The Core Composition of Terrestrial Planets

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The Core Composition of Terrestrial Planets:

A Study of the Ternary Fe-Ni-Si System

Elizabeth Wann

UCL

A thesis submitted to University College London for the degree of

Doctor of Philosophy.

August 2015 2 I, Elizabeth Wann, conﬁrm that the work presented in this thesis is my own.

Where information has been derived from other sources, I conﬁrm that this has been

indicated in the thesis.

3 Abstract

The exact composition of the cores of terrestrial planets is not known, but it is generally agreed that they are composed of iron alloyed with a fraction of nickel plus a small percentage of a light element, likely Si, S, O, C or H. Silicon has long been a popular choice and is still regarded as a very likely candidate, based on density deficit and cosmochemical arguments. Although much work has been carried out on the Fe-Si system, studies on the Fe-Ni-Si system have only recently been carried out. The majority of studies have concentrated on specific candidate core compositions, based on core formation models or matching the observed density deficit. This can be problematic when core formation models depend on core composition. In this thesis, the Fe-Ni-Si system is investigated as a whole, starting with the end-member binary systems, FeSi and NiSi. This provides a more methodical approach to solving the core composition problem.

Both ab initio calculations and high-pressure, high-temperature experiments have been used in this work. Ab initio calculations at 0 K were used to ﬁnd the transition pressure of the "-FeSi to CsCl-FeSi phase transition, and also to test the stability of newly discovered NiSi-structured phases in FeSi. Lattice dynamics calculations at high temperatures and pressures have been carried out to determine the Clapeyron slope of the "-FeSi to CsCl transition, in both FeSi and NiSi systems. Laser-heated diamond anvil cell experiments were used to measure the melting curves of NiSi and the

Fe-FeSi eutectic, and in-situ neutron di↵raction experiments were used to determine the equation of state of MnP-structured NiSi at high-pressure and high-temperature.

Finally, X-ray di↵raction experiments were used to measure the thermal expansion of a range of (Fe,Ni)Si alloys.

4 Acknowledgements

This PhD would not have been possible without the support and advice of my supervisors, Prof. Lidunka Voˇcadlo and Prof. Ian Wood. Their help and expertise were invaluable and I feel extremely grateful to have had such excellent supervisors. I am particularly grateful to Lidunka for her kindness and patience and whose encouragement pushed me forward to accomplishing this work.

I would also like to thank Dr. Oliver Lord, for all his guidance with the diamond anvil cell experiments, and Dr. Benjam´ıMartorell for his helpful advice with VASP. Thank you also to Prof. John Brodholt, Prof. David Dobson and Prof. Dario Alf´e,as well as Dr. Simon Hunt and Dr. Alex Lindsay-Scott for their help and input during my

PhD.

Finally, my thanks go to my friends and family for their support and encouragement.

To Becky, Adam, Jo and Amanda for sharing the highs and lows of my PhD and to

Amy for making each conference more fun and enjoyable. To my family, for their words of encouragement and for always having faith in me. And lastly to Stephen, for being a constant source of support and for being there for me always.

This PhD was funded by the Science & Technology Facilities Council, and made use of the facilities of HECToR and ARCHER (http://www.archer.ac.uk), the UK’s national high-performance computing service, which was provided by UoE HPCx Ltd at the

University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Oce of Science and Technology through EPSRC’s High End Computing Programme.

5 6 Table of Contents

Abstract 4

Acknowledgements 5

Table of Contents 7

List of Figures 13

List of Tables 19

1 Introduction 21

1.1 Introduction ...... 21

1.2 The Earth’s Inner Core ...... 23

1.3 Seismology ...... 25

1.3.1 Anisotropy and Layering of the Inner Core ...... 26

1.4 The Terrestrial Planets ...... 28

1.4.1 The Moon ...... 28

1.4.2 Mars ...... 29

1.4.3 Mercury ...... 33

1.4.4 Venus ...... 37

1.4.5 Summary of Terrestrial Planetary Cores ...... 38

1.5 The Case For Silicon ...... 38

1.5.1 Silicon in the Earth’s Inner Core ...... 39

1.5.2 Seismological Evidence for Silicon ...... 40

1.5.3 The E↵ect of Si on the Structure of Iron ...... 41

1.6 E↵ect of Nickel ...... 45

1.7 Investigations on the Fe-Ni-Si System ...... 47

2 Computational Methods 49

2.1 Ab initio ...... 49

2.1.1 The Schr¨odinger Equation ...... 50

2.1.2 The Born-Oppenheimer Approximation ...... 50

7 2.1.3 The Many Electron Problem ...... 51

2.1.4 Independent Electrons ...... 51

2.1.5 Indistinguishable Electrons ...... 52

2.1.6 Self-Consistency ...... 53

2.1.7 The Hartree Method ...... 54

2.1.8 Hartree-Fock Theory ...... 55

2.2 Density Functional Theory (DFT) ...... 56

2.2.1 Hohenberg-Kohn Theorems ...... 56

2.2.2 Kohn-Sham Electrons ...... 57

2.2.3 LDA and GGA ...... 58

2.3 Plane Waves and Basis Sets ...... 58

2.4 k-point Sampling ...... 60

2.5 Pseudopotentials ...... 60

2.6 Projector Augmented Wave Method ...... 61

2.7 Ab initio Packages ...... 62

2.8 Static Calculations ...... 63

2.8.1 Geometry Optimisations ...... 63

2.8.2 Birch-Murnaghan Equation of State ...... 65

2.9 High Temperature Calculations ...... 66

2.9.1 Molecular Dynamics ...... 66

2.9.2 Lattice Dynamics ...... 67

2.9.3 Gibbs Free Energy ...... 68

2.9.4 Phase Diagram ...... 71

2.10 Summary ...... 72

3 Experimental Methods 73

3.1 High Pressure Experiments ...... 73

3.1.1 Shock Experiments ...... 74

3.1.2 Compression Experiments ...... 74

3.1.3 The Multi-Anvil Press (MAP) ...... 75

8 3.1.4 The Diamond Anvil Cell (DAC) ...... 77

3.2 O✏ine DAC Melting Experiments ...... 78

3.2.1 Sample Preparation ...... 79

3.2.2 Pressure Determination ...... 82

3.2.3 Heating the Cell ...... 83

3.3 X-Ray and Neutron Di↵raction ...... 85

3.3.1 Bragg’s Law ...... 86

3.3.2 Di↵use Scattering ...... 88

3.3.3 Analysis of Powder Di↵raction Data ...... 89

3.3.3.1 Rietveld Method ...... 89

3.3.3.2 Le Bail Method ...... 90

3.3.3.3 GSAS ...... 91

3.4 Thermal Expansion ...... 91

3.5 Synchrotron Experiments ...... 92

3.5.1 Introduction ...... 92

3.5.2 In situ Experiments ...... 93

3.5.3 ESRF DAC Experiments ...... 94

3.5.4 NSLS MAP Melting Experiments ...... 95

3.5.5 ISIS Equation of State Measurements ...... 95

3.6 Summary ...... 97

4 The Calculated FeSi Phase Diagram 99

4.1 Introduction ...... 99

4.2 The FeSi Phase Transition at 0 K ...... 100

4.3 Static FeSi Calculations ...... 102

4.3.1 VASP Calculations ...... 103

4.3.2 CASTEP Calculations ...... 104

4.3.3 Abinit Calculations ...... 104

4.4 Di↵erences in Transition Pressure ...... 105

4.5 The FeSi Phase Transition at High Temperatures ...... 107

9 4.6 Experiments on the FeSi Phase Transition ...... 107

4.7 Lattice Dynamics Calculations of FeSi ...... 109

4.8 FeSi Phase Diagram ...... 113

4.9 Conclusions ...... 115

4.10 Further Work ...... 115

5 Calculated Stabilities of NiSi-structured Phases in FeSi 117

5.1 Introduction ...... 117

5.2 NiSi Phases ...... 119

5.2.1 The MnP Phase ...... 119

5.2.2 The ‘anti-MnP’ Phase ...... 120

5.2.3 The Pbma-I Phase ...... 120

5.2.4 The WC Structure ...... 121

5.2.5 The Pmmn Phase ...... 121

5.3 VASP Calculations ...... 122

5.4 Conclusions ...... 124

6 The Calculated "-FeSi CsCl Phase Transition in NiSi 127 ! 6.1 Introduction ...... 127

6.2 The "-FeSi CsCl Phase Transition ...... 127 ! 6.3 Lattice Dynamics Calculations ...... 129

6.4 Conclusions ...... 134

7 NiSi Melting 137

7.1 Introduction ...... 137

7.2 Methods ...... 137

7.2.1 O↵-line LH-DAC Melting Experiments ...... 138

7.2.2 In situ LH-DAC Melting Experiments ...... 138

7.2.3 In situ MAP Experiment ...... 139

7.3 The Melting Curve of NiSi ...... 140

7.4 Conclusions ...... 143

10 8 Fe-FeSi Eutectic Melting 145

8.1 Introduction ...... 145

8.2 The Fe-Si Phase Diagram ...... 145

8.3 The Melting Curves of Fe and FeSi ...... 148

8.4 Experimental Methods ...... 150

8.5 The Fe-FeSi Eutectic Melting Curve ...... 152

8.6 Conclusions ...... 156

9 Equation of State for MnP-NiSi 159

9.1 Introduction ...... 159

9.2 The MnP Phase ...... 159

9.3 Neutron Di↵raction Experiments ...... 160

9.4 Birch-Murnaghan Equation of State ...... 164

9.5 Cell Parameters of MnP-NiSi ...... 167

9.6 Fractional Co-ordinates of Ni and Si ...... 170

9.7 Conclusions ...... 172

10 Thermal Expansion of (Fe,Ni)Si Alloys 173

10.1 Introduction ...... 173

10.2 Experimental Details ...... 173

10.3 Results ...... 174

10.4 Conclusions ...... 183

11 Conclusions 185

11.1 Summary of Results ...... 185

11.2 Implications for the Cores of the Terrestrial Planets ...... 186

Appendix A Convergence Tests 191

A.1 VASP Convergence Tests ...... 191

A.2 CASTEP Convergence Tests ...... 192

A.3 Abinit Convergence Tests ...... 193

11 Appendix B Further Work on FeSi 194

B.1 Crystal Structure of the "-FeSi Phase ...... 194

B.2 Equations of State ...... 194

References 199

12 List of Figures

1.1 Comparison of the terrestrial planets ...... 22

1.2 Three-shell model of the Earth ...... 23

1.3 Sound velocities as a function of density for metallic elements ...... 24

1.4 The PREM model ...... 26

1.5 Diagrams of upper and lower inner core of the Earth showing ...... 27

(a) non-uniform layering of inner core ...... 27

(b) observed seismic anisotropy ...... 27

1.6 Data collected by MGS of Mars’ magnetic ﬁeld ...... 30

1.7 Three possible core models for Mars ...... 31

1.8 Areotherm of Mars ...... 32

1.9 Di↵erentiation model of Mars ...... 33

1.10 Comparison of internal structures of the Earth and Mercury ...... 34

1.11 Melting curves of Fe-S alloys ...... 35

1.12 Two possible snowing core models for Mercury ...... 36

1.13 ‘Hit and run’ simulation of the formation of Mercury ...... 37

1.14 V values against density for a range of iron alloys ...... 41

1.15 Premelting e↵ects of VP and VS observed in hcp-Fe ...... 42 1.16 Structure of "-FeSi ...... 43

1.17 Phase diagram of Fe-7.9Si ...... 44

1.18 Phase diagram of Fe-3.4Si ...... 44

1.19 Phase diagrams of Fe, Fe-16Si, Fe-9Si and FeSi ...... 45

1.20 Seismic wave velocities for a variety of Fe-Ni-Si alloys ...... 46

1.21 Phase diagrams for Fe-5Ni, Fe-15Ni and Fe-20Ni ...... 47

2.1 Representation of the Aufbau principle ...... 53

2.2 Di↵erence between a pseudopotential and an all-electron potential wave-

function...... 61

2.3 Potential energy as a function of atomic separation ...... 64

13 2.4 Phase transition of iron (bcc hcp) ...... 65 ! 2.5 Di↵erences between lattice dynamics and molecular dynamics ...... 67

2.6 Total Helmholtz free energies of CsCl-FeSi ...... 70

2.7 Pressure against volume values for CsCl-FeSi ...... 70

3.1 Schematic of a double-stage light-gas gun ...... 74

3.2 The ﬁrst multi-anvil press ...... 75

3.3 Diagram of truncated cubic anvils around octahedral sample cell . . . . 76

3.4 Schematic of D-DIA multi-anvil press ...... 77

3.5 Schematic of a diamond anvil cell ...... 77

3.6 Diagram of a screw-type diamond anvil cell ...... 79

3.7 Princeton-type diamond anvil cell; used for melting experiments . . . . 79

3.8 Fe-FeSi disc; sample used for melting experiments ...... 81

3.9 Image showing cut-out Fe-FeSi discs ...... 81

3.10 Plot showing ruby ﬂuorescence peak ...... 82

3.11 Setup for measuring ruby ﬂuorescence at Bristol ...... 83

3.12 Setup of melting equipment at Bristol ...... 83

3.13 Diagrammatic representation of spectroradiometry ...... 85

3.14 Laser power versus temperature graph ...... 85

3.15 Diagrammatic representation of Bragg’s law ...... 87

3.16 Diagram showing systematic absences of fcc structure ...... 88

3.17 X-ray di↵raction pattern showing di↵use scattering ...... 89

3.18 Neutron di↵raction pattern of epsomite analysed by the Rietveld method 90

3.19 Screenshot of graphical user interface for GSAS ...... 91

3.20 Diagram of synchrotron at ESRF ...... 93

3.21 Set-up of anvils in the MAP, showing X-ray transparent anvils . . . . . 94

3.22 Image of hydraulic press installed at beamline X17B2, NSLS ...... 96

4.1 Structure of ...... 100

(a) CsCl-FeSi ...... 100

(b) "-FeSi ...... 100

4.2 Plots showing the "-FeSi CsCl-FeSi transition of ...... 101 !

14 (a) Voˇcadlo et al. (1999) ...... 101

(b) Caracas and Wentzcovitch (2004) ...... 101

4.3 Plot of enthalpy against pressure showing phase transition of "-FeSi to

CsCl-FeSi as calculated by VASP ...... 103

4.4 Plot of enthalpy against pressure showing phase transition of "-FeSi to

CsCl-FeSi as calculated by CASTEP ...... 104

4.5 Plot of enthalpy against pressure showing phase transition of "-FeSi to

CsCl-FeSi as calculated by Abinit ...... 105

4.6 Phase diagram of FeSi from LH-DAC experiments ...... 108

4.7 Phase diagrams of FeSi determined experimentally by ...... 109

(a) Fischer et al. (2013) ...... 109

(b) Geballe and Jeanloz (2014) ...... 109

4.8 Graphs of thermodynamic properties for CsCl-FeSi and "-FeSi at 200 K

ofthe ...... 112

(a) Helmholtz free energy ...... 112

(b) Pressure ...... 112

4.9 Phase diagram of FeSi with calculated "-FeSi CsCl boundary . . . . . 114 ! 4.10 Calculated phonon dispersion curve of "-FeSi ...... 114

5.1 Calculated phase diagram of NiSi at 0 K ...... 118

5.2 Structure of the MnP phase ...... 119

5.3 Structure of the P bma-I phase ...... 120

5.4 Structure of the WC phase ...... 121

5.5 Structure of the P mmn phase ...... 122

5.6 Plots of enthalpy against pressure for di↵erent FeSi structures ...... 123

5.7 Phase diagram at 0 K of di↵erent transition metal silicides ...... 125

6.1 Phase diagram of NiSi at 0 K, with highlighted "-FeSi to CsCl-FeSi phase

transition ...... 128

6.2 Graphs of thermodynamic properties for "-FeSi and the CsCl phase at

500 K of the ...... 132

15 (a) Helmholtz free energy ...... 132

(b) Pressure ...... 132

6.3 Phase diagram of NiSi with calculated phase boundary of the "-FeSi ! CsCl phase transition ...... 134

7.1 Screencaps of X-ray videography ...... 139

(a) before melting ...... 139

(b) after melting ...... 139

7.2 Temperature versus laser power plot of NiSi and corresponding XRD

patterns showing appearance of di↵use scattering ...... 140

7.3 Di↵use scattering of NiSi seen in XRD pattern from NSLS ...... 141

7.4 Melting curve of NiSi ...... 142

7.5 XRD patterns showing disappearance of P mmn peaks ...... 143

8.1 Phase diagrams of Fe-Si ...... 146

(a) at ambient pressure ...... 146

(b) at 21 GPa ...... 146

8.2 Phase diagrams of ...... 147

(a) Fe-9Si and Fe-16Si ...... 147

(b) Fe-18Si ...... 147

8.3 Fe-Si phase diagrams at 50, 80, 125 and 145 GPa ...... 148

8.4 Melting curve of FeSi ...... 149

8.5 Melting curve of Fe ...... 150

8.6 Laser power versus temperature plot of the Fe-FeSi eutectic at 44 GPa . 151

8.7 Screenshot of 2D analysis software used in melting experiments . . . . . 152

8.8 The Fe-FeSi eutectic melting curve ...... 155

9.1 Fitted neutron di↵raction pattern of MnP-NiSi ...... 161

9.2 The Equation of State of MnP-NiSi ...... 165

9.3 Comparison of computational and experimental EoS of MnP-NiSi . . . . 167

9.4 Relative cell parameters of MnP-NiSi ...... 169

16 (a) as a function of pressure ...... 169

(b) as a function of temperature ...... 169

9.5 Development of the MnP-NiSi structure with pressure ...... 170

9.6 Experimental and calculated fractional co-ordinates of Ni and Si of MnP-

NiSi for ...... 171

(a) Ni x ...... 171

(b) Ni z ...... 171

(d) Si z ...... 171

10.1 X-ray di↵raction pattern of (Fe0.9Ni0.1)Si at 300°C ...... 174

10.2 Thermal expansion of (Fe0.9Ni0.1)Si, (Fe0.8Ni0.2)Si and FeSi ...... 177 10.3 Comparison of unit cell volumes of FeSi with that of Voˇcadlo et al. (2002)178

10.4 Thermal expansion coecient by numerical di↵erentiation ...... 180

(a) for (Fe0.9Ni0.1)Si, (Fe0.8Ni0.2)Si and FeSi ...... 180 (b) for FeSi, from neutron di↵raction ...... 180

10.5 XRD patterns of FeSi ...... 181

(a) showing the appearance of additional peaks at 900°C ...... 181

(b) indexed to identify extra peaks ...... 181

10.6 Volume ratios of (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si to FeSi as a function of temperature ...... 182

11.1 Adiabats of Mercury and Mars plotted alongside the Fe-FeSi eutectic

melting curve ...... 188

A.1 VASP convergence tests for the ...... 191

(a) k-point grid of CsCl-FeSi ...... 191

(b) k-point grid of "-FeSi ...... 191

A.2 CASTEP convergence tests for the ...... 192

(a) k-point grid of CsCl-FeSi ...... 192

(b) k-point grid of "-FeSi ...... 192

17 (c) plane wave cut-o↵ energy ...... 192

A.3 Abinit convergence tests for the ...... 193

(a) k-point grid of CsCl-FeSi ...... 193

(b) k-point grid of "-FeSi ...... 193

B.1 Graphs of experimentally measured EoS for ...... 196

(a) CsCl-FeSi ...... 196

(b) "-FeSi ...... 196

B.2 Graphs of calculated EoS for ...... 198

(a) CsCl-FeSi ...... 198

(b) "-FeSi ...... 198

18 List of Tables

2.1 The pressure and Gibbs free energy of CsCl-FeSi at selected temperatures

and volumes ...... 71

4.1 The calculated transition pressures for the "-FeSi CsCl-FeSi phase ! transition ...... 106

4.2 Calculated transition pressures for "-FeSi CsCl-FeSi ...... 113 !

5.1 Sizes of k-point grids used for the NiSi-structured phases in FeSi . . . . 122

5.2 Fitting parameters for the third-order BMEoS ﬁts of the NiSi-structured

phases in FeSi ...... 124

6.1 NiSi phases found by Wood et al. (2013) ...... 129

6.2 Calculated transition pressures for the "-FeSi CsCl phase transition ! inNiSi...... 133

7.1 Fitting parameters for the Simon-Glatzel equation of NiSi ...... 141

8.1 Melting temperatures from Fe-FeSi melting experiments ...... 153

8.2 Melting Points Attributed to Fe-FeSi Eutectic Melting ...... 154

9.1 Room temperature lattice cell parameters of NaCl and NiSi ...... 162

9.2 The high-temperature, high-pressure lattice cell parameters of NaCl and

NiSi ...... 163

9.3 Fitting parameters for the room temperature Birch-Murnaghan equation

of state of MnP-NiSi ...... 165

9.4 Fitting parameters for the thermal Birch-Murnaghan equation of state

of MnP-NiSi ...... 166

10.1 Unit Cell Volumes of FeSi, (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si ...... 174 10.2 Thermal expansion co-ecients for the (Fe,Ni)Si alloys ...... 179

B.1 Fractional co-ordinates of "-FeSi at room pressure ...... 194

19 B.2 The calculated ﬁtting parameters for the third-order BMEoS ﬁts of CsCl-

FeSi and "-FeSi ...... 197

B.3 The experimental ﬁtting parameters for the third-order BMEoS ﬁts of

CsCl-FeSi and "-FeSi ...... 197

20 CHAPTER 1

Introduction

The stars began to burn through the sheets of clouds

— Mary Oliver

The study of the planets has been driven by a desire to understand the natural world, and the planets closest to our own have held a particular interest. These terrestrial planets are very similar to the Earth in composition and structure, and research into them can inform research into the Earth. Despite this, a lot of questions about Earth and the terrestrial planets are still unanswered. In particular, there is still a lot unknown about the cores of these planets. A relative lack of data on the core, combined with the extreme conditions under which the cores are found, have hindered advances, and the exact composition of the cores of terrestrial planets is still unknown. Deter- mining the composition, structure and dynamics of the planetary cores would aid our understanding of many other processes of the planets, answering questions about the core dynamo and formation of the planets.

1.1 Introduction

The terrestrial planets are so named because of their likeness to the Earth. Mars, Venus and Mercury have a similar composition to the Earth, having formed at roughly the same time nearly four and a half billion years ago. Sharing a similar bulk composition and conditions of formation mean that the internal structure of the terrestrial planets share similarities. As Figure 1.1 shows, each planet possesses a dense metallic core surrounded by a silicate mantle and a thin crust making up the surface of the planet.

The core formed when the much denser metal sank to the centre, separating from the

21 1.1. Introduction lighter silicate material. From consideration of the bulk composition, it is known that the Earth’s core is made up of mainly iron alloyed with a fraction of nickel along and a small percentage of a light element (Birch, 1952). The identity (or identities) of the light element, however, is still unknown.

Figure 1.1: Although the terrestrial planets vary in size they all have the same structural features, with the core making up a di↵erent proportion of each planet (from NASA (2015)).

Research on the composition of the cores of the terrestrial planets has mainly focused on binary Fe-X systems, where the X is a light element such as S, Si, O or C (Lord et al.,

2009; Mookherjee et al., 2011; Asanuma et al., 2008). However, recent experiments and calculations have found that under some conditions relevant to planetary cores, nickel has a signiﬁcant e↵ect on the physical properties of the iron system (Mao et al.,

2006; Voˇcadlo et al., 2006; Martorell et al., 2013b), and therefore cannot be left out of investigations on the properties of the core. Nickel has long been accepted to be present in the cores of terrestrial planets, from consideration of the Fe/Ni ratio in chondritic meteorites (Birch, 1952; McDonough and Sun, 1995). There have been a number of studies investigating the ternary Fe-Ni-X systems (Antonangeli et al., 2010; Narygina et al., 2011; Sakai et al., 2011). However, these studies have tended to focus on speciﬁc candidate core compositions within the ternary system. Since the precise proportions in the planetary cores are not known deﬁnitively, these studies have an inherent limitation.

More useful, instead, would be a fuller understanding of the ternary system, ideally with data and calculations mapping the entire phase diagram of the ternary system. This is the approach adopted in this work, used to study the Fe-Ni-Si system. Silicon has long been considered a likely light element in the Earth’s core, having ﬁrst been suggested as a candidate by Birch (1952) due to its abundance in the Earth and the fact that only a small percentage is required to lower the density suciently (see Section 1.2).

22 Chapter 1. Introduction

Silicon is still widely considered as a possibility for the light element present in the cores of the terrestrial planets (see Section 1.5) and it is for this reason that this work focuses on the Fe-Ni-Si system.

1.2 The Earth’s Inner Core

Of all the terrestrial planets, the Earth’s core is most understood, primarily because of the ability to directly observe the Earth’s core using seismic waves. The Earth’s inner core was ﬁrst discovered by Inge Lehmann in 1936 (see Figure 1.2). Lehmann noticed abnormally large amplitudes in the P-wave shadow zone, which could only be explained by the presence of a solid inner core. Assuming a simple three-shell model of the Earth, Lehmann calculated travel times for these waves, ﬁnding that they matched extremely well with those observed (Bolt, 1987), proving her hypothesis.

Figure 1.2: Lehmann’s theory and proof of the three-shell model of the Earth explaining the apparently anomalous seismic waves. Shown are the PKIIKP and PKiKP seismic waves. Figure taken from Bolt (1987).

After the initial discovery of the Earth’s inner core, the focus turned to what the core was made of. Measurements carried out by Birch (1937) revealed that the velocity of seismic waves was a↵ected by the density of the material. A comparison of the seismic wave velocities of the Earth’s core and that of iron showed a mismatch (see Figure 1.3) meaning that the Earth’s core cannot be made of just iron. Further investigation revealed that the density of the core was approximately 10 wt.% lower than that expected for pure iron (Birch, 1964). This density deﬁcit could only be explained by the presence

23 1.2. The Earth’s Inner Core of light elements, such as silicon, sulphur or oxygen, in the core.

Figure 1.3: Sound velocities as a function of density for metallic elements and the Earth’s mantle and core from Birch (1937). The closest match for the density of the core is iron, but a core consisting only of iron would not produce the correct sound velocity.

Conﬁrming the identity of the light element is dicult since there is no way to directly sample the Earth’s core. However, there are some pieces of information that can aid in identiﬁcation such as, for example, details about core formation. It is generally agreed that the core formed during early accretion of the Earth, therefore the light element in the core must have been able to partition into liquid iron at relatively low pressures.

In addition, the light element must have been suciently abundant during this time so that even after volatilisation by the heat generated during accretion it did not escape completely. The current status of the Earth’s core must also be taken into account. For example, the continuing crystallisation of the inner core is thought to drive convection in the outer core, creating the geodynamo responsible for the Earth’s magnetic ﬁeld.

For this to occur some of the light element must be released into the melt during crystallisation. Details such as these all provide clues as to the identity of the light element in the core, but for direct observables, we must turn to seismology.

24 Chapter 1. Introduction

1.3 Seismology

Other than satellite data, seismology o↵ers the only way of directly observing the

Earth’s core. Earthquakes produce seismic waves that travel through the Earth, known as body waves. Measuring these waves gives information about the material they have travelled through. The two most useful waves for observing the deep interior of the

Earth are the P- and S-waves. P-waves can travel through both solids and liquids and are observed ﬁrst, while the secondary S-waves can only travel through solids.

The velocities of the P- and S-waves are dependent on the density of the material. The velocity of the P-wave, vp, is given by

4 K + 3 µ vp = (1.1) s ⇢ where K is the bulk modulus, µ is the shear modulus and ⇢ is the density of the material.

The S-wave velocity, vs, is dependent only on the shear modulus and density

µ v = (1.2) s ⇢ r

Therefore, measurements of the seismic velocities can yield information about important physical properties of the material the waves are travelling through. In addition to the body waves, seismic activity also causes the whole Earth to oscillate. These free oscillation waves are known as normal modes and can provide invaluable information on the whole Earth, including the Earth’s core.

From the seismic wave data collected, a seismic model can be generated. One of the ﬁrst such models, the Preliminary Earth Reference Model (PREM), created by Dziewonski and Anderson (1981), is still widely used today. PREM combined a large data set, generating a reference model for the Earth of bulk elastic properties as a function of depth

25 1.3. Seismology

(see Figure 1.4). Although such a model does not give direct information on the composition of the Earth, the model can be used for comparison with candidate materials since it is derived from observed data. Suggested compositional models for the Earth can therefore be tested for robustness against seismic models such as PREM.

Figure 1.4: The PREM model of Dziewonski and Anderson (1981).

1.3.1 Anisotropy and Layering of the Inner Core

One of the key results to come from seismological studies of the Earth’s core is the observed anisotropy of the inner core. It has been observed that P-waves travel about

3 % faster along the polar axis than in the equatorial plane (Poupinet et al., 1983;

Beghein and Trampert, 2003; Creager, 1992; Oreshin, 2004). Originally, this anisotropy was thought to extend across the whole of the inner core, with cylindrical symmetry that has its symmetry axis aligned with the Earth’s rotational axis. However, recent seismological observations have shown that the inner core is not uniformly anisotropic

(Morelli et al., 1986; Woodhouse et al., 1986; Song and Helmberger, 1998; Ishii and

Dziewoski, 2003). Instead, the inner core is split into two parts, consisting of an irregularly shaped central region, termed the lower inner core, which exhibits seismic anisotropy, and an upper layer, which is isotropic or weakly anisotropic, surrounding this (see Figure 1.5a). The most recent seismic studies (Wang et al., 2015) indicate

26 Chapter 1. Introduction that, contrary to previous work, the fast direction in the lower inner core lies in the equatorial plane (see Figure 1.5b). Hemispherical variations in anisotropy have also been observed seismically (Wang et al., 2015; Sun and Song, 2008) with the western region exhibiting stronger anisotropy at greater depths compared to the eastern region

(see Figure 1.5b).

(a) (b)

6*$

-*$ "YJT -*$

Figure 1.5: Illustrations of the upper inner core and lower inner core showing (a) the non- uniform layering seen in the inner core and (b) the seismic anisotropy observed. Blue lines depict the magnitude of anisotropy and alignment of the fast direction. The upper inner core (UIC) is only weakly anisotropic, with the fast direction aligned North-South, but showing hemispherical variations – stronger anisotropy is observed at greater depths in the western part. Recent seismic studies by Wang et al. (2015) suggest that the lower inner core (LIC) has the fast direction aligned equatorially. Figure (a) after Song and Helmberger (1998) and Figure (b) after Wang et al. (2015).

The isotropic, or weakly anisotropic, upper layer is seen to vary in thickness from between 100 to 400 km (Shearer, 1994; Tanaka and Hamaguchi, 1997; Song and Helm- berger, 1998; Ishii and Dziewoski, 2003; Ouzounis and Creager, 2001; Sun and Song,

2008). This observed layering raises the question as to whether there are two composi-

tionally or structurally distinct areas in the inner core. The presence of seismic isotropy

and anisotropy could give information about the structure of the material in the core.

Seismic isotropy would be exhibited by a phase with an intrinsically low anisotropy, or

alternatively it could be explained by having a random orientation of crystals of any

phase, regardless of any intrinsic anisotropy, such as the body-centred cubic (bcc), face-

27 1.4. The Terrestrial Planets centred cubic (fcc) or hexagonally-close packed (hcp) structures. On the other hand, anisotropy may come from the inherent structure of a phase, or when there is preferred orientation of aligned crystals, either of the same phase as in the upper isotropic layer, or an entirely di↵erent phase of iron. Therefore the correct composition for the inner core must also have the right structure to match the anisotropy seen in the seismic data.

Since the addition of light elements and nickel to iron a↵ects the structure adopted (see

Section 1.5) it is important to understand the Fe-Ni-Si system as a whole rather than just the pure iron system.

1.4 The Terrestrial Planets

Other than the Earth and the Moon, seismic data are unavailable for the terrestrial planets, although there have been plans for seismic stations on Mars and Venus. Despite this, exploratory missions to the planets, such as the MESSENGER mission to Mercury and the Mars Global Surveyor, have yielded data from direct observations of the planets.

In addition, since formation for the terrestrial planets followed a similar process as the

Earth (Morbidelli et al., 2012), our understanding of the deep interior of the Earth and its core can inform our understanding of the terrestrial planets and can be used to provide a preliminary model for their composition and structure.

