we limit our activity-composition model to

the 1073 – 2673 K interval. We have extracted the excess free energies of mixing from

the Monte Carlo energies and fitted them with a Redlich-Kister polynomial of the 6th

order. Polynomials of lower order were not able to describe the curvature of the free

energy isotherms in the vicinity of the order/disorder transition.


The fitting was performed separately with the data for 0 GPa and 20 GPa. The excess free

energies were calculated with respect to the ideal one-site molecular solid solution. Our

activities thus describe the mixing effect scaled to a single octahedral site. Table 5 plots

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

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Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

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Deleted: 0…¶¶Deleted: Figure 8. The configurational entropy of the Deleted: ¶

Deleted: Figure 10. The temperature – composition phase Deleted: S

Deleted: olid black curve shows the result for 20 GPa, while the Deleted: designate

Deleted: represent the experimental results of Heinemann et Deleted: have

Deleted: showed

Deleted: n

Deleted: cubic and tetragonal symmetries, respectively. ¶

Deleted: (Gasparik, 2003).

Deleted: by

Deleted: Table 5. Coefficients of the Redlich-Kister polynomial

... [104]

... [99]

... [103]

... [100]

... [119]

... [101]

... [122]

... [118]

... [121]

... [102]

... [98]

... [93]

... [125]

... [94]

... [126]

... [97]

... [127]

... [123]

... [128]

... [124]

... [129]

... [105]

... [130]

... [106]

... [120]

... [107]

... [132]

... [108]

... [133]

... [109]

... [134]

... [110]

... [135]

... [111]

... [136]

... [112]

... [137]

... [113]

... [138]

... [114]

... [139]

... [115]

... [95]

... [116]

... [96]

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the coefficients of the Redlich-Kister polynomial for the calculation of the excess free

energy of the solid solution in the ranges of 1073-2673 K and 0 - 20 GPa. The Akn set

gives the coefficients corresponding to 0 GPa and the Bkn set gives the difference between

the Redlich-Kister fit at 20 GPa and 0 GPa. The free energies of mixing corresponding to

any pressure of interest in the range of 0-20 GPa can be interpolated using the Akn and Bkn

coefficients with the equation

∑ −−=7

1

11221excess

=n

n

n )x(xCxxG (13)

where

( )∑ −3

1

120/=k

k

knknn TPB+A=C (14)

and where T is measured in K and P in GPa. Equations 13, 14 will probably be

reasonably accurate at pressures slightly above 20 GPa. Figure 12 plots the activity-

composition relations at 20 GPa. The activities of pyrope and majorite are very much

perturbed by the cubic/tetragonal transition and by the miscibility gap, which develops

along the transition line. Comparing the Figures 10 and 11, one observes that the simple

polynomial equation is not enough robust for the description of the narrow gap at the

high temperatures. This drawback of the model would not be significant in the majority

of petrological applications though.

2.8. Equilibrium volume

The relaxed volumes of the 260 configurations have been calculated using GULP through


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energy minimization calculations. The excess volumes have been cluster expanded with

an equation analogous to Equation 5. This has made it possible to find a set of pair and

quaternary volumetric constants, which approximates the volume of any configuration.

These constants have been used to monitor the equilibrium volume during the course of

the Monte Carlo simulations and to calculate the average volumes along the binary at

different pressures and temperatures. Fig. 13 shows the result for 20 GPa and 2273 K.

The shape of the volume isotherm compares well with the measurements of Heinemann

et al. [11] performed on a series of samples synthesized at 19 GPa and 2273 K. The

predicted and experimental curves both show that the cubic/tetragonal transition leads to

a slight, but noticeable, increase in the volume. Heinemann et al. [11] observed the break

at xmaj=0.8, which is in very good agreement with the calculated value of xmaj=0.75.



3. Discussion

The developed statistical-thermodynamic model helps in understanding the driving forces

which make the tetragonal majorite more stable than other possible ordered structures

with the same composition. Table 4 shows that the strongest ordering pair interactions

occur at the 1st, 2nd and 3rd neighbor distances. The electrostatic origin of these forces

becomes clear when comparing the values of the Js corresponding to Al-Si, Al-Mg and

Mg-Si interactions. Obviously, the strength of the interaction correlates with the charge

difference within the pairs of cations: the Mg-Si interaction is always stronger than those


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Deleted: Figure 12. The variation of equilibrium volume along the binary at 2273 K and 20 GPa predicted with the volumetric cluster expansion.¶¶

aa

¶Figure 13. The arrangement of octahedral cations in I41/a (a) and P 4 32 majorites.¶Deleted: to

... [140]

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of Al-Si and Al-Mg. The arrangement of the octahedral cations can be easily visualized

as a BCC lattice, with two dissimilar cubic clusters centred on Si and Mg, respectively

(Fig. 14a). Although it is possible to design a structure (Fig.14b) with an even larger

number of the 2nd Mg-Si pairs than in I41/a majorite, it will have a too low number of the

3rd-neighbor Mg-Si pairs. Our calculations show that this structure, with a symmetry of P

41 32, is significantly less stable than the tetragonal majorite. The tetragonal majorite is

certainly the best choice considering the advantage of having the maximum number of

the third-neighbor Mg-Si pairs and a not too low a number of the second-neighbor pairs.

However, as it was noted above, the optimum configuration of the octahedral cations can

have a different relationship with regard to the underlying sublattice of the Si[4] atoms.

The stabilization of the tetragonal majorite relative to the alternative C/2c phase probably

occurs due to the interactions of the octahedral cations with the Si[4] atoms. In order to

understand these interactions, it is convenient to visualize the BCC lattice as a

superposition of the near-neighbor four-atom clusters of octahedral cations. One quarter

of these clusters contain a Si[4] atom in the middle and the other three quarters are vacant.

Our simulations suggest (Fig. 8) that the MgMgMgMg, SiSiSiSi and MgMgSiSi have a

greater affinity to the Si[4] atoms than MgMgMgSi, SiSiSiMg and MgSiMgSi clusters.

Since the centering of the first three clusters with a Si[4] atom is consistent with the four-

fold symmetry, the I41/a majorite is more stable than the C2/c majorite.

The present calculations constrain the phase diagram and the activity-composition

relations in the pyrope - majorite system at the conditions which cover the probable

stability range of majorite in the transition zone. The main question is how accurate these

calculations are. The predicted degree of LRO in tetragonal majorite (Fig. 7) is in good

agreement with the data of Phillips et al. [10] obtained on a sample prepared at 2000 °C


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and 17.7 GPa. The sample of Angel et al. [9] synthesized at 1800 °C and 17 GPa falls

slightly outside the trend. However, the authors concede that the result of their X-ray site

occupancy refinement “must be treated with extreme caution because the scattering

factors of Mg and Si are very similar.” From bond lengths considerations they conclude

that the degree of Mg, Si order is probably larger than that obtained from site refinement.

