INVESTIGATIONS INTO THE STRUCTURE AND PROPERTIES OF ORDERED PEROVSKITES, LAYERED PEROVSKITES, AND DEFECT PYROCHLORES.

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Meghan C. Knapp

*****

The Ohio State University 2006

Dissertation Committee: Approved by Professor Patrick M. Woodward, Advisor

Professor Claudia Turro ______Professor Terry Gustafson Advisor Graduate Program in Chemistry Professor Mohit Randeria

ABSTRACT

The work described in this thesis explores the effects of chemical substitution on the

structures and properties of perovskites, layered perovskites, and defect pyrochlores.

Layered perovskites, particularly of the variety K2NiF4, the n = 1 Ruddlesden-Popper

structure, were studied to determine the factors that drive octahedral tilting distortions. It

was determined that these structures, which are more inherently strained than perovskites,

are influenced by the bonding environment around the anions and the A-cation as well as

the electrostatic interactions between layers. The effects of cation ordering on the

symmetry of Ruddlesden-Popper structures are also presented. Dion-Jacobson structures

were also analyzed, and it was found that the trends that govern the behavior of

Ruddlesden-Popper structures were not applicable. When n = 1 for Dion-Jacobson

structures, the weak inter-layer interactions make the parent structure prone to tilting and

plane slippage. This stoichiometry has several competing structures, many of which are

observed for AMO4 compounds with highly covalent M-O interactions.

Stoichiometric perovskites with multiple A-cations rarely exhibit layered ordering of the

A-cations. Double perovskites having two A-cations and two M-cations with the formula

AA'MM'O6 (A= Na, K, Li, A' = La, M = Mg, Sc M' = W, Nb, Sb, Te, or when M = M', M =

Ti, Zr) were studied to determine the driving force for layered ordering of A-site cations.

ii It was determined that such ordering is cooperative with the displacement of d0 transition

metals from the M-cation site, which allows for relief of the bonding strain on the intra-

layer oxygen ions. This represents a novel way to propagate cation displacements, i.e.

via the ordering of the A-cation which works synergistically with the M-site cation

displacements. Such displacements can produce desirable dielectric properties, and these

properties can be further enhanced by the use of an A-cation with a stereo-chemically

active lone pair. As such, analogous compounds were prepared where A' = Bi3+. It was

found that when perovskites were formed, no layered ordering of the A-cations was

produced. When M' was a main group element, namely Sb5+, the perovskite phase and

the defect pyrochlore phase were observed to be competitive. The dielectric properties of

these materials were tested and it was found that the bismuth structure containing Nb5+ had the highest dielectric constant.

iii

DEDICATION

For Samantha and Charlotte You Can

iv

ACKNOWLEDGMENTS

My time at Ohio State has been a fascinating period of intellectual development and personal growth. This work would not have been possible without the help and support of many people. I would like to acknowledge the people who helped me with the acquisition and treatment of my data: Dr. Gordan Renkes for his assistance with X-ray powder diffraction; Dr. Judith Stalick for use of the BT-1 line at NIST, Dr. Cameron

Begg for SEM elemental analysis, Dr. Harold Stokes for help with ISOTROPY, and Dr.

Suzanna Garcia-Martin for SAED analysis of NaLaMgWO6.

I would also like to acknowledge Dr. Bruce Bursten for his mentorship in my

career development, both as a teacher and a scientist. I had the opportunity to TA for

honors chemistry under Bruce and to discuss teaching methodology. He has provided

both guidance and encouragement in exploring my options for teaching at the collegiate

level. I have benefited from his contagious enthusiasm for Group Theory and his

dedication to his craft.

I would like to thank the members of the Woodward group, both past and present

for their help with my academic career. Paris Barnes, Hank Eng, and Mike Lufaso,

taught me all about solid state synthesis and X-ray powder diffraction. Young-il Kim

taught me about the measurement and treatment of impedance data. Fus, Matt, Rebecca,

v and Harry helped me to learn our craft the best way there is: by teaching it. Fus also taught me to use CASTEP… three times! All the Woodward group members provided

feedback and support for my preparations for presentations, candidacy, and graduation.

Of course, it is important for me to acknowledge my advisor, Dr. Pat Woodward.

As with all good advisors, he supported me in my research by constructively critiquing my analysis and forcing me to think more critically about my work. He has always been

available for discussion, but has given me a great deal of freedom in the conduction of my research. Additionally, Pat has served as a role model of a parent-scientist, being able to advance in his career while staying actively involved in his kids’ lives. Without his

support, completion of my degree would not have been possible while still being the kind

of mother I want to be to my girls.

Special thanks go to my family for their support of my decision to return to

school. In particular my parents, Penny and Jerry, seemed to know more than I did how

much I needed this. My husband Peter has gone above and beyond what most women

expect of their spouses in support of me. He has listened to and read countless

presentations of my research, and advised me on mathematical calculations. He has

taken over almost all aspects of maintaining our home. He has been a stellar parent at all

times, and has managed on his own on the numerous occasions that I have traveled to

conferences. Peter has managed my stress far better than I have, and been the primary

witness to my tears in the past few months, even as he has had his own stresses. Without

his love and support, I would never have considered returning to school, let alone had the

confidence to complete my degree.

vi

VITA

28 October 1975...... Born – Indianapolis, Indiana

8 May 1998...... Bachelor’s of Science Peabody College at Vanderbilt University.

1998 – 2001...... Science Teacher. The Columbus City Schools.

2001 – 2002...... University Distinguished Fellow. The Ohio State University.

2002 – 2005...... Teaching/Research Asst. The Ohio State University.

2003 – 2004...... NSF GK-12 Fellow. The Ohio State University.

2005 – 2006...... University Distinguished Fellow. The Ohio State University.

PUBLICATIONS

Research Publications

1. Knapp, Meghan C.; Woodward, Patrick M. A-site cation ordering in AA'BB'O6 perovskites. Journal of Solid State Chemistry (2006), 179(4), 1076-1085.

FIELDS OF STUDY

Major Field: Chemistry

vii

TABLE OF CONTENTS

P a g e

Abstract...... ii

Dedication...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables...... xii

List of Figures ...... xiv

Abbreviations...... xvi

Chapters:

1. INTRODUCTION...... 1

1.1. Background, properties and applications of perovskites and related structures. . .1 1.1.1. Historical Background 1.1.2. Properties of Ordered Perovskites 1.1.3. Properties of Layered Perovskites 1.1.4. Properties of Pyrochlores 1.2. Introduction to the Perovskite Structure...... 5 1.2.1. General Description of the Perovskite Structure 1.2.2. Octahedral Tiliting Distortions and Notations 1.3. Layered Perovskites...... 6 1.3.1. The Ruddlesden-Popper and Dion-Jacobson Structures 1.3.2. Notations for Tilting Distortions 1.4. Perovskites with A- and B-site Cation Ordering...... 11 viii 1.5. Pyrochlores ...... 15 1.5.1. General Description of the Pyrochlore Structure 1.5.2. Defect Pyrochlore Structure 1.6. References...... 18

2. GROUP THEORETICAL ANALYSIS OF LAYERED PEROVSKITES ...... 20

2.1. Introduction...... 20

2.2. Distortions in Ruddlesden-Popper Structures ...... 20 2.2.1. Symmetry Allowed Distortions 2.2.2. Octahedral Tilting Distortion 2.3. Distortions in n = 1 Dion-Jacobson Structures...... 37 2.3.1. Symmetry Allowed Distortions 2.3.2. Octahedral Tilting Distortions 2.4. Conclusions...... 48

2.5. References...... 48

3. EVALUATION OF OCTAHEDRAL TILTING DISTORTION TRENDS IN RUDDLESDEN-POPPER AND DION-JACOBSON STRUCTURES...... 50

3.1. Introduction...... 50

3.2. Ruddlesden-Popper Structures...... 50 3.2.1. Lattice Strain and Bond Valence Sums 3.2.2. Measures of Strain 3.2.3. Known n=1 Ruddlesden Popper Structures 3.2.4. Modeling of RP n = 1 Structures Using Lattice Eneries 3.3. Dion-Jacobson Structures ...... 71 3.3.1. Lattice Strain 3.3.2. Known n = 1 Dion Jacobson Structures 3.3.3. Structures with n > 1 3.4. Conclusions...... 71

3.5. Referencecs...... 73

4. STRUCTURAL STUDIES OF NaLaMM'O6 PEROVSKITES AND THE INFLUENCE OF CATION SUBSTITUTION ON THE LAYERED ORDERING OF SODIUM AND LANTHANUM ...... 85 ix 4.1. Introduction ...... 85

4.2. Effects of distortions cation ordering on X-ray powder diffraction...... 86 4.2.1. M-site Cation Ordering 4.2.2. A-site Cation Ordering 4.2.3. Distortions and Group Theoretical Analysis 4.3. Experimental ...... 93 4.3.1. Synthesis 4.3.2. Structural Characterization (XRD, NPD, SAED) 4.3.3. Data Analysis 4.4. Results...... 97

4.4.1. NaLaMgWO6 4.4.2. NaLaMgTeO6 4.4.3. NaLaScNbO6 4.4.4. NaLaScSbO6 4.4.5. NaLaTi2O6 and NaLaZr2O6 4.4.6. KLaMgWO6 and LiLaMgWO6 4.5. Discussion...... 113

4.6. Conclusion...... 115

4.7. References...... 115

5. STRUCTURAL STUDIES OF NaBiMM'O6 STRUCTURES AND THE INFLUENCE OF THE STEREO-ACTIVE LONE PAIR ON ORDERING AND THE STABILITY OF THE PEROVSKITE structure...... 118

5.1. Introduction...... 118

5.2. Introduction to Dielectrics...... 119

5.3. Experimental...... 120 5.3.1. Synthesis 5.3.2. Structural Characterization 5.3.3. Dielectric Measurements 5.4. Results...... 122

5.4.1. Comparison of NaLaScNbO6 and NaBiScNbO6 5.4.2. Comparison of NaLaScSbO6 and NaBiScSbO6 5.4.3. Other NaBiMM’O6 structures 5.4.4. Dielectric Measurements 5.5. Discussion...... 134 x 5.5.1. Relative Stability of Pyrochlore and Perovskite Structures 5.5.2. Comparison of Dielectric constants 5.6. Conclusion ...... 134

5.7. References...... 134

List of References...... 136

Appendix A. Additional Tables of Tilting Distortions...... 152

A.1 Low Symmetry Tilting Distortions for Ruddlesden-Popper Structures A.2. Tilting Distortions for RP with n>1 A.3 Tilting Distortions for DJ with n > 1

xi

LIST OF TABLES

Table Page

2.1 Tilt Systems in n=1 Ruddlesden-Popper Structures ...... 28

2.2 Sr-O bond lengths for 10° tilt in each systems ...... 31

2.3 Interlayer anion distances ...... 36

2.4 Space Groups Resulting From Cation Ordering in n = 1 Ruddlesden- Popper Structures ...... 35

2.5 Tilt Systems in n=1 Dion-Jacobson Structures ...... 38

2.5 Tilt Systems in n=1 Dion-Jacobson Structures...... 42

3.1 Comparison of Analogous Perovskite and Ruddlesden-Popper Structures ...... 58

3.2 Indicators of strain in RP structures ...... 60

3.3 Room temperature Ruddlesden-Popper structural data...... 62

3.4 High and low temperature Ruddlesden-Popper structures...... 70

3.5 Lattice energy calculations for Ca2TiO4 ...... 72

3.6 Lattice energy calculations for Ca2RuO4...... 72

3.7 Theoretical and empirical DJ lattice constants ...... 74

3.8 Room temperature Dion-Jacobson structures...... 74

3.9 Compounds exhibiting plane slippages...... 79

xii 4.1 Synthetic conditions...... 94

4.2 Refined structural parameters for NaLaMM'O6 compounds...... 100

4.3 M-cation displacements and tilting angles...... 111

4.4 Pawley fit lattice constants for KLaMgWO6 and LiLaMgWO6...... 112

5.1 Pawley fit lattice parameters for NaBiMM'O6 compounds...... 125

5.2 Dielectric data for NaBiMM'O6 compounds...... 125

A.1 Additional tilting distortions for n = 1 RP structures...... 153

A.2 Symmetry of n = 1 DJ structures with combinations of simple tilts . . . 154

A.3 Symmetry of n = 1 DJ structures with n>1 ...... 159

xiii

LIST OF FIGURES

Figure Page

1.1 The Perovskite Structure ...... 2

1.2 In-Phase and Out-of-Phase Tilting ...... 7

1.3 Th n = 1 and n = 2 Dion-Jacobson Structures ...... 9

1.4 The Ruddlesden Popper Structure...... 9

1.5 Parallel and Anti-parallel Tilting Distortions in DJ Structures ...... 10

1.6 Parallel and Anti-parallel Tilting Distortions in RP Structures...... 12

1.7 Perovskite with M-site Cation Ordering ...... 13

1.8 Perovskite with A-site Cation Ordering...... 14

1.9 Pyrochlore structure as derived from Fluorite...... 16

1.10 The B2O6 and A2O networks separated ...... 17

2.1 Comparison of the RP and Ln2CuO4 structures...... 22

− − 0 2.2 The RP structure with apa0 c tilting...... 24

2.3 A-site environments for RP structures in each tilt system...... 32

2.4 Comparison of sublayer orientation...... 36

2.5 A-site environments for DJ structures in each tilt system...... 45

3.1 Sr2TiO4 and lattice strain...... 56

xiv 3.2 Competing A2BX4 structures...... 66

3.3 Symmetry allowed shift of A-cations...... 67

3.4 Determination of tolerance for DJ structures...... 77

3.5 Competing ABX4 structures...... 80

4.1 Anion environments for perovkites with A-site ordering ...... 89

4.2 Phase diagram for tilting distortions in A and M-site ordered perovskites ...... 92

4.3 XPD pattern for NaLaMgWO6 ...... 98

4.4 NPD pattern for selected regions of NaLaMgWO6...... 103

4.5 XPD patterns for NaLaMM'O6...... 104

5.1 XPD pattern for NaBiScNbO6...... 123

5.2 XPD pattern for NaBiScSbO6 ...... 126

5.3 XPD patterns of NaBiScSbO6 while heating ...... 127

5.4 XPD pattern for NaBiGaSbO6...... 129

5.5 Bode Plot for NaAMM’O6 compounds ...... 133

xv

LIST OF ABBREVIATIONS

β beta (the angle between the a and c axes)

BVS Bond Valence Sum

°C degrees Celsius

calcd calculated

DJ Dion-Jacobson

ε dielectric permittivity

g gram(s)

h hour(s)

IR infrared

κ dielectric constant

μ micro

M moles per liter

MHz megahertz min minute(s) mol mole(s)

NPD Neutron powder diffraction ppm parts per million

xvi RP Ruddlesden-Popper rt room temperature

XPD X-ray powder diffraction

xvii

CHAPTER 1

INTRODUCTION

1.1 Background, properties and applications of perovskites and related structures

The perovskite structure has shown great flexibility of composition, incorporating nearly every member of the periodic table. The basic perovskite framework (see figure 1.1), a 3-dimensional array of corner sharing octahedra, is also the basis for related structures, known collectively as layered perovskites, which include two dimensional layers of corner sharing octahedra separated by layers of cations. The perovskite and layered perovskite structures can incorporate ions of a variety of sizes and charges, as this framework is flexible, allowing for subtle distortions that ease the bond strains created by size mismatch. However, the perovskite structure does compete with other structures of similar stoichiometry, particularly the defect pyrochlore structure. Perovskites and layered perovskites are of great interest for the wide variety of useful properties that they exhibit. Each of these properties is influenced by the structure, as subtle changes alter symmetry considerations, bond overlap, and band energy levels. Understanding and predicting the structure of these compounds is essential for the intelligent design of new and useful materials. In this work, various aspects of the structure of perovskites and layered perovskites have been explored. It has been observed that in layered perovskites, the undistorted, or aristotype structure, is most commonly observed, whereas this is not the case for perovskites.1 To understand this phenomenon, in Chapter 2 the possible distortions were determined and compared in relationship to various aspects of structure, 1

Figure 1.1: The polyhedral representation of the ternary perovskite structure, with the B- or M-cations in the octahedra.