1.4.1 The Moon

Although there is no seismic data available for the terrestrial planets, there is seismic data available for the Moon. This data was gathered from the four-station seismic network established on the moon during the Apollo lunar landing missions (Nakamura et al., 1982). Using the collected seismic data, along with other measurements from the

Apollo missions such as remote sensing, surface exploration and sample return, some key observations about the Moon have been made. Anderson et al. (1970) first proposed that a large part of the Moon was initially molten, from which mafic cumulates crys- tallised out. These then make up the mantle, and plagioclase flotation cumulates rise to form the crust. This hypothetical structure of the Moon is now generally accepted as correct (Wieczorek, 2006). However, the structure of the lunar core is controversial.

28 Chapter 1. Introduction

Although indirect geophysical observations support the existence of a metallic core in the Moon (Wieczorek, 2006; Konopliv, 1998; Williams et al., 2001), key features such as the radius, composition and whether the core is partially molten or solid, remain unclear. Additionally, there has only been one seismic study on the lunar core (Weber et al., 2011). This study suggested that the lunar core consists of a solid inner and

ﬂuid outer core, much like Earth, with the liquid outer core making up approximately

60 % by volume of the total core. As with the Earth, a precise determination of the composition and structure of the lunar core is necessary for an understanding of the

Moon as a whole, and is especially important for understanding the dynamo.

1.4.2 Mars

Out of all the terrestrial planets, Mars has been subject to the most missions, many of which have received signiﬁcant media coverage. These have included rovers, including the 1997 Mars Pathﬁnder Sojourner and NASA’s current Curiosity rover (Bell, 2012), as well as several orbiters of the planet. Much media attention has focused on the rovers, which are on the lookout for evidence of extra-terrestrial life, but the orbiters provide invaluable data and insight into the deep interior of the planet.

The Mars Global Surveyor (MGS) was an orbiter launched by NASA in 1996, staying in orbit four times longer than originally planned and becoming one of the most productive missions to Mars. In addition to the much reported discovery of water on Mars – and therefore the possibility of life – the MGS also provided magnetic ﬁeld observations of the whole planet. It was found that Mars does not have a global magnetic ﬁeld like the

Earth. Instead, localised areas of strongly magnetised ancient crust was detected by the

MGS (see Figure 1.6). The crustal magnetic ﬁeld detected is much stronger than that of the Earth, with the strongest magnetisations an order of magnitude larger than the

Earth’s. The large areas of magnetised crust suggest that the magnetisation measured is thermal remanence of a global magnetic ﬁeld obtained during cooling through the blocking temperature.

For a global magnetic ﬁeld to have been present on Mars, a dynamo must have once operated, that has now ceased to be active. Since Mars was geologically active only

29 1.4. The Terrestrial Planets

Figure 1.6: Data collected by the MGS of Mars’ magnetic ﬁeld. The Southern hemisphere contains strong localised anomalies. Figure from Stevenson (2001).

during its early history, the early chemical and isotopic signatures of its di↵erentiation process remain and therefore research into Mars can shed light onto the di↵erentiation process for the Earth. Using data from the MGS, Yoder et al. (2003) calculated the tidal Love number, k ,tobe0.153 0.017, indicating at least a partially liquid core 2 ± since the Love number is greater than 0.10, the value required for a liquid core to be ⇠ present. However, this does not answer the question of whether a solid inner core is present. For core convection to occur, and therefore generate the core dynamo, there either has to be large heat flow, which can be dicult to obtain, or the presence of a solid inner core. There are three possible core models that satisfy the original presence of a core dynamo that has since ceased to operate (see Figure 1.7). The first is of a completely liquid core, where the large heat flow is obtained from the monotonic cooling of the Martian mantle. Core convection eventually stops when mantle cooling stops. The second model is one of a solid inner core inside a liquid outer core, such as in the Earth, but with a stagnant lid. Convection then occurs as in the Earth, with the solid inner core growing to the point where the outer core is too small to sustain a core dynamo. The third model is the plate tectonic model, exactly as on the Earth, but with the core dynamo ceasing when plate tectonics ceases.

30 Chapter 1. Introduction

Figure 1.7: Illustrations of three possible core models that ﬁt the current data obtained about Mars. Figure from Stevenson (2001).

The presence of at least a partially liquid core indicates that the Martian core contains a proportion of a light element. The estimated temperature proﬁle of Mars, known as the areotherm, is lower than the melting curve of pure iron (see Figure 1.8), and since light elements are known to depress the iron liquidus (Fei, 1997; Fischer et al.,

2012; Terasaki et al., 2011), it follows that the Martian core must be made up of some percentage of a light element. This would have implications for the state of the core, since di↵erent light elements a↵ect the melting curve by di↵erent amounts. For example, sulphur depresses the melting point of iron very strongly (Fei, 1997), whereas

silicon has a much smaller e↵ect (Fischer et al., 2013). Which light element is present,

and in what quantities, would inﬂuence core formation and whether a solid inner core

is present. Nickel, too, is also likely to a↵ect the melting curve of iron, although the

e↵ect may not be large (Lord et al., 2014).

Other than the Earth, Mars is the only planet for which material is available for direct

analysis in the laboratory, although data from Martian missions shows that meteorites

do not sample all the rocks on Mars. Martian meteorites, named SNC meteorites for

the three areas where these meteorites were found – Shergotty, Nakhla and Chassigny

31 1.4. The Terrestrial Planets

Figure 1.8: A graph showing the estimated temperature proﬁle of Mars, known as the areotherm, determined through consideration of thermal evolution models of Mars based on data captured by the Mars Global Surveyor (Fei and Bertka, 2005).

– all have iron contents less than that of chondritic meteorites, implying that the core consists primarily of iron, as for the Earth. However, SNC meteorite studies also indicate that the Martian mantle is far richer in iron than the Earth’s mantle, in the form of FeO in the mantle (Halliday et al., 2001). This implies that Mars has a smaller metallic core than the Earth, with recent estimates putting the core radius at 1794

65 km (Rivoldini et al., 2011). The smaller metallic core and higher Fe content in ± the Martian mantle indicate that core formation occurred in oxidising conditions. A possible model of core formation in Mars is shown in Figure 1.9.

Finally, being much smaller than the Earth, the pressure within Mars at any given depth is only about one third that of the Earth. This has implications for the mineral assemblages found in Mars, which may be quite di↵erent to those found in the Earth.

For example, the presence of light elements, even in small amounts, have a greater e↵ect at lower temperatures and pressures than at the core conditions of the Earth (see

Section 1.5).

32 Chapter 1. Introduction

Figure 1.9: A possible model of the main di↵erentiation events occurring on Mars, including the evolution of a Martian magma ocean. The metallic core, shown in black, is formed by segregation from the silicate portion of the mantle. The magma ocean then crystallises from the bottom up (shown in the middle image). The dense Fe-rich cumulates that result are found on top of the less dense, Mg-rich cumulates, which results in mantle overturn as seen in the picture on the right. Diagram from Mezger et al. (2012).

1.4.3 Mercury

There have been only two spacecraft sent to Mercury, the recent MESSENGER mission and the previous Mariner 10 mission. The Mariner 10 spacecraft ﬂew past the planet three times in 1974 and 1975, gathering data on less than half the surface of the planet.

The MESSENGER mission however, orbited Mercury 4105 times and has returned over

10 terabytes of data on the planet. The flybys of Mariner 10 established the existence of an intrinsic magnetic field in Mercury; data from MESSENGER confirmed the dipolar nature of the magnetic field (Anderson et al., 2008). The dipolar nature of Mercury’s magnetic field suggests that a core dynamo, such as that in the Earth, is present in the planet. However, Mercury has a much weaker magnetic field than the Earth, which is dicult to explain with an Earth-like dynamo, and various dynamo models have been proposed (Heimpel et al., 2005; Wicht et al., 2007; Schubert and Soderlund,

2011). These models have reproduced the weak magnetic field but not the observed asymmetry of the magnetic field. Recent work suggests that in order to produce the asymmetry observed in the magnetic field, a dynamo driven by thermal buoyancy, in

33 1.4. The Terrestrial Planets addition to local excess equatorial heat ﬂow at the core-mantle boundary, is required

(Cao et al., 2014). This is in contrast to the Earth’s dynamo, which is primarily driven by chemical buoyancy as well as the release of latent heat of fusion due to inner core growth.

Mercury also has a very large inner core (see Figure 1.10). Moment of inertia data measured by MESSENGER reveal that Mercury is extremely dense (Smith et al., 2012).

Most recent analysis of MESSENGER data puts the radius of the core of Mercury at

2020 30 km, over 80% of the radius of the planet, with an average core density of ± 6980 280 kg m-3 (Hauck et al., 2013). If such a large core were made entirely of iron ± then it would currently be solid, based on thermal history considerations (Siegfried and Solomon, 1974), which is at odds with the core dynamo model of Mercury and observations of at least a partially liquid core from libration measurements (Margot et al., 2007). For a liquid core to exist, the iron must be alloyed with a proportion of light element, although, as in the Earth, the identity of this light element is unknown.

Figure 1.10: A comparison of the internal structure of the Earth and Mercury from Stevenson and David (2012). Note that Mercury’s core makes up the majority of the interior, unlike the Earth, which has a mantle consisting of two-thirds of the total planetary mass.

A lot of work has focused on sulphur being the light element in Mercury’s core due to its cosmochemical abundance and the fact that sulphur strongly depresses the melting curve of iron. However, the highly reducing conditions suggested by the high abundance

34 Chapter 1. Introduction of sulphur and low abundance of iron in the silicate part of Mercury would indicate that silicon is more likely to alloy with iron in the core either with or instead of sulphur

(Malavergne et al., 2010). A Fe-Si-S alloy would also work with the buoyancy dynamo model suggested by Cao et al. (2014). Experiments have shown that the Fe-S-Si alloy exhibits liquid immiscibility at pressures less than 15 GPa (Morard and Katsura, 2010).

A core made of a Fe-S-Si alloy would therefore segregate into a S-rich liquid which ﬂoats to the top of the core, giving the necessary buoyancy to generate the core dynamo.

Research on core composition and theories of core formation and evolution are mutually dependent on each other – understanding core formation, as well as how the core is currently evolving, can aid in determining the core composition and vice versa. As with the other terrestrial planets, it is believed that the metallic core formed from gravitational separation of the more dense iron from the silicate part. One suggested model for core evolution in Mercury is the ‘snowing core’ model. Chen et al. (2008) found that the melting curves of Fe-S alloys exhibit two inﬂection points – at 14 GPa and 21 GPa – that coincide with phase transitions to Fe3S2 and Fe3Srespectively (see Figure 1.11). The presence of these inﬂections mean that the the cooling adiabat

Figure 1.11: Melting curves measured by Chen et al. (2008) for a range of Fe-S alloys. All exhibit two inﬂections (at 14 GPa and 21 GPa) allowing the adiabat to cross the liquidus more than once for di↵erent sulphur concentrations. of Mercury’s core may intersect the melting curve at multiple points. It is therefore

35 1.4. The Terrestrial Planets possible for solid to precipitate at a range of depths within Mercury’s core, depending on the sulphur content. At shallow depths, solid precipitation is in the form of ‘snow’

– solid precipitates out, sinks and accumulates at the centre to form the core. Chen et al. (2008) estimate the present-day temperature of Mercury’s core to be between

1700 and 1900 K, with an adiabatic gradient of 11 K/GPa. Assuming this, there are three possible core states for Mercury – a Ganymede-like state, a double-snow state

(see Figure 1.12) or an Earth-like state. If the core contains 7 – 8 wt. % sulphur, solid forms at shallow depths (Ganymede-like state); a core containing 8 – 10 wt. % sulphur would have solid forming at low and high pressures simultaneously (double-snow state); if the core contains 12 wt. % sulphur, freezing occurs at high pressure only (Earth-like state). These core formation models, however, are dependent on Mercury’s core being solely composed of a binary Fe-S alloy. Other light elements, as well as the addition of nickel, are likely to a↵ect the melting curve.

Figure 1.12: Two of the possible suggested states of Mercury’s core, as proposed by Chen et al. (2008). The double-snow state has two distinct zones where solid precipitation (in yellow) occurs, whereas the Ganymede-like state only has solid precipitation occurring at shallower depths.

More recently, it has been suggested that the ‘hit and run’ model which is commonly accepted for Moon formation, can also be used to explain the formation of Mercury

(Asphaug and Reufer, 2014). This theory suggests that proto-Mercury had its mantle stripped in a high-speed collision with a larger target planet, which could possibly explain the large iron core found in Mercury. Numerical hydrocode simulations show that this type of collision can produce a Mercury-like planet (see Figure 1.13).

36 Chapter 1. Introduction

Figure 1.13: One solution of the numerical hydrocode simulations carried out by Asphaug and Reufer (2014) of two planetary bodies after a collision, showing a Mercury-like remnant many hours after colliding.

1.4.4 Venus

Unlike Mercury, Venus has been explored by over 40 spacecraft and the Magellan mission has managed to map approximately 98 % of the planet. However, although there have been multiple orbiters, atmospheric probes and landers that have explored

Venus (Sengupta, 2010), the lack of a bulk of comprehensive data on Venus has inhibited the research e↵ort into understanding Venus, particularly its deep interior.

Venus is very similar to the Earth in terms of size, density, mass and bulk composition during formation. However, there are some key di↵erences between Venus and the

Earth. For example, Venus lacks plate tectonics and also has no measurable magnetic

ﬁeld. It has been suggested that the lack of plate tectonics is the cause of the lack of magnetic ﬁeld. Nimmo (2002) suggests that in the absence of plate tectonics, the mantle cannot cool fast enough to provide a driving force for core convection and generate a dynamo. The lack of plate tectonics could also point to the absence of a solid inner core, since the inability to take heat away from the core quickly enough

37 1.5. The Case For Silicon would result in a higher core temperature at the same pressures of the Earth, meaning the inner core cannot solidify. Calculations of the tidal Love number, k2, using data from Magellan and the Pioneer Venus Orbiter also reveal that the core must be at least partially molten (Konopliv and Yoder, 1996). However, problems with the slow spin rate of Venus combined with its dense atmosphere means that measurements which would shed light on the interior of Venus are not well constrained (Mocquet et al.,

2011). Therefore the most conservative model for the deep interior of Venus is still only a scaled version of the Earth’s deep interior. Further data on Venus is required to draw deeper conclusions on its history and the state of its core.

1.4.5 Summary of Terrestrial Planetary Cores

It can be seen from these discussions that there are still many aspects that remain unknown about the cores of terrestrial planets. It is also clear that a deeper understanding of core composition would help resolve a lot of the mysteries surrounding the cores of these planets. Although the current focus has been on sulphur as the light element in the cores of the terrestrial planets, there has been no evidence to deﬁnitively rule out other light elements. Indeed, silicon is now considered likely in Mercury’s core (see

Section 1.4.3). In any case, a larger database of knowledge on the ternary Fe-Ni-X systems likely to make up the cores of terrestrial planets would be highly useful, especially since current theories may need to be adapted as newer data on the planets becomes available.

1.5 The Case For Silicon

From the discussions so far, it is clear that despite major advances in research on the terrestrial planets, a lack of data makes it dicult to ﬁrmly establish the identity of the light element in the planetary cores. In this section, the evidence for silicon as the light element is considered. Although this has been discussed with regards to the Earth’s core – since there is greater availability of data on the Earth – the similarities between the Earth and the terrestrial planets, as well as the Moon, mean that the evidence presented here is also widely applicable to the other terrestrial planets.

38 Chapter 1. Introduction

1.5.1 Silicon in the Earth’s Inner Core

Silicon has long been considered a potential light element in the Earth’s core. Since the core is surrounded by a silicate mantle, the required abundance of silicon for it to be in the core is satisﬁed. In addition, the mantle is ‘missing’ a small percentage of silicon if the Earth is to have a chondritic composition (MacDonald and Knopo↵ , 1958;

Ringwood, 1961). Having silicon in the core would solve this problem.

In addition to being abundant enough, the light element must be congruent with core formation. However, a core formation model is still not agreed upon. There is still de- bate surrounding whether the core formed in reducing or oxidising conditions. Previous work has suggested that core formation could only occur with little or no oxygen present

(Wood et al., 2006; Huang et al., 2011). A highly reducing environment would result in a signiﬁcant proportion of silicon being incorporated into the core (Malavergne et al.,

2004). More recent work has suggested more oxidising conditions for core formation

(Siebert et al., 2013). However, this does not rule out silicon being in the core; it has been found that the presence of oxygen inﬂuences silicon partitioning, and causes more silicon to partition into the metal (Tsuno et al., 2013). Therefore it seems likely that silicon would be present in the core regardless of whether the core formed in reducing or oxidising conditions.

Another method for determining the composition of the inner core is to consider the partitioning of Fe-X alloys between the inner and outer core. If the solid inner core and liquid outer core are in thermodynamic equilibrium at the inner core boundary (ICB), then the chemical potentials of each species will be equal at that point. The ratio of the concentrations of elements in the liquid and solid phase is then fixed, which in turn fixes the density ratios. Since the inner core and outer core densities must match that determined by seismic data, the proportion of the light element in the core can be determined. Chemical potential calculations carried out by Alfèet al. (2002) show that both silicon and oxygen partitions preferentially from solid to liquid, but only weakly so for silicon and much more strongly for oxygen. A binary system of either Fe/Si or Fe/O is ruled out because the calculated percentages of either light element does

39 1.5. The Case For Silicon not match the ICB density discontinuity seen seismically – the percentage of silicon is too low and the percentage of oxygen too high. However, a mixture of silicon and oxygen together can account for the seismic data, but since oxygen partitions strongly to liquid, this requires less than 1 % oxygen in the solid inner core, and 8 % in the liquid outer core. In contrast, there would be about 8 % silicon in the inner core and 10

% in the outer core. Previously, the presence of both silicon and oxygen was thought to be incongruous since silicon is a known reducing agent. However, the work of Takafuji

(2005); Ricolleau et al. (2011); Tsuno et al. (2013) suggests that this incompatibility may not be as problematic as once thought. Their work indicates that a small increase in temperature, from 3000 K to 3500 K, is enough to raise the levels of silicon and oxygen in liquid iron to above that required to compensate for the density deﬁcit seen seismically.

1.5.2 Seismological Evidence for Silicon

In addition to inferred density measurements, the velocities of seismic waves can be matched to those of mineralogical models in order to identify the light element in the core. Seismic wave velocities can be measured experimentally or calculated using computational techniques for di↵erent candidate compositions. Experimental and computational work both show that the shear wave velocity of pure hcp-Fe is higher than that seen seismically (Antonangeli et al., 2004; Voˇcadlo, 2007). Calculations on

Fe0.875Si0.125 show that a small amount of silicon reduces the shear wave velocity by 15 % in the hcp phase of iron at 360 GPa (Tsuchiya and Fujibuchi, 2009). These calculations found a linear relationship between seismic velocity and density – behaviour which is also observed in FeSi experimentally and computationally (Badro et al., 2007;

Voˇcadlo, 2007). However, recent work by Mao et al. (2012) suggests that a power-law function may be more suitable for describing the velocity-density relationship. Mao et al. (2012) also analysed the V ⇢ data for a wide range of iron alloys (see Fig- ure 1.14). From this, it can be seen that there are many possible compositions that would satisfy the seismically measured velocity data, meaning that using this method alone is not enough to determine the composition of the Earth’s core.

40 Chapter 1. Introduction

Figure 1.14: Graph showing V, the bulk sound velocity, reproduced from Mao et al. (2012). Solid lines, except the black and grey lines, are the V of Fe alloys calculated from static compression results at 300 K. Blue line indicates hcp-Fe, orange line indicates Fe3S, violet line indicates Fe0.85Si0.15, dark cyan line indicates Fe3C, red solid and dashed lines indicate FeO in the B1 and B8 phase, respectively, navy blue line indicates FeSi, purple line indicates FeS in the IV phase, magenta and olive circles stand for V of Fe0.9O0.08S0.02 and Fe0.925O0.053S0.022 from shock-compression studies respectively. For references and further details see Mao et al. (2012).

Very recent work has suggested that pre-melting behaviour in pure iron is sucient to explain the discrepancy in VS between mineralogy and seismology. Calculations carried out by Martorell et al. (2013a) show that above 6600 K, some of the elastic constants of hcp-Fe decrease with temperature, resulting in both VP and VS decreasing to values that match those seen seismically (see Figure 1.15). The density deﬁcit still requires a small percentage of light element, which would likely decrease the melting temperature, bringing the softening e↵ect to a temperature more reasonable for core conditions. In any case, it is clear that although much work has been done to pin down the seismic behaviour of iron and iron alloys, more research is necessary to fully understand the behaviour of seismic waves in mineralogical models.

1.5.3 The E↵ect of Si on the Structure of Iron

It has been found, in both experimental and computational studies, that the addition of a light element increases the stability field of the bcc structure of iron (Côtéet al., 2008;

Lin et al., 2002; Sata et al., 2008; Voˇcadlo et al., 1999). The stable high pressure phase

41 1.5. The Case For Silicon

Figure 1.15: Calculated VP and VS velocities for hcp-Fe as a function of T/TM and simulation temperature at 360 GPa. The grey band represents minimum and maximum melting temperatures for hcp-Fe. Figure from Martorell et al. (2013a). of FeSi is known to have the CsCl structure, which itself is topologically equivalent to the bcc structure. The low-pressure phase of FeSi is the "-FeSi structure, a slightly modiﬁed form of an idealised seven-fold co-ordinated structure (see Figure 1.16). The pressure at which the "-FeSi structure transforms to the high-pressure CsCl-FeSi structure has been investigated experimentally and computationally. In computational studies, the phase transition is found to occur at 13 GPa (Voˇcadlo et al., 1999), 20 GPa (Zhang and Oganov, 2010) and 40 GPa (Caracas and Wentzcovitch, 2004). The reason for the discrepancy between these studies is not known deﬁnitively, and has been investigated in Chapter 4. Experimentally, the transition to CsCl-FeSi has been seen at 24 GPa, at temperatures greater than 1950 50 K (Dobson et al., 2002), although the experimental ± phase boundary of this transition is also in dispute (see Chapter 4).

As noted above, experiments have found that even a small percentage of silicon increases the stability ﬁeld of the bcc phase of iron. Lin et al. (2002) found that 7.9 wt.% Si can stabilise the bcc phase up to about 40 GPa, forming a large two phase region of bcc

+ hcp (see Figure 1.17). This two phase region, consisting of a silicon-rich bcc phase and a silicon-poor hcp phase, suggests that the inner core may consist of a mixture of

42 Chapter 1. Introduction

Figure 1.16: A diagram of the "-FeSi structure that is stable at low pressures. Silicon is represented by small light grey spheres and iron by darker spheres. The structure is formed from small displacement of the Fe and Si atoms from the ideal seven-fold co-ordinated structure, resulting in one slightly shorter Fe-Si bond, three intermediate length bonds and three slightly longer bonds. From Voˇcadlo et al. (1999).

two iron phases. Further work with an Fe0.85Si0.15 sample found the same two phase mixture persists to 150 GPa and 3000 K, but between 170 GPa and 240 GPa, only the hcp phase is stable (Lin et al., 2009). However, experiments carried out by Asanuma et al. (2008) with a lower percentage of silicon – 3.4 wt.% Si – found that the fcc phase is stabilised instead at high temperatures and pressures (see Figure 1.18), creating a two phase fcc and hcp stability ﬁeld.

Computer simulations have also found that silicon stabilises certain structures in iron.

Calculations carried out at 0 K by Côtéet al. (2008) found that even small amounts of silicon (3.2 % Si) stabilises the bcc phase over the hcp phase. An increasing amount of silicon has the e↵ect of increasing the stabilising e↵ect of bcc. However, lattice dynamics calculations on iron alloyed with silicon found that the addition of about 7 wt.% Si to iron stabilises the fcc phase at high temperature (Côtéet al., 2010), in agreement with

Asanuma et al. (2008); whereas calculations by Zhang and Oganov (2010) on a range of FexSi(1-x) phases, using a structure searching method, ﬁnd that only the CsCl phase of FeSi is stable at high pressures, with all other silicides decomposing to Fe + FeSi.

More recent experiments by Fischer et al. (2012, 2013) have found that iron with 9 wt.% Si creates a two phase CsCl-FeSi and hcp stability region at high pressures and

43 1.5. The Case For Silicon

Figure 1.17: Phase diagram measured by Lin et al. (2002) showing the di↵erent stability ﬁelds for bcc, hcp and fcc structured iron in the presence of 7.9 wt.% Si.

Figure 1.18: The phase diagram measured by Asanuma et al. (2008) of iron with 3.4 wt.% Si showing the fcc phase persisting at high temperatures and pressures compared to the pure Fe phase diagram.

44 Chapter 1. Introduction temperatures, and increasing the amount of Si to 16 wt.% increases this two phase stability ﬁeld (see Figure 1.19).

Figure 1.19: Phase diagrams for a range of compositions in the Fe-FeSi systems showing the stability ﬁeld of the B2 (CsCl-FeSi phase) and hcp phase increasing with an increasing amount of silicon added. Figure from Fischer et al. (2013).

1.6 E↵ect of Nickel

Previously it was generally assumed that since the amount of nickel alloyed to the iron in the core was small it wouldn’t have a signiﬁcant e↵ect on the properties of the core.

Additionally, nickel has a similar density to iron, making the two indistinguishable seismically. Indeed, experiments carried out by Antonangeli et al. (2010) to measure the seismic velocities of Fe0.89Ni0.04Si0.07 found that the addition of a small percentage of nickel has negligible e↵ects on the seismic velocities (see Figure 1.20).

Experiments on Fe0.78Ni0.22 also show no signiﬁcant deviations from the elastic properties of pure iron (Kantor et al., 2007). Computational studies have shown that the negligible e↵ect of nickel is due to the high temperature of the system rather than chemical similarities between iron and nickel as previously thought (Martorell et al.,

2013b), indicating that more work needs to be done to fully understand the e↵ects of nickel. Experiments have also shown that nickel inﬂuences the structure adopted by

45 1.6. E↵ect of Nickel

Figure 1.20: Seismic wave velocities for a range of Fe-Ni-Si alloys. Adding a signiﬁcant percentage of Ni to both pure-Fe and Fe-Si appears to have a negligible e↵ect on the seismic wave velocity. From Antonangeli et al. (2010). iron. Mao et al. (2006) carried out laser heated DAC experiments on Fe-Ni alloys and found that nickel has the e↵ect of stabilising the fcc phase to lower temperatures and higher pressures (see Figure 1.21). Experiments by Komabayashi et al. (2012) on Fe-Ni alloys with up to 15 wt.% Ni show the same e↵ect on stability of fcc. In addition, experiments that reached higher temperatures and pressures – 340 GPa and 4700 K – on iron alloyed with 10 wt.% Ni found that the hcp phase was most stable throughout, including up to core conditions (Tateno et al., 2012).

Ab initio calculations, too, have shown that a small amount of nickel stabilises the hcp structure over the bcc structure by about 20 GPa (Voˇcadlo et al., 2006). More recent calculations have shown that the fcc phase is stabilised by the addition of nickel at core conditions (Cˆot´eet al., 2012).

46 Chapter 1. Introduction

Figure 1.21: Phase diagram for a range of Fe-Ni alloys, showing that an increasing amount of nickel has the e↵ect of increasing the stability of the fcc phase to higher pressures and lower temperatures Mao et al. (2006).

1.7 Investigations on the Fe-Ni-Si System

Although a large body of work exists on the Fe-Ni-Si system, more research is required to fully define the e↵ects of nickel and silicon on the properties of iron. One common aspect of the work carried out so far is the focus on specific compositions that the cores of terrestrial planets are thought to consist of. The basis for picking out candidate compositions is either from core formation models or matching the density deficit from seismic data. Core formation models are in part reliant on mineralogical models, which introduces circularity to the problem of determining composition. Using the density

47 1.7. Investigations on the Fe-Ni-Si System deficit is also problematic since there is not a unique core composition which satisfies this (see Section 1.5.1). Therefore as more data is collected from exploratory missions to the terrestrial planets, as well as continued analysis of seismic data on the Earth, it seems likely that the arguments for these specific core compositions studied so far may change.

Rather than focus on speciﬁc compositions, this work has instead focused on understanding the Fe-Ni-Si ternary system as a whole. This approach provides a more methodical way of researching how nickel and silicon a↵ects iron and its properties. The two endmembers, FeSi and NiSi, have been studied using both computational and experimental techniques. The phase transition from "-FeSi to CsCl-FeSi has been investigated computationally at 0 K (Chapters 4) with additional calculations performed to ascertain whether any of the numerous structures recently found to be stable in

NiSi (Voˇcadlo et al., 2012) are also stable in FeSi (Chapter 5). In addition, lattice dynamics calculations were carried out to determine the phase boundary of the "-FeSi

CsCl transition at high temperatures for both FeSi and NiSi (Chapters 4 and 6). ! Experimentally, the melting curve of NiSi (Chapter 7) and the the eutectic melting curve of the Fe-FeSi system (Chapter 8) have both been investigated, as have the equation of state of MnP-NiSi (Chapter 9) and the thermal expansions of di↵erent Fe-Ni-Si alloys (Chapter 10). The melting curves have been determined by laser-heated DAC experiments, and the thermal expansions and equation of state measurements measured using di↵raction techniques. The next two chapters explain the methodology behind this work, with computational techniques discussed in Chapter 2 and experimental methods in Chapter 3.

48 CHAPTER 2

Computational Methods

Tonight the giant galaxies outside Are tiny, tiny on my windowpane

— Gjertrud Schnackenberg

As seen in the previous chapter, computational methods are an important tool in investigating the cores of terrestrial planets, and are a necessary complementary technique to experimental methods, providing an alternative method of investigating materials at high temperatures and pressures – conditions that are dicult to reach and maintain in experiments. In this work, ab initio calculations have been used to investigate the phase transition from "-FeSi to CsCl-FeSi at 0 K and at high temperatures and pressures using lattice dynamics, as well as to calculate the stability of NiSi structures in the FeSi system at 0 K. The "-FeSi CsCl phase transition has also been studied ! using ab initio lattice dynamics calculations in the NiSi system.

2.1 Ab initio

Ab initio simulations employ quantum-mechanical theory in order to determine the energy of a system accurately from its wavefunction. The wavefunction, (r), of a system contains all the information of that system. It is not observable directly, but the wavefunction multiplied by its complex conjugate gives the probability of an electron at a point, r. The following sections outline the basics of an ab initio calculation, starting with the Schr¨odinger equation.

49 2.1. Ab initio

2.1.1 The Schr¨odinger Equation

The Schr¨odinger equation links the total energy of the system with the wavefunction through the Hamiltonian, Hˆ

Hˆ =E (2.1)

where is the wavefunction and E is the energy. The Hamiltonian describes the potentials in terms of the nuclei and electrons

1 1 Z 1 Z Z Hˆ = 2 2 µ + + µ ⌫ (2.2) 2M 5µ 2 5i r r R µ µ i,µ ij µ,⌫>µ µ⌫ X Xi Xi,µ i,j>iX X

The Greek labels, µ and ⌫, refer to the nuclei and labels i and j refer to the electrons, with M being the mass, R being the distance and Z being the atomic number of the nucleus, all in atomic units. Thus the ﬁrst two terms make up the kinetic energy of the nuclei and the electrons respectively with the last three terms corresponding to the potential energy.

Solving Schr¨odinger’s equation gives the total energy of the system, from which all relevant physical properties may be calculated. The computational power required to solve

Schr¨odinger’s equation increases exponentially with the number of electrons, making it impossible to calculate exact solutions for systems with more than two electrons.

However, with a number of simpliﬁcations, it is possible to ﬁnd the total energy to a very high degree of accuracy.

2.1.2 The Born-Oppenheimer Approximation

The ﬁrst simpliﬁcation is the Born-Oppenheimer approximation, which is an adiabatic approximation and provides a way of decoupling the dynamics of the atomic nuclei and the electrons. The Born-Oppenheimer approximation exploits the huge di↵erence in mass between the nucleus and the electrons. Even considering the lightest element,

50 Chapter 2. Computational Methods the hydrogen nucleus is about 1800 times heavier than an electron. It is therefore safe to assume that the nuclei move much more slowly compared to the electrons, and thus the dynamics of the nuclei can be decoupled from that of the electrons. This immediately simplifies the problem since the nuclei can be fixed at certain positions in space and Schrödinger’s equation solved just for the electrons in a potential generated by the stationary nuclei. The nuclear dynamics can be solved classically; however, the electronic problem requires more thought.