The comparison of our results with those of Phillips et al. [10] and Angel et al. [9] is

valid only if the experimental data essentially represent equilibrium properties, i.e. if the

majorites were synthesized within the stability field of the tetragonal phase. However,

Hatch and Ghose [12], Heinemann et al. [11] and Tomioka et al. [15] argue that the

ordering to the tetragonal phase occurs during quenching such that the cubic/tetragonal

boundary would lie significantly lower than the temperature of synthesis, and certainly

significantly lower than our transition temperature (~3000 °C). The authors have based

their arguments on the observation of merohedral and pseudomerohedral twins generated

by the loss of symmetry elements due to the ordering transition. Such transformation

twins suggest that the majorites first crystallized with cubic symmetry. However, we will

argue that the mere occurrence of transformation twins may not necessarily imply that the

crystals have passed a transformation boundary during quenching. In fact, the metastable

formation of a higher symmetry disordered structure within the stability field of the low

temperature ordered form is not an unusual occurrence; it is to be expected if the starting

materials are reactive [51].

Two examples may be given. The first one concerns the 1/1 IC transition in Ca-rich

plagioclases. The ordering transition leads to an alternation of the Al and Si atoms and

thus produces superstructure reflections (so-called b-reflections) which may be used to

image type b anti-phase domains in the TEM. Note that in contrast to the ordering



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transition in majorite which is translationengleich, but klassenungleich and as such is

associated with the development of twins, the anorthite transition is klassengleich, but

translationenungleich so that anti-phase domains are generated. In natural anorthites (An

> 95%), although b-reflections are observed, no type b anti-phase domains can be

imaged. Obviously, these anorthites directly crystallized with the ordered stable 1I

structure. In contrast, anorthite samples synthesized below the melting temperature do

show type b domains although they crystallized in the stability field of the 1I phase [52,

53]. In the experiments of Kroll and Müller [53], the domains strongly increased in size

with annealing time. Therefore, they certainly did not develop during the quench lasting

only for seconds. This suggests that the synthetic anorthites first crystallized in a higher

symmetry phase ( 1C ) and only then developed the 1I -type ordering at the annealing

temperature. It is now agreed that also in natural anorthites with An-contents less than

95% showing type b domains nucleation of the domains occurred in a metastable 1C

albite matrix leading to the stable 1I structure without crossing a transformation

boundary (Smith and Brown [54], p.89).

Another well-documented example is the case of Mg-cordierite which when

synthesized from glass always forms the hexagonal high form (no long range order)

within the stability field of the orthorhombic ordered structure [55-58]. The first-formed

hexagonal phase has no domain structures. Continued annealing within the orthorhombic

stability field produces a modulated structure and ultimately a twinned orthorhombic

fully Al,Si ordered cordierite.

The discussion suggests that it is worth reconsidering the arguments of Hatch and

Ghose [12] and Heinemann et al. [11]. Although the occurrence of twins suggests the



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Deleted: Let us consider a phenomenon observed in the albite-anorthite system. It is well established (Carpenter and McConnell, 1984; Carpenter, 1991)

that the 1/1 IC order/disorder transition in pure anorthite would occur at about 2000 °C, a temperature significantly above the melting temperature (1553 °C). In contrast to the order/disorder transition in majorite, which is translationengleich, but klassenungleich and as such is associated with the development of twins, the anorthite transition is klassengleich, but translationenungleich such

Deleted: o

Deleted: that anti-phase domains are generated. The

1/1 IC transition leads to an alternation of Al and Si atoms and thus produces superstructure reflections h+k=odd, l=odd (so-called b-reflections) which may be used to image type b anti-phase domains in the TEM. In natural anorthites (An > 95%), although b-reflections are observed, no type b anti-phase domains can be imaged. Obviously, these anorthites directly

crystallized with the 1I anorthite structure. However, anorthite samples synthesized below the melting temperature do show type b anti-phase domains although they crystallized in the stability field of

Deleted: (1989)

Deleted: (1997)

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existence of a cubic precursor, it is clear that the presence of twinning is not an indication

that the material had ever been within the stability field of the high symmetry phase. We

suggest that the cubic phase orders to the stable tetragonal phase immediately after

crystallization. The tetragonal phase is characterized by the prevalence of MgMgMgMg,

SiSiSiSi and MgMgSiSi tetrahedral clusters centred on Si[4] atoms. Presumably, during

the initial stages of growth the clusters form with equal probability around both the Si[4]

sites and the vacant sites leading to cubic symmetry and only then their proportion

changes in favour of the Si sites leading to tetragonal symmetry.

One can argue that the empirical potential calculations might not be sufficiently

accurate for predicting energy differences between configurations with different states of

order, and thus the calculated transition temperatures could be significantly off. The

comparison of our SLEC and ab initio calculations shows that the ab initio results suggest

a larger difference in the energies of cubic and tetragonal majorites. If we assume that the

ab initio values represent the correct answer, then the real energetic difference between

ordered and disordered majorites should be even larger than that predicted by the SLEC

and thus the cubic/tetragonal transition would occur at an even higher temperature than

we have here predicted. However, we observe that the present SLEC results are in very

good agreement with the data of Heinemann et al. [11], assuming that these data

constrain true equilibrium. The predicted enthalpies of mixing are in good agreement

with the available calorimetric data of Akaogi et al. [4] and Yusa et al. [5]: Fig. 6 predicts

that the enthalpy of mixing of a Maj60 sample synthesized at 1273 K is about 11 kJ

kJ/mol per 12-oxygen formula unit, which compares well with the value of 12 ± 3.0

kJ/mol, estimated for this composition by Yusa et al. [5]. This agreement suggests that

the present model is reasonably correct.


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The present results might be questioned based on the lack of the evidence for a

miscibility gap in phase equilibrium studies. However, the miscibility gap could be

detected in a phase equilibrium study only if the both exsolved phases are stable at the

experimental conditions. Our phase equilibrium calculations for the pyrope-majorite

binary system (Fig. 15) suggest that the miscibility gap does not affect the topology of the

pressure-composition diagram at temperatures lower than 1600 °C. In these calculations,

aided with the Thermo-Calc program [59], we have used the thermodynamic database of

Fabrichnaya et al. [3] and the presently developed activity-composition model for the

garnet phase. The gap is absent in the 1773 K diagram because it occurs at compositions

that are not stable with respect to the other high-pressure mineral assemblages, such as

pyrope-rich garnet + clinopyroxene, garnet + wadsleyite + stishovite and garnet +

ilmenite. In the 2073 K set the gap is not revealed because at temperatures above 1873 K

our Redlich-Kister model does not reproduce the small inflection in the free energy of the

solid solution associated with the transition, although this inflection is present in the

Monte Carlo results. A narrow gap separating the cubic and tetragonal phases could be

possibly observed experimentally at about 1800 °C, where the majoritic garnet becomes

stable over a wide range of compositions. However, the experiments at such conditions

are extremely challenging.


