HIGH-PRESSURE STUDIES OF ANALOGS WITH

APPLICATIONS TO MATERIALS SCIENCE AND

GEOSCIENCE

Camelia Veronica Stan

A DISSERTATION

PRESENTED TO THE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE

BY THE DEPARTMENT OF CHEMISTRY

ADVISER: THOMAS S. DUFFY

MAY 2016

© Copyright by Camelia Veronica Stan, 2016. All rights reserved.

Abstract

Analog materials can provide insights to understanding inaccessible phase transitions and novel material properties. The work presented here examines both structural and chemical variation of analogs in order to probe the physics and chemistry of materials at high pressure and temperature conditions. The experiments reported here used the diamond anvil cell to achieve pressures up to 1.5 Mbar and temperatures as high as 2500 K, in conjunction with synchrotron x- ray diffraction and Raman spectroscopy. The work is divided into three major projects.

Polymorphism in AX2-type compounds was explored using PbF2 as an archetype for phases with highly coordinated cations. These materials are of interest due to their potential technical applications and, in the case of SiO2, their geophysical relevance. Compression studies at room temperature revealed an unusual isosymmetric phase transition in PbF2, and the combination of experimental and theoretical approaches explored the mechanism of the phase transformation that had hitherto not been understood. High-pressure-temperature experiments led to the discovery of a new phase transition in this material, thus furthering our understanding of phase transformation pathways in AX2 materials.

Garnets are important compounds in geoscience and materials science. The high-pressure behavior of Y3Fe5O12 has been controversial due to conflicting reports regarding its high- pressure polymorphs. Here I show that the high-pressure phase is a perovskite with a

3+ (Y0.75Fe0.25)FeO3 composition. In addition, I also identify a spin transition in the octahedral Fe site at 45-51 GPa. Comparison with other perovskite-structured orthoferrites shows that the volume discontinuity associated with the spin transition is controlled by the size of the cation occupying the distorted dodecahedral site.

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Silicate perovskites and post-perovskites are the dominant mineral phases in the Earth’s lower mantle. The effect of incorporation of Fe2+ on the perovskite and post-perovskite structures, and on the phase transition between them, is investigated using the analog system

(Mg,Fe)GeO3. At high pressures and temperatures, single-phase perovskites and post-perovskites were be synthesized for Mg-rich compositions with Mg# ≥ 78. Iron is found to lower the perovskite-post-perovskite phase boundary, and to affect a modest decrease of bulk modulus and a modest increase in volume of perovskites.

iv

Acknowledgements

In some ways, this is the hardest part to write, because it requires raw emotional honesty about six years’ worth of despair and satisfaction and work and elation. A part of me feels like I will never properly be able to express my gratitude to everyone who has been here for me along the way both when everything felt grueling and when everything felt great. Everything I write feels insufficient and I am humbled in the face of your help, your love, and the magnitude of your impact on who I am as a scientist, a friend, a daughter, a person.

Certainly, I would never have finished this work if it weren’t for the tireless help of my advisor, Tom Duffy. I hounded you for over a month to let me join the lab, was incredibly awkward countless times, and was probably way more forward with my thoughts and opinions than was necessary (as I probably am right now). Despite this, you soldiered on along with me, providing much needed oversight, encouragement, and advice. I take great joy in the way our working relationship has developed over the course of my time at Princeton, and I admire your dedication to your students and the level of interest you took to ensure I, and my colleagues, were doing the best work we could do. I will be forever grateful. Thank you.

Speaking of colleagues, I have had many excellent ones throughout the years. Thank you (in chronological order) to Suki Dorfman, Raj Dutta and Sally Tracy for working alongside me all these years. Thank you in particular to Greg Finkelstein and June Wicks. You have taught me so much, and have certainly made me into a better scientist. More importantly, you have spent countless nights babysitting DACs with me at the synchrotron, which is sort of like a friendship trial-by-fire. Much of my success depended on you, and I wouldn’t have made it through to the end without either of you. Thank you also to our colleagues at the synchrotron, particularly

Clemens Prescher, whose stoic German sense of humor was always greatly appreciated. Thank v you to colleagues in the department of Geosciences and Chemistry who have guided and supported me through the minutiae of life.

I am very grateful to the friends who have been here along the way. This might get a little wordy, and is also in no particular order. Aaron, thank you for over a decade of mental and emotional support; I know we can always rely on one another through thick and thin (friendship is the gift that keeps on giving) and your advice, resilience, patience and strength have been a boon and a comfort throughout the years. Allison, thank you for over a decade of snarking (in many ways just as important as anything else in a friendship). Jack and Laura, I will always appreciate your generosity, hospitality, and honesty. Bailey, Harry, Michelle: thank you for befriending and including me. Kristen: I am so glad we became close friends, and I am so grateful to you for being an island of day to day calm in the turbulent ocean of graduate life.

Nick and Shoham: thank you so much for all of the extracurricular activities we did together that made life bearable. Ethan, Calvin, Andy, Kate: it is refreshing to have had friends like you for so long, who anchor me in reality; talking to you feels like going home. And to Earl, without whom

I would have lost my mind years ago, who always listens to my trials and tribulations, who I admire, cherish, look up to and respect, and who means the world to me: you have taken on the role of colleague, confidante, and partner, made me a better scientist, revitalized me when I flagged, shared in my successes, calmed me when I stormed. There are not enough words to express my gratitude. Thank you.

I want to thank my Romanian family who has been there for me, very loudly, throughout the past six years, especially my grandmother, Victoria, and my grandfather, Ion. And, I am very grateful to Gabi and Vio Enciu. Thank you for treating me as though I was your daughter, for

vi taking me in and feeding me, doing construction projects with me, cooking with me, and talking to me until the wee hours of morning.

Finally, I would like to thank my parents, Irina and Ovidiu Stan, for always being there for me. We come from a culture that very greatly values family ties, and it’s difficult to express everything they mean to me. Thank you for the food and wine, for calling me and comforting me, for taking care of me, for encouraging me and pushing me to do more and be more, and for years and years of love and support that have allowed me to become the person I am today. I am nothing without you.

vii

Table of Contents

Abstract iii

Acknowledgements v

Chapter 1: Introduction 1 1.1 Analogs in Materials Science 1

1.1.1 The AX2 system 1 1.1.2 The perovskite family of compounds 4 1.2 Analogs in Geoscience 4 1.3 Experimental Methodology 9 1.3.1 Generating high pressures with the diamond anvil cell 9 1.3.1.1 Diamonds 11 1.3.1.2 Insulating pressure media 11 1.3.1.3 Gaskets 13 1.3.2 Laser heating 14 1.3.3 Synchrotron X-Ray diffraction 15 1.3.2.1 Synchrotron operation 15 1.3.2.2 X-ray diffraction 16 1.3.4 Raman spectroscopy 18 1.3.5 Pressure determination 21 1.3.5.1 Differential pressure 23

Chapter 2: High-Pressure Polymorphism of PbF2 to 75 GPa 25

2.1 Abstract 25 2.2 Introduction 26 2.3 Methods 29 2.4 Results and Discussion 31

2.4.1 Room-temperature compression of PbF2 31

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2.4.1.1 X-ray diffraction 31 2.4.1.2 Raman spectroscopy 38 2.4.2 Isosymmetric phase transition 41 2.4.3 Room-temperature equation of state 44 2.4.4 High pressure-temperature behavior 45 2.5 Conclusion 50 2.6 Acknowledgements 51 2.S Supplementary Material 52 2.S.1 Starting material 52 2.S.2 Effects of sample pre-compression on stress state 53

Chapter 3: High-Pressure Phase Transition in Y3Fe5O12 61 3.1 Abstract 61 3.2 Introduction 62 3.3 Methods 65 3.4 Results/Discussion 67 3.4.1 Pressure-induced amorphization 67 3.4.2 High-pressure and -temperature experiments 70 3.4.3 Equation of state and spin transition 76 3.5 Conclusion 84 3.6 Acknowledgments 84

Chapter 4: Synthesis and Equation of State of Perovskites and Post-Perovskites in (Mg,Fe)GeO3 86 4.1 Abstract 86 4.2 Introduction 87 4.3 Methods 90 4.3.1 Pyroxene synthesis 90 4.3.2 High-pressure experiments 91 4.4 Results and Discussion 92

ix

4.4.1 Synthesis of germanate pyroxene starting materials 92 4.4.2 Perovskite and post-perovskite synthesis at high pressures 100 4.4.3 Perovskite – post-perovskite phase relationships 109 4.4.4 Equations of state 114 4.4.4.1 Perovskite 114 4.4.4.2 Post-perovskite 120 4.5 Conclusion 123 4.6 Acknowledgments 124 4.7 Supplementary Material 125

Appendix A: First-Principles Study of the High-Pressure

Isosymmetric Phase Transition in PbF2 130 A.1 Abstract 130 A.2 Introduction 131 A.3 Methods 132 A.4 Results and Discussion 134 A.4.1 Structural evolution under pressure 134 A.4.2 Bonding and polyhedral analysis 140 A.4.3 Elasticity 147 A.4.4 Comparison with metal dioxides 151 A.5 Conclusion 153

References 155

x

List of Figures

1.1. Phase relationships in the AX2 material system 2 1.2. Mineralogical profile of the Earth’s mantle 5 1.3. Schematic diagram of the diamond anvil cell 10 1.4. Sample loading within the diamond anvil cell 12 1.5. Typical sample setup at the synchrotron 15 1.6. Bragg’s Law 17 1.7. Representation of the diamond anvil cell Raman system 20

2.1. Fluorite, cotunnite, Co2Si and Ni2In structures 27

2.2. Coordination and distortion between cotunnite, Co2Si and Ni2In structures 28

2.3. PbF2 diffraction at room temperature upon compression 32

2.4. PbF2 orthorhombic lattice parameters upon compression 33

2.5. PbF2 orthorhombic unit cell volume upon compression 34

2.6. PbF2 Raman spectra upon compression 39

2.7. PbF2 Raman shift as a function of pressure 40

2.8. Lattice parameter ratios in AX2 materials 42 2.9. Diffraction upon heating at 25.9 GPa 46 2.10. Diffraction upon heating at 34.0 GPa 47 2.11. Diffraction upon heating at 43.0 GPa 48

2.S.1. PbF2 diffraction at room temperature and pressure 52

2.S.2. Diffraction of PbF2 using different initial sample compaction methods 55

2.S.3. Diffraction of PbF2 using different pressure media 56

2.S.4. Volume and lattice parameters of PbF2 upon compression and decompression 58

2.S.5. Lattice parameter ratios in PbF2 upon heating 59

3.1. Garnet and perovskite structures in Y3Fe5O12 63

3.2. Y3Fe5O12 starting material characterization 66

3.3. Room temperature amorphization of Y3Fe5O12 68

3.4. Synthesis of Y3Fe5O12 perovskite 71

3.5. Phase identification in Y3Fe5O12 perovskite 72

3.6. (Y0.75Fe0.25)FeO3 perovskite diffraction patterns at selected pressures 74

xi

3.7. Compression of Y3Fe5O12 78 3.8. Volume discontinuity in rare earth perovskites across the Fe3+ spin transition 80 4.1. Perovskite and post-perovskite structures 88

4.2. (Mg,Fe)GeO3 orthopyroxene diffraction 93 4.3. Synthesized sample pellets 95 4.4. Volume and lattice parameter dependence on Mg# of pyroxenes 97 4.5. Raman spectra of germanate pyroxenes 99

4.6. (Mg,Fe)GeO3 perovskite synthesis at high pressure 101

4.7. (Mg,Fe)GeO3 post-perovskite synthesis at high pressure 103

4.8. Rietveld refinement of (Mg0.78Fe0.22)GeO3 perovskite 107

4.9. Rietveld refinement of (Mg0.78Fe0.22)GeO3 post-perovskite 108

4.10. MgGeO3 phase distribution in pressure and temperature space 110

4.11. (Mg0.78Fe0.22)GeO3 phase distribution in pressure and temperature space 111

4.12. Stability of post-perovskite in MgGeO3 versus (Mg0.78Fe0.22)GeO3 112 4.13. Unit cell volumes and lattice parameters of germanate perovskites 115 4.14. Octahedral tilt and lattice parameter ratios in perovskites 117

4.15. MgGeO3 perovskite unit cell parameters as a function of pressure 119

4.16. MgGeO3 post-perovskite unit cell parameters as a function of pressure 121

4.S.1. GeO2 impurity in pyroxene starting materials 125

4.S.2. Impurities in (Mg0.61Fe0.39)GeO3 perovskite 126

4.S.3. Impurities in (Mg0.48Fe0.52)GeO3 post-perovskite 127 4.S.4. Full width at half maximum of Au (111) in perovskite 128 4.S.5. Lattice parameter ratios in post-perovskites 129

A.1. Equilibrium structure of α-PbF2 135

A.2. Relative lattice parameters of PbF2 as a function of pressure 136 A.3. Lattice parameter ratios of several cotunnite-like compounds 139

A.4. Changes in PbF2 atomic positions with pressure 141 3 A.5. Total electron density (e/Å ) of PbF2 in the (010) plane 142

A.6. Atomic level structural evolution of PbF2 with compression 143

A.7. Interatomic distances in PbF2 as a function of pressure 144

A.8. Elastic constants in PbF2 as a function of pressure 148 xii

A.9. Polycrystalline elastic moduli of PbF2 150

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List of Tables

2.1. Unit cell parameters of PbF2 during room-temperature compression 36

2.2. Raman peaks assignments and 1-bar frequencies for α-PbF2 38

2.3. Equation of state parameters for PbF2 44

2.S.1. Ionic radii and phase transition pressures in AX2 materials 53

2.S.2. Summary of PbF2 X-ray diffraction experiments 54

2.S.3. Lattice parameter and volume changes in PbF2 upon compression 60

3.1. Lattice parameters and d spacing of (Y0.75Fe0.25)FeO3 perovskite at high pressure 75

3.2. Unit cell parameters of (Y0.75Fe0.25)FeO3 perovskite as a function of pressure 77 3.3. Equation of state properties in orthoferrite perovskites 82 4.1. Composition of germanate pyroxenes 94

4.2. Unit cell parameters and volumes of (Mg,Fe)GeO3 pyroxenes 96

4.3. MgGeO3 Raman peak positions 98

4.4. d spacing of selected (Mg,Fe)GeO3 perovskites and post-perovskites 102 4.5. Perovskite and post-perovskite unit cell parameters at select pressures 104 4.6. Lattice parameters and atomic positions from Rietveld refinement 105 4.7. Select bond lengths and site volumes from Rietveld refinement 106 4.8. Equation of state parameters in perovskite and post-perovskite 120

A.1. Lattice parameters and atomic positions of α-PbF2 at ambient conditions 137

A.2. Deformation in PbF2 as a function of pressure 146

A.3. Elastic constants of cotunnite-type PbF2 at 0 GPa 149 A.4. Elastic anisotropy in lead fluoride 152

xiv

CHAPTER 1

Introduction

The exploration of material properties through chemical analogs has a long history in chemistry, materials science, and geoscience. Often the substitution of an element (or functional group) within a structure can help elucidate the function of another element or group, or has affected the crystal structure or magnetic or electronic properties (e.g. dopants causing luminescence).1–3 Similarly, the use of structurally similar materials as analogs has enabled experiments that are otherwise untenable, particularly those at inaccessible pressure or temperature conditions.4 The use of analogs thus has broad application to the understanding of the physics and chemistry of solid state materials.

1.1 Analogs in Materials Science

1.1.1 The AX2 system

One of the simplest families of compounds is the AX2/A2X-type, which includes a wide variety of oxides, hydrides, halides, sulfides, phosphides, as well as intermetallic compounds.

Members of this family have a wide range of applications in materials science, and include SiO2, which is the archetypal silicate in geoscience. It has been recognized since the early 1960s that these compounds experience complex polymorphism with variable transition pathways, spanning a wide range of cation coordination environments (Figure 1.1).5–8 These transition pathways can be accessed by the application of pressure, temperature, or both. Dioxides, such as TiO2 and

SiO2, have relatively low cation coordination numbers (4- and 6-fold) at ambient conditions due to their smaller radii, but transform to a sequence of highly coordinated phases with

1

5 Figure 1.1. Crystal structure map for phases in the AX2 system based on the works of Vegas and

Haines and Léger,7 and on the results of Chapter 2. Different structure types are indicated in purple. CN refers to coordination number about the A cation. PbF2 pathway is highlighted by red labels. 2 compression. Actinide and lanthanide dioxides are technologically important materials due to their use as catalysts, oxygen storage devices, and as potential nuclear fuels. A number of these

9 10 11 oxides such as CeO2, PuO2, ThO2, and UO2 have also been shown experimentally to adopt a cotunnite-type (9-fold, Pnma) structure upon compression, while theoretical studies have predicted possible further transformations to post-cotunnite structures at pressures over 1 Mbar which have not yet been reached experimentally in these materials.12–14 The use of alkaline earth fluorides as analogs for oxides under high degrees of compression provides an illustrative example of the structural relationships between AX2 compounds, albeit at vastly different pressure conditions. Difluorides such as PbF2, SrF2, CaF2, etc., adopt an 8-fold coordinated fluorite-type structure (Fm-3m) at room temperature and pressure, and undergo phase transitions to the cotunnite-type phase at relatively modest pressures. Further transformation to an 11-fold coordinated structure (Ni2In-type, P63/mmc) has also been observed in difluorides at sub- megabar pressures, but has not been achieved in oxides due to the high pressures necessary to effect such transitions.

Knowledge of the behavior of highly coordinated AX2 compounds at lower pressure may give insights into the possible transformation pathways that may occur at higher pressure conditions. Ultimately, the use of highly-coordinated AX2 materials as analogs for simple oxides such as SiO2 may help us understand the behavior of this fundamental oxide at the extreme pressures (up to 1 TPa) that may be found in the interiors of large terrestrial-type extrasolar

15 planets. In Chapter 2 of this thesis, lead fluoride, PbF2, is studied at high pressure and both ambient and high temperature to explore polymorphism in highly coordinated (9- to 11-fold)

AX2 compounds.

3

1.1.2 The perovskite family of compounds

The behavior of perovskites has been of great interest in materials science due to their wide range of interesting properties, particularly ferroelectricity, superconductivity, and optical properties, as well as for their use in chemical catalysis.16 For instance, the discovery of perovskite-based high temperature superconductors in the Ba-La-Cu-O system17 lead to a wide interest in perovskite-type superconductors using substitutional chemistry.18–20 Other perovskite compositions have found recent technological applications as photovoltaics.21,22 The ability of a variety of compounds to adopt the perovskite structure at ambient conditions or under pressure, and the properties of those perovskites, has long been of fundamental interest in materials science. This is particularly true in the case of materials with interesting mechanical, optical, or electronic properties, such as the rare earth garnets (e.g. yttrium iron garnet, Y3Fe5O12). The rare earth garnets have applications as solid-state lasers,23,24 among other things. Several of these have been found to transform to perovskite-structured compounds at high pressure.25–27 The study of Fe3+-containing perovskites under compression can also lead to insights into pressure- induced spin transitions.28 We explore the structural phase transition from garnet to perovskite in a ferric iron containing composition, Y3Fe5O12, in Chapter 3.

1.2 Analogs in Geoscience

The Earth’s mantle makes up the largest volumetric portion of the planet and is host to a wide variety of different mineral phases (Figure 1.2) that play a defining role in the planet’s dynamics. Many studies have been performed on the composition expected to make up the bulk

29–36 of the mantle (a relatively SiO2-poor peridotitic composition), and in combination with seismic data place constraints on its density,37,38 pressure,38 and temperature profiles with depth

4

Figure 1.2. Mineral phases and their relative fraction in the Earth’s mantle as a function of depth after Trønnes.39 5 in the Earth.30,33,34,40,41 The mantle can be divided broadly into an upper and lower mantle based on a major seismic discontinuity at 660 km depth. Mineralogically, the main difference between the two regions is the preponderance of silicates with silicon in four-fold coordination by oxygen in the upper mantle and six-fold coordination in the lower mantle. The upper mantle has several seismic discontinuities that correlate with phase transitions of the underlying mineralogy: at 410 km depth, corresponding to the olivine (Mg2SiO4) to wadsleyite (β-Mg2SiO4) phase transition, and at 660 km, corresponding to the ringwoodite (γ-Mg2SiO4) to bridgmanite ((Mg,Fe)SiO3) plus ferropericlase [(Mg,Fe)O] phase transition and defining the upper to lower mantle boundary.39

Subducting slabs with more SiO2-rich basaltic composition undergo a related set of phase transitions, with the upper-to-lower mantle discontinuity characterized by a garnet-bridgmanite transition, with a depth closer to 700 km (Figure 1.2).39

The perovskite-structured mineral bridgmanite [(Mg,Fe)SiO3] is of great geophysical interest as it forms the bulk of the Earth’s lower mantle.42 For bridgmanite, a wide variety of studies have been performed, focusing on the elasticity,43–45 sound velocities,33,46,47 and elemental partitioning with other phases,48 as well as the effect of the inclusion of the major elements Fe2+, Fe3+ and

Al3+ on physical properties such as bulk modulus and elasticity.49–55

At higher pressures, corresponding to those near the Earth’s core-mantle boundary region,

56 bridgmanite transforms to (Mg,Fe)SiO3 post-perovskite (Cmcm CaIrO3-type structure). Since the discovery of post-perovskite ten years ago, many studies have been performed to examine the relative properties of these two structures and their importance for interpretation of geophysical observations of the lower mantle. In particular, the perovskite-post-perovskite phase transition may be mainly responsible for the D’’ discontinuity in the Earth, which is a seismic discontinuity that has been detected at ~60-300 km above the core-mantle boundary.57–67 The D’’ region (i.e.

6 the region between the core-mantle boundary and the D’’ discontinuity) is characterized by large lateral heterogeneity and a strong thermal gradient, as well as a global variation in the depth of the discontinuity. Understanding the effect of Fe on the perovskite-post-perovskite phase transition is important in interpreting whether the topography of D’’ is due to global chemical and/or temperature variations. Existing studies do not currently provide a consensus, suggesting that Fe2+ either increases68,69 or decreases55,67,70 the transition pressure. Fe2+ incorporation into the two structures also leads to changes in density, thermal conductivity, and elastic properties, and potential spin transitions in Fe2+ can further lead to changes in physical or chemical properties of perovskite and post-perovskite.71,72 Consequently, understanding the effect of Fe on the compression behavior of the two structures, as well as Fe partitioning between the two as well as with the ferropericlase phase, (Mg,Fe)O, is of particular interest. However, the pressure- temperature conditions for MgSiO3 post-perovskite synthesis can be difficult to achieve experimentally (125 GPa, 2500 K)56 in a reliable manner, making this a good candidate system for studies employing isostructural analogs.

Historically, early studies of the mineralogy of the interior of the earth used analogs due to technological constraints on achievable experimental conditions. In particular, the use of germanate analogs enabled identification and study of major crystal structures of the Earth’s upper mantle in the late 1950s and early 1960s.35,73–75 Germanium, in the same group of the periodic table as silicon, has a larger radius (0.53 Å in Ge4+ vs 0.4 Å in Si4+ when in 4-fold coordination).76 Radius ratio rules then suggest that germanates should achieve higher coordination at lower pressures. Pyroxenes with composition M2Ge2O6 (where M is a metal such

75 73,74 as Mg, Fe, Ca, etc.) and spinels or olivines with composition M2GeO4 were used to explore pyroxene and olivine phase transition pathways. Ultimately, this early analog-based research

7 lead to the initial interpretation of mineralogy in the mantle, as well as the attribution of discontinuities in the upper mantle to high-pressure phase transitions (Figure 1.2)., In later years, germanate analogs have continued to find uses in studying a wide range of mantle and crustal properties and dynamics due to the strong similarities in physical and chemical properties between silicates and germanates. Recently, germanates have been used to employed as silicate analogs in studies of equations of state,77,78 shear strength,79 rheology and deformation behavior,80,81 earthquake mechanisms,82 structural and magnetic properties,83–88 spectroscopic properties,89 and the dynamics of liquids and glasses.90–94

The perovskite-post-perovskite phase transition is now recognized to occur over a wide range

77,86,95 of compositions at high-pressure and -temperature conditions, including MgGeO3,

96 88 88 97 98 99,100 101 102,103 NaIrO3, MnGeO3, CaGeO3, CaRuO3, CaRhO3, CaPtO3, CaSnO3, NaMgF3,

104 NaZnF3, etc. In the case of (Mg,Fe)GeO3, the perovskite phase can be synthesized at similar conditions to the formation of silicate perovskite (25-28 GPa, 1300-1700 K),105,106 while the post-perovskite phase can be synthesized at much lower pressure conditions (~65 GPa)95 than its silicate counterpart (~125 GPa). Because of the lower pressures required, larger sample sizes can be used, which leads to a wider variety of accessible experiments. The larger sample sizes can enable better sample preparation, and higher data quality due to the larger analytical volumes. A major advantage of is that the compressed sample chamber experiences lower thermal gradients due to thicker insulation, and better pressure calibration (as the accuracy of calibration decreases with pressure). Consequently, the lower transition pressures of the germanates make this system more amenable to the study of several aspects of the perovskite-post-perovskite phase transition.