2 including those commonly considered for predicting the perovskite structure (geometric and electrostatic considerations). In Chapter 3, structural distortions of Ruddlesden- Popper and Dion-Jacobson structures are compared with the analysis in Chapter 2. In examining literature on perovskites with more than one cation at a particular site, it was noted that while rock-salt ordering of the M-cations is very common, ordering of the A-cations is fairly uncommon. A layered ordering of these cations is observed primarily for non-stoichiometric perovskites (those with anion vacancies). Ordering in a stoichiometric perovskite has only been reported in conjunction with ordering of the M- cations.2, 3 This phenomenon was explored in Chapter 4 to determine the driving forces for layering of the A-cations. The layering of the A-cations is of interest as it could lead to desirable electronic properties. The dielectric constants and ionic conductivity of some of the materials synthesized were measured, and the effect of using an A-cation with a stereo-chemically active lone pair was examined in Chapter 5. In the process, it was determined that replacing the La3+ cation with the similarly sized Bi3+ cation altered the stability of the perovskites structure with respect to the pyrochlore structure. 1.1.1 Historical Background

The mineral perovskite, CaTiO3, was discovered by German chemist and mineralogist Gustav Rose in 1839, who named it for Russian dignitary Lev Alexeievich Perovsky.4 Since then, the name “perovskite” has been applied to the many compounds, synthetic and natural, that have similar structure and stoichiometry. Much of the early work on synthetic perovskites was done by V.M. Goldschmidt5, 6 who developed the principal of the tolerance factor,5 as well as other principals in use today. Natural perovskites make up much of the Earth’s mantle (50-90%) in the form of

MgSiO3. The dense packing of the perovskite structure makes it ideal for high pressure environments like this. It is believed to be the most abundant mineral within the Earth. The perovskite structure is also a part of many materials whose properties make them useful in industry, from ferroelectric behavior to superconductivity and colossal magnetoresistance. The properties of different types of perovskites and perovskite-like structures are discussed below.

3 1.1.2 Properties of ordered perovskites Ordered perovskites are of great interest due to the flexibility that they introduce into the composition. Having multiple cations at a particular site allows for ions of greater or lesser oxidation state to be incorporated. Ordering of these cations affects the symmetry of the structure, and further the bulk properties of the material. One of the

most interesting perovskites is BaTiO3, whose ferroelectric nature has gained it attention as a high κ dielectric. Substitutions at both the A- and B-sites have been made in an attempt to make these materials more technologically useful. There is a peak in the dielectric constant at a resonant frequency, and the dielectric constant also tends to be variable with temperature. It has been found that ordering of the cations affects the dielectric loss of a material, as well as the smoothness of the phase transition with temperature.7, 8 For these reasons, perovskites have been explored as possible dielectric relaxors.9, 10 1.1.3 Properties of layered perovskites Layered perovskites, such as Ruddlesden-Popper and Dion-Jacobson structures have been studied for a variety of applications. Their layered nature lends them to intercalation and possibly solid state proton conductivity.11 More interesting though have been studies on their useful bulk properties. Some of the earliest high temperature 12 superconductors, La2CuO4, had the RP structure. Studies of DJ structures containing manganese have investigated their ability to demonstrate colossal magnetoresistance.13 The Auruvillius structure, which is structurally related to the RP structure, has been incorporated into multiferroic materials.14 1.1.4 Properties of pyrochlores The pyrochlore structure is interesting in its highly symmetric and complex structure. Recent studies of bismuth containing pyrochlores have investigated its dielectric properties. Pyrochlores are an interesting choice for dielectric studies because the cubic symmetry and geometric frustration do not lend themselves to permanent

polarization. However, the pyrochlore Bi1.5ZnNb1.5O7, or BZN, has been investigated as a microwave dielectric.15, 16 The polarizable Bi3+ ion and the d0 Nb+ cation each are

4 susceptible to displacements under electric fields. These distortions are not permanent, so ferroelectric polarization is not seen. Research has also been done on an interesting property of some defect pyrochlores in which, under certain conditions, the crystal structure will expand under high pressures. Pressure Induced Volume Expansion, or PIE, was described by Barnes,17 where the open channel structure allows solvent molecules to be incorporated into the structure in under high pressure, thus expanding the crystal while reducing the size of the overall crystal-solvent system.

1.2 Introduction to the perovskite structure

1.2.1 General descriptions of perovskites structure

The simple, or ternary, perovskites have the stoichiometry ABX3 (also noted as

AMX3), where X is an anion and A and B are cations. Typically, the A-cation is a large soft cation with low valency, often an alkali or alkaline earth metal. The B-cation, in contrast, tends to be smaller and more highly valent. Almost every element on the periodic table, aside from the noble gases, has been incorporated into the perovskite structure. The structure can be thought of as a three dimensional network of corner

sharing BX6 octahedra (figure 1.1). The network forms a cubic array in which the B-X-B

bond angles are ideally 180°. The A-cations sit in a cube of 8 BX6 octahedra and are coordinated by 12 anions. The ideal, undistorted structure, or aristotype, is cubic and has the space group Pm3m (221). The structure can be described with either the A-cation or the B-cation at the origin. When the B-cation is at the origin, it has the Wyckoff position 1a (0,0,0). The A-cation has the Wyckoff position 1b (½, ½, ½), and the X-anion the position 3d (½, 0, 0). 1.2.2 Octahedral tilting distortions and notations For the aristotype, the size of the unit cell is determined by the B-X bond length,

RB-X, as one unit cell is twice this distance. However, the face diagonal is of length

2RA-X, or twice the A-X bond length. For perovskites then, RA-X and RB-X are related by:

5 R τ = A−X 2RB− X This is known as the Goldschmidt tolerance factor5. When τ = 1, the ideal geometry can be achieved, minimizing bond strain. However, if τ < 1, the A-cation is too small for the

cubo-octahedral cavity created for it by the BX3 network. Without some sort of distortion, the A-cation will tend to be under-bonded, having a total bond valence sum less than its oxidation state. One of the most common types of distortions is for the octahedral network to buckle, allowing the individual octahedra to rotate without significantly changing size or disconnecting from one another. If an octahedron rotates along the c-axis, it will force neighboring octahedra in the a-b plane to rotate in the opposite fashion to maintain connectivity. However, neighboring octahedra along the c-axis are not affected by this rotation, and as such may rotate in the same direction (in- phase) or the opposite direction (out-of phase) (See figure1.2). A number of systems of notation have been developed to describe these types of distortions. The most commonly used are those developed by Glazer18 and Aleksandrov19, 20. Glazer notation uses three letters with superscripts to indicate the three axes: a#b#c#. The superscript may be + to indicate in-phase tilting or – to indicate out-of- phase tilting. When tilting is equal about two different axes, the same letter is used for both. For example, a −b + a − describes a distortion in which there are out-of-phase rotations of equal magnitude about the a and c-axes, and an in-phase distortion about the b-axis. In Aleksandrov notation, the in-phase distortions are signified by a Greek letter Ψ, and the out-of-phase distortions are signified by a Greek letter Φ. Differences in magnitude are indicated by numerical subscripts. The tilt system described in Glazer

notation above would be shown as Φ1Ψ2Φ1 in Aleksandrov notation. Throughout this work, Glazer notation and variations on it will be used.

1.3 Layered perovskites

1.3.1 The Ruddlesden-Popper and Dion -Jacobson structures The Dion-Jacobson structure is a layered structure made up of two dimensional perovskite like slabs separated by A-cations. The DJ structure has the stoichiometry

6 (a)

(b)

Figure 1.2: a) In-phase tilting and b) out-of-phase tilting

7 AnBnX3n+1 in which n refers to the number of perovskite-like layers per slab (see figure

1.3). The simplest structures have the stoichiometry ABX4, with n = 1. When n = ∞, the perovskite structure is achieved. In the inter-layer region, the A-site cation is coordinated to 8 anions, rather than 12 as is seen in the perovskite structure. This interlayer region resembles the cesium chloride structure in this respect. Like the DJ structure, the Ruddlesden-Popper structure (RP) consists of perovskite-like slabs separated by A-cations. However, the RP structure has the

stoichiometry An+1BnX3n+1. The simplest structures have the stoichiometry A2BX4, with n = 1. As can be seen in figure 1.4, the perovskite-like layers are offset from one another by half a unit cell in the a and b directions. This creates a body centered structure with the perovskite slabs being related to one another by a screw axis. The c-axis is determined by the spacing of 3 slabs. Where the interlayer region of the DJ structures can be thought of as a cesium chloride (CsCl) arrangement of AX, the cation environment in RP structures more resembles the rock salt (NaCl) structure. However, with the A-site at the boundary between the two layers, the A-cation is coordinated by nine anions, four from the perovskite layer and five from the rock-salt layer. 1.3.2 Notations for tilting distortions A system of describing octahedral tilting distortions based on the Glazer 21 notation was developed by Bulou et al. to describe distortions of ABX4 structures. With slight modification this can be applied to the RP structures as well. Because the aristotypes of layered perovskites are tetragonal, the c-axis is always unique. That is, the c-axis is not symmetrically related to the other two. Therefore, the third indicator for the distortion is always written as c, even when there is no distortion along any axis: a 0 a 0c 0 . Additionally, the layered perovskites distinguish themselves from three dimensional perovskites in that rotations about the a or b-axes do not force rotations along the c-axis in the next layer. Therefore, these rotations may be either parallel, in which the rotations are the same from one layer to the next (see figure 1.5a), or anti-parallel, as observed with perovskites (figure 1.5b). As such, the subscripts p and a, respectively, may be added to the first two indicators. In the case of RP structures there is a further complication. The perovskite layers do not stack such that the octahedra are aligned.

8

Figure 1.3: The n=1 and n=2 DJ structures. Heavy lines indicate the cell edges.

Figure 1.4 The n = 1 RP structure. 9

(a) (b) Figure 1.5 Tilting in the DJ systems (a) parallel and (b) anti-parallel.

10 First of all, this means that the labels “parallel” and “antiparallel” are arbitrary. Figures 1.6a and b illustrate the tilting distortions that correspond to each of these labels. (Note that this is not the same labeling that Aleksandrov uses20). Further details on the unique aspects of tilting in the layered perovskites are given in Chapter 2.

1.4 Perovskites with A- and M-cation ordering

To distinguish between the layered perovskites and the perovskites with A-cation layering, discussions of ordering in the perovskite structure included here and in chapters 4 and 5 will use the letter M to refer to the 6-coordinate cation in perovskites. This will be true regardless of whether there is ordering at this site or not. Quaternary perovskites with two different M-site cations are well known. If these cations are significantly different in size and charge, they may order in a regular fashion. The most common type of ordering seen for these perovskites involves a rock salt arrangement of the cations, in which the M-site cations, M and M', alternate along each of the three cell axes (see figure 1.7). This causes a doubling of the a lattice parameter, so these perovskites are often referred to as “double perovskites”. This new larger cell has face centering, and the new space group is Fm3m , with the A- and M-cations remaining on fixed Wyckoff positions (8d, ¼ ¼ ¼ , 8a 0, 0, 0, and 8b ½ ½ ½ ). The anion moves to the 24e site (x, 0, 0), and thus has the ability to shift along the cell edge between the two different M-cations. This is an electrostatically favored arrangement in which the anion can move closer to the smaller, more highly charged M-cation, making room for the larger, lower valent cation. It would seem then that ordering of the A-cations would also favor a rock salt arrangement. This too would produce a face centered cell with doubled cell edges and Fm3m symmetry. However, in this case the anion is on a fixed position (24d). This is because the anion is coordinated to four A-cations rather than two as with the M-cations. The like A-cations are at 180° to one another, so a shift of the A-cation does not compensate for differences in size of the two types of A-cations. What is more typically observed is a layered arrangement of the two different types of the A-cations (Figure 1.8).

11

Figure 1.6: Tilting in RP structures (a) parallel, and (b) anti-parallel along the [110]T − − 0 − − 0 axis. For the a p a p c tilt system, the tilts in both layers are identicle. For the aa aa c tilt system, the top layer tilts in the opposite sense from the bottom layer.

12

Figure 1.7: The double perovskite structure, Fm3m , with rock salt ordering of the M- site cations.

13

Figure 1.8: Perovskite with layered ordering of the A-site cations, P 4 mmm .

14 1.5 Pyrochlores

1.5.1 General description of pyrochlore structure The pyrochlore structure is a complex arrangement of two different types of cations, again called A and B, and two different types of anions, X and X'. The

stoichiometry of a pyrochlore is A2B2X6X', where the B-cation is in a pseudo-octahedral environment and the A-cation has 8-fold coordination. There are a few ways to visualize the pyrochlore structure. Symmetrically, it is similar to the fluorite structure, and thus compounds with the fluorite and pyrochlore structures have very similar diffraction patterns. The fluorite structure, AX2, is shown in figure 1.9a. Each cation is coordinated to eight anions, surrounding it as a cube. In the pyrochlore structure, there is an ordered arrangement of the two cations, and one eighth of the anions are removed, figure 1.9b. These anion vacancies are directly coordinated only to the B-cations. The remaining anions surrounding this cation, ¾ of the total, shift along the edges of the coordination cube to form a trigonal antiprism. While technically not octahedral by symmetry, these pseudo-octahedra will be referred to as octahedra for the remainder of this work. The A- cation maintains the 8-fold coordination, but rather than having a cubic arrangement of anions, the shift in the anions surrounding the B-cation causes them to form a puckered ring around the A-cation. Only the remaining 1/8 of the anions are on a fixed position, still 180° apart. Alternatively, this structure can be described as two interpenetrating networks.

The BX6 octahedra form a corner sharing network, in which groups of four octahedra

from a tetrahedral cluster, and these clusters form channels through which the A2X' network penetrates (See figure 1.10). The A-X interaction is ignored because the A-X bond distance is significantly longer than the A-X' bond distance. The A2X' network is 22 identical to the network in anticrystobalite Cu2O. The A-cation is in linear

coordination, while the X' is in the center of an A4 tetrahedron. This is an important way of viewing the pyrochlore structure because it shows the independence of the two

networks. The BX6 network is structurally stable, with the A-cations serving primarily for charge balance.