2.1.3 The Many Electron Problem

The Schrödinger equation can be solved exactly when only one or two electrons are considered. However, the complexity of the problem grows exponentially as the number of electrons increases, since the wavefunction exists in 3N dimensions, where N is the number of electrons. In a first-principles simulation, the wavefunction requires an infinite space of several thousand dimensions. As a first step, we can consider the electrons as independent. From this first consideration, a density related method can be developed (see Section 2.2).

2.1.4 Independent Electrons

One of the most obvious simpliﬁcations is to consider electrons that do not interact with each other, instead seeing an external potential generated by the other electrons. For a non-interacting system, the electronic Hamiltonian is the sum of the Hamiltonians of the single particles, hˆ

Hˆ = hˆ + hˆ + + hˆ (2.3) 1 2 ··· N where the Hamiltonian for the single particle is given by

1 hˆ = 2 +V ext(r) (2.4) 2 5 and V ext(r) is the external potential felt by the electron at position, r.

51 2.1. Ab initio

Solving Schr¨odinger’s equation for a single particle, with wavefunction n(r), gives the energy, ✏n

hˆ n(r)=✏n n(r) (2.5)

The energy of the whole system, E, is therefore

E = ✏ni (2.6) Xi since the many-particle wavefunction is

(r r )= (r ) (2.7) 1 ··· N ni i Yi

2.1.5 Indistinguishable Electrons

In quantum formalism, each particle is required to have a label even though they are indistinguishable. Because of this, the wavefunction used must contain the same physical information when the labels are exchanged. This can be achieved by using an antisymmetric wavefunction. Any wavefunction may be antisymmetrised in the following way

1 (r1,r2) (r1,r2) (r2,r1) (2.8) ! p2

However, this produces a wavefunction that is not physical. This can easily be seen when considering the wavefunction for a system consisting of two electrons

= (r1) (r2) (2.9)

52 Chapter 2. Computational Methods

Antisymmetrising this produces

1 (r1) (r2) (r2) (r1)) (2.10) p2

This equals zero since (r1)= (r2) which is not a physical result. In order to produce a physically meaningful result, the single particle wavefunction must be di↵erent.

However, so far the spin of the electron has been ignored, if this is taken into account now then two electrons may share the same function if they are of di↵erent spin. This explanation is now easily recognisable as the Pauli Exclusion Principle, which states that two electrons cannot occupy the same energy state if they have the same spin. This also leads to the well-known Aufbau picture of electronic energy levels, with electrons occupying the lowest possible energy levels ﬁrst (see Figure 2.1).

Figure 2.1: Electronic energy levels filled according to the Aufbau principle, so that the lowest energy orbitals are filled first and electrons fill an orbital with the same spin number until it is full before beginning to fill with the opposite spin number.

2.1.6 Self-Consistency

To solve the Schrödinger equation, we require the wavefunction to define the Hamil- tonian; however the wavefunction is obtained as a result of solving the Schrödinger equation. This produces a self-consistency issue. The equation must therefore be solved iteratively by

i. First proposing trial wavefunctions

ii. Building the Hamiltonian

iii. Calculating the eigenfunctions by solving the Schr¨odinger equation

53 2.1. Ab initio

iv. Iterate using the output of the previous cycle as input

This loop is then repeated until convergence is reached.

2.1.7 The Hartree Method

The previous section has only considered independent electrons within an external potential generated by the nuclei but neglects electron-electron repulsion. Hartree introduced a method to include this repulsion term using a mean ﬁeld approximation.

The electrons are regarded as independent but each feeling an electrostatic repulsion of electron density from the other electrons

1 hˆ = 2 +V ext(r)+V H (r) (2.11) i 2 5 i where hˆ is the one particle Hamiltonian, 1 2 is the kinetic energy term, V ext(r)is i 2 5 H the potential due to the nuclei and Vi (r) is the Hartree potential – the potential due to other electrons. The potential terms are linked to the charge density by the following equation

⇢(r0) V (r)= dr0 (2.12) r r Z | 0| where ⇢(r) is the charge density and is given by

(r) (2.13) | i | Xi

The Hartree potential is then given by

2 H nj(r0) 3 V (r)= | | d r0 (2.14) i r r j=i Z 0 X6 | |

Hartree’s method only partially used the indistinguishability of electrons – in this

54 Chapter 2. Computational Methods method, the electronic states are ﬁlled according to Pauli’s exclusion principle, which is a direct result of the indistinguishability of electrons. However, this method does not take into account the requirement for the wavefunction to be antisymmetric.

2.1.8 Hartree-Fock Theory

Given a trial, many-electron wavefunction of the form

( r , )= ( r , ) (2.15) { i i} i { i i} Yi where is the total wavefunction and is the wavefunction of a single electron, with ri being the position of the electron and i being the spin, the Variational Principle applies

( r , ) Hˆ ( r , ) = ( r , ) Hˆ ( r , ) (2.16) h { i i} | | { i i} i h i { i i} | | i { i i} i Yi Yi Yi Yi

This can be generalised to take into account the need for an anti-symmetric wavefunction if the above trial wavefunction is replaced by the Slater determinant

1 ( ri,i )= 1 2 N (2.17) { } pNi k ··· k

This is the Hartree-Fock method, and it results in the following single particle Hamil- tonian

1 hˆ = 2 +V ext(r)+V H (r)+V X (r) (2.18) i 2 5 i i

The only di↵erence between this and the Hamiltonian from Hartree’s original method

X is the addition of the exchange term, Vi (r). The Hartree-Fock method lacks accuracy because it still uses independent particles. The electron correlation is the deviation of the Hartree-Fock approximation from the exact solution for both the energy and the

55 2.2. Density Functional Theory (DFT) wavefunction. For a very small number of electrons, the electron correlation may be calculated. Again, however, as the number of electrons increases, the complexity of the problem increases very quickly and the computational power required to calculate the electron correlation increases exponentially. Using a density-related method simpliﬁes this problem.

2.2 Density Functional Theory (DFT)

DFT is a method of approximating the electron correlation, and is one of the most common methods used when implementing ab initio technique in calculations today.

Two assumptions are made

1. The ground state properties of a many-electron system are uniquely deter-

mined by an electron density that depends on only 3 spatial co-ordinates.

2. The non-interacting electrons are in an e↵ective, self-consistent potential

(all other electrons are seen not as individual particles but a collective

potential)

2.2.1 Hohenberg-Kohn Theorems

Density Functional Theory was formalised through the two Hohenberg-Kohn theorems.

The ﬁrst theorem states that for a system of interacting particles, the total energy of the system is a unique functional of the electron density and is always larger or equal to the ground state energy. This may be written as

E[⇢(r)] = d3rV ext(r)⇢(r)+F [⇢(r)] (2.19) Z where F [⇢(r)] is a universal functional. The second theorem states that the ground state energy, EGS, can be obtained via the Variational Principle

E[⇢(r)] = d3rV ext(r)⇢(r)+F [⇢(r)] E (2.20) GS Z

56 Chapter 2. Computational Methods and therefore the ground state energy corresponds to the ground state density. Thus, by minimising the density, the ground state energy may be found. Finding the ground state energy therefore becomes a trivial problem were it not for the fact that F [⇢(r)] is unknown.

2.2.2 Kohn-Sham Electrons

In order to put the theory of DFT into practice, we require a mathematical form for the functional F [⇢(r)]. Kohn and Sham rewrote the unknown functional, F [⇢(r)], and therefore the functional E[⇢(r)], by re-introducing the idea of auxiliary electrons (also known as Kohn-Sham electrons). These are electrons that do not interact with each other but with the e↵ective potential, giving the correct density and energy. Rewritten, this gives the following equation

1 E [⇢(r)] = T [⇢] d3r⇢(r) V ext(r)+ (r) + E [⇢] (2.21) 0 2 XC Z  where

E[⇢(r)] is the energy as a function of density

T0[⇢] is the kinetic energy for a system with no electron-electron interactions

⇢(r)isthedensity

V ext(r) is the external potential

(r) is the Hartree potential

EXC[⇢] is the exchange-correlation energy

By re-writing the equation using auxiliary electrons, Kohn and Sham split the unknown functional, F [⇢(r)], into its known and unknown parts, thereby improving the accuracy of the calculation. Only the exchange and correlation term, Exc[⇢] is now unknown and this is much smaller in comparison to the kinetic energy and the electrostatic interaction between the core-valence and valence-valence electrons.

57 2.3. Plane Waves and Basis Sets

2.2.3 LDA and GGA

There are two main methods for approximating the exchange energy – the local density approximation (LDA) and the generalised gradient approximation (GGA). The local potential, VXC is deﬁned generally as

E [⇢] V = XC (2.22) XC ⇢(r)

The LDA method deﬁnes the local potential as the variation of the exchange energy with the density at that point. However, VXC depends on density everywhere – it is a function of density. This is taken into account by the GGA method, which includes how the density changes at the point considered.The LDA method has well documented drawbacks, such as an overestimation of binding energies (van de Walle and Ceder,

1999), and the wrong ground state may be predicted to be stable. Other approaches are also now in use, such as GGA + U (Jain et al., 2011). The GGA method predicts the correct ground state of iron, unlike the LDA method (Stixrude et al., 1994). Therefore only the GGA implementation has been considered and used in the work presented in this thesis.

2.3 Plane Waves and Basis Sets

To implement DFT, a basis set is required to represent the wavefunction. The wavefunction, nk(r), can be written as a summation of basis sets

nk(r)= cµkµ(r) (2.23) µ X where µ(r) are the basis functions and cµk are the complex Fourier coecients. For atoms, the wavefunction is relatively simple and can be represented by atomic orbitals but the wavefunction for a solid is much more complex and would require an almost inﬁnite number of basis functions to describe it. However, in a periodic system such as a crystal, plane waves can be used instead as the basis function. Plane waves are

58 Chapter 2. Computational Methods preferred over atomic orbitals since the basis set is complete and orthogonal, leaving to computationally fast calculations where convergence can be readily demonstrated.

Since the ions in a crystal are arranged periodically, the external potential experienced by the electrons is also periodic, with a periodicity equal to the unit cell length. Bloch’s theorem states that for such a system, the wavefunction can be written as the product

ik.r of a plane wave, e , and a periodic function, unk(r)

ik.r nk(r)=e unk(r) (2.24)

Expanding the periodic function gives

(iG.r) unk(r)= cn,Ge (2.25) XG where G are the reciprocal lattice vectors deﬁned by G.l =2⇡m where l is a lattice vector of the crystal and m is an integer. The wavefunction then becomes

i(k+G).r nk(r)= cn,k+Ge (2.26) XG which is a sum of the plane waves. For a fully accurate wave function, an inﬁnite number of plane waves is required which isn’t computationally possible. However, the coecients c become smaller as G 2 becomes larger and therefore an energy cut- n,G | | o↵, Ec, can be set so that a calculation includes all plane waves up to the cut-o↵. The energy cut-o↵ is deﬁned as

~2G2 E = max (2.27) c 2m

The cut-o↵ energy is di↵erent for each system and convergence tests must be carried out to determine the best cut-o↵ energy for each calculation.

59 2.4. k-point Sampling

2.4 k-point Sampling

A periodic system in real space is also periodic in reciprocal space, therefore a unit cell in real space also exists in reciprocal space. Typically, the smallest unit cell centred around G = 0 is considered. This is known as the ﬁrst Brillouin zone, and can be mapped out by a series of continuous k-points, with the electronic potential of the solid being contributed to by the occupied states at each k-point (Chadi and Cohen,

1973). The set of k-points is very dense and e↵ectively an inﬁnite number of k-points is required to calculate the wavefunction. However, the electronic wavefunctions at k-points that are close together are virtually identical and therefore one k-point can be used to describe the electronic wavefunctions for a region of reciprocal space. Hence, k-space is discretised and the electronic potential can be calculated at a ﬁnite number of k-points. The error resulting from this approximation can be decreased by increasing the number of k-points used.

To determine the number of k-points required, convergence tests are run, similar to those used to determine an appropriate plane wave cut-o↵ energy. In the calculations carried out here, a Monkhorst-Pack grid (Monkhorst and Pack, 1976) is used, which distributes the k-points sampled evenly through the Brillouin zone, with each k-point given equal weight.

2.5 Pseudopotentials

Using a full potential requires a lot of computing power because of complexities around the nuclei. Tightly bound core electrons, an oscillating wave function and an electron- nucleus potential that diverges to all require a large number of plane waves to 1 model accurately. To reduce computational costs, pseudopotentials can be used instead

(Phillips, 1958; Yin and Cohen, 1982).

A pseudopotential removes the core electrons and replaces interactions between the core and valence electrons with an e↵ective potential instead. This is possible because only valence electrons are involved in bonding and core electrons are chemically inert.

60 Chapter 2. Computational Methods

Since the core electrons are removed, their wavefunction doesn’t need to be calculated.

In addition, the pseudo-wavefunction is smoothly varying. To ensure that the pseudopotential is norm-conserving and gives the same results as an all-electron calculation, the pseudopotential must match the all-electron wavefunction in the region outside of the core (see Figure 2.2).

Figure 2.2: Figure from Payne et al. (1992) showing an all-electron potential (solid line) and a pseudopotential (dashed line), along with their corresponding wavefunctions. The two wavefunctions match at the core radius, labelled as rc.

2.6 Projector Augmented Wave Method

Although the pseudopotential method, combined with a plane-wave basis set, allows for formal simplicity, pseudopotentials become ‘hard’ – that is they require either a very large or complicated basis set of plane waves to maintain accuracy. The projector augmented wave (PAW) method maintains the formal simplicity of the pseudopotential method, while extending the augmented-wave approach to access full wavefunctions and therefore determine the potential from the full charge density (Bl¨ochl, 1994). The

PAW method derives from the augmented wave method, which splits the complicated wavefunction seen in Figure 2.2 into parts consisting of a partial-wave expansion within an atom-centred sphere and envelope functions outside the spheres. The PAW method

61 2.7. Ab initio Packages uses the method of partial waves combined with projectors to describe the wavefunction. Partial waves are constructed following the pseudopotential approach described in Section 2.5. Further details on this method are found in Bl¨ochl (1994).

2.7 Ab initio Packages

An ab initio calculation is carried out by a program which implements the steps described in the previous sections to calculate the energy of the system. There are many computational packages available to do this. The packages used in this work are VASP

(Kresse and Furthmuller, 1996a,b), CASTEP (Clark et al., 2005) and Abinit (Torrent et al., 2008). VASP, the Vienna ab initio Simulation package, and CASTEP are very similar since VASP was developed from an early version of CASTEP. The input files for VASP and CASTEP are therefore also very similar, although VASP requires four input files instead of the two required by CASTEP. These input files allow many input variables to be set, including the size of the k-point grid, the psuedopotential used and the plane wave energy cut-o↵. Any variables that are not specified are set to de- fault values. Unlike CASTEP and VASP, Abinit is an open source code released under general public license. The input files for Abinit are of a slightly di↵erent format to

CASTEP and VASP but allow for the same input variables to be speciﬁed.

All three packages feature a wide range of capabilities, and each is suitable for the calculations carried out in this work, which involve geometry optimisation at 0 K (Sec- tion 2.8) and phonon calculations for lattice dynamics (Section 2.9.2). The three packages use the same method as detailed above to solve Schr¨odinger’s equation, but di↵er in the pseudopotentials used. Since the accuracy di↵ers between pseudopotentials, some ab initio packages may be more accurate than others.

Ab initio calculations require a huge amount of computational power and are usually run on a supercomputer. The calculations done in this work made use of HECToR, the UK national supercomputing service, and ARCHER, the supercomputer that su- perseded HECToR.

62 Chapter 2. Computational Methods

2.8 Static Calculations

Static calculations are one of the more straightforward ab initio calculations since there are no temperature e↵ects that need to be taken into account. Static calculations di↵er from 0 K calculations in that zero point motion is not included. One important aspect of studying the cores of terrestrial planets is ﬁnding out which structure is adopted at core conditions, for example, the structure in the Earth’s core must exhibit anisotropy to match the anisotropy seen seismically. Geometry optimisations can be used to determine the most stable structure at 0 K at various pressures.

2.8.1 Geometry Optimisations

The purpose of a geometry optimisation calculation is to establish the most energetically favourable structure at a given pressure. The initial starting geometry is speciﬁed by the user in the input ﬁle and the total energy calculated. The forces on the ions, FI , can then be calculated, from the equation

@E FI = (2.28) @RI

where I is the ion, E is the energy of the system and RI is the position of the ion. This force is a restoring force, pushing the ions towards their equilibrium position at the energy minima (see Figure 2.3). A new ionic conﬁguration is generated using these forces and the process repeated. This repeats until either the number of maximum steps or the force tolerance for the ion is reached. Both quantities are user speciﬁed, so it is important to allow for enough steps for the calculation to converge to the lowest energy structure.

The pressure for the calculation is easily controlled by specifying and ﬁxing the volume of the unit cell. This results in a range of energy-volume values from which the pressure can be derived, using an equation of state such as the Birch-Murnaghan Equation of

State (BMEoS). This is the equation used for the calculations carried out in this work

(see Section 2.8.2).

63 2.8. Static Calculations

Figure 2.3: Graph showing the potential energy of a system as a function of atomic separation. Short-range repulsion dominates when atoms are close together and Coulombic attraction dominates when they are further apart. Also shown is the potential energy of a simple harmonic oscillator – when the separation between atoms are small, it can be seen that the harmonic approximation is very good. However as the separation increases, as in the case at very high temperatures, the potential energy becomes more anharmonic and the approximation breaks down. Figure from Voˇcadlo and Dobson (1999).

By carrying out geometry optimisation calculations for a range of candidate structures of a system, and ﬁtting the results using the BMEoS, the enthalpies of the di↵erent structures can be calculated from the equation

H = E + PV (2.29)

where H is the enthalpy, E is the internal energy, P is pressure and V is volume.

Static calculations are e↵ectively 0 K calculations and so the Gibbs free energy, G, is equal to the enthalpy, H (see Section 2.9.3). In this way, the relative stabilities of di↵erent phases, and therefore the phase transitions between them, can be obtained

(see Figure 2.4). This has been done to ﬁnd the "-FeSi CsCl-FeSi phase transition ! in the FeSi system (see Chapter 4). Geometry optimisation calculations have also been carried out on the NiSi-structured phases in the FeSi system (see Chapter 5).

64 Chapter 2. Computational Methods

-6.8 HCP BCC

-7.0

-7.2

-7.4

-7.6

G per atom /eV

-7.8

-8.0

-8.2 0 2 4 6 8 10 12 14 16 18 20 Pressure /GPa

Figure 2.4: Geometry optimisation calculations were carried out on two structures in the iron system – bcc (body centred cubic) and hcp (hexagonal close packed) – at a range of volumes. Fitting the resulting enthalpy-volume values to the third order Birch-Murnaghan equation of state gives Gibbs free energy, or enthalpy at 0 K, values as a function of pressure. Plotting these for both structures shows that the most stable phase at low pressures is bcc and a phase transition to the hcp structure occurs at around 8 GPa.

2.8.2 Birch-Murnaghan Equation of State

The Birch-Murnaghan Equation of State (BMEoS) relates the volume, V , of a system to the pressure, P , and is derived from a Taylor series expansion of the energy in the

ﬁnite strain(Murnaghan, 1937; Birch, 1947). The volume derivative of this then gives the pressure, as shown below. The following is the third order BMEoS used in this work, after Poirier (2000);

7 5 2 3K0 V 3 V 3 3 V 3 P = 1+ K0 4 1 (2.30) 2 V V 4 0 V ✓ 0 ◆ ✓ 0 ◆ ⇢ ✓ 0 ◆

where K0 is the bulk modulus at P = 0, K0 is the ﬁrst derivative of the bulk modulus with respect to pressure, evaluated at P = 0, and V0 is the volume of the unit cell at zero pressure. Integrating this gives the relation between the internal energy of the system and the volume;

65 2.9. High Temperature Calculations

E(V )=E1 + E2 + E3

where

1 1 9 1 V 3 V 3 E = K V 0 0 1 4 0 0 2 V V ✓ ◆  ✓ ◆ ✓ ◆ 1 1 9 V0 V0 V0 3 V0 3 E = K (K0 4)V 3 +3 2 16 0 0 0 V V V V ✓ ◆ ✓ ◆✓ ◆ ✓ ◆ ✓ ◆ 9 E = E(V ) K V (K0 6) (2.31) 3 0 16 0 0 0 ✓ ◆

This integrated form of the BMEoS can be ﬁtted to the energy-volume values obtained from the ab initio calculations, thereby obtaining values for V0, K0 and K0 . Using these, the corresponding pressure for each volume can be calculated. Additionally, V0,

K0 and K0 can easily be compared to those obtained experimentally or by di↵erent calculations.

2.9 High Temperature Calculations

So far, the discussion has considered static calculations, e↵ectively at 0 K, only. In order to incorporate temperature e↵ects, the movements of the ions with temperature have to be considered. There are two methods for doing this, either using molecular dynamics or lattice dynamics.

2.9.1 Molecular Dynamics

In molecular dynamics, the movement of each atom is considered individually. The force on the atom is calculated, ab initio, by di↵erentiating the potential energy with respect to atomic separation (see Section 2.8.1). Using Newton’s second equation of motion, F = ma,whereF is the force on the atom, m is the mass and a is acceleration, a series of mathematical relationships can be obtained that relate the derivative of the potential energy to the changes in atomic position as a function of time. The path of each atom can be calculated using these relations to generate a series of time-dependent atomic trajectories with the positions and velocities of the atoms evolving with each

66 Chapter 2. Computational Methods time step. This method of introducing temperature to the calculation makes it well suited to modelling liquids as well as solids. However, calculation of free energies using this method is not trivial, resulting in a very high computational cost and it is, therefore, not used in this work. Instead, lattice dynamics has been used to model temperature e↵ects.

2.9.2 Lattice Dynamics

Instead of considering each atomic motion individually as in molecular dynamics, lattice dynamics considers the vibrations of the atoms as a collective motion (see Figure 2.5).

Under harmonic conditions, the movement of atoms are co-ordinated and can be described by phonons, a quantisation of the vibrations of the atoms similar to a photon, with an associated wavelength and frequency. Describing the movement of atoms using phonons allows the vibrational free energy to be calculated using the phonon frequencies.

Figure 2.5: At low temperatures (left), atoms vibrate harmonically (see Figure 2.3). At high temperatures (right), motion becomes anharmonic, which causes the phonons to interact with each other, resulting in phonon frequency becoming dependent on temperature. Figure from Voˇcadlo and Dobson (1999).

Atomic vibrations with a small enough amplitude are very well described by a harmonic approximation. Using the Taylor expansion, the potential energy function, U, can be expanded around the energy minimum, Eperf , at which the atoms are in their equilibrium position;

1 U = E + u u (2.32) harm perf 2 ls↵,l0t ls↵ l0t lsX↵,l0t

67 2.9. High Temperature Calculations where Uharm is the harmonic part of the potential energy, uls is the displacement of atom s in unit cell l, ↵ and are Cartesian components and ls↵,l0t is the force constant matrix given by the double derivative of U with respect to the atomic displacements.

The force constant matrix, Fls↵, describes the relationship between the forces and displacements, and is used to calculate the phonon frequencies;

@U Fls↵ = = ls↵,l0tul0t (2.33) @uls↵ Xl0t

To calculate the force constant matrix, the small-displacements or frozen phonon method is used. When the displacements of the atoms are small enough, the relationship between the displacement and force is linear. This proportionality can be exploited to calculate the force constant matrix. Each atom in turn must be displaced slightly from its equilibrium position, keeping the other atoms ﬁxed at their equilibrium positions.

The force resulting from this gives one element of the force constant matrix. If this is repeated for all atoms, the complete matrix may be obtained. Symmetry considerations mean that the number of displacements needed are signiﬁcantly reduced for high symmetry systems. The phonon frequencies can then be calculated using the force constant matrix, since the frequencies are the eigenvalues of the dynamical matrix, which itself is dependent on the force constant matrix. These steps are all implemented in the program Phon (Alf`e, 2009), which has been used in this work.

2.9.3 Gibbs Free Energy

To construct the phase diagram of a system, the most stable structure at any given pressure and temperature must be determined. For any system, the most stable structure is that which has the lowest Gibbs free energy, G. The Gibbs free energy is given by

G = U + PV TS = F + PV (2.34) where U is the internal energy, P is the pressure, V is the volume, T is temperature,

68 Chapter 2. Computational Methods

S is enthalpy and F is the Helmholtz free energy. The total Helmholtz free energy,

Ftotal, is a function of volume and temperature (see Figure 2.6), and can be split into two parts

Ftotal(V,T)=Fperfect (V,0) + Fvib(V,T) (2.35)

where Fperfect (V,0) is the Helmholtz free energy of the system at 0 K and Fvib(V,T)is the vibrational free energy due to temperature. Fperfect (V,0) is obtained through static calculations (see Section 2.8). Fvib(V,T) can be written in terms of temperature and phonon frequencies

! ~!i ~ i F (V,T)=k T +ln 1 e 2kB T (2.36) vib B 2k T i  B X where !i is the phonon frequency, kB is the Boltzmann constant, ~ is Planck’s constant divided by 2⇡, and T is the temperature. Therefore, once the phonon frequencies have

been obtained using lattice dynamics, the vibrational free energy can be calculated and

hence the total Helmholtz free energy can be obtained (see Figure 2.6).

The pressure is given by the derivative of the free energy with respect to volume at a

constant temperature

@F P = (2.37) @V ✓ ◆

Therefore fitting the F V curve and calculating the derivative of the fit will give the pressure. A sixth-order polynomial is used to fit the F V plots. Di↵erentiating the polynomial then gives the pressure (see Figure 2.7). The Birch-Murnaghan equation of state may also be used, but a polynomial fit has been used since the di↵erence between the two is very slight for the values considered here.

Since both volume and temperature are input variables, and therefore user deﬁned, for

69 2.9. High Temperature Calculations

-4 200 K 1500 K 2000 K

-5

-6

-7

Helmholtz Free Energy /eV per atom

-8

7.0 8.0 9.0 10.0 11.0 12.0 3 Volume /Å per atom

Figure 2.6: Plot of the total Helmholtz free energy as a function of temperature and volume for the CsCl structure of FeSi, with a sixth-order polynomial ﬁtted to the F V values.

400 200 K 1500 K 2000 K

300

200

Pressure /GPa Pressure

100

7.0 8.0 9.0 10.0 11.0 12.0 3 Volume /Å per atom

Figure 2.7: The pressures obtained from ﬁtting a sixth order polynomial to the graphs in Figure 2.6 plotted as a function of volume. each value of volume and temperature, there is now a corresponding pressure value and total Helmholtz energy value. It is now possible to obtain the Gibbs free energy as a

70 Chapter 2. Computational Methods function of volume and temperature, since

G(V,T)=F (V,T)+P (V,T)V (2.38)

Obtaining G(P, T) is then a matter of interpolation of the values already obtained. For each temperature and volume ‘pair’ there is a corresponding pressure and Gibbs free energy. The dependencies of all these variables mean that interpolation is possible by

‘matching up’ the pressure and Gibbs free energy, ﬁnally giving the Gibbs free energy as a function of pressure and temperature (see Table 2.1).

Table 2.1: The pressure and Gibbs free energy of CsCl-FeSi at selected temperatures and volumes

Pressure /GPa G /eV per atom Temperature /K Volume 200 1000 3000 200 1000 3000 /A˚3 per atom 8.2 98 104 120 -1.66 -1.57 -1.87 9.2 46 51 67 -4.47 -4.40 -4.74 9.4 38 43 58 -4.97 -4.89 -5.23 9.6 29 35 50 -5.44 -5.36 -5.70 10.0 15 21 36 -6.50 -6.21 -6.55

2.9.4 Phase Diagram

Once the Gibbs free energies for the di↵erent structures in a system have been calculated, the phase diagram can be constructed, showing the regions of stability for the di↵erent structures. The Clausius-Clapeyron slope can then be calculated for any phase transitions, useful for comparison between di↵erent calculations or with experimental results. The Clausius-Clapeyron relation is given by

dP dS = (2.39) dT dV where P is pressure, T is temperature, dS is the di↵erence in entropy across the phase boundary and dV is the di↵erence in volume.

71 2.10. Summary

An accurate phase diagram is necessary for determining which structure is likely to be stable at core conditions – leading to knowledge of core composition and evolution – and is therefore an important and essential tool for the study of the cores of terrestrial planets. Calculations can also be compared to experimental data and provide complementary information, such as whether any of the phase transitions are kinetically inhibited. In this work, the phase diagrams for FeSi and NiSi have been determined using the lattice dynamics technique detailed here.

2.10 Summary

This chapter has described the theory of the ab initio calculations used in this work, as well as the techniques used to determine the phase diagram of a system at 0 K and also at high temperatures. Chapter 3 explores the theory and techniques used in the experimental work undertaken in this thesis.

72 CHAPTER 3

Experimental Methods

Two ﬂoors below your ﬁngertips still pinch the last one-hundredth of an inch.

— Simon Armitage

Experiments provide another way of investigating systems relevant to the cores of the terrestrial planets. Computational and experimental techniques are complementary and the two can be used in tandem to reinforce any conclusions drawn as well as providing di↵erent insights into the system in question.

One challenge that is faced when studying the cores of terrestrial planets experimentally arises from the conditions at which the cores are found. Being at the centre of the planet, the inner core is subject to the highest pressures and temperatures of the planet, with pressures reaching up to 360 GPa and temperatures of around 6000 K in the Earth’s inner core (Stevenson, 1981). Experiments to probe matter at core conditions therefore need to be designed to reach high pressures and temperatures, which require specially designed equipment.

3.1 High Pressure Experiments

High pressures in experiments are achieved by compression, either dynamic or static.

In static compression experiments, the sample is literally compressed between two or more anvils. The sample is subject to high levels of pressure as the anvils are brought closer together. Shock experiments, which are dynamic, use a high pressure shock wave to pressurise the sample. Both methods are described here (shock experiments are described only brieﬂy since this method is not used in this work).

73 3.1. High Pressure Experiments

3.1.1 Shock Experiments

In shock experiments, high pressures are achieved by the generation of a high-pressure shock wave, produced when a projectile is shot at the sample (see Figure 3.1).

Figure 3.1: Figure from Voˇcadlo and Dobson (1999) showing a schematic of a double-stage light-gas gun. Igniting the charge forces the piston along the ﬁrst stage barrel, compressing the hydrogen gas, which in turn forces the rupture disc to break. The expansion of the gas causes the projectile to accelerate down the second stage barrel and hit the sample. The shock wave generated by this collision produces ultra-high pressures as well as adiabatically heating the sample.

The faster the projectile is travelling, the greater the pressure generated. Since the projectile can be made to travel extremely quickly, the pressure generated can be very high. Pressure is then determined from the P T t path associated with the shock wave and the thermodynamical relationship sampled is deﬁned by the Hugoniot (Steinle-

Neumann et al., 2002). The temperature is measured using optical pyrometry (Yoo et al., 1993) or using a thermodynamic model (Brown and McQueen, 1986). One advantage of shock experiments is the ability to measure Vs and Vp – the set-up of the experiment results in a longitudinal sound wave travelling through the impactor and sample (Steinle-Neumann et al., 2002).

3.1.2 Compression Experiments

The other method of generating high pressures is to compress the sample statically.

The two most common pieces of equipment used in static compression experiments for high pressures greater than 5 GPa are the multi-anvil press (MAP) and the diamond- anvil cell (DAC). As suggested by the name, the MAP requires several anvils, arranged geometrically around the sample, whereas the DAC only uses two anvils in the form of two diamonds facing opposite each other. In both cases, the anvils used are made of hard materials, with tungsten carbide regularly used in the MAP. The pressures that can be reached in the DAC are much higher than in the MAP, with maximum pressures

74 Chapter 3. Experimental Methods for the MAP reaching about 100 GPa (Liebermann, 2011) and that of the DAC being around 300 GPa (Asanuma et al., 2011), although the more common pressure limits are around 25 GPa and 130-150 GPa for the MAP and DAC respectively. However, the DAC is much smaller than the MAP, which means that sample sizes are also much smaller, measuring tens of microns in the DAC and a few millimetres in the MAP.

3.1.3 The Multi-Anvil Press (MAP)

The ﬁrst multi-anvil press was built by Tracy Hall in 1958 (Liebermann, 2011), who developed the tetrahedral-anvil device (see Figure 3.2). Four hydraulic rams each drove a piston that compressed a tetrahedron cell assembly. It was able to reach pressures of

10 GPa and temperatures over 3000 K. In addition, it was possible to perform in situ

X-ray di↵raction on the sample.