While a number of studies have focused on the properties of MgGeO3 perovskite and post- perovskite, particularly on its crystal structure and phase transitions at high pressure,77,78,86,95

8

107–109 only very limited studies have been performed on the iron endmember, FeGeO3, and no studies have been performed on intermediate compositions in the (Mg,Fe)GeO3 system. Our work on synthesis of a wide variety of (Mg,Fe)GeO3 compositions, and our analysis of the properties of perovskites and post-perovskites with these composition, is presented in Chapter 4.

1.3 Experimental Methodology

1.3.1 Generating high pressure with the diamond anvil cell

The main goal of this work is to explore structural polymorphism as a function of pressure and temperature variation, particularly the application of pressures of 10s of GPa in conjunction with heating to above 1000 K. Achieving such high pressures has been the subject of technological development for decades. There are two primary approaches being currently used for static high-pressure generation: the multi-anvil apparatus and the diamond anvil cell (DAC).

The multi-anvil apparatus uses a hydraulic press to drive a set of 6-8 anvils together to generate pressures on samples up to millimeters in size. Typical pressures reached by these devices are up to 25 GPa, making them largely unsuitable for studies of the lower mantle which extends from a pressure range of 24-135 GPa

The diamond anvil cell is able to achieve higher pressures than the multi-anvil press at the price of smaller sample volumes. The DAC is an opposed-anvil device that uses two flat diamonds to apply pressure to a sample. The first DAC design was reported by Weir et al.110 in

1959, and has undergone continuous evolution in the succeeding decades. While opposed anvil devices (such as piston-cylinder presses) had been used to generate pressures of a few GPa, the development of the DAC was revolutionary in allowing much higher pressures reaching up to a

Mbar and beyond with a small, hand-held device. The transparency of diamond across large

9

Figure 1.3. Schematic illustration of a diamond anvil cell (not to scale). Approximate size

of cell is 3.5 cm high by 4.5 cm wide. Diamonds are approximately 1.5-2.5 mm high.

regions of the electromagnetic spectrum further offered the ability to probe samples in situ with spectroscopy or X-rays. A basic diagram of a symmetric-type DAC is presented in Figure 1.3.

The procedure for a typical DAC experiment is outlined as follows: natural, gem-quality diamonds (~1/4 carat) are attached to rigid backing plates (termed “seats”) and aligned in a piston-cylinder device such that their opposed faces (called the culets) are parallel and overlap precisely. The sample is embedded within a pressure-transmitting medium and placed on the culet surface. A gasket/confining ring is placed around the sample/pressure medium, and prevents sample extrusion as pressure increases. The opposing diamond anvil is then placed on top to close the cell, and pressure is provided by driving the anvils together with one of a variety 10 of mechanisms that depends on the particular DAC design. A typical drive mechanism, such as that employed by the symmetric DACs used in the experiments presented here, involves pairs of right- and left-handed screws and Belleville washers that can be tightened to apply controlled loads to the anvils (Figure 1.3).

Due to the transparency of the diamonds to a wide range of photon wavelengths, a variety of experiments such as X-ray diffraction, Raman and infrared (IR) spectroscopy, Mössbauer, X-ray absorption spectroscopy, etc. are feasible, limited only by the small sample size and sample geometry. More detailed explanations of the experimental setup used in this work follow below.

1.3.1.1. Diamonds

For most of the experiments reported here, diamond culets vary between 500 and 100 μm in diameter. Sizes such as these enable experiments with maximal pressures between 30 GPa to in excess of 150 GPa. In some cases, a two-stage beveled design, with the outer culet size being

300 μm and the inner culet size being 100 μm, was used. Such a design allows for higher experimental pressure ranges compared to a non-beveled diamond. The diamond backing plates where chosen from among different materials depending on both the experiments being performed as well as the pressure range desired. These ranged between steel, tungsten carbide

(WC, stronger than steel, able to withstand higher pressures) and cubic boron nitride (cBN, also stronger than steel, and partially X-ray transparent).

1.3.1.2 Insulating pressure media

Since pressure on the sample is applied uniaxially (i.e., by advancing the two diamonds toward each other together via screw turns, Figure 1.3), a pressure gradient develops across the

11 sample as well as differential/shear stress. Consequently, pressure media ranging from weak solids (NaCl, KCl, KBr, etc.), liquids (methanol, ethanol, silicone oil, etc.) or gases (Ne, Ar, He, etc.) can be used to ensure a hydrostatic or quasi-hydrostatic pressure distribution within the sample chamber. The hydrostatic limits of a wide variety of materials have been investigated

Figure 1.4. Typical DAC loading style for a 200-μm culet viewed from above and from

the side. a. “sandwich” loading, employed when a solid such as NaCl is used as a thermal

insulator and pressure medium. Exposed diamond surface is shown in blue. When the cell

is closed, the gasket collapses further and the insulator fills the chamber. b. “tripod”

loading, employed in conjunction with gas loading. As the cell is pressurized, the gas

undergoes a gas → liquid → solid transition, leading to significant gasket collapse. The

ruby is used both as a physical separator between the sample and the diamond and as a

pressure marker when trapping gas within the sample chamber.

12 extensively in the literature.6 For most of the experiments performed here, Ne or NaCl were used as pressure transmitting media. The loading procedure for these two media is very different. In the case of Ne, the sample is placed within a cell, and the cell is assembled, after which Ne is pumped within the DAC at elevated pressure (up to ~0.3 GPa) and the cell is closed remotely to trap the gas inside. In the case of NaCl, the sample is embedded within a NaCl disk which is placed within the DAC confining ring, followed by closure of the DAC. For several of the experiments performed here, laser heating (described below) is employed to achieve temperatures in the 1000-2500 K range. Since diamonds are efficient thermal conductors and large in size relative to the sample, uniform heating can only be achieved if the sample is well isolated from the diamonds. The pressure transmitting media described above (Ne, NaCl) can also serve as insulation media for heating experiments, although sufficient care must be taken during loading to ensure that the sample is completely isolated from the diamonds (Figure 1.4).

Failure to insulate the sample can lead to difficulties in achieving high temperatures at available laser powers and in uneven heat distribution, resulting in temperature gradients.111 These can compromise experiments in a number of ways by, for example, resulting in chemical heterogeneities due to diffusion or non-equilibrium behavior.112

1.3.1.3 Gaskets

Samples are placed within gaskets/confining rings that in this work were typically made out of steel or Re metal, although other compositions such as Be, cBN or epoxy can be used depending on the experimental conditions. These gaskets were preindented to ~30-35 um thickness between the previously-aligned diamonds, and then drilled to create sample chambers whose size depended on the diamond culet width and the experimental conditions. For instance,

13 we have empirically observed that when using a Ne pressure medium, gasket hole sizes should be 55-60% of the culet size due to allow for contraction of the hole upon increasing pressure as

Ne undergoes phase transitions from gaseous to liquid to the solid state. In contrast, samples loaded with NaCl require a gasket hole size that is 35-50% of the culet size, due to the relatively smaller volume contraction experienced by this medium as pressure increases.

1.3.2 Laser heating

Another important aspect of our experiments is the system for achieving high temperatures in compressed samples. The schematic in Figure 1.5 includes a diagrammatic view of the laser heating system used in this work.113 The laser system consists of two diode-pumped single-mode ytterbium fiber lasers (λ = 1064 μm), each with output power up to 100 W. The IR beams are directed through beam-shaping optics to produce a flat top shape with a spot size of approximately 10-20 μm. Temperature is recorded using spectroradiometry and is determined by fitting the radiated spectrum to the Planck radiation function:

c1ε(λ) Iλ = , 5 c2 λ [exp (λT) - 1] where Iλ is the spectral intensity, c1 = 2πhc, c2 = hc/k , ε is the emissivity, λ is the wavelength, and T is the temperature.114 As can be seen, the spectral radiation intensity is proportional to the emissivity, which is a measure of the ability of a sample to thermally radiate. In a black body radiator ε = 1, but most materials, including those measured here, function as grey body emitters, with ε < 1. Additionally, ε is sample dependent as well as dependent on wavelength.115 In these experiments, ε is considered to be constant, as wavelength dependences of emissivity are not well constrained at high pressures and temperatures.114

14

1.3.3 Synchrotron X-Ray diffraction

Most of the experiments performed in this work use the DAC with synchrotron X-ray diffraction as the primary diagnostic. Here an explanation of the basics of X-ray diffraction and synchrotron operation are presented.

1.3.3.1 Synchrotron Radiation Facilities

A synchrotron is a high-energy particle storage ring where a guiding magnetic field is synchronized to a particle beam confined within a ring. At the Advanced Photon Source, where the experiments presented here were performed, the synchrotron takes the form of a storage ring, where electrons are kept in radial acceleration for prolonged periods. The electrons are initially accelerated using a linear accelerator, then injected into a booster synchrotron, which brings their speed to >99.999999% that of the speed of light, prior to being injected into the main storage

Figure 1.5. Schematic of the laser-heated diamond anvil cell synchrotron experiment

based on Prakapenka et al.113

15 ring. X-rays are emitted when the electrons are radially accelerated by one of several methods, and the emitted X-rays are directed along beamlines located tangentially to the ring. Because of the energy loss due to X-ray emission, several radio-frequency acceleration cavities are located around the ring to replenish electron energy. A schematic diagram of the geometry employed in these experiments can be seen in Figure 1.5.

1.3.3.2 X-ray diffraction

Since the wavelength of X-rays is on the atomic size scale, X-ray diffraction is well suited to probing the structure of crystalline solid materials. The basic principle of X-ray diffraction is that

X-rays will coherently diffract when the distance between the crystal lattice planes fulfills the following relationship (Bragg’s law):

nλ = 2dhklsin(θ), where λ is the X-ray wavelength, dhkl is the interplanar distance, and θ is the angle of the diffracted beam relative to the crystal lattice plane (Figure 1.6). The intensity of the diffracted beam of the plane hkl is given by: p I = k hkl (LPA)λ3F2 exp⁡(-2M - 2M ), hkl 1 v2 hkl t s where k1 is an instrument scaling constant, phkl is the multiplicity of the crystal plane, v is the volume of the unit cell, LPA is the Lorentz polarization and absorption factor, λ is the X-ray wavelength, Fhkl is the structure factor, and exp⁡(-2Mt⁡- 2Ms) is a temperature attenuation factor due to lattice thermal vibrations and static displacement.116 In randomly oriented polycrystalline samples, diffracted intensity forms cones at 2θ angles relative to the incoming beam which are then collected by a 2D detector, producing concentric diffraction rings.

16

Figure 1.6. Schematic representation of Bragg’s law conditions.

The dhkl spacing can be calculated from the 2θ angle of the diffracted beam. It is also dependent on the size of the unit cell and can be found through a mathematical relationship dependent on the lattice parameters, the lattice angles, and the Miller indices. The general form of this expression is:

1 [h2b2c2sin2(α) + k2a2c2sin2(β) + l2a2b2sin2(γ) + m + n + o] = , d2 a2b2c2[1 - cos2(α) - cos2(β) - cos2(γ) + 2cos(α)cos(β)cos(γ)] where m, n, and o can be represented by:

m = 2hkabc2[cos(α)cos(β) - cos(γ)],

n = 2kla2bc[cos(β)cos(γ) - cos(α)],

o = 2hlab2c[cos(α)cos(γ) - cos(β)].

This relationship becomes significantly simplified once symmetry considerations are taken into effect. For instance, in a cubic unit cell a = b = c and α = β = γ = 90º, and the d spacing expression described above simplifies to:

17

1 h2 + k2 + l2 = . d2 a2

As mentioned above, symmetry considerations greatly influence the diffraction pattern, and the structure factor, Fhkl, is dependent on the symmetry (i.e. periodicity) of the unit cell such that:

N

Fhkl = ∑ exp[2πi(hun + kvn + lwn)]fn , 1

th where N is the total number of atoms in the unit cell, (un, vn, wn) give the location of the n atom or ion in the unit cell, and fn is the atomic scattering factor, which is dependent on the number of electrons, the wavelength, and the scattering angle.116 The intensity of the beam is proportional

2 2 * * to|Fhkl| , where |Fhkl| = Fhkl∙Fhkl and Fhkl is the complex conjugate of Fhkl. Because of this relationship between the structure factor and the intensity of the beam, atomic positions can also

* be fit, and some lattice planes experience extinction, such that Fhkl = Fhkl. This enables the unit cell lattice parameters and the symmetry to be determined from the diffracted pattern.

Additionally, since the peak intensity is ultimately related to atomic scattering factors, the atomic positions can be refined.

1.3.4 Raman spectroscopy

Raman spectroscopy is based on the inelastic scattering of light by optic vibrational modes.117,118 Atomic bonds have non-zero polarizability when in an oscillating electric field such that the induced dipole moment of a bond, μind, can be described by:

1 μ = α ε cos(2πνt) + ∆αε {cos[2π(ν + ν )t] + cos[2π(ν - ν )t]}, ind 0 0 2 0 0 0 where α0 is the equilibrium polarizability of the molecule, ε0 is the maximum value of the electric field, ν is the incoming light source frequency, t is time, α is the polarizability of the 18 bond, and ν0 is the natural frequency (i.e. the frequency of the bond vibration). In terms of the above equation, Δα must be a non-zero term for Raman scattering to occur. The first term of the equation describes elastic Raleigh scattering at frequency ν, while the second term describes the inelastic Raman scattering with frequencies of (ν - ν0) (Stokes) and (ν + ν0) (anti-Stokes). Since the scattering effect is due to photon interactions with the electron cloud about the bond, the symmetry of the vibrational mode will directly affect the shape and polarizability of the electrons, and thus only certain vibrational modes can produce Raman scattering. Only 1 in 106-

107 photons will be Raman scattered, and the intensity of the scattered light [in this case, of the

Stokes-shifted component, I(νS), which is the one measured here] is described by:

4 2 νS 2 δα I(νS) = 3 Q0I0 | | , 12πϵ0c δQ where ϵ0 is the permittivity in vacuum, c is the speed of light, I0 is the intensity of the incident light, and Q is the nuclear motion described by:

Q(t) = 2Q0cos(ν0t + ϕ),

119 where Q0 is the amplitude of nuclear motion and ϕ is the phase of the nuclear mode vibration.

The intensity of the scattered light is directly correlated to the polarizability as a function of atomic motion. Raman spectroscopy can thus provide information about symmetry and bonding characteristics within a compound, and changes in the Raman spectrum as a function of pressure or temperature can indicate changes in bond strength and the occurrence of phase transitions.

In the case of a Raman measurement, monochromatic laser light with frequency ν is directed onto a sample, and scattered light with frequency ν ± ν0is collected by a detector. Because the

Raman effect is a scattering phenomenon, the angle at which light can be collected is independent of that of the incident laser. The experiments described in this thesis were

19

Figure 1.7. Schematic of a typical Raman spectrometer in backscatter geometry.

performed on a Horiba LabRam HR Evolution instrument, using a 532-nm diode pumped solid state laser (Figure 1.7). Scattered radiation is collected from the backscattered photons and resolution of the instrument depends on the groove density and focal length of the diffraction grating as well as the width entrance slit to the spectrometer, and is ~1 cm-1 for the work presented here. The detector is calibrated against a known standard, such as silicon, which relates the spatial dispersion on the detector to specific wavenumber values.

20

1.3.5 Pressure determination

One final consideration is that of pressure determination. Direct determination of pressure through applied force on the DAC culet is not possible. As a result, calibrated standards are used.

Pressure is then determined through the use of a pressure marker within the sample chamber, ideally one mixed directly within the sample volume of interest. An ideal pressure standard is compressible, inert, and does not undergo phase transitions over the desired pressure and temperature range of an experiment.

Calibration of pressure markers has to be performed through independent measurements of the pressure or compressibility and the volume or density of the marker. Two different approaches exist: shock experiments, where the density, pressure, and internal energy of a material along a Hugoniot curve is determined, or through the determination of adiabatic elastic properties using either ultrasonic interferometry or Brillouin scattering combined with density measurements such as those obtained from X-ray diffraction (e.g. Zha et al.120). Both approaches suffer from some limitations. In the case of shock experiments, corrections to account for thermal effects in shock loading are required (e.g. Brown and McQueen121). Elastic property measurements are primarily hampered by experimental difficulties, and the fact that equations of state determined through such measurements cannot reliably be extrapolated to pressures beyond the range of measurement.120 High-pressure equation of state determinations are often performed by cross-calibrating several primary standards simultaneously, and different sets of equations of state have been determined at high pressure using the DAC.122–128

In this work, high-pressure experiments were performed using X-ray diffraction and Raman spectroscopy. In the case of X-ray diffraction experiments, pressure determination is typically performed using the equation of state of a simple cubic material such as Pt, Au, or MgO.122–126

21

The unit cell volume of the pressure standard is calculated from its measured diffraction pattern, and then the pressure is calculated using an equation of state such as the third order Birch-

Murnaghan equation of state, which takes the following form:

7 5 2 3 V 3 V 3 3 V 3 P(V) = K [( 0) - ( 0) ] {1 + (K' - 4) [( 0) - 1]} , 2 0 V V 4 0 V

' where P is the pressure, K is the bulk modulus, K is the pressure derivative of the bulk modulus,

V is the unit cell volume and the subscript 0 refers to ambient pressure.

A commonly used optical pressure scale is the ruby fluorescence scale.127–130 This method is based on using a laser to excite the R1-R2 fluorescence bands of ruby, whose frequency is known to be a strong function of pressure. Broadening and overlap of the R1 and R2 bands is observed under non-hydrostatic conditions, and ruby fluorescence measurements can be sensitive indicators of stress conditions at high pressure. One main advantage of this technique is that a strong fluorescence signal can be detected with a basic laser system setup, and thus it is a relatively inexpensive lab-based method of pressure determination.

One problem that can arise is that many equations of state are empirically derived through diamond anvil cell experiments. For instance, in the case of the ruby fluorescence scale, an empirical equation describes the relationship between the position of the R1 and R2 lines as a function of pressure. While calibration of the ruby fluorescence scale has been performed using the volumes of several metals whose primary equations of state had been previously determined by shock compression experiments, the ruby fluorescence scale is not itself a primary scale, and several independently derived equations of state exist.1,129,130 Thus, pressure scales used in different studies may not always be directly comparable. In fact, this is one of the motivations behind out approach to Chapter 4, where we use consistent pressure scales, pressure media, and

22 loading techniques, so that experiments on different compositions can be directly compared to one another.

1.3.5.1 Differential pressure

As stated before, one challenge in high pressure studies that must be overcome is the difficulty inherent in generating hydrostatic stress conditions in the cell. At high pressures, and depending on the pressure medium and experimental conditions employed, differential (i.e. non- hydrostatic) stress is inevitably generated to some degree. Quantifying the amount of differential stress in a given experiment provides valuable information about hydrostaticity, gives an insight into the robustness of equations of state determined during the experiment, and allows for comparison of equations of state across different experiments experiencing similar stress conditions.

In the simple case of a cubic material, differential stress will be observed as a variation in the

125,131–133 measured lattice parameter, am, as a function of the lattice plane being measured . The differential stress, t, can be determined from the relationship between the measured lattice parameter, am(hkl), of a cubic material and the set of elastic relationships:

2 am(hkl) = M0 + M1[3(1 - 3sin θ)Γ(hkl)], where

αt 1 2 1 M0 = ap {1 + ( ) (1 - 3sin θ) [S11⁡- S12⁡- (1 - )]} , 3 2GV α

apαtS M = - , 1 3

h2k2 + k2l2 + l2h2 Γ(hkl) = 2 , (h2 + k2 + l2)

23

1 1 S = ⁡- , C11-C12 2C44 and ap is the hydrostatic lattice parameter, Sij are elastic compliances, Cij are elastic stiffnesses, and GV is the Voigt bound on the shear modulus. Here, α is a parameter that defines the stress condition on the iso-stress (Reuss-like) – iso-strain (Voigt) grain boundary condition continuum.

The value of α typically lies between 0.5 and 1, where 1 indicates the Reuss condition and assumes that both the pressure marker and the sample are experiencing the same pressure (stress) condition. Values for Cij can be determined from high-pressure extrapolations of ultrasonic measurements using a form such as:

1 3 0 dCij V Cij = Cij + P ( ) , dP V0

0 where Cij are the atmospheric pressure value. The value of t can be determined through the simplification M0 ≈ ap, so that:

3M t ~ - 1 , αM0S and M0 and M1can be calculated as the intercept and slope of a plot of am(hkl) versus

3(1-3sin2θ)Γ(hkl). In a cubic material such as Au, differential stress in He-loaded experiments can reach up to 0.9 GPa at pressures of 116 GPa.125 In our experiments, the value of t ranges to as high as 2.6 GPa at pressures of 150 GPa (see Chapter 4), and was used as a criterion to determine suitability of data for equation of state determination.

24

CHAPTER 2

High-Pressure Polymorphism of PbF2 to 75 GPa

Portions of this work were presented as:

Stan C. V., Prakapenka V. B., Duffy T. S. (2012) Post-Cotunnite Phases in the AX2 System: Compression of PbF2 to

80 GPa, Abstract MR23B-2401 presented at 2012 Fall Meeting, AGU, San Francisco, CA, 3-7 Dec.

Stan C. V., Dutta R., Prakapenka V. B., Duffy T. S. (2015) High-pressure isosymmetric phase transformation in

PbF2, Abstract presented at 2015 COMPRES Annual Meeting, Colorado Springs, CO, 6-9 Jul.

2.1 Abstract

The high-pressure behavior of lead fluoride PbF2 is of interest for understanding phase transformations in cotunnite-structured materials under compression. Here PbF2 was examined at ambient temperature up to 75 GPa by X-ray diffraction and to 39 GPa using Raman spectroscopy. In addition, simultaneous high-pressure high-temperature X-ray diffraction experiments were performed up to 65 GPa and 2430 K. During room-temperature compression, no discontinuous change in the X-ray diffraction pattern or volume discontinuity was observed, but the lattice parameters display highly anomalous trends between 10-22 GPa with enhanced compressibility along the a direction and reduced or even negative compressibility along b and c.

Our Raman and X-ray results are consistent with an isosymmetric phase transition in orthorhombic PbF2 from a 9-coordinated cotunnite-type phase to a 10-coordinated Co2Si-type phase with the transition occurring over a broad pressure interval rather than abruptly as previously suggested. The characteristic values for the cell constants a/c and (a+c)/b which are used to distinguish among cotunnite-, Co2Si-, and Ni2In-type phases require modification based

25 on our results. An equation of state fit yields a bulk modulus, K0, of 72(3) GPa for cotunnite, and

3 an ambient-pressure volume, V0, of 182(2) Å , and K0 = 81(4) GPa for Co2Si-type phase when fixing the pressure derivative of the bulk modulus, K0’ at 4. Upon heating above 1200 K at pressures above 26 GPa, PbF2 partially transforms to the hexagonal Ni2In-type phase but wholly or partially reverts back to Co2Si-type phase upon temperature quench. From 43-65 GPa, nearly complete transformation to the Ni2In-type PbF2 is observed at high temperature, but the material partially transforms back to the orthorhombic phase upon temperature quench. Our results show that high-pressure behavior of PbF2 is distinct from that of the alkaline earth fluorides with similar ionic radii.

2.2 Introduction

The high-pressure behavior of compounds in the AX2 family has attracted much attention due to their extensive polymorphism, highly coordinated structures, and diverse transformation

7 pathways. The difluoride family (AF2, A = Ca, Sr, Ba, Pb, etc.) has potential technical applications134–137 which have motivated a number of studies of their high-pressure behavior.138–

149 The alkaline earth fluorides, CaF2, SrF2, and BaF2, undergo transformations from the ambient fluorite structure to the cotunnite-type and Ni2In-type structures at high pressures and temperatures (Figure 2.1).138,149,150 These difluorides can serve as analogs for various dioxides including those of group 4 (TiO2, ZrO2), group 14 (SiO2, GeO2, SnO2, PbO2) and the lanthanides and actinides (CeO2, ThO2, PuO2, and UO2), all of which have been shown or predicted to adopt the cotunnite and related structures at high pressures.12,13,15,151–157 The expected transformation of

SiO2 to the cotunnite structure at pressure exceeding 750 GPa is of potential importance for understanding the interior of large extrasolar planets.15,158

26

Figure 2.1. a. Fluorite, b. cotunnite, c. Co2Si and d. Ni2In structures. Ni2In is viewed

down the c axis; all others are viewed down the b axis.

Dorfman et al.149 performed powder X-ray diffraction experiments on alkaline earth difluorides and showed that the transition pressures from fluorite (cubic, Fm3̅m, Z=4) to cotunnite (orthorhombic, Pnma, Z=4) to Ni2In-type (hexagonal, P63/mmc, Z=2) correlate with the cationic radii of these materials (Table 2.S.1). Lead fluoride, PbF2, has a comparable cationic radius to the alkaline earth metals but a rather different electronic configuration. PbF2 can be synthesized in either the fluorite β-PbF2 or cotunnite α-PbF2 phase at ambient conditions and

143,159 adopts the cotunnite structure at pressures exceeding 0.4 GPa. Cotunnite-type PbF2 was 27

Figure 2.2. Projections of the a. cotunnite b. Co2Si and c. Ni2In structures along the

orthorhombic ac plane. Triangular prisms are denoted by triangles on structure.