15 (a) (b)

Figure 1.9 (a) The fluorite structure. (b) The fluorite structure with ordering of the cations. (c) The pyrochlore structure.

16

Figure 1.10 The B2O6 octahedral network and A'2O network.

17 1.5.2 Defect Pyrochlore structure In the defect pyrochlore structure, some or all of the X' anions are missing, thus forming a pyrochlore with the stroichiometry A2B2X7-y, or A2B2X6 in the case of no X'

anions. In this case the network of BX6 octahedra remains in tact. This stoichiometry, it should be noted, is analogous to the ABX3, or AA'MM'X6 stoichiometry of the perovskite, or double perovskite. The majority of compounds with this stoichiometry do adopt the

perovskite structure. There are a few examples of compounds having the A2B2X6 defect 23 pyrochlore structure (BiBO6 and PbSnO3 ). Note that for these compounds, the closely coordinated X' anions are missing, so the A-cations are surrounded only by a puckered ring of 6 X-anions. The electrostatic implications of the removal of these anions will be discussed further in chapter 5.

1.6 References

1.Lufaso, M. W.; Woodward, P. M., Prediction of the cyrstal structures of perovskites using the software program SPuDS. Acta Crystallographica 2001, B57, 725-738.

2.Arillo, M. A.; Gomez, J.; Lopez, M. L.; Pico, C.; Veiga, M. L., Structural and electrical characterization of new materials with perovskite structure. Solid State Ionics 1997, 95, 241-248.

3.Sekiya, T.; Yamamoto, T.; Torii, Y., Cation ordering in sodium lanthanum magnesium tungstate (NaLa)(MgW)O6) with the perovskite structure. Bulletin of the Chemical Society of Japan 1984, 57, (7), 1859-62.

4.Navrotsky, A.; Weidner, D. J., Preface. In Perovskite: A Structure of Great Interest to Geophysics and Material Science, Navrotsky, A.; Weidner, D. J., Eds. American Geophysical Union: Washington D.C., 1989; Vol. 45, p xi.

5. Goldschmidt, V. M., Naturwissenschafsen 1926, 14, 477-485.

6.Bhalla, A. S.; Guo, R.; Roy, R., The perovskite structure - a review of its role in ceramic science and technology. Materials Research Innovations 2000, 4, (1), 3-26.

7.Davies, P. K.; Tong, J.; Negas, T., Effect of ordering-induced domain boundaries on low-loss Ba(Zn1/3Ta2/3)O3-BaZrO3 perovskite microwave dielectrics. Journal of the American Ceramic Society 1997, 80, (7), 1727-1740.

8.Setter, N.; Cross, L. E., The role of B-site cation disorder in diffuse phase transition behavior of perovskite ferroelectrics. Journal of Applied Physics 1980, 51, (8), 4356-60.

18 9.Scarisoreanu, N.; Dinescu, M.; Craciun, F.; Verardi, P.; Moldovan, A.; Purice, A.; Galassi, C., Pulsed laser deposition of perovskite relaxor ferroelectric thin films. Applied Surface Science 2006, 252, (13), 4553-4557.

10.de Los S. Guerra, J.; Lente, M. H.; Eiras, J. A., Microwave dielectric dispersion process in perovskite ferroelectric systems. Applied Physics Letters 2006, 88, (10), 102905/1-102905/3.

11.Venkataraman, T.; Weppner, W., Journal of Materials Chemistry 2001, 11, 636-639.

12.Bednorz, J. G.; Mueller, K. A., Zeitschrift fuer Physik B: Condensed Matter 1986, 64, (2), 189-193.

13.Moritomo, Y.; Asamitsu, A.; Kuwahara, H.; Tokura, Y., Giant Magnetoresistance of manganese oxides with a layered perovskite structure. Nature 1996, 380, 141-144.

14.Porob, D.; Yang, C.; Maggard, P., Hydrothermal synthesis and characterization of Bi4Ti3O12.nBiMO3 (M=Fe, Mn, Co; n=1,2) series of compounds. Abstracts, 56th Southeast Regional Meeting of the American Chemical Society, Research Triangle Park, NC, United States, November 10-13 2004, GEN-520.

15.Levin, I.; Amos, T. G.; Nino, J. C.; Vanderah, T. A.; Randall, C. A.; Lanagan, M. T., Structural Study of an Unusual Cubic Pyrochlore Bi1.5Zn0.92Nb1.5O6.92. Journal of Solid State Chemistry 2002, 168, (1), 69-75.

16.Tagantsev, A. K.; Lu, J.; Stemmer, S., Temperature dependence of the dielectric tunability of pyrochlore bismuth zinc niobate thin films. Applied Physics Letters 2005, 86, (3), 032901/1-032901/3.

17.Barnes, P. W. Exploring structural changes and distortions in quaternary perovskites and defect pyrochlores using powder diffraction techniques. The Ohio State University, Columbus, 2003.

18. Glazer, A. M., Acta Crystallographica 1972, B28, 3384-3392.

19.Aleksandrov, K. S.; Beznosikov, B. V., Crystal chemistry and phase transitions in halides with perovskite structure. Fazovye Perekhody V Kristallakh. 1975, 68-129.

20.Aleksandrov, K. S., Structural phase transitions in layered perovskite-like crystals. Kristallografiya 1995, 40, (2), 279-301.

21.Bulou, A.; Fourquet, J. L.; Leble, A.; Nouet, J.; De Pape, R.; Plet, F., Structural phase transitions in the tetrafluoroaluminates MAlF4 with M = NH4, Rb, K, Tl. Studies in Inorganic Chemistry 1983, 3, (Solid State Chem.), 679-82.

22.Subramanian, M. A.; Aravamudan, G.; Rao, G. V. S., Oxide pyrochlores - a review. Progress in Solid State Chemistry 1983, 15, (2), 55-143. 19 23.Morgenstern-Badarau, M. I.; Michel, M. A., Pyrochloretype compound of the formula Pb2Sn2O6.xH2O. Annales de Chimie (Paris, France) 1971, 6, (2), 109-24.

20

CHAPTER 2

GROUP THEORETICAL ANALYSIS OF LAYERED PEROVSKITES

2.1 Introduction

The relationship between perovskites and the layered perovskites was described in chapter one, and is illustrated in figures 1.3 and 1.4. Group theoretical analysis of the perovskite structure has been extensively covered in literature.1-6 However, because of the differing aristotype and octahedral connectivity, the group theoretical analysis of RP and DJ structures, while similar in method, must be completed independently. The tilting distortions described in section 2.2.2 and 2.3.2 lend insight into the possible modes for relieving bond strain. More recent work has been done on RP structures with multiple A- site cations7-11 or multiple anions.12, 13 As such, there is a need to understand the symmetry changes caused by ordering of these ions within the structure.

2.2 Distortions in Ruddlesden-Popper Structures

The Ruddlesden-Popper (RP) structure, being a layered structure, has multiple modes of distortion while maintaining the same basic connectivity. Some types of distortions do not alter the symmetry of the structure; rather they serve only to adjust bond lengths to more appropriately allow the constituent ions relieve bond strain. These are described in section 2.2.1. The layered structure also results in two different types of lattices competing to determine the mutually compatible cell dimensions. In the case of the RP structure, the perovskite layer and the rock salt layer may have different ideal repeat distances. An example described by Brown14 involves

La2NiO4. The ideal repeat distance for the perovskite-like nickel oxide layer is twice the

21 ideal nickel oxide bond distance at 4.12Å. The LaO layer, with the NaCl structure, has an ideal repeat distance of √2 times the ideal lanthanum oxide bond distance, or 3.65Å. In cases like this where the repeat distances are not the same, or nearly the same, lattice strain is introduced which may be relieved through any combination of buckling of the perovskite-like slabs, introduction of defects, oxidation changes, or phase transitions14. The symmetry lowering effect of buckling the perovskite like layers, which can be visualized as rotations of the BX6 octahedra, is described in section 2.2.2. Defects and oxidation state changes often occur simultaneously, as the charge of a transition metal B- site cation may adjust to accommodate the loss of an anion.

Phase changes are seen most clearly with the Ln2CuO4 structures (Ln = Rare 3+ Earth Metal). While La forms the distorted RP structure La2CuO4, other rare earth metals, such as Pr, Nd, Sm, Eu, and Gd, form a related layered structure. This structure

also has a tetragonal aristotype in the space group I4/mmm. However, there are no CuO6 octahedra. In this case, the nominally axial oxygen anions move into the interlayer region so that the AO layer has a CsCl rather than NaCl arrangement.15, 16 This leaves the Cu2+ in a square planar environment (see figure 2.1) 2.2.1 Symmetry allowed distortions The aristotype perovskite structure has only one variable, the size of the lattice. The lattice constant is determined by the B-X bond length, which in turn determines the A-X bond length. Changes in the relative bond lengths, spacing, or shape of the octahedra cause changes in the symmetry of the structure itself. However, this is not necessarily the case for layered perovskites. The RP lattice is defined by two lattice parameters (a and c) and ½ of the anions as well as the A-site cations sit on a position with one free positional parameter. This means that the layers can separate from one another as the c-axis elongates with respect to the a-axis. The a-axis can expand or contract to accommodate lattice size preferences, and the c-axis can compensate for some of these changes.

Further, the BX6 “octahedra” are technically square bi-pyramids. They can be elongated or compressed along the c-axis without altering the symmetry of the structure. The axial anions, which help define the interlayer space, and the equatorial anions, which share a plane with the B-site cations, have different bonding environments. The axial

22

(a) (b)

Figure 2.1 Comparison of the (a)K2NiF4 and (b) Ln2CuO4 structures.

23 anions are coordinated to only one B-site cation, and five of the lower valence A-site cations. The equatorial anions,Xeq, are coordinated to two B-site cations and four A-site cations. As such, conditions exist to favor D4h distortions of the octahedra. The equatorial anions surround the B-site cation in the same plane, and the B-Xeq bond distance determines the a lattice parameter. The axial anions, Xax, cap the B-site cations and sit roughly in the same plane with the A-site cations. They are symmetrically fixed in the x and y directions, but are free to move in the z direction, thus allowing for the expansion or compression of the pseudo-octahedra. 2.2.2 Octahedral Tilting Distortions In-phase and out-of-phase octahedral tilting distortions for perovskites have been described in Chapter 1. Similar octahedral tilting distortions are possible in layered perovskites, though they are more complex. As with perovskites, these distortions necessitate symmetry changes. While tilts along any axis will cause cooperative tilts of adjacent octahedra in the perpendicular a and b directions, this is not the case in the c direction. Along the c-axis, octahedra are not connected via an anion shared between to B-site cations. As such, the tilting in one layer does not force tilting in the next layer, leading to more complex possibilities for tilting combinations. As shown in figures 1.5 and 1.6, the layer can be parallel or anti-parallel with one another. Another complexity of tilting in RP structures is that the symmetry of the system includes a screw axis. This allows for distortions in which tilting about one axis in one layer is followed by tilting about a different axis in the next layer. That is, while the notation a −b0c 0 would imply tilting about the a-axis in every layer, these structures have tilting about the a-axis in one layer followed by tilting of equal magnitude about the b- axis in the next layer. For these types of structures, the subscript 0 is used in place of p or

− − 0 a. So for example, the tilting in figure 2.2 is described as a p a0 c . There is an out of phase tilt in the a-direction in the bottom and top layers, with no tilt in the b-direction. The middle layer has no tilt in the a direction and an out of phase tilt in the b direction. Additionally, there are certain distortions in which the tilting in one layer is not repeated in the adjacent layer at all, but rather two layers away. This type of distortion is noted with the subscript s for “super-anti-parallel”. With a super-anti-parallel rotation

24

a) b)

− − 0 Figure 2.2: An RP compound with a p a0 c tilting, two views of the same structure: a) looking down the a-axis, b) looking down the b-axis.

25 about the c-axis, which causes a doubling of this axis, the octahedra in one layer will be rotated as a0a0c-, but in the next layer, there will be no rotation at all, and in the following layer there will be a rotation about c that is anti-parallel with the rotation in the first layer.

0 0 Such a tilting scheme would be described as a a cs . 2.2.2.1 Previous work on tilting The symmetry changes that correspond to octahedral tilting distortions have been described previously. Work on classification of tilting in A2BX4 structures was done by Dorian Hatch and Harold Stokes17, 18 and analysis of phase transitions was done by Aleksandrov et al.19 Aleksandrov later performed a more complete analysis of both RP and DJ structures with n > 1.20 While the works by Hatch and Stokes enumerate the possible distortions, they do not correlate these to tilt systems, so practical application of their tables is difficult. Furthermore, the tables in their works include duplications of symmetrically equivalent distortions. Aleksandrov does not include systems with super- anti-parallel tilting, though these structures have been observed. Also, Aleksandrov

makes a misleading distinction between cp and ca tilts in A2BX4 structures, though they are symmetrically equivalent. Additionally, he distinguishes between in-phase and out- of-phase tilts about the c-axis for all RP structures. However, when n is odd, there are different irreducible representations responsible for tilts in adjacent layers within a slab. In-phase and out of phase tilts about the c-axis are symmetrically equivalent, as adjacent layers are not required by symmetry to have identical tilts, or to have rotations in the same or opposite directions. The work described below clarifies some of these issues in order to produce a more useful and accurate analysis of tilting in RP structures. 2.2.2.2 Group Theoretical Analysis The analysis of tilting in RP and DJ structures was performed using the software ISOTROPY developed by Hatch and Stokes,21 which is now available online (http://stokes.byu.edu/isotropy.html). This program will lists subgroups of a parent space group that are related through irreducible representations (irreps). Symmetry changing distortions of a unit cell cause the loss of certain symmetry elements. The remaining elements determine the irrep for the distortion, and so the distortions are said to be a “basis” for the irrep. Irreps are indicated by a letter, a numerical subscript and,

26 sometimes, a superscript. The letter can be thought of as the direction of the distortion in reciprocal space. In the simplest case, an X indicates a distortion along one of the cell axes. The possible directions (and thus letters) are different for the different Bravais Lattices. Because different kinds of distortions are possible in each direction, they are distinguished by subscript numbers. A superscript + or – is an indication of whether the distortion will affect the inversion center. Some distortions can happen in more than one direction. For example, in perovskites the octahedral tilting distortion can occur along the a-, b-, or c-axis, or any combination of these axes in equal are unequal magnitude. Irreps based on such distortions are said to be multidimensional and have multiple direction vectors. A three dimensional irrep will have three direction vectors, (a, b, c). ISOTROPY is capable of analyzing a variety of distortions, including vector distortions such as atomic displacement or polyhedral rotations, and ordering of constituent atoms, vacancies, spins, or charges. Each distortion that destroys different symmetry elements of the parent, or aristotype, space group is the basis for a different irrep. Irreps may be considered in isolation or combination. Thus in the following analysis, tilts about the a- or b-axis can be combined with tilts about the c-axis. For more details on the theory behind ISOTROPY, see Stokes and Hatch, 2003.22 In the parent space group of RP structures I4/mmm, there are six irreps that are related to octahedral tilting. Of these, four are responsible for the distortions observed in

+ + the literature. The X 3 and X 4 irreps create out-of-phase distortions along the [100]T and [010]T axes (here the subscript T indicates that the tilting is in reference to the aristotype axes, though the resulting space groups may be orthorhombic or monoclinic with the defining a, b, and c-axes altered). In each case the tilt along the [100]T, or a- axis, in one layer is observed along the [010]T, or b-axis, in the next layer, owing to the screw axis of the parent space group. The primary difference between the two irreps is

+ the direction of tilt in the [010]T-axis with respect to the [100]T-axis, with the X 3 irrep

+ − − 0 being arbitrarily designated as parallel and the X 4 as anti-parallel. The a p a0 c

+ + + distortion shown in figure 2.2 is a basis for the X 3 irrep. Both the X 3 and the X 4

+ irreps are two dimensional, so use of both direction vectors in the X 3 irrep (a,a) would

27 − − 0 result in the tilt system a p a p c , with tilting of equal magnitude about the a and b axes in every layer.