Figure 3.2: Figure from Liebermann (2011) showing the ﬁrst multi-anvil press.

Since the ﬁrst tetrehedral-anvil device, many di↵erent geometries for the MAP have been developed, the common ones being the cubic-anvil and octahedral-anvil presses.

Cubic-anvil apparatus are similar to tetrahedral-anvil presses, but use six orthogonal pistons to compress a cubic cell volume instead of four. The octahedral-anvil press,

75 3.1. High Pressure Experiments or ‘Kawai-type’ multi-anvil press, uses a more complex system consisting of a sphere cut across three planes to produce eight wedge-shaped anvils, which, when brought together, press down on an octahedral space in which the cell assembly is placed (Kawai et al., 1970). Further modifications lead to the current design, which features two stages of anvils - the first consisting of six hardened steel anvils which press onto a second set of eight tungsten-carbide anvils, with modified corners which form the octahedral space in which the sample sits (see Figure 3.3).

Figure 3.3: A diagram showing the truncated tungsten-carbide cubic anvils around the octahedral sample cell on the left, and on the right, the arrangement within the steel sphere inside the press. Figure from Liebermann (2011).

Solid-media, high-pressure apparatus are inherently non-hydrostatic and the set-up of the multi-anvil press is designed to minimise deviatoric stress. However, the fact that deviatoric stresses are present in the MAP can be exploited to perform deformation experiments, which provide an insight into how materials behave when not under hydrostatic conditions. The D-DIA apparatus was designed to facilitate this. Figure 3.4 shows a schematic of the D-DIA apparatus, which is the same type of equipment installed at the X17B2 beam line at NSLS (National Synchrotron Light Source) (Hunt et al., 2012).

In order to reach the highest possible pressures in the multi-anvil press, sintered diamond anvils, less than 10 mm in length, have to be used (Kunimoto and Irifune, 2010).

However, even with these specialised anvils, the pressures reached are limited to around

100 GPa. To reach higher pressures through compressional techniques, the diamond

76 Chapter 3. Experimental Methods

Figure 3.4: Figure from Liebermann (2011) showing a schematic of a D-DIA multi-anvil press. anvil cell must be used.

3.1.4 The Diamond Anvil Cell (DAC)

The Diamond Anvil Cell (DAC) works through the same principle as the multi-anvil press, but with a two-anvil system rather than a multi-anvil system. The sample is placed between two opposing diamonds, with their tips truncated to leave a ﬂat culet

(see Figure 3.5). When force is applied to the two diamond anvils, the sample is pressurised.

Figure 3.5: A schematic of the DAC from Jayaraman (1983) showing the two opposing diamond anvils pressing down on the sample surrounded by a metal gasket.

A metal gasket, typically made of rhenium or stainless steel, is used to contain the

77 3.2. O✏ine DAC Melting Experiments sample within the sample chamber as well as preventing the two diamonds from coming into direct contact with each other. The pressure medium surrounds the sample in the sample chamber and transmits the applied force to the sample, helping to produce hydrostatic pressure on the sample. Typically the pressure medium is a soft un-reactive material, which remains soft and compressible at high pressures, such as NaCl or KCl.

Gases such as helium or nitrogen may also be used.

There are two broad categories of DACs, screw-type and gas-membrane type. These describe the two methods by which force is applied to the diamonds, and within both categories there are a wide range of di↵erent DAC constructions as well (Jayaraman,

1983). There are advantages and disadvantages associated with both types of DACs – the screw-type are more capable of providing a stable pressure, especially when cells are held at pressure for a prolonged period of time, since temperature ﬂuctuations may a↵ect gas expansion. However, gas-membrane type cells provide a more precise method of achieving the right pressure as well as allowing for remote pressure control (Kantor et al., 2012). Both types of DAC have been used in this work, though primarily the screw-type has been used.

3.2 O✏ine DAC Melting Experiments

The transparency of the diamonds allows the sample to be heated easily using a laser (see Figure 3.6). Experiments that require simultaneous high-pressure and high- temperature are therefore relatively straightforward to carry out in a DAC.

It is possible to determine melting in a Laser Heated-DAC (LH-DAC) without using in situ X-ray di↵raction, and therefore melting curves can be measured in experiments away from a synchrotron. O↵-line LH-DAC experiments have been carried out at the School of Earth Sciences, University of Bristol to determine the melting curves of NiSi and the Fe-FeSi eutectic. This section describes the methodology of these experiments.

78 Chapter 3. Experimental Methods

Figure 3.6: A diagram of a screw-type DAC from Voˇcadlo and Dobson (1999) showing the possible pathways for X-ray di↵raction as well as laser-heating.

3.2.1 Sample Preparation

The DACs used in the o↵-line melting experiments were Princeton-type symmetric

DACs (see Figure 3.7). Diamonds mounted in the DACs had culets with diameters of

200 – 300 µm; for experiments requiring higher pressures, diamonds with the smaller culets were used.

Figure 3.7: One of the Princeton-type DACs used to carry out the melting experiments. The cells are screw-type DACs, where force is applied to the diamonds by gradually tightening the screws (see Section 3.1.4)

The ﬁrst step is to clean the diamonds. A strip of alumina abrasive paper is run over the two diamonds to dislodge any dirt attached to the diamonds and a cotton bud soaked

79 3.2. O✏ine DAC Melting Experiments in acetone is used to remove any dislodged dirt. As can be seen in Figures 3.5 and

3.6, a metal gasket is used in the DAC to prevent the sample and pressure media from escaping during compression. For these experiments, both rhenium and stainless steel gaskets were used. The method for making both are the same; ﬁrst the gasket material is compressed to 50 µm thickness. Next a hole, roughly one-third the diameter of the ⇠ culet, is drilled out using an automated UV (266 nm) laser ablation unit (New Wave

Research, LUV series). This creates a cylindrical sample chamber. The hole is cleaned under a microscope, using a needle to remove the burrs produced during cutting. The gasket is then placed in an ultrasonic bath for 20 minutes to remove the loosened ⇠ burrs.

To make the samples used in the NiSi melting experiments, thin foils were created by compressing a small amount of the NiSi powder between two large diamonds, with a culet diameter of 300 µm, compressed by hand. For the eutectic melting experiments, discs of FeSi surrounded by Fe were made. The discs were created by ﬁrst drilling small holes, approximately 40 µm in diameter, into a piece of iron metal, then packing the holes with FeSi. The entire piece is then compressed in the DAC by hand to ensure the holes were fully packed. Discs measuring 80 µm in diameter were cut out, ensuring that the FeSi was positioned centrally in the discs (see Figures 3.8 and 3.9).

The pressure media used for the experiments were discs of either NaCl or KCl, each with a thickness of 10 µm. These were made in a similar way to the samples – sheets ⇠ of NaCl or KCl were created between the large 300 µm culet diamonds, and the laser ablation unit was used to cut 80 µm discs out of it.

To load the cell, the gasket is mounted in the DAC using small pieces of modelling clay. A piece of the pressure medium is placed in the sample chamber, followed by the sample, then another piece of pressure medium is placed on top of the sample.

This arrangement means the pressure medium sandwiches the sample and encases it completely, ensuring that a quasi-hydrostatic pressure is applied to the sample. A small amount of ruby is placed around the sample on top of the pressure medium for use in determining the pressure of the sample (see Section 3.2.2).

80 Chapter 3. Experimental Methods

Figure 3.8: The Fe-FeSi disc, showing the darker FeSi packed inside the hole drilled out of the Fe metal piece. This results in a clear boundary between FeSi and Fe, which is then heated in order to melt at the eutectic.

Figure 3.9: Image showing the seven Fe-FeSi discs after being cut out using the laser ablation unit. The bright outline is due to transmitted light passing through the metal. Note that the discs are not fully cut out from the iron metal to prevent the discs from ‘jumping’ and getting lost.

Both NaCl and KCl absorb moisture readily, therefore the cell is heated in a neon environment after loading to dry the pressure media. The cell is closed and partway tightened, leaving a small gap to ensure neon is allowed through to prevent oxidation of the sample. The cell is heated for approximately an hour and then closed shut, after which it is ready to be laser-heated.

81 3.2. O✏ine DAC Melting Experiments

3.2.2 Pressure Determination

Ruby fluorescence and Raman spectroscopy can both be used to determine the pressure of the cell. Ruby fluoresces under laser light (see Figure 3.10). It has been observed that the resulting ruby fluorescence peak shifts in relation to pressure (Forman et al., 1972).

The peak shift can be used to determine pressure inside the DAC using a calibration scale (Mao et al., 1986).

Figure 3.10: A plot showing the ruby ﬂuorescence peak at around 700 nm.

Raman spectroscopy can also be used to determine pressure (Hanﬂand and Syassen,

1985). As with ruby fluorescence, it has been found that the Raman profile of diamond is related to pressure – specifically the sharp high-frequency edge of the Raman peak is linearly dependent on pressure. Raman spectroscopy therefore provides another method of pressure determination. The advantage of this method is that it requires no additional material near the sample, since it is the Raman spectrum of the diamond surface that is measured. Both Raman spectroscopy and ruby fluorescence techniques have been used in this work. Figure 3.11 shows the setup used at Bristol to measure ruby fluorescence.

82 Chapter 3. Experimental Methods

Figure 3.11: The setup used for measuring ruby ﬂuorescence at Bristol. The DAC can be seen in the holder to the right of the image. A series of mirrors and lenses guide the laser towards the DAC and also guides the ﬂuorescence from the ruby back towards the detector where it can be measured.

3.2.3 Heating the Cell

The sample is heated using a laser in a double-sided heating geometry, the speciﬁcs of which follow Lord et al. (2009). The equipment for this is shown in Figure 3.12. Both sides of the DAC are monitored with a camera and detector. First the laser spots must be matched optically so that the laser is heating the same spot on both sides. Next the lasers are coupled so that the heating on both sides is even.

Figure 3.12: Setup of the equipment used at Bristol for the melting experiments.

The temperature of the sample can be determined from thermal radiation emitted from the sample as a consequence of laser heating. It is known that the electromagnetic radi-

83 3.2. O✏ine DAC Melting Experiments ation emitted by a black body, an idealised material that absorbs all incident radiation, is given by Planck’s law and the intensity of the radiation, I, is given by

2 5 1 I(,T )=(2⇡c h)✏ exp (hc/k)/T ) 1 (3.1) ⇥ ⇤ where ✏ is the emissivity, h is Planck’s constant, k is Boltzmann’s constant, is the wavelength, T is the temperature and c is the velocity of light. For a black body, the emissivity, ✏, is equal to 1. However, the black body is idealised and no material absorbs all radiation in reality. Despite this, Heinz and Jeanloz (1987) found that materials can be treated as grey bodies – samples with an emissivity that is independent of wavelength but is not equal to 1 – by comparing the intensities of thermal radiation at two wavelengths, so that the emissivity cancels out. This concept is used for determining temperatures using spectroradiometry, which involves measuring the emitted intensity of the radiation at a variety of wavelengths, resulting in greater accuracy and precision.

The more information is collected through spectroradiometry, the less noisy the data is.

To carry out spectroradiometry, the heated spot is sampled using a narrow slit (see

Figure 3.13). As stated in Walter and Koga (2004), temperature is more conveniently calculated using Wien’s approximation to Planck’s law

! J =ln(✏) (3.2) T where J is the normalised intensity, and ! is the normalised wavelength. There are therefore two unknowns, T and ✏. The intensity is measured at each wavelength, with

T and ✏ allowed to vary in order to determine the temperature. For, 2D images, a single

CCD camera is used to collect images of the laser-heated spot sampled at four di↵erent wavelengths (670, 700, 800 and 900 nm). The four images are then combined by spatial correlation, resulting in four intensity-wavelength data points at each pixel.

The temperature can therefore be continually monitored as the laser power is increased.

84 Chapter 3. Experimental Methods

Figure 3.13: Diagrammatic representation of the principles of spectroradiometry. The image of the laser heated spot is magniﬁed and focused onto the slit of the spectrograph. The light is dispersed over the CCD, consisting of 1024 pixels. Figure from Walter and Koga (2004)

It has been observed that an abrupt change in slope in the temperature versus laser power function (see Figure 3.14) is a good indicator of melting occurring (Lord et al.,

2009, 2014). Therefore, using this metric, the melting of the sample can be detected without the need of in situ techniques such as X-ray di↵raction (see Section 3.5).

Figure 3.14: A typical laser power versus temperature graph showing melting occurring at 2697 50 K where the discontinuity in the plot occurs. The plot is for melting of FeSi at 22 GPa.± From Lord et al. (2010).

3.3 X-Ray and Neutron Di↵raction

The method of determining melting in Section 3.2 relied on previous experiments showing deﬁnitively that melting of a sample occurred at the same time as an abrupt change in slope is seen in the power versus temperature graph. Direct observation of melting

85 3.3. X-Ray and Neutron Di↵raction relies on a method of ascertaining whether the sample has turned liquid, which results in a di↵erent arrangement of atoms from that of the solid state. X-ray di↵raction and neutron di↵raction can be used to do this.

Both X-ray di↵raction and neutron di↵raction are techniques that can yield information about a material at the atomic level, including quantities such as the cell parameter, atomic co-ordinates and preferred orientation. There are advantages to both X-ray and neutron di↵raction. The intensity of the di↵raction peak is not dependent on the atomic number with neutron di↵raction, which is not the case with X-ray di↵raction, where the amount of scattering increases as the atomic number increases. However, neutron di↵raction requires a much larger sample than X-ray di↵raction.

X-ray and neutron di↵raction are powerful tools useful for analysis of a sample, and both follow the same theory of di↵raction. For simplicity, only X-rays have been referred to but the same principles apply equally to neutron di↵raction.

3.3.1 Bragg’s Law

X-ray di↵raction of a material results in a di↵raction pattern, which, for a crystalline material, consists of a series of distinct di↵raction peaks. These peaks can be envisaged as being produced by constructive interference between reflected waves. An X-ray reaching a plane of atoms in a sample is partially reflected o↵ the plane. If two reflected

X-rays are coherent and have a path di↵erence of an integer number of wavelengths, then constructive interference will occur and a peak will be seen. A simple consideration of this geometrically (see Figure 3.15) results in Bragg’s Law

2d sin ✓ = (3.3)

where d is the spacing between two atomic planes, ✓ is the angle of reﬂection and

is the wavelength. The left hand term, 2d sin ✓, therefore corresponds to the path di↵erence (see Figure 3.15).

A powder di↵raction pattern will show peaks at a variety of ✓ values (more commonly

86 Chapter 3. Experimental Methods

Figure 3.15: A diagrammatic representation of the requirements for constructive interference to occur. X-rays 1 and 2 must be coherent, and the path di↵erence, xyz, must be equal to an integer number of wavelengths. Bragg’s Law can therefore be derived using simple trigonometry, using ✓, the angle of reﬂection of the X-ray, and d, the spacing between the two atomic planes. Figure from West (2014).

plotted as 2✓ values since this is the value that is measured). It can be shown that the path di↵erence, d, is related to the cell parameter, a, and the Miller indices, hkl, of a unit cell. For a simple cubic structure, this relationship is

a d = (3.4) ph2 + k2 + l2

The 2✓ values can therefore be related to the cell parameters by substitution of Equa- tion 3.4 into Equation 3.3. For a cubic structure, the lattice type can be determined by systematic absences. These are absences of certain hkl values associated with a { } particular lattice type due to destructive interference occurring for these reﬂections.

For example, in the face-centred cubic (fcc) structure, the 200 di↵raction peaks are present but the 100 peaks are not because of destructive interference of adjacent planes of atoms (see Figure 3.16). This can be seen mathematically, resulting in only peaks with indices that are either all odd or all even being present for the fcc phase. Similarly, for body-centred cubic (bcc) structures, the indices must add up to an even number for a peak to be seen.

In addition, the intensity of a peak is proportional to the structure factor, which itself is dependent on the types of atom and their positions in the unit cell. The structure factor, Fhkl, is given by

87 3.3. X-Ray and Neutron Di↵raction

Fhkl = fi exp 2⇡i(hxi + kyi + lzi) (3.5) Xi

where fi is the scattering factor for atom i, with fractional co-ordinates xi,yi,zi.There- fore a di↵raction pattern contains all the necessary information required for identifying a particular phase, as well as providing pertinent information about other properties, such as cell parameters.

3.3.2 Di↵use Scattering

Melting can be detected by the presence of di↵use scattering, which appears as a consequence of liquid in the sample. Bragg’s law only applies in crystalline materials, since it relies on parallel planes of atoms for the X-rays to interfere either constructively or destructively. Once the material starts to melt, Bragg’s law no longer fully applies

– peaks are still seen since some of the material is still present in the solid crystalline phase – but the melt produces a broad di↵use peak underneath the sharp peaks (see

Figure 3.17).

Figure 3.16: A diagram showing the systematic absence of the 100 reﬂections from the fcc structure. On the left is the fcc structure, with 100 and 200 planes highlighted. X-rays reﬂecting o↵ the 100 planes have a path di↵erence{ of} exactly{ half} a wavelength meaning these waves interfere destructively.{ }

88 Chapter 3. Experimental Methods

Figure 3.17: X-ray di↵raction pattern taken of NiSi showing the appearance of di↵use scattering. As temperature increases, the broad di↵use peak becomes more intense as the sample continues to melt further. Experiment carried out at the National Synchrotron Light Source, New York (see Chapter 7).

3.3.3 Analysis of Powder Di↵raction Data

Samples used for X-ray di↵raction and neutron di↵raction may either be a single crystal or a powder. In general, powder di↵raction is more convenient than single crystal di↵raction since it can be hard to grow single crystals, especially for materials undergo- ing phase transformations at high pressure. Powder di↵raction, however, poses a problem due to overlapping peaks from di↵erent Bragg reﬂections. Rietveld (1967) devised a method to separate out the peaks so that powder di↵raction patterns with overlapping peaks could be interpreted accurately. This is known as the Rietveld method.

3.3.3.1 Rietveld Method

When analysing some neutron powder di↵raction patterns, Rietveld (1967) noticed that di↵raction peaks were Gaussian (see Figure 3.18) and that a peak made up of several overlapping peaks can be described as the sum total of the Gaussian functions of the individual Bragg peaks. The precise peak position, intensity and shape in a given X- ray or neutron di↵raction pattern are determined by a variety of factors either from the experiment, for example beam characteristics and experimental set-up, as well as from the sample, such as the fractional atomic co-ordinates and unit cell structure (see

89 3.3. X-Ray and Neutron Di↵raction

Section 3.3.1). In the Rietveld method, a theoretical line profile is calculated using all these parameters and least-squares refinement is used in an iterative process to refine the parameters so that the properties of interest can then be extracted (Rietveld,

1969).

Figure 3.18: Part of the neutron powder di↵raction pattern of epsomite, MgSO4.7H2O, from Fortes et al. (2006), analysed using the Rietveld method. Observed data are shown as points and the calculated di↵raction pattern as a line; the lower graph shows the di↵erence between the observed and calculated di↵raction patterns. The marks, ‘ ’, show the calculated peak positions of the Bragg reﬂections. |

3.3.3.2 Le Bail Method

The Le Bail method is an extension of the Rietveld method. Unlike the Rietveld method, the Le Bail method doesn’t require a structural model, instead the starting intensities are obtained by fitting the powder di↵raction data with only the cell parameters constrained (Le Bail et al., 1988). These intensities, which are sensitive to atomic positions, are freely adjusted to get the best fit to the data and then fed back into the Rietveld algorithm which continues as before with a least squares refinement.

This method is much quicker than the Rietveld method since the structural factors are never reﬁned.

90 Chapter 3. Experimental Methods

3.3.3.3 GSAS

There are many di↵erent programs used to carry out analysis of powder di↵raction data. GSAS, the General Structure Analysis System, (Larson and von Dreele, 1994), is used here. GSAS is capable of carrying out both Le Bail and Rietveld methods of analysis, both of which are used in this work.

Figure 3.19: The graphical user interface developed by Toby (2001) for GSAS. The Le Bail or Rietveld method can be chosen by toggling the radio buttons in the main window.

3.4 Thermal Expansion

One of the applications of X-ray powder di↵raction is to determine thermal expansion of a material. The sample is heated or cooled in a series of steps, and at each step a powder pattern is taken. Indexing the patterns yields the cell parameters (see Section 3.3.1) which can then be used to determine the value for thermal expansion of the sample.

Further analysis gives other properties, such as the Gruneisen parameter.

In this work, the thermal expansion of a range of (Fe,Ni)Si samples have been measured in this way. These experiments were carried out using a PANalytical, X’pert PRO

91 3.5. Synchrotron Experiments

MPD, high resolution powder di↵ractometer, in the X-Ray di↵raction laboratory at

UCL. The di↵ractometer produces X-rays using a cobalt anode and an incident beam

Ge monochromator. This results in a single Co K↵1 line source which is capable of producing very sharp di↵raction peaks.

3.5 Synchrotron Experiments

The thermal expansion experiments described in Section 3.4 were carried out in a university laboratory. However, many in situ high-pressure and high-temperature experiments are carried out at synchrotron facilities, which provide sources of neutrons or intense beams of X-rays. The details of these experiments are described here.

3.5.1 Introduction

Synchrotrons are cyclical particle accelerators, capable of producing high energy radiation in the form of X-rays or neutrons. A linear accelerator (Linac) is ﬁrst used to accelerate the charged particle, either electrons for X-ray production, or H– ions for neutron production. Electrons are produced by an electron gun, similar to a cathode ray tube, while H– ions are produced using an electric discharge. Once accelerated in the Linac, the electrons or H– ions are injected into a storage ring where magnets are used to maintain and focus their path around the ring (see Figure 3.20).

In synchrotron light sources, an X-ray beam is produced as the electrons follow their curved path. Insertion devices, such as an undulator or a wiggler, can be placed in the straight sections of the synchrotron to produce a much narrower, more parallel X-ray beam. Undulators and wigglers consist of a number of small magnets which direct the electrons along a path that zigzags. This produces X-rays at each bend which, due to the position of the magnets, interfere constructively creating a more brilliant beam of radiation than a X-ray beam produced using a single magnet. Beamlines for di↵erent experiments are built o↵ the storage ring where X-rays are produced.

To produce neutrons, a beam of protons is needed. These are produced by stripping the electrons from the H– ions just before they enter the synchrotron. The protons

92 Chapter 3. Experimental Methods

Figure 3.20: A diagram of the synchrotron at ESRF (European Synchrotron Radiation Facil- ity) from the ESRF website. A short Linac (linear accelerator) accelerates the electrons before they are injected into the storage ring where X-rays are produced. X-rays are produced and guided o↵ the storage ring to separate beamlines. are then accelerated to speeds close to the speed of light, and are ﬁred at a tungsten target, driving neutrons from the nuclei of the tungsten atoms. This process is known as spallation and produces an extremely intense neutron pulse. The neutrons travel very quickly and to be useful must be slowed down in hydrogenous moderators. Neutron beams are then directed to di↵erent target stations where they can be used to carry out in situ neutron di↵raction and/or spectroscopy.

3.5.2 In situ Experiments

High-pressure and high-temperature in situ experiments are possible at synchrotron facilities, using both the DAC and the MAP. Synchrotron experiments are signiﬁcantly easier in the DAC, since it is small and easily transportable and also possesses transparent anvils. With the MAP, care must be taken to choose anvils that are transparent to the type of beam used. For example, if using X-rays, the anvils must be made out of

X-ray transparent material, such as boron nitride or the X-ray beam must be directed along the anvil gaps. Only those anvils which are in the path of the X-ray need to be

X-ray transparent, as shown in Figure 3.21. The MAP has the added diculty of its size, and therefore presses must be installed at beamlines at synchrotron sources, unlike

93 3.5. Synchrotron Experiments the DAC, which can be easily transported. Another large volume press, the relatively easy to transport Paris-Edinburgh cell, which operates under the same principles of compression as the MAP and DAC, can also be used for in situ experiments.

Figure 3.21: Set-up of anvils in the MAP, showing X-ray transparent cubic boron nitride anvils in pale grey and tungsten carbide anvils in dark grey (Hunt et al., 2012).

Melting experiments have been carried out at two light synchrotrons, the ESRF (Euro- pean Synchrotron Radiation Facility) in France and the NSLS (National Synchrotron

Light Source) in the US. Equation of state measurements were also done at ESRF, as well as at the neutron source ISIS at the Rutherford Appleton Laboratory (RAL) in

Oxfordshire. The details of these experiments are described here.

3.5.3 ESRF DAC Experiments

The European Synchrotron Radiation Facility (ESRF) is the most powerful synchrotron radiation source in Europe, and has a number of beamlines suited to a wide range of

X-ray techniques. The ID27 beamline is speciﬁcally designed for use in high-pressure experiments with both diamond anvil cells and multi-anvil presses and uses a monochro- matic X-ray beam. In this work, the melting experiments were carried out at beamline

ID27 using only DACs. These are gas membrane driven – pressure is generated by allowing more gas into the bellows of the DAC, forcing the diamonds closer together and increasing the pressure on the sample. This method gives a more precise way for applying pressure, and also means that pressure can be controlled remotely, unlike the screw-type Princeton cells. This is beneﬁcial for in situ XRD experiments since it removes the need to interrupt the experiment and unlock the hutch to increase pressure.

In situ laser heating is also available at ID27, meaning the sample can be continuously

94 Chapter 3. Experimental Methods monitored by XRD while heating.

Loading of the cell and sample preparation follows the same procedure as for o↵-line

DAC experiments described in Section 3.2, and temperature determination is also the same (see Section 3.2.3). Pressure measurements are made by analysis of XRD patterns of a pressure standard, such as NaCl, instead of ruby ﬂuorescence or Raman spectroscopy. The beneﬁt of using NaCl as the pressure standard is that it is already present as the pressure medium. Melting can be detected using both the change in slope in the laser power versus temperature function, as described in Section 3.2.3, and also by the appearance of di↵use scattering in di↵raction patterns (see Section 3.3.2).

Part of the melting curve of NiSi was measured at the ID27 beamline at ESRF.

3.5.4 NSLS MAP Melting Experiments

As discussed in Section 3.1.3, multi-anvil presses may be installed at beamlines to facilitate in situ XRD experiments at high pressure. This is the case on the X17B2 beamline at NSLS (see Figure 3.22), which uses polychromatic X-rays that are produced by a 4.2 T super-conducting wiggler. The high energy X-rays produced have a critical energy of 22 keV.

A cubic DIA-type apparatus is used at the beamline. The sample chamber is cylindrical and consists of three compartments separated by rhenium foil. One compartment contained the NiSi sample, another the NaCl pressure standard, and the third elemental

Ni and Si. The sample was compressed to a pressure of 7 GPa and then heated to 1523

K in 50-100 K steps. Each sample was analysed by XRD at each step and radiographic

X-ray images were also taken periodically of the sample chamber. Melting of NiSi was determined by di↵use scattering appearing in the XRD patterns as well as X-ray videography of convection occurring in the sample chamber.

3.5.5 ISIS Equation of State Measurements

As described in Section 2.8.2, the Equation of State (EoS) is an equation that links the volume of a sample to the pressure the sample is under. It is therefore possible to measure the equation of state of a sample at a high pressure beamline, since the XRD

95 3.5. Synchrotron Experiments

Figure 3.22: The 1100 ton hydraulic press installed at the X17B2 beamline at NSLS. patterns will give the cell parameters (see Section 3.3), and di↵raction patterns can be collected at di↵erent pressures in situ. Neutron di↵raction can be used in the same way as X-ray di↵raction to determine cell parameters. While X-rays interact with the electron cloud surrounding the atom, neutrons interact with the nucleus of the atom, and scatter o↵ the nuclei. This key di↵erence means there is no atomic form factor needed to define the shape of the electron cloud, which removes one cause of decrease in intensity as a function of scattering angle. Therefore neutron di↵raction patterns show very well defined, intense peaks even at high angles (or small d-spacings). ISIS is a pulsed neutron source and so the time-of-flight method may be used to collect powder di↵raction patterns. In this method, the resolution of the di↵ractometer is almost

d t constant for di↵erent d-spacings, with d equal to t (where t is the flight time), This means that accurate cell parameters may be determined from strong reflections with large d-spacings. This makes time-of-flight neutron di↵raction ideal for measuring equations of state.

The Pearl beamline at ISIS, the neutron source at Rutherford Appleton Laboratory, is a high pressure powder di↵raction beamline, designed speciﬁcally to carry out in situ

96 Chapter 3. Experimental Methods experiments of materials at high pressure. A Paris-Edinburgh pressure cell capable of reaching pressures of up to 25 GPa is used. Because of the in situ capabilities of the beamline, it is possible to take di↵raction patterns at di↵erent pressures and temperatures. Analysing the data (see Section 3.3.3) allows for the equation of state to be deﬁned for the sample. The thermal equation of state has been measured for

MnP-structured NiSi at the Pearl beamline.

3.6 Summary

This chapter has discussed the experimental methods and techniques used in this work to determine the melting curves of NiSi and the eutectic Fe-FeSi, as well as the thermal expansion of a range of (Fe,Ni)Si alloys and measurements of the equation of state of MnP-NiSi. Together with Chapter 2, all the methodology used in this work has been covered. The following chapters document the results and conclusions of the experiments and calculations carried out in this work.

97 98 CHAPTER 4

The Calculated FeSi Phase Diagram

I run through snow and turn around just to make sure I’ve got a past

— Jeffrey McDaniel

4.1 Introduction

As discussed in Chapter 1, the likely presence of nickel and silicon in the cores of terrestrial planets makes understanding the Fe-Ni-Si ternary system important for determining core composition. As one of the endmembers of the Fe-Ni-Si system, understanding the FeSi system is an important ﬁrst step in understanding the ternary system. Exper- iments and calculations both show that two stable structures exist in the FeSi system

–the"-FeSi and the CsCl-phase (Voˇcadlo et al., 1999; Dobson et al., 2002). However, the pressure at which the phase transition occurs is not agreed upon, either between experiments or calculations. Calculations have been carried out here at 0 K using three di↵erent ab initio packages – CASTEP, VASP and Abinit – to determine the transition pressure and to determine the reason for the disagreement between previous 0 K calculations. However, so far, there have been no calculations on the FeSi system at temperatures above 0 K. Here, both static and lattice dynamics calculations have been performed to determine the phase boundary of the "-FeSi to CsCl-FeSi phase transition at 0 K and at high temperatures. From these calculations the Clapeyron slope of the "-

FeSi to CsCl-FeSi transition can be determined, and hence deﬁne the high temperature

FeSi phase diagram.

99 4.2. The FeSi Phase Transition at 0 K

4.2 The FeSi Phase Transition at 0 K

There are two stable phases in the FeSi system – the CsCl-FeSi phase and the "-FeSi phase. The CsCl-FeSi phase has a simple structure topologically equivalent to the body-centred cubic (bcc) structure of iron, with a central Fe atom surrounded by eight

Si atoms, shown in Figure 4.1a, while the "-FeSi structure, shown in Figure 4.1b, is less trivial, though still possessing cubic symmetry (space group P 213). The "-FeSi structure is a semi-conductor and the CsCl phase is metallic.

(a) (b)

Figure 4.1: (a) The CsCl-structured FeSi, showing the central (purple) Fe atom surrounded by eight (blue) Si atoms at the vertices of the cubic unit cell; and (b) The low pressure ✏- FeSi structure. The purple Fe atom is surrounded by seven blue Si atoms with bonds of three di↵erent lengths; the slightly shorter bond (pink and grey), three intermediate bonds (green) and three slightly longer bonds (mauve).

In the "-FeSi structure, the Fe atom is surrounded by seven Si atoms, forming three groups of bonds – one slightly shorter Fe-Si bond, three slightly longer Fe-Si bonds, and three Fe-Si bonds of intermediate length (see Figure 4.1b). As noted by Voˇcadlo et al. (1999), the phase transition from the "-FeSi phase to the CsCl-FeSi phase can be envisaged as a displacement of the atoms in the 111 direction. h i

Both calculations and experiments ﬁnd only one phase transition in the FeSi system, that of "-FeSi CsCl-FeSi (Voˇcadlo et al., 1999; Dobson et al., 2002; Fischer et al., ! 2013; Caracas and Wentzcovitch, 2004). However, there is no agreement between ex-

100 Chapter 4. The Calculated FeSi Phase Diagram periments and calculations, or indeed between experimental studies, on the transition pressure (see Chapter 4 for further details). Calculations at 0 K indicate a range of transition pressures for the "-FeSi CsCl-FeSi phase transition: Voˇcadlo et al. (1999) ! ﬁnd a transition pressure of 13 GPa using GGA with VASP, Caracas and Wentzcovitch

(2004) ﬁnd transition pressures of either 30 GPa using LDA or 40 GPa using GGA, both with Abinit (see Figure 4.2), while Zhang and Oganov (2010) ﬁnd the transition occurring at 20 GPa using evolutionary simulations with GGA in VASP.