Orthorhombic unit cells are represented by dashed boxes. Red dashed lines indicate the

bonds between the central Pb atom and the triangular prism caps. The structures are

related through a rotation (cotunnite → Co2Si) followed by a distortion (Co2Si → Ni2In)

of the triangular prisms around the b axis to bring them in alignment relative to the c axis.

suggested to undergo an isosymmetric phase transition to a “Co2Si-like” phase [orthorhombic,

Pnma, Z=4, Figure 2.1 (c)] at ~10 GPa at room temperature.144 Its behavior at higher pressures and temperatures is poorly constrained at present.

The cotunnite-type, Co2Si-type, and Ni2In-type structures occur over a wide range of compositions and are closely related structurally.160 Each is based on a distorted hexagonal close 28 packed arrangement of anions with cations in face-sharing polyhedra forming corrugated sheets.

The cation coordination polyhedra are multi-capped trigonal prisms, where the number of caps increases with coordination number in sequence. Thus, the cotunnite-type phase (9-fold coordination) is tri-capped, Co2Si (10-fold coordination) is tetra-capped and Ni2In (11-fold coordination) is penta-capped (Figure 2.2). All caps are located along the rectangular faces of the trigonal prisms, which form corrugated layers that are stacked along the b direction. While the

144 cotunnite-type and a Co2Si-like structure have been reported previously in PbF2 and

161 theoretical calculations suggest that Ni2In-type PbF2 should be stable at higher pressures,

Ni2In-type PbF2 has not been synthesized to date, and no simultaneous high-pressure and - temperature experimental studies have been reported.

Here we investigate the behavior of PbF2 using both powder X-ray diffraction and Raman spectroscopy in order to better understand the high-pressure behavior of this fundamental material. In a separate work (Appendix A), we have performed related theoretical calculations to further explore the changes in bonding and structure in this material.

2.3 Methods

Polycrystalline PbF2 (Alfa Aesar, >99% purity) was examined by X-ray diffraction at ambient conditions and indexed as orthorhombic cotunnite-type with the unit cell a = 6.4472(3)

Å, b = 3.9019(2) Å, c = 7.6514(3) Å and V = 192.48(1) Å3 (Figure 2.S.1). The sample was ground to micron-sized grains and mixed with 10-15 wt.% platinum (Aldrich, 99.9% purity, 0.5–

1.2 μm particle size) which was used as a pressure calibrant and laser absorber. The mixture was pressed into a thin layer and loaded into piston-cylinder type diamond anvil cells using either

NaCl or Ne as pressure-transmitting media (Table 2.S.2). In one experiment, a small foil of Pt

29 metal was loaded adjacent to pure PbF2 powder, and in another, 5 wt.% graphite was mixed in the sample as a laser absorber instead of Pt. This strategy was employed in order to eliminate overlap between the sample and Pt peaks in the diffraction patterns between 7-15 GPa. Diamond anvils with culet sizes between 500 μm and 200 μm were mounted on tungsten carbide or cubic boron nitride seats. Sample chambers were formed by drilling holes in Re gaskets that were pre- indented to 20-30 μm thickness. The holes were approximately half the diameter of the diamond culet.

Angle-dispersive X-ray diffraction experiments were performed at beamline 13-ID-D

(GSECARS) of the Advanced Photon Source APS, Argonne National Laboratory. A monochromatic X-ray beam with λ = 0.3344 Å was focused using Kirkpatrick-Baez mirrors to a size of approximately 4 μm x 4 μm onto the sample. Diffracted X-rays were recorded using a

MarCCD detector. The distance and orientation of the detector was determined using CeO2 or

LaB6 as standards. The two-dimensional CCD images were radially integrated using the

162 163 programs FIT2D or DIOPTAS to produce conventional one-dimensional diffraction patterns.

High-temperature experiments were carried out using a double-sided laser heating system consisting of two near-infrared diode-pumped single-mode ytterbium fiber lasers. Beam-shaping optics were used to produce a flat-topped laser profile with a spot size of ~24 μm. Sample temperatures were measured from both sides by spectroradiometry. During these experiments, the incident laser power was adjusted to keep the measured temperature difference from the two sides of the cell to less than 100 K. Further details of the X-ray diffraction and laser heating systems at 13-ID-D are reported elsewhere.113

X-ray diffraction peaks were fit using non-linear least squares to background-subtracted pseudo-Voigt line profiles. Lattice parameters were obtained from measured peak positions using

30

164 the program UNITCELL. Pressure was determined from the equation of state of platinum using its (111) diffraction line.123 Equation of state fits were performed using the EosFit7 program suite.165

Raman spectra were acquired using a Horiba LabRam HR Evolution system using a 532-nm laser (Ventus, Laser Quantum) and an 1800-g/mm diffraction grating. The diamond anvils in these experiments had 300-μm culets and were selected for low-fluorescence. Loose PbF2 powder was loaded with a small ruby crystal into a hole in a Re gasket. Neon served as the pressure-transmitting medium. Pressure was determined using the ruby fluorescence scale.130

2.4 Results and Discussion

2.4.1 Room-temperature compression of PbF2

2.4.1.1 X-ray diffraction

X-ray diffraction experiments were carried out at ambient temperature to 75 GPa. The diffraction peaks evolved under compression towards smaller d-spacings but no new peaks were observed over the measured pressure range (Figure 2.3a). An impurity peak belonging to Pb metal was also seen with varying intensity in some patterns, but disappeared above 18 GPa. A notable feature of the dataset was that the (200) and (211) diffraction lines of PbF2 exhibit a remarkably rapid shift in d-spacing as a function of pressure relative to other diffraction lines, indicating a highly anisotropic compressibility (Figure 2.3b).

The lattice parameters and volume of PbF2 were fit using an orthorhombic cell over the measured pressure range (Figures 2.4 and 2.5, Table 2.1). The a lattice parameter exhibits very high compressibility below 22 GPa. In the 10-22 GPa range, there is a distinct change in the

31

Figure 2.3. a. Representative X-ray diffraction patterns of PbF2 as a function of pressure.

Red tick marks indicate cotunnite-/Co2Si-type diffraction peak locations, black tick

marks indicate Pt peaks. Miller indices (hkl) of selected PbF2 peaks are shown. Asterisk

(*) tracks the (200) peak of PbF2. Dagger (†) indicates peak from metallic Pb impurity. b.

Measured d-spacings of selected PbF2 peaks up to 75 GPa. The (200) peak is shown in

blue to highlight its unusual compressibility.

compressibility of the b and c lattice parameters whereby b axis exhibits little compression while the c axis compresses weakly at first and then expands slightly, resulting in a small net compression over this pressure range (Figure 2.4, Table 2.S.3). Despite the large amount of

32

Figure 2.4. Lattice parameters as a function of pressure for PbF2. The dashed red lines are from theoretical calculations. Dashed black vertical lines mark the transition region.

33

Figure 2.5. Unit cell volume of PbF2 as a function of pressure. Solid lines represent the

equation of state fits. Symbols are the same as in Figure 2.5.

anisotropy in the axial compressibilities, the pressure-volume curve exhibits a generally smooth variation over the experimental pressure range without any sharp discontinuity. The variation in cell volume and lattice parameters is reversible upon decompression with little hysteresis (Figure

2.S.4). The anomalous compression behavior observed here is generally consistent with prior

144,166 experiments on PbF2 covering more limited pressure ranges (Figures 2.4, 2.5). The slightly higher volumes reported by Haines et al.144 above 12 GPa are likely the result of differential stress due to their use of silicone grease as a pressure medium (see supplementary material). To explain the anomalous compression behavior, Haines et al.144 proposed an isosymmetric phase 34 transition between 10 and 13 GPa from cotunnite-type PbF2 to a “Co2Si-like” structure, accompanied by a ~2% volume discontinuity. Our results are broadly consistent with those of

Haines et al.,144 indicating a change in compressibility beginning near 10-13 GPa; however we observe a second compressibility change near 22 GPa which was not recognized by Haines et al.144 as it is close to the upper pressure limit of their experiment. From 22 GPa to the maximum pressure of 75 GPa, all lattice parameters exhibit smooth variation (Figure 2.4). Our study demonstrates that there is a continuous variation of lattice parameters between 10-22 GPa and, in contrast to Haines et al.,144 no detectable volume discontinuity (Figure 2.5). In a separate work

(Appendix A), we have also carried out theoretical calculations of the high-pressure behavior of

PbF2 using density functional theory. The theoretical calculations yield volumes and axial compression behavior that are consistent with our experimental results (Figures 2.4, 2.5).

Haines et al.144 reported other changes in the diffraction patterns between 10-13 GPa including the sudden appearance of peak broadening and preferred orientation. In contrast, we do not observe any abrupt peak broadening or significant changes in relative peak intensities over the course of the experiment. The behavior reported by Haines et al.144 is likely due to their use of silicone grease as a pressure medium which becomes strongly non-hydrostatic near 12 GPa.167

The anisotropic compression in PbF2 is similar to behavior reported in the cotunnite-type

149 phase of the alkaline earth fluorides CaF2, SrF2 and BaF2. A negative compressibility along the c axis and very high compressibility along a is observed in these compounds. However, this change in compressibility is part of a continuous transition to the hexagonal Ni2In phase in the alkaline earth fluorides, whereas under quasi-hydrostatic conditions PbF2 does not transform to the Ni2In-type structure to at least 75 GPa without the application of heating. The unusual

35

Table 2.1. Unit cell parameters of PbF2 during room-temperature compression.

Pressure (GPa) a (Å) b (Å) c (Å) V (Å3)

0 6.4472(3) 3.9019(2) 7.6514(3) 192.48(1) 0.6 6.444(2) 3.899(1) 7.652(2) 192.23(6) 1.3 6.437(5) 3.889(3) 7.642(4) 191.3(2) 2.7 6.3814(8) 3.8670(6) 7.5984(8) 187.51(3) 2.7 6.379(1) 3.8681(8) 7.596(1) 187.40(4) 2.7 6.3789(5) 3.8679(3) 7.5964(4) 187.43(2) 2.8 6.381(1) 3.8686(9) 7.593(1) 187.44(4) 3.0 6.363(4) 3.859(3) 7.585(4) 186.2(1) 3.0 6.363(2) 3.859(2) 7.583(2) 186.18(7) 3.2 6.360(1) 3.8583(9) 7.583(1) 186.09(4) 3.8 6.333(1) 3.8447(8) 7.564(1) 184.16(4) 3.8 6.332(3) 3.845(2) 7.563(3) 184.2(1) 3.9 6.332(1) 3.8443(7) 7.5636(9) 184.12(3) 5.1 6.292(3) 3.827(2) 7.537(2) 181.52(9) 5.1 6.289(1) 3.8280(9) 7.536(1) 181.40(4) 5.2 6.251(2) 3.820(2) 7.527(2) 179.73(7) 5.3 6.286(3) 3.827(2) 7.532(3) 181.2(1) 5.3 6.246(1) 3.8169(8) 7.522(1) 179.35(4) 5.5 6.2429(8) 3.8152(6) 7.5196(7) 179.10(3) 5.6 6.236(2) 3.813(1) 7.518(2) 178.74(6) 6.4 6.240(3) 3.812(2) 7.510(3) 178.63(9) 6.4 6.242(2) 3.809(1) 7.512(2) 178.60(7) 6.7 6.235(5) 3.806(3) 7.511(5) 178.2(2) 7.0 6.207(3) 3.798(2) 7.494(3) 176.69(9) 7.0 6.208(2) 3.797(1) 7.496(2) 176.69(6) 8.7 6.164(3) 3.784(3) 7.475(3) 174.4(1) 8.7 6.164(2) 3.783(2) 7.477(2) 174.36(8) 9.9 6.108(2) 3.769(1) 7.457(2) 171.66(6) 11.5 6.046(5) 3.762(4) 7.445(5) 169.3(2) 13.5 5.970(2) 3.752(2) 7.431(2) 166.47(7) 15.2 5.910(2) 3.750(2) 7.421(2) 164.48(7)

36

Table 2.1, continued.

Pressure (GPa) a (Å) b (Å) c (Å) V (Å3)

15.8 5.846(7) 3.745(6) 7.421(7) 162.5(1) 17.9 5.85(3) 3.73(2) 7.42(4) 162(1) 19.0 5.713(5) 3.740(4) 7.424(5) 158.6(2) 19.4 5.618(3) 3.746(3) 7.433(4) 156.4(1) 22.3 5.507(2) 3.732(2) 7.438(3) 152.86(8) 26.2 5.442(3) 3.714(3) 7.433(4) 150.2(1) 27.0 5.378(2) 3.694(2) 7.426(3) 147.52(8) 29.3 5.330(5) 3.670(4) 7.417(7) 145.1(2) 30.0 5.365(4) 3.664(4) 7.399(6) 145.4(2) 31.7 5.28(1) 3.65(1) 7.42(2) 142.8(5) 34.6 5.250(8) 3.623(7) 7.39(1) 140.6(3) 38.0 5.216(6) 3.590(5) 7.360(8) 137.8(2) 38.3 5.206(4) 3.597(3) 7.355(5) 137.7(1) 39.6 5.196(3) 3.587(3) 7.346(4) 136.9(1) 43.0 5.166(4) 3.569(4) 7.319(6) 134.9(1) 47.3 5.135(6) 3.546(6) 7.286(9) 132.7(2) 48.5 5.12(1) 3.55(1) 7.29(2) 132.4(5) 51.2 5.111(7) 3.529(7) 7.26(1) 130.9(3) 54.4 5.088(9) 3.516(9) 7.24(1) 129.5(3) 57.9 5.07(1) 3.50(1) 7.21(2) 128.0(4) 61.8 5.05(1) 3.49(1) 7.19(2) 126.6(4) 63.3 5.04(1) 3.48(1) 7.18(2) 126.1(5) 64.6 5.03(1) 3.48(1) 7.17(2) 125.6(5) 65.9 5.03(2) 3.47(1) 7.17(2) 125.1(5) 67.4 5.02(1) 3.47(1) 7.16(2) 124.6(5) 68.4 4.990(5) 3.46(1) 7.164(7) 123.7(2) 70.8 4.99(1) 3.46(1) 7.15(2) 123.4(5) 73.0 4.972(4) 3.446(4) 7.135(6) 122.2(1) 74.9 4.957(5) 3.437(5) 7.126(9) 121.4(2)

37 compressional behavior seen here is similar to that reported in theoretical calculations for cotunnite-structured lanthanide and actinide dioxides.12–14,156

2.4.1.2 Raman spectroscopy

-1 -1 Raman spectra for PbF2 were collected over a spectral range of 50 cm to 1000 cm in ~1-2

GPa steps up to 39 GPa, as well as upon decompression (Figure 2.6). At ambient conditions, our

159,168 results are consistent with previous measurements of α-PbF2. Mode assignments for the

159 room-pressure PbF2 spectrum based on the work of Kessler et al. are listed in Table 2.2. A total of 18 Raman-active modes are expected for this structure, of which thirteen are observed at ambient pressure. The pressure dependence of mode frequencies is shown in Figure 2.7. At

Table 2.2. Raman peaks assignments and 1-bar frequencies for α-PbF2 at ambient

conditions.

Symmetry Raman Shift (cm-1) This work Kessler et al. (1974)

1 1 1 2 - 48 A1g , B1g , B3g , B1g 2 56 62 A1g 1 84 81 B2g 3 116 124 A1g 3 146 127 B1g 2 146 151 B3g 4 167 172 A1g 4 173 174 B1g 2 5 208 211 B2g , A1g 5 3 224 228 B1g , B3g 6 244 241 B1g 6 250 250 A1g 3 - 253 B2g 38

Figure 2.6. Representative Raman spectra of PbF2. Peak identifications follow Kessler et

al.159 Below 120 cm-1, the intensity of the spectra were scaled by a factor of 1/2 at 15.4

GPa and 1/5 at 39.0 GPa.

pressures below about 10 GPa, the modes generally exhibit a smooth increase with pressure.

Between 10-21 GPa, a number of changes are observed including strong intensity changes, emergence of new peaks, and nonlinearities in pressure dependence of some peaks. Above about

21 GPa, the Raman spectra again display generally smooth pressure dependence. Peak position are largely reversible upon decompression with little hysteresis.

2 1 Upon compression, the low-frequency modes A1g and B2g modes displayed a strong increase in intensity as a function of pressure (Figure 2.6). We also observe the appearance of 39

Figure 2.7. Raman frequencies in PbF2 as a function of pressure. Dashed vertical lines

mark transition region.

two new modes, ν1 and ν2 in this spectral region above 10 GPa. A new mode ν4 emerges above ~14 GPa at 122 cm-1 and steadily increases in intensity with increasing pressure. The

3 intensity of the A1g mode steadily increases up to 10 GPa, at which point it decreases sharply.

3 It also overlaps with the neighboring B1g peak above 8 GPa and up to 14 GPa, at which point

4 2 these two split again. Simultaneously, the A1g peak and the B3g peak merge, and this new

3 merged peak approaches B1g up to 20 GPa before splitting off from it (Figure 2.7).

40

-1 Another peak with very low intensity emerges at 362 cm at pressures above ~10 GPa (ν3).

The intensity of this peak steadily increases, and it shifts positively with increasing pressure throughout the entire pressure range. This mode is not reported in the work of Lorenzana et al.168 due to the more restricted frequency range covered in the prior study.

The number of observed Raman modes at high pressure ranges from about 11 to 13 which is compatible with expectations for the Co2Si-type structure (18 Raman-active modes). Overall, our

Raman spectra are consistent with previous work168 although we are able to observe a number of additional features due to the higher resolution and wider spectral range of our study. The Raman results are compatible with the X-ray diffraction results and theoretical calculations which show that PbF2 undergoes continual structural modification over the 10-22 GPa pressure range.

2.4.2 Isosymmetric phase transition

The cotunnite-type and Co2Si-type phases are orthorhombic structures with space group

Pnma and are isopuntal in that both have the same occupation of Wyckoff sites. The two structures are distinguished through the cation coordination which is nine-fold in cotunnite and ten-fold in the Co2Si-type phase. Our density functional theory calculations of electron density and Pb-F bond distances demonstrate that PbF2 transforms to the Co2Si-type structure above 21

GPa (Appendix A). This is consistent with our experimental measurements. The uniform compression behavior observed below 10 GPa and above 22 GPa represents compression of the cotunnite, and Co2Si-type phases, respectively. The behavior observed between 10-22 GPa is an intermediate region of continuous structural modification in which a 10th F ion approaches the coordination polyhedron and finally bonds to the central Pb ion, producing structural

41

Figure 2.8. Selected AX2 compounds plotted as a function of selected lattice parameter

11,169–218 ratios. Diamonds shows data from binary compounds at room pressure. PbF2 data

are contoured by color to indicate pressure. The lattice parameters ratios defining the

Co2Si-type field have been expanded (from dashed to solid boundary) to encompass

lattice parameter ranges observed in PbF2. Transition region denotes the pressure range

from 10-22 GPa where anomalous changes in lattice parameter and volume

compressibility are observed (Figures 2.4, 2.5).

rearrangements and anomalous compressibility. Empirical support for this is provided by consideration of lattice parameter ratios as discussed below. 42

It was shown by Jeitschko171 and Léger et al.148 that the three structures under consideration

(cotunnite, Co2Si, Ni2In) can be distinguished by characteristic values of the unit cell ratios a/c

148,171 and (a+c)/b. For the Ni2In-type structure, this requires transformation from a hexagonal to an orthorhombic cell as discussed later. The empirically observed ratios range from a/c = 0.80-

0.90 and (a+c)/b = 3.3-4.0 for cotunnite-type and 0.66-0.74 and 3.1-3.3 in the case of the Co2Si- type phase.144,148,171 We have surveyed the more recent literature and have compiled additional data for a wide range of compositions that adopt these structures (Figure 2.8). These compositions include phosphides, hydrides, sulfides, oxides, and intermetallics11,144,149,169–218 and

149 are mostly at ambient pressure but data for high-pressure SrF2 is also included. The bounds established by Jeitschko 171 are largely confirmed although there are several exceptions

197,198 201 202 183 213 149 [Yb2(Pb,Ga), Pu2Pt, Ca2Hg, Rh2Ta, Zr2Al, and SrF2 ] that fall outside the established boundaries to varying degrees. In particular, there is evidence that the (a+c)/b ratio defining the Co2Si-type phase may extend to larger values.

Our measured PbF2 lattice parameters as a function of pressure are in general agreement with previous data144 as well as theoretical calculations (Appendix A). Upon compression, the lattice parameter ratios of PbF2 begin to evolve from the cotunnite field toward the Co2Si field

(Figure 2.8). The anomalous lattice compressibility observed from 10-22 GPa generally corresponds to the region where the lattice parameter ratios adopt values that are intermediate between the cotunnite and Co2Si regions. Above 22 GPa, the a/c ratio becomes consistent with

Co2Si values, but the (a+c)/b ratio at these pressures falls in the range of 3.50-3.55, which is

144,148,171 larger than the previously defined boundary of the Co2Si-type phase. Consequently, we propose to extend the boundary describing Co2Si-type phases including those under compression to values of (a+c)/b from 3.1 to 3.55 (Figure 2.8). Alkaline earth difluorides such as SrF2 also

43 evolve with application of pressure. Above 20 GPa, the lattice parameters of SrF2 move out of the cotunnite stability field, and by 28.4 GPa adopt lattice parameter ratios consistent with the

149 Co2Si-type phase. Unlike PbF2, SrF2 appears to evolve toward the Ni2In-type field with

149 increasing pressure, eventually transforming the Ni2In-type phase.

2.4.3 Room-temperature equation of state

The PbF2 cotunnite and Co2Si-type compression data were fit using a Birch-Murnaghan equation of state (Figure 2.6). Results are reported in Table 2.3. The ambient pressure volume,

V0, for the cotunnite-type phase was taken from experimental X-ray data. Fitting the data between 0 and 10 GPa yields a value for the isothermal bulk modulus of K0 = 72(3) GPa when the pressure derivative of the bulk modulus, K0’, was fixed at 4. Our bulk modulus for cotunnite

144,166 structured PbF2 is similar to those reported in earlier studies of PbF2 but is greater than the

219 value of K0 = 62(1) GPa obtained for the fluorite-type phase of PbF2. It is also similar to

149,172 values reported for the alkaline earth fluorides CaF2, SrF2, and BaF2. A fit to our

Table 2.3. Equation of state parameters for PbF2.

3 Reference V0 (Å ) K0 (GPa) K0’ Cotunnite-type This study 192.48(1)* 72(3) 4* Haines et al. (1998) 192.6(3) 66(7) 7(3) Dutta et al. (Appendix A) 180.82 68.0

Co2Si-type This study 182(2) 81(4) 4* Dutta et al. (Appendix A) 171* 95.2(8) 4* Asterisk (*) indicates fixed values.

44

3 compression data for the Co2Si-type phase between 22-74 GPa yields V0 = 182(2) Å and K0 =

81(4) GPa when K0’= 4 (Table 2.3). The K0 value calculated here is somewhat lower than obtained from extrapolation of high-pressure theoretical calculations. The experimental results indicate that there are only modest changes in bulk compressibility between the fluorite, cotunnite, and Co2Si-type phases of PbF2.

2.4.4 High pressure-temperature behavior

PbF2 samples were laser heated to 1000-2400 K at pressures from 16-64.5 GPa. Up to 25

GPa, there was no change in the PbF2 diffraction pattern upon heating to 1400 K, but a new peak was observed that we attribute to the reaction of impurity Pb with the platinum pressure standard,

220 forming a face-centered cubic Pt3Pb compound. At 25.9 GPa, new, weak peaks were observed within 40 seconds after the beginning of heating. These peaks are consistent with the Ni2In-type phase of PbF2. As far as we are aware, this is the first study to report synthesis of the Ni2In-type phase in PbF2. The Ni2In-type peaks grew in intensity with additional heating over ~6-9 minutes

(Figure 2.10). Peaks corresponding to the Co2Si-type phase remained present together with

Ni2In-type peaks throughout the heating cycle. A diffraction pattern recorded immediately upon temperature quench showed only peaks corresponding to the Co2Si-type phase while those of the

Ni2In-type phase had disappeared (Figure 2.9). These observations contrast with the work of

161 Costales et al., who find that the Ni2In-type phase should be stable at 21.64 GPa and 300 K.

At 30.1 GPa, growth of the Ni2In phase together with continued persistence of peaks of the low-temperature phase was also observed starting from 1250 K, the lowest temperature measured. The intensity of the Ni2In-type peaks grew compared to those of the Co2Si-type phase as a function of increasing temperature up to 2430 K, at which point the Ni2In-type peaks were

45

Figure 2.9. High pressure-temperature X-ray diffraction patterns for PbF2 from heating

experiment at 25.9 GPa. Tick marks indicate expected peak locations for Co2Si-type

phase (red, a = 5.329(5) Å, b = 3.788(5) Å, c = 7.363(7) Å), Ni2In-type phase (green, a =

4.042 Å, c = 5.285 Å), platinum (black), and neon (blue). Asterisks (*) indicate new

Ni2In-type peaks emerging upon heating and daggers (†) mark the location of Pt3Pb alloy

peaks.

much more intense than those of the Co2Si-type. However, even after hearing for over 20 minutes, the Co2Si-type peaks did not completely disappear. Upon temperature quench, the

46

Figure 2.10. X-ray diffraction pattern from heating experiment at 34.0 GPa. Ni2In

becomes the dominant phase at high temperature, although Co2Si-type peaks do not

completely disappear. Upon quench (upper trace), the two weak Ni2In peaks are retained

with low intensity at 2θ = 5.6° and 6.6°. The high-temperature lattice parameters of the

Ni2In-type phase are: a = 4.001(1) Å, c = 5.182(2) Å.