4. Conclusions

Atomistic simulations have allowed the investigation of the behaviour of mixing


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functions in the pyrope-majorite system in great detail and thus to calculate the activity-

composition relations consistently with the cubic/tetragonal transition. The model

predicts that the transition occurs through a miscibility gap, which widens with the

decrease in the temperature. The calculations suggest that the gap does not affect the

topology of phase relations below 1600 °C, where the majorite-rich phase is not stable

with respect to other high-pressure mineral assemblages. However, if the mantle

geotherm rises above 1800 °C in the transition zone, it can cross the cubic/tetragonal

transformation boundary and the associated with the transformation narrow miscibility

gap. The heterogeneity of the garnet phase related to the miscibility gap and the twins

related to the cubic/tetragonal transformation might significantly alter rheological

properties of the lowest interval of the transition zone. These conclusions should be tested

for systems containing Fe and Ca.

Acknowledgements

The majority of the results included in this publication have been developed as

illustration materials for the lecture course “Introduction to Computer Simulations of

Minerals” read during the Spring semester 2005 at the University of Münster by AP and

VVL. Ulrik Hans, Jürgen Hansmann, Elis Hoffman, Arne Janßen and Dominik

Niedermeier are thanked for asking tough questions and for the help in the computation.

The support of the Deutsche Forschungsgemeinschaft (grant Wi 1232/14-2) is greatly

acknowledged. JDG would like to thank the Government of Western Australia for

funding under the Premier’s Research Fellowship program.

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References

[1] G. Fiquet. Mineral phases of the Earth's mantle. Z. Kristallogr., 216 248 (2001).

[2] T. Gasparik. Phase diagrams for geoscientists. An atlas of the Earth’s interior.

Springer-Verlag, Berlin, Heidelberg (2003)

[3] O.B. Fabrichnaya, S.K. Saxena, P. Richet, E.F. Westrum.. Thermodynamic data,

models and phase diagrams in multicomponent oxide systems. Springer-Verlag,

(2004).

[4] M. Akaogi, A. Navrotsky, T. Yagi, S. Akimoto. Pyroxene-garnet transformation:

Thermochemistry and elasticity of garnet solid solutions, and applications to a

pyrolite mantle. In High-Pressure Research in Mineral Physics. M.H. Manghnani and

Y. Syono (Eds.). pp. 251-260, Am. Geophys. Union, Washington, D.C. (1987).

[5] H. Yusa, M. Akaogi, E. Ito. Calorimetric study of MgSiO3 garnet and pyroxene: Heat

capacities, transition enthalpies, and equilibrium phase relations in MgSiO3 at high

pressures and temperatures. J. Geophys. Res., 48, 6453 (1993).

[6] M. Akaogi, S. Akimoto. Pyroxene-garnet solid solution equilibria in the systems

Mg4Si4O4-Mg3Al2Si3O12 and Fe4Si4O4-Fe3Al2Si3O12 at high pressures and

temperatures. Phys. Earth Planet. Interiors., 15, 90 (1977).

[7] M. Kanzaki. Ultrahigh-pressure phase relations in the system. Phys. Earth Planet.

Interiors, 49, 168 (1987).

[8] T. Kato, M. Kumazawa. Garnet phase of MgSiO3 filling the pyroxene-ilmenite gap at

very high temperatures. Nature, 316, (1985).

[9] R.J. Angel, L.W. Finger, R.M. Hasen, M. Kanzaki, D.L. Weidner, R.C. Liebermann,


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Deleted: , M.…, A.… T.,…and , S., 1987

Deleted: : M.H. Manghnani and Y. Syono (Editors).

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Deleted: ¶

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... [150]

... [151]

... [149]

... [145]

... [147]

... [146]

... [153]

... [148]

... [144]

... [154]

... [152]

Page 53 of 116


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31

D.R. Veblen. Structure and twinning of single crystal MgSiO3 garnet synthesized at

17 GPa and 1800 °C. Am. Mineral., 74, 509 (1989).

[10] B.L. Phillips, D.A. Howell, R.J. Kirkpatrick, T. Gasparik. Investigation of cation

order in MgSiO3-rich garnet using 29Si and 27Al MAS NMR spectroscopy. Am.

Mineral., 77, 704 (1992).

[11] S. Heinemann, T. Sharp, F. Seifert, D.C. Rubie. The cubic-tetragonal phase

transition in the system majorite (Mg4Si4O12) - pyrope (Mg3Al2Si3O12), and garnet

symmetry in the Earth's transition zone. Phys. Chem. Minerals, 24, 206 (1997).

[12] D.M. Hatch, S. Ghose. Symmetry analysis of the phase transition and twinning in

MgSiO3 garnet: implications to mantle mineralogy. Am. Mineral., 74, 1221 (1989).

[13] Y. Wang, T. Gasparik, R.C. Liebermann. Modulated microstructure in synthetic

majorite. Am. Mineral., 78, 1165 (1993).

[14] T. Gasparik. Phase relations in the transition zone. J. Geophys. Res., 95, 15751

(1990).

[15] N. Tomioka, K. Fujino, E. Ito, T. Katsura, T. Sharp, T. Kato. Microstructures and

structural phase transition in (Mg,Fe)SiO3 majorite. Eur. J. Mineral. 14, 7 (2002).

[16] R.J. Harrison, S.A.T. Redfern, J. Smith. The effect of transformation twins on the

seismic frequency mechanical properties of polycrystalline Ca1-xSrxTiO3 perovskite.

Am. Mineral., 88, 574 (2003).

[17] M.A. Carpenter, J.D.C. McConnell. Experimental delineation of the

1/1 IC transformation in intermediate plagioclase feldspars. Am. Mineral., 69, 112

(1984).

[18] E.R. Myers, V. Heine, M.T. Dove. Some consequences of Al/Al avoidance in the

ordering of Al/Si tetrahedral framework structures. Phys. Chem. Minerals, 25, 457


Formatted: Font: TimesNew Roman, German Germany

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... [160]

... [163]

... [161]

... [158]

... [162]

... [164]

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32

(1998).

[19] M.T. Dove. Computer simulations of solid solutions. In Solid Solutions in Silicate

and Oxide Systems of Geological Importance. EMU Notes in Mineralogy, 3. Ch.

Geiger (Ed.), pp. 245-249, Eötvös University Press (2001)

[20] A. Bosenick, M.T. Dove, E.R. Myers, E.J. Palin, C.I. Sainz-Diaz, B.S. Guiton, M.C.

Warren, M.S. Craig, S.A.T. Redfern. Computational methods for the study of

energies of cation distributions: applications to cation-ordering phase transitions and

solid solutions. Mineral. Mag., 65, 193 (2001).

[21] Warren, M.S., M.T. Dove, E.R. Myers, A. Bosenick, E.J. Palin, C.I. Sainz-Diaz,

B.S. Guiton, M.C. Craig, S.A.T. Redfern. Monte Carlo methods for the study of

cation ordering in minerals. Mineral. Mag., 65, 221 (2001).