(a) (b)

Figure 4.2: Graphs of (a) enthalpy versus volume for the ideal and real "-FeSi structures and CsCl-FeSi, with a common tangent connecting "-FeSi and CsCl-FeSi indicating a transition pressure of 13 GPa (Voˇcadlo et al., 1999); and (b) di↵erence in enthalpy plot for "-FeSi (B20 structure) and CsCl-FeSi (B2 structure) showing the transition occurring at 30 GPa using LDA and 40 GPa using GGA (Caracas and Wentzcovitch, 2004).

In contrast to Voˇcadlo et al. (1999) and Caracas and Wentzcovitch (2004) – both of whom calculated and compared enthalpies of the two structures at a range of pressures to ﬁnd the transition pressure – Zhang and Oganov (2010) use an evolutionary crystal structure prediction method to ﬁnd the transition pressure. This method generates random structures which are then structurally optimised to produce the next generation of structures.

The cause of the di↵erence in transition pressure between the three calculations is not clear. The most obvious culprit for the di↵erence in transition pressure between Zhang and Oganov (2010) and that of Voˇcadlo et al. (1999) and Caracas and Wentzcovitch

(2004) is the di↵erence in method used. However, it is unclear why there is such a large discrepancy between the results of Voˇcadlo et al. (1999) and Caracas and Wentzcovitch

101 4.3. Static FeSi Calculations

(2004) since both use the same method for calculating transition pressure. Caracas and

Wentzcovitch (2004) ﬁnd a di↵erence in transition pressure of 10 GPa when using LDA compared to GGA, which is not surprising, since LDA has been seen to underestimate the transition pressure by about this amount for other materials (Wentzcovitch et al.,

2013). However, this does not explain the di↵erence of nearly 30 GPa from Voˇcadlo et al. (1999), whose calculations also used the GGA implementation. There is also a di↵erence in pseudopotentials used between the two calculations, which may account for some of the di↵erence, which is the reason that Caracas and Wentzcovitch (2004) cite as the cause of the discrepancy. Another di↵erence is in the ab initio package used.

Voˇcadlo et al. (1999) used VASP while Caracas and Wentzcovitch (2004) used Abinit.

However, since all ab initio programs are based on solving the same equations (see

Chapter 2), this is unlikely to be a cause of the di↵erence. However, for a complete analysis, all three ab initio packages – VASP, CASTEP and Abinit – have been used to determine the phase transition at 0 K.

4.3 Static FeSi Calculations

Static calculations have been carried out on FeSi to determine the transition pressure of the "-FeSi to CsCl-FeSi phase transition, using the method described in Section 2.8.

Three di↵erent ab initio packages have been used – VASP, CASTEP and Abinit. The internal energies of both structures have been calculated for a range of unit cell volumes.

The resulting E V values were ﬁtted to a third order Birch-Murnaghan Equation of State (BMEoS, see Section 2.8.2), which gives the equation of state parameters as well as the corresponding enthalpy and pressure values for the di↵erent volumes speciﬁed.

This allows for the ﬁtted pressure to be checked against the pressure calculated by the ab initio package. Plotting enthalpy against pressure for both structures produces a phase diagram for FeSi at 0 K, showing the pressures at which the two di↵erent phases are stable, and therefore giving the transition pressure for the phase transition. Con- vergence tests were carried out for each set of calculations, so that the resulting energy calculated would have an error of less than 0.001 eV per atom, and all calculations were spin-polarised to account for magnetism, with initial magentic moments speciﬁed

102 Chapter 4. The Calculated FeSi Phase Diagram on the Fe atoms in a ferromagnetic structure.

4.3.1 VASP Calculations

The ﬁrst set of ab initio calculations were carried out using VASP, the same package used by Voˇcadlo et al. (1999). Ultra-soft GGA pseudopotentials (Bl¨ochl, 1994; Kresse,

1999) were used, with 14 electrons treated as valence for Fe and 4 valence electrons for Si. The PBE functional was used for these calculations (Perdew et al., 1996).

Convergence tests were carried out on the plane wave energy cut-o↵ and size of the k-point grid (see Figures A.1a, A.1b and A.1c in Appendix A.1). The k-point grids used were of size 17 17 17 and 9 9 9 for the CsCl-FeSi phase and the "-FeSi ⇥ ⇥ ⇥ ⇥ phase respectively, and the plane wave cut o↵ energy used was 600 eV. As described above, the phase transition was determined from a plot of enthalpy against pressure, shown in Figure 4.3. This shows the phase transition occurring at a pressure of 10.61

GPa.

Figure 4.3: A plot of enthalpy against pressure for the two structures, CsCl-FeSi in black, and "-FeSi in red, calculated using VASP. The phase transition occurs at 10.61 GPa.

103 4.3. Static FeSi Calculations

4.3.2 CASTEP Calculations

CASTEP is another ab initio package with much of the same capabilities as VASP.

The PBE functional was used with the GGA implementation of the pseudopotential, with 8 electrons treated as valence in Fe and 4 in Si (Perdew et al., 1996). The same convergence tests were carried out in CASTEP as with VASP, following the same criteria for error (see Figures A.2a and A.2b in Appendix A.2). The k-point grid size chosen was 11 11 11 for CsCl-FeSi and 7 7 7 for "-FeSi. A plane wave cut-o↵ ⇥ ⇥ ⇥ ⇥ energy of 2000 eV was used. The same method of relaxing the ions was chosen for the

CASTEP calculations, and the calculated energies were ﬁtted to the same third-order

BMEoS as in the VASP calculations. Figure 4.4 shows the plot of enthalpy against pressure, showing that the phase transition occurs at 4.31 GPa.

Figure 4.4: A plot of enthalpy against pressure for the two structures, CsCl-FeSi in black, and "-FeSi in red, calculated using CASTEP. The phase transition occurs at 4.31 GPa.

4.3.3 Abinit Calculations

The last ab initio package used was Abinit, the program used by Caracas and Wentz- covitch (2004). The PBE functional (Perdew et al., 1996) was used with the GGA implementation for the pseudopotentials, with 8 valence electrons in the Fe pseudopo-

104 Chapter 4. The Calculated FeSi Phase Diagram tential and 4 in Si. As with the CASTEP and VASP calculations, convergence tests were carried out (see Figures A.3a, A.3b and A.3c), and a 19 19 19 k-point grid ⇥ ⇥ was used for the CsCl-FeSi phase and a 13 13 13 k-point grid was used for "-FeSi. ⇥ ⇥ A plane wave cut-o↵ energy of 32 Hartree was used. Again a plot of enthalpy against

pressure was plotted to determine the transition pressure (see Figure 4.5). The phase

transition according to these calculations occurs at 3.02 GPa.

Figure 4.5: A plot of enthalpy against pressure for the two structures, CsCl-FeSi in black, and "-FeSi in red, calculated using Abinit. The phase transition occurs at 3.02 GPa.

4.4 Di↵erences in Transition Pressure

The calculations carried out on the FeSi phase transition show small variations between the di↵erent ab initio packages used, as expected. These are summarised in Table 4.1.

The largest variation is between the VASP and Abinit calculations, with a di↵erence in transition pressure of 7.59 GPa. This is in comparison to a di↵erence of 27 GPa between the calculations of Caracas and Wentzcovitch (2004) and Voˇcadlo et al. (1999).

The small di↵erences in transition pressure of a few GPa can be explained from looking at the enthalpy-pressure plots. The enthalpy against pressure graphs for CsCl-FeSi and "-FeSi share very similar gradients, and it is therefore not inconceivable for small

105 4.4. Di↵erences in Transition Pressure

Table 4.1: The calculated transition pressures for the "-FeSi CsCl-FeSi phase transition ! ab initio Program Transition Pressure /GPa VASP 10.61 Castep 4.31 Abinit 3.02

variations to cause a change in transition pressure of a few GPa. This is the most likely cause of the di↵erence in transition pressure observed here and also between the VASP calculations carried out here and those of Voˇcadlo et al. (1999). In addition, since the calculations of Voˇcadlo et al. (1999), VASP has undergone several updates and both pseudopotentials and functionals have been improved, which likely would also cause slightly di↵erent results. Similarly, di↵erences of under 10 GPa between the transition pressures calculated by VASP, CASTEP and Abinit in this work are likely to be because of small di↵erences between the di↵erent ab initio packages.

The most likely cause of the di↵erence in transition pressure of roughly 30 GPa between

Caracas and Wentzcovitch (2004) and Voˇcadlo et al. (1999) is the pseudopotential used. Caracas and Wentzcovitch (2004) used a pseudopotential which had a separate f electron channel, unlike the pseudopotentials used here and the pseudopotential used by Voˇcadlo et al. (1999). Since the pseudopotentials used here are very similar to that of Voˇcadlo et al. (1999) and because the transition pressures are very similar to Voˇcadlo et al. (1999) it would appear that it is a di↵erence in pseudopotential that is the cause of the discrepancy in transition pressure at 0 K. In addition, the "-FeSi phase was found to be a semi-conductor and the CsCl phase found to be metallic, in agreement with previous studies (Voˇcadlo et al., 1999).

Further work has been carried out on the FeSi static calculations for comparison with experimental results. This includes a comparison of the crystal structure of the "-FeSi phase as well as comparisons of the calculated equations of state and experimentally measured equations of state (see Appendix B for further details).

106 Chapter 4. The Calculated FeSi Phase Diagram

4.5 The FeSi Phase Transition at High

Temperatures

The following section details the high temperature calculations carried out to determine the phase boundary between "-FeSi and CsCl-FeSi at high temperatures and pressures.

The results have been compared to experimental measurements of the phase boundary, which are described in the following section.

4.6 Experiments on the FeSi Phase Transition

The "-FeSi CsCl-FeSi phase transition was ﬁrst discovered from ab initio calculations ! at 0 K (Voˇcadlo et al., 1999). There have since been several experimental studies investigating the phase transition of FeSi at high pressures and temperatures. The high- pressure CsCl-FeSi phase was ﬁrst observed experimentally by Dobson et al. (2002) in a multi-anvil press experiment. This phase was synthesised from a starting mixture of

Fe and Si powders. Analysis of the quenched sample by X-ray di↵raction and electron microprobe indicated that the CsCl-FeSi phase was stable from 24 GPa and 1950 50 ± K.

The results from Dobson et al. (2002) agree well with LH-DAC experiments carried out by Lord et al. (2010), who mapped the "-FeSi to CsCl-FeSi phase transition using synchrotron X-ray di↵raction techniques. They ﬁnd that the phase boundary has a shallow negative Clausius-Clapeyron slope, with the equation of the linear phase boundary being

P = 0.0551T + 121.6 (4.1) where P is pressure in GPa and T is temperature in Kelvin. As seen in Figure 4.6, all their experiments are satisﬁed by this line, with the exception of one point at 13 GPa and 1800 K which Lord et al. (2010) attribute to a failed reversal experiment. The slope of the transition, 55 MPa/K, is quite similar to the slope of the corresponding

107 4.6. Experiments on the FeSi Phase Transition transition in NiSi, found to be 67 MPa/K (see Chapter 6 and Lord et al. (2014)). Also of note is the apparent persistence of "-FeSi as a metastable phase at high pressures. As seen in Figure 4.6, none of the experiments show the transition running to completion

(divided squares). The exception is an experiment at 46 GPa and 1830 K (not shown in Figure 4.6).

Figure 4.6: Phase diagram of FeSi as determined by Lord et al. (2010) from LH-DAC experiments. Melting experiments are shown as crosses and in situ XRD experiments are shown as squares (ﬁlled for "-FeSi and open for CsCl-FeSi, a mixture of both phases shown as divided squares). Also plotted are MAP experiments in circles (Dobson et al., 2002). The negative Clapeyron slope of the "-FeSi to CsCl-FeSi phase boundary is clearly shown. Figure after Lord et al. (2010).

More recently, further in situ LH-DAC experiments have been carried out on FeSi.

These experiments, however, contradict the negative Clapeyron slope found by Lord et al. (2010). Both Fischer et al. (2013) and Geballe and Jeanloz (2014) ﬁnd a vertical phase boundary between the two phases, although the two disagree on the transition pressure; Fischer et al. (2013) puts the phase boundary at 42 GPa, while Geballe and

Jeanloz (2014) ﬁnd the transition occurring at 30 GPa (see Figure 4.7). Both studies observe a two-phase stability ﬁeld, with Fischer et al. (2013) reporting a surprisingly large two-phase region for FeSi between 14 and 42 GPa. Geballe and Jeanloz (2014)

ﬁnd a much smaller two-phase region, the size of which varies depending on the pressure

108 Chapter 4. The Calculated FeSi Phase Diagram medium used – in argon, the two-phase stability ﬁeld exists between 23 and 30 GPa; in neon, between 30 and 32.3 GPa.

(a) (b)

Figure 4.7: Phase diagrams determined experimentally by (a) Fischer et al. (2013) and (b) Geballe and Jeanloz (2014). Both find that the phase boundary is vertical. Fischer et al. (2013) find the phase transition to be complete at 42 GPa, with a two phase region consisting of both phases from 14 GPa, while Geballe and Jeanloz (2014) find a transition pressure of 30 GPa and observe a much smaller two phase region, whose size varies depending on whether argon or neon is used as the pressure medium. In (a) purple crosses denote "-FeSi and CsCl-FeSi and red triangles for CsCl-FeSi; in (b) red symbols are for "-FeSi and blue symbols are for CsCl-FeSi.

It is clear that there is an underlying problem with the experimental studies, resulting in a range of possible phase diagrams of FeSi. At least one of the problems appears to be the metastable persistence of "-FeSi, which is common to all four experimental studies. One method of investigating this problem is to determine the phase boundary computationally, using calculations which only take into account the free energy of the structure to determine stability. This is the approach adopted here. Lattice dynamics calculations have been used to calculate the free energies of the "-FeSi and

CsCl-FeSi structures, from which the high temperature FeSi phase diagram can be constructed.

4.7 Lattice Dynamics Calculations of FeSi

The Gibbs free energy is required to determine which phase is stable at a given pressure and temperature. This is obtained following the method detailed in Section 2.9.3. The

109 4.7. Lattice Dynamics Calculations of FeSi

Gibbs free energy, G, can be written as

G = U + PV TS = F + PV (4.2) where U is the internal energy, P is the pressure, V is the volume, T is temperature, S is enthalpy and F is the Helmholtz free energy. To obtain the Helmholtz free energy, static calculations were ﬁrst carried out to obtain Fperfect (V,T), the Helmholtz free energy at 0 K, but with electronic free energy included to account for electronic temperature e↵ects. The program Phon was then used to generate a 2 2 2 supercell as well ⇥ ⇥ as the required displacements needed to calculate Fvib(V,T), the vibrational Helmholtz free energy. Combining the two gives the total Helmholtz free energy, Ftotal(V,T), as a function of volume and temperature

Ftotal(V,T)=Fperfect (V,T)+Fvib(V,T) (4.3)

The pressure can then be derived from the Helmholtz free energy, with respect to volume

@F P = (4.4) @V ✓ ◆T

Finally, the Gibbs free energy can be obtained from

G(V,T)=F (V,T)+P (V,T)VT (4.5)

G(P, T) can then be obtained by interpolation of the G(V,T) and P (V,T) values (see

Section 2.9.3). Figure 4.8 shows typical graphs obtained at each step of this analysis for both structures at 200K.

Unit cell sizes corresponding to approximately 8 – 12.5 A˚3 per atom were chosen to

110 Chapter 4. The Calculated FeSi Phase Diagram encompass a range of pressures from 0 to 400 GPa for both structures, and the calculations were carried out at a range of temperatures, from 500 to 3000 K, in 500 K steps, with additional calculations carried out at 200 and 400 K. However, only temperatures up to 2000 K were used in the analysis once it was clear temperatures above 2000 K exceeded the melting temperature of FeSi.

From this analysis, a series of G P graphs can be plotted for each temperature (see Figure 4.8c). Since the same temperature values were chosen for both structures, at each temperature there exists two G P graphs – one for "-FeSi and one for CsCl-FeSi. The point at which the two lines cross gives the transition pressure at that particular temperature. Repeating this process across all temperatures gives the transition pressure for each temperature value considered.

111 4.7. Lattice Dynamics Calculations of FeSi

(a)

(b)

(c)

Figure 4.8: A typical set of graphs obtained at 200 K of (a) Helmholtz free energy; (b) Pressure; and (c) Gibbs free energy; for CsCl-FeSi and "-FeSi at 200 K. The transition pressure at this temperature is 12.7 GPa.

112 Chapter 4. The Calculated FeSi Phase Diagram

4.8 FeSi Phase Diagram

Lattice dynamics calculations indicate that the phase boundary between "-FeSi and

CsCl-FeSi is very steep with a negative Clapeyron-slope. Figure 4.9 shows the calculated phase boundary along with the results from the experimental studies; the calculated transition pressures are also listed in Table 4.2. As discussed above, the 0 K value of the transition pressure is still in dispute, and therefore may be at a value higher than that shown here, in keeping with the negative Clapeyron slope. In addition, the phonon dispersion curve for "-FeSi is shown in Figure 4.10.

Table 4.2: Calculated transition pressures for "-FeSi CsCl-FeSi ! Temperature /K Transition Pressure /GPa 200 12.7 300 12.5 400 12.5 500 12.4 1000 11.4 1500 10.2 2000 8.3

From Figure 4.9, it can be seen that the calculated phase boundary is lower in pressure than in all experiments, but matches best with the boundary deﬁned by Fischer et al. (2013), at 14 GPa. However, this boundary is not well deﬁned by Fischer et al.

(2013) – their lowest pressure experiments already show the presence of a two phase mixture. The Clapeyron slope of the calculated phase boundary matches best with the experimental phase boundaries deﬁned by Fischer et al. (2013) and Geballe and

Jeanloz (2014), both of whom report a vertical phase boundary, in sharp contrast to

Lord et al. (2010) who observes a much shallower Clapeyron slope.

113 4.8. FeSi Phase Diagram

Figure 4.9: Phase diagram of FeSi, showing the calculated transition pressures (in red diamonds; filled for lattice dynamics calculations and open for static calculations). Also plotted are the results from experimental studies. Blue squares from Lord et al. (2010) and circles from Dobson et al. (2002) (filled indicate "-FeSi structure only, open indicate CsCl-FeSi and half filled indicate a mixture of two phases. The phase boundary (blue line) as determined by Lord et al. (2010) has also been plotted. Also shown is the melting curve (pink line) of FeSi as measured by Lord et al. (2010). Phase diagram of Geballe and Jeanloz (2014) is shown as shaded regions (red area for "-FeSi, purple area for mixture of two phases and blue area for CsCl-FeSi. Phase boundaries of Fischer et al. (2013) plotted as straight black lines; phases annotated on the graph in black.

Γ P N H Γ N

Figure 4.10: The calculated phonon dispersion curve of "-FeSi along the path P N H N, where is [000], P is [ 1 1 1 ], N is [00 1 ] and H is [ 1 1 1 ]. 4 4 4 2 2 2 2

114 Chapter 4. The Calculated FeSi Phase Diagram

4.9 Conclusions

Since the calculated phase boundary is at lower pressures and temperatures than the experimentally observed phase boundaries, it seems likely that the phase transition is kinetically inhibited in experiments. Lattice dynamics calculations do not take into account any kinetic e↵ects; the stable structure is deﬁned as the lowest energy structure at a speciﬁed pressure and temperature. Therefore it is feasible for kinetic inhibition to distort the phase boundary measured experimentally, pushing the phase boundary to higher temperatures. Further evidence of this can be seen in the experiments themselves

– each of the experimental studies reports some degree of metastable persistence of "-

FeSi, with Lord et al. (2010) ﬁnding only one instance where the transition reaches completion to yield pure CsCl-FeSi at 46 GPa and 1830 K. Di↵erences in the size of the two-phase stability ﬁeld observed by experiments may then be due to experimental di↵erences that a↵ect whether complete transformation could take place, for example, whether the sample was held at high temperature and pressure for long enough.

However, kinetic inhibition alone may not be sucient to explain the large di↵erence seen between experiments. Another credible explanation is the possibility of Si di↵usion during heating. Dobson et al. (2002) used electron microprobe analysis to determine the exact stoichiometry of their samples, but this was not done in the LH-DAC experiments. It is possible that the stoichiometry of the samples di↵ered suciently in the

LH-DAC experiments which, when combined with the e↵ect of kinetic inhibition, are enough to explain the large discrepancies between the experiments. Giving credence to this argument is the fact that small changes in composition a↵ect the structure adopted by RuSi, an analogue material of FeSi; excess Ru favours the CsCl-structure whereas samples deﬁcient in Ru show a mixture of "-FeSi and Ru2Si3 (Voˇcadlo et al., 2000).

4.10 Further Work

It is clear that more work is required to deﬁnitively explain the di↵erences between experiments and calculations, as well as between the experiments themselves. Compar-

115 4.10. Further Work ison between the calculations and experiments indicate that there is a high activation energy barrier between "-FeSi and CsCl-FeSi. As noted by Voˇcadlo et al. (1999), the transformation path from "-FeSi to CsCl-FeSi would necessitate this, especially if the pathway requires a mid-transformation to the NaCl-structure to facilitate the transition. Evidently, the FeSi system is more complicated than ﬁrst expected.

Further calculations can help shed light on the enigmas of FeSi. Molecular dynamics can reveal what is happening at the atomic level during the phase transition, which would not only provide information on the transformation pathway, but also on the kinetics involved in the phase transition. Given the surprising complexity of the FeSi system, these calculations are also likely to reveal new information about the phase transition, which may further help explain the di↵erences seen. In addition to molecular dynamics calculations, lattice dynamics calculations on FeSi with varying stoichiometries could be carried out. This would determine whether silicon di↵usion in the LH-DAC experiments plays any part in the di↵erences seen.

These calculations have provided another piece of the puzzle concerning the FeSi phase diagram but more work needs to be done before all the di↵erences can be explained.

116 CHAPTER 5

Calculated Stabilities of NiSi-structured Phases in FeSi

and all the gold in the sun fell upon me.

— Sheryl Luna

5.1 Introduction

Only one phase transition has been observed in the FeSi system – the "-FeSi to CsCl-

FeSi transition discussed in the previous chapter. In contrast, the NiSi phase diagram is more complicated. Both calculations and experiments indicate that there is a range of previously unseen stable phases in NiSi (Voˇcadlo et al., 2012; Lord et al., 2012;

Wood et al., 2013). However, whether these new phases are stable or metastable in

FeSi, but have yet to be seen experimentally or computationally, is unknown. In this work, calculations at 0 K have been carried out to determine whether any of the NiSi- structured phases are also stable in FeSi.

Calculations on the NiSi system at 0 K were initially carried out by Voˇcadlo et al.

(2012), showing that a range of structures were stable with di↵erent phases coming into stability as pressure increased. The sequence of stable structures observed was as follows: MnP P4/nmm (or the CuTi phase) Pbma-I Pnma-III (FeB) CsCl, ! ! ! ! with transitions occurring at 23, 61, 168 and 247 GPa respectively. Following this, experiments by Wood et al. (2013) found a new stable phase in experiments carried out in the MAP. This phase, the Pmmn phase, hadn’t originally been considered in the calculations by Voˇcadlo et al. (2012), but further ab initio calculations revealed that

117 5.1. Introduction the Pmmn phase was more stable than either of the P4/nmm, Pbma-I or Pnma-III

(FeB) phases, resulting in a much simpler phase diagram at 0 K (see Figure 5.1). The new phase stability sequence in NiSi therefore became: MnP Pmmn CsCl, with ! ! transitions occurring at 21 and 264 GPa.

Figure 5.1: Phase diagram of NiSi based on 0 K ab initio calculations, showing the original NiSi phase diagram with a range of stable phases in black (Voˇcadlo et al., 2012). Newer calculations by Wood et al. (2013) indicate that the Pmmn phase (in red) is more stable between 21 GPa and 264 GPa. The new sequence of stable phases in NiSi is therefore MnP Pmmn CsCl. ! !

Despite the simpler revised phase diagram, it should be noted that this is calculated at

0 K. The di↵erences in enthalpy between the stable and unstable phases are very small

(see Figure 5.1), and therefore it is possible that an increase in temperature would bring the unstable phases into stability and change the phase diagram. Indeed, experiments have shown that NiSi with the "-FeSi structure possesses a ﬁeld of stability at high temperature (Lord et al., 2012). In this chapter, calculations have been carried out to determine whether any of these new structures in NiSi are stable in FeSi, using the same method as in Chapter 4. The following sections describe the structures of the phases considered, the results of the calculations and ﬁnally a discussion of the results.

118 Chapter 5. Calculated Stabilities of NiSi-structured Phases in FeSi

5.2 NiSi Phases

Some of the stable phases seen in Voˇcadlo et al. (2012) were a result of spontaneous transformations into more thermodynamically stable phases after ionic relaxation of the starting structure. In order to replicate their calculations as closely as possible, the same starting structures have been used here, with the exception of the NiAs, ‘anti’-

NiAs and NaCl structures, which were all found to be unstable by a large margin in

NiSi. Therefore the structures that have been considered are the MnP, ‘anti-MnP’,

Pbma-I and WC structures, as well as the new Pmmn phase found by Wood et al.

(2013).

5.2.1 The MnP Phase

The MnP structure (see Figure 5.2) is orthorhombic with space group P nma.In

NiSi, the cell parameters are found experimentally to be a =5.18 A,˚ b =3.34 A˚ and ˚ 1 c =5.61 A, with Ni and Si atoms situated on special 4c positions at (x, 4 ,z), where x 0.008,z 0.188 for Ni and x 0.679,z 0.917 for Si (Toman , 1951). These are ⇠ ⇠ ⇠ ⇠ the starting values used for the FeSi atoms, with Fe replacing Ni.

Figure 5.2: Projection of the MnP structure of FeSi viewed down the [001] direction. The purple atoms are Fe and blue are Si. The ‘anti-MnP’ phase is the same but with Fe and Si atoms positions swapped around.

As described in Voˇcadlo et al. (2012), the MnP-structure is derived from a distortion of the NiAs structure, which consists of close-packed layers following an ABACA pattern

– the Ni atoms form the A layer and the As atoms make up the B and C layers. In the

MnP structure, the close-packed layers are o↵set from each other such that the Fe and Si

119 5.2. NiSi Phases atoms are both six-fold co-ordinated with the other atoms. The six Si atoms surround the Fe atom at the vertices of an octahedron whilst the Si atoms are surrounded by six

Fe atoms forming a trigonal prism.

5.2.2 The ‘anti-MnP’ Phase

The ‘anti-MnP’ structure is the same as the MnP structure but with the Fe atoms switched with the Si atoms (see Figure 5.2). It therefore shares the same space group,

P nma, as the MnP-structure, as well as the same cell parameters and fractional co- ordinates, but with the Fe and Si atom positions switched (see Section 5.2.1).

5.2.3 The Pbma-I Phase

The calculations carried out by Voˇcadlo et al. (2012) found that the MnP-phase un- dergoes two spontaneous transformations to form the Pnma-II phase and then the

Pnma-III phase with increasing pressure (see Figure 5.1). These two phases belong to the Pnma space group and with each phase transformation, the atoms move closer to forming a close-packed layer. The Pnma-III phase is essentially hexagonally close- packed, but with the symmetry broken since the layers of atoms consist of both atom types instead of one (see Voˇcadlo et al. (2012) for more details). The Pbma-I phase, named after its space group, is formed by switching around the places of half the Ni and Si atoms in the Pnma-III phase.

Figure 5.3: The Pbma-I phase as shown in Voˇcadlo et al. (2012), with blue spheres representing Ni atoms (replaced with Fe atoms here) and darker Si atoms. Also shown are the pseudo- 1 3 hexagonal cells at y = 4 and y = 4 showing the pseudo-close-packed layers of Fe and Si atoms.

120 Chapter 5. Calculated Stabilities of NiSi-structured Phases in FeSi

Figure 5.3 shows the almost hexagonal ‘cells’ in the Pbma-I phase. The calculated cell parameters of this phase are a =4.802 A,˚ b =4.461 A,˚ c =4.2 A,˚ with Ni and Si atoms situated on special 4d positions at (x, y, 1 ), where x 0.609,y 0.07 for Ni and 4 ⇠ ⇠ x 0.112,y 0.930, at P =0(Voˇcadlo et al., 2012). These were the starting values ⇠ ⇠ used for reﬁnement for the FeSi calculations, again with Fe substituting for Ni.

5.2.4 The WC Structure

The WC phase consists of layers of atoms of one type stacked in an ABAB arrangement, as in the hexagonally close-packed arrangement, and is shown in Figure 5.4. The space group of the WC structure is P 6¯m2. The Fe atoms are found at (0, 0, 0) and Si atoms 1 2 1 ˚ ˚ at ( 3 , 3 , 2 ). The cell parameters at P = 0 are a = b =2.91 A, c =2.84 A(Leciejewicz, 1961).

Figure 5.4: The WC phase, with Fe atoms in purple and Si atoms in blue. The Si atom is at 1 2 1 3 , 3 , 2 . 5.2.5 The Pmmn Phase

The Pmmn phase is orthorhombic, with space group P mmn. The Ni and Si atoms are

1 1 found on special 2a positions at ( 4 , 4 ,z), with z =0.3442 for Ni and z =0.8423 for Si, and cell parameters of a =3.27 A,˚ b =3.03 A,˚ c =4.70 A˚ at 1300 K and 17.5 ⇠ GPa (Wood et al., 2013). The Pmmn phase is very similar to the P4/nmm phase, or

CuTi phase, that Voˇcadlo et al. (2012) found to be stable in their calculations. Like the

CuTi phase the Pmmn phase has atoms that are six-fold coordinated with each other.

121 5.3. VASP Calculations

This results in a distorted octahedron, with one half longer than the other half in the

[001] direction (see Figure 5.5). As Wood et al. (2013) point out, the Pmmn structure can be viewed as a distorted form of NaCl, from which it can transform in a series of continuous steps.

Figure 5.5: The Pmmn phase, as shown in Wood et al. (2013), with blue atoms representing Ni (replaced with Fe here) and black atoms being Si. It can be seen that the co-ordination of the atoms results in a series of edge-sharing distorted octahedra.

5.3 VASP Calculations

The method used to determine stability of the structures described in Section 5.2 is the same as in Chapter 4 (see Section 2.8 for further details). As before, convergence tests were carried out to determine the k-point grid and energy cut-o↵ value required for each structure (see Table 5.1).

Table 5.1: Sizes of k-point grids used for the NiSi-structured phases in FeSi

Structure k-point grid CsCl 17 17 17 ⇥ ⇥ "-FeSi 9 9 9 ⇥ ⇥ MnP 7 7 7 ⇥ ⇥ Anti-MnP 7 7 7 ⇥ ⇥ Pbma-I 7 7 7 ⇥ ⇥ WC 15 15 15 ⇥ ⇥ Pmmn 9 9 9 ⇥ ⇥

As for the calculations in Chapter 4, the criteria for the convergence tests were an error of less than 0.001 eV per atom. The calculated energy-volume values were ﬁtted to a third-order Birch-Muraghan Equation of State (see Section 2.8.2) to obtain the enthalpy-pressure values. A plot of the enthalpy di↵erences, relative to that of CsCl-

122 Chapter 5. Calculated Stabilities of NiSi-structured Phases in FeSi

FeSi, shows that the only two phases that are stable are the "-FeSi and CsCl-FeSi phases

(see Figure 5.6). The remaining structures all have a much greater enthalpy di↵erence, of more than 0.1 eV per atom at all pressures, meaning that even at high temperatures, these phases are unlikely to become stable. The enthalpy curve for the MnP-structure, however, shows a rapid decrease as pressure approaches 0. Extrapolating the enthalpy- pressure curves past 0 GPa to negative pressures show that the MnP curve crosses over the "-FeSi curve at around 34 GPa (see Figure 5.6).

Figure 5.6: Plots of enthalpy against pressure for the di↵erent FeSi structures, normalised against CsCl-FeSi, showing transitions from MnP to "-FeSi at 34 GPa and from "-FeSi to CsCl-FeSi at 11 GPa.