Ni2In-type peaks once again immediately disappeared and only the Co2Si-type peaks were observed. At 34 GPa, the Ni2In-type peaks were again much stronger than the Co2Si-type peaks during heating but upon temperature quench the Co2Si phase returned and only weak peaks of

47

Figure 2.11. X-ray diffraction pattern from heating experiment at 43.0 GPa. Nearly

complete transformation to Ni2In is observed at high temperature but a mixture of both

phases is observed upon quench. Ni2In is fit with a = 3.942(1) Å and c = 5.044(2) Å.

the Ni2In-type phase were observed (Figure 2.10). At pressures above 43 GPa, nearly single- phase Ni2In-type was synthesized at 1400 K, partially reverting to Co2Si-type on temperature quench (Figure 2.11).

The laser-heating experiments show that with increasing pressure from 25.9-64.5 GPa, greater amounts of the Ni2In-type phase are observed at temperatures above 1200 K. The volume

48 differences between the Co2Si- and Ni2In type phases appear to be small (~1%), with Ni2In volumes being consistently larger at room temperature and consistently smaller at high temperature compared to those of the coexisting Co2Si-type phase. Upon temperature quenching, mixtures of Co2Si-type and Ni2In-type are observed, with increasing amount of the Ni2In-type being retained upon quench at higher pressures. These observations suggest a shallow, negative

Clapeyron slope for the Co2Si-Ni2In-type phase boundary. Similar results were reported previously for SrF2 in which the Ni2In phase was observed at both high temperature and upon quench at 36 GPa, but the Ni2In phase existed only at high temperature and reverted to the

149 cotunnite phase upon quench at 28 GPa. Additionally, the extensive coexistence region for the

Co2Si- and Ni2In-types at both high temperature and upon quench suggests the two phases are nearly equally stable and that energy differences between them are small.

We do not observe the transformation in PbF2 when starting from the cotunnite-type structure, as the Ni2In-type phase did not form at high temperatures below 25.9 GPa, a pressure that lies above the cotunnite-Co2Si-type phase transition region in PbF2. It should also be noted that in experiments conducted under higher degrees of differential stress, evidence for the Ni2In- type phase could be observed in PbF2 at ambient temperature (Figure 2.S.2). This suggests that the transition is sensitive to differential stress, which is consistent with the close structural similarity of the phases (Figure 2.2) and provides further evidence for a small energy difference between them.

To enable comparison to the lattice cell parameter ratios of cotunnite- and Co2Si-type PbF2, the Ni2In-type structure can be represented by an orthorhombic unit cell according to Léger et

148 al.: a = ch, b = ah, and c = √3ah, where h refers to the hexagonal lattice parameters, and the

49 orthorhombic lattice parameters are given without subscripts. In the coordinates of Figure 2.9, the Ni2In-type phase then describes a straight line given by:

a 1 a⁡+⁡c ⁡=⁡ ⁡-⁡1. c √3 b

Based on our literature survey, Ni2In-type compounds adopt orthorhombic a/c values of 0.69 to

0.775. At high temperature, the unit cell of the PbF2 hexagonal phase falls as expected on this line (Figure 2.9). The Ni2In-type alkaline earth fluorides, CaF2, SrF2 and BaF2 exhibit slightly

149 smaller orthorhombic a/c values than Ni2In-type PbF2. It is notable that the PbF2 lattice parameter ratios evolve with pressure away from the Ni2In stability region consistent with the stability of the Co2Si-type PbF2 to higher pressure than in the alkaline earth fluorides.

2.5 Conclusion

The behavior of PbF2 under compression was investigated in the diamond anvil cell to by X- ray diffraction and Raman spectroscopy to pressures of 75 GPa at room temperature and by X- ray diffraction to 64.5 GPa and temperatures up to 2430 K by laser heating. At room temperature, we observe highly anisotropic compression with maximum compressibility along the a axis, together with minimal or even negative compressibility along the b and c axes at 10-

22 GPa. This behavior is consistent with an isosymmetric orthorhombic Pnma-Pnma phase transition from the 9-coordinated, cotunnite-type phase to a 10-coordinated, Co2Si-type phase occurring over a broad transition region without any detectable volume discontinuity. Our X-ray diffraction results are supported by observations of changes in the Raman spectrum over this interval. Above 22 GPa, the Co2Si-type phase exhibits smooth compression behavior to our maximum pressure of 75 GPa. The equations of state of the cotunnite-type and Co2Si-type phases of PbF2 have been constrained. 50

The high-pressure behavior of PbF2 is also examined in the context of lattice parameter systematics for AX2 compounds. The transition region from 10-22 GPa corresponds to lattice parameter ratios intermediate between those characteristic of the cotunnite-type phase and the

Co2Si-type phase. We show that existing systematics for the Co2Si phase need to be revised to account for the high-pressure behavior of PbF2.

Upon laser heating above 25.9 GPa, the Co2Si-type phase begins to transform to the hexagonal, 11-coordinated, Ni2In-type. This phase disappears completely upon temperature quench, suggestive of a negative Clapeyron slope for the phase transition. Upon heating at higher pressures, the fraction of the Ni2In-type phase continues to grow until almost complete transformation is observed at 43 GPa. Upon temperature quench, the sample either completely or partially converted back to Co2Si-type throughout the pressure range of the experiment (up to

65.4 GPa). PbF2 shows distinct differences from the behavior of the alkaline earth fluorides and provides an example of a novel pathway for transformations among highly coordinated AX2 compounds.

2.6 Acknowledgements

We thank Greg Finkelstein and Earl O’Bannon for experimental assistance and helpful discussion. This work was supported by the NSF. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by

Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-

06CH11357. We acknowledge the support of GeoSoilEnviroCARS (Sector 13), which is supported by the National Science Foundation - Earth Sciences (EAR-1128799), and the

Department of Energy, Geosciences (DE-FG02-94ER14466).

51

2.S Supplementary Material

2.S.1 Starting material

A slightly compacted pellet composed of a mixture of PbF2 and Pt was loaded inside a Re gasket. The starting material was analyzed by X-ray diffraction at GSECARS using a wavelength of 0.3344 Å, and by Raman spectroscopy. The X-ray diffraction pattern and Raman spectrum

(Figure 2.S.1) are consistent with the orthorhombic cotunnite-type structure and there is no evidence for the fluorite-type phase.

Figure 2.S.1. (a). Cotunnite-type PbF2 diffraction pattern measured at ambient conditions.

All peaks can be assigned to PbF2 (black) or Pt (yellow). The PbF2 unit cell parameters

are: a = 6.4472(3) Å, b = 3.9019(2) Å, c = 7.6514(3) Å and V = 192.48(1) Å3.

52

Table 2.S.1. Ionic radii and phase transition pressures for several difluorides. Ionic radii

are from Shannon.76 CN refers to the coordination number of the cation. Phase transition

pressures are from Dorfman et al.149

r (Å) r (Å) r (Å) r (Å) P (GPa) P (GPa) Cation tr tr

(CN 8) (CN 9) (CN 10) (CN 11) Fluorite to cotunnite Cotunnite to Ni2In Ca2+ 1.12 1.18 1.23 - 9 79 Sr2+ 1.26 1.31 1.36 - 5 36 Ba2+ 1.42 1.47 1.52 1.57 3 12 Pb2+ 1.29 1.35 1.4 1.45 0.4

2.S.2 Effects of sample pre-compression on stress state

In some experiments, we observed significant broadening of X-ray diffraction peaks, in some cases accompanied by other evidence of non-hydrostatic behavior. In order to understand the origin of these differences, a few different sample loading types were examined (Table 2.S.2). In experiment 0812-09x1, two different PbF2 samples were loaded in the same chamber. One was a

10-μm thick compressed pellet, while the other was loose, uncompressed powder (Figure 2.S.2).

X-ray diffraction peaks from the compressed pellet were broader than those of the loose powder over the entire pressure range of the experiment. Peak broadening can have several causes, but the most likely one is that the sample is experiencing local non-hydrostatic stress due to grain-to- grain contacts. The resulting range of stresses experienced by grains leads to peak broadening. In the case of the compressed pellet, the pressure transmitting medium is less effective in reducing these intergrain stresses. Notably, evidence for formation of the Ni2In-type phase was observed above 38 GPa upon non-hydrostatic compression at room temperature (Figure 2.S.2b). The difference between the Co2Si and Ni2In structures consists of a 7° distortion of the triangular 53

Table 2.S.2. Summary of PbF2 X-ray diffraction experiments.

Experiment Culet Size (μm) Gasket P Medium P Std P Range (GPa) Temp (K) 1211-10x1 200 Re NaCl Pt 0-68 300-2200 1211-10x2 200 Re Ne Pt + ruby 0-64 300-2300 1211-16x1 500 Re NaCl Pt 0-23 300-2300 0212-12x1 200 Re Ne Pt + ruby 0-78 300-1000 0212-17x1 300 Re Ne Pt + ruby 0-60 300 0812-03x1 200 Re Ne Pt + ruby 0-80 300 0812-09x1 200 Re Ne Pt + ruby 0-80 300 0812-12x1 200 Re Ne Pt + ruby 0-70 300-2430 0615-04x1 200 Re Ne Pt + ruby 25 300-1435 0615-04x2 200 Re - - 25-43 300-1470 0615-12x1 200 Re Ne Pt + ruby 0-10 300 0615-12x2 200 Re Ne Pt + ruby 0-30 300

prisms within the structure (Figure 2.2). Thus, differential stress upon the Co2Si may result in a metastable transition to Ni2In as a result of shearing within the structure.

At low pressures (~3 GPa), there is further evidence of anomalous behavior is observed in the two-dimensional diffraction patterns of samples loaded in NaCl, providing further evidence for non-hydrostaticity (Figure 2.S.3a, b). In order to understand this, a brief explanation of the experimental diffraction conditions is necessary. The experimental geometry is such that the X- ray beam is parallel to the direction of compression along the diamond anvil cell. If the unit cells experience no differential stress, then the diffracted signal should take the form of concentric circles. Upon transformation to φ-2θ space, where φ is the azimuthal angle, the diffracted signal should take the form of straight lines (as in Figure 2.S.3c). If the sample is experiencing differential stress, then differential strain leads to differences in the unit cell parameters that

54

Figure 2.S.2. Comparison of X-ray diffraction patterns for packed versus loose starting material at 4 GPa (top) and 80 GPa (bottom). Dagger indicates the location of impurity peaks due to Pb metal. Asterisks indicate the location of the Ni2In-type phase peaks.

55

Figure 2.S.3. Radially integrated PbF2 diffraction patterns. a. 3 GPa, loaded in NaCl (B1

phase). b. 27 GPa, loaded in NaCl (B2 phase is dominant). c. 3 GPa, loaded in Ne.

depend on the nature of the stress distribution. In effect, the diffraction volume sampled is experiencing an uneven load, where crystallites will contract at a greater rate based on their orientation relative to the applied stress direction. In terms of observables, that would mean that the diffraction rings take the shape of ellipses, and upon radial integration of the 2D data curved lines would be observed instead of straight. This can be observed primarily in samples that are measured in a radial geometry, where the direction of the diffraction beam is perpendicular to the direction of loading. However, the geometry in the experiments done here is such that the loading direction is coaxial with the incoming X-ray beam, suggesting that the simple case of a uniaxially applied load does not entirely describe what is happening in PbF2.

When the sample is loaded using NaCl as the pressure transmitting medium, the PbF2 diffraction patterns display nonhydrostatic behavior at both 3 and 27 GPa (Figure 2.S.3a, b). At

27 GPa, when NaCl transforms to the B2 phase, the shape of the PbF2 diffraction lines becomes straighter than that observed at 3 GPa, likely due to release of shear stress as a result of the volume discontinuity at the B1-B2 transition in NaCl. The same anomalous behavior is not

56 observed when using a Ne pressure transmitting medium, which is liquid and therefore purely hydrostatic at 3 GPa (Figure 2.S.3c). It is unusual for this amount of deviation from hydrostaticity to be observed at such low pressures in the axial geometry. This suggests that great care must be taken when selecting loading conditions for this material.

57

Figure 2.S.4. Volume and lattice parameter evolution as a function of pressure in PbF2 comparing compression and decompression experiments. Dashed lines mark the transition region.

58

Figure 2.S.5. PbF2 lattice parameter ratios at room and high temperature from this study.

59

Table 2.S.3. Percentages changes in lattice parameters and volume over selected pressure ranges.

Pressure Range (GPa) Parameter % Change a -5.2 b -3.3 0-10 c -2.5 V -10.7

a -9.8

10-22 b -1.0 c -0.3 V -11.0

a -10.0

22-75 b -7.9 c -4.2 V -20.6

60

CHAPTER 3

High-Pressure Phase Transition in Y3Fe5O12

Portions of this work were presented as:

Stan C. V., Wang J., Zouboulis I. S., Prakapenka V. B., Duffy T. S. (2014), High-Pressure Phase Transition In

Yttrium Iron Garnet, Abstract MR33A-4345 presented at 2014 Fall Meeting, AGU, San Francisco, Calif., 15-19

Dec.

Stan, C. V.; Wang, J.; Zouboulis, I. S.; Prakapenka, V.; Duffy, T. S. High-Pressure Phase Transition in Y3Fe5O12. J.

Phys.: Condens. Matter 2015, 27 (40), 405401.

3.1 Abstract

Yttrium iron garnet (YIG, Y3Fe5O12) was examined up to 74 GPa and 1800 K using synchrotron X-ray diffraction in a diamond anvil cell. At room temperature, YIG remained in the garnet phase until abrupt amorphization occurred at 51 GPa, consistent with earlier studies. Upon laser heating up to 1800 K, the material transformed to a single-phase orthorhombic GdFeO3- type perovskite of composition (Y0.75Fe0.25)FeO3. No evidence of decomposition of the sample was observed. Both the room-temperature amorphization and high-temperature transformation to the perovskite structure are consistent with the behavior of other rare earth oxide garnets. The perovskite sample was compressed between 28-74 GPa with annealing to 1450-1650 K every 3-5

GPa. Between 46 and 50 GPa, a 6.8% volume discontinuity was observed without any accompanying change in the number or intensity of diffraction peaks. This is indicative of a high-spin to low-spin electronic transition in Fe3+, likely in the octahedrally coordinated B-site of

61 the perovskite. The volume change of the inferred spin transition is consistent with those observed in other rare earth ferric iron perovskites at high pressures.

3.2 Introduction

Rare earth garnets have wide application as ferrite, laser, and luminescent materials.221,222

The garnet structure of these materials has the general formula X3Y2Z3O12 where X, Y, and Z are dodecahedral, octahedral, and tetrahedral sites, respectively (Figure 3.1a). The dodecahedral site is occupied by a rare earth cation while the octahedral and tetrahedral sites are occupied by trivalent Al, Ga, Sc, or Fe. The high-pressure behavior of compounds in this family has attracted considerable attention. At ambient temperature, pressure-induced amorphization has been observed in various rare earth garnets above 50 GPa including yttrium iron garnet (YIG),

223 224 Y3Fe5O12 , europium gallium garnet (EGG), Eu3Ga5O12, gadolinium gallium garnet (GGG),

27,225 26,225 Gd3Ga5O12, and gadolinium scandium gallium garnet (GSGG), Gd3Sc2Ga3O12. At high pressure-temperature (P-T) conditions, EGG, GGG, and GSGG all transform to the perovskite

26,27,224 structure. On the other hand, yttrium aluminum garnet (YAG), Y3Al5O12, is reported to retain the garnet structure to at least 120 GPa and 700 K,1,226 although a mechanical instability is predicted theoretically at 108 GPa.227 The broad stability region for this material has contributed to its utility as an optical pressure sensor in diamond anvil cell experiments.1,228–231

High-pressure studies of YIG at ambient temperature have identified several interesting phenomena including pressure-induced amorphization, magnetic collapse, a spin crossover transition, and metallization.223,232,233 X-ray diffraction measurements revealed the occurrence of amorphization above 50 GPa, which was irreversible upon decompression back to ambient pressure.223,233 At 48 GPa, Mössbauer spectroscopy indicated that YIG also undergoes a

62

Figure 3.1. a. Yttrium iron garnet and b. yttrium iron perovskite crystal structures

63 magnetic collapse from a ferrimagnetic to a non-magnetic state at nearly the same pressure as the

X-ray amorphization, with the simultaneous disappearance of magnetic moments in both the octahedral and tetrahedral sublattices.232 The magnetic transition was irreversible down to 1 bar.

Abrupt changes in the quadrupole splitting and isomer shift were also observed indicating the likely presence of a spin transition accompanying the amorphization.232 Finally, it was shown that the optical absorption edge vanishes in this pressure range, indicating transition to a metallic state.233 Thus, YIG garnet simultaneously experiences structural amorphization, magnetic collapse, metallization, and a probable spin transition at 50 GPa.

High pressure-temperature studies of YIG were conducted in the 1960s but were restricted to very low pressures and yield contradictory results. One study reported the decomposition of YIG to YFeO3 and Fe2O3 based on examination of samples recovered from compression to 0.4 GPa

234 and 1123 K. Here Fe2O3 was in the hematite structure and YFeO3 was an orthorhombic perovskite (Figure 3.1b). However, another study reported the synthesis of a single-phase ferrimagnetic perovskite (Y3,Fe)Fe4O12 or (Y0.75Fe0.25)FeO3 after compressing YIG to 2.52 GPa and 1473 K.235 A related Mössbauer study demonstrated the existence of two distinct iron sites in the high-pressure YIG phase, supporting the observation of a single-phase perovskite.236 The differing results obtained in these low-pressure studies have been attributed to different pressure transmitting media and to the fact that reduction of a small fraction of the Fe3+ to Fe2+ may favor formation of the single perovskite phase over the decomposition reaction.237

In the present investigation we carried out X-ray diffraction experiments on Y3Fe5O12 over a wide pressure-temperature range (28-75 GPa, up to 1800 K) using the laser-heated diamond anvil cell (DAC) to determine the phase stability, equation of state, and structural evolution of

Y3Fe5O12 at high pressures.

64

3.3 Methods

A polycrystalline Y3Fe5O12 sample (Alfa Aesar, 99.9%) was characterized at ambient conditions by X-ray diffraction and Raman spectroscopy. The unit cell parameter was found to be a = 12.37391(3) Å, consistent with literature values.235 High-pressure experiments were carried out with a diamond anvil cell using 200-μm diameter culets. The sample was ground to a few micron grain size and mixed with 10 wt.% gold powder (Goodfellow Corp., 99.95%), which served as a laser absorber and pressure calibrant. Compressed pellets (5-10 μm thick) of this mixture were embedded between layers of NaCl, which acted as a thermal insulator and quasi- hydrostatic pressure medium. Samples were placed within 100-μm diameter holes drilled in Re gaskets that had been pre-indented to 30 μm in thickness.

Angle-dispersive x-ray diffraction was performed at beamline 13-ID-D of the Advanced

Photon Source (APS), Argonne National Laboratory. .X-rays with a wavelength of 0.3344 Å were directed through the diamond anvils and the diffracted signal was collected with a CCD detector (MarCCD). The detector position and orientation were calibrated using a CeO2 standard.

High temperature experiments were carried out by double-sided laser heating using two diode- pumped single-mode ytterbium fiber lasers (λ = 1064 μm), each with output power up to 100 W.

Beam-shaping optics were used to produce a flat-topped laser profile with a spot size of ~24 μm.

Sample temperatures were measured from both sides by spectroradiometry. Details of the X-ray diffraction and laser heating systems at 13-ID-D are reported elsewhere.113 X-ray diffraction patterns were typically collected for 10-15 s. The resulting two-dimensional CCD images were

162 integrated to produce 1-D diffraction patterns using the program FIT2D. Peak positions were determined by fitting background-subtracted Voigt line shapes to the data. Lattice parameters

65

Figure 3.2. a. Rietveld-fitted X-ray diffraction pattern of the Y3Fe5O12 garnet starting material. b.

Raman spectrum of the garnet starting material. Assignments have been made based on

Shimada.238

66

164 were refined using the program UNITCELL using at least six diffraction lines from each phase observed. The pressure was calculated using the (111) diffraction peak of gold and its thermal equation of state from.123

Two separate experiments were performed. In the first, the sample was compressed at room temperature to 51 GPa. The sample was then laser heated at 1550-1700 K for 5-10 minutes. The sample was further compressed to 74 GPa in ~3 GPa steps and laser heated to 1400-1800 K for

5-10 minutes at each step. In the second experiment, the sample was compressed to ~30 GPa, and laser heated for 5 minutes to 1550 K. The synthesized sample was then compressed in 2-4

GPa steps up to 67 GPa with heating for 5 minutes at 1450-1550 K every 5 GPa.

3.4 Results/Discussion

3.4.1 Pressure-induced amorphization

At room temperature and pressure, Raman and X-ray diffraction of Y3Fe5O12 garnet show excellent agreement with literature (Figure 3.2). The garnet can be fit with a = 12.37391(3) Å, or

V = 1894.615(12) Å3. The O position has been refined to x = 0.0283(2), y = 0.0566(3) and z =

0.6503(3). The unit cell size is slightly smaller than that reported in previous measurements, which have a = 12.38.235,239 However, technological advances in the intervening years may account for the increase in precision reported here. The garnet Raman spectrum also shows good agreement with prior studies.238,240 Room-temperature diffraction patterns recorded up to 50.6

GPa consisted of diffraction peaks that could all be assigned either to Y3Fe5O12 garnet, the NaCl insulator (B1 or B2 phase) or the gold pressure standard (Figure 3.3). The diffraction peaks remained generally sharp but exhibited modest broadening with increasing pressure, likely as a result of increasing differential stress. Upon further increase of pressure from 50.6 to 51.1 GPa,

67

Figure 3.3. Room-temperature X-ray diffraction patterns of yttrium iron garnet (YIG)

between 35.1 and 51.1 GPa. Calculated peak locations for YIG, Au, and NaCl (in B1 or

B2 phase) are shown as ticks below each diffraction pattern. Selected (hkl) values for

YIG peaks are indicated.

the Y3Fe5O12 garnet diffraction peaks completely disappeared (Figure 3.3). This indicates that the sample undergoes an abrupt structural transition to a disordered or amorphous state over a narrow pressure range, consistent with previous work.223

Pressure-induced amorphization (PIA) has also been reported in other rare-earth garnets, including EGG, GGG and GSGG, at 85 GPa, 84(4) GPa, and 62(2) GPa, respectively.26,27,224,225

EGG, GGG and GSGG all transform to a perovskite structure at high pressure-temperature 68 conditions.26,27,224 PIA is commonly observed in framework structures, such as the garnet structure, which can be envisioned as a framework of alternating corner-sharing tetrahedra and octahedra. In EGG, GGG and GSGG, the amorphization has been proposed to be related to instabilities in this corner-sharing network with the increase in pressure.224 The amorphization pressure has been correlated with the crystal field strength (CFS) of the garnets,225 which has been found empirically to increase as a function of decreasing unit cell size.241 It has also been proposed that amorphization in these materials is driven by low- or room-temperature kinetic hindrance of a crystalline phase transition.224 For instance, in YIG, the local atomic structure in the amorphous state was found to be dominated by iron-oxygen FeO6 complexes with disordered orientation of local axis.223 Furthermore, all Fe3+ ions in the amorphous YIG structure were found to be octahedrally coordinated,223 which means that the Fe3+ from the garnet tetrahedral site must undergo a coordination change. If the kinetically hindered transition involves coordination changes, then the d orbitals and their CFS would play a key role in controlling the amount of energy required to undergo the transition, and would act as mediators of the amorphization pressure. In contrast to YIG, the amorphization occurs over a broader pressure range for EGG, GGG and GSGG, with pronounced peak broadening beginning at 75 GPa for

EGG, 74(3) GPa for GGG and at 55.5(5) GPa for GSGG. That is, the amorphization occurs over a range of 7-10 GPa in these materials, in contrast to <1 GPa for YIG. The amorphization is irreversible in GGG and GSGG, as the amorphous phase is retained upon decompression to 1 bar,225 while the decompression behavior of amorphous EGG has not been reported.

Pressure-induced amorphization has received considerable attention and has been reported in a wide range of materials.242–244 This phenomenon occurs when pressure induces instabilities in the crystal lattice but there is insufficient energy to permit the formation of a more stable

69 crystalline phase, resulting in the loss of long-range order.244 Two different approaches to understanding this type of amorphization have been proposed, one thermodynamic and one mechanical. In the thermodynamic case, the amorphization phenomenon is similar to melting, but occurs below the crystalline-glass phase transition for the material. In the mechanical case, the structure becomes amorphous when its mechanical stability is violated. The mechanical stability can be tested by examining the Born stability criteria.244 In the case of a cubic crystal, such as garnet, these can be described by relationships among the elastic constants:

C11+2C12>0; C44>0; C11-C12>0.

The high-pressure elastic constants of YIG can be estimated using the ambient pressure

245 values (C11 = 269 GPa, C12 = 107.7 GPa, C44 = 76.4 GPa ), and the measured pressure

246 derivatives of the elastic constants (dC11/dP = 6.22, dC12/dP = 4.01, dC44/dP = 0.41 ) through the following relationship:247

1 3 0 dCij V Cij=Cij+P ( ) ( ) . dP V0

At 50 GPa, all the Born criteria remain satisfied and no elastic softening is observed for YIG and thus there is no evidence for a mechanical instability.