+ The X 2 irrep is related to tilts along the [001]T axis. As indicated above, for n = 2k+1 structures (having an odd number of layers per perovskite slab), there is no distinction between in-phase and out-of-phase, and the parallel and anti-parallel systems

0 0 are symmetrically equivalent for n = 1. The tilt system a a c p is a basis for this irrep. No superscript is used as in the n = 1 structure; there is only one layer, and so there are no

other layers with which to be “in-phase” or “out-of-phase”. The P4 irrep also creates tilts

along the [001]T axis. However, alternating layers have no tilts and layers separated by

0 0 one length of c have anti-parallel tilts. These are the “super-anti-parallel” tilts ( a a cs ). This subtle difference creates a more highly symmetric structure with both the A- and B- cation sites having higher symmetry than in any other structure but the aristotype.

+ The P5 and N1 irreps create, respectively, in-phase and out-of-phase tilts along the a- and b-axes that are super-anti-parallel. These more complex tilt systems create multiple cation-sites and are not observed in reported structures. In an extended structure, it is usually preferable for like ions to have identical or similar bonding environments23. The ideal position for one An+ cation is also ideal for all other An+ cations, and it is undesirable to have an arrangement with the same cation in different positions with large differences in energy. At the same time, it is electrostatically favored for ions to be surrounded by a symmetric arrangement of ions of the opposite charge. In

+ the tilt systems resulting from the P5 and N1 irreps, the cations sit on Wyckoff positions with very low symmetry, largely 1 (no symmetry) or 1 (containing only and inversion center). These tilt systems are not favored for real structures, and the related irreps will not be discussed further in this work, though a table for these notations appears in the appendix (Table A.1).

+ + + The tilt systems that result from irreps X 2 , X 3 , X 4 , and P4 for n = 1 RP structures are listed in table 2.1. The isotropy subgroup (new space group) for each

28 Tilt Space Unit- Direction Vectors Cation Anion Positions System Group cell Positions size + + + X 2 X 3 X 4

0 0 0 a a c I4/mmm a=ap none none none A-cation Xax (139) c=c0 (4e) 0, 0, z (4e) 0, 0, z z≈0.35 z≈0.15 B-cation Xeq (2a) 0, 0, 0 (4c) ½, 0, 0 − − 0 a a c P42/ncm a≈√2ap none (a,0) none A-cation Xax p 0 (138) c≈c0 (8i) (8i) x'≈0, z'≈z x'≈0, z'≈z B-cation Xeq (4d) (4a) (4e) z'≈0 − − 0 a a c P42nnm a≈√2ap none none (a,0) A-cation Xax a 0 (134) c≈c0 (8m) (8m) x'=1/2, z'=1/2+z x'=1/2, z'=1/2+z B-cation Xeq (4e) (4d) (g) z'=0 − − 0 a a c Cmca a≈√2ap none (a,a) none A-cation Xax p p (64) b≈c0 (8f) y'=z, z'=0 (8f) y'=z, z'=0 c≈√2ap B-cation Xeq (4a) (8e) y'=0 − − 0 a a c Cccm a≈c0 none none (a,a) A-cation Xax a a (66) b≈√2ap (8l) (8l) c≈√2ap x'=1/4+z, y'=1/4 x'=1/4+z, y'=1/4 B-cation Xeq (4e) (8g) x'=3/4 − − 0 a b c Pccn a≈√2ap none (a,b) none A-cation Xax p p (56) b≈√2ap (8e) (8e) c≈c0 x'=0, y'=0, z'=z x'=0, y'=0, z'=z B-cation Xeq (4a) (4c) z'=0 (4d) z'=1/2 − − 0 a b c Pnnn a≈√2ap none none (a,b) A-cation Xax a a (48) b≈√2ap (8m) (8m) x'=0, y'=0, z'=z x'=0, y'=0, z'=z c≈c0 B-cation Xeq (4f) (4k) z'=0 (4l) z'=-1/2 0 0 a a c Cmca a≈c0 (a,a) none none A-cation Xax p (64) b≈√2ap (8d) x'=z (8d) x'=z c≈√2ap B-cation Xeq (4a) (8f) y'=-3/4, z'=1/4

Table 2.1: Possible octahedral tilting distortions for RP structures. (continued) 29 Table 2.1: continued

Tilt Space Unit- Direction Vectors Cation Anion

System Group cell + + + Positions Positions size X 2 X 3 X 4 − − + a a c Pbca a≈√2ap (c,-c) (a,a) none A-cation Xax p p a (61) b≈ c0 (8c) 0, z, 0 (8c) 0, z, 0 c≈√2ap B-cation Xeq (4a) 0,0,0 (8c) ¼, 0, ¼ − − + a a c P21/a a≈√2ap (c,c) (a,a) none A-cation Xax p p p (14) b≈√2ap (4e) 2z, 0, -z (4e) 2z, 0, -z c≈c0 B-cation Xeq γ≈90° (2a) 0,0,0 (4e) 0, ¾, ¼ − − a b c P21/c a≈c0 (c,c) (a,b) none A-cation Xax p p p (14) b≈√2ap (4e) z, 0, 0 (4e) z, 0, 0 c≈√2ap (4e) ½+z, ½, 0 (4e) ½ +z, ½, 0 γ≈90 B-cation Xeq (2a) 0,0,0 (4e) 0, ¾, ¼ (2d) ½, 0, ½ (4e) ½, ¼, ¼ − − a a c Pccn a≈c0 (c,-c) none (a,a) A-cation Xax a a a (56) b≈√2ap (8e) z, 0, 0 (8e) z, 0, 0 c≈√2ap B-cation Xeq (4a) 0,0,0 (8e) 0, ¾, ¼ − − a a c C2/c a≈c0 (c,c) none (a,a) A-cation Xax a a p (15) b≈√2ap (8f) ¼ -z, ¼, 0 (8f) ¼ -z, ¼, 0 c≈√2ap B-cation Xeq γ≈90 (4c) ¼, ¼, 0 (8f) ¼, ½, ¾ − − a b c P2/n a≈c0 (c,c) none (a,b) A-cation Xax a a p (13) b≈√2ap (4g) z, 0, z (4g) z, 0, z c≈√2ap (4g) z+ ½, ½, (4g) z+ ½, ½, z γ≈90° z Xeq B-cation (4g) ¾, ¼, 0 (2a) 0,0,0 (4g) ¾, ¾, 0 (2b) ½, ½, 0 − − 0 a b c P42/n a≈√2ap none (a,0) (0,b) A-cation Xax p a (86) c≈c0 (8g) 0, ½, ½ (8g) 0, ½, ½ +z +z Xeq B-cation (4e) 0, ½, 0 (4c) ¼, ¼, ¼ (4f) 0, 0, 0 0 0 a a c I41/acd a≈√2ap P4 (a,0) A-cation Xax s (142) c≈2c0 (16d) 0,0, (16d) 0,0, 3/8+1/2z 3/8+1/2z B-cation Xeq (8a) 0,0,0 (16f) ¼, ¼, ¼

30 distortion is given, along with the tilt system which is described using the modified Bulou notation.24 It should be noticed that for these irreps, tilting about the a- or b-axis is always out-of-phase. The direction vectors of the irrep used for each tilt system is indicated, with the top portion of the table showing tilt systems in which only one irrep is used, and the bottom portion where they are used in combination. Finally, the Wyckoff positions in the new space group are given (always assuming the first setting, unless otherwise indicated), with approximate position given for the free positional parameters. The exact position depends on the degree of atomic displacement or octahedral tilting. 2.2.2.3 Cation and Anion Environments To understand the types of tilting observed in RP structures it is necessary to examine the A-cation environment. This was done mathematically starting with an undistorted structure. The literature values for the lattice parameters and ionic positions 25 of Sr2TiO4 were used. For each tilt system, a ten degree tilt was applied. Only systems with one distinct magnitude of tilt were considered. Fortunately, nearly all examples of distorted structures fall into these tilt systems. Where applicable, the A-cation was shifted to idealize its valence. The results of this analysis appear in table 2.2. Ions separated by a distance greater than 2.9Å are not considered to be bonded, as this distance exceeds 5% of the Sr-O bond length in undistorted Sr2TiO4. The A-cation is surrounded by nine anions. (The labeling of the anions as axial or equatorial remains in reference to the B-site cation throughout this chapter and the next).

In the aristotype, four of these, Xax, form a square roughly in the same x-y plane as the

A-cation, four Xeq form a square below the cation, and one, which will be referred to as “capping”, sits at the apex (see figure 2.3a). As described in section 2.2.1, this environment is somewhat malleable even without octahedral tilting. A shift of the A- cation along the c-axis reduces the bonding to the capping anion, which is typically the shortest A-X bond. Elongation of the c-axis by separation of the layers causes the capping Xax anion to shift away from the A-cation. This has the effect of decreasing the overall bonding between the layers. Such a separation is less likely when the cations and anions have higher oxidation states. Distortion of the octahedral shape by altering of the

B-Xax to B-Xeq ratio is a mechanism for compensating for different preferred repeat lattice

31 Tilt Coordination System Space Group number Bond Lengths BVS CN Peak equitorial base 0 0 0 a a c I4/mmm (139) 9 2.558 1.927 4x2.748 4x2.666 0 0 a a c p Cmca (64) 7 2.318 2.08 4x2.709 2x2.598, 2x3.055 P4 /ncm − − 0 2 a p a0 c (138) 8 2.335 2.093 2x2.617, 2x2.883 1x2.456, 2x2.799, 1x3.141 − − 0 a p a p c Cmca (64) 6 2.311 2.242 1x2.483, 2x2.747, 1x2.938 2x2.516, 2x 3.113 a − a −c 0 a 0 P42nnm (134) 8 2.389 2.011 2x2.697, 2x2.823 1x2.392, 2x2.814, 1x3.216 − − 0 aa aa c Cccm (66) 7 2.389 2.207 1x2.75, 2x2.803, 1x 2.672 2x2.408, 2x3.260 a 0 a 0c s I41/acd (142) 7 2.318 2.08 4x2.709 2x2.598, 2x3.055

Table 2.2: Bond lengths in Sr2TiO4 for 10-degree tilt in each system.

32 (a) (b)

(c) (d)

(e) (f)

0 0 0 0 0 0 0 Figure 2.3: The A-site cation environment in the (a) a a c , (b) a a c p or a a cs , − − 0 − − 0 − − 0 − − 0 (c) a p a0 c , (d) aa a0 c , (e) a p a p c , and (f) aa aa c tilt systems. The A-site cation is shown as a small black circle. Anions with bonds that are within 5% of the length of the 25 Sr-O bond in the undistorted SrTiO3 perovskite (2.76Å) are shaded light gray. Those with shorter bonds (<2.62Å) are dark gray, and anions with longer bonds (>2.90Å) are white. 33 distances in the perovskite and rock salt layers. By altering the octahedra, an average lattice distance between the two preferred ones is achieved without reducing the symmetry of the structure. In the typical case, the octahedra are elongated along the c- axis, bringing the base Xeq anions closer to the A-cation. If all the octahedra are elongated along the c-axis and thus compressed along the a-axis to preserve bonding at the B-site, the Xeq are inherently over-bonded. Alternatively, lengthening of the a-axis will cause a decreasing of bonding at each of the ion sites. Tilts about the c-axis

0 0 0 0 Tilting along the c-axis, as with a a c p or a a cs (see figure 2.3b), preserves more symmetry than with the other tilt systems, and yet is observed less often. The local environments for these two tilt systems are identical, but the latter has greater symmetry. Long range forces may influence the preference for one over the other. In this case, the coplanar Xax-anions are drawn in closer to the A-site cation, while still maintaining a square planar environment. Two of the base Xeq-anions also move closer, while two move farther away, forming an elongated diamond shape below the anion. This type of tilting creates a smaller environment for the A-cation with an increased valence sum. However, the impact on the bond valence sum (BVS) for the A-cation is smaller for tilting about the c-axis than for other tilting systems. The Xeq-anions are bonded to four

A-cations. While the Xeq-anion shifts away from two of these A-cations, it does so by shifting towards the other two. The overall effect is to increase bonding at the A-site and most of the anion sites. Tilts about a single axis

− − 0 Tilt systems with tilts along only one axis in each layer, as with a p a0 c and

− − 0 aa a0 c , have six types of A-X bonds. Two of the coplanar Xax-anions move closer to the

A-cation, while two move farther away. At the base, one Xeq anion moves down and away, while the opposite anion moves up and toward the A-cation (see figure 2.3c and

2.3d). However, since the A2BX4 structures have only one perovskite-like layer per slab, each of these two Xeq anions is moving toward one A-site cation and away from another in a different layer. The remaining base anions are essentially unmoved. The capping anion moves toward the “far” Xax anions in the case of the parallel tilt and toward the 34 “close” Xax anions in the case of the anti-parallel tilt. The bonding increases at both the anion and the A-cation sites. However, the reduction in symmetry will allow the A-cation to move away from the Xeq that has moved towards it. This will allow a slight reduction in the BVS at the Xeq. Tilts about the a- and b-axes

− − 0 − − 0 Structures with tilts about the [110]T axes, a p a p c and aa aa c , also have six types of bonds (see figures 2.3e and 2.3f). In this case, one of the coplanar Xax-anions moves toward the A-cation while the opposing anion moves farther away. The remaining two anions move in the direction of the former. At the base, the Xeq anions move in pairs with two moving up toward the A-cation while the other two move down and away, towards the A-cation in the next layer. The overall effect is to slightly increase the bonding at the Xeq sites. The capping anion will move in the direction of the close base anions in the case of the parallel tilt and the far base anions in the case of the anti-parallel tilt. As can be seen from figures 2.3c, 2.3d, 2.3e, and 2.3f, parallel and anti-parallel tilts create similar local environments, so the reason for the parallel tilting preference is not immediately evident. Calculated bond valence sums around the A-cation are nearly identical. Also, when comparing the O-O bond distances within a layer, there is no distinction between parallel and anti-parallel tilts, as the distinction between the tilts is between layers, not within layers. However, the parallel tilts create interlayer anion- anion distances of about 3.7Å, whereas the anti-parallel tilts create distances ranging from 3.2Å-4.2Å (see table 2.3). An important clue may be the arrangement of the AO layer. In the undistorted RP structure, this layer is analogous to the rock salt structure, with two sub-layers, a top and bottom. Figure 2.4 shows a side view of structures with

− − 0 − − 0 the a p a p c and aa aa c tilt systems and the two sub-layers as viewed down the [001]T- axis for these parallel and the anti-parallel tilts. The sub-layers themselves are identical. However, the anions from the top sub-layer move in the same direction at those from the

− − 0 − − 0 bottom sub-layer in the aa aa c tilt system, which in the a p a p c tilt system, these anions move in opposite directions, creating longer anion-anion distances. This means that parallel tilts will be electrostatically favored. A similar argument can be made for the 35 Tilt wyc System Space Group site interlayer O-O distances 0 0 0 a a c I4/mmm (139) e, c 4x2.693 P4 /ncm − − 0 2 a p a0 c (138) i, e, a 4x3.707 a − a −c 0 a 0 P42nnm (134) m, g, d 2x3.344, 2x4.038 − − 0 a p a p c Cmca (64) f, e 2x3.662, 2x3.723 − − 0 aa aa c Cccm (66) l, g 1x3.196, 2x3.662, 1x4.184 0 0 a a c p Cmca (64) d, f 4x3.662 a 0 a 0c s I41/acd (142) d, f 4x3.662

Table 2.3: Interlayer anion distances for Sr2TiO4 with ten degree tilt.