[22] U. Becker, A. Fernandez-Gonzales, M. Prieto, R. Harrison, A. Putnis. Direct

calculation of thermodynamic properties of the barite/celestite solid solution from

molecular principles. Phys. Chem. Minerals, 27, 291 (2000).

[23] U. Becker, K. Pollock. Molecular simulations of interfacial and thermodynamic

mixing properties of the grossular-andradite garnets. Phys. Chem. Minerals, 29, 52

(2002).

[24] V.L. Vinograd, M.H.F. Sluiter, B. Winkler, A. Putnis, U. Hålenius, J.D. Gale, U.

Becker.. Thermodynamics of mixing and ordering in the pyrope-grossular solid

solution. Mineral. Mag., 68, 101 (2004).

[25] J.D. Gale. Empirical potential derivation for ionic materials. Phil. Mag. B, 73, 3.

(1996).

[26] J.D. Gale. GULP – a computer program for symmetry adapted simulations of solids.

J. Chem. Soc.: Faraday Trans., 93, 629 (1997).


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Formatted

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Formatted: Font: Bold,Italic



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Deleted: ¶Bass, J.D., 1995. Elasticity of Minerals, Glasses, and Melts. In: Th.J. Ahrens (Editor) Mineral Physics & Crystallography. A Handbook of Physical Constants. Am. Geophys. Union, AGU Reference Shelf Ser., 2: 45-63. ¶¶

Deleted: , U.

Deleted: A.,

Deleted: M.

Deleted: , R.

Deleted: and

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Deleted: 2000.

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Deleted: 2002.

Deleted: :

Deleted: -64

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... [168]

... [166]

... [165]

... [167]

... [169]

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33

[27] M.J. Sanders, M. Leslie, C.R. Catlow. Interatomic potentials for SiO2. Journal of the

Chem. Soc., Chem. Communications, 19: 1271 (1984).

[28] A. Patel, G.D. Price, M.J. Mendelsson. A computer-simulation approach to modelling

the structure, thermodynamics and oxygen isotope equilibria of silicates. Phys. Chem.

Minerals, 17, 690 (1991).

[29] B. Winkler, M.T. Dove, M. Leslie. Static lattice energy minimization and lattice

dynamics calculations on aluminosilicate minerals. Am. Mineral., 76, 313 (1991).

[30] V.L. Vinograd, M.H.F. Sluiter, B. Winkler, A. Putnis, J.D. Gale. Thermodynamics

of mixing and ordering in silicates and oxides from static lattice energy and ab initio

calculations. In First-Principles Simulations: Perspectives and Challenges in Mineral

Sciences. Deutsche Gesellschaft für Kristallographie. Berichte aus Arbeitskreisen der

DFK, 14. M. Warren, A. Oganov and B. Winkler (Eds), pp.143-151 (2004).

[31] J.D. Bass. Elasticity of Minerals, Glasses, and Melts. In: Mineral Physics &

Crystallography. A Handbook of Physical Constants. Am. Geophys. Union, AGU

Reference Shelf Ser., 2. Th.J. Ahrens (Editor), pp. 45-63 (1995).

[32] R.T. Downs, M. Hall-Wallace. The American Mineralogist Crystal Structure

Database. Am.Mineral.,88, 247.

[33] R.E.G. Pacalo, D.L. Weidner. Elasticity of majorite MgSiO3 tetragonal garnet. Phys.

Earth. Planet. Interiors, 99, 145 (1997).

[34] S.V. Sinogeikin, J.D. Bass. Elasticity of majorite and majorite-pyrope solid solution

to high pressure: Implications for the transition zone. Geophys. Res. Lett. 29(2), 4-1

(2002).

[35] P. Hohenberg, W. Kohn. Inhomogeneous electron gas. Phys. Rev. 136: B864.

(1964).

Formatted

Formatted

Formatted


Double


New Roman

Formatted: Indent:

Before: 0 pt, Hanging:

18 pt, Line spacing:

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Formatted

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... [174]

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For Peer Review O

nly

34

[36] W. Kohn, L.J. Sham. Self-consistent equations including exchange and correlation

effects. Phys. Rev. 140: A1133 (1965).

[37] R.G. Parr, W. Yang. Density-functional theory of atoms and molecules. Oxford

University Press (1989).

[38] E.S. Kryachko, E.V. Ludena. Energy density functional theory of many-electron

systems. In Understanding Chemical Reactivity, 4, Kluwer Academic Publishers,

Dordrecht (1990).

[39] R.O. Jones, O. Gunnarsson. The density functional formalism, its applications and

prospects. Rev. Mod. Phys., 61, 689 (1989).

[40] J.P. Perdew, K. Burke, M. Ernzerhof. Generalized gradient approximation made

simple. Phys. Rev. Lett., 77, 3865 (1996).

[41] T.C. Leung, C.T. Chan, B.N. Harmon. Ground-state properties of Fe, Co, Ni and

their monoxides: results of the generalized gradient approximation. Phys. Rev. B44:

2923 (1991).

[42] J. Goniakowski, J.M. Holdender, L.N. Kantarovich, M.J. Gillan, J.A. White.

Influence of gradient corrections on the bulk and surface properties of TiO2 and SnO2.

Phys.Rev. B53, 957 (1996).

[43] D.R. Hamman. Generalized gradient theory for silica phase transitions. Phys. Rev.

Lett. 76, 660 (1996).

[44] D.Vanderbilt. Soft self-consistent pseudopotentials in a generalized eigenvalue

formalism. Phys. Rev. B41, 7894 (1990).

[45] G. Kresse, J. Hafner. Norm-conserving and ultrasoft pseudopotentials for first-row

and transition elements. J. Phys.: Condens. Matt. 6, 8245 (1994).

[46] C.A. McCommon, N.L. Ross. Crystal chemistry of ferric iron in (Mg,Fe)(Si,Al)O3

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted

Formatted: Complex Script

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... [175]

... [176]

... [181]

... [177]

... [182]

... [178]

... [183]

... [179]

... [184]

... [180]

Page 57 of 116


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For Peer Review O

nly

35

majorite with implications for the transition zone. Phys. Chem. Minerals, 30, 206

(2003).

[47] V.L. Vinograd. Configurational entropy of binary silicate solid solutions. In: Solid

Solutions in Silicate and Oxide Systems of Geological Importance. EMU Notes in

Mineralogy, 3. Ch. Geiger (Ed.), pp. 245-249 (2001).

[48] S.D. Fleming, A.L. Rohl. GDIS: A visualization program for molecular and periodic

systems. Z. Kristallogr. 220, 580 (2005).

[49] V.L. Vinograd, A. Putnis. The description of Al,Si ordering in aluminosilicates using

the cluster variation method. Am. Mineral., 84, 311 (1999).