3.4.2 High-pressure and -temperature experiments

At 51 GPa the amorphous sample was heated to 1650 K for 5-10 minutes. New diffraction peaks quickly appeared after three minutes of heating. The new peaks can be indexed to an orthorhombic unit cell and the peak positions match the GdFeO3-type perovskite phase (space group Pbnm) (Figure 3.4). All other observed peaks can be identified as either NaCl or Au. No further changes in the diffraction pattern were observed after 30 minutes of heating at 1375-1775

70

Figure 3.4. X-ray diffraction pattern of Y3Fe5O12 at 49.2 GPa, quenched after heating at

1650 K and 51.1 GPa. The pattern can be fit to an orthorhombic unit cell with a =

4.822(6) Å, b = 5.127(5) Å and c = 6.991(4) Å. The tick pattern beneath shows the

expected peak locations for the Pbnm perovskite structure (black). Au (yellow) and NaCl

(blue) peak locations, as well as Miller indices (h k l) for selected perovskite peaks.

K or upon quenching to room temperature. The pressure measured upon quenching is reduced slightly to 49.2 GPa, likely as a result of relaxation of differential stress. The sample was then further compressed to 74 GPa in 2-3 GPa steps with annealing for 5-10 minutes to 1550-1650 K at every other step, and the perovskite diffraction peaks continued to be observed up to the highest pressure measured.

A second YIG sample was compressed to 30.7 GPa at room temperature followed by laser heating at 1550 K for 5 minutes. The YIG peaks disappeared rapidly after heating began, and a

71

Figure 3.5. (Y0.75Fe0.25)FeO3 perovskite (YIP) diffraction upon quench. At 28.2 GPa, YIP

peaks are fit to an orthorhombic Pbnm unit cell with a = 5.017(3) Å, b = 5.333(2) Å and c

= 7.288(2) Å. Tick marks show expected peak locations for other candidate phases

248 including Rh2O3-type Fe2O3. At 64.8 GPa, the YIP unit cell is fit with a = 4.753(4) Å,

249– b = 5.059(3) Å, c = 6.900(4) Å. There is no evidence for peaks of CaIrO3-type Fe2O3.

251

new set of diffraction lines appeared. These new peaks can also be indexed as orthorhombic perovskite (Figure 3.5). This indicates that the perovskite phase is stabilized at high temperatures well below the room-temperature amorphization pressure, suggesting that the abrupt amorphization is not simply related to a solid-solid phase boundary. This sample was then 72 further compressed to 64 GPa in 2-3 GPa steps, with annealing every 5 GPa to 1450-1550 K.

Again, only peaks from orthorhombic perovskite could be observed over this range.

The orthorhombic ABO3 perovskite structure consists of 8-fold coordinated A cations located within a framework of octahedrally coordinated B cations (Figure 3.1b). In the case of YIG, two different garnet-to-perovskite synthesis pathways have been proposed, leading to differences in the A and B site occupancy. YIG could transform into a single-phase perovskite of composition

3+ 3+ 3+ (Y0.75Fe0.25)FeO3, where Y and Fe share the A site in 3 : 1 stoichiometry and Fe occupies

235 the B site. Alternatively, YIG could undergo disproportionation into Fe2O3 and YFeO3- perovskite, where Y3+ alone occupies the A site and Fe3+ occupies the B site.234 At room

3+ 3+ 76 pressure, Fe in 8-fold coordination is expected to have a radius 20% smaller than that of Y , so the inclusion of Fe3+ into the A site would lead to a small reduction in the unit cell volume in

(Y0.75Fe0.25)FeO3 versus YFeO3, but the diffraction peak positions of YFeO3 and

(Y0.75Fe0.25)FeO3 would be expected to be similar.

In the case of disproportionation, however, additional peaks associated with Fe2O3 would also be present. While there are conflicting reports on the high-pressure phase transition in Fe2O3 at room temperature,249,252–254 previous high-temperatures studies consistently find a phase transition from hematite to an Rh2O3(II)-type structure (orthorhombic, Pbcn) at pressures of

248,249,251–254 above ~25 GPa. The Rh2O3(II)-type structure contains only one crystallographically distinct cation site and has an X-ray diffraction pattern that is indistinguishable from the orthorhombic Pbnm perovskite structure. We do not observe evidence of either the hematite or

Rh2O3(II)-type structures of Fe2O3 in our diffraction pattern at 28.2 GPa (Figure 3.5). The expected unit cell parameters for the Fe2O3 high-pressure phase at 27 GPa and 300 K are approximately a = 4.918 Å, b = 5.071 Å, and c = 7.187 Å.248 These are not consistent with the

73

Figure 3.6. (Y0.75Fe0.25)FeO3 perovskite diffraction patterns at selected pressures. All

peaks can be indexed as perovskite (YIP), Au or NaCl in the B1 or B2 phase. The d

spacings corresponding to selected perovskite peaks for each of these diffraction patterns

are reported in Table 3.1.

orthorhombic unit cell parameters we measured at 28.2 GPa: a = 5.017(3) Å, b = 5.333(2) Å, c =

7.288(2). The intense (112) peak (I/I0 = 100) of the Rh2O3(II)-type structure, expected to be at 2θ

= 7.61º, is not observed, nor is there evidence for other low-angle peaks of this structure. Since all observed peaks can be accounted for by a single perovskite-type phase, Au, or NaCl, we conclude that our diffraction patterns are consistent with formation of a single-phase yttrium iron perovskite (YIP) with stoichiometry (Y0.75Fe0.25)FeO3. No additional peaks indicative of ordering 74

Table 3.1. Comparison of observed and calculated d spacings for the perovskite phase of

(Y0.75Fe0.25)FeO3 at selected pressures.

Pressure (GPa) h k l dobs (Å) dcalc (Å) Δd (Å) 28.2 GPa: a = 5.017(3) Å, b = 5.333(2) Å, c = 7.288(2) Å and V = 195.0(1) Å3 1 1 1 3.26749 3.26593 0.00156 0 2 0 2.66608 2.66542 0.00066 1 1 2 2.58085 2.58120 -0.00035 1 1 3 2.02239 2.02448 -0.00209 2 2 0 1.82557 1.82606 -0.00049 0 0 4 1.82557 1.82430 0.00127 0 2 3 1.79626 1.79671 -0.00045 2 2 1 1.77223 1.77144 0.00079

47.9 GPa: a = 4.841(6) Å, b = 5.192(4) Å, c = 7.042(6) Å and V = 177.0(2) Å3 1 1 1 3.16459 3.16224 0.00235 0 2 0 2.59713 2.59541 0.00172 1 1 2 2.49954 2.49667 0.00287 1 1 3 1.95611 1.9569 -0.00079 2 2 0 1.76840 1.7693 -0.00090 0 0 4 1.76200 1.76155 0.00045 0 2 3 1.74035 1.7415 -0.00115 2 2 1 1.71627 1.71603 0.00024

64.8 GPa: a = 4.753(4) Å, b = 5.059(3) Å, c = 6.900(4) Å and V = 165.9(1) Å3 1 1 1 3.09485 3.09531 -0.00046 0 2 0 2.52836 2.52989 -0.00153 1 1 2 2.44564 2.44376 0.00188 1 1 3 1.91510 1.91539 -0.00029 2 2 0 1.73129 1.73189 -0.00060 0 0 4 1.72335 1.72413 -0.00078 0 2 3 1.70239 1.70136 0.00103 2 2 1 1.68012 1.67973 0.00039

75 of Y3+ and Fe3+ along the A site are present, and thus we conclude the YIP is an A-site disordered perovskite. At higher pressure and temperature conditions (P > 50 GPa, T ~ 2000 K),

249–251 Fe2O3 is reported to adopt a CaIrO3-type post-perovskite structure (orthorhombic, Cmcm), although one study reports an alternative orthorhombic high-pressure phase for Fe2O3 above 44

GPa.248 Again, peaks indicating the presence of the post-perovskite phase, particularly the highest intensity (022) peak at 2θ = 7.48º, or those of the alternative orthorhombic high-pressure phase248 were not observed in our sample, indicating that the sample remains a single-phase perovskite throughout the pressure and temperature range examined in this study (Figure 3.5).

Representative diffraction patterns across our pressure range are shown in Figure 3.6, and comparisons of the measured and calculated d spacings of YIP at these pressures are recorded in

Table 3.1. The garnet to perovskite phase transition is accompanied by a ~8% decrease in volume, ΔV, which is consistent with the ~6-10% ΔV observed in the garnet-perovskite transition in EGG, GSGG, and GGG.26,27,224

3.4.3 Equation of state and spin transition

The unit cell volume of Y3Fe5O12 garnet was measured up to 51.1 GPa and the results are consistent with those reported by Gavriliuk et al.223 (Figure 3.7a). Fitting a third-order Birch-

Murnaghan equation of state to the P-V data results in an isothermal bulk modulus, KT0, of

204(5) GPa with pressure derivative KT0' = 3.1(3). V0 was fixed at to our refined value of

3 1894.61(1) Å . When KT0' is held fixed at 4, a bulk modulus of 189(1) GPa is obtained. These values differ from those reported by ultrasonic elasticity measurements which report values for

245,255–258 the adiabatic bulk modulus, KS0, of 159-163 GPa. This difference is likely the result of

76 differential stress during room-temperature compression in an NaCl medium which is known to yield an overestimate of the bulk modulus in the axial diamond anvil cell geometry.259

Table 3.2. Measured unit cell parameters of (Y0.75Fe0.25)FeO3 perovskite as a function of pressure.

Pressure (GPa) a (Å) b (Å) c (Å) V (Å3) 2.0a 5.27 5.57 7.59 222.8 28.2 5.017(3) 5.333(2) 7.288(2) 195.0(1) 29.4 5.009(7) 5.327(1) 7.277(1) 194.2(1) 30.3 4.999(6) 5.321(1) 7.274(2) 193.5(1) 32.8 4.992(6) 5.317(4) 7.255(5) 192.6(2) 33.5 4.984(2) 5.303(1) 7.242(2) 191.5(1) 35.5 4.980(4) 5.289(3) 7.222(4) 190.2(1) 37.5 4.959(4) 5.281(3) 7.203(4) 188.7(1) 40.2 4.943(3) 5.276(2) 7.192(3) 187.6(1) 42.4 4.932(5) 5.261(3) 7.175(4) 186.2(2) 44.3 4.923(6) 5.241(4) 7.150(5) 184.5(2) 46.2 4.87 (1) 5.207(7) 7.083(9) 179.7(3) 47.9 4.841(6) 5.192(4) 7.042(6) 177.0(2) 49.7 4.82(2) 5.12(2) 7.01(2) 174.4(7) 50.7 4.803(5) 5.118(3) 6.965(4) 171.2(1) 53.9 4.782(3) 5.099(2) 6.953(3) 169.5(1) 56.5 4.776(1) 5.089(1) 6.942(1) 168.7(1) 58.8 4.761(5) 5.081(3) 6.928(4) 167.6(1) 61.4 4.746(2) 5.086(1) 6.914(2) 166.9(1) 63.8 4.748(5) 5.075(4) 6.898(5) 166.2(2) 64.8 4.753(4) 5.059(3) 6.900(4) 165.9(1) 66.5 4.740(3) 5.060(2) 6.896(3) 165.4(1) 69.9 4.723(2) 5.055(1) 6.885(2) 164.4(1) 71.5 4.727(5) 5.039(4) 6.880(5) 163.9(2) 73.8 4.71(1) 5.042(8) 6.87(1) 163.2(4) aShimada et al.235

77

Figure 3.7. a. Measured volumes (per ABO3 unit) of Y3Fe5O12 garnet and perovskite. The

Eastman et al.258 equation of state is derived from ultrasonic measurements corrected to

isothermal conditions. The Shimada et al.235 data point was collected at 2 GPa, and is

used in the equation of state fit for the low-pressure data. Equations of state (solid lines)

have been extrapolated where shown as dashed lines. b. Lattice parameters of the

perovskite phase as a function of pressure. Colors are the same as those in (a.). Equation

of state fits are based on the parametric form of the third-order Birch-Murnaghan

equation of state.260

Room-temperature equation of state data were obtained for YIP from 28.2-74.0 GPa (Table

3.2). Representative diffraction patterns over this range are shown in Figures 3.4-3.6. The 78 measured unit cell volumes and individual lattice parameters of YIP are shown as a function of pressure in Figure 3.7. Remarkably, we observe a large (6.8%) reduction in volume between 46 and 50 GPa without the appearance of any new peaks or other changes in the diffraction pattern

(Figure 3.6). The same behavior can also be observed in the individual lattice parameters (Figure

3.7b). There is a 2.7%, 2.4% and 2.5% reduction in the a, b, and c lattice parameters over this pressure range, indicating a nearly isotropic volume reduction. These changes are indicative of an isosymmetric phase transition that we attribute to a high-spin to low-spin transition in yttrium iron perovskite as discussed below.

Pressure-induced spin transitions have been reported in a number of Fe-bearing phases in

261–263 3+ recent years. Andradite (Ca3Fe2Si3O12), a calcium silicate garnet with Fe in the octahedral site, has been reported to undergo a spin transition at 60-70 GPa at room temperature, resulting in a 2.5% volume collapse.262 Spin transitions have also been observed in a number of perovskites28,71,264–267 and perovskite-related structures268–271 with Fe3+ in octahedral

3+ coordination. The Fe -bearing rare earth oxide perovskites, NdFeO3, LaFeO3, PrFeO3, LuFeO3 and EuFeO3 were all found to undergo high-spin to low-spin transitions with a volume change of

28,264,265,272 2.8-6.5% at 30-60 GPa (Table 3.3). NdFeO3 was examined using optical spectroscopy and X-ray diffraction, which revealed a volume discontinuity concurrent with an electronic

3+ 272 transition that the authors interpreted as being due to a spin transition in Fe . LuFeO3,

LaFeO3, PrFeO3 and EuFeO3 were studied by high-pressure X-ray diffraction, Raman spectroscopy, and Mössbauer spectroscopy, and a volume discontinuity was observed in all four.

This was concurrent with a transition to a paramagnetic state and a significant change in the quadrupole splitting in the Mössbauer spectra, indicative of a spin transition in all of these compounds as well.28,264,265,273 The percentage volume change across the spin transition in these

79

Figure 3.8. Percentage volume discontinuity in rare earth perovskites across the spin

transition as a function of A cation radius. Ionic radii are taken from Shannon.76

Measured volume discontinuities are from this work and Rozenberg et al.,28 Gavriliuk et

al.272 and Xu et al.273 For YIP, the mean ionic radius is determined from the proportions

of Y3+ and Fe3+ in the A site, i.e. in 3 : 1 stoichiometry. The solid line shows the linear fit

to the data.

perovskites is inversely correlated to the ionic radius of the rare earth cation in the 8-fold A site

(Figure 3.8). This largely reflects the relative sizes of the A and B sites: the relative volume collapse in the octahedral B site is larger when the A site is smaller. Thus YIP, for which the A

80 site is occupied by yttrium with a relatively small radius (and Fe3+), exhibits a relatively large volume change, as expected from this trend.

Equations of state were fit to YIP both above and below the spin transition using the third- order Birch-Murnaghan equation (Figure 3.7a). KT0' was fixed to 4.0 in both cases. The high-spin

3 phase can be fit with V0 = 223(1) Å and KT0 = 161(9) GPa using data from this study only,

3 or V0 = 225.2(8) Å and KT0 = 150(4) GPa when including a low-pressure datum from Marezio

235 3 et al. For the low-spin phase, we obtain V0 = 197(1) Å and KT0 = 272(11) GPa. These results are preliminary, as the volumes have been constrained over only limited ranges of compression.

Nevertheless, the results suggest that the compressibility of the high-spin perovskite phase is similar to that of the garnet phase (based on ultrasonic elasticity data) but that there is a large decrease in compressibility across the spin transition. This is broadly consistent with static compression results for other rare earth ferric iron perovskites which may exhibit as much as a

30-40% increase in bulk modulus across the spin transition (Table 3.3).

In (Y0.75Fe0.25)FeO3, iron occupies 100% of the B sites as well as a 25% of the A sites. Thus, a spin transition could occur in either or both sites. While additional studies such as Mössbauer spectroscopy are necessary to address this question, insight can be obtained from consideration of the high-pressure spin transition in Fe-bearing silicate perovskites. In the case of

(Mg,Fe)(Al,Si,Fe)O3, the behavior of Fe cations depends on site occupancy and valence.50,261,274,275 For ferric iron, both theoretical calculations and experiments indicate that

Fe3+ in the A site remains in the high spin state until pressures above 100 GPa, whereas Fe3+ in the B site undergoes a pressure-induced spin transition at ~20-70 GPa.50,275–277 This is likely related to the longer Fe-O distances in the more highly coordinated A site. The almost spherical ligand field of the high-spin state of the A-site Fe3+ ion is caused by the full occupancy of its 3d

81

Table 3.3. Comparison of properties for rare earth orthoferrite perovskites (AFeO3).

High Spin Low Spin A3+ Cation

Reference Composition 3 3 ΔP (GPa) ΔV (%) Radius K (GPa) K (GPa) V (Å ) T0 V (Å ) T0 0 0 (Å)

This study (Y0.75Fe0.25)FeO3 223(1) 161(9) 197(1) 272(11) 46-50 6.8 0.959

Rozenberg et al. LuFeO 218.4(4) 241(5) 199.4(9) 313(10) 47-58 6.5 0.977 3 EuFeO 230.7(2) 241(2.5) 214.4(6) 339(7) 46-51 4.7 1.066 (2005) 3 PrFeO 239.3(3) 274(5) 218.6(1.2) 312(7) 33-58 4.0 1.126 3

PrFeO 35-50 2.8 1.126 Xu et al. (2001) 3 LaFeO 35-50 3.0 1.160 3 Gavriliuk et al. (2003) NdFeO 244(4) 239(4) 35-42 4.0 1.109 3 A3+ cation radius is from Shannon.76 In the case of YIP, the value has been calculated based on the radii of Y3+ (1.019 Å) and (high spin) Fe3+ (0.78 Å). V0: zero-pressure volume KT0: isothermal bulk modulus ΔP: pressure range of high spin to low spin transition ΔV: change in volume across the transition.

82 orbitals, having the 3-dimensional representation eg. It appears that this charge distribution maintains its spherical symmetry in the pressure range of this experiment. By contrast, the octahedral coordination of the Fe3+ ions occupying the B-sites leads to a 3-dimensional representation t2g of the corresponding orbitals. These orbitals direct themselves away from the 6

O2- ions linked to each B-site Fe3+ ion. This orientation favors the pressure-dependent shortening

263 the Fe-O bonds in the FeO6 octahedra. Thus, by analogy with silicate perovskite, we expect the spin transition is occurring in the B site while the fraction of Fe3+ in the A site likely remains high spin. This is supported by the magnitude of the volume change of the transition which is consistent with the trend in other rare earth perovskites with Fe3+ confined only to the B site

(Figure 3.8).

This pressure-instigated breakdown of Hund’s rule of maximum spin forces energetically lower orbitals to become occupied by a pair of electrons of opposite spin, leading to a downsizing of the whole crystal structure without any symmetry change. The pressure-actuated close-packing arrangement should also take into account the orientation and occupancy of the valence electron orbitals at the relevant crystal structure sites. Thus pressure is a powerful tool for the controlled alteration of the physical and chemical properties of materials. Be it subtle, gradual, or abrupt (phase-transitional), pressure variation affects the short- as well as the long- range behavior of atoms or ions constituting or embedded in the structure under investigation.

The high-pressure high-to-low-spin phase transition of YIP described above is a manifestation of the application of quantum mechanical principles, or, more accurately, of the pressure-mediated breach of these principles, in the macroscopic scale.

83

3.5 Conclusion

Static compression experiments were performed in the laser-heated diamond anvil cell on yttrium iron garnet (Y3Fe5O12, YIG) to a maximum pressure of 74 GPa and temperature of

~1800 K. At room temperature, we confirmed the sudden onset of amorphization in YIG near 51

GPa. Upon laser heating of either the garnet structure at 28 GPa or pressure-amorphized material at 51 GPa, Y3Fe5O12 transforms into an orthorhombic perovskite structure (GdFeO3-type). Due to the absence of diffraction peaks that would indicate the presence of iron oxide (Fe2O3), the composition of the synthesized perovskite (YIP) was inferred to be (Y0.75Fe0.25)FeO3, in agreement with other low-pressure studies.235,236 The room-temperature equation of state of yttrium iron perovskite was measured from 28 to 74 GPa. The sample was observed to undergo an isosymmetric volume change of 6.8% between 46 GPa and 50 GPa. The persistence of the

YIP peaks and the pressure range and volume change of the discontinuity are consistent with a high-spin to low-spin transition in Fe3+ in the B site of yttrium iron perovskite. Our results are consistent with the observed behavior of spin transitions in other ferric iron-containing perovskites with Fe3+ occupying the octahedral B site.263 Additional studies, such as X-ray emission spectroscopy and Mössbauer spectroscopy, are needed to further characterize the high- pressure transition in this material.

3.6 Acknowledgements

We benefitted from discussions with Gregory Finkelstein, June Wicks and Earl O’Bannon. This work was supported by the National Science Foundation (NSF) and the Carnegie-DOE Alliance

Center. The experiments were performed at GeoSoilEnviroCARS of the Advanced Photon

Source, Argonne National Laboratory. Use of the APS, an Office of Science User Facility, was

84 supported by the U.S. Department of Energy (DOE) under Contract No. DE-AC02-06CH11357.

GeoSoilEnviroCARS (Sector 13) is supported by the NSF - Earth Sciences (EAR-1128799), and the DOE, Geosciences (DE-FG02-94ER14466). I.S.Z. acknowledges support from the National

Technical University of Athens during his sabbatical stay at Princeton University

85

Chapter 4

Synthesis and Equation of State of Perovskites and Post-Perovskites in

(Mg,Fe)GeO3

Portions of this work were presented as:

Stan C. V., Krizan J., Cava R. J., Prakapenka V. B., Duffy T. S. (2014), Synthesis and compression of perovskites and post-perovskites in the (MgxFe1-x)GeO3 System, Abstract presented at 2014 COMPRES Annual Meeting,

Skamania, WA , 16-19 Jun.

Stan C. V., Krizan J., Cava R. J., Prakapenka V. B., Duffy T. S. (2015), Synthesis and equation of state of perovskites and post-perovskites in the (MgxFe1-x)GeO3 System, Abstract presented at 2015MRS Annual Meeting, Boston, MA,

29 Nov – 4 Dec

4.1 Abstract

Germanates can serve as effective analogs for the high-pressure behavior of silicates.

Pyroxenes in the (Mg,Fe)GeO3 system were synthesized with Mg# = 100, 92, 78, 61, 48 and 0 at

1273 K using ceramic sintering in vacuo. The samples were characterized by powder X-ray diffraction, inductively coupled plasma mass spectrometry (ICP-MS), electron microprobe analysis, and Raman spectroscopy. All samples exhibit orthorhombic symmetry except Mg#0, which crystallized as a mixture of ortho- and clinopyroxene. High-pressure experiments were performed on samples with Mg# ≥ 48 using the laser-heated diamond anvil cell and synchrotron

X-ray diffraction. All compositions were shown to transform to the perovskite (~30 GPa and

~1500 K) and post-perovskite (> 55 GPa, ~1600-1800 K) structures. Compositions with Mg# >

78 formed single-phase perovskite and post-perovskite. . Incorporation of Fe into the perovskite

86 structure causes a decrease in octahedral distortion relative to the cubic ideal perovskite.

Additionally, it leads to a modest decrease in bulk modulus (K0) and a modest increase in zero pressure volume (V0), and lowers the perovskite to post-perovskite phase transition pressure by

~10 GPa in the case of Mg#78 versus Mg#100. These novel high-pressure and -temperature analog phases can be of use for further investigation of the effect of Fe on the behavior of perovskites and post-perovskites, including studies of site occupancies, spin state, and partitioning behavior.

4.2 Introduction

The Earth’s lower mantle is the volumetrically the largest part of the planet, and is a key for understanding its structure and dynamics. Bridgmanite, (Mg,Fe)SiO3, is expected to be the major mineral phase of the Earth’s lower mantle. It adopts the GdFeO3-type orthorhombic perovskite structure (Pbnm, Figure 4.1), consisting of a network of corner-shared SiO6 octahedra (B site) with Mg and Fe cations occupying a distorted dodecahedral site (A site). At high pressures and temperatures, bridgmanite undergoes a phase transition to the orthorhombic CaIrO3-type (post- perovskite, Cmcm, Figure 4.1) structure which has been proposed as the main component of the core-mantle boundary region, known as the D’’ layer. The post-perovskite structure consists of layers of corner- and edge-sharing SiO6 octahedra (B site) along b intercalated with Mg/Fe cations occupying distorted dodecahedral sites (A site). The bridgmanite-post-perovskite phase transition may be responsible for the seismic discontinuity observed at the uppermost boundary of the D’’ layer, and the presence of post-perovskite in D’’ could explain important properties of this layer such as its seismic anisotropy.56,59,278

87

Figure 4.1. Orthorhombic perovskite (Pbnm) (left) and post-perovskite (Cmcm) (right)

structures of ABO3. Yellow spheres: A cation. Purple octahedra: B site in octahedral

coordination. The perovskite structure is made up of a network of corner-linked

octahedra with A cations in distorted dodecahedral sites. The post-perovskite structure

features layers of corner- and edge-sharing octahedra and distorted dodecahedral A-sites.