36

a) (b)

c) d)

Figure 2.4: Orientation of the sub-layers of the rock salt ordered interlayer (a) and (b) − − 0 − − 0 show the side view with octahedra for the a p a p c and aa aa c tilt systems respectively. The box indicates the portion shown in (c) and (d). These show the rock salt ordered inter layer space only for the (c)parallel and (d) anti-parallel tilt systems, looking down the [001]T-axis. The A-cation is shown in black, while the Xax-anion from the top sub- layer is shown in white and the Xax-anion from the bottom sub-layer is shown in gray.

37 anions in the tilt systems with tilts about the [100]T axis. As a note, the opposite will be true for RP structures with an even number of layers (n = 2k). 2.2.2.4 Ordering effects Thus far, only structures containing one type of anion, one type of B-site cation and one type of A-site cation have been considered. As described in chapter 1, it is known that the ordering of cations can influence the properties of perovskites. In layered perovskites, mixing of the A-site cations, particularly with the cations Sr2+ and La3+ has been explored in order to improve to properties of RP type superconductors.9, 26-28 Recently oxy-halide RP structures have also been studied.12, 13 In general, the structural characterization of these materials has not taken ordering of the cations into account, and little work has been done to predict the structures that could result from ordering of these ions. The NaLnTiO4 compounds have been structurally characterized with ordering of the A-site cations.29-31 However, much of the synthetic work has involved multiple cations on both the A and B-sites.32, 33 Some difficulties have arisen in the structural characterization of these types of materials. For example, the reported structure of 34 La2Li1/2M1/2O4 included very low occupancies on unusual sites. Perhaps a group theoretical analysis can provide insight into a more appropriate assignment of the atoms in these types of structures. The results of this analysis are shown in table 2.4. There are four irreps related to ordering of the B-site cations, each of which also allows for ordering of the A-site cations by creating multiple A-cation sites. Within the perovskite layer, rock-salt ordering of the

+ B-sites is induced by the irreps X 1 and P1, with the difference being how the cations are ordered from layer to layer. This is the type of ordering described by Abou-Warda et

34 + al. Analogous ordering is seen on the A-sites. The ordering induced by the X 1 irrep preserves the mirror plane through the offset layer, while the ordering from P1 destroys this mirror plane by alternating the cation sites across the central RP plane, and thus

+ causes a doubling of the c-axis. With the irrep M 1 , alternating perovskite layers have different B-cation sites. Similarly, each perovskite-like layer is bounded by a different

+ A-cation site. Various kinds of ordering are possible with the four dimensional irrep N1 , depending upon which direction vectors are used. The direction vector that minimizes 38 the number of A- and B-cation sites is (a,0,a,0), which corresponds to the tilt system

+ + 0 asa a0 c (super-anti-parallel in-phase tilts about one axis, which rotates 90° from layer to layer). That is, introduction of multiple cation sites destroys the requirement of no tilting to preserve symmetry. In this case both the A and B-cation sites are ordered

+ + perpendicular to the [100]T axis. Neither of the ordering patterns induced by M 1 and N1 are favored electrostatically for cations with different oxidation states. These ordering patterns place the more highly charged cations in close proximity without alternating them with the cations of lower valence. In addition to the irreps discussed above, there are four irreps related to A-cation

− ordering only. In the irrep Γ3 , the sub-layers of the A-X interlayer alternate between two different cation sites. This irrep preserves the dimensions of the aristotype cell. The

− ordering induced by the irreps X 2 and P3 is like that for P1. However, without ordering

− of the B-site cations, the unit cell in X 2 does not double. The difference between P1 and

P3 is the way the ordering in the interlayer region translates from cell to cell. Finally,

− ordering of the A-sites from one layer to the next is a basis for the M 3 irrep. Again, without alternating the A-cations of different charges, this irrep is not electrostatically favored, unless it can be compensated for by ordering of the anions on the Xax-site.

2.3 Distortions in n = 1 Dion-Jacobson Structures

The types of distortions seen in Dion-Jacobson (DJ) structures are analogous to those seen in RP structures, so only brief commentary will be made in this section. One driving force for distortions described above is the lattice strain caused by the difference in preferred repeat distance between the perovskite-like layers and the interlayer region. A similar argument can be made for the Dion-Jacobson structures. However, while the preferred repeat distance in the perovskite layer is analogous in DJ and RP structures, this is not true of the interlayer region. Instead of being coplanar with the axial anions, the A- site cation lies between two layers, coordinated to eight axial anions. As such, the preferred repeat distance for the A-X layer is √3 (rather than √2) times the A-X bond length. 39 Space Group Irrep Cation Ordering Lattice Cation Anion Vectors positions positions + Cmmm X B 2d rock salt a≈c0 g, x'=z g, x'=z (65) 1 A 110 layer h, x'= 1/2+z h, x'= 1/2+z b≈√2ap B

c≈√2ap a, c n, y'=3/4 z'=1/4 I41/amd P1 B 2d rock salt a≈√2ap e, z'=- e, z'=- (141) A 110 layer, 1/8+1/2z 1/8+1/2z c≈2c0 double c e, e, z'=3/8+1/2z z'=3/8+1/2z B g, x'=1/4 a,b + P4/mmm M B Layered a≈ap g, z'=z g, z'=z (123) 1 A layered w/i h, z'= 1/2+z h, z'= 1/2+z b≈ap RS preserve B

mirror c≈c0 a, d f,e + B 100 layer a≈2a I41/amd N1 p h, y'=0, h, y'=0, A 100 layer b≈2a (141) p z'=1/2z z'=1/2z c≈2c0 h, y'=-1/2, h, y'=-1/2, z'=1/2+1/2z z'=1/2+1/2z B c, d e, z'=0 f, x'=3/4 e, z'=1/2 − P4/nmm M A layered a≈ap c, z'=1/4-z c, z'=1/4-z (129) 3 c, z'=1/4+z c, z'=1/4+z b≈ap B

c≈c0 c, z'=1/4 f, z'=3/4 P A 110 layer, I41/amd 3 a≈√2ap e,z'=1/8+1/2z e,z'=1/8+1/2z double c c≈2c (141) 0 e, z'=1/8-1/2z e, z'=1/8-1/2z B e, z'=1/8 g, x'=1/4 − A layer w/I RS, a≈a I4mm Γ3 p a, z'=z a, z'=z no mirror b≈a (107) p a, z'=-z a, z'=-z c≈c0 B a, z'=0 b, z'=0 − A 110 layer no a≈√2a Cmcm X 2 p c, y'=-1/4-z c, y'=-1/4-z mirror b≈ c (63) 0 c, y'=-1/4+z c, y'=-1/4+z c≈√2ap B c, y'=-1/4 d

Table 2.4: Symmetry of Ruddlesden-Popper structures with cation ordering. 40 2.3.1 Symmetry allowed distortions The tetragonal symmetry of the parent space group, P4/mmm, will allow for elongation or compression of the BX6 octahedra. In the DJ structure, the axial anions are coordinated by one B-site cation and only four A-site cations, which are typically of lower valence than the B-site cations. Thus compression of the octahedra is expected. The interlayer region can also contract, shortening the c-axis. Unlike in RP structures, this does not serve to relieve the bond strain of an unusually short A-X bond, as none exists. Rather, the anions are pulled away in the x-y plane and drawn closer along the z- axis such that the square prism coordination environment of the A-cation is axially compressed.

This change in bonding at the Xax-site is important in understanding the distortions seen in DJ structures with multiple layers in each perovskite-like slab. In DJ structures with n > 1, the B-site cation is coordinated by at most one of these under- bonded anions, rather than two as in the n = 1 case. This means that the ideal translation of the perovskite like layer is contracted, closer to that of a perovskite. 2.3.2 Octahedral tilting distortions Octahedral tilting distortions would seem to be the preferred method of dealing with bond and lattice strain, since the symmetry allowed distortions may be less effective than in RP structures. Classification of these distortions into tilt systems is somewhat simpler than for RP structures, and here the Bulou notation is used for n = 1 DJ structures. When n > 1, a superscript is added for the c-axis to indicate in-phase or out- of-phase distortions. As with perovskites, within each layer of the DJ structure distortions can be in-phase or out-of-phase, as indicated by the superscript. Because all of the layers line up along the c-axis, designation of parallel and anti-parallel is more intuitive as parallel tilts will be in the same direction from one layer to the next. With anti-parallel tilts in the opposite direction, doubling of the c-axis is required. 2.3.2.1 Previous work on tilting

Analysis of the symmetry changes in ABX4 structures was published by Bulou et al.24 and Debliek et al. 35 Later Aleksandrov published tables of distortions for DJ

41 structures with multiple layers per perovskite slab.20 These papers include numerous duplications that are symmetrically equivalent. Aleksandrov and Bartolome36 also published work specifically on layered perovskites with even numbered layers, as out-of- phase distortions in these structures inherently destroy the mirror plane perpendicular to the c-axis. 2.3.2.2 Results of Group Theoretical analysis Group theoretical analysis of the DJ structures was performed using ISOTROPY,

+ + − as described above. Parallel tilts are induced by the irreps M 2 , M 5 , and X 2 . Because the tilting in these systems remains consistent from one slab to the next, these tilt systems

+ do not cause the c-axis to double. M 2 corresponds to in-phase tilting about the c-axis,

0 0 + − + resulting in the tilt system a a c p . X 2 and M 5 correspond respectively to in-phase and out-of-phase distortions along the a- or b-axis. Both of these irreps are two dimensional with the distinction between the (a,0) and (a,a) direction vectors being whether the tilt is only about the a-axis or both the a and b-axes. The tilt systems resulting from the use of

+ 0 0 − 0 0 only one direction vector are a p b c and a pb c , respectively. There are likewise three irreps related to anti-parallel tilts. In-phase tilting about

0 0 + − the c-axis giving rise to the a a ca tilt system is a basis for the A4 tilt system. For n = 1 structures, the superscript on the c-axis is meaningless. When n > 1 other irreps are involved, as described below. Unlike the situation for RP structures, there is a meaningful distinction between parallel and anti-parallel tilting about the c-axis. For

+ − tilting along the a- and b-axes, the irreps R1 and A5 are used for in-phase and out-of- phase distortions respectively. Again, both of these irreps are two dimensional, allowing for tilting about one or both of the a and b-axes. The resulting symmetry changes from each of these irreps are shown in table 2.5, and those from combinations are shown in table A.2. Only simple combinations of tilting were considered. For example, in-phase and out-of-phase tilting about the same axis was not considered for any of the combinations. Such combinations are very low in symmetry and were excluded as per the discussion by Howard and Stokes4 on perovskites. 42 When n > 1, each perovskite-like slab has more than one layer. While the parent space group, P4/mmm (123) remains unchanged, the Wyckoff position of the B-site changes. In an n = 1 structure the B-site is the fixed Wyckoff position 1d ( ½, ½, ½). While this position remains for some of the B-sites in odd layered DJ structures, it is eliminated in even layered DJ structures. Additional layers cause B-sites to appear along the vertical cell edges at the 2h Wyckoff position (1/2, 1/2, z). Accordingly, the equatorial anions have the position 4i (0, ½, z) instead of 2e (0, ½, ½). With the positions not being related by symmetry, combinations of irreps must be considered for tilting in odd layered DJ structures, and the tilting about the c-axis must always be considered to be out-of-phase. When there is an even number of layers, both out-of-phase and in-phase tilting distortions can be considered, because all of the B-sites are on the 2h position.

+ In-phase distortions about the c-axis are still a basis for the M 2 irrep, but out-of-phase

− distortions are basis for the irrep M 4 . Interestingly, in odd layered structures (which are

+ always out-of-phase) the M 2 irrep is always the corresponding one. Since the B-site at 1d is distinct from the B-sites at 4h, these latter sites must be in-phase with one another, as with the in-phase tilt for even layered DJ structures! Additional irreps must be considered for tilting about the a- and b-axes. Tables resulting from the group theoretical analysis of multi-layered DJ structures are presented in the appendix (table A.3). 2.3.2.3 Cation and Anion environments

With forty-eight possible combinations of simple tilts in the ABX4 structures, the possible environments are too numerous for all to be explored them in depth. The discussion below considers a few example tilt systems, including those most commonly found in literature. Only tilt systems including tilts of a single magnitude are considered, and thus combinations of different irreps are not included in the discussion. In the aristotype structure, all of the ions sit on mirror planes perpendicular to the c-axis. Destruction of these mirror planes allows the cations to shift relative to both the anions and to one another. It is important to consider the bonding environments of both the cations and the anions in the aristotype before looking at how tilting distortions will affect this environment. As with the RP structure, the B-cations are surrounded by two axial anions (Xax) and four equatorial anions (Xeq) and the equatorial anions are 43

Space Tilt System Unit-cell Irrep Cation Anion Group size Positions Positions 0 0 0 P4/mmm a a c a=ap A (1a) Xax (123) c=c0 0,0,0 (2e) 0, ½, ½ B (1d) Xeq ½, ½, ½ (2h) ½, ½, z (z≈0.2) P4/mbm 0 0 a≈√2a + A (2a) X (4e) a a c p p M 2 ax (127) c≈c0 (a,a) 0, ½, 0 0,0, z B (2b) Xeq (4h) 0, 0, ½ ¼, ¾, ½ Pmma + 0 0 a≈2a − A (2e) X (4j) a p b c p X 2 ax (51) b≈ap (a,0) ¼, 0, 0 0, ½, z c≈c0 B (2d) Xeq(2f) 0, ½, ½ ¼, ½, ½ (2c) 0,0, ½ Cmma − 0 0 a≈2a + A (4e) X (8m) a pb c p M 5 ax (67) b≈2ap (a,a) ¼, ¼, 0 0, 0, z c≈c0 B (4d) Xeq (4g) 0, 0, ½ 0, ¼, ½ (4b) ¼, 0, ½ P4/nmm + + 0 a≈2a − A (2c) X (8j) a p a p c p X 2 ax (129) c≈c0 (a,a) 0, ½, 0 0, ½, z (2a) 0,0,0 Xeq (8i) B (4e) 0, 0, ½ ¼, ¼, ½ Pmna − − 0 a≈√2a + A (2a) X (4h) a p a p c p M 5 ax (53) b≈c0 (a,0) 0,0,0 0, -z, ½ c≈√2ap B (2c) Xeq (4g) ½, ½, 0 ¼, ½, ¼ Pmmn + + 0 a≈2a − A (2a) X (8g) a p bp c p X 2 ax (59) b≈2ap (a,b) 0,0, 0 0,0, z c≈c0 (2b) 0, ½, 0 Xeq (4e) B (4d) 0, 0, ½ ¼, ¼, ½ (4f) ½, 0, ½ P2/c − − 0 a≈√2a + A (2a) X a pbp c p M 5 ax (13) b≈c0 (a,b) 0, 0, 0 c≈2ap B (2b) Xeq (4f) 0,0, ½ ½, ¼, ¼ (4e) 0, ¼, ¼

Table 2.5: Octahedral tilting distortions for n = 1 DJ structures.