The values for the parameters obtained from ﬁtting the Birch-Murnaghan Equation of

State for each structure have been listed in Table 5.2. This shows that almost all the structures considered here have higher V0 and lower K0 values than the only two stable structures above 0 GPa – CsCl-FeSi and "-FeSi. The K0 values of anti-MnP, P bma-I, WC and P mmn are very similar, di↵ering only by 2 GPa, and are signiﬁcantly lower

than either that of CsCl or for "-FeSi. The MnP structure has a slightly lower K0 value at 213 GPa. In contrast, the K0 values for all structures, including "-FeSi and

123 5.4. Conclusions

CsCl-FeSi, are fairly similar, varying only by 0.3.

Table 5.2: Fitting parameters for the third-order BMEoS ﬁts of the NiSi-structured phases in FeSi

Parameters CsCl "-FeSi MnP Anti- P bma-I WC P mmn MnP 3 -1 V0 / A˚ atom 10.52 11.01 11.52 10.86 11.04 11.30 11.11 K0 /GPa 236 231 213 222 220 200 222 K0 4.4 4.2 4.1 4.1 4.1 4.3 4.2

V0 is the volume, K0 is the bulk modulus, and K0 is the derivative of K0, all at P =0

5.4 Conclusions

The calculations carried out here indicate that there are only two stable phases in the

FeSi system, the "-FeSi and CsCl-FeSi phases. The energy gap separating the phases means that, despite these calculations being carried out at 0 K, it is unlikely that any other phases will become stable at high temperatures since the energy gap to be overcome is so large.

Extrapolation of the H P curves to negative pressures show that the MnP-phase becomes stable at 34 GPa. Taking this into account, the stability sequence of FeSi then becomes MnP "-FeSi CsCl-FeSi. This stability sequence is also observed in ! ! the RuSi, OsSi and CoSi systems, obtained from ab initio static calculations, similar to those carried out here (Hernandez et al., 2015). As with FeSi, Hernandez et al. (2015)

ﬁnd the MnP structure is only stable at negative pressures at 0 K, with transition pressures of 15, 6.3 and 10 GPa respectively for each of RuSi, OsSi and CoSi (see Figure 5.7), indicating that at 0 K, the MnP phase is stabilised as one descends Group

8. A similar pattern is seen with the Group 9 elements. Their calculations did not extend past 300 GPa, but it is possible that the same MnP "-FeSi CsCl-FeSi ! ! stability sequence is seen with the Group 9 elements, with CsCl-FeSi becoming stable above 300 GPa.

However, this does not help explain the di↵erence in phase stability between FeSi and

NiSi. The CoSi system displays the same phase stability sequence as FeSi at 0 K, rather

124 Chapter 5. Calculated Stabilities of NiSi-structured Phases in FeSi

Figure 5.7: Figure from Hernandez et al. (2015) showing the 0 K phase diagrams of various transition metal silicides, as found from static ab initio calculations. than the MnP P mmn CsCl sequence seen in NiSi. As with FeSi, Hernandez et al. ! ! (2015) found that the P mmn structure is also never the most thermodynamically stable phase in CoSi.

It is still unclear why the FeSi and NiSi systems behave so di↵erently, making it dif-

ﬁcult to predict which phases would be stable in a Fe-Ni-Si system – a composition more likely to be found in the cores of terrestrial planets than either pure FeSi or pure

NiSi. Although at high pressures, the CsCl-structured phase is stable for both FeSi and

NiSi, at low pressures, the two systems have di↵erent stable phases, with "-FeSi being stable in FeSi and the MnP and P mmn phase stable in NiSi. This complicates the issue further when considering the cores of smaller terrestrial planets such as Mercury and Mars, where core pressures are much lower than in Earth. It is therefore important to understand exactly how and why phase stability di↵ers in the FeSi and NiSi systems, in order to be able to predict how a Fe-Ni-Si ternary phase diagram might look.

One possible method of investigating this is to determine phase diagrams for various

(Fe,Ni)Si alloys at a range of x values, which may be helpful in gaining an insight into how the addition of Ni into the FeSi system a↵ects phase stability. Of interest would be whether the addition of Ni is enough to bring either the MnP or P mmn phase into stability.

125 126 CHAPTER 6

The Calculated "-FeSi CsCl ! Phase Transition in NiSi

Do not make things too easy. There are rocks and abysses in the mind As well as meadows.

— Martha Baird

6.1 Introduction

The "-FeSi to CsCl phase transition is not only seen in the FeSi system, but is also present in the NiSi system. As with the phase transition in FeSi, calculations on the phase transition in NiSi at 0 K have been carried out previously (Voˇcadlo et al.,

2012), but not at high temperatures. The boundary of this phase transition at high temperatures has been calculated here, following the same lattice dynamics method as used for the FeSi system (see Chapter 4).

6.2 The "-FeSi CsCl Phase Transition !

As discussed in Chapter 5, the phase diagram of NiSi at 0 K is more complex than FeSi.

Calculations show that the stable high pressure phase in the NiSi system is also CsCl, but that this structure comes from a transformation from the Pmmn phase, with the phase stability sequence being MnP Pmmn CsCl (Wood et al., 2013). However, ! ! as noted by Wood et al. (2013), these calculations were carried out at 0 K, and the very small energy di↵erences between the di↵erent phases mean the phase diagram is liable to change at high temperatures. This is the case seen experimentally. Laser- heated DAC experiments carried out by Lord et al. (2012) show that at high pressure

127 6.2. The "-FeSi CsCl Phase Transition ! and temperature, there is also a phase transition from the "-FeSi to the CsCl phase.

Calculations by Voˇcadlo et al. (2012) show that at 0 K this transition occurs at 168

GPa (see Figure 6.1).

Figure 6.1: The phase diagram of NiSi at 0 K from calculations carried out by Voˇcadlo et al. (2012), showing the transition (highlighted in red) from "-FeSi (dotted line) to the CsCl phase (solid horizontal line) at 168 GPa.

In the LH-DAC experiments performed by Lord et al. (2010), NiSi samples were compressed and laser heated to pressures of 124 GPa and temperatures of 2700 K. The sample was then quenched and analysed by X-ray di↵raction. Annealing the sample then quenching ensured that any phase transitions were captured and maintained in the XRD patterns. These experiments allowed Lord et al. (2012) to bracket the low pressure phase transition from the MnP phase to the "-FeSi phase at 12.5 4.5 GPa ± and 1550 150 K and the higher pressure phase transition from "-FeSi to the CsCl- ± structured phase at 46 3 GPa and 1900 150 K. More recent work detected the ± ± triple point between "-FeSi, CsCl and liquid phases at 28.5 1.3 GPa and 2165 61 ± ± K, as well as providing an additional bracket for the "-FeSi/CsCl phase boundary at

58.5 1 GPa and 1715 35 K (Lord et al., 2014). This leads to an experimental ± ± Clapeyron slope for the "-FeSi to CsCl transition of 67 MPa/K, comparable to that found for FeSi of 55 MPa/K (Lord et al., 2010).

Further high pressure experiments showed the existence of the P mmn phase. Wood

128 Chapter 6. The Calculated "-FeSi CsCl Phase Transition in NiSi ! et al. (2013) carried out three MAP experiments, ﬁnding a mixture of three phases in all (see Table 6.1), helping to map the lower pressure end of the NiSi phase diagram.

Additional high pressure experiments in the MAP (Dobson et al.,submitted)further constrain the phase boundaries between the lower pressure phases.

Table 6.1: NiSi phases found by Wood et al. (2013)

Pressure /GPa Temperature /K Phases Found 14.0 1 1270 50 MnP + "-FeSi ± ± 17.5 1 1310 20 Pmmn + "-FeSi ± ± 17.5 1 1223 50 Pmmn + "-FeSi ± ± The di↵erent phases found in the MAP experiments of Wood et al. (2013). These are also plotted in Figure 6.3.

As a result of these high pressure experiments, the phase diagram of NiSi is fairly well deﬁned, with the exception of the "-FeSi to CsCl-NiSi phase transition. Here, this phase boundary has been calculated using lattice dynamics, helping complete the

NiSi phase diagram at high pressures. Since these calculations follow the same method as those covered in Chapter 4 and are on the same "-FeSi CsCl phase transition, ! but in the NiSi system rather than the FeSi system, a direct comparison of the two is possible.

6.3 Lattice Dynamics Calculations

As with the FeSi system, the phase boundary between the "-FeSi and CsCl-structured phase of NiSi was calculated using the lattice dynamics method (see Section 2.9.2).

In order to facilitate comparison between the two sets of calculations, parameters for the NiSi calculations, such as temperature and pressure ranges, were chosen to be as similar as possible to those chosen for FeSi. Unit cell volumes corresponding to between

6 and 11 A˚3 per atom were chosen, which give pressures of roughly 0 to 400 GPa, and temperatures ranged from 0 to 3000 K, increasing in 500 K steps, with additional calculations carried out at 300 and 750 K. As with the FeSi calculations (see Chapter 4), only temperatures up to 2000 K were used in the analysis due to temperatures above this exceeding the melting temperature of NiSi. As with the FeSi calculations, the PBE functional was used with the GGA implementation of the pseudopotential (Perdew

129 6.3. Lattice Dynamics Calculations et al., 1996). The same convergence criteria of < 0.001 eV per atom was chosen, resulting in k-point grids of 17 17 17 for the CsCl structure and 9 9 9 for the ⇥ ⇥ ⇥ ⇥ "-FeSi structure. Energy cut-o↵s of 800 eV were used.

The method used here was the same as that of the FeSi calculations in Chapter 4.

As with the FeSi calculations, the total Helmholtz free energy was ﬁrst obtained by separately calculating Fperfect (V,T) and Fvib(V,T), the Helmholtz free energy at 0 K and the vibrational Helmholtz free energy respectively (see Sections 2.8.1 and 2.9.2 for further details). Static calculations at 0 K are performed to obtain Fperfect (V,T). To calculate F (V,T), the program Phon was used to generate a 2 2 2 supercell along vib ⇥ ⇥ with the required displacements for the Fvib(V,T) calculations (see Section 2.9.3 for further details). Adding together Fperfect (V,T) and Fvib(V,T) gives the total Helmholtz free energy, Ftotal, as a function of volume, V , and temperature, T ,

Ftotal(V,T)=Fperfect (V,T)+Fvib(V,T) (6.1)

Third order polynomials were then ﬁtted to the F V values for each temperature, and from this, the pressure, P , was obtained

@F P = (6.2) @V ✓ ◆T

The Gibbs free energy, G, is then found from

G(V,T)=F (V,T)+P (V,T)VT (6.3)

Interpolation of the G(V,T) and P (V,T) values allows the Gibbs free energy as a function of pressure and temperature – G(P, T) – to be obtained. Figure 6.2 shows typical F V , P V and G P graphs for the CsCl and "-FeSi structures at 500 K. As with the FeSi calculations, the above analysis results in a series of G P curves, such that at each temperature there is a G P curve for both "-FeSi and CsCl-FeSi

130 Chapter 6. The Calculated "-FeSi CsCl Phase Transition in NiSi !

(see Figure 6.2c). The point at which these two curves cross determines the transition pressure at that temperature. Therefore the transition pressure for each temperature chosen can be found.

131 6.3. Lattice Dynamics Calculations

(a)

(b)

(c)

Figure 6.2: Graphs of (a) Helmholtz free energy; (b) Pressure; and (c) Gibbs free energy; for the "-FeSi and CsCl structures in NiSi at 500 K. The transition occurs at 143.2 GPa

132 Chapter 6. The Calculated "-FeSi CsCl Phase Transition in NiSi !

The transition pressures at the temperatures considered are listed in Table 6.2 and are shown in Figure 6.3, where the calculated transition pressures are plotted as red diamonds.

Table 6.2: Calculated transition pressures for the "-FeSi CsCl phase transition in NiSi ! Temperature /K Transition Pressure /GPa 0 158.0 300 151.1 500 143.2 750 131.9 1000 118.2 1250 102.6 1500 82.2 2000 24.8

Also shown in Figure 6.3 is the second order polynomial that was ﬁtted to these points.

This ﬁt had the equation

T = 0.089P 2 +2.0281P + 1988.6 (6.4) where T is in Kelvin and P is in GPa. Although thermodynamics requires a func-

dT tion which describes the dP boundary as inﬁnite at T = 0, the polynomial adopted seems to be adequate above around 300 K. Figure 6.3 shows that the calculated phase boundary agrees remarkably well with the experimentally constrained phase boundary

(Lord et al., 2012, 2014). Although ideally, more experiments need to be carried out to conﬁrm the calculated phase boundary between the two phases, it would appear that the agreement between the calculations and experiments is very good. Additionally, there is very good agreement of the transition pressure at 0 K between the calculations carried out here and that of Voˇcadlo et al. (2012).

133 6.4. Conclusions

Figure 6.3: The phase diagram of NiSi, showing the calculated phase boundary in red (ﬁlled red diamonds from this study; open red diamond from Voˇcadlo et al. (2012)). The melting curve of NiSi is shown in blue (see Chapter 7 and Lord et al. (2014)) and phase boundaries of the MnP, P mmn and "-FeSi phases in black (Dobson et al., submitted). MAP experiments (Wood et al., 2013) are in purple and in situ DAC experiments (Lord et al., 2012) are in green. The "-FeSi + CsCl-FeSi + liquid triple point is shown as a blue triangle (Lord et al., 2014). The "-FeSi to CsCl phase transition is drawn as a green line, dashed to show extrapolation. See graph for all other symbols.

6.4 Conclusions

The lattice dynamics calculations carried out here indicate very good agreement with experimental ﬁndings. To fully test the lattice dynamics method in NiSi, the phase boundaries between the lower pressure phases – the MnP, "-FeSi and Pmmn phases – could be calculated to see whether there is good agreement at low pressures with experiments. This would be a particularly rigorous test since these boundaries have been well constrained experimentally by MAP experiments (Dobson et al.,submitted).

This result yet again raises the question as to why the NiSi system di↵ers so much from the FeSi system. Despite there being four stable phases in NiSi, in contrast to just

134 Chapter 6. The Calculated "-FeSi CsCl Phase Transition in NiSi ! two in FeSi, determining the NiSi phase diagram has proved relatively straightforward, using both computational and experimental methods. If the underlying problem with

FeSi is due to kinetic inhibition, it seems odd that the NiSi system seemingly does not su↵er from this problem at all. Unlike FeSi, the experiments and calculations for

NiSi show very good agreement, suggesting that NiSi may be less sensitive to small changes in stoichiometry. As with FeSi, molecular dynamics simulations would provide additional information on the NiSi system and could also reveal any other potential causes for the di↵erences between the two systems, such as di↵erences in magnetic spin. Calculations for both FeSi and NiSi were spin-polarised with no total magnetic moment present after relaxation in either case for both structures. However, di↵erences in the magnetic structure were not considered – an investigation into this can help rule out di↵ering magnetic structures playing a part in the di↵erence seen between the two systems.

The NiSi lattice dynamics calculations again show that there are some fundamental di↵erences between the NiSi and FeSi systems, making a full understanding of both systems necessary in order to fully understand the ternary Fe-Ni-Si system.

135 136 CHAPTER 7

NiSi Melting

Death is coming and you must build a starship to take you to Venus.

— D. Nurkse

7.1 Introduction

As discussed previously, in order to understand the Fe–Ni–Si ternary system, the endmember binary systems must ﬁrst be understood. Here, the melting curve of NiSi has been measured using a combination of di↵erent experiments. The majority of melting points were determined by o↵-line LH-DAC experiments, carried out at the University of Bristol. In addition, in situ XRD experiments were carried out at two synchrotron sources at a range of pressures. The in situ XRD experiments help provide conclusive evidence of melting of NiSi, and were used as a test for robustness of the o↵-line LH-

DAC experiments. These experiments were led by Oliver Lord and were primarily used to learn the techniques involved; the results have recently been published (Lord et al.,

2014).

7.2 Methods

Three di↵erent types of experiments have been used to measure the melting curve of

NiSi – o↵-line laser-heated diamond anvil cell (LH-DAC) experiments, carried out at the University of Bristol; in situ XRD experiments in the LH-DAC performed at the

ID27 beamline at ESRF, which used X-ray di↵raction to determine melting; and a

MAP experiment carried out at the X17B2 beamline at NSLS, using both in situ XRD

137 7.2. Methods and X-radiography to determine melting. The majority of the NiSi melting curve was measured using o✏ine LH-DAC experiments, with the two in situ experiments used to conﬁrm the melting curve at selected pressures. The methods of these experiments are described in detail in Chapter 3; an overview is provided here.

7.2.1 O↵-line LH-DAC Melting Experiments

The o↵-line LH-DAC melting experiments were carried out at the University of Bris- tol using Princeton-type symmetric DACs. These had diamond anvils with culet sizes of 200 or 250 µm diameter depending on the pressure required – higher pressure experiments required a smaller culet size. Rhenium gaskets were used in the DAC, and

NaCl was used as a pressure medium and to provide thermal insulation. The sample chamber contained a small foil of NiSi sandwiched between two discs of NaCl and a small amount of ruby was placed between the sample and NaCl on one side to allow for pressure determination using ruby ﬂuorescence (see Section 3.2.1). Once set up, the cell was heated in a nitrogen atmosphere at 120°C to remove any adsorbed water.

Heating of the sample was carried out in a double-sided heating geometry, using two 100

WdiodepumpedTEM00 (Gaussian mode) ﬁbre lasers, operating at 1070 nm. Further details on heating the sample are given in Section 3.2.3.

7.2.2 In situ LH-DAC Melting Experiments

LH-DAC melting experiments were also carried out at the European Synchrotron Radi- ation Facility (ESRF) on beamline ID27. These experiments followed the same sample chamber set-up in the DAC as the o↵-line LH-DAC experiments carried out at the

University of Bristol, with the exception that some experiments were carried out in gas membrane driven DACs rather than Princeton-type DACs, which allowed pressure to be controlled remotely. Further details are given in Section 3.5.3.

As with the o↵-line LH-DAC experiments, laser heating was carried out using a double sided geometry, with heated spots measuring 20 to 30 µm in diameter. A small area of the sample, measuring 2 2 µm centred on the heating spot was used to collect reﬂected ⇥ light to be analysed by spectroradiometry (see Section 3.2.3 for further details).

138 Chapter 7. NiSi Melting

A3 3 µm X-ray beam, with wavelength 0.3738 A,˚ was used to capture in situ ⇥ XRD patterns of the sample during heating. The X-ray beam was co-aligned with the laser-heated spot to ensure XRD patterns were taken of the heated area and the di↵racted X-rays were collected using a MAR345 CCD with exposure times of 2 to 10 seconds.

7.2.3 In situ MAP Experiment

In addition to the LH-DAC melting experiments, a melting experiment was also carried out in the MAP at the X17B2 beamline at the National Synchrotron Light Source

(NSLS). The details of this experiment are given in Section 3.5.4. As with the ESRF experiments, in situ XRD from an incident white X-ray beam was used to monitor the sample during heating. A 10 element energy-dispersive detector was used to measure the di↵racted X-rays. In addition to this, X-ray videography was also used, with a sampling rate of 10 frames per second. Using this, melt was additionally conﬁrmed by the presence of convection in the sample (see Figure 7.1).

(a) (b)

Figure 7.1: X-ray videography images showing the foil marker placed on top of the NiSi falling through the molten sample once melt occurs. On the left is the sample before melting, and on the right is the sample after melting.

139 7.3. The Melting Curve of NiSi

7.3 The Melting Curve of NiSi

Melting in the o↵-line LH-DAC experiments was determined by an abrupt change in slope of the temperature versus laser power function (see Section 3.2). For the in situ

LH-DAC experiments carried out at the ESRF, the appearance of di↵use scattering in the XRD patterns of the sample was used to determine melting instead (see Sec- tion 3.3). Since the in situ experiments are also laser heated, the ‘change in slope’ criterion of melting can be veriﬁed against the XRD patterns showing di↵use scattering. Comparison of the two shows that the two methods for determining melting agree very well (see Figure 7.2).

Figure 7.2: Melting of NiSi at 29 GPa from ESRF detected at 2180 K by (a) temperature versus laser power plot, with crossed circles indicating a mix of Pmmn and CsCl-structured phases; and (b) XRD patterns showing di↵use scattering occurring at the same temperature. XRD patterns colour-coded according to points in (a). Open circles indicate ﬁrst appearance of di↵use scattering. From Lord et al. (2014).

In addition to the X-ray videography discussed above, the appearance of di↵use scattering was also used to detect melt in the in situ MAP experiments at NSLS (see

Figure 7.3).

The melting temperatures determined from the three experiments are plotted in Fig- ure 7.4. Since it was not possible to determine thermal pressure in the o↵-line LH-DAC experiments – which make up the bulk of the melting data – none of the melting points have been plotted with thermal pressure included, although analysis of the in situ ex-

140 Chapter 7. NiSi Melting

Figure 7.3: The appearance of di↵use scattering seen in XRD patterns collected from experiments at NSLS, showing melting occurring at 1425 K. These experiments were at 5.5 GPa. periments carried out at ESRF indicate that the thermal pressure is approximately 6

GPa. The melting curve has been found from a ﬁt to the Simon-Glatzel equation

P P 1/C T = M ref T (7.1) M A +1 ⇥ 0 

where TM is the melting temperature in Kelvin at pressure PM in GPa, T0 is the melting temperature at Pref, the reference pressure, and A and C are ﬁtting parameters. Both

Pref and T0 were held for the fits; the resulting fitting parameters are listed in Table 7.1. There were too few points to fit for the solidus of the "-FeSi phase, and so the melting curve for this phase has been approximated.

Table 7.1: Fitting parameters for the Simon-Glatzel equation of NiSi

Phase T 0 /K Pref /GPa A /GPa C MnP 1275 0 10 5 4 1 ± ± CsCl 2165 28.5 12.8 6.1 4.7 1.4 ± ± T0 and Pref have been ﬁxed for both ﬁts; T0 for MnP-NiSi from Massalski et al. (1990)

In situ XRD also detected the broad two-phase region of the P mmn and "-FeSi phases;

Figure 7.5 shows XRD patterns taken during heating showing P mmn peaks at 1500

K at 29 GPa. As the sample is heated, these peaks start decreasing in intensity and eventually disappear, indicating a complete phase transition to the "-FeSi phase (see

141 7.3. The Melting Curve of NiSi also Figure 7.2).

Figure 7.4: The melting curve of NiSi, shown in blue, as fitted to the Simon-Glatzel equation (see Table 7.1). Thermal pressure is not included as this was not measured for the o↵-line experiments that make up the bulk of the data. Also shown are the phase boundaries between the MnP, "-FeSi and CsCl phases, with the red line representing the calculated phase boundary between "-FeSi and CsCl (see Chapter 6 for details). The "-FeSi melting curve is not well defined and more experiments are needed to fit this more accurately. Notably there is a lack of points around the triple point which makes this part of the melting curve dicult to fit.

142 Chapter 7. NiSi Melting

Figure 7.5: XRD patterns from LH-DAC experiments carried out at ESRF at 29 GPa, showing the disappearance of peaks associated with the Pmmn phase (indicated by arrows) on heating. This agrees with the ﬁndings of Dobson et al. (submitted).

7.4 Conclusions

In these experiments, the melting curve of NiSi has been mapped out using a combination of o↵-line LH-DAC experiments and high pressure in situ XRD experiments.

This reveals a relatively smooth melting curve despite covering three di↵erent phases in the pressure range. The results also show very good agreement between the in situ

XRD experiments and the o↵-line experiments, lending credence to the o↵-line LH-

DAC measurements, which do not rely on direct observation of melting. Additionally, the presence of a phase transition between Pmmn and "-FeSi was also detected. This phase transition was evidenced by the disappearance of the Pmmn peaks on heating, as seen in Figure 7.5.

The NiSi melting curve and phase transition between "-FeSi and CsCl phases resembles that for FeSi, but o↵set by about 20 GPa (Lord et al., 2010). However, the FeSi experiments of Lord et al. (2010) do not agree with others (see Chapter 4), which complicates the issue. Regardless, it is clear that the NiSi system is much less complicated than the FeSi system, despite exhibiting a greater number of polymorphs.

143 144 CHAPTER 8

Fe-FeSi Eutectic Melting

And the Moon and the Stars and the World

— Charles Bukowski

8.1 Introduction

Accurately measured melting curves are required to construct any phase diagram. Al- though the melting curves of FeSi and a variety of Fe-Si alloys have already been measured and reported (Lord et al., 2010; Fischer et al., 2013; Asanuma et al., 2010), the phase diagram of Fe-Si at high pressures is still incomplete. Determining the melting curve of the Fe-FeSi eutectic at a range of pressures would provide an important anchoring point in the high-pressure Fe-Si phase diagram, which in turn would aid the construction of the ternary Fe-Ni-Si phase diagram at high pressures. Here, o↵-line LH-

DAC melting experiments have been conducted at the University of Bristol to measure the Fe-FeSi eutectic melting curve.

8.2 The Fe-Si Phase Diagram

Various experiments have investigated the Fe-Si alloys, revealing a fairly complex phase diagram, with two eutectic points between Fe and FeSi at ambient pressure (see Fig- ure 8.1a). However, at high pressures the Fe-Si phase diagram is liable to change – the

Fe-Si phase diagram at 21 GPa (Kuwayama and Hirose, 2004) shows the eutectic point shifted closer to Fe from its ambient pressure composition (see Figure 8.1).

Although there has only been one eutectic melting experiment at 21 GPa (Kuwayama and Hirose, 2004), there have been a number of melting experiments on a range of

145 8.2. The Fe-Si Phase Diagram

(a)

(b)

Figure 8.1: The phase diagram of Fe-Si (a) at ambient pressure calculated by MTDATA from experiments collated by the National Physical Laboratory (Davies et al., 2002) with the two eutectic points indicated by red arrows; and (b) at 21 GPa, from high pressure MAP experiments (Kuwayama and Hirose, 2004). The FeSi composition corresponds to 33.46 wt.% Si.

Fe-Si alloys. The melting curves of Fe-9Si and Fe-16Si have been measured by Fischer et al. (2012, 2013) and Asanuma et al. (2010) investigated the melting curve of Fe-18Si

(see Figure 8.2). These melting curves may still prove useful for comparison to the

Fe-FeSi eutectic melting curve, since a fraction of the melt obtained would have an eutectic composition, in accordance with the lever rule. In addition, the melting curves

146 Chapter 8. Fe-FeSi Eutectic Melting of two Fe-Ni-Si alloys have been measured by Morard et al. (2011). They determined the melting curves of 80Fe-5Ni-15Si and 85Fe-5Ni-10Si, ﬁnding that the e↵ect of 5 wt.

% Ni has a negligible e↵ect on the melting curve of the Fe-Si alloys.

(a)

(b)

Figure 8.2: Phase diagrams of (a) Fe-9Si and Fe-16Si (Fischer et al., 2013), showing di↵erent regions of stability for the fcc, hcp, B2 (CsCl) and D03 (BiF3) phases, as well as the melting curve for both. Symbols are deﬁned as for Figure 8.3; and (b) Fe-18Si (Asanuma et al., 2010). Solid circles are the lower bound of the melting curve, found using the change of laser heating eciency, and open inverted triangles are the upper bound, determined by the quench texture of the samples. Filled triangles are for the bcc phase.

In addition to the Fe-Si phase diagram at 21 GPa, Fischer et al. (2013) have constructed the Fe-Si phase diagram at a range of pressures based on their own LH-DAC experiments as well as past experimental studies (see Figure 8.3). These phase diagrams have fairly well deﬁned boundaries between di↵erent structured phases, but there is a notable lack of points constraining the melting curves, in particular the eutectic point,

147 8.3. The Melting Curves of Fe and FeSi highlighting the need for an accurately measured eutectic melting curve.

Figure 8.3: Fe-Si phase diagrams at pressures of A) 50 GPa, B) 80 GPa, C) 125 GPa, and D) 145 GPa from Fischer et al. (2013). Phase diagrams constructed using data for pure-Fe (Komabayashi and Fei, 2010; Ma et al., 2004); Fe-3.4Si (Asanuma et al., 2008); Fe-9Si (Fischer et al., 2013); 85Fe-5Ni-10Si and 80Fe-5Ni-15Si (Morard et al., 2011); Fe-16Si (Fischer et al., 2012); Fe-18Si (Asanuma et al., 2010) and FeSi (Fischer et al., 2013; Lord et al., 2010). Points from these experiments are plotted on the phase diagrams. Grey crosses indicate melting; green crosses indicate fcc+hcp; filled blue circles for fcc+B2 (CsCl phase); open blue circles for metastable fcc+hcp+B2; filled pale blue diamonds for hcp; open red triangles for D03 (BiF3- type) structure; filled red triangles for B2 phase; filled orange squares for hcp+B2.

8.3 The Melting Curves of Fe and FeSi

The melting curves of both Fe and FeSi have been measured experimentally and are well documented. Lord et al. (2010) and Santamar´ıa-P´erez and Boehler (2008) both measured the melting curve of FeSi using LH-DAC experiments but with di↵ering results. The two experiments used di↵erent criteria to determine melting; Lord et al.

(2010) used a discontinuity in the laser power versus temperature function to pinpoint melting, while Santamar´ıa-Pérez and Boehler (2008) used the appearance of ‘speckle’ as an indication of melt motion. This di↵erence resulted in a significantly higher melting curve determined by Lord et al. (2010) compared to that of Santamar´ıa-Pérez and

Boehler (2008) (see Figure 8.4). However, Lord et al. (2010) veriﬁed their melting

148 Chapter 8. Fe-FeSi Eutectic Melting curve with in situ XRD on several melting experiments, ﬁnding a very good match with melting temperatures determined using the discontinuity in laser power versus temperature. Additionally, the ‘speckle’ technique has recently been discounted for determining melting (Anzellini et al., 2013), as described below. Therefore for comparison purposes in this work, the FeSi melting curve of Lord et al. (2010) has been used.

Figure 8.4: Plot of the melting curve of FeSi from LH-DAC experiments (shown as crosses). Also plotted is the melting curve of Santamar´ıa-P´erez and Boehler (2008), shown as a dashed line. Figure from Lord et al. (2010)

In the case of iron, there have been many experimental studies to determine the melting curve, each giving di↵erent results. At conditions matching the inner core boundary of

Earth, a wide range of melting temperatures have been measured, from 4850 to 7600 K

(Anzellini et al., 2013). In particular, experiments by Boehler (1993) measure a much lower melting temperature than both shock experiments and ab initio calculations

(Nguyen and Holmes, 2004; Alf`eet al., 2002). However, recent experiments carried out by Anzellini et al. (2013) appear to have reconciled the di↵erent results for the Fe melting curve. Using in situ fast X-ray di↵raction, they were able to show that the lower melting curve observed by Boehler (1993) corresponded to ‘fast recrystallisation’ of Fe whereby single-crystal spots appear and disappear at each XRD exposure (see

Figure 8.5). This would explain the ‘speckle’ observed by Boehler (1993), which they had attributed to melt motion.

The new melting curve measured by Anzellini et al. (2013) is in good agreement with

149 8.4. Experimental Methods

Figure 8.5: The melting curve of pure Fe as measured by Anzellini et al. (2013). Also shown is the boundary between fast and slow re-crystallisation, as well as the phase boundary between -Fe and "-Fe. Figure from Anzellini et al. (2013). both shock experiments and ab initio calculations of the melting of iron when extrapolated to high pressures.

8.4 Experimental Methods

O↵-line DAC melting experiments, carried out at the University of Bristol, were used to measure melting at the Fe-FeSi eutectic. As described in Section 3.2, melting is determined by the presence of a abrupt change in slope in the laser power versus temperature function. A typical heating graph is shown in Figure 8.6 where the change in slope is clearly seen occurring at 2832 K.

Preparation of the DAC and sample followed the details given in Section 3.2.1. The sample was prepared by cutting holes, roughly 40 µm in diameter, into a piece of Fe metal. These were then ﬁlled in with FeSi powder and the entire assembly compressed slightly in a DAC to ensure the hole was fully ﬁlled with FeSi. Each packed hole was then cut out ensuring sucient Fe metal remained surrounding the FeSi-packed hole.

This results in a roughly 80 µm diameter disc composed of a circle of FeSi surrounded by Fe metal.

150 Chapter 8. Fe-FeSi Eutectic Melting

Figure 8.6: A typical laser power vs temperature plot taken at 44 GPa, showing the change in slope indicative of melting at 2832 84 K. Red and blue circles indicate temperatures generated from 1D spectrometry analysis from± the left and right hand side of the sample, and green circles are temperatures as measured from 2D spectrometry. As can be seen, the temperatures from each line up very well.