[50] E.R. Myers. Al/Si ordering in silicate minerals. Ph.D. thesis, Univ. of Cambridge.

(1999).

[51] M.A. Carpenter, A. Putnis. Cation order and disorder during crystal growth: some

implications for natural mineral assemblages. In Advances in Physical Geochemistry,

4, A.B.Thompson, D.C.Rubie (Eds.) Springer-Verlag (1985).

[52] M.A. Carpenter. Mechanisms and kinetics of Al-Si ordering in anorthite: I.

Incommensurate structure and domain coarsening. Am. Mineral., 76, 1110 (1991).

[53] H. Kroll, W.F. Müller. X-ray and electron-optical investigation of synthetic high-

temperature plagioclases. Phys. Chem. Minerals, 5, 255 (1980).

[54] J.V. Smith, W.L. Brown. Feldspar minerals. 1. Crystal structures, physical, chemical

and microtextural properties. Springer-Verlag, Berlin, Heidelberg (1988).

[55] A. Putnis. The distortion index in anhydrous Mg-cordierite. Contrib. Mineral.

Petrol., 74, 135 (1980).

[56] A. Putnis, D.L. Bish. The mechanism and kinetics of Al,Si ordering in Mg-

cordierite. Am. Mineral., 68, 60 (1983).

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Deleted: 1991.

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Deleted: -

Deleted: 1119.

Deleted: ¶Carpenter, M.A., and McConnell,

Deleted: , H.

... [193]

... [194]

... [204]

... [195]

... [219]

... [192]

... [220]

... [205]

... [191]

... [206]

... [202]

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... [198]

... [209]

... [199]

... [210]

... [200]

... [211]

... [201]

... [212]

... [185]

... [213]

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36

[57] A. Putnis, C.A. Fyfe, G.C. Gobbi. Al,Si ordering in cordierite using "magic angle

spinning" NMR spectroscopy. I: 29Si spectra of synthetic cordierites. Phys. Chem

Minerals, 12, 211 (1985).

[58] A. Putnis, E.S. Salje, S.A.T. Redfern, C.A. Fyfe, H. Strobl. Structural states of

Mg-cordierite I: order parameters from synchrotron X-ray and NMR data. Phys.

Chem. Minerals, 14: 446 (1987).

[59] B. Sundman, B. Jansson, J-O. Andersson. The Thermo-Calc databank

system. Calphad, 9, 153 (1985).

Figure captions

Fig. 1. A plot of the predicted interatomic distances against the experimental data of

Angel et al. [9].

Figure 2. The feed-back algorithm which improves the accuracy of the cluster expansion

in predicting ground state configurations.

Figure 3. Correlation between the energies (a) and enthalpies (b) calculated with GULP

and predicted with the pair cluster expansion. The enthalpies were calculated at 20 GPa.

Figure 4. The I 41/a (a) and C 2/c (b) majorites viewed along the 4-fold axis of the

tetragonal phase. The tetragonal symmetry is lost when MgMgMgMg and SiSiSiSi

tetrahedral clusters move away from the centering Si[4] atoms. The image is prepared

using the GDIS software [48].

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Figure 5. Correlation between the excess enthalpies calculated with GULP at 20 GPa and

predicted with the J-Q expansion. A similarly good correlation is observed for the

energies calculated at zero pressure.

Figure 6. Enthalpy of disorder predicted with Monte Carlo simulation (solid lines). The

dashed line shows the enthalpy at the complete disorder or infinitely high temperature.

Circles correspond to the excess energies of randomized configurations calculated with

GULP. All values are per formula unit with one octahedral atom (6 oxygens).

Figure 7. The temperature dependence of the long-range order parameter at 0 pressure

and 20 GPa as calculated from Monte Carlo simulations (circles). The cross and star

denote the experimental data.

Figure 8. The probability difference between two different four-atom clusters predicted

with the Monte Carlo simulations. Below the transition temperature the proportion shifts

in favour of the clusters consistent with the 4-fold symmetry.

Figure 9. The configurational entropy of the octahedral site (per 1 mole of octahedral

atoms) calculated using the method of thermodynamic integration. Dashed line shows the

ideal entropy.

Figure 10. The free energy of mixing and calculated using the method of thermodynamic

integration.


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Figure 11. The temperature – composition phase diagram calculated based on the results

of the Monte Carlo simulations. (All other non-garnet phases are suppressed). The solid

black curve shows the result for 20 GPa, while the dashed line shows the phase

boundaries at zero pressure. The squares represent the experimental results of Heinemann

et al. [11]: open and filled squares correspond to the samples, which after the quench

showed cubic and tetragonal symmetries, respectively.

Figure 12. The activity – composition relations in the pyrope – majorite system consistent

with the excess free energies described by Equations 13, 14.

Figure 13. The variation of equilibrium volume along the binary at 2273 K and 20 GPa

predicted with the volumetric cluster expansion.

Figure 14. The arrangement of octahedral cations in I41/a (a) and P 41 32 (b) majorites.

Figure 15. The phase relations in the pyrope-majorite system at (a) 1773 K and (b) 2073

K.


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these pressure/temperature conditions



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Table 1. The empirical interatomic potentials used in the present study. The

notation [4] and [6] refers to the coordination number of the associated species.

Buckingham

Interaction A (eV) ρB (Å) C (eV* Å6)

Al(core)-O(shell) 1262.2081 0.286370 0.0

Mg(core)-

O(shell)

1432.8544 0.277265 0.0

Si(core)-O(shell) 1073.4668 0.298398 0.0

O(shell)-O(shell) 598.8996 0.314947 26.89746

Spring

Interaction K (eV* Å-2 )

O(core)-O(shell) 56.5598

Three-body

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Interaction kθQ (eV*degreerad-2 ) θ G ( degree)

O(shell)-Si[4]- O(shell) 0.77664 109.47

O(shell)-Si[6]- O(shell) 2.2955 90.0

O(shell)-Al[4]- O(shell) 1.2883 109.47

O(shell)-Al[6]- O(shell) 1.8807 90.0

Note: The charges on the oxygen core and shell are 0.746527 and -2.446527, respectively. Cutoff distance

for the Buckingham potentials is 12 Å.

Table 2. Structural data and elastic constants for majorite


Subscript


English U.S.






















as calculated with the SLEC and DFT GGA in

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comparison with available experimental data.