44 Table 2.5 continued

Space Tilt System Unit-cell Irrep Cation Anion Group size Positions Positions I4/mcm 0 0 a≈√2a − A (4b) X (8q) a a ca p A4 ax (140) c≈2c0 (a) 0, ½, ¼ ½+½z, ¼, ½ B (2c) Xeq (4h) 0, 0, 0 ¼, 0, ½ (4e) ¼, ¼, 0 Cmmm + 0 0 a≈2c + A (2a) X (4j) aa b c 0 R1 ax (65) b≈2ap (a,a) 0, 0, 0 0, ½, z c≈ ap (2b) ½, 0, 0 Xeq(2f) B (4f) ¼, ½, ½ ¼, ¼, ½ (2c) 0,0, ½ Fmmm − 0 0 a≈2a − A (8h) X (16n) aa b c p A5 ax (69) b≈2ap (a,a) 0, ¾ , 0 ¾, 0, ½ + ½ z c≈2c0 B (8d) Xeq (8i) ¼, 0, ¼ 0, 0, ¼ (8f) ¼, ¼, ¼ I4/mmm + + 0 a≈2a + A (2a) X (16m) aa aa c p R1 ax (139) c≈2c0 (a,0) 0,0,0 ¼, ¼, ½z (2b) 0,0, ½ Xeq (16n) (2c) 0, ½, 0 0, ¼, ¼ B (8f) ¼, ¼, ¼ Imma − − 0 a≈ 2c − A (4e) X (8i) aa aa c 0 A5 ax (74) b≈√2ap (a,a) 0,1/4, ¼ ½ - ½ z, ¼, ¾ c≈√2ap B (4d) Xeq (8f) ¼, ¼, ¾ ¼, 0, 0 Immm + + 0 a≈2a + A (2a) X (16o) aa ba c p R1 ax (71) b≈2ap (a,b) 0,0, 0 ¼, ¼, ½z c≈2c0 (2b) 0, ½, ½ Xeq (8l) (2c) ½, ½, 0 0, ¼, ¼ (2d) ½, 0, ½ (8m) ¼, 0, ¼ B (8k) ¼, ¼, ¼ C2/m − − 0 a≈2a − A (4i) X (8j) aa ba c p A5 ax (12) b≈2c0 (a,b) ¾, 0, 0 ¾, ½+½z, ½ c≈√2ap B (4f) Xeq (4g) ¼, ¼, ½ 0, ¼, 0 (4h) 0, ¼, ½

45 coordinated by two B-cations and four A-cations. Unlike the anions in the RP structure, however, the DJ axial anions are coordinated by five, not six, cations: four A-cations and only one B-cation. Because of this the axial anions relax toward the B-cations, which in turn causes the equatorial anions to move away from the B-cations. The A-cation is coordinated by eight axial anions in a much more symmetrical environment than the nine-fold coordination of the RP structure. These anions form a square prism around the

A-cation. As the square base of the prism expands due to the lengthening of the B-Xeq bonds, the interlayer spacing can decrease to balance the bonding at the A-site. The eight equatorial anions contribute minimally to the bonding of the A-cation. Because of the relaxation mechanisms, the structure itself does not create inherently over- or under- bonded sites. Tilts about the c-axis

Considering the ABX4 system, each perovskite-like slab contains only one layer of octahedra. In this case, there is no in-phase or out-of-phase tilting. However, from one

0 0 0 0 slab to the next, the tilts can be either parallel or anti-parallel ( a a c p or a a ca ). For either tilt system, there is a constriction of the a-axis that leads to a shortening of all eight

A-Xax bonds. Four equatorial anions are drawn closer to the A-cations, while four move farther away. This has the effect of slightly increasing the bonding at the A-cation and

Xax sites, without much impact at the B-cation and Xeq sites. The main difference between the two systems is the symmetry at the A-site. For the parallel tilt, the A-site cation sits on a fixed position of three orthogonal mirror planes. The more distant equatorial anions move away in the same direction both above and below the A-cation (see figure 2.5a). The anti-parallel tilt has the mirror planes replaced by a four-fold roto-inversion center and a two-fold axis along the c- and a-axes respectively. In this case the equatorial anions below the A-cation move away from it orthogonally to the equatorial anions above it (see figure 2.5b). Parallel tilts about the a- and b-axes There are four simple cases of parallel tilts about the a and b-axes. There could

+ 0 0 − 0 0 be a tilt about one axis that is in-phase ( a p b c ) or out-of-phase ( a p b c ), or tilts about

+ + 0 − − 0 both the a- and b-axes that are in-phase ( a p a p c ) or out-of-phase ( a p a p c ). None of 46 (a) (b)

(c) (d)

0 0 Figure 2.5: A-site environment for simple tilts in n = 1 DJ structures: a) a a c p 0 0 + 0 0 − 0 0 (P4/mbm), b) a a ca (I4/mcm), c) a p b c (Pmma), d) a pb c (Cmma), e) + + 0 − − 0 a p a p c (P4/nmm), f) a p a p c (Pmna).(continued)

47 Figure 2.5 continued

e) f) g)

+ + 0 these tilt systems will cause the doubling of the c-axis, and with the exception of a p a p c , which is tetragonal, they are all orthorhombic. In the case of a tilt about the single axis, four axial anions will move toward the A-cation and four will move away. For the in-phase tilt, the closer anions will all be on one side of the A-cation and the farther anions will be on the other side. In contrast, the out-of-phase tilt will create a trapezoid of axial anions above and below the A-cation (see figures 2.5c and d). Two of the distant equatorial anions will move toward the A-cation and two will move away, again with the closer anions on the same side for the in-phase tilt and opposite sides (above and below) for the out-of-phase tilt. Either of these distortions will increase the bonding at the A-site while only minimally impacting bonding at either the B-site or the anion sites. Again, symmetry is the main difference between the in-phase and out-of-phase distortions. From the description above and figures 2.5c and d, it can be seen that in-phase tilting increases the bonding on one side of the A-cation while decreasing the bonding on the other side, while the out-of-phase tilting balances the long and short bonds on either side of the A-cation. This imbalance 48 in the in-phase tilting destroys the mirror plane perpendicular to the [001]T axis and allows the A-cation to shift toward the further axial anions. When there are tilts of equal magnitude about the a- and b-axes, an in-phase tilt creates a tetragonal cell of higher symmetry than the orthorhombic out-of-phase cell. However, the in-phase tilt creates two different A-sites. In the first site, all of the axial anions on one side of the A-cation move toward it, while on the other they all move away (see figure 2.5e). Again, this destroys the mirror plane and the A-cation can shift toward the further anions. The second site is a fixed position with all of the axial anions moving slightly away from the A-cation. However, compression of the a-axis due to tilting may in fact cause the A'-Xax bond length to shorten. The overall impact is an increase in bonding at both sites, but with a much larger impact on one site than the other. There is minimal impact on the bonding at the B-sites and the anion sites. Because of the creation of multiple A-cation sites, this tilt system will accommodate ordering of two types of A- cations without additional changes in symmetry. Out-of-phase tilting causes an overall lower symmetry cell, but with only one type of A-cation. This type of tilting will again increase the bonding at the A-cation site, with minimal impact on bonding at either the B-cation site or the anion sites. The major difference between this tilt system and the one described above is that there is only one A-cation position. There is a less dramatic change in bonding at the A-cation site as

+ + 0 opposed to the less fixed A-cation position in the a p a p c tilt system. Anti-parallel Tilt Systems The DJ structure distinguishes itself from the RP structure in that as the layers come together in the DJ structure, the axial anions are touching. Electrostatically, this is not favorable, particularly for small A-site cations. Anti-parallel rotations as a general rule will shift the anions in the same direction both on top and below the A-site cation, not allowing these anions to separate from one another. In contrast, the parallel tilts in general have the anions moving in different directions. As will be discussed in chapter 3, for ABX4 systems without plane slippages, anti-parallel tilting is not observed.

49 2.4 Conclusions

Understanding the changes in bonding environments that can happen as a result of distortions of a structure is important for being able to predict the structures that may result. A group theoretical analysis of specific types of distortions can aid in predicting the symmetry and changes in bonding. Assuming the interactions between the anions and cations are largely ionic, this model can elucidate some of the causes for particular distortions observed in RP and DJ structures. The layered perovskites with the K2NiF4 and the ABX4 DJ structure produce complicated distortions due to the flexible interlayer region. In the case of the RP structures it is possible from this simple qualitative analysis to understand how the changes in bonding environments works with the inherent lattice strain in the structure. The more flexible environment around the DJ structure allows for an aristotype that does not inherently have lattice strain, and has many more possible octahedral tilting distortions than are possible in the RP structures. The implications of these observations for the structures observed in RP and DJ compounds are explored in greater detail in Chapter 3.

2.5 References

1. Glazer, A. M., Classification of tilted octahedra in perovskites. Acta Crystallographica, Section B: Structural Crystallography and Crystal Chemistry 1972, 28, (Pt. 11), 3384-92.

2. Glazer, A. M., Simple ways of determining perovskite structures. Acta Crystallographica, Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography 1975, A31, (6), 756-62.

3. Howard, C. J., Structures and phase transitions in perovskites - a group-theoretical approach. Acta Crystallographica, Section A: Foundations of Crystallography 2005, A61, (1), 93-111.

4. Howard, C. J.; Stokes, H. T., Group-Theoretical Analysis of Octahedral Tilting in Perovskites. Acta Crystallographica 1998, B54, 782-789.

5. Woodward, P. M., Octahedral Tilting in Perovskites. I. Geometrical Considerations. Acta Crystalografia 1997, B53, 32-43.

50 6. Woodward, P. M. Structural distortions, phase transitions, and cation ordering in the perovskite and tungsten trioxide structures. 1997.

7. Khasanova, N. R.; Kovba, M. L.; Putilin, S. N.; Antipov, E. V.; Lebedev, O. I.; Tandeloo, G. V., Synthesis, structure and properties of layered bismuthates: (Ba, K)3Bi2O7 and (Ba, K)2BiO4. Solid State Communications 2002, 122, 189-193.

8. Oyanagi, H.; Ihara, H.; Matsubara, T.; Matsushita, T.; Hirabayashi, M.; Tokumoto, M.; Murata, K.; Terada, N.; Senzaki, K.; et al., Local structure in orthorhombic and tetragonal barium yttrium copper oxide (Ba2YCu3O7-y): the role of oxygen vacancies for high Tc superconductivity. Japanese Journal of Applied Physics, Part 2: Letters 1987, 26, (7), L1233-L1236.

9. Takeda, Y.; Kanno, R.; Sakano, M.; Yamamoto, O.; Takano, M.; Bando, Y.; Akinaga, H.; Takita, K.; Goodenough, J. B., Crystal chemistry and physical properties of lanthanum strontium nickel oxide (La2-xSrxNiO4, 0≤ x≤ 1.6). Materials Research Bulletin 1990, 25, (3), 293-306.

10. Tezuka, K.; Inamura, M.; Hinatsu, Y.; Shimojo, Y.; Morii, Y., Crystal Structures and Magnetic Properties of Ca2-xSrxMnO4. Journal of Solid State Chemistry 1999, 145, (2), 705-710.

11. Yoshii, S.; Murata, K.; Sato, M., Anomalous transport behavior in geometrically frustrated system R2-xBixRu2O7 (R=rare earth, Y). Journal of Physics and Chemistry of Solids 2000, 62, (1-2), 331-336.

12. Kriworuschenko, B.; Kahlenberg, V., On the symmetry of the n = 1 Ruddlesden- Popper phase Ca2FeO3Cl. Crystal Research and Technology 2002, 37, (9), 958-963.

13. Tobias, G.; Beltran-Porter, D.; Lebedev, O. I.; Van Tendeloo, G.; Rodriguez- Carvajal, J.; Fuertes, A., Anion Ordering and Defect Structure in Ruddlesden-Popper Strontium Niobium Oxynitrides. Inorganic Chemistry 2004, 43, (25), 8010-8017.

14. Brown, I. D., The Chemical Bond in Inorganic Chemistry: the bond valence model. Oxford University Press: Oxford; New York, 2002; Vol. 12, p 278.

15. Uzumaki, T.; Hashimoto, K.; Kamehara, N., Raman Scattering and X-ray diffraction study in layered cuprates. Physica C 1992, 202, 175-187.

16. Uzumaki, T.; Kamehara, N.; Niwa, K., Crystal structure and Madelung potential in R2-xCexCuO4-d (R = praseodymium, neodymium, samarium, europium and gadolinium) system. Japanese Journal of Applied Physics, Part 2: Letters 1991, 30, (6A), L981-L984.

17. Hatch, D. M.; Stokes, H. T., Physical Review B 1987, 35, 8509-.

18. Hatch, D. M.; Stokes, H. T.; Aleksandrov, K. S.; Misyul, S. V., Phase transitions in the perovskitelike A2BX4 structure. Physical Review B 1989, 39, (13), 9282-9288. 51 19. Aleksandrov, K. S.; Beznosikov, B. V.; Misyul, S. V., Successive Phase Transitions in Crystals of K2MgF4-Type Structure. Phys. status solidi 1987, 104, 529-543.

20. Aleksandrov, K. S., Structural phase transitions in layered perovskite-like crystals. Kristallografiya 1995, 40, (2), 279-301.

21. Stokes, H. T.; Hatch, D. M. ISOTROPY, 2002.

22. Stokes, H. T.; Hatch, D. M., Symmetry of Possible Average Mulidomain Sturctures at a Phase Transition. Ferroelectrics 2003, 292, 59-63.

23. Pauling, L., The principles determining the structure of complex ionic crystals. Journal of the American Chemical Society 1929, 51, 1010-26.

24. Bulou, A.; Fourquet, J. L.; Leble, A.; Nouet, J.; De Pape, R.; Plet, F., Structural phase transitions in the tetrafluoroaluminates MAlF4 with M = NH4, Rb, K, Tl. Studies in Inorganic Chemistry 1983, 3, (Solid State Chem.), 679-82.

25. Lukaszewicz, Struktura krystaliczna tytanianow strontu alpha-(Sr O)2 (Ti O) and (Sr O)3 (Ti O2)2. Angewandte Chemie 1958, 70, 320.