Princeton-type symmetric DACs with stainless steel gaskets were used for the melting experiments. A hole drilled through the centre of the gasket formed the sample chamber, in which the sample is placed in-between two discs of KCl, which act as both pressure medium and thermal insulation. A small amount of ruby was placed on top of the entire sample assembly, for use in pressure determination by ruby ﬂuo- rescence (see Section 3.2.2). Further details on preparation of the DAC are given in

Section 3.2.1.

The sample was heated in a double-sided heating geometry, with both sides of the DAC monitored by a camera and a detector. Both 1D and 2D spectroradiometry were used to determine the temperature of the sample. In 1D spectroradiometry, a narrow slit samples the heated spot. The temperature is then found from ﬁtting the Wien function to the spectra of the incandescent light generated by heating (see Section 3.2.3 for further details). In contrast, 2D spectroradiometry samples the entire heated spot. A single CCD camera is used to collect images of the laser-heated spot sampled at four di↵erent wavelengths (670, 700, 800 and 900 nm). The four images are then combined

151 8.5. The Fe-FeSi Eutectic Melting Curve by spatial correlation, resulting in four intensity-wavelength data points at each pixel.

The temperature can then be determined at each pixel by ﬁtting to the Wien function, as with 1D spectroradiometry. A 2D heat map of the laser-heated spot can therefore be generated (see Figure 8.7).

Figure 8.7: Screenshot of the 2D analysis used in these experiments, showing the 2D temperature map of the sample at 93 GPa, as well as the intensity of the radiation measured and the error associated. In this particular example, there is a high error associated with the temperature reading on the lower right hand corner of the hotspot. This is reﬂected in the much higher temperature recorded on the temperature map, which is likely false.

8.5 The Fe-FeSi Eutectic Melting Curve

The melting temperatures obtained from the melting experiments are plotted in Fig- ure 8.8 and listed in Table 8.1, with a pressure o↵set of 5 % to account for thermal pressure. This ﬁgure is an estimate since it is not directly measurable in the o↵-line

DAC experiments, and was arrived at after discussion with Dr Oliver Lord. Compari- son of the melting points from these experiments with the melting curves of Fe and FeSi indicate that experiments 2, 3A and 4 are likely to be Fe melting and experiments 3B,

5A, 5B and 6A can be attributed to FeSi melting (see Figure 8.8 and Table 8.1). There- fore only experiments 1, 6B and 7 have been used to determine the Fe-FeSi eutectic melting curve.

152 Chapter 8. Fe-FeSi Eutectic Melting

Table 8.1: Melting Points from This Study

Experiment Pressure /GPa Melting Temperature /K 1 21 2093 131 ± 2 28 2610 107 ± 3A 38 2738 158 ± 3B 38 3165 128 ± 4 44 2832 84 ± 5A 53 3241 159 ± 5B 76 3483 196 ± 6A 16 2689 74 ± 6B 74 3096 108 ± 7 93 3237 174 ± Highlighted experiments attributed to Fe-FeSi eutectic melting. Thermal pressure has been included in all pressure values.

In order to ﬁt the eutectic melting curve, additional data collected by Dr Oliver Lord

(pers. comm.) were used in conjunction with the melting points obtained here (see

Table 8.2). These data were obtained from two o↵-line LH-DAC experiments (dark red circles in Figure 8.8) and from in situ LH-DAC experiments performed on beamline

12.2.2 at ALS (Advanced Light Source), Berkeley (open red circles). Both experiments used the set-up described here, with the exception that Al2O3 was used as a pressure medium rather than KCl. In addition, two o↵-line LH-DAC experiments on 85Fe-5Ni-

10Si, shown as red squares in Figure 8.8, were also used (Lord et al., 2014).

The Simon–Glatzel equation (Simon and Glatzel, 1929) was used to ﬁt the melting

curve to the combined data

T = P /A +1 1/C T (8.1) m m ⇥ 0 ⇥ ⇤ where Tm is the melting temperature at pressure Pm and T0 is the ambient pressure melting temperature. A and C are fitting parameters. With T0 fixed at 1473 K (Lacaze and Sundman, 1991), the fit gives values of 21.659 4.72 for A and 2.02 0.25 for C. ± ± This melting curve is plotted in Figure 8.8 as a red line. Also shown are the eutectic melting points determined by Fischer et al. (2013) from their analysis of Fe-Si melting experiments (purple diamonds). The Simon–Glatzel equation was fitted to these points

153 8.5. The Fe-FeSi Eutectic Melting Curve

Table 8.2: Melting Points Attributed to Fe-FeSi Eutectic Melting

Experiment Pressure /GPa Melting Temperature /K 21 2093 131 ± This study 74 3096 108 ± 93 3327 174 ± Kuwayama and Hirose (2004) 21 2093 30 ± 11 1746 50 Lord, o↵-line DAC ± 10 1765 50 ± 11 1850 100 ± 18 2050 100 Lord, ALS ± 23 2028 100 ± 36 2200 100 ± 85Fe-5Ni-10Si 49 2700 118 ± (Lord et al., 2014) 51 2810 106 ± Lacaze and Sundman (1991) 0 1473 Thermal pressure of 5% included in pressures for this study, Lord’s o↵-line DAC experiments and the 85Fe-5Ni-10Si experiments.

too, again with T held at 1473 K. This ﬁt gave values of 10.448 2.57 for A and 2.73 0 ± 0.25 for C , and is shown in Figure 8.8 as a dashed purple line. Also plotted are the ± melting curves of FeSi (green line) and Fe (blue line) – with the hcp/fcc phase boundary also shown – and the melting curves of a range of Fe-Si alloys, as well as the eutectic melting curve of Fischer et al. (2013), obtained from their Fe-Si phase diagrams (see

Figure 8.3).

Figure 8.8 shows that there is very good agreement between the eutectic melting curve found here and that of Fischer et al. (2013). The eutectic melting point measured experimentally in the MAP by Kuwayama and Hirose (2004) also agrees perfectly with eutectic melting found here at 21 GPa (Experiment 1; see Figure 8.8 and Table 8.1).

At low pressures – below 80 GPa – the eutectic melting curve is also deﬁned by several of the Fe-Si melting experiments. The melting temperatures of Fe-9Si (Fischer et al.,

2013), as well as those of Fe-18Si (Asanuma et al., 2010) bracket the eutectic melting curve at pressures below 80 GPa, with the melting curve of 85Fe-5Ni-10Si matching almost perfectly with the eutectic melting curve. However, several of the Fe-9Si melting temperatures deviate from the eutectic melting curve, starting at about 45 GPa and leaving the eutectic melting curve completely above 80 GPa. The good agreement between the eutectic melting curve and the Fe-Si alloys plotted in Figure 8.8 indicate

154 Chapter 8. Fe-FeSi Eutectic Melting

Figure 8.8: The Fe-FeSi eutectic melting curve (in red) plotted against data for various Fe-Si alloys. Filled red circles are eutectic melting experiments performed at Bristol (bright and pale red for this study – bright red for eutectic melting and pale red for Fe or FeSi melting; dark red for experiments of Oliver Lord), open red circles are eutectic melting experiments carried out by Oliver Lord at ALS. Experiments from this study have been labelled (see Table 8.1). Purple triangles indicate the upper and lower bounds of Fe-9Si melting from the appearance of di↵use scattering (Fischer et al., 2013) and purple diamonds are eutectic melting points from Fischer et al. (2013), with melting curve (dashed purple line) ﬁtted through the points. All other symbols and lines, see legend on graph.

155 8.6. Conclusions that the eutectic composition has a silicon content close to those of the Fe-Si alloys, that is, between 9 and 18 wt. %.

8.6 Conclusions

The results from these experiments show that, as expected, the eutectic melting curve lies below the melting curves of both Fe and FeSi. The good agreement between the eutectic melting curve and that obtained from the Fe-Si phase diagrams of Fischer et al. (2013) indicates that despite the relatively few data points around the melting curves (see Figure 8.3), the phase diagrams at high temperatures are indeed quite robust.

The Fe-Si phase diagrams also indicate that the eutectic point moves closer to pure-Fe as pressures increase, providing a possible explanation as to why the melting temperatures of Fe-9Si corresponds to eutectic melting at lower pressures but not at higher pressures. At low pressures, between around 20 and 60 GPa, it is likely that the eutectic composition is very close to 9 or 10 wt.% Si, as demonstrated by the close match to the eutectic melting curve of both the Fe-9Si melting points and the 85Fe-5Ni-10Si melting curve at lower pressures. However, as pressure increases – and the eutectic point moves closer to pure Fe and further away from 9 wt.% Si – the mismatch between Fe-9Si and the eutectic increases. Since Fischer et al. (2013) uses the appearance of di↵use scattering to determine melting, it is possible that at higher pressures the amount of eutectic melt never builds up enough to cause di↵use scattering to be seen at eutectic melting temperatures; instead the melting temperatures at high pressures likely correspond to the solidus. In contrast, Asanuma et al. (2010) use a di↵erent method to determine melting – quench texture and a change in laser heating eciency – which could possibly explain why their melting temperatures match the eutectic melting curve well, despite containing a higher Si content, up to about 80 GPa.

To fully verify the Fe-Si phase diagrams of Fischer et al. (2013), the compositions of the samples must be known. This could be carried out with an electron microprobe analysis of the quenched samples used here. Unfortunately, time did not allow for composition

156 Chapter 8. Fe-FeSi Eutectic Melting analysis to be carried out in this study. Nevertheless, this work has helped consolidate the high pressure Fe-Si phase diagrams constructed by Fischer et al. (2013). A deeper understanding of the Fe-Si phase diagram can, in turn, aid our understanding of the Fe-

Ni-Si ternary phase diagram. Although experiments on Fe-Ni-Si (Morard et al., 2011) indicate that a small percentage of nickel has a negligible e↵ect on the melting curve of the Fe-Si alloy, the e↵ect of nickel cannot be ruled out completely. The melting curve of

NiSi is notably lower than that of FeSi (see Chapter 7 and Lord et al. (2014)), and the exact percentage of nickel in the cores of terrestrial planets is still not known exactly, therefore building an accurate Fe-Ni-Si phase diagram is still important. Determining the Fe-Si phase diagram is an important and necessary ﬁrst step in doing this.

157 158 CHAPTER 9

Equation of State for MnP-NiSi

Thought followed thought – star followed star

— Emily Bronte¨

9.1 Introduction

As seen in previous chapters, the NiSi phase diagram features several stable phases at high pressure and temperature. The behaviour of these phases provides information which can be used when considering the Fe–Ni–Si system and the stable phases in that system. One useful tool is the equation of state, which mathematically links the state variables of a phase. Here, the equation of state of the MnP-NiSi phase has been measured using neutron di↵raction experiments carried out at ISIS.

9.2 The MnP Phase

In NiSi, the MnP phase is the stable structure at room temperatures and pressures.

Calculations carried out by Wood et al. (2013) show that at 0 K, the MnP phase is stable from 0 to 21 GPa, when the Pmmn phase becomes stable. The lower pressure bounds of the MnP phase are not fully deﬁned, although multi-anvil press (MAP) experiments indicate the existence of a MnP–Pmmn phase transition at high temperature (Wood et al., 2013). High pressure LH-DAC experiments also show that at 12.5 4.5 GPa and ± 1550 150 K, the MnP phase transforms to the "-FeSi phase (Lord et al., 2012). ±

Structurally, the MnP phase is orthorhombic, with a = b = c. At ambient pressure, 6 6 these values are found experimentally to be a =5.1752,b =3.3321,c =5.6094 in A˚3

(Rabadanov and Ataev, 2002), which matches closely to those obtained from computer

159 9.3. Neutron Di↵raction Experiments simulations of NiSi (Voˇcadlo et al., 2012). Further details about the structure of the

MnP phase can be found in Chapter 5.

Although the equation of state of MnP-NiSi has been calculated at 0 K (Voˇcadlo et al.,

2012) and an approximate equation of state of MnP-NiSi has been reported experimentally by Lord et al. (2012), the equation of state at high temperatures has not been measured. Here, the equation of state of MnP-NiSi has been measured experimentally at both room temperature and high temperatures. This has then been compared to the computationally calculated EoS, as well as the experimentally measured EoS of other stable NiSi phases.

9.3 Neutron Di↵raction Experiments

Experiments were carried out at the Pearl beamline at ISIS, the neutron source at the

Rutherford Appleton Laboratory. As described in Section 3.3, neutron di↵raction can be used to determine a host of structural parameters, such as the cell parameters, which is what will be measured here. Two sets of experiments have been carried out, one at room temperature and one at high temperature. Both sets of data have been used in the

fit to obtain the Birch-Murnaghan Equation of State (BMEoS). Two fits were carried out, an isothermal fit using only the room temperature data, and a fit to determine the thermal equation of state using combined data from both experiments.

The Pearl beamline at ISIS allows for in situ neutron di↵raction to take place at high temperatures and pressures. A Paris-Edinburgh pressure cell equipped with sintered diamond anvils was used to reach pressures up to 11 GPa in these experiments. The exact pressure within the cell is determined through the use of di↵raction patterns of NaCl, which is mixed in with the sample. NaCl has a very well deﬁned EoS, and therefore the unit cell parameters can be used to ﬁnd the pressure of the cell. The sample temperature in the high-temperature experiments were determined by neutron absorption resonance radiography (Stone et al., 2005a,b)

The program GSAS was used to ﬁt the di↵raction patterns collected using the Rietveld method (see Section 3.3.3.3 for further details). Figure 9.1 shows a typical ﬁtted di↵rac-

160 Chapter 9. Equation of State for MnP-NiSi tion pattern. In addition to the NaCl and NiSi peaks, diamond peaks are also seen – these are from the diamond sintered anvils used in the press.

Figure 9.1: One of the ﬁtted neutron di↵raction patterns collected of MnP-NiSi, showing observed (points in pink), calculated (blue line) and di↵erence (lower trace) of the di↵raction pattern. Also shown are tick marks marking the position of the Bragg peaks for the three phases seen (NiSi in orange, NaCl in green and diamond peaks in red). This di↵raction pattern was collected at room temperature under a load of 150 bar, which translated to a pressure of roughly 1 GPa.

In the ﬁrst experiment carried out at the Pearl beamline at ISIS at room temperature, pressure was gradually increased in steps of roughly 1 GPa, from 0 to 11.27 GPa, with a di↵raction pattern taken at each pressure. The second experiment was carried out at high-temperature and high-pressure, from pressures of 0 to 6.65 GPa and temperatures from 293 to 1003 K. The sample was brought to the required pressure than subjected to heating, with di↵raction patterns taken at each temperature point. The load on the sample was then increased, with pressure increased in approximately 1 GPa steps, followed by heating to the required temperature. The cell parameters of NaCl and

NiSi found from analysis of the room temperature experiments are listed in Table 9.1;

Table 9.2 lists those for the high pressure experiments.

161 9.3. Neutron Di↵raction Experiments

Table 9.1: Lattice cell parameters for NaCl and MnP-NiSi obtained at room temperature at di↵erent pressures

NaCl NiSi P /GPa a /A˚ a /A˚ b /A˚ c /A˚ V/A˚3 0.00 5.640(1) 5.184(1) 3.334(1) 5.618(1) 97.08(2) 1.36 5.545(2) 5.170(3) 3.328(2) 5.597(3) 96.29(7) 2.29 5.492(3) 5.160(3) 3.324(3) 5.584(4) 95.78(7) 3.24 5.445(3) 5.151(4) 3.322(3) 5.568(4) 95.26(7) 3.75 5.421(2) 5.144(3) 3.320(2) 5.561(3) 94.97(5) 4.12 5.405(8) 5.139(8) 3.320(7) 5.557(10) 94.82(17) 4.57 5.387(3) 5.135(3) 3.317(3) 5.549(4) 94.52(7) 5.64 5.345(3) 5.123(4) 3.314(3) 5.533(4) 93.95(7) 6.95 5.300(3) 5.110(4) 3.312(3) 5.513(4) 93.29(7) 8.04 5.266(2) 5.097(3) 3.311(3) 5.497(4) 92.76(6) 8.92 5.241(4) 5.090(4) 3.308(3) 5.486(5) 92.36(8) 9.91 5.214(3) 5.079(5) 3.308(4) 5.470(5) 91.89(9) 11.27 5.180(3) 5.063(4) 3.305(4) 5.449(5) 91.18(8) Pressures calculated using the equation of state for NaCl determined by Decker (1971)

162 Chapter 9. Equation of State for MnP-NiSi

Table 9.2: High-temperature, high-pressure lattice cell parameters of NaCl and NiSi

NaCl NiSi T /K P /GPa a /A˚ a /A˚ b /A˚ c /A˚ V/A˚3 293.2 0.00 5.632(4) 5.180(5) 3.333(3) 5.614(5) 96.94(1) 293.2 0.70 5.582(5) 5.175(7) 3.327(4) 5.615(8) 96.68(2) 441.2 0.42 5.638(5) 5.196(6) 3.322(4) 5.631(7) 97.20(1) 567.0 0.36 5.664(5) 5.230(6) 3.302(4) 5.659(6) 97.76(1) 698.5 0.43 5.698(7) 5.277(7) 3.260(5) 5.706(8) 98.17(2) 293.2 1.32 5.541(5) 5.192(8) 3.294(5) 5.614(8) 96.01(2) 293.2 2.21 5.489(4) 5.181(8) 3.288(5) 5.601(8) 95.41(2) 452.6 2.43 5.501(4) 5.195(7) 3.260(4) 5.614(8) 95.96(2) 616.6 2.52 5.523(5) 5.231(7) 3.274(5) 5.645(8) 96.66(2) 869.7 2.45 5.573(8) 5.335(6) 3.185(4) 5.741(7) 97.57(1) 992.2 1.59 5.632(10) 5.370(7) 3.175(4) 5.773(8) 98.42(2) 293.2 3.19 5.439(5) 5.181(9) 3.269(6) 5.596(10) 94.76(2) 405.6 3.46 5.442(6) 5.190(9) 3.272(6) 5.600(10) 95.11(2) 603.7 3.86 5.450(5) 5.217(8) 3.247(6) 5.645(10) 95.63(2) 838.2 4.12 5.472(6) 5.326(6) 3.164(4) 5.723(6) 96.43(1) 1003.6 4.16 5.495(6) 5.359(6) 3.147(3) 5.753(6) 97.03(1) 293.2 4.20 5.394(4) 5.176(9) 3.266(6) 5.582(10) 94.38(2) 395.2 4.44 5.396(5) 5.186(10) 3.266(6) 5.588(10) 94.66(2) 575.2 4.78 5.404(5) 5.209(9) 3.252(7) 5.611(10) 95.04(2) 838.5 5.35 5.414(6) 5.315(7) 3.153(4) 5.708(7) 95.68(1) 956.8 5.46 5.425(7) 5.353(7) 3.127(4) 5.739(7) 96.07(1) 293.2 5.26 5.352(4) 5.177(10) 3.254(6) 5.573(11) 93.86(2) 573.8 5.95 5.357(5) 5.200(9) 3.248(6) 5.591(11) 94.42(2) 791.5 6.40 5.365(5) 5.310(6) 3.141(3) 5.694(6) 94.98(1) 951.8 6.65 5.374(6) 5.351(6) 3.109(4) 5.729(6) 95.30(1) Pressures calculated using the equation of state for NaCl determined by Decker (1971) and temperatures determined by neutron absorption resonance radiography (Stone et al., 2005a,b)

163 9.4. Birch-Murnaghan Equation of State

9.4 Birch-Murnaghan Equation of State

A third order Birch-Murnaghan Equation of State (BMEoS) has been used here to ﬁt the experimental data. The exact form of this equation is given in Section 2.8.2, and relates the volume of a system to the pressure. Having a well-determined equation of state allows for easy comparison to other structures and phases, as well as providing a quantitative insight into the system’s structure when subjected to pressure and temperature.

The experimental data obtained from ISIS were ﬁtted using the program EOS-FIT

(Angel, 2000; Angel et al., 2014) which uses a least-squares algorithm to obtain the correct parameters. Uncertainties in the volumes were taken into account in the ﬁt by weighting each point. For the thermal equation of state, thermal expansion is modelled at ambient pressure by

V0(T )=V0Tr exp ↵(T )dT (9.1)  ZTr where V0(T ) is the volume at P = 0 and Tr is the reference temperature (here taken to be 298 K) at which the volume is V0Tr . The thermal expansion coecient, ↵(T ), is given by

↵(T )=↵0 + ↵1T (9.2)

where ↵0 and ↵1 are parameters of the thermal expansion coecient. Separate ﬁts were carried out for the room temperature data (see Table 9.3) and the combined data

(both room temperature and high temperature, see Table 9.4).

For the room temperature data, two ﬁts were carried out, one with all three parameters

– V0, K0 and K0 – varying and one fit with K0 fixed at 4. The values of the parameters obtained from these fits are shown in Table 9.3. The di↵erence between V0 and K0 between Fits 1 and 2 are small, showing that holding K0 doesn’t a↵ect the fit very

164 Chapter 9. Equation of State for MnP-NiSi much. Figure 9.2 shows the third-order Birch-Murnaghan Equation of State (BMEoS) plotted against the volume-pressure data, illustrating the good agreement of the BMEoS with the data.

95 3 94

Volume /Å 93

0.0 2.5 5.0 7.5 10.0 12.5

Pressure /GPa

Figure 9.2: A graph showing how the unit cell volume of MnP-NiSi decreases with pressure at room temperature. The points were ﬁtted to a third-order Birch-Murnaghan Equation of State (BMEoS), which is shown on the graph (see also Table 9.3).

Table 9.3: Fitting parameters for the room temperature Birch-Murnaghan equation of state of MnP-NiSi

Fit 1 Fit 2 3 -1 V0 / A˚ atom 12.1355(8) 12.1363(10) K0 /GPa 165(1) 160.8(6) K0 2.9(3) 4* Fit 1 with all parameters varying; Fit 2 with * value ﬁxed

Both the room temperature and high temperature data were used to fit the thermal equation of state. As with the isothermal BMEoS, several fits were carried out but with di↵erent parameters held. Table 9.4 shows the values of these parameters, with the fixed parameters indicated by a star. All parameters were allowed to vary in Fit 1, resulting in a low value of ↵0 and a high value of ↵1. For Fits 2 and 3, ↵0 was fixed at 3.6 10 5 K-1, the average of the experimentally determined values of 3.5 10 5 K-1 ⇥ ⇥ (Perrin et al., 2008) and 3.7 10 5 K-1 (Wilson and Cavin, 1992) for ↵ . Additionally, ⇥ 0

165 9.4. Birch-Murnaghan Equation of State

↵1 was fixed at 0 for the second fit (see Table 9.4), since both studies by Perrin et al. (2008) and Wilson and Cavin (1992) indicate that, to a good approximation, ↵(T )does not vary with temperature. Allowing ↵1 to vary, as in the third fit in Table 9.4, shows that ↵1 does indeed have a value of 0 within error. However, allowing ↵1 to vary has

@K the e↵ect of increasing the error on the @T value. The three ﬁts show good agreement @K between the V0, K0 and K0 values, but ↵0, ↵1 and @T show a high error when allowed @K to vary in Fit 1. Similarly, @T has a high error in Fit 3, where ↵1 has been allowed to vary. Fixing both ↵0 and ↵1, as in Fit 2, using values obtained from previous studies (Wilson and Cavin, 1992; Perrin et al., 2008) appears to yield the most robust value

@K for @T .

Table 9.4: Fitting parameters for the thermal Birch-Murnaghan equation of state of MnP-NiSi

Fit 1 Fit 2 Fit 3 3 V0 / A˚ 12.1349(20) 12.1346(21) 12.1345(21) K0 /GPa 160.6(35) 159.9(37) 159.8(37) K0 3.9(9) 4.1(9) 4.1(9) 5 -1 ↵0 / 10 K 2.2(6) 3.6* 3.6* ⇥ 5 -2 ↵1 / 10 K 2.5(11) 0* 0.08(32) @K ⇥ @T / GPa/K -0.024(10) -0.021(4) -0.024(10) Fit 1 with all values varying; Fits 2 and 3 with * values ﬁxed.

The ﬁtting parameters obtained from the BMEoS are broadly in line with those of other stable phases in the NiSi system. Lord et al. (2012) measured isothermal EoS parameters for the "-FeSi and CsCl-structured phases, ﬁnding V0 to be 12.15 and 10.87

3 A˚ per atom, with K0 values of 165 and 240 GPa for the "-FeSi and CsCl phases respectively when K0 is ﬁxed at 4. Comparison of these parameters with those obtained for the MnP phase (see Table 9.3), shows that the V0 and K0 values of the MnP phase are very similar to those of the "-FeSi phase. This is perhaps not surprising, since the

MnP phase transforms to the "-FeSi phase at high temperature.

The experimental BMEoS found here have also been compared to the calculated BMEoS

(Voˇcadlo et al., 2012). Figure 9.3 shows the two BMEoS plotted against V/V0. It has been well documented that the GGA implementation for nickel results in an overestimation of the volume (Zhang et al., 2014). The BMEoS has therefore been plotted

166 Chapter 9. Equation of State for MnP-NiSi against V/V0 rather than V to account for this. As can be seen from Figure 9.3, after the over-estimation of volume is taken into account, there is excellent agreement between the experimental and computational equations of state.

Figure 9.3: Comparison of the BMEoS obtained from calculations (Voˇcadlo et al., 2012) and from the room temperature experiments here, plotted against V/V0,showingexcellent agreement between the experimental EoS and the computational EoS.

9.5 Cell Parameters of MnP-NiSi

Investigating how the individual cell parameters behave under pressure and temperature can give useful insights into the way the structure is changing at these conditions.

The relative cell parameters have been used instead of the actual cell parameters in order to see the di↵erences between them more clearly. These are given by

a a b b c c 0 , 0 , 0 (9.3) a0 b0 c0

where a, b and c are the cell parameters and a0, b0 and c0 are the cell parameters at 0 GPa. Figure 9.4 shows the behaviour of the cell parameters as a function of pressure and temperature. Figure 9.4a shows the experimental relative cell parameters at room temperature plotted against their computational equivalent at 0 K (Voˇcadlo et al.,

167 9.5. Cell Parameters of MnP-NiSi

2012), showing excellent agreement between calculations and experiments. It can be seen from Figure 9.4a that all the cell parameters decrease with pressure, however, the c and a cell parameters decrease much quicker. In addition, both a and c decrease fairly linearly with pressure whereas b decreases much slower and is much sti↵er than either a or c. Conversely, from Figure 9.4b, it can be seen that the b cell parameter exhibits negative thermal expansion while both a and c show positive thermal expansion. The a and c cell parameters increase at almost exactly the same rate as a function of temperature. The positive thermal expansion in a and c is mirrored by b in the opposite direction, resulting in an overall positive thermal expansion of the volume. Previous results from Wilson and Cavin (1992) are also plotted in Figure 9.4b. This shows a larger change in cell parameter, however, Wilson and Cavin (1992) used X-ray di↵raction which is known to be less accurate in measuring cell parameters than the neutron di↵raction technique used here.

168 Chapter 9. Equation of State for MnP-NiSi

(a)

0.000

-0.005

-0.010

-0.015

-0.020

Relative Cell Parameter

-0.025 a b -0.030 c

0.0 2.5 5.0 7.5 10.0 Pressure /GPa

(b)

0.15

0.10

0.05

0.00

Relative Cell Parameter

-0.05 a b c

300 400 500 600 700 800 900 Temperature /K

Figure 9.4: The relative cell parameters obtained experimentally (symbols as on graph) (a) as a function of pressure, with a straight line fitted to a and c values and a second order polynomial fitted to b values, plotted against the calculated cell parameters (dashed lines) from Voˇcadlo et al. (2012); and (b) as a function of temperature at ambient pressure, with a second- order polynomial fitted to a, b and c values, with previous experimental values plotted as open symbols (Wilson and Cavin, 1992).

169 9.6. Fractional Co-ordinates of Ni and Si

The observed behaviour of the axes can be explained by considering the MnP-NiSi

1 3 structure. As pressure increases, the atoms lying in planes y = 4 and y = 4 move closer together to become pseudo-close packed planes made up of equal numbers of Ni and Si atoms (see Figure 9.5), therefore causing a contraction in the a and c axes. Since it is energetically more favourable to adopt a close-packed type structure, the b axis stays relatively unchanged during compression. Under heating, the planes of atoms move away from a close-packed type arrangement, causing the a and c axes to expand while the b axis shrinks, as seen in Figure 9.4b.

1 Figure 9.5: Figures of NiSi viewed along [010], with planes of atoms at y = 4 showing how the structure of (a) MnP-NiSi changes with pressure to (b) Pnma-II and then, with further 3 pressure, to (c) Pnma-III. Figure (d) is as for (c) but including atoms at y = 4 , and with 1 3 outlines to show the pseudo-cells produced by ABAB stacking of the y = 4 and y = 4 layers. Ni atoms are shown as pale blue spheres and Si atoms as smaller dark spheres. Figure from Voˇcadlo et al. (2012).

9.6 Fractional Co-ordinates of Ni and Si

The fractional co-ordinates of Ni and Si are shown in Figure 9.6, with the experimentally measured x and z co-ordinates of the two atoms at room temperature plotted against those calculated by Voˇcadlo et al. (2012).

170 Chapter 9. Equation of State for MnP-NiSi

(a) (b)

0.014 Ni x Ni z 0.189

0.188 0.012

0.187

0.010 0.186 Fractional Co-ordinate Fractional Co-ordinate

0.185 0.008

0.184

11.25 11.50 11.75 12.00 12.25 11.25 11.50 11.75 12.00 12.25 3 3 Volume /Å per atom Volume /Å per atom

Si x Si z 0.684 0.920

0.682

0.918

0.680

0.916 0.678

0.914 Fractional Co-ordinate 0.676 Fractional Co-ordinate

0.674 0.912

0.672 0.910

11.25 11.50 11.75 12.00 12.25 11.25 11.50 11.75 12.00 12.25 3 3 Volume /Å per atom Volume /Å per atom

Figure 9.6: Fractional co-ordinates of NiSi obtained from the XRD data, shown in red and pink, plotted against the calculated fractional co-ordinates (Voˇcadlo et al., 2012), shown in blue: (a) x co-ordinate of Ni; (b) z co-ordinate of Ni; (c) x co-ordinate of Si and (d) z co-ordinate of Si.

These graphs show that there is reasonable agreement between the calculated and experimentally determined co-ordinates. The z co-ordinate of both Ni and Si show the same upwards trend seen in the calculations, although the experimentally determined z co-ordinate of Ni is slightly lower than the calculated co-ordinate for all values. There is less agreement in the x co-ordinates of Ni and Si, with the experimentally determined

Ni x co-ordinate appearing to slightly increase with volume while the experimentally determined Si x co-ordinate appears to decrease with volume. However, the small number of data points and the small changes in the values of the co-ordinates mean that it is not possible to establish a trend since the apparent correlation may be a result of scatter. Nevertheless, the co-ordinates of both Ni and Si show fair agreement with those obtained from calculations.

171 9.7. Conclusions

9.7 Conclusions

Both the thermal EoS and isothermal EoS have been determined for the MnP-NiSi phase using neutron di↵raction data from ISIS. These ﬁts show that physically, the MnP phase of NiSi behaves quite similarly to the other NiSi phases in terms of reaction to pressure and temperature, having a very similar bulk modulus to the other NiSi phases.

The MnP phase has a very similar sti↵ness to the other phases, which is in keeping with the small energy di↵erences between these phases (Voˇcadlo et al., 2012).

The neutron di↵raction experiments also reveal the behaviour of the individual cell parameters, showing that while the a and c axes both expand on heating, the b axis in MnP-NiSi contracts rather than expands, in line with observations from previous experimental studies (Wilson and Cavin, 1992; Perrin et al., 2008). On compression, however, the b axis does not expand but instead contracts, albeit at a slower rate than both a and c axes. As shown above, this is likely a result of a move towards a pseudo close-packed arrangement of atoms on compression.

172 CHAPTER 10

Thermal Expansion of (Fe,Ni)Si Alloys

And what if the sky here is no di↵erent And it is my eyes that have been sharpening themselves?

— Sylvia Plath

10.1 Introduction

Thermal expansion is an important physical property that quantiﬁes how a material behaves as temperature increases. Here, the thermal expansion of three (Fe,Ni)Si alloys have been measured using X-ray di↵raction of the samples. Using a range of (Fe,Ni)Si alloys means that the e↵ect of nickel on thermal expansion can be examined, providing additional information relating to the Fe–Ni–Si ternary system.