The incorporation of Fe2+ into perovskite and post-perovskite and its influence on the phase boundary and transition width are of interest for understanding the complex structure of the D’’ region.279 In particular, the observed topography on the D’’ discontinuity may be due to temperature and/or chemical (i.e., Fe content) variations. However, existing studies have reached conflicting conclusions about the effect of Fe2+ on the phase boundary. Some studies suggest that the perovskite to post-perovskite transition pressure increases with Fe content, indicating the

88

Fe2+ stabilizes perovskite to higher pressures.68,69 Other studies suggest that Fe-rich post- perovskite is stable at lower pressures than MgSiO3 post-perovskite, and thus Fe stabilizes post- perovskite relative to bridgmanite.55,67,70 Theoretical studies also suggest that the presence of Fe will lower the transition boundary.58,280 Additionally, Fe content may affect density, thermal conductivity, and elastic properties, all of which are important for understanding the seismic properties and geodynamic behavior of the deep mantle. The possibility of a spin transition in

Fe2+ or Fe3+ in perovskite or post-perovskite at high pressures could also lead to significant changes in physical and chemical properties.71,72

Experimental studies of the bridgmanite to post-perovskite transition in (Mg,Fe)SiO3 are hampered by the extreme pressure-temperature conditions required (~125 GPa, >2000 K). At such conditions, a number of experimental difficulties are encountered, such as difficulty in maintaining uniform P-T conditions and uncertainties in pressure calibration. An alternative approach is to study the transition at lower pressures in an analog system. Germanates have long been recognized as good analogs for silicates. Ge is a Group 14 metalloid with similar chemical behavior to Si, with a [Ar]3d104s24p2 electron configuration compared to [Ne]3s23p2 in Si. Both form ions with a 4+ charge when bonding with O. Radius ratio considerations (Ge4+ = 0.53 Å versus Si4+ =0.4 Å)76 suggest that germanates will achieve higher coordination environments at lower pressures than corresponding silicates.

For many years germanates have been employed as silicate analogues, leading to studies of structure and phase transitions,4,73,106,281,282 rheological properties,79 magnetic and structural properties of various pyroxene phases,83–85 and behavior of glasses and melts.90–92 The magnesian end-member, MgGeO3, exhibits a similar phase diagram to the corresponding silicate

77,78,95,283 MgSiO3, including formation of perovskite and post-perovskite phases. Compressional

89 and rheological properties of the germanate Mg end-member post-perovskite phase have been investigated.67,77,80 Its perovskite to post-perovskite phase transition boundary was found to be near 63 GPa and 1800 K, well below the conditions required in MgSiO3 (~125 GPa, >2000

56,77,95,284 K). The Fe end member of the germanate system, FeGeO3, was reported to decompose into oxides rather than form the perovskite phase at high pressures in early studies.4,285 However, more recent work provides some evidence that FeGeO3 perovskite and post-perovskite may exist.107–109

To date, no high-pressure studies have been reported on intermediate compositions in the

(Mg,Fe)GeO3 system. Here we synthesized a suite of (MgxFe1-x)GeO3 pyroxenes over a wide range of compositions. We report the equations of state and phase diagrams for the perovskite and post-perovskite phases for selected compositions.

4.3 Methods

4.3.1 Pyroxene synthesis

Polycrystalline samples of (Mgx,Fe1-x)GeO3 pyroxene with nominal compositions of x = 1,

0.9, 0.8, 0.6, and 0.5 were prepared by solid-state ceramic sintering in evacuated silica tubes

286 following the method of Redhammer et al. For each composition, GeO2, MgO, Fe2O3 and Fe were mixed in stoichiometric proportions and then ground under ethanol, dried, pelletized, and placed into silica tubes. The silica tubes were evacuated, welded shut, and heated to 1023 K for 3 days. After the initial heating, the samples were reground, placed into new silica tubes and heated to 1223 K, a procedure which was repeated 3-4 times over the course of 30 days.

Reaction completeness was evaluated between heating steps via powder X-ray diffraction.

Characterization of the final synthesis products was carried out using high-resolution

90 synchrotron powder X-ray diffraction at beamline 11-BM at the Advanced Photon Source (APS),

Argonne National Laboratory.

Chemical analysis of the samples was performed by ICP-MS techniques (Galbraith

Laboratories). Major cation abundances (Mg, Fe, and Ge) were determined to ~0.5% precision.

The samples were also examined by electron microprobe analysis at the Geophysical Laboratory of the Carnegie Institution for Science. Raman spectroscopy was also performed on the samples using a Horiba LabRam HR Evolution system with a 532-nm laser (Ventus, Laser Quantum) and an 1800-g/mm grating.

4.3.2 High-pressure experiments

High-pressure experiments were carried out in symmetric-type diamond anvil cells (DACs) using 100-300-μm diameter culets. Germanate samples were ground to a few micron grain size and mixed with 10 wt.% gold powder (Goodfellow Corp., 99.95% purity), which served as a laser absorber and pressure calibrant. The mixture was pressed into a thin layer and loaded into symmetric type DACs using Ne as a quasi-hydrostatic pressure-transmitting medium. Diamond anvils were mounted on tungsten carbide or cubic boron nitride backing plates. Sample chambers were formed by drilling holes in Re gaskets that were pre-indented to 20-30 μm thickness. The hole diameters were approximately half the radius of the corresponding diamond culet.

Angle-dispersive X-ray diffraction was performed at beamline 13-ID-D of the Advanced

Photon Source (APS), Argonne National Laboratory. X-rays with a wavelength of 0.3344 Å were directed through the diamond anvils and the diffracted signal was collected with a CCD detector (MarCCD). The detector position and orientation were calibrated using a LaB6 standard.

High-temperatures were achieved by double-sided laser heating using two diode-pumped single-

91 mode ytterbium fiber lasers (λ = 1064 μm) each with output power up to 100 W. Beam-shaping optics were used to produce a flat-topped laser profile with a spot size of ~24 μm. The power on each side was independently controlled to achieve uniformity of temperature across each sample.

Temperatures were measured from both sides of the sample by spectroradiometry. Details of the

X-ray diffraction and laser heating systems used here are reported elsewhere.113

X-ray diffraction patterns were typically collected for 30-120 s. In some cases, data collection was accompanied by a ± 4° rotation in ω to reduce preferred orientation effects. The resulting two-dimensional CCD images were integrated to produce 1-D diffraction patterns using

163 162 the programs DIOPTAS and FIT2D. Peak positions were determined by fitting background- subtracted Voigt line shapes to the data. Lattice parameters were refined using the program

164 UNITCELL using at least six diffraction lines from each phase observed. For selected diffraction

287,288 patterns, LeBail and Rietveld refinements were performed using GSAS and EXPGUI software.

Pressure was determined using the (111) diffraction peak of gold and the thermal equation of state from Fei et al.123

4.4 Results and Discussion

4.4.1 Synthesis of germanate pyroxene starting materials

(Mg,Fe)GeO3 pyroxene synthesis was completed after approximately 30 days. The final compositions determined by ICP-MS and microprobe, are Mg#100, Mg#92, Mg#78, Mg#61,

Mg#48 and Mg#0, where Mg# = Mg/(Mg+Fe)*100 (Table 4.1). Microprobe analysis across several spots in selected samples showed little deviation, indicating that the elemental distribution was homogeneous. The measured compositions of the Mg#78 and Mg#48 samples

92

Figure 4.2. X-ray diffraction patterns of (Mg,Fe)GeO3 orthopyroxenes at ambient conditions.

Compositions are given as Mg# = (Mg / (Mg+Fe)*100). Tick marks are based on LeBail-fit unit cells for each composition. Rutile-type GeO2 is observed in Mg#78 and Mg#48 samples.

Diffraction was collected using a wavelength of 0.413828 Å. Detectors covering an angular range are scanned over a 34º 2 range, with data points collected every 0.001º 2 and a scan speed of 0.01º/s. 93 yield an (Mg+Fe)/Ge ratio lower than the expected pyroxene value, suggesting the presence of excess GeO2.

Ambient-pressure X-ray diffraction patterns for all compositions are shown in Figure 4.2.

With the exception of Mg#0, the samples can be indexed as orthopyroxenes (space group Pbca).

Pyroxenes are single-chain silicates with the tetrahedral sites being occupied by Ge and the two metal sites being occupied by Mg or Fe. The Mg#0 sample was found to be a mixture of orthopyroxene and clinopyroxene (C2/c) (Figure 4.2). However, Redhammer et al.286 reported the formation of single-phase clinopyroxene in Mg#0 when heated at 1223 K. This discrepancy may be due to poor temperature stability over the heating period in our sample, as temperatures

286 in excess of 1323 K are reported to lead to recovery of FeGeO3 orthopyroxene. Quenching rate may also be a factor. In silicates, the orthorhombic phase is thermodynamically stable only at high temperature. If the same is true in germanates, then the combination of high temperatures and a rapid temperature quench may prevent FeGeO3 from completely converting from the orthorhombic to the monoclinic phase, leading to a mixture of both. Further tests of quench rate are needed to clarify this point.

Table 4.1. Composition of germanate pyroxenes (wt. %) from microprobe measurements.

Sample MgO FeO* SiO2 GeO2 Composition

Mg#100 27.05 0.00 0.08 72.87 Mg0.974Ge1.011O3

Mg#92 24.94 3.97 0.41 70.69 (Mg0.910Fe0.081)Ge0.994O3

Mg#78 20.48 10.08 0.16 69.28 (Mg0.771Fe0.213)Ge1.005O3 a Mg#61 16.26 18.68 0.00 65.06 (Mg0.608Fe0.392)1.067GeO3

Mg#48 11.74 23.01 0.15 65.01 (Mg0.470Fe0.515)Ge1.002O3

Mg#0 0.09 41.27 0.29 58.35 Fe1.013Ge0.983O3 a. ICP-MS analysis. *All iron analyzed as FeO

94

Figure 4.3. Synthesized samples a. before and b. after final heating. Some reaction with

the inner tube wall can be observed after heating.

Lattice parameters for all compositions were determined via LeBail refinement of the X-ray data, and are reported in Table 4.2. The lattice parameters and volumes of the endmember compositions are similar to those previously reported.286,289–291 The Mg#78 and Mg#48 samples were found to also contain rutile-type GeO2, consistent with their slightly elevated Ge/(Mg+Fe) ratio. The amount of GeO2 is expected to be minor based on relative intensities of diffraction 95

Table 4.2. Unit cell parameters and volumes of (Mg,Fe)GeO3 pyroxenes

Composition Space Group Lattice Parameters Volume (Å3) a (Å) b (Å) c (Å) β (º) This work Mg#100 Pbca 18.80672(6) 8.95884(3) 5.34236(2) 90 900.114(4) Mg#92 Pbca 18.82050(6) 8.97963(3) 5.35009(2) 90 904.171(4) Mg#78 Pbca 18.83568(3) 8.99572(1) 5.36050(1) 90 908.286(2) Mg#61 Pbca 18.86520(5) 9.03705(2) 5.37502(1) 90 916.365(3) Mg#48 Pbca 18.87622(2) 9.05077(1) 5.38607(1) 90 920.180(1) Mg#0 Pbca 18.93718(6) 9.15075(3) 5.42555(2) 90 940.190(4) C2/c 9.79167(3) 9.13552(3) 5.20024(2) 101.815(1) 455.316(2)

Mg#100290 Pbca 18.661 8.954 5.346 90 893.27 Mg#100289 Pbca 18.8099(12) 8.9484(8) 5.3451(4) 90 899.62(12) Mg#0286 C2/c 9.7993(2) 9.1491(2) 5.1991(1) 101.891(2) 456.12(8) Mg#0291 C2/c 9.58 9.16 5.21 102.33 445.34

96

Figure 4.4. a. Volume and b. lattice parameter dependence of silicate and germanate

orthopyroxenes on Mg#. Values are normalized relative to the magnesian end-member.

Silicate data are from Turnock et al.292

peaks (Figure 4.S.1) and on the Ge/(Mg+Fe) ratio. While some corrosion was observed on the inside of the silica tubes after synthesis (Figure 4.3), as also reported by Redhammer et al.,286 no evidence of cristobalite-type oxide was found in our samples, and the sample pellets did not appear to have reacted with the side walls of the tubes. This is supported by the low or absent silica content in the samples (Table 4.1).

The unit cell volumes and lattice parameters of (Mg,Fe)GeO3 orthopyroxene display a linear dependence on Mg# (Figure 4.4), a similar trend as observed in silicate orthopyroxenes.292 97

Table 4.3. MgGeO3 Raman peak positions compared with those found by Ross and Navrotsky.

282 br: broad peak. w: weak peak. sh: shoulder. st: strong peak.

Ross and Navrotsky (1988) (cm-1) This Study (cm-1) 107 109 118 st 118 130 131 138 w 141 w 145 w 147 w 157w 156 165 165 181 181 190 190 206 206 226 w 226 w 236 234 268 266 276 sh 278 293 293 327 326 344 344 362 sh 362 396 396 402 401 408 sh 408 422 424 br 445 445 460 br 477 479 498 br 498 br, w 541 543 573 st 575 st 711 712 745 w 746 784 785 798 w 799 sh, br 821 sh 830 830 864 862 880 st 881 st 903 sh 903 sh

98

Figure 4.5. a. Raman spectra of germanate pyroxenes. b. Positions of major Raman peaks

as a function of Mg#. Lines are linear fits to the data.

However, the germanates, with the exception of the c axis, exhibit a slightly smaller relative dependence on the volume and lattice parameters than the corresponding silicates.292 The similarity in c axis expansion between germanates and silicates as a function of Fe suggests that this lattice direction is more affected by distortion of the metal site due to substitution than it is by changes in tetrahedral site volume. Due to the larger ionic radius of Ge4+, the volume and lattice parameter increase relative to the magnesian end-member will be smaller in germanates due to the larger overall volume of the germanate unit cell (900.0 Å3 and 830.4 Å3 respectively for the Mg end-member). 99

Raman spectra for all compositions are shown in Figure 4.5a. Raman frequencies of the

Mg#100 sample match those reported by Navrotsky and Ross282 (Table 4.3). The mode shifts of the orthopyroxenes exhibit a linear frequency increase dependence on the Mg# (Figure 4.5b), similar to what has been observed in synthetic silicate orthopyroxenes.293 Raman peaks for our samples broaden as a function of Fe substitution, with the Mg#100 sample showing the sharpest peaks. This may be indicative of cation disorder in the Mg/Fe crystallographic sites. Broad peaks observed in Mg#0 may be due to the overlap of ortho- and clinopyroxene modes.282 The increase in broadening is accompanied by a decrease in measured intensity. Another Raman study on silicate pyroxenes indicates that some intensity discrepancies occur between natural and synthetic silicate samples, but does not report broadening on the level observed in the germanates.294 Line broadening was observed in Fe-containing synthetic silicate olivine compositions, which were observed to decompose somewhat under applied laser power.295 It is possible that one or more of these factors contributes to the broadening observed in the germanate pyroxene samples.

4.4.2 Perovskite and post-perovskite synthesis at high pressures

Samples with Mg#100-48 compositions were compressed to 30-40 GPa and subsequently heated. Prior to heating, sample peaks were broad and no longer corresponded with expected orthopyroxene positions. After less than 5 minutes of heating to 1500-1800 K, all compositions exhibited transformation to a GdFeO3-type (Pbnm) perovskite structure (Figure 4.6). All observed peaks can be indexed to perovskite, Au, Ne, or GeO2. Observed d spacings for these diffraction patterns are listed in Table 4.4 and lattice parameters and pressures are listed in Table

4.5. In the case of Mg#61 and Mg#48, peaks attributable to Au, Ne and GeO2 as well as a few

100

Figure 4.6. a. X-ray diffraction patterns for Mg#78 composition showing synthesis of perovskite by laser heating at 32 GPa. Ne, Au, and GeO2 are present both before and after heating. b. Representative X-ray diffraction patterns for perovskites in the (Mg,Fe)GeO3 system at room temperature after laser heating near 32 GPa. Expected peak locations are shown by tick marks underneath each pattern. Selected Miller indices (hkl) are indicated. Q = 2π/d is used to normalize data collected using disparate wavelengths. Pressures are listed in Table 4.5. c. d-spacings of major perovskite peaks from b. as a function of composition. Linear fits are shown. 101

Table 4.4. Observed and calculated d spacing for selected peaks (hkl) of the perovskite (perovskite) and post-perovskite (post- perovskite) phases.

hkl Mg#100 perovskite Mg#92 perovskite Mg#78 perovskite Mg#48 perovskite d (Å) d (Å) res (Å) d (Å) d (Å) res (Å) d (Å) d (Å) res (Å) d (Å) d (Å) res (Å) obs calc obs calc obs calc obs calc 111 3.06970 3.07201 -0.00232 3.07612 3.07783 -0.00171 3.08209 3.08387 -0.00177 3.09452 3.09515 -0.00063 020 2.48024 2.48475 -0.00450 2.48272 2.48624 -0.00351 2.48466 2.48801 -0.00335 2.4909 2.49064 0.00025 112 2.43423 2.43045 0.00378 2.43774 2.43846 -0.00072 2.44213 2.43983 0.00231 2.45118 2.44738 0.0038 200 2.36857 2.37392 -0.00535 2.37415 2.37700 -0.00285 2.38370 2.38864 -0.00494 2.3997 2.40487 -0.00518 130 1.56223 1.56404 -0.00181 1.56366 1.56509 -0.00143 1.56453 1.56692 -0.00238 1.56731 1.56953 -0.00223 114 1.54421 1.53830 0.00592 1.54828 1.54554 0.00274 1.54423 1.54423 0.00533 1.54998 1.54814 0.00183 222 1.53569 1.53601 -0.00032 1.54062 1.53892 0.00170 1.54230 1.54193 0.00036 1.54823 1.54758 0.00066 131 1.52358 1.52516 -0.00157 1.52557 1.52656 -0.00099 1.52694 1.52811 -0.00117 1.53222 1.5307 0.00152 310 1.50348 1.50799 -0.00451 1.50974 1.50985 -0.0011 1.51405 1.51666 -0.00261 1.52721 1.52615 0.00106

Mg#100 post-perovskite Mg#92 post-perovskite Mg#78 post-perovskite Mg#48 post-perovskite

d (Å) d (Å) res (Å) d (Å) d res (Å) d (Å) d (Å) res (Å) d (Å) d (Å) res (Å) obs calc obs calc obs calc obs calc 020 4.22704 4.22495 0.00209 4.20264 4.2026 -0.00005 4.22242 4.22080 0.00162 4.24206 4.25136 0.00929 (Å) 002 3.21271 3.21360 -0.00089 3.19496 3.1956 -0.00070 3.20572 3.20746 -0.00174 3.22052 3.21957 -0.00096 9 022 2.55684 2.55778 -0.00094 2.54309 2.54387 -0.00071 2.55501 2.55376 0.00125 2.56647 2.56663 0.00016 110 2.49221 2.49474 -0.00253 2.47847 2.47850 -0.00003 2.48647 2.48808 -0.00160 2.50010 2.49669 -0.00341 040 2.11467 2.11248 0.00219 2.10155 2.10130 0.00020 2.10976 2.11040 -0.00064 2.12691 2.12568 -0.00123 112 1.97627 1.97063 0.00564 1.95819 1.95845 -0.00030 1.96588 1.96593 -0.00005 1.97021 1.97297 0.00276 132 1.64437 1.64499 -0.00062 1.63522 1.63539 -0.00008 1.64204 1.64176 0.00028 1.64939 1.64945 0.00006 113 1.62581 1.62532 0.00049 1.61596 1.61560 0.00035 1.62251 1.62169 0.00081 1.62773 1.62760 -0.00013 004 1.60587 1.60680 -0.00093 1.59792 1.59781 0.00009 1.60324 1.60373 -0.00049 1.61019 1.60978 -0.00041 3 102

Figure 4.7. a. Synthesis of CaIrO3-type post-perovskite in Mg#78 compositions by laser heating at 85 GPa. b. X-ray diffraction patterns for (Mg,Fe)GeO3 post-perovskites at room temperature after laser heating. Expected peak locations are shown below each pattern. Au peaks have been truncated for clarity. Select Miller indices, (hkl), are indicated. Asterisks denote diffuse peaks corresponding to single perovskite grains; these peaks disappear with annealing at higher pressures.

103 peaks at high d spacing of unknown origin were also observed (Figure 4.S.2). While GeO2 was detected in the Mg#48 orthopyroxene starting material, it was not in the Mg#61 sample. The presence of GeO2 after synthesis of a perovskite phase may indicate a maximum in the solubility

296 of iron in the perovskite structure at this pressure as is also observed for silicates. GeO2 peaks are not observed in diffraction patterns of compositions with Mg# > 61.

Studies on perovskite synthesis in Fe-bearing silicates show the presence of a SiO2 phase.52,297–300 This is likely due to a disproportionation of iron that follows the mechanism Fe2+

→ Fe0 + Fe3+, which is coupled with a substitution of the form Mg2+ + Si4+ → (vacancy, Fe3+) +

Fe3+.297–299 Typically in these silicates Fe metal was not observed under diffraction, although it was detected using Mossbauer and electron energy loss spectroscopy (EELS), and Fe3+ was found in both cation sites of the perovskite although preferentially in the octahedral site. Such a mechanism could explain the presence of GeO2 peaks in germanate samples. A companion study dedicated to determining the oxidation state of Fe in Mg#78 perovskite at various pressures is

Table 4.5 Perovskite and post-perovskite unit cell parameters at select pressures.

a (Å) b (Å) c (Å) V (Å3) P (GPa) Perovskite Mg#100 4.743(9) 4.966(9) 6.89(2) 162.4(5) 32.0 Mg#92 4.753(7) 4.970(8) 6.89(1) 162.9(4) 31.2 Mg#78 4.772(6) 4.974(7) 6.91(1) 164.0(3) 31.7 Mg#48 4.810(4) 4.981(4) 6.924(6) 165.9(2) 31.7 Post -perovskite Mg#100 2.611(3) 8.450(9) 6.427(5) 141.8(2) 80.7 Mg#92 2.6938(4) 8.405(1) 6.3913(7) 139.34(5) 88.4 Mg#78 2.604(1) 8.442(3) 6.415(2) 141.0 (1) 85.3 Mg#61 2.621(6) 8.44(3) 6.421(8) 142.1(4) 78.6 Mg#48 2.621(2) 8.512(4) 6.458(4) 144.1(1) 76.3 104

currently in preparation.301

In the next set of experiments, previously unheated samples with composition between

Mg#100-48 were first compressed to 70-80 GPa and then heated to ~1500-1900 K. Prior to

heating, only weak broad peaks were observed in the starting material. Within 5 minutes of

heating, these peaks disappeared, and peaks corresponding to the post-perovskite phase appeared

(Figure 4.7). Peaks for compositions with Mg# ≥ 78 can be indexed as post-perovskite, Au, or

Ne. GeO2 in the pyrite (Pa-3) phase may also be present, but its diffraction peaks coincide with

either post-perovskite or Au throughout most of the analytical pressure range. In some instances

for the Mg#61 and Mg#48 compositions additional peaks grew in upon prolonged heating but

these were not consistently observed (Figure 4.S.3). In the case of Mg#48, synthesis of the

sample from compressed amorphous starting material at 103 GPa yielded a post-perovskite

diffraction pattern free from these additional peaks, suggesting that, in compositions with higher

Table 4.6 Lattice parameters and atomic positions from Rietveld refinement of Mg#78

perovskite and post-perovskite.

Perovskite – 40 GPa Post-perovskite – 89.1 GPa a (Å) b (Å) c (Å) V (Å3) a (Å) b (Å) c (Å) V (Å3) 4.7282(2) 4.9437(3) 6.8803(6) 160.83(1) 2.5980(8) 8.420(2) 6.399(2) 140.01(5) x y z F x y z F Mg 0.528(1) 0.5667(8) 0.25 0.78 0.0 0.251(2) 0.25 0.78 Fe 0.528(1) 0.5667(8) 0.25 0.22 0.0 0.251(2) 0.25 0.22 Ge 0.5 0.0 0.5 1 0.0 0.0 0.0 1 O1 0.126(3) 0.456(3) 0.25 1 0.0 0.937(4) 0.25 1 O2 0.185(2) 0.189(2) 0.559(2) 1 0.0 0.629(2) 0.442(4) 1 χ2 2.1 3.2

105

Fe content, post-perovskite becomes more stable at higher pressure-temperature conditions.

Rietveld refinements were performed on Mg#78 perovskite and post-perovskite samples

(Figs. 4.8, 4.9). LeBail refinement was performed first to determine lattice and profile parameters. Profile parameters were constrained to be the same across all relevant phases. Initial lattice parameters were based on peak fitting. The background was manually fit for each pattern using a Chebyshev polynomial with 12-14 parameters and then refined. Once a good fit was found in this manner, Rietveld refinement performed. First, phase fractions were allowed to refine. The atomic positions of Mg and Fe were constrained to be the same throughout, and

Table 4.7. Select bond lengths and site volumes from Rietveld refinement.