26. Butaud, P.; Segransan, P.; Berthier, C.; Berthier, Y.; Paulsen, C.; Tholence, J. L.; Lejay, P., Temperature dependence of lanthanum spin-lattice relaxation time in the high- Tc superconductor lanthanum strontium copper oxide (La1.85Sr0.15CuO4). Physical Review B: Condensed Matter and Materials Physics 1987, 36, (10), 5702-4.

27. Yanson, I. K.; Rybal'chenko, L. F.; Fisun, V. V.; Bobrov, N. L.; Obolenskii, M. A.; Brandt, N. B.; Moshchalkov, V. V.; Tret'yakov, Y. D.; Kaul, A. R.; Graboi, I. E., Point- contact spectroscopy of high-temperature superconductor lanthanum strontium copper oxide (La1.8Sr0.2CuO4). Fizika Nizkikh Temperatur (Kiev) 1987, 13, (5), 557-60.

28. Renker, B.; Apfelstedt, I.; Kuepfer, H.; Politis, C.; Rietschel, H.; Schauer, W.; Wuehl, H.; Gottwick, U.; Kneissel, H.; et al., Magnetic properties of the high-Tc superconductor lanthanum strontium copper oxide (La1.85Sr0.15CuO4). Zeitschrift fuer Physik B: Condensed Matter 1987, 67, (1), 1-7.

29. Itoh, M. In Effect of ordering on the properties in perovskite-related bulk crystal, First symposium on Atomic-scale Surface and Interface Dynamics, Tokyo, 1997; Tokyo, 1997.

30. Toda, K.; Kameo, Y.; Kurita, S.; Sato, M., Crystal structure determination and ionic conductivity of layered perovskite compounds NaLnTiO4 (Ln = rare earth). Journal of Alloys and Compounds 1996, 234, (1), 19-25.

31. Zhu, W. J.; Feng, H. H.; Hor, P. H., Synthesis and characterization of layered titanium oxides NaRTiO4 (R = La, Nd and Gd). Materials Research Bulletin 1996, 31, (1), 107-11. 52 32. Ebbinghaus, S.; Reller, A., Structure and thermochemical reactivity of La2-xSrxCu1-yRuyO4-d. Solid State Ionics 1997, 101-103, (Pt. 2), 1369-1377.

33. Ganguly, R.; Rajagopal, H.; Sequeira, A.; Yakhmi, J. V., Structural features of La1.85Sr0.15CuO4 as influenced by substitution of Zn: neutron diffraction studies. Journal of Superconductivity 2000, 13, (1), 163-170.

34. Abou-Warda, S.; Pietzuch, W.; Berghofer, G.; Kesper, U.; Massa, W.; Reinen, D., Ordered K2NiF4 structure of the solids La2Li1/2M1/2O4 (M(III) = Co, Ni, Cu) and the bonding properties of the MO6 polyhedra in various compounds of this type. Journal of Solid State Chemistry 1998, 138, (1), 18-31.

35. Debliek, R.; Van Tandeloo, G.; Van Linduyt, J.; Amelinckx, S., Acta Crystallographica Structural Science 1985, 41, 319-322.

36.Aleksandrov, K. S.; Bartolome, J., Octahedral tilt phases in perovskite-like crystals with slabs containing an even number of octahedral layers. Journal of Physics: Condensed Matter 1994, 6, 8219-8235.

53

CHAPTER 3

EVALUATION OF OCTAHEDRAL TILTING DISTORTION TRENDS IN RUDDLESDEN-POPPER AND DION-JACOBSON STRUCTURES

3.1 Introduction

The Ruddlesden-Popper and Dion-Jacobson structures were discussed in Chapter 2, along with the distortions that can alter the bond or lattice strain in these structures. Octahedral tilting distortions were described in great detail as the buckling of the perovskite layers is presumed to be a primary mode of relaxation for these structures. The argument was made that understanding the symmetry and bonding changes could enable prediction of the distorted structure based on the knowledge of the sizes and charges of the constituent ions. In this chapter, the reported structures in literature are compared to the tables presented in Chapter 2 to test the assumptions and predictions made.

3.2 Ruddlesden Popper Structures

3.2.1 Lattice Strain and Bond Valence Sums When an ionic model is considered for an extended solid, the bonding environments for each of the constituent ions can be considered according to Pauling’s Rules.1 Pauling’s second rule states that the bond valence sum of each ion should be equal to its oxidation state. Oxygen, which typically has an oxidation state of two, tends to form two single bonds or one double bond in molecular compounds. This can be stated mathematically as follows:

54 n BVSi = ∑ sij (equation 3.1) j=1 where sij is the valence of a bond between atoms i and j. Conversely, the valence of a bond to a symmetrically coordinated ion can be calculated by dividing the oxidation state of that ion by the number of equivalent bonds. Pauling extended this principle to extended solids where bond valences are not necessarily, or even usually, whole numbers. A tetrahedrally coordinated Cr (VI) ion would have individual bond valences of 6/4 or 1.5vu. An alternative method for calculating bond valences is more empirical in nature. Given a typical bond length R0 between two ions with a valence of one, the valence of a bond of length Rij, in an actual crystal is calculated by

sij = exp[(R0 − Rij ) / b] (equation 3.2) where b is an empirically determined constant typically found to be between 0.35 and 0.40. For the calculations used here, the value 0.37 was used, which is the most commonly used value. Crystal structures are often described in terms of connected polyhedra. This is because ions tend to arrange themselves symmetrically in order to minimize the energy from electrostatic interactions. Pauling further stated that crystals tend to avoid edge or face sharing polyhedra. That is, two different polyhedra tend to share no more than one anion. One last point of relevance that Pauling makes is that ions of a single type tend to have very few chemically different coordination environments. This ensures that for a given ion, there are not large differences in the energies of the different sites.

If we consider a typical RP structure, such as Sr2TiO4, we can see how these rules are applied. The structure is made up of corner sharing TiO6 octahedra, and there is only one chemically unique coordination environment for either the Ti4+ or the Sr2+. The BVS for strontium and for oxygen should equal 2, while for titanium it should equal 4. Due to the flexibility of the structure within the aristotype space group, it is possible for all of these conditions to be met without any symmetry changing distortions. However, in chapter 2 the concept of lattice strain was introduced. The preferred translational distance for a perovskite layer is not necessarily equal to that of the rock salt SrO layer. This is illustrated in figure 3.1. So alleviating all bond strain may introduce lattice strain.

55

Figure 3.1: Sr2TiO4 with the aristotype RP structure. Lattice strain is produced by a difference in the ideal repeat distance between the perovskite-like layer (2RTiO) and the SrO layer (2RSrO/√2).

56 If this is the case, it seems reasonable that the actual crystal would adopt some intermediate structure that would minimize both lattice and bond strain.

Both the ABX3 stoichiometry of perovskites and the A2BX4 stoichiometry of RP structures will accommodate ions with the charges A+, B2+, X- or A2+, B4+, X2-, so a comparison can be made between the lattice constants of perovskites and the analogous RP structure. In a perovskite, the A-cation is coordinated by twelve anions, whereas in the RP structure it is only coordinated by nine. It seems likely that the average A-X bond valence in the RP structure would be greater than that of the perovskites. That is, the A- site cation would prefer a shorter translational distance in the RP structure, and to relieve bond strain, the RP structures would be likely to have a smaller a lattice constant. The known lattice constants of several perovskites and RP structures can be seen in table 3.1. Looking at the undistorted structures (in bold), the oxides do indeed tend to have smaller a lattice parameters in the RP structure than the perovskite structure, though the difference is quite small. Fluorides have comparable lattice constants for the perovskite and RP structures, with a slight tendency for the RP structure to be slightly larger, though the tolerance factors for the fluorides tend to be higher as well. 3.2.2 Measures of Strain There are several ways to quantify the strain in an observed structure. The Global Instability Index (GII) takes into account the difference between the oxidation state (ideal BVS) and the actual BVS for each ion. Detailed descriptions of these strain indices are given by Brown.2 This is summed over the entire structure. The Bond Strain Index (BSI) is a calculation of the difference between the observed valence of the bonds and the theoretical valence from the coordination environment. In either case, indices near zero imply stable structures, while indices greater that about 0.2 imply a structure unlikely to be stable. The BSI and GII for several known halides and oxides with the RP structure are shown in table 3.2. It can be seen that the strain in the RP structures is quite high. Another way at looking at structure stability is with the tolerance of the structure as determined by relative ionic size. The tolerance factor for perovskites was described in section 1.1.2. Because of the altered bonding environment, as well as the flexibility of the aristotype structure, this tolerance factor is not exactly applicable to RP structures. A

57 Perov- Tilt ar τ BVS RP Tilt a τRP BVS RXax skite X Xax RXeq Xeq a 0 0 0 b 0 0 0 SrTiO3 a a a 3.901 1.00 2.090 Sr2TiO4 a a c 3.884 0.961 1.797 0.986 2.328 c 0 0 0 c 0 0 0 SrVO3 a a a 3.8425 1.02 2.166 Sr2VO4 a a c 3.837 0.977 1.745 1.034 2.359 SrCrO3 DNE -- 1.00 -- Sr2CrO4 not RP-- 0.964 d e 0 0 0 SrMnO3 NP -- 1.03 -- Sr2MnO4 a a c 3.787 0.993 1.807 1.036 2.42 f 0 0 0 SrCoO3 a a a 3.835 1.05 1.970 Sr2CoO4 DNE -- 1.010 g 0 0 0 h 0 0 0 SrFeO3 a a a 3.851 1.04 1.998 Sr2FeO4 a a c 3.864 0.997 1.707 1.009 2.23 i - + - SrZrO3 a b a 5.9763 0.95 Sr2ZrO4* DNE -- 0.909 -- -- 5.8171 8.2048 j - + - SrNbO3 a b a 5.6894 0.97 Sr2NbO4* DNE -- 0.931 -- -- 8.0684 5.6944 k 0 0 0 SrMoO3 a a a 3.9651 0.96 2.162 Sr2MoO4 DNE -- 0.928 -- -- l - + - m 0 0 0 SrRuO3 a b a 5.567 0.99 Sr2RuO4 a a c 3.873 0.952 1.708 1.068 5.5304 2.351 7.8446 n - + - o 0 0 0 SrSnO3 a b a 5.7089 0.96 Sr2SnO4 a a c 4.053 0.919 1.685 1.012 5.7034 2.16 8.0648 p 0 0 SrIrO3 NP -- 0.97 -- Sr2IrO4 a a cs 3.886 0.935 1.768 1.089 2.187 q - + - r SrPbO3 a b a 5.8627 0.90 -- Sr2PbO4 Not -- 0.862 -- -- 5.9544 RP 8.3293 - + - CaTiO3 a b a 0.95 -- Ca2TiO4* DNE -- 0.908 -- -- - + - s 0 0 CaMnO3 a b a 0.98 -- Ca2MnO4 a a cs 0.938 - + - r CaSnO3 a b a 0.90 -- Ca2SnO4 Not 0.868 RP t 0 0 0 u 0 0 0 KMgF3 a a a 3.989 1.04 1.075 K2MgF4 a a c 3.98 1.00 0.925 1.007 1.123

Table 3.1: Comparison of analogous perovskite and Rudlesden-Popper Strucutres. References to structural parameters: a. Abromov et al., 1995; b. Lukaszewicz, 1958; c. Range et al., 1991; d. Kuroda et al., 1981; e. Bouloux et al., 1981; f. Bezdicka et al., 1993; g. Hodges et al., 2000; h. Dann et al., 1991; i. Kennedy and Howard, 1999; j. Hannerz et al., 1999; k. Liu et al., 1992; l. Jones et al. 1989; m. Huang et al.1994; n Green et al., 2000; o. Kennedy, 1997; p. Shimura et al. 1995; q. Fu and Idjo, 1995; r. Toemel, 1969; s. Tezuka et al. 1999; t. Muradyan et al., 1984; u. Babel and Herdtweck, 1982; v. Kijima et al., 1983; w. Yeh et al., 1993; x. Hoppe and Homann, 1969; y. Wu and Hoppe, 1984; z. Welsch and Babel, 1991; aa. Schroetter and Mueller.

58 Table 3.1 (cont’d)

Perov- Tilt ar τ BVS RP Tilt a τRP BVS RXax skite X Xax RXeq Xeq v 0 0 0 w 0 0 0 KNiF3 a a a 4.0115 1.03 1.069 K2NiF4 a a c 4.013 0.99 0.951 0.986 1.135 x KHgF3 a −b + a − 0.89 K2HgF4 DNE 0.86

y 0 0 0 y 0 0 0 RbMgF3 a a a 4.01 1.10 1.275 Rb2MgF4 a a c 4.055 1.06 1.074 0.983 1.15 y z 0 0 0 RbNiF3 NP 1.09 Rb2NiF4 a a c 4.086 1.05 1.052 0.982 1.179 y 0 0 0 aa 0 0 0 RbHgF3 a a a 4.47 0.94 1.076 Rb2HgF4 a a c 4.56 0.91 0.882 0.975 1.112

59 Compound τRP BSI GII

K2MgF4 0.980 0.092 0.089 K2MnF4 0.971 0.136 0.136 K2CoF4 1.003 0.111 0.100 K2NiF4 1.025 0.247 0.097 K2ZnF4 1.015 0.183 0.089 Rb2MgF4 1.025 0.228 0.195 Rb2CoF4 1.049 0.414 0.155 Rb2NiF4 1.072 0.604 0.198 Rb2HgF4 0.959 0.281 0.153 Cs2CaF4 0.978 0.994 0.062 Cs2AgF4 1.112 0.810 0.116 Cs2HgF4 1.014 0.825 0.132 Rb2MnBr4 0.947 0.551 0.280 K2MgCl4 0.936 0.408 0.120 Rb2CrCl4 0.965 0.365 0.116 Rb2MnCl4 0.960 0.291 0.113 Rb2CdCl4 0.925 0.768 0.147 Cs2CrCl4 1.013 0.190 0.106 Cs2CdCl4 0.972 0.280 0.111 Sr2TiO4 0.962 0.323 0.250 Sr2VO4 0.977 0.166 0.247 Sr2MnO4 0.994 0.174 0.296 Sr2FeO4 0.997 0.246 0.274 Sr2RuO4 0.984 0.184 0.247 Sr2SnO4 0.919 0.655 0.252 Ba2SnO4 0.988 0.431 0.185 Ba2PbO4 0.904 1.333 0.640

Table 3.2: Indicators of strain in several reported RP structures.