Cell parameters

XRD SLEC DFT-GGA

a (Å) 11.501a; 11.481b;

11.519c

11.494 11.670

c (Å) 11.48a; 11.406b;

11.420c

11.392 11.561

a/c 1.0018 1.0089 1.0094

Volume (Å3) 1518.494 a 1505.040 1574.474

Atomic coordinates

XRDa SLEC

Core

DFT-GGA

x 0.1253 0.1256 0.1287

y 0.0112 0.0120 0.0135

Mg1 16f

z 0.2587 0.2623 0.2665

x 0.0000 0.0000 0.0000

y 0.2500 0.2500 0.2500

Mg2 8e

z 0.6258 0.6227 0.6235

x 0.0000 0.0000 0.0000

y 0.0000 0.0000 0.0000

Mg3 8c

z 0.5000 0.5000 0.5000

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x 0.0000 0.0000 0.0000

y 0.2500 0.2500 0.2500

Si1 4a

z 0.3750 0.3750 0.3750

x 0.0000 0.0000 0.0000

y 0.2500 0.2500 0.2500

Si2 4b

z 0.8750 0.8750 0.8750

x 0.1249 0.1261 0.1254

y 0.0065 0.0116 0.0116

Si3 16f

z 0.7544 0.7560 0.7568

x 0.0000 0.0000 0.0000

y 0.0000 0.0000 0.0000

Si4 8d

z 0.0000 0.0000 0.0000

x 0.0282 0.0257 0.0268

y 0.0550 0.0603 0.0588

O1 16f

z 0.6633 0.6685 0.6691

x 0.0380 0.0429 0.0447

y 0.9529 0.9540 0.9565

O2 16f

z 0.8562 0.8610 0.8611

x 0.2195 0.2248 0.2234

y 0.1023 0.1079 0.1069

O3 16f

z 0.8021 0.8070 0.8053

x 0.2150 0.2145 0.2117

y 0.9106 0.9161 0.9165

O4 16f

z 0.7000 0.7013 0.7028

x 0.9412 0.9353 0.9372

y 0.1617 0.1629 0.1638

O5 16f

z 0.4680 0.4690 0.4678

x 0.8960 0.8980 0.8977

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y 0.2080 0.2102 0.2153 O6 16f

z 0.7851 0.7821 0.7833

Interatomic distances

XRDa

(Å)

SLEC (Å)

Core

DFT GGA

(Å)

Mg1-O1 2.1209 2.0588 2.0940

Mg1-05 2.1303 2.0762 2.1344

Mg1-04 2.1533 2.1169 2.1631

Mg1-04 2.2500 2.2275 2.2179

Mg1-03 2.2668 2.2576 2.2701

Mg1-02 2.3319 2.3869 2.4922

Mg1-03 2.4334 2.4240 2.5286

Mg1-06 2.5823 2.6385 2.7487

Mg2-O5 2.1841 2.1481 2.1877

Mg2-O6 2.2379 2.2281 2.2372

Mg2-O1 2.3066 2.2651 2.3143

Mg2-O2 2.3848 2.3992 2.4702

Mg3-O4 1.9756 2.0355 2.0665

Mg3-O1 2.0049 2.0419 2.0809

Mg3-O5 2.0127 2.0663 2.0956

Si1-O5 1.6213 1.6446 1.6437

Si2-O6 1.6520 1.6384 1.6478

Si3-O1 1.6254 1.6225 1.6243

Si3-O4 1.6372 1.6259 1.6281

Si3-O3 1.6424 1.6705 1.6593

Si3-O2 1.6567 1.6845 1.6910

Si4-O2 1.7916 1.7420 1.7628

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Si4-O6 1.7931 1.7835 1.8114

Si4-O3 1.8348 1.7909 1.8152

Elastic constants

Observed (GPa) SLEC (GPa)

C11 286.4d 295.10

C44 85.0d 85.23

C66 93.2d 93.66

C12 83.0d 112.7

C16 1.4d 14.72

Bulk modulus 159.8d; 166.0e 170.14

References: aAngel et al., 1989; bKato & Kumazawa, 1985; cHeinemann et al. 1997; dPacalo & Weidner,

1997; eSinogeikin & Bass, 2002;.


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,



the difference


energies obtained with the








Highlight


Font: Times New Roman, Italic, Highlight




Highlight




4


16


56


Font: Times New Roman, Italic

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the


inaccuracy of modeling interatomic interactions with


or


approximation


Usually, the


s






It is possible that this overestimation could alter the relative energies of the calculated

structures.


uncertainties




Highlight




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The a/c ratios and atomic coordinates calculated with the DFT GGA and SLEC are

consistent with each other and compare well with the measurements of Kato &

Kumazawa (1985) and Heinemann et al. (1997). The a/c ratio of Angel et al. (1989) is in

disagreement with the majority of more recent structural determinations (e.g.

McCommon and Ross, 2003).


The atomic coordinates and bond distances predicted both by SLEC and DFT GGA

are in a very good agreement with each other and with the results of Angel et al. (1987).


W


Font: Italic








initio calculations of the lattice energies of selected configurations.

Total energies

Structure Description Empirical

potentials

Ab initio

A Pyrope, d3Ia -247.9798 -4114.7275

B Majorite, I41/a -252.0903 -4797.7550

C Majorite, P4132 -252.0783 -4797.5994

D Majorite50, 3Ia -250.0309 -4556,1669

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Energy differences

Reaction Empirical

potentials

Ab initio

D-(A+B)/2 0.0508 0.0744

C-B 0.1054 0.1556

Note: Values are in eV per octahedral atom





























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zero K


(1989)



In


has


first


has


ing


,


the next set of GULP

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,


t


its


constrain


a


Our


, however,


with GULP


d


.

Pairwise energies

Mg-Al Al-Si Mg-Si Mg-Al Al-Si Mg-Si Distance

(Å) 0 GPa 20 GPa

4.9511 -13.849 -12.332 -32.350 -21.125 -21.611 -54.338

5.7178 -17.769 -16.637 -46.718 -20.852 -18.233 -51.553

8.0883 -9.239 -8.113 -16.804 -11.700 -9.697 -24.627

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9.4852 -6.203 -4.772 -8.486 -7.239 -4.259 -13.335

9.9072 -4.526 -3.923 -5.208 -7.152 -3.093 -8.592

11.4400 -2.321 -3.344 -5.148 -5.540 -6.009 -7.435

12.4676 -2.948 -0.165 0.664 -7.252 -3.375 -1.886

12.7916 -2.093 -1.855 -1.742 -4.913 -1.385 -2.286

The elastic term

Parameter 0 GPa 20 GPa

A12 610.85 906.51

A21 640.00 899.48

Quaternary energies

Cluster type 0 GPa 20 GPa

MgMgMgSi -3.8054 -4.0740

MgSiMgSi -12.7228 -14.4620

MgSiSiSi -3.3496 -4.3289

AlAlAlSi 2.7180 1.0271

AlAlSiSi 5.0292 3.2307

AlSiAlSi -2.0462 -3.1154

AlSiSiSi 1.6866 2.3576

AlAlAlMg -0.8564 -0.0710

AlAlMgMg 0.0363 2.2000

AlMgAlMg -6.4528 -6.8823

AlMgMgMg -1.7643 -2.1405

MgAlSiSi 2.1632 1.0320

MgSiAlSi -7.2655 -4.7004

AlSiMgMg -0.4698 0.0186

AlMgSiMg -5.8033 -7.8190

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MgSiAlAl 0.0 0.0

MgAlSiAl 0.0 0.0

MgMgSiSi 0.0 0.0

MgMgMgMg 0.0 0.0

SiSiSiSi 0.0 0.0

AlAlAlAl 0.0 0.0

Note: the values are in kJ per mole of octahedral cations, or per 1.5 moles of Si[4] atoms.