10.2 Experimental Details

Three di↵erent samples were used in this experiment with varying levels of nickel.

The compositions investigated here were (Fe0.9Ni0.1)Si, (Fe0.8Ni0.2)Si and FeSi. These samples were made by Dr Simon Hunt at Birmingham University using argon arc- melting. Chemical analysis found all these to be single phase and of the composition required within a margin of 5 %, with the material adopting the cubic "-FeSi structure in each case. To prepare the samples for XRD analysis, each were ground into a powder in a pestle and mortar under ethanol. Grinding the samples also ensured clear XRD patterns could be obtained. The powdered sample was then placed in a holder for

173 10.3. Results the X-ray di↵ractometer, which was equipped with an Anton Paar HTK 1200 N high- temperature stage and an Oxford Cryosystems PheniX ‘Front Loader’ low-temperature stage. X-ray di↵raction patterns were then taken at each temperature step as the temperature was increased. Figure 10.1 shows a typical XRD pattern obtained. The range covered was between 40 to 1273 K, with the temperature being increased in 20 degree steps. Each XRD pattern was analysed using the program GSAS to obtain the unit-cell volume of the sample (see Section 3.3.3.3 for further details).

Figure 10.1: A typical XRD pattern collected for these experiments. This was collected at 300°C from the (Fe0.9Ni0.1)Si sample. The pattern was analysed and ﬁtted using the GSAS program.

10.3 Results

The unit cell volumes found from analysis of the XRD patterns of the three samples are listed in Table 10.3 and plotted in Figure 10.2. This shows that each sample expands as temperature increases, indicating positive thermal expansion throughout for all three samples.

Table 10.1: Unit Cell Volumes of FeSi, (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si

Unit cell volumes /A˚3 T /K FeSi (Fe0.9Ni0.1)Si (Fe0.8Ni0.2)Si 40 89.609(2) 89.624(2) 89.578(1)

Continued on next page

174 Chapter 10. Thermal Expansion of (Fe,Ni)Si Alloys

Table 10.1 – Continued from previous page

Unit cell volumes /A˚3 T /K FeSi (Fe0.9Ni0.1)Si (Fe0.8Ni0.2)Si 60 89.613(1) 89.620(2) 89.584(1)

80 89.627(2) 89.641(2) 89.591(2)

100 89.650(1) 89.673(1) 89.613(1)

120 89.677(1) 89.699(1) 89.649(1)

140 89.726(1) 89.743(1) 89.688(1)

160 89.781(1) 89.794(1) 89.741(1)

180 89.842(1) 89.847(1) 89.787(1)

200 89.911(1) 89.914(1) 89.850(1)

220 89.981(2) 89.970(1) 89.908(1)

240 90.054(2) 90.048(1) 89.970(1)

260 90.135(2) 90.118(1) 90.033(1)

280 90.216(2) 90.179(1) 90.097(1)

300 90.289(2) 90.252(1) 90.161(1)

298 90.310(1) 90.266(1) 90.168(1)

313 90.391(1) 90.349(1) 90.247(1)

333 90.503(1) 90.452(1) 90.339(1)

353 90.605(1) 90.549(1) 90.425(1)

373 90.703(1) 90.643(1) 90.512(1)

393 90.793(1) 90.728(1) 90.599(1)

413 90.892(1) 90.815(1) 90.681(1)

433 90.982(1) 90.897(1) 90.763(1)

453 91.074(1) 90.975(1) 90.843(1)

473 91.157(1) 91.059(1) 90.928(1)

493 91.237(2) 91.146(1) 91.008(1)

513 91.326(1) 91.226(1) 91.086(1)

533 91.409(1) 91.310(1) 91.166(1)

553 91.497(1) 91.389(1) 91.247(1)

573 91.581(1) 91.467(1) 91.323(1)

593 91.661(2) 91.549(1) 91.401(1)

613 91.740(1) 91.628(1) 91.475(1)

633 91.829(2) 91.708(1) 91.554(1)

653 91.908(2) 91.788(1) 91.637(1)

673 91.994(2) 91.873(1) 91.716(1)

Continued on next page

175 10.3. Results

Table 10.1 – Continued from previous page

Unit cell volumes /A˚3 T /K FeSi (Fe0.9Ni0.1)Si (Fe0.8Ni0.2)Si 693 92.073(2) 91.955(1) 91.801(1)

713 92.154(2) 92.034(1) 91.880(1)

733 92.231(2) 92.115(1) 91.959(1)

753 92.321(2) 92.202(1) 92.043(1)

773 92.406(1) 92.283(1) 92.125(1)

793 92.491(1) 92.364(1) 92.214(1)

813 92.575(2) 92.443(1) 92.298(1)

833 92.659(1) 92.534(1) 92.384(1)

853 92.746(2) 92.616(1) 92.471(1)

873 92.836(2) 92.704(1) 92.561(1)

893 92.917(2) 92.791(1) 92.647(1)

913 93.013(2) 92.881(1) 92.739(1)

933 93.107(2) 92.971(1) 92.818(1)

953 93.201(2) 93.062(1) 92.909(1)

973 93.290(2) 93.151(1) 92.999(1)

993 93.381(2) 93.242(1) 93.089(1)

1013 93.480(2) 93.334(1) 93.175(1)

1033 93.565(2) 93.428(1) 93.273(1)

1053 93.658(2) 93.522(1) 93.368(1)

1073 93.755(2) 93.615(1) 93.462(1)

1093 93.847(2) 93.709(1) 93.559(1)

1113 93.950(2) 93.798(1) 93.664(1)

1133 94.056(2) 93.904(1) 93.786(1)

1153 94.169(2) 94.020(1) 93.903(1)

1173 94.279(2) 94.146(1) 94.013(1)

1193 94.391(3) 94.265(1) 94.122(2)

1213 94.489(3) 94.371(2) 94.231(2)

1233 94.621(4) 94.479(2) 94.338(2)

1253 94.723(4) 94.585(2) 94.446(3)

1273 94.815(5) 94.690(2) 94.549(3)

The results shown in Figure 10.2 indicate that adding nickel to FeSi causes a decrease in volume, most visible at high temperatures, following a fairly uniform trend between

FeSi, (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si.

176 Chapter 10. Thermal Expansion of (Fe,Ni)Si Alloys

95 FeSi

(Fe0.9Ni0.1)Si

(Fe0.8Ni0.2)Si 94

93 3

Volume /Å

0 250 500 750 1000 1250 Temperature /K

Figure 10.2: Graph showing how the volume of the unit cell increases with temperature for the three samples. It can be seen that the addition of nickel has the e↵ect of decreasing thermal expansion.

However, slightly odd behaviour for all three samples can be seen in Figure 10.2 between about 300 and 900 K. At 300 K, there is a discontinuity in the slope of all three curves, especially noticeable for the (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si samples. This is likely caused by a mismatch between the low- and high-temperature experiments. However, between 300 and 900 K, the data suggests that there is an initial increase, followed by a decrease, in the thermal expansion coecient (see Figure 10.2). This type of behaviour was not seen in earlier studies of FeSi (Voˇcadlo et al., 2002). Figure 10.3 shows the unit cell volume of FeSi obtained here with those of Voˇcadlo et al. (2002), whose neutron powder di↵raction experiments were carried out on a di↵erent sample of FeSi, much larger than the sample used here. It can be seen that the two sets of measurements agree very well below room temperature, but not in the range from

300 to 900 K as the neutron powder di↵raction data doesn’t exhibit the anomalous behaviour seen in the samples here. In addition, earlier experiments at UCL using a very similar experimental set up for the X-ray di↵raction study (Lindsay-Scott et al.,

2007) as well as very recent studies on MgGeO3 compounds using exactly the same high-temperature apparatus (James Santangeli, pers. comm.) show no such anomalies,

177 10.3. Results suggesting that the anomalous behaviour observed here might instead be a property of these speciﬁc (Fe,Ni)Si samples. Although this may be an indication that the results here need to be treated with some caution, the e↵ect of this behaviour appears to be similar for all three samples and so should not a↵ect the conclusions drawn on the e↵ect of Ni on the thermal expansion of FeSi.

95 FeSi (This study) FeSi (Vocadlo et al., 2002)

93 3

Volume /Å

0 250 500 750 1000 1250 Temperature /K

Figure 10.3: Comparison of the unit cell volume of FeSi from this study and from neutron di↵raction experiments (Voˇcadlo et al., 2002), showing a slight ‘bump’ between 300 and 900 K. Note the excellent agreement below 300 K.

The thermal expansion coecient has also been determined by ﬁtting the data to the equation

V (T )=VTr exp ↵(T )dT (10.1)  ZTr

where VTr is the volume at the reference temperature, Tr, and ↵ is the thermal expansion coecient, given by

↵(T )=↵0 + ↵1T (10.2)

178 Chapter 10. Thermal Expansion of (Fe,Ni)Si Alloys where ↵0 and ↵1 are parameters of ↵(T ). For comparison with thermal coecients determined by Voˇcadlo et al. (2002) and Uchida et al. (2001), the reference temperature was chosen to be 300 K. Only the high temperature data (above 300 K) were used for this ﬁt, both for comparison purposes and also because this equation does not describe thermal expansion accurately at low temperatures.

The fitted parameters are listed in Table 10.2, along with values obtained at ambient pressure by Voˇcadlo et al. (2002) and Uchida et al. (2001) for FeSi and "-Fe (hcp-Fe) respectively. As expected, the fitted VTr values decrease as nickel content increases. Comparison of the fitted parameters obtained here for FeSi with that of Voˇcadlo et al.

(2002) shows fair agreement between the two. Both the volume of the unit cell and the ↵0 values are slightly higher than that of Voˇcadlo et al. (2002), but not by a large amount. The thermal expansion coecient parameters for hcp-Fe at ambient pressure are also listed in Table 10.2 (Uchida et al., 2001), showing that Fe has a similar ↵0 value to (Fe0.8Ni0.2)Si, but a much higher ↵1 value than all (Fe,Ni)Si samples.

Table 10.2: Thermal Expansion Coecients for the FeSi, (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si

This work a. b. FeSi (Fe0.9Ni0.1)Si (Fe0.8Ni0.2)Si FeSi Fe ˚3 VTr /A 90.42(1) 90.37(1) 90.26(1) 90.33(1) 22.6(4) 5 1 ↵0 /10 K 4.40(7) 4.12(7) 3.96(6) 3.75(9) 3.98(24) 8 2 ↵1 /10 K 1.1(1) 1.5(1) 1.8(1) 1.4(1) 5.07(88) a. Voˇcadlo et al. (2002); b. Uchida et al. (2001)

In addition to ﬁtting the data, the thermal expansion coecient has also been calculated by numerical di↵erentiation of the cell volume data, following the equation

1 dV ↵(T )= (10.3) V dT

Figure 10.4 shows the thermal expansion coecients found from this, along with those of FeSi obtained from neutron powder di↵raction experiments of Voˇcadlo et al. (2002).

Figure 10.4a shows that – disregarding scatter and the region of about 100 K above room temperature – there is very good agreement between the thermal expansion coecient

179 10.3. Results obtained for FeSi here and that of Voˇcadlo et al. (2002). It can also be seen that up to temperatures of about 650 K, the thermal expansion coecient decreases as nickel content increases.

(a)

(b)

Figure 10.4: (a) Graph showing the thermal expansion coecient calculated from numerical di↵erentiation for all three samples; (b) The thermal expansion coecient for FeSi obtained through numerical di↵erentiation (points) by Voˇcadlo et al. (2002); the line shows the expansion coecient resulting from a ﬁt to the V (T ) data and is shown as a dashed red line in (a).

Although there is a fair amount of scatter in the data, all three samples follow the same trend and show a similar shape (see Figure 10.4a). The peak in the expansion coe-

180 Chapter 10. Thermal Expansion of (Fe,Ni)Si Alloys cient for all samples at around 300 K, probably due to the issues with the experiment discussed above. There is also some scatter seen at high temperatures. Examination of the XRD patterns shows new peaks appearing at high temperatures that are not attributed to the sample (see Figure 10.5), which have a↵ected the ﬁt and is likely the cause of the scatter seen at high temperatures. As shown in Figure 10.5, these peaks start to appear at 900°C and are attributed to a mixture of (Fe,Mg)SiO4,Fe3Si and

Fe5Si3. These new phases probably formed as a result of reaction between the FeSi and MgO at high temperatures, and are seen in the XRD patterns of all the (Fe,Ni)Si samples, appearing at around the same temperature for each.

(a)

(b)

Figure 10.5: XRD patterns of FeSi showing (a) the appearance of additional peaks at 900°C and (b) the XRD pattern taken at room temperature after heating, with additional peaks iden- tiﬁed as (Fe,Mg)SiO4 (green indicators); Fe3Si (grey indicators) and Fe5Si3 (brown indicators). The peaks attributed to FeSi and MgO are shown by red and blue indicators respectively.

181 10.3. Results

Probably the most e↵ective way to demonstrate the e↵ect of the addition of Ni on the thermal expansion of FeSi is to visualise the data as volume ratios of (Fe,Ni)Si to FeSi, or

V x (10.4) VFeSi

where Vx is the volume of the (Fe,Ni)Si sample and VFeSi is the volume of the FeSi sample. This ratio has been plotted against temperature in Figure 10.6. By using the ratio of the volumes the e↵ect of nickel on the volume of FeSi can be visualised more easily.

(Fe0.9Ni0.1)Si 1.0000 (Fe0.8Ni0.2)Si

0.9995

0.9990

0.9985

Ratio of Volumes 0.9980

0.9975

0.9970

0 250 500 750 1000 1250 Temperature /K

Figure 10.6: The volume ratios, Vx , of (Fe Ni )Si and (Fe Ni )Si to FeSi. From VFeSi 0.9 0.1 0.8 0.2 the graph, it can be seen that the addition of nickel has the e↵ect of decreasing volume. The volume of both (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si decrease uniformly with respect to FeSi from about 100 to 750 K, where the volume ratio appears to start to plateau.

Figure 10.6 shows that at temperatures below 100 K there is little change in the volume ratio of both samples, with the volume of (Fe0.9Ni0.1)Si almost equal to that of FeSi, with a ratio close to 1, at low temperatures. As temperature increases, the volume ratio decreases linearly for both (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si until about 750 K. At this temperature both volume ratios start to plateau, indicating that the e↵ect of nickel on

182 Chapter 10. Thermal Expansion of (Fe,Ni)Si Alloys

FeSi becomes saturated at this temperature.

10.4 Conclusions

X-ray di↵raction patterns of FeSi, (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si have been collected at a range of temperatures, from 40 K to 1273 K. Analysis of the XRD patterns show that the addition of nickel has the e↵ect of decreasing the volume of the unit cell as temperature increases. Comparing the volume ratios of (Fe0.9Ni0.1)Si and (Fe0.8Ni0.2)Si to FeSi show that the addition of nickel causes a roughly linear decrease in volume from

FeSi with temperature, from about 100 K until about 750 K when the volume ratio

ﬂattens out for both (see Figure 10.6). The e↵ect of nickel in decreasing the volume may be related to the Invar e↵ect (van Schilfgaarde et al., 1999) which is related to the magnetic structure. Further work on the magnetic structure in these alloys are required to explore this possibility.

The thermal expansion coecient has also been obtained, both from fitting the data and by numerical di↵erentiation. Only data above 300 K was used for the fit; for a complete analysis, the low temperature data must also be fitted to an appropriate thermal expansion model. Unfortunately the limitations of the data sets did not allow for this. However, results from numerical di↵erentiation, show good agreement between the data obtained here and that of Voˇcadlo et al. (2002), as do the fitted thermal expansion coecients.

Despite these results showing a deﬁnite decrease in volumes with added nickel, it should be noted that the e↵ect is very small at the temperatures investigated, with the maximum decrease being 0.3 % when 20 wt. % Ni is added, occurring at about 1000 K. This would suggest that, at least when considering thermal expansion, the e↵ect of nickel on density is negligible at core conditions for the terrestrial planets. However, although the temperature range covered is close enough to core temperatures for Venus, Mercury and Mars for extrapolation of the results to be feasible, the temperature of the Earth’s core, where temperatures reach close to 6000 K, is too high for extrapolation to be reasonable. It should also be remembered that the "-FeSi structure is not the stable

183 10.4. Conclusions form of (Fe,Ni)Si at high pressure. One method of investigating thermal expansion at the conditions of the Earth’s core would be to carry out ab initio calculations. If these also covered the range measured experimentally here, then these calculations could be tested for robustness by comparison to the experimental results. In addition, these experiments were carried out at room pressure, and therefore do not account for the e↵ect of nickel at pressures found in the cores of any of the terrestrial planets. Therefore despite there being only a small e↵ect of nickel at the temperatures investigated here, these results are not necessarily conclusive enough to conﬁrm this e↵ect at planetary core conditions.

Subsequent to the writing of this thesis and the oral examination, it has been reported to me (IG Wood, pers. comm.) that the origin of the anomalies in the thermal expansion curves, shown in Fig. 11.2 and Fig. 11.3, almost certainly arise from an unexpectedly large o↵set between the temperature of the sample and the temperature recorded by the thermocouple in the sample holder when the Anton Paar HTK 1200N high-temperature stage is operated with the sample chamber under vacuum.

184 CHAPTER 11

Conclusions

Regret none of it, not one of the wasted days you wanted to know nothing when the lights from the carnival rides were the only stars you believed in

— Dorianne Laux

As discussed in Chapter 1, the aim of this work has been to further our understanding of the Fe-Ni-Si system, with the view that this would in turn aid our understanding of the core composition of the Earth and other terrestrial planets. Here, the results of the calculations and experiments and the implications these have on the cores of the terrestrial planets are discussed.

11.1 Summary of Results

Both ab initio calculations and high-pressure, high-temperature experiments have been carried out to investigate the Fe-Ni-Si system. The majority of the investigations concentrated on the two binary endmember systems – the FeSi and NiSi system. By establishing the phase diagram and behaviours of these two systems, a solid foundation is built on which further research into the Fe-Ni-Si system can be based.

Ab initio calculations were carried out to determine the stability ﬁelds of various structures in FeSi and NiSi. In Chapter 4, the "-FeSi CsCl phase transition was inves- ! tigated at 0 K and at high temperatures and pressures. In Chapter 5, calculations revealed that none of the newly discovered structures in NiSi were stable in FeSi at

0 K, although it was found that the MnP structure would become stable at negative pressures. Chapter 6 described the calculations carried out to determine the phase

185 11.2. Implications for the Cores of the Terrestrial Planets boundary of the "-FeSi CsCl phase transition at high temperatures and pressures in ! the NiSi systems.

High-pressure, high-temperature experiments have also been carried out in this work.

The melting curves of NiSi (Chapter 7) and of the Fe-FeSi eutectic (Chapter 8) were measured using laser-heated diamond anvil cell (LH-DAC) experiments, and in Chap- ter 9, both the thermal and room temperature Equations of State of MnP-NiSi were determined from neutron di↵raction experiments. Additionally, the e↵ect of nickel on the thermal expansion of FeSi was investigated in Chapter 10 using X-ray di↵raction experiments.

11.2 Implications for the Cores of the Terrestrial

Planets

Although pressures and temperatures at the Earth’s core are predicted to be in the region of 6000 K and 330 GPa (Anzellini et al., 2013), the cores of the terrestrial planets and the Moon are thought to be at much lower temperatures and pressures.

The core of Mars is estimated to be at pressures of between 24 and 42 GPa and at temperatures of between 2000 and 2600 K (Fei and Bertka, 2005), while the core of

Mercury is thought to be at pressures of between 8 and 40 GPa, with temperatures of

1700 to 2200 K (Chen et al., 2008). The core of the Moon is thought to be between 5 and

6 GPa and 1300 to 1900 K (Wieczorek, 2006). These lower pressures and temperatures have implications on the stable structure adopted by FeSi and NiSi. According to both calculated and experimental phase diagrams of FeSi and NiSi, the CsCl structure is stable in both systems at the high temperatures and pressures of the Earth’s core.

However, at the milder conditions of the terrestrial planetary cores, the situation is less clear cut. The lower core pressures of Mercury falls across the calculated phase boundary of the "-FeSi CsCl phase transition in FeSi and the MnP "-FeSi phase ! ! boundary in NiSi. The higher pressures in the core of Mars means that the CsCl phase is stable in FeSi, but crosses the "-FeSi CsCl phase boundary in NiSi. !

Although NiSi is unlikely to occur in the core, a Fe-Ni-Si alloy is a realistic composition

186 Chapter 11. Conclusions of the planetary cores and the core of the Moon. While new phases are unlikely to come into stability with the combination of the two FeSi and NiSi phase diagrams, it is possible that the phase boundaries and melting curves shift although not drastically.

Therefore, study of the FeSi and NiSi phase diagrams can indicate which phases are likely to be stable in the core of the terrestrial planets; it is a deﬁnite possibility that the low pressure "-FeSi phase, common to both FeSi and NiSi, is found in the cores of small terrestrial planets. As such, having well deﬁned phase diagrams for both FeSi and NiSi is very important for research into the cores of the terrestrial planets.

The low core pressures of Mercury and the Moon mean that it is important to understand the behaviour of the low pressure phases found in NiSi. Of particular importance is the MnP phase, since calculations indicate that the MnP phase would also be stable in FeSi, albeit at negative pressures. Since the exact behaviour of the Fe-Ni-Si system is not known it is not unfeasible for the MnP phase to exhibit stability with the addition of Ni into FeSi. Therefore understanding how the MnP phase behaves under compression and on heating, as has been done here with neutron di↵raction experiments, is very important.

Despite the cores of Mercury, Mars and the Moon being at lower pressures and temperatures, the estimated temperatures are still fairly high, as would be expected. It is therefore important and necessary to determine the melting curves of possible core compositions. Measuring the melting curves of the endmember systems, such as has been done here with the melting curve of NiSi and the eutectic Fe-FeSi melting curve, establishes the boundaries of melting, and helps constrain the phase diagram. Accu- rately determined melting curves are of particular importance for the core of Mercury, where a snowing core model has been proposed for core growth. Having accurate melting curves of the binary endmember systems of the ternary Fe-Ni-Si system can also aid in establishing whether a pattern can be observed in melting across the Fe-Ni-Si system. The eutectic melting curve is of particular importance since this can provide a vital anchoring point in the Fe-Si phase diagram.

Further implications can be inferred by comparison of the Fe-FeSi eutectic melting curve

187 11.2. Implications for the Cores of the Terrestrial Planets and the adiabats of Mars and Mercury. Figure 11.1 shows the adiabats of Mercury and Mars as calculated by Chen et al. (2008) and Fei and Bertka (2005) respectively overlying the Fe-FeSi eutectic melting curve measured in Chapter 8. This indicates that a ‘double snow’ state would exist for Mars, due to the two points where the adiabat of Mars crosses over the melting curve. In comparison, Mercury’s adiabat only crosses the melting curve once, and at a fairly low pressure, supporting the proposed snowing core model.

2400 Fe-FeSi Melting Curve Mercury Adiabat Mars Adiabat

2200

2000

1800

Temperature /K

1600

1400 0 4 8 12 16 20 24 28 32 36 40 Pressure /GPa

Figure 11.1: Adiabats of Mars (blue) and Mercury (pink), determined by Chen et al. (2008) and Fei and Bertka (2005) respectively, plotted over the Fe-FeSi melting curve in red (see Chapter 8 for further details.

Although the focus of this work has been on the phase diagrams of the endmember systems of the Fe-Ni-Si ternary system, it is equally important to consider how the two FeSi and NiSi systems come together to form the Fe-Ni-Si ternary system, since a

Fe-Ni-Si alloy is a more realistic composition for the core. To that end, it is important to understand how the addition of Ni a↵ects FeSi. As core conditions of the terrestrial planets are under high temperatures, it is important to understand how a material reacts under heating, hence an understanding of the e↵ect of Ni on thermal expansion of FeSi is crucial when investigating the cores of the terrestrial planets.

The aim of this thesis has been to investigate the Fe-Ni-Si ternary system, in order to

188 Chapter 11. Conclusions provide us with additional insight into the core composition of the terrestrial planets.

Calculations have given possible explanations for the di↵erences observed both computationally and experimentally in the phase diagram of FeSi, as well as showing the di↵erences between the FeSi and NiSi systems, and experiments have helped constrain the phase diagrams of the endmember systems, and have given an insight into the behaviour of the Fe-Ni-Si alloys. In tandem, these provide a solid platform for further research into the core composition of the terrestrial planets.

189 190 APPENDIX A

Convergence Tests

Convergence tests for the calculations carried out in Chapter 4, with conditions for convergence of H<0.001 eV per atom for all calculations.

A.1 VASP Convergence Tests

(a) (b)

(c)

Figure A.1: Graphs showing enthalpy values calculated using VASP for (a) CsCl-FeSi for di↵erent sized k-point grids, showing convergence at 17 17 17; (b) "-FeSi for di↵erent sized k-point grids, showing convergence at 5 5 5; (c) di↵erent⇥ plane⇥ wave cut-o↵ energies, showing convergence at 600 eV. ⇥ ⇥

191 A.2. CASTEP Convergence Tests

A.2 CASTEP Convergence Tests

(a) (b)

(c)

Figure A.2: Graphs showing enthalpy values calculated using VASP for (a) CsCl-FeSi for di↵erent sized k-point grids, showing convergence at 11 11 11; (b) "-FeSi for di↵erent sized k-point grids, showing convergence at 5 5 5; (c) di↵erent⇥ plane⇥ wave cut-o↵ energies, showing convergence at 1800 eV. ⇥ ⇥

192 Appendix A. Convergence Tests

A.3 Abinit Convergence Tests

(a) (b)

(c)

Figure A.3: Graphs showing enthalpy values calculated using VASP for (a) CsCl-FeSi for di↵erent sized k-point grids, showing convergence at 13 13 13; (b) "-FeSi for di↵erent sized k-point grids, showing convergence at 9 9 9; (c) di↵erent⇥ plane⇥ wave cut-o↵ energies, showing convergence at 34 Hartree. ⇥ ⇥

193 APPENDIX B

Further Work on FeSi

B.1 Crystal Structure of the "-FeSi Phase

To determine how well the calculations match with experiments on the "-FeSi phase, the calculated fractional co-ordinates at P = 0 have been compared with fractional co- ordinates measured at room pressure by neutron di↵raction (Wood et al., 1996). These values are shown in Table B.1.

Table B.1: Fractional co-ordinates of "-FeSi at room pressure

Wood et al. (1996) VASP CASTEP Abinit xFe 0.1359(3) 0.1352 0.1344 0.1378 xFe – 0.0007 0.0015 -0.0019 xSi -0.1581(4) -0.1595 -0.1592 -0.1573 xSi – 0.0014 0.0011 -0.0008 xFe and xSi are the fractional co-ordinates of Fe and Si. xFe and xSi are di↵erences between the calculated and experimental co-ordinates.

From Table B.1, it can be seen that all the calculated fractional co-ordinates are in good agreement with the experimentally measured co-ordinates of Wood et al. (1996).

The fractional co-ordinate of Fe is most closely matched by VASP calculations, but that of Si is best matched by Abinit calculations. However, all calculated fractional co-ordinates are within 0.002 of the experimental fractional co-ordinates.

B.2 Equations of State

The equations of state (EoS) of both phases of FeSi have been measured experimentally in high-pressure experiments. The computational equations of state were determined by

ﬁtting to energy and volume values. The pressure from this ﬁt could then be compared to that obtained by the ab initio package.

194 Appendix B. Further Work on FeSi

Figure B.1 shows the EoS for di↵erent experiments for both CsCl-FeSi and "-FeSi, with a dotted line showing extrapolation for pressures greater than the highest pressure reached in the experiment. From Figure B.1a, it can be seen that the EoS measured at the highest pressure, that of Sata et al. (2010), matches best with the calculated EoS at high pressures. However, the worst match is that of Lord et al. (2010), which predicts a higher pressure than both the calculated EoS and other experimental EoS at high pressures, though agrees well with Dobson (2003) at low pressures. The experimental

EoS were measured at room temperature, compared to the calculated EoS, which was

obtained at 0 K. However, it can be seen from Figure B.1a that this temperature

di↵erence appears to have a negligible e↵ect on the EoS when compared with the

experimental EoS. At roughly 50 GPa, all but the EoS of Lord et al. (2010) match very

well.

The same trend can be seen for "-FeSi in Figure B.1b, where the calculated EoS matches

almost perfectly with the high pressure experimental EoS measured by Lin (2003).

Both Guyot et al. (1997) and Ross (1996) carried out experiments that only reached pressures of about 10 GPa, which means that the EoS has to be extrapolated a lot to reach high pressures. However, it can also be seen that there is a fair amount of mismatch even at low pressures although Guyot et al. (1997) and Ross (1996) agree very well with each other. This shows that high-pressure data are necessary for an EoS that is accurate over a wide range of pressure. Although less pronounced, this can also be seen in Figure B.1a – the EoS of Sata et al. (2010), whose experiments reached 180

GPa, agrees the best with the calculated EoS.

However, there are also discrepancies between the calculated EoS themselves. Fig- ure B.2 shows the EoS calculated using the di↵erent ab initio packages as well as the two calculated by Caracas and Wentzcovitch (2004) and Voˇcadlo et al. (1999) for both

CsCl-FeSi and "-FeSi. The EoS parameters obtained from these ﬁts are also shown in

Tables B.2 and B.3. As expected, there is also some variation between the calculated

EoS of the di↵erent ab initio packages, likely due to the variations in the pseudopotentials used for each package, as discussed in Chapter 4.

195 B.2. Equations of State

(a)

VASP Calculations Dobson et al. (2003) 400 Ono et al. (2007) Sata et al. (2010) Lord et al. (2010)

300

200

Pressure /GPa Pressure

100

7.0 8.0 9.0 10.0 11.0 12.0 3 Volume /Å per atom

(b)

400 VASP Calculations Lin et al. (2003) Guyot et al. (1997) Ross (1996)

300

200

Pressure /GPa Pressure

100

6.0 7.0 8.0 9.0 10.0 11.0 12.0 3 Volume /Å per atom

Figure B.1: Comparison of the Equations of State for (a) CsCl-FeSi; and (b) "-FeSi, showing the EoS for the VASP calculations with experimentally measured EoS. The experiments reached di↵erent pressures, extrapolations of the EoS beyond the highest pressure reached is shown by a dotted line.

196 Appendix B. Further Work on FeSi

Table B.2: The calculated ﬁtting parameters for the third-order BMEoS ﬁts of CsCl-FeSi and "-FeSi

CsCl-FeSi VASP CASTEP Abinit a. b. 3 -1 V0 / A˚ atom 10.52 10.17 10.64 10.69 10.61 K0 /GPa 236 251 270 220 226 K0 4.4 4.5 4.0 4.8 5.4

"-FeSi VASP CASTEP Abinit a. b. 3 -1 V0 / A˚ atom 11.01 10.71 11.26 11.27 11.11 K0 /GPa 231 233 242 221 227 K0 4.2 4.4 3.9 4.2 3.9

V0 is the volume, K0 is the bulk modulus, and K0 is the derivative of K0, all at P = 0. a. Voˇcadlo et al. (1999) b. Caracas and Wentzcovitch (2004)

Table B.3: The experimental ﬁtting parameters for the third-order BMEoS ﬁts of CsCl-FeSi and "-FeSi

CsCl-FeSi Dobson (2003) Ono et al. (2007) Sata et al. (2010) 3 -1 V0 / A˚ atom 10.87 10.66 10.69 K0 /GPa 184 225 200 K0 4.2 4.0 4.2 Lord et al. (2010) 11.09 222 4.2

"-FeSi Lin (2003) Guyot et al. (1997) Ross (1996) 3 -1 V0 / A˚ atom 11.27 11.30 11.30 K0 /GPa 185 172 176 K0 4.8 4.0 4.0 V0 is the volume, K0 is the bulk modulus, and K0 is the derivative of K0, all at P = 0. a. Voˇcadlo et al. (1999) b. Caracas and Wentzcovitch (2004)

197 B.2. Equations of State

(a)

(b)

Figure B.2: Comparison of the calculated Equations of State for (a) CsCl-FeSi; and (b) "-FeSi, showing the calculated EoS from VASP, CASTEP and Abinit compared to those calculated by Voˇcadlo et al. (1999) and Caracas and Wentzcovitch (2004).

198 References

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