60 variety of analogous tolerance factors have been developed for RP structures.3-5 One of the more useful of these was worked out by Poix.4

ψ A t POIX = (Equation 3.3) 2β B

In this formula, ψA is the weighted average bond length of the 9 A-X bonds, and βB is the weighted average of the 6 B-X bond lengths. Poix used experimental data to determine an average ψA and βB for each cation. However, his work from 1980 does not include data for many of the cations included in the tables here. By its empirical nature, this tolerance factor is less predictive than one that can be determined based on known cation- anion distances. The A-site cation is 9 coordinate, so the ideal translational distance in the rock salt layer is √2 times the bond length for an An+-X bond of valence n/9. As such, when calculated for this work, bond lengths were determined theoretically from valences n m n+ m+ of /9 and /6 for A and B cations. n R0 AX − 0.37 *ln( ) RAX 9 t RP = = (Equation 3.4) 2R m BX 2[R − 0.37 *ln( )] 0BX 6 6, 7 R0 was determined from previously published tables as indicated in table 3.3. This is similar to the Goldschmidt8 tolerance factor used for perovskites, except that the A-X bond distance in equation (1.1) is calculated for twelve-fold coordination of the A- cation. This is significant because it means that for RP structures, the preferred translation distance in the A-X layer is smaller than for perovskites, and that a given A- cation will have a lower tolerance in the RP structure than in the perovskite structure. For example, for strontium titanate, the Goldschmidt tolerance is 1.00, while the RP tolerance 0.961, and for calcium titanate, the Goldschmidt tolerance is 0.946, while the RP tolerance is 0.908. As can be seen from T\table 3.3, undistorted structures typically have a tolerance in the range of 0.95-1.0, while distorted structures have theoretical tolerance near or below 0.9.

61 Dev. Vsum Vsum Vsum Vsum RBXax From Formula tPOIXt tPOIX ap C A B Xax Xeq RBXeq plana rity a0a0c0 Halides (121) a K2MgF4 0.999 0.986 3.980 13.179 1.076 1.945 0.925 1.123 1.007 0.0040 b K2MnF4 0.942 0.965 4.171 13.259 0.904 2.059 0.806 1.127 1.012 0.0159 a K2CoF4 0.969 0.978 4.073 13.087 1.011 2.077 0.901 1.148 0.994 0.0090 c K2NiF4 0.990 0.989 4.013 13.088 1.071 1.978 0.916 1.144 0.986 0.0030 d K2ZnF4 0.979 0.979 4.058 13.109 1.020 1.991 0.892 1.124 0.999 0.0084 † e Rb2MgF4 1.059 1.008 4.055 13.790 1.305 1.839 1.074 1.150 0.983 -0.0048 † e Rb2CoF4 1.027 0.996 4.149 13.696 1.225 1.939 1.017 1.177 0.977 0.0013 † e Rb2NiF4 1.050 1.001 4.086 13.708 1.304 1.854 1.052 1.179 0.982 -0.0010 † f Rb2HgF4 0.911 0.962 4.560 13.757 0.860 2.267 0.882 1.112 0.975 0.0181 † g Cs2CaF4 0.988 0.994 4.495 14.907 0.989 2.026 0.920 1.081 0.995 0.0000 † f Cs2HgF4 0.964 0.996 4.625 14.518 0.932 2.309 0.995 1.091 0.933 0.0012 †‡ h Cs2AgF4 1.012 0.986 4.581 14.192 1.112 1.835 0.912 1.117 0.929 0.0100 †‡ i Rb2MnBr4 0.925 0.995 5.370 17.320 0.713 2.594 0.916 1.094 0.964 0.0000 j K2MgCl4 0.901 0.964 4.940 15.580 0.955 2.055 0.838 1.144 1.008 0.0165 k Rb2MnCl4 0.965 0.978 5.050 16.140 1.010 2.138 0.913 1.167 0.988 0.0097 l Rb2CrCl4 0.981 0.974 5.135 15.773 1.044 1.887 0.855 1.133 0.949 0.0165 m Cs2YbCl4 0.887 0.972 5.418 17.276 0.772 2.256 0.785 1.115 0.999 0.0126 n Cs2TmCl4 0.933 0.968 5.440 17.259 0.771 2.208 0.789 1.087 1.006 0.0140 o Cs2CdCl4 0.972 0.990 5.260 16.888 0.953 2.171 0.906 1.133 0.957 0.0060 p Cs2CrCl4 1.013 0.990 5.215 16.460 1.090 1.758 0.962 1.008 0.953 0.0030 Oxides (242) q Sr2TiO4 0.974 0.984 3.884 12.600 1.943 4.363 1.797 2.328 0.986 0.0070 r Sr2VO4 0.978 0.969 3.837 12.569 2.132 3.944 1.745 2.359 1.034 0.0133 s Sr2MnO4 0.995 0.970 3.787 12.496 2.291 3.873 1.807 2.420 1.036 0.0130 t Sr2FeO4 0.998 0.970 3.864 12.387 2.158 3.557 1.707 2.230 1.009 0.0143 u Sr2RuO4 0.953 0.960 3.873 12.732 2.011 4.096 1.708 2.351 1.068 0.0149 v Sr2SnO4 0.920 0.961 4.053 12.585 1.732 4.227 1.685 2.160 1.012 0.0150 w Ba2PbO4 0.904 4.302 13.261 1.769 5.426 1.890 2.592 0.998 0.0169 x Ba2SnO4 0.978 0.975 4.141 13.283 1.989 3.840 1.728 2.181 1.000 0.0099 y Ba2ZrS4 0.919 0.969 4.785 15.964 2.130 5.858 1.898 3.160 1.068 0.0129 y Ba2HfS4 0.927 0.962 4.834 15.842 1.986 5.030 1.715 2.786 1.049 0.0178 Lanthanides (322) z La2NiO4 0.893 0.918 3.869 12.600 2.716 2.268 1.652 2.197 1.166 0.0412 aa Nd2NiO4 0.875 0.908 3.854 12.214 2.669 2.302 1.715 2.105 1.173 0.0462

Table 3.3: Room temperature RP structural data. References: a. Babel and Herdtweck, 1982; b. Loopstra et al., 1968; c. Yeh et al., 1993; d. Herdtweck and Babel, 1980; e. Welsch and Babel, 1991; f. Schroetter and Mueller, 1992; g. Jo et al., 1996, h. Odenthal et al., 1974; i. Goodyear et al., 1979; j Gibbons et al., 1975; k Goodyear et al., 1977; l. Muenninghoff et al., 1980; m. Gaebell and Meyer, 1984; n. Schleid and Meyer, 1994; o. Siegel and Gebert, 1964; p. Seifert and Klatyk, 1964; q. Lukaszwicz, 1958; r. Range et al., 1991; s. Bouloux et al., 1981; t. Dann et al., 1991; u. Huang et al.,1994; v. Kennedy, 1997; w. Rosseinsky and Prassides, 1991; x. Green et al.,1996; y. Chen and Eichhorn, 1991; z. Takeda et al.,1990; aa. Nishijima et al.,1991. R0 data from Brown and Altermatt, 1985 unless indicated: † Brese and O’Keefe, 1991; ‡I.D. Brown

62 Table 3.3 (cont’d): References a. Kovba, 1971; b Kovba et al.,1961; c. Shimura et al., 1995; d. Vogt and Buttrey, 1996; e. Tezuka et al., 1999; f. Lehmann and Mueller- Buschbaum, 1980; g. Rial et al., 1993; h. Saez-Puche et al.,1989; i. Green et al., 1996; j. Tuilier et al., 1992; k. Witteveen et al., 1974; l. Kidaka et al., 1983.

RBXax Formula tPOIXt tPOIX ap c V A V B V Xax V Xeq RBXeq Uraniles (162) K2UO4 a 0.997 0.993 4.335 13.130 1.053 5.775 1.896 2.045 0.909 Rb2UO4 b 1.029 1.023 4.345 13.830 1.186 6.210 2.156 2.135 0.878 Cs2UO4 b 1.090 1.064 4.380 14.790 1.228 6.073 2.292 1.973 0.871 0 0 a a cs Sr2IrO4 c 0.935 0.942 3.886 12.897 2.115 4.771 1.768 2.187 1.089 Sr2RhO4 d 0.992 0.957 3.854 12.853 2.110 3.757 1.874 1.553 1.051 Ca2MnO4 e 0.942 0.962 3.669 12.065 2.063 4.243 1.792 1.969 1.041 0 apapc La2CoO4 f 0.869 0.954 3.881 12.548 2.592 2.731 1.749 2.545 1.046 La2NiO4 g 0.885 0.925 3.873 12.636 2.774 2.171 2.136 1.989 1.146 Pr2NiO4 h 0.873 0.917 3.883 12.220 2.748 2.084 3.141 1.713 1.141 Sr2SnO4 i 0.919 0.960 4.049 12.583 1.873 4.847 1.525 2.599 1.012 La2CuO4 j 0.979 0.910 3.801 13.157 2.743 1.400 1.164 1.956 1.257 Rb2CuCl4 k 1.081 0.993 5.086 15.534 1.719 1.802 1.458 1.163 0.796 K2CuF4 l 1.119 0.983 4.175 12.734 1.080 1.061 0.767 0.843 0.929

63 3.2.3 Known n=1 Ruddlesden Popper Structures 3.2.3.1 Comparison of Perovskites and RP structures An interesting comparison can be made between perovskite and RP structures. As illustrated in table 3.1, there exist for many of the perovskites, analogous n = 1 RP compounds. By and large RP structures favor the aristotype, whereas the perovskites tend to distort, primarily through octahedral tilting. It has been shown that most perovskites adopt a distorted structure, primarily of the tilt system a-b+a-.9 This might lead the observer to conclude that the RP structure is inherently more stable due to its ability to distort without changes in symmetry. However, careful examination of table 3.1 will reveal that in fact there are many examples of perovskites for which there is no analogous RP structure, and few examples of RP crystals for which there is not a reported analogous perovskite. As the Goldschmidt tolerance factor drops below unity,

ABX3 compounds tend to adopt distorted structures, such that many of the perovskites in - + - table 3.1 with τ < 0.96 are of the tilt system a b a . However, as τRP falls below unity, the A2BX4 compounds largely remain in the aristotype until τRP < 0.95, at which point the compound may adopt an alternate structure, or not exist at all. It is interesting to note, however, that there are a few groups of ions that will form the perovskite structure and the RP structure with n > 1 (with more than one perovskite like layers per slab), but do not for the K2NiF4 structure. Examples include Ca-Ti-O and Sr-Zr-O. Another striking difference between the perovskite and RP structures in table 3.1 is in the BVS of the anions. Though the lattice constants do not differ greatly between analogous structures, the BVS of the perovskite anions tend to be close to their oxidation states (1 for halides and 2 for oxides). However, a different story is illustrated with the two anions in the RP structures, where the equatorial anions tend to be over-bonded. This is particularly true of the oxides. So the perovskite structure’s inherent flexibility leads it to be more stable in the distorted state than distorted RP structures. Reasons for this lack of stability and its relationship to the bonding at the Xax site are explored further in the next section. It should be noted that most of the examples of RP structures in tables 3.1 and 3.3 contain transition metal B-site cations. There do exist a few competing structures for the

64 stoichiometry A2BX4. The olivine structure (figure 3.2a) is adopted by a number of compounds, notably Mg2SiO4 and Ca2SiO4. These compounds have a τRP of 0.88 and

0.96 respectively. Ca2SnO4, Ca2PbO4, and Sr2PbO4 adopt the strontium plombate structure (figure 3.2b).10 These compounds have tolerance factors below 0.85. The polyhedral connectivity of these alternate structures is interesting. In the case of the strontium plombate structure, the B-cation octahedra are edge-sharing rather than corner sharing. This arrangement is usually preferred for small, highly electronegative cations. While lead is more electronegative than the transition metals, it is not smaller. The olivine structure has the B-site cation in isolated tetrahedra. Again, the lower coordination number is preferred by small electronegative cations. The A-cations are also in a lower coordination environment. Here there are two crystallographically distinct sites, one of edge sharing octahedra and one of octahedra coordinated to the other polyhedral sites, but isolated from like sites. These arrangements would not be preferred by the larger cations which follow the ionic modeling described here. 3.2.3.2 Affect of Octahedral Tilting on BVS In chapter 2, the affect of octahedral tilting on the bonding environments at each of the ions was considered. As shown in tables 3.1 and 3.2, RP structures tend to have an inherent over-bonding at the Xeq-site. Of the tilt systems described in section 2.2, the ones involving rotations about the c-axis will serve to increase the bonding at both the A and Xeq sites. Tilts about a single axis perpendicular to c will generally increase the bonding of the Xeq site, but the increased bonding can be compensated for by a shift of the A-site cation. Tilts about both the a- and b-axes have less of an impact on the Xeq site, though it still increases the bonding here. So any sort of tilting mechanism aggravates the over-bonding on the Xeq site. Unless the A-cation is unusually small, giving the compound a very low tolerance factor, the aristotype structure appears to be the preferred structure. In fact, an examination of table 3.3 reveals that by and large most room temperature RP compounds do adopt the aristotype structure, even when the tolerance is as low as 0.95.

In each tilt system, the A-site cation is able to shift along the [001]T axis (see figure 3.3). However, only the symmetry reduction brought about by the tilts along the

65 (a)

(b)

Figure 3.2: a) The olivine structure. b) the strontium plombate structure.

66

Figure 3.3: Symmetry allowed shift of the A-cations in n = 1 RP structures.

67 a- and b-axes will allow for shifts in these directions. A shift perpendicular to the layering may help relieve the over-bonding at the Xeq site, but brings the A-site cation very close to the capping anion. A shift along the [100]T or [110]T axis allows the Xeq bond strain to be relieved while avoiding a shorter capping A-Xax bond. 3.2.3.3 Comparison of structures in ambient conditions A look at reported structures shows the actual bonding environments at these sites, and indicates the bond strain that needs to be relieved. Table 3.3 shows a summary of reported n = 1 RP structures at room temperature. This table is separated in to types of compounds, generally by the oxidation states of the constituent ions. The oxidation states of the A,B, and X ions, respectively, are given in parentheses next to the subsection heading. Ideally, the BVS of each ion would equal its oxidation state. The A and B-site cations can be over or under-bonded. However, in general, the axial anion, Xax, in each structure tends to be under-bonded while the equatorial anion, Xeq, tends to be over- bonded. This is the scenario that might be predicted for a contraction of the a lattice parameter to alleviate bond strain. Additionally, there is a tendency for the A-site cation to move out of the plane of axial anions toward the perovskite layer. This is because the

A-Xax distance to the capping anion is unusually short without such a displacement. This distortion is shown in the column “deviation from planarity”, calculated as zA - zXax - 0.5. If the A-cation and the axial anion sit in the same plane, this equals zero. A positive deviation indicates a movement of the A-cation toward the perovskite layer, while a negative value indicates a shift into the rock salt layer.

With the exception of K2CuF4 and Rb2CuCl4, there are no reported examples of distorted halides under any conditions. The copper structures represent a special case that is discussed below. There are a few Alkaline-Earth oxides that are distorted at room temperature, and these have the unusual super-anti-parallel tilting distortion about the c-axis. The exception to this is Sr2SnO4, which has been reported in the aristotype and

− − 0 the a p a p c tilt systems. This is the same tilt system adopted by many of the rare earth oxides at room temperature. It has been noted above that the fluorides do not tend to contract relative to the analogous perovskite structure. This would imply that the octahedra are more regular for 68 halides than oxides, or tend to be compressed rather than elongated along the c-axis. In table 3.3, the shape of the octahedra can be concluded from the ratio of B-Xax to B-Xeq.

For most of the halides, the value of B-Xax / B-Xeq is less than one, indicating a compressed octahedron. The over-bonding at the Xeq–site for these structures is due entirely to the out of plane shift of the A-site cation, as shown by the deviation from planarity, or a small inter-layer spacing. In contrast, the rare-earth oxides tend to have elongated octahedra. Furthermore, there is a