Perovskite Post-perovskite a b (Mg0.78Fe0.22)GeO3 (Mg0.95Fe0.05)SiO3 (Mg0.78Fe0.22)GeO3 MgGeO3 40 GPa ambient 89.1 GPa 89.5 GPa A site Volume (Å3) 16.893 17.619 15.948 14.098 A-O1 (Å) 1.9811(1) 1.986(10) 2.034(30) ×2 1.850 ×2 A-O1 (Å) 2.0568(1) 2.068(9) A-O2 (Å) 1.96639(9) ×2 2.093(7) ×2 2.217(25) ×2 2.176 ×2 A-O2 (Å) 2.23985(9) ×2 2.330(8) ×2 2.06(2) ×4 2.017 ×4 (x2) A-O2 (Å) 2.4327(2) ×2 2.377(7) ×2 Average (Å) 2.1645 2.207 2.093 2.015

B site Volume (Å3) 7.996 7.62 6.701 7.570 B-O1 (Å) 1.8332(1) ×2 1.804(2) ×2 1.686(10) ×2 1.757 ×2 B-O2 (Å) 1.80464(8) ×2 1.761(6) ×2 1.734(4) ×4 1.803 ×4 B-O2 (Å) 1.81656(9) ×2 1.806(6) ×2 Average (Å) 1.8181 1.790 1.718 1.787 a. Jephcoat et al. 302 b. Kubo et al. 86 106

Figure 4.8. Rietveld refinement of Mg#78 perovskite at 40 GPa. Pressure was determined

from Raman spectroscopy of diamond.303

302 initial fractional coordinate guesses were made based on (Mg0.95Fe0.05)SiO3 perovskite or

86 86 MgGeO3 post-perovskite. Thermal displacement parameters were based on Kubo et al. and were kept fixed throughout the refinement. The Mg:Fe ratio was fixed to 0.78:0.22. Fractional coordinates of all atoms were refined. Preferred orientation was significant in the case of perovskite, and 6th-order spherical harmonics in a cylindrical geometry were applied during the refinement. Final lattice parameters and atomic fractional coordinates for perovskite and post- perovskite are listed in Table 4.6. No prior reports of Rietveld refinement on any germanate perovskite composition exist at the time of this publication. Comparison to bridgmanite302

(silicate perovskite) shows that the germanate B site volume is significantly larger and shows 107

Figure 4.9. Rietveld refinement of Mg#78 post-perovskite at 89 GPa.

more regular octahedra (Table 4.7). This result is expected due to the larger ionic radius of Ge4+ relative to that of Si4+. Differences in pressures make it more difficult to draw comparisons about the nature of the A site in germanate perovskite, but we note that the germanate sample has slightly higher distortion in A relative to the silicate, with a wider spread in calculated bond lengths.

At comparable pressures, Mg#78 post-perovskite has a smaller B site and larger A site volume than Mg#100.86 At ambient conditions, the ionic radius of Fe2+ is larger than that of

2+ Mg , and so the increase in A site volume is expected. Oddly, in (Mg,Fe)SiO3 post-perovskite inclusion of 9 mol% Fe2+ into the A site leads to virtually no volume change.304,305 A decrease is observed in the B site volume as a function of Fe2+ incorporation in both silicate and germanate 108 post- perovskite,304 but the B site decreases to a greater extent per mole of Fe2+ in germanates than it does in silicates. In fact, Fe2+ does not affect unit cell volume relative to the respective magnesian endmembers for the silicate or germanate post-perovskites considered here, which is an unexpected observation.

4.4.3 Perovskite – post-perovskite phase relationships

The perovskite to post-perovskite transition boundary was investigated for compositions

Mg#78 and Mg#100. Previously unheated samples were compressed at room temperature to pressures near the expected phase boundary, then heated between 1000 and 2600 K. Consistent with earlier studies,54,55,306 we found that both perovskite and post-perovskite could be synthesized rapidly from the disordered starting material (especially at pressures well above or below the phase boundary), but once one phase formed, the transformation to the other was very sluggish unless a large over- or under-pressure was applied. The observed phases as a function of pressure and temperature are shown in Figure 4.10 for Mg#100 and Figure 4.11 for Mg#78.

In Mg#100, the phase boundary is univariant, but we observe a wide pressure range over which both perovskite and post-perovskite coexist due to the sluggish transition kinetics. The first appearance of post-perovskite in this study corresponds well with the observations of Hirose et al.95 and Ito et al.284 (Figure 4.10). In an early study of the transition, Hirose et al.95 placed the boundary where the first observance of post-perovskite is recorded. The work of Ito et al.284 was performed more rigorously, with prolonged heating over a wide array of pressure conditions in a multi-anvil press. The two studies are generally consistent but the work of Ito et al.284 places the boundary at slightly higher pressure-temperature conditions, with a shallower Clapeyron slope.

Our data are consistent with this boundary. Heating for 10-15 minute intervals at a fixed pressure

109

Figure 4.10. Mg#100 phases in pressure and temperature space. Open symbols:

perovskite present. Closed symbols: post-perovskite present. Where open and closed

symbols overlap, both phases are present. Lines indicate the perovskite-post-perovskite

phase boundary.

and temperature did not significantly alter the ratio of perovskite:post-perovskite in the Mg#100 sample, in accordance with prior observations of slow reaction kinetics.284 The presence of small amounts of perovskite in the sample even at high pressures suggests either the existence of a temperature gradient, or that the transition from starting material to perovskite is kinetically more accessible. Coexistence of perovskite and post-perovskite to high pressures was also observed by

110

Figure 4.11. Mg#78 phases in pressure and temperature space. Open symbols: perovskite

present. Closed symbols: post-perovskite present. Where open and closed symbols

overlap, both phases are present. Lines indicate the Mg#100 perovskite-post-perovskite

phase boundary for comparison.

Hirose et al.95 and Ito et al.,284 suggesting that kinetics may play a more important role than temperature gradients in this system.

In (Mg,Fe)SiO3, the perovskite to post-perovskite phase transition will be divariant with a coexistence region in pressure-temperature space. Indeed, we observe the presence of both phases over a range of a wide range 54 GPa – 79 GPa, and 1350 K – 2575 K (Figure 4.11) but this may reflect kinetics as discussed above. Notably, the appearance of post-perovskite peaks 111

Figure 4.12. Radially integrated a.-d. Mg#100 and e.-h. Mg#78 perovskite (pv) diffraction patterns collected at 1700-1800 K and different pressures. Perovskite, post-perovskite (ppv), gold

(Au), and GeO2 positions, as well as pressures, are indicated. 112 occurs at significantly lower pressure than in Mg#100 samples. This indicates that the presence of Fe lowers the perovskite to post-perovskite phase transition pressure in germanates, similar to observations in silicates.55,58,67,70,280 Our lower boundary for the perovskite-post-perovskite two phase loop is indicated by a solid black line in Figure 4.11, assuming a similar Clapeyron slope as that determined by Ito et al.284 We find that the inclusion of Fe reduces the lower boundary of the divariant loop by ~9.5 GPa relative to that of Mg#100. We were unable to reliably determine the upper boundary of the two phase loop in Mg#78. We observe a single post-perovskite in

Mg#78 at pressures above 80 GPa.

In an additional experiment, Mg#100 and Mg#78 samples were loaded adjacent to each other within the same cell and perovskite was synthesized in each composition at 30 GPa. These samples were then compressed to 74 GPa with heating to ~1700-1800 K every 5 GPa. At ~1700

K above 55.7 GPa, we observe evidence for the incipient formation of post-perovskite in Mg#78 but not in Mg#100 (Figure 4.12). With increasing pressure, the intensity of the post-perovskite peaks grows significantly in Mg#78. Post-perovskite is not observed in Mg#100 below 74.0 GPa, at which point weak diffraction spots associated with this phase begin to appear. X-ray diffraction on the temperature-quenched sample demonstrated that the region between the two laser heating spots contained only unreacted starting material, demonstrating that the two samples did not chemically interact. This supports our conclusion that the presence of Fe enhances post-perovskite stability in germanates.

113

4.4.4 Equations of state

4.4.4.1 Perovskite

The equations of state of Mg#100 and Mg#78 perovskite were measured in the following manner. Both compositions were loaded adjacently into a diamond anvil cell and compressed to

~30 GPa, similar to the experiment presented in the prior section. Perovskite was synthesized at this pressure and ~1600 K, and the samples were then compressed up to 50 GPa in 2-3 GPa steps with annealing for 3-5 minutes at ~1500 K every ~5 GPa. Above 50 GPa, the samples were compressed in 2-3 GPa steps without annealing in order to prevent conversion to the post- perovskite phase (as observed in the previous section). Unit cells volumes and lattice parameters for Mg#78 and Mg#100 are presented in Figure 4.13. Unreacted starting material was found between the two samples, indicating that no chemical interaction between the two samples occurred and the resulting perovskites have the same composition as the respective starting materials. It was not possible to measure differential stress using the method of Singh et al.125,131,132 (see Chapter 1) due to the low peak intensities of Au and peak overlap with perovskite. Instead, we analyzed the full width at half maximum (FWHM) of the Au (111) reflection as a function of pressure in both Mg#100 and Mg#78 (Fig. S4.4). In Mg#100, the peak width increases at pressures above 50 GPa, the pressure at which the sample was no longer annealed. In Mg#78, there does not appear to be as abrupt an increase in peak width. This suggests that the Mg#100 sample may be subject to greater differential stress than the Mg#78.

However, the amount is difficult to quantify.

Incorporation of iron into the perovskite unit cell leads to an increase in the a axis, a smaller increase in the c axis, and little change in the b axis (Figure 4.13). This effect is similar to observations in silicates, where the a and c axes respond to Fe substitution more strongly than

114

Figure 4.13. a. Unit cell volumes and b. lattice parameters of Mg#78 and Mg#100 perovskite measured upon compression. Lattice parameters were determined using LeBail refinement. Lines indicate Birch-Murnaghan equation of state fits with K0’ = 4. 115 the b axis, which appears to be almost insensitive to Fe content.55 The addition of Fe into the orthorhombic lattice can be examined by its effect on the distortion of the perovskite away from ideal cubic symmetry. Assuming regular octahedra, the tilt, Φ, i.e. of the octahedra about the cubic crystallographic <111> axis101,307,308 is defined as:

a2√2 Φ = cos-1 ( ). bc

In an ideal cubic perovskite, Φ = 0. In both germanate and silicate perovskites, Φ increases as a function of both pressure and Mg# (Figure 4.14a). This indicates that the structure becomes increasingly distorted with the application in pressure,101 but distortion decreases with incorporation of Fe.

Another metric used to characterize non-cubic distortion is the lattice parameter ratios a/c and b/c (Figure 4.14b). In a cubic perovskite that is described in terms of an orthorhombic lattice, both of these ratios are equal to 1/√2⁡(dashed line on figure). In both germanates and silicates, there is a dependence of a/c on the composition as it approaches the ideal value with increasing

Fe content. There appears to be little dependence of a/c on pressure in silicate perovskites although there is some evidence for an increase in distortion at higher pressures for higher iron contents. This is not observed in the germanates over the measured pressure range.

Conversely, the ratio b/c has less dependence on composition in both silicates and germanates, but a stronger dependence on pressure, with increasing pressure leading to greater distortion. It is clear from this analysis that, of the three lattice parameters, the a dimension shows the strongest dependence on Fe content in both silicate and germanate compositions, and all three parameters contribute to the pressure-induced distortion of the perovskite structure.

116

Figure 4.14. a. Octahedral tilt, Φ, and b. lattice parameter ratios as a function of pressure in (Mg,Fe)GeO3 and (Mg,Fe)SiO3

55 perovskite. Dashed line corresponds to 1/√2, the value for a cubic perovskite. (Mg,Fe)SiO3 perovskite data is from Dorfman et al.

The dashed line in b shows lattice parameter ratios for an ideal cubic perovskite. 117

Additionally, the a and c axes compress at a similar rate, while the b axis is less compressible, leading to the trend observed in Figure 4.14b.

Perovskite unit cell volumes upon compression were fit using the 3rd order Birch-Murnaghan equation of state, which takes the form:

7 5 2 3 V 3 V 3 3 V 3 P(V) = K [( 0) - ( 0) ] {1 + (K' - 4) [( 0) - 1]} , 2 0 V V 4 0 V

' where P is the pressure, K is the bulk modulus, K is the pressure derivative of the bulk modulus,

V is the unit cell volume and the subscript 0 refers to ambient pressure. K0 and V0 were fit by

' constraining K0 = 4 or the literature value of 3.7, and results of the fit are presented in Table 4.8.

The equation of state of Mg#100 was previously measured by Runge et al.,78 whose values agree with those found in this study within one standard deviation, but our data exhibits less scatter and covers a wider pressure range than the previous work (Figure 4.15). K0 exhibits a 7.3% decrease between Mg#100 and Mg#78, while the fit value of V0 increases by 1.6%. In silicate perovskites, equation of state measurements indicate that V0 only increases by 0.8%, and K0 remains statistically the same within measurement error across a broad range of Mg-Fe substitution.55

Overall, the germanate perovskites are more compressible than the silicates, with overall lower

K0 values across all compositions (e.g. 221 GPa versus 255 GPa for Mg#100 germanate versus silicate perovskite), and higher V0 (e.g. 182.6 versus 162.3 for Mg#100 germanate versus silicate perovskite).52,55 Germanate perovskites exhibit a stronger effect of Fe substitution on the equation of state than silicates for all compositions.

118

Figure 4.15. Mg#100 perovskite a. unit cell volume and b. lattice parameters measured upon compression. Lines indicate Birch-

Murnaghan equation of state fits with K0’ = 4. 119

Table 4.8. (Mg,Fe)GeO3 Equation of state parameters in perovskite and post-perovskite

from experimental studies.

V (Å3) K (GPa) K ’ 0 0 0 Perovskite This study , Mg#100 182.6(5) 221(4) 4* 181.7(4) 236(4) 3.7* This study , Mg#78 185.5(3) 205(3) 4* 184.6(3) 219(3) 3.7* Runge et al. (2006), Mg#100 182.9(3) 216(3) 4* 182.2(3) 229(3) 3.7* 180.16(5) 221.3(7) 3.90a

Post-perovskite This study, Mg#100 175.9(3) 241(3) 4* 177.5(3) 216(3) 4.4* Kubo et al. (2006), Mg#100 175.9(5) 245(5) 4* 179.2(7) 207(5) 4.4* 178.0 201.9 4.3a Hirose et al. (2005), Mg#100 183.1 ± 0.8 192 ± 5 4*

a. theoretical calculation * value fixed during fit

4.4.4.2 Post-perovskite

The post-perovskite equation of state was measured for Mg#100 in two separate runs. In both runs, the samples the samples were first compressed without heating to 75-80 GPa, at which point it was transformed to post-perovskite by heating to ~1800 K for 2-5 minutes. In the first run, the sample was then compressed to 130 GPa in 2-5 GPa steps with annealing for 2-5 minutes to 1800 K every 5 GPa. In the second run, the sample was decompressed to 29 GPa, but 120

Figure 4.16. MgGeO3 post-perovskite a. unit cell volume and b. lattice parameters upon compression and decompression. Lines indicate Birch-Murnaghan equation of state fits with K0’ = 4. 121 no annealing was performed below 50 GPa to prevent conversion to perovskite. Nevertheless, some perovskite peaks were evident upon decompression. Differential stress was estimated from the measured lattice strain of Au using the method outlined in Chapter 1. Our results indicate that differential stresses were between 0.5-1.3 GPa, consistent with the expectation that annealing reduces differential stress, and within the same range as prior work on this composition.77

Third order Birch-Murnaghan equation of state fits of the unit cell volumes as a function of

' pressure are reported in Table 4.8. K0 was fixed in these fits to 4 (i.e., a second-order Birch-

Murnaghan equation of state) or 4.4, the values determined for silicate post-perovskites.77 The volume and lattice parameter measurements as well as the calculated K0 and V0 of Mg#100 match prior studies,77,86 but the current work exhibits less scatter throughout the pressure range measured (Figure 4.16). The equation of state calculated by Kubo et al.77,86 includes data collected on compression and decompression with a mixture of loading conditions, which may explain the scatter observed in that data set. The lattice parameter ratios of the two data sets also exhibit similar behavior (Figure 4.S.5), with b/a showing a greater response to pressure than c/a.

This effect is more pronounced in germanates than in silicates, and is due to greater compressibility along the b axis (the direction of layer stacking in the post-perovskite structure).

The Mg#100 post-perovskite V0 is 3.7% smaller than that of perovskite, while K0 is 9% larger.

This relationship is different than what is observed in silicates, where MgSiO3 shows no change in V0 (depending on study sampled) and a decrease in K0 between the perovskite and post- perovskite.55,67,305,309–314

122

4.5 Conclusion

Orthopyroxene, perovskite and post-perovskite germanates were synthesized with compositions of (MgxFe1-x)GeO3 where x = 100 – 48. Ceramic sintering in vacuo was used to synthesize orthopyroxenes that show a linear dependence between unit cell size and Raman shift versus Fe content. Compositions were confirmed using microprobe analysis and showed relatively good sample purity.

High-pressure experiments were carried out in the laser-heated diamond anvil cell coupled with x-ray diffraction at the Advanced Photon Source. Germanate perovskites were synthesized at pressures above 30 GPa and temperatures from 1500-1800 K in all compositions. Post- perovskites could be synthesized above 70 GPa and 1500-1900 K. However, peaks indicative of possible decomposition were observed in Mg#61 and 48 perovskites and post-perovskites, indicating that these compositions may not be able to fully incorporate large amounts of Fe in their crystal structures. We find that compositions with Mg# ≥ 78 form single phase perovskites and post-perovskites.

We find that our Mg#100 composition undergoes the perovskite-post-perovskite phase transition at pressure and temperature conditions consistent with prior work.95,284 However, we observe a broad perovskite-post-perovskite coexistence region in this composition due to sluggish conversion kinetics between the two phases. In Mg#78, the phase transition is first observed at pressures ~10 GPa below its appearance in Mg#100, indicating that Fe substitution leads to a decrease of the phase boundary, but we were unable to constrain the upper boundary of the two-phase loop.

The equations of state of Mg#100 and Mg#78 perovskite, and Mg#100 post-perovskite were measured. Fe substitution into the perovskite A site leads to a decrease in distortion of the

123 perovskite relative to the cubic ideal, similar to observations in silicate perovskites.55 This is manifested mainly through an increase in the a lattice parameter of the perovskite unit cell relative to b and c. We find that Fe substitution leads to a smaller bulk modulus and larger volume as a function of Fe content in perovskites, unlike observations in silicate perovskites where Fe has no effect on bulk modulus.55 The Mg#100 post-perovskite equation of state is in good agreement with prior studies.77 Overall, we find evidence that germanate perovskites and post-perovskites with Mg# ≥ 78 function as good models for corresponding silicates when analyzing structural behavior and phase transition thermodynamics.

4.6 Acknowledgements

Drs. Gregory Finkelstein, June Wicks, Jason Krizan, John Armstrong, and Katherine Crispin provided experimental assistance with material synthesis, characterization, and high-pressure experiments. This work was supported by the NSF. Use of the Advanced Photon Source, an

Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of

Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No.

DE-AC02-06CH11357. We acknowledge the support of GeoSoilEnviroCARS (Sector 13), which is supported by the National Science Foundation - Earth Sciences (EAR-1128799), and the Department of Energy, Geosciences (DE-FG02-94ER14466).

124

4.7 Supplementary Material

Figure 4.S.1. GeO2 rutile impurity in Mg#78 and Mg#48 orthopyroxene starting materials. The angular range has been reduced to highlight the GeO2 peaks of maximum intensity relative to those of the corresponding orthopyroxene.

125

Figure 4.S.2. Diffraction pattern after synthesis of perovskite in Mg#61 sample. Asterisks indicate unidentified peaks.

126

Figure 4.S.3. Mg#48 post-perovskite after synthesis at 75 GPa.

127

Figure 4.S.4. Full width at half maximum (FWHM) of the Au (111) diffraction peak as a function of pressure in perovskite equation of state measurements.

128

Figure 4.S.5. Lattice parameter ratios in the germanate and silicate post-perovskites.

129

APPENDIX A

First-Principles Study of the High-Pressure Isosymmetric Phase Transition in

PbF2

R. Dutta1,*, C. V. Stan2, C. E. White3,4 and T. S. Duffy1

1Department of Geosciences, Princeton University, NJ-08544, USA.

2Department of Chemistry, Princeton University, NJ-08544, USA.

3Department of Civil and Environmental Engineering, Princeton University, NJ-08544, USA.

4Andlinger Center for Energy and the Environment, Princeton University, NJ-08544, USA.

A.1 Abstract

Using first-principles calculations we examine the behavior and mechanism of the isosymmetric phase transition in orthorhombic lead fluoride. We confirm that cotunnite- structured α-PbF2 (Pnma) undergoes a transition to a Co2Si-type structure (Pnma) at high pressures. Our calculations also reveal the detailed atomic rearrangements associated with the development of an extra Pb-F bond in the high-pressure phase. At P > 9 GPa, both the lead and fluorine ions begin to reorient themselves as an additional fluorine atom, initially outside the 9- fold coordination polyhedron of cotunnite, approaches it. The transition regime from 9 to 21 GPa is marked by continuous atomic movements leading to anomalous behavior of the lattice parameters and compressibilities. At P ~21 GPa, the transformation to the Co2Si-type structure is completed with an increase in the coordination number from 9 to 10. The lattice parameters and unit cell volume exhibit smooth variation above this pressure. We have used the atomic positions, valence electron densities, and elastic moduli to explore the behavior and mechanism of this phase transition. 130

A.2 Introduction

There has been considerable interest in the optical and electronic properties of lead fluoride

315–320 due to its technological applications as a fast ion conductor and scintillating material. PbF2 is a member of the AX2 system whose high-pressure behavior has attracted strong interest due to the many phase transitions and diverse transformation pathways available for these compounds.7

At ambient conditions PbF2 crystallizes in the 8-fold coordinated cubic fluorite structure (β-

PbF2,⁡퐹푚3̅푚). At 0.4 GPa, it undergoes a phase transition to an orthorhombic 9-fold coordinated cotunnite structure (α-PbF2, Pnma), which remains metastable when quenched to ambient

321 pressure for T < 610 K. Several metal dihalides (AX2) also adopt the cotunnite structure, and exhibit varied polymorphism upon compression with or without the application of high temperatures. For instance, PbCl2 was found experimentally to transform from cotunnite into

322 monoclinic P21/a structure, or by theory to transition to an orthorhombic 10-fold coordinated

323 Co2Si-type and then to a monoclinic P21/n structure. The difluorides CaF2, SrF2 and BaF2 undergo a transition from cotunnite to 11-fold coordinated Ni2In-type structure (P63/mmc) at

149,324 pressures that scale with cation size. SiO2, an AX2 material of importance in geoscience and planetary science, is predicted to adopt the cotunnite structure at ultrahigh pressures of ~700

GPa,15,325,326 which would correspond to conditions deep in the interior of large, earth-like

327 extrasolar planets. Additionally, a number of metal dioxides including TiO2, ZrO2, HfO2,

SnO2, PbO2, CeO2, PuO2, UO2 and ThO2 adopt the cotunnite or related structure at elevated pressures, leading to phases with very high bulk moduli or unusual compressibilities.12,13,151,156,328–330 Difluorides have been found to be good analogs for understanding the high-pressure behavior of oxides due to the lower pressures at which the cotunnite phase can be reached in these materials.149

131

The high-pressure post-cotunnite phase of PbF2 has been somewhat controversial. In a high- pressure X-ray diffraction study, Haines et al.144 suggest an isosymmetric phase transition to a

“Co2Si-related” structure at 12.9 GPa. Their structural analysis is based solely on empirical observations of characteristic lattice parameter ratios for the cotunnite-type, Co2Si-type and

171,324 Ni2In-type phases. For these three phases, the empirical ranges for the orthorhombic ratios a/c and (a+c)/b are reported to be 0.80-0.90, 3.3-4.0, 0.66-0.74, 3.1-3.3, and 0.70-0.78, 2.9-3.2 respectively.171,324 On the other hand, quantum mechanical calculations based on the ab initio perturbed ion method predict that the post-cotunnite phase in PbF2 is the Ni2In-type structure.161,320 A joint experimental and theoretical study discovered a structural transformation at 14.7 GPa, but the structure of the high-pressure phase could not be determined.168 In a related

331 experimental study, we have compressed PbF2 to 75 GPa at room and high temperature. Our experimental results331 are qualitatively similar to those of Haines et al.144 at room temperature but differ in some important ways. We observe a region of anomalous compressibility from approximately 10-22 GPa featuring high compressibility in the a direction but little or even negative compressibility along b and c. To better understand the structural response of this material, we have carried out first-principles calculations based on density functional theory. We base our study on bonding and coordination numbers rather than ratios of lattice constants to distinguish among the high-pressure phases.