Figure 2. Correlation between the energies calculated with GULP and

predicted with the pair cluster expansion.

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a bSi[4] Si[6]

Mg[6]

a bSi[4] Si[6]

Mg[6]

Figure 3. The I 41/a (a) and C 2/c (b) majorites viewed along the 4-fold

axis of the tetragonal phase. The tetragonal symmetry is lost when

MgMgMgMg and SiSiSiSi tetrahedral clusters move away from the

centering Si[4] atoms. The image is prepared using the GDIS software

(Fleming and Rohl, 2005).












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Figure 4. Correlation between the energies calculated with GULP and

predicted with the J-Q expansion.

Figure 5. Enthalpy of disorder predicted with Monte Carlo simulation (solid lines). The

dashed line shows the enthalpy at the complete disorder or infinitely high temperature.

Circles correspond to the excess energies of randomized configurations calculated with

GULP. All values are per


formula unit with one octahedral atom (6 oxygens).


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Figure 6. The temperature dependence of the long-range order parameter

at 0 pressure and 20 GPa as calculated from Monte Carlo simulations

(circles). The cross and star denote the experimental data.

Figure 7. The probability difference between two different four-atom

clusters predicted with the Monte Carlo simulations. Below the transition

temperature the proportion shifts in favor of the clusters consistent with

the 4-fold symmetry.

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(Myers, 1999; Dove, 2001)


λ−


Font: Symbol




Note that the E0 term, i.e. the energy of the disordered state, contributes only to the F0

term, but is excluded from the integral.


change


due to the ordering


When varies between 0 and 1, λ

E∆ , varies from the state of complete disorder to the

state of equilibrium order at a given temperature.


at the


of






0

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Figure 8. The configurational entropy of the octahedral site (per 1 mole

of octahedral atoms) calculated using the method of thermodynamic

integration. Dashed line shows the ideal entropy.

Figure 9. The free energy of mixing and ordering calculated using the

method of thermodynamic integration.


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1000

1500

2000

2500

3000

3500

4000

Tem

pera

ture,

K

Mole fractionof majorite

I 41/aI a 3 d

Pyr Maj

0.0 0.2 0.4 0.6 0.8 1.01000

1500

2000

2500

3000

3500

4000

Tem

pera

ture,

K


I 41/aI a 3 dI a 3 d

0.0 0.2 0.4 0.6 0.8 1.01000

1500

2000

2500

3000

3500

4000

Tem

pera

ture,

K



Pyr Maj

0.0 0.2 0.4 0.6 0.8 1.01000

1500

2000

2500

3000

3500

4000

Tem

pera

ture,

K




Figure 10. The temperature – composition phase diagram calculated based on the results

of the Monte Carlo simulations. (All other non-garnet phases are suppressed). The s


olid black curve shows the result for 20 GPa, while the dashed line shows the phase

boundaries at zero pressure. The squares


represent the experimental results of Heinemann et al. (1997): open and filled squares

correspond to the samples, which after the quench


cubic and tetragonal symmetries, respectively.

Figure 11. The activity – composition relations in the pyrope – majorite

system consistent with the excess free energies described by Eqns. 13, 14.


Font: Italic


Indent: First line: 0 pt, Line spacing: Double


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Highlight






Table 5. Coefficients of the Redlich-Kister polynomial for the excess free energy in

the pyrope – majorite solid solution (all values are in kJ/mole of octahedral atoms)

N A1n A2n*10-3 A3n*10-6

1 20.7761 2.91736 -2.3053

2 -11.1357 19.77231 -5.979

3 -17.4504 22.02424 -4.9062

4 14.7575 -24.45104 8.7821

5 40.7301 -36.20679 8.1251

6 -13.7634 19.75529 -6.5512

7 -17.2006 10.69285 -1.4385

N B1n B2n*10-3 B3n*10-6

1 -0.6976 2.1641 -0.3205

2 3.1557 -0.0637 0.1517

3 -0.2216 9.8304 -3.9812

4 -16.0881 20.3068 -5.5879

5 -24.0941 1.1400 4.6094

6 23.0833 -26.5699 7.2069

7 37.4186 -16.7439 0.0





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Figure 12. The variation of equilibrium volume along the binary at 2273

K and 20 GPa predicted with the volumetric cluster expansion.

a ba b

Figure 13. The arrangement of octahedral cations in I41/a (a) and P 41 32

majorites.


that anti-phase domains are generated. The 1/1 IC transition leads to an alternation

of Al and Si atoms and thus produces superstructure reflections h+k=odd, l=odd (so-

called b-reflections) which may be used to image type b anti-phase domains in the TEM.

In natural anorthites (An > 95%), although b-reflections are observed, no type b anti-

phase domains can be imaged. Obviously, these anorthites directly crystallized with the

1I anorthite structure. However, anorthite samples synthesized below the melting

temperature do show type b anti-phase domains although they crystallized in the stability

field of the ordered 1I phase as did the natural samples (Kroll and Müller, 1980;

Carpenter, 1991). In the experiments of Kroll and Müller, glasses of anorthite

composition were crystallized at 1430 °C in runs lasting from 5 min to 16 days. Type b

domains could be imaged in all samples, the domain sizes strongly increased with time.

Therefore, the domains certainly did not develop during the quench which lasted only for

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seconds.

This suggests that the anorthites first crystallized in a higher symmetry phase ( 1C )

and only then developed the 1I -type ordering at the annealing temperature. Kroll and

Müller suggested that the domains nucleated independent of each other during growth of

the crystals immediately behind the growth front and subsequently coarsened with

annealing time. Their experiments show that the synthetic anorthites indeed had a higher-

symmetry precursor which in the feldspar case is definitely metastable. It formed in the

stability field of the 1I phase. It is now agreed that also in natural anorthites with An-

contents less than 95% showing type b domains nucleation of the domains occurred in a

metastable 1C albite matrix leading to the stable 1I anorthite structure without

crossing a transformation boundary (Smith and Brown, 1988, p.89).

The metastable formation of a higher symmetry disordered structure within the

stability field of the low temperature ordered form is not an unusual occurrence, it is to be

expected if the starting materials are reactive (Carpenter and Putnis, 1985). Another well-

documented example is the case of Mg-cordierite which when synthesized from glass

always forms the hexagonal high form (no long range order) within the stability field of

the orthorhombic ordered structure (Putnis, 1980; Putnis and Bish, 1983; Putnis et al.,

1985, 1987). The first-formed hexagonal phase has no domain structures. Continued

annealing within the orthorhombic stability field produces a modulated structure and

ultimately a twinned orthorhombic fully Al,Si ordered cordierite.


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Figure 14. The phase relations in the pyrope-majorite system at (a) 1773 K




Font: Italic


Font: Bold




English U.S.


, M.


, A.


T.,


and

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, S., 1987


.,


251-260.






English U.S.






Font: Times New Roman, German Germany


German Germany




German Germany




German Germany




German Germany




, D.R.


, 1989.

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Font: Times New Roman, Italian Italy


Italian Italy




Italian Italy






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Bass, J.D., 1995. Elasticity of Minerals, Glasses, and Melts. In: Th.J. Ahrens (Editor)

Mineral Physics & Crystallography. A Handbook of Physical Constants. Am.

Geophys. Union, AGU Reference Shelf Ser., 2: 45-63.












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Bosenick, A., Dove, M.T., Myers, E.R., Palin, E.J., Sainz-Diaz, C.I., Guiton, B.S.,

Warren, M.C., Craig, M.S., and Redfern, S.A.T., 2001. Computational methods for

the study of energies of cation distributions: applications to cation-ordering phase

transitions and solid solutions. Min. Mag., 65:193-219.




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Carpenter, M.A., and McConnell, J.D.C., 1984. Experimental delineation of the

1/1 IC transformation in intermediate plagioclase feldspars. Am. Mineral., 69: 112-

121.

Carpenter M.A. and Putnis A., 1985. Cation order and disorder during crystal growth:

some implications for natural mineral assemblages. In: Advances in Physical

Geochemistry, Vol.4, Eds: A.B.Thompson and D.C.Rubie. Springer-Verlag.

Hatch, D.M., and Ghose, S., 1989. Symmetry analysis of the phase transition and

twinning in MgSiO3 garnet: implications to mantle mineralogy. Am. Mineral.,

74:1221-1224.

Page 109 of 116


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Dove, M.T., 2001. Computer simulations of solid solutions. In: Ch. Geiger (Editor).

Solid Solutions in Silicate and Oxide Systems of Geological Importance. EMU Notes

in Mineralogy, 3: 245-249.

Fabrichnaya, O.B., Saxena, S.K., Richet, P., and Westrum, E.F., 2004. Thermodynamic

data, models and phase diagrams in multicomponent oxide systems. Springer-Verlag,

198pp.

Fiquet, G., 2001. Mineral phases of the Earth's mantle. Z. Kristallogr., 216:248-271.

Fleming, S.D. and Rohl, A.L. 2005. GDIS: A visualization program for molecular and

periodic systems. Z. Kristallogr. 220, 580-584.

Gale, J.D., 1996. Empirical potential derivation for ionic materials. Phil. Mag. B, 73, 3-

19.

Gale, J.D., 1997. GULP – a computer program for symmetry adapted simulations of

solids. J. Chem. Soc.: Faraday Trans., 93: 629-637.

Gasparik, T., 1990. Phase relations in the transition zone. J. Geophys. Res., 95: 15751-

15769.

Gasparik, T., 2003. Phase diagrams for geoscientists. An atlas of the Earth’s interior.

Page 110 of 116


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Springer-Verlag, Berlin, Heidelberg. 462pp.

Hamman, D.R., 1996. Generalized gradient theory for silica phase transitions. Phys. Rev.

Lett. 76: 660-663.

Harrison, R.J., Redfern, S.A.T., and Smith, J., 2003. The effect of transformation twins

on the seismic frequency mechanical properties of polycrystalline Ca1-xSrxTiO3

perovskite. Am. Mineral., 88: 574-582.

Hatch, D.M., and Ghose, S., 1989. Symmetry analysis of phase transition and twinning in

MgSiO3 garnet: implications to mantle mineralogy. Am. Mineral., 74: 1221-1224.

Heinemann, S., Sharp, T., Seifert, F., and Rubie, D.C., 1997. The cubic-tetragonal phase

transition in the system majorite (Mg4Si4O12) - pyrope (Mg3Al2Si3O12), and garnet

symmetry in the Earth's transition zone. Phys. Chem. Minerals, 24: 206-221.

Hohenberg, P., Kohn, W., 1964. Inhomogeneous electron gas. Phys. Rev. 136: B864-

B871.

Jones, R. O., Gunnarsson, O., 1989. The density functional formalism, its applications

and prospects. Rev. Mod. Phys., 61: 689-746.

Page 111 of 116


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Kato, T., and Kumazawa, M., 1985. Garnet phase of MgSiO3 filling the pyroxene-

ilmenite gap at very high temperatures. Nature, 316: 803-805.

Kanzaki, M., 1987. Ultrahigh-pressure phase relations in the system. Phys. Earth Planet.

Interiors, 49: 168-175.

Kohn, W., Sham, L.J., 1965. Self-consistent equations including exchange and

correlation effects. Phys. Rev. 140: A1133-A1138.

Kresse, G., Hafner, J. 1994. Norm-conserving and ultrasoft pseudopotentials for first-row

and transition elements. J. Phys.: Condens. Matt. 6: 8245-8257.






German Germany















Page 112 of 116


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

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For Peer Review O

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,














Sinogeikin, S.V., and Bass, J.D., 2002. Elasticity of majorite and majorite-pyrope solid

solution to high pressure: Implications for the transition zone. Geophys. Res. Lett.

29(2):4-1- 4-4.

Smith, J.V., Brown, W.L. 1988. Feldspar minerals. 1. Crystal structures, physical,

chemical and microtextural properties. Springer-Verlag, Berlin, Heidelberg. 828pp.


Indent: Before: 0 pt, Hanging: 22.5 pt, Line spacing: Double




Swedish Sweden


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Page 114 of 116


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

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1985.






Tomioka, N., Fujino, K., Ito, E., Katsura, T., Sharp, T., and Kato, T., 2002.

Microstructures and structural phase transition in (Mg,Fe)SiO3 majorite. Eur. J.

Mineral. 14: 7-14.


Font: (Default) Times New Roman


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Indent: Before: 0 pt, First line: 0 pt, Line spacing: Double


Vanderbilt, D., 1990. Soft self-consistent pseudopotentials in a generalized eigenvalue

formalism. Phys. Rev. B41: 7894-7895.

Vinograd, V.L., and Putnis, A., 1999. The description of Al,Si ordering in

aluminosilicates using the cluster variation method. Am. Mineral., 84: 311-324.

Vinograd V.L., Sluiter, M.H.F., Winkler, B., Putnis, A., and Gale, J.D., 2004.

Thermodynamics of mixing and ordering in silicates and oxides from static lattice energy

and ab initio calculations. In: M. Warren, A. Oganov and B. Winkler (Editors). First-

Page 115 of 116


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

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Principles Simulations: Perspectives and Challenges in Mineral Sciences. Deutsche

Gesellschaft für Kristallographie. Berichte aus Arbeitskreisen der DFK, 14: 143-151.









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Thermodynamics of pyrope - majorite, Mg3Al2Si3O12 ...

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