A.3 Methods

The total energy calculations were carried within the framework of density functional theory

332,333 334 (DFT) as implemented in the CASTEP code. We used a kinetic energy cutoff of 400 eV for the plane wave basis set, while the self-consistency convergence was set to 10-6 eV. The

132 electron-ion interactions were treated using ultrasoft pseudopotentials335 with valence configurations of 2s22p5 and 5d106s26p2 for F and Pb respectively. The exchange-correlation potentials for the electron-electron interactions were treated in the local density approximation

(LDA).336 The Brillouin zone was sampled using a Monkhorst-Pack337 grid, with a separation of

0.07/ Å between the k-points. All structural optimizations were carried out using the Broyden–

Fletcher–Goldfarb–Shanno (BFGS) technique and was considered complete when the forces on atoms were less than 0.01 eV/Å and the energy change was 5 x 10-6 eV/ Å.338

Both lattice parameters and atomic positions were relaxed at each pressure step with unconstrained symmetry. The elastic tensor (of which there are nine independent components in case of orthorhombic symmetry) was calculated using Hooke’s law (휎푖푗 = 퐶푖푗푘푙휀푘푙), where 휎푖푗,

휀푘푙 and 퐶푖푗푘푙 are the stress, strain and elasticity tensors respectively. To calculate the full elastic constant tensor we used three positive and three negative strains with magnitudes of ± 0.003, ±

0.002 and ± 0.001.

The distortion of coordination polyhedra was determined using the circumscribed sphere approach of Makovicky and Balić-Žunić339 and the continuous symmetry measure (CSM) approach of Pinsky and Anvir.339,340 The former assumes that the ligands of both the real and ideal polyhedra lie on a common circumsphere with a radius equal to the average distance of the centroid of the polyhedron to the ligands.341 The centroid is calculated by minimizing its distance from the vertices, the expression of which is given by:

2 2 2 C= ∑ (di - ∑ di /n) ,

푡ℎ where 푑푖 is the distance from the centroid to the 푖 vertice of a polyhedron with n sides. In the

CSM method, a regular polyhedron is embedded within the real (distorted) one using least

133 squares minimization. The CSM is the normalized root mean square deviation from the closest structure. The expression is given by:

N 2 ∑ |Q -Pk| CSM=min k=1 k x 100, N 2 ∑k=1|Qk-Q0| where 푄푘 and 푃푘 are the co-ordinates of the distorted and ideal polyhedra respectively. 푄0 is the coordinate vector of the center of mass of the relevant polyhedron.

A.4 Results and Discussion

A.4.1 Structural evolution under pressure

Here we restrict our attention solely to the cotunnite and the post-cotunnite phases of lead fluoride. The ambient-pressure fluorite structure has not been considered, i.e., we have taken orthorhombic cotunnite (Pnma) to be the ambient-pressure phase. The cotunnite structure is composed of corrugated tricapped trigonal PbF9 prisms (TTP) which share edges and faces

(Figure A.1).342,343 The trigonal faces of the TTPs are perpendicular to the b-direction, forming corrugated sheets of face sharing TTPs stacked along the a-axis and sharing faces and edges along c. The cotunnite structure can be described completely using nine degrees of freedom, i.e., the three lattice constants, a, b, c and six atomic coordinates (Pbx, Pbz, F1x, F1z, F2x and F2z, where the subscripts indicate the non-symmetry-constrained coordinate directions of the respective atoms). In Table A.1, we have compared the values obtained from our theoretical calculations at ambient pressure for the nine degrees of freedom to other results from experiment and theory.144,166,344 LDA, as expected, underestimates the lattice parameters in comparison to

134

Figure A.1. Equilibrium structure of α-PbF2 (cotunnite, Pnma). Pb, F1 and F2 are the lead

and two crystallographically distinct fluorine atoms with coordinates (Pbx, 0.25, Pbz),

(F1x, 0.25, F1z) and (F2x, 0.75, F2z), respectively.

the experimental values and the differences are 2.2%, 2.6% and 1.2% for a, b and c respectively.

Considering the overbinding inherent in the local density approximation345 our results are consistent with experimental data.

135

Figure A.2 shows the change in lattice parameters of PbF2 as a function of pressure. The compressional regime can be broadly divided into three sections. In the low-pressure range (0-9

GPa), the behavior is monotonic whereby all three lattice constants decrease nearly linearly with pressure. The pressure range from 9 – 21 GPa defines a transition region in which the response of the lattice parameters to compression is anomalous in the sense that a decreases with a steep gradient, while b and c slightly increase with pressure. Above 21 GPa, the lattice parameters decrease monotonically to the highest pressure of our calculation, 70 GPa. These observations are consistent with experimental observations from X-ray diffraction (Figure A.2).144,331 Changes

Figure A.2. Relative lattice parameters of PbF2 as a function of pressure. Triangles

indicate the experimental lattice parameters.331 136

Table A.1. Lattice parameters and atomic positions of α-PbF2 at ambient conditions. All atoms are at y = 0.2500.

Reference Lattice Parameters (Å) Atomic positions

a b c Pbx Pbz F1x F1z F2x F2z This studya 6.4472(3) 3.9019(2) 7.6514(3) This studyb 6.298 3.800 7.556 0.25469 0.11149 0.86023 0.06450 0.46994 0.84235 Boldrini and Loopstra342,c 6.440 3.899 7.651 0.2527 0.1042 0.8623 0.0631 0.4622 0.8457 Haines et al.144,b 6.444 3.900 7.648 Dubinin et al.344,b 6.314 3.812 7.558 a. X-ray diffraction experiment b. theoretical calculation c. neutron diffraction experiment

137 in the Raman spectra are also observed beginning from ~10 GPa.168,331 In our calculations the structures were completely relaxed at each pressure step with unconstrained symmetry. Within the pressure range considered here (0-70 GPa), the structures maintained the same symmetry

(orthorhombic, Pnma). However, the lattice parameters evolved continuously through the anomalous region mentioned above. This suggests that PbF2 undergoes an isosymmetric phase transition (Pnma → Pnma) consistent with earlier observations.144 In subsequent sections, we explore the evidence for this structural change and its mechanism in more detail.

In earlier work,144 systematics of lattice parameter ratios were used to infer that the high- pressure phase is a “Co2Si-related” structure, as the a/c and (a+c)/b values just above the transition pressure (0.773, 3.521) and that of the high-pressure phase (0.729, 3.431) are outside the expected range of Co2Si-type or Ni2In-type structures. The Ni2In-type structure can be described by a unit cell that is half the size of the orthorhombic cell with a c/b ratio of √3. This relationship is not satisfied in either our theoretical calculations or the experimentally determined values (Figure A.3).331 Our calculations give (a+c)/b = 3.521, 3.451 and a/c = 0.764, 0.720 at 12

GPa and 21 GPa respectively. These values are in excellent agreement with the experimental

331 171 data, but are beyond the empirical limits of Co2Si-related structures as defined by Jeitschko

(Figure A.3).

A more direct approach would avoid empirical systematics for phase identification and instead examine the underlying bonding and coordination numbers. The Co2Si-type phase is isopuntal with the cotunnite-type phase, having the same space group and the same occupied

Wyckoff positions but different unit cell ratios and an increase in coordination from 9- to 10-fold about the central (A) atom.170 The analysis described in detail below shows that the high- pressure phase of PbF2 does indeed have a Co2Si-type structure. The empirical boundaries

138

Figure A.3. Lattice parameter ratios of several cotunnite-like compounds. The boxes

delineate empirically based boundaries for phases based on the a/c and (a+c)/b ratios.171

Color data points and lines (and pressure scale) have been used for the experimental and

theoretical data reported in this study. The dotted black lines are polynomial fits to the

12,13 AO2 data of Song et al. Arrows indicate the direction of increasing pressure. The

solid black line shows the linear relationship guiding Ni2In-type compounds.

defining phases based on lattice parameter ratios of Jeitschko171 should be modified with the

331 Co2Si field expanded to a wider range of lattice constant ratio. Our calculations, combined

331 161 with experimental data, also show that the Ni2In-type phase predicted by Costales et al. is 139 not the stable post-cotunnite phase in lead fluoride at 0 K as the high-pressure phase is 10- coordinated and not 11. Experimentally, the Ni2In-type phase is observed at pressures up to 65

GPa only under high temperatures (T > 1250 K) or high differential stress.331

A similar isosymmetric phase transition has also been predicted in theoretical calculations12–

14,156 for the cotunnite phase of the actinide oxides UO2, PuO2, CeO2 and ThO2, but at much higher pressures (e.g. 106 – 160 GPa in CeO2). Thus we suggest that PbF2 can act as a structural analog of these phases and would be useful for experimental validation of their properties at much lower pressures.

A.4.2 Bonding and polyhedral analysis

The structural distinction between the cotunnite, Co2Si- and Ni2In-type structures has not always been clear in the literature.160 Prior studies have used lattice parameter ratios for different compositions at ambient pressure to distinguish between the three structure types (Figure

A.3).144,171,324 In addition to the lattice parameter ratios we can distinguish the cotunnite-type,

Co2Si-type and Ni2In-type structures based on their coordination numbers of 9, 10 and 11 respectively. To get a better understanding of the bonding behavior, the change with pressure of the atomic positions for the six symmetrically unconstrained coordinates is shown in Figure A.4.

We again observe three distinct regions, where the low-pressure regime (0-9 GPa) is marked by a smooth continuous increase/decrease, followed by an anomalous zone (9-21 GPa) and then another region of monotonic increase or decrease (> 21 GPa). There are two symmetrically distinct fluorine atoms in the cotunnite/Co2Si structure which are designated F1 and F2. A clear change in the slope is evident especially in case of Pbx, F2x, F2z in the transition region. In the transition regime, the atomic coordinates of the second fluorine atom (F2) exhibit a greater

140

Figure A.4. Change in atomic positions with pressure. Pb, F1 and F2 are the lead and two

flourine atoms respectively. Subscripts indicate coordinate directions.

change in position with respect to those of the first one, with F2z showing the largest displacement.

Figure A.5 shows our calculations of the total electron densities along the (010) plane at different pressures. In an electron density map, sharing of the charge density indicates a covalent bond, while an ionic bond can be identified from the complete transfer of charge density to the anion. The presence of a bond can be identified from the existence of a charge density valley between two atoms. Figure A.5a shows that there is no sharing of valence charge density at ambient conditions between Pbd and F1c (crystallographically equivalent to Pba and F1b2) while there is clear evidence for the presence of charge sharing at 25 GPa (indicated by the white box in Figure A.5b). Up to 21 GPa, there is no evidence of charge sharing between the two concerned

141

3 Figure A.5. Total electron density (e/Å ) of PbF2 in the (010) plane at a. 0 GPa and b. 25

GPa. The white box indicates the Pbd-F1c bond (crystallographically equivalent to Pba-

F1b2) which, when formed, indicates transition to the Co2Si-type phase.

Pb and F atoms. This confirms that the 10th Pb-F bond does not exist in the low-pressure region, while 10-coordinated Co2Si-type structure is the stable phase in the high-pressure region (>21

GPa). As expected, the Pb–F bonds appear to be partially ionic and partially covalent.320

Figure A.6 shows the geometry of bonding and atomic displacements in lead fluoride at 0 and 70 GPa. As mentioned previously, the cotunnite structure contains two crystallographically distinct fluorine atoms, F1 and F2, at (F1x, 0.25, F1z) and (F2x, 0.75, F2z) respectively. For simplicity, we discuss changes in the structure relative to Pba. Pba is bonded to 4 F1 atoms

(2×F1a, F1b, F1d) and 5 F2 atoms (F2a, 2×F2b2, 2×F2d). The Co2Si- and Ni2In-type structures are marked by extra Pb-F1 (Pba-F1b2) and Pb-F2 (Pba-F2c) bonds, respectively. From the calculated

142 atomic positions, the variation of bond lengths can be determined as a function of pressure

(Figure A.7a). As with the lattice parameters, the lengths of all 9 Pb-F bonds in the cotunnite structure decrease (or, for Pba-F2a, increase) smoothly up to a pressure of 9 GPa. With increasing pressure, the longest bond, Pba-F2d, shows a marked reduction in length. Therefore, the fluorine atom which is diametrically opposite (F2a) to F2d, moves away from Pba, leading to an increase

th in the Pba-F2a bond length. Also over the same pressure region, a 10 fluorine atom, F1b2, which initially lies well outside the TTP, exhibits a dramatic reduction in distance relative to Pba, eventually approaching the TTP (Figure A.7a). As F1b2 moves inward, F2a and F2b2 experience a strong repulsion and swing away from the incoming F atom (Figure A.6). The two opposing

Figure A.6. Atomic level structural evolution of PbF2 with compression. Pb atoms are in

gray, F atoms are in orange. The different shades of orange indicate the two different

crystallographic sites of F (F1 and F2). Arrows indicate the direction of atomic

movement.

143 forces along with the movement of the central lead atom balance each other giving rise to F2b2 showing relatively low compressibility in the transition interval. After the formation of the Pb-

F1b2 bond, as seen from the electron density calculations (at P = 25 GPa, Figure A.5), all the

Figure A.7. a. Calculated interatomic distances in PbF2 as a function of pressure. b.

Difference in bond length between the longest bond of the 9-fold coordinated structure

and the 10th and 11th bonds and difference between the maximum and minimum bond

lengths in 9-fold coordinated structure.

144 bonds show a monotonic decrease with pressure including Pb-F2a. Figure A.7a compares the compression along hypothetical 10th and 11th bonds as a function of pressure. It can be seen that the slope of the 10th bond is much steeper than that of the 11th bond, especially in the transition region, indicating its tendency to form a bond. As can be seen in Figure A.6, this motion is along the ac plane, and the F2b2 – Pba – F2a angle widens such that the F2b2 – Pba and the – Pba – F2a approach the c axis. This atomic displacement leads to an increase in the c lattice parameter with simultaneous decrease in the a lattice parameter in the transition region (Figure A.2).

Figure A.7b shows the difference between the longest and shortest Pb-F bonds for the 9-fold coordinated cotunnite TTP, and the difference between the longest bond of a hypothetical 10- or

11-fold coordinated structure and that of the TTP. For 9-fold coordination, the difference between the longest and shortest Pb-F bonds is initially ~ 0.5 Å, which declines strongly with pressure to ~0.2 Å at 15 GPa, after which it starts to increase as a function of pressure. If the 10th and 11th nearest neighbors are included, we see that the difference in bond length between the

10th or 11th bond and the longest bond in the TTP initially increases, but above 9 GPa starts to decrease as the outer fluorine atoms begin to move closer due to increasing pressure. In fact, the

th difference between the length of the 10 bond (Pba-F1b2) and that of the next longest Pb-F bond decreases very rapidly as a result of a strong movement of the 10th most distant F toward the coordination sphere about Pb. Qualitatively, we can say that if the difference between the 10th

th (Pba-F1b2) or 11 (Pba-F2c) bond length and that of the longest bond of the TTP is less than difference between the longest and shortest TTP bonds, then the 10th/11th bond should exist, and the structure is exhibiting 10/11-fold coordination. At P ~30 GPa, we see that the 10th bond satisfies this criterion, while the 11th bond does not satisfy it any pressure (up to 70 GPa). This provides an estimate of the upper limit of the cotunnite-Co2Si transition pressure or the pressure

145 above which the Co2Si structure is stable. This pressure is close to, although slightly above, the upper range of the region where the lattice parameters exhibit anomalous behavior.

We have also calculated the polyhedral distortion of the low-pressure phase to further describe the structural changes (Table A.2). On compression, the average distance of the fluorine atoms (푑퐹) from the centroid of the TTP decreases, while the central lead atom moves towards the centroid leading to a decrease of variance of the fluorine atom distances. CSM measures the distortion of the relevant polyhedron with respect to an embedded ideal polyhedron (TTP in this case). The CSM of the TTP under ambient conditions is 0.781 (ideal value = 0), indicating a slight distortion. With increasing pressure the CSM increases, with a sharp increase near the start of the transition region. This supports our previous observation of the anisotropic increase of the

Table A.2. Deformation of the tri-capped trigonal prism polyhedron in PbF2 as a function

of pressure. dF is the average distance of the fluorine atoms from the centroid of the TTP,

σ is the standard deviation of the fluorine distances with respect to the mean, dPb is the

distance of the central lead atom from the centroid and CSM is the continuous symmetry

measure.

Pressure (GPa) dF (Å) σ (Å) dPb (Å) CSM

0 2.598 0.140 0.254 0.781

6 2.537 0.109 0.137 0.787

9 2.517 0.100 0.107 0.887

12 2.501 0.085 0.085 1.389

15 2.490 0.079 0.085 1.920

146 bond lengths in that region. In summary, our theoretical electron density calculations provide direct evidence for the formation of a 10-fold coordinated Co2Si-type structure in PbF2 above 21

GPa (Figure A.5). The anomalous changes in lattice compressibility and atomic positions at ~9-

21 GPa (Figures A.2, A.4) can be understood as a transitional region of structural rearrangement as the 10th fluorine atom approaches and then joins the coordination polyhedron (Figure A.7).

This transition interval also corresponds to a region where the calculated and measured lattice parameter ratios evolve from values characteristic of cotunnite to those of the Co2Si-type structure (Figure A.3). Above 21 GPa, all structural parameters display smooth trends with further compression up to 70 GPa, consistent with compression of a single homogeneous phase.

A.4.3 Elasticity

An isosymmetric phase transition is often accompanied by anomalies in the pressure dependence of the elastic constants.346 We therefore calculated the single-crystal elastic moduli and also verified the elastic stability of the phases. In Table A.3, we compare our calculated single-crystal elastic constants (at 0 GPa) with other theoretically calculated (SVWN,347 LDA functional) values from the literature.348 To the best of our knowledge no experimental data are available to validate our results. The two theoretical studies are in fair agreement with each other.

Figure A.8 shows the evolution of the elastic constants with pressure. As expected the elastic constants increase with pressure, but in the transition region (9-21 GPa), we can clearly see anomalous behavior. In the transition region, C11 shows a discontinuity, C22 displays an abrupt stiffening while C33 exhibits a nearly flat slope in the transition regime. The shear elastic

147

Figure A.8. Elastic constants of PbF2 as a function of pressure. Top: longitudinal moduli.

Middle: off-diagonal moduli. Bottom: shear moduli.

148

Table A.3. Elastic constants (in GPa) of cotunnite-type PbF2 at 0 GPa.

348 Cij This Study (LDA) Erk et al. (SVWN)

C11 95.8 107

C22 99.5 119

C33 110.3 130

C44 8.7 6.74

C55 18.9 24.4

C66 31.3 36.9

C12 50.3 60.5

C13 47.2 56.5

C23 57.9 64.5

constant C44 also shows an abrupt stiffening in the intermediate zone while C55 displays a nearly flat slope. In contrast, all the elastic constants show nearly linear behavior in both the low- (<9

GPa) and high-pressure (>21 GPa) regimes.

The aggregate bulk and shear moduli calculated from the elastic constants are shown in

Figure A.9. The elastic anisotropy of a crystal can been expressed in terms of various anisotropy

349,350 351 factors. In this study we have used the percent anisotropy in bulk (AK) and shear (AG) to define the polycrystalline elastic anisotropy. These are given by:

KV-KR AK= , KV+KR

GV-GR AG= , GV+GR

149

Figure A.9. Variation in polycrystalline elastic moduli (bulk and shear) as a function of

pressure.

where K and G are the bulk and shear moduli and the subscripts V and R stand for the Voigt and

Reuss bounds.352,353 In the 9-21 GPa transition region (Figure A.9), we see that the difference between the Voigt and Reuss bounds of the bulk modulus reaches a maximum, but there are only slight differences in the bounds outside this region. The same trend, although not as prominent, is also apparent in the shear modulus, although in this case the anisotropy grows at higher pressures in the Co2Si-type phase. AK and AG have been calculated to quantify the variation in anisotropy with pressure. It is seen that the anisotropy exhibits a sharp increase in the transition region and

150 decreases in both the low- and high-pressure regimes (Table A.4). The overall trend of the elastic moduli is consistent with the behavior of the lattice parameters shown in Figure A.2.

Finally, we checked the Born stability criterion to determine if there is any evidence for an elastic instability.354 The orthorhombic elastic stability criterion under high pressure is given by:

Bij> 0,

B11 + B22 - 2B12 > 0,

B11 + B33 - 2B13 > 0,

B22 + B33 - 2B23 > 0,

B11 + B22 + B33 + 2B12 + 2B13 + 2B23 > 0,

355,356 where Bij= Cij - P for i = j and Bij= Cij + P for i ≠ j. All the orthorhombic elastic stability criteria are fulfilled up to the highest pressure considered in this study (70 GPa). Thus, we conclude that the isosymmetric phase transition is not driven by an elastic instability.

A.4.4 Comparison with metal dioxides

Several theoretical studies have recently reported the compression behavior to megabar pressures

12–14,156 of several AO2 materials including CeO2, ThO2, UO2, and PuO2. These first-principles calculations based on density functional theory show that all four compounds undergo isostructural phase transitions starting from an orthorhombic cotunnite-type structure with continuous evolution of lattice parameters. CeO2 and ThO2 isosymmetrically transform from a 9- coordinated cotunnite-type Pnma to an unspecified 10-coordinated type at pressures above 1

Mbar.13 Both materials exhibit anomalous changes in lattice parameters similar to what we observe in PbF2, with enhanced compression along a and reduced or negative compression along b and c. The transition region ranges from 106-160 GPa in CeO2 and from 80-130 GPa in ThO2. 151

Table A.4. Elastic anisotropy in lead fluoride. AK and AG are the percent anisotropy in

bulk and shear respectively.

Pressure (GPa) AK AG

6 0.037 0.0995

12 0.120 0.101

25 0.015 0.020

Figure A.3 shows the variation in lattice parameter ratios for these phases with compression. The lattice parameter ratios evolve away from cotunnite values and approach but do not cross into the

Co2Si-type field. The high-pressure Pnma-phase for these materials is described as an orthorhombic distortion of the Ni2In-type (P63/mmc) structure, which is consistent with the final value of lattice parameter ratios in Figure A.3. Since Pnma is a subgroup of the hexagonal

P63/mmc space group, the hexagonal structure can also be represented in an orthorhombic

331 setting according to:⁡푎0 = 푐ℎ,⁡⁡푏0 = 푎ℎ , and 푐0 = √3푎ℎ, where h and o refer to the hexagonal and orthorhombic lattice parameters respectively.

12 In case of UO2 and PuO2, the pressure-enthalpy curve of the Pnma has 3 distinct regions,

Pnma-I or cotunnite, followed by Pnma-II at P > 90 GPa and Pnma-II (degenerates to Cmcm).

The Pnma-II phase has a Co2Si-type structure and was identified by the anomalous behavior of the lattice parameters (the same as PbF2 in Figure A.2). The lattice parameter ratios of UO2 and

PuO2 are consistent with the Co2Si range at high pressures, while CeO2 and ThO2 approach the region, but do not actually pass through it (Figure A.3). The ultra-high-pressure phase of all the four compounds resembles an orthorhombic distortion of the Ni2In-type: the lattice parameter ratios approach the line that defines the Ni2In-type structure but do not fall directly on it at any

152

331 pressure calculated. UO2 and PuO2 form centrosymmetric structures (either Cmcm or Cmc21, with 11-fold coordination) which is a supergroup of Pnma at P ~ 120 GPa, while CeO2 and ThO2 transform into a distorted Ni2In-type phase with Pnma space group (the Co2Si-type and centrosymmetric structures have higher enthalpy than the Pnma phase) at P > 160 GPa and > 130

GPa respectively. Unlike the oxides, with increasing pressure the lattice parameter ratios of PbF2 remain within the Co2Si field to the uppermost limit of this study (70 GPa using DFT and 75

331 GPa experimentally ). The hexagonal Ni2In-type phase is formed only on heating or under non- hydrostatic conditions.331 Experimental studies have not yet verified the isosymmetric phase transition in the actinide and lanthanide oxides and are required to confirm the lattice parameter trends observed theoretically. Our results show that isosymmetric phase transitions may be an important aspect of high-pressure behavior in AX2 materials.

A.5 Conclusion

In this study we have shown that α-PbF2 undergoes a pressure-induced isosymmetric phase transition to a post-cotunnite Co2Si-type structure through consideration of atomic distances and bonding changes with compression. The variation of the lattice parameters with pressure is characterized by two monotonic compression regions where the cotunnite and post-cotunnite structures are stable (below 9 GPa and above 21 GPa, respectively) with an intermediate anisotropic compression zone between them. This intermediate zone is characterized by

th continuous atomic rearrangements whereby a 10 fluorine atom, which distinguishes the Co2Si structure from the cotunnite structure, approaches the cotunnite coordination polyhedron and then finally establishes a bond at ~21 GPa. Our calculated lattice parameters and pressure ranges for the stability fields of the respective phases are in very good agreement with the most recent

153 experimental study.331 The bond formation is also confirmed through calculated valence electron densities. We conclude that the post-cotunnite phase in PbF2 is a Co2Si-type structure under static conditions. The Ni2In-type phase (coordination number = 11) is not stable at 0 K up to the peak pressure considered in this study (70 GPa). The anomalous behavior of the lattice constants in the intermediate transition zone is supported by strong elastic anisotropy in this zone. Based

331 on these calculations, we can justify the empirical extension of the Co2Si stability field in the empirical lattice parameter ratio diagram of Jeitschko171 to include a wider range of values. The post-cotunnite phase in PbF2 can be used to provide insights into the high-pressure behavior of isostructural AX2 compounds such as actinide and lanthanide oxides.

*Corresponding author: [email protected]. Reprinted with permission from R. Dutta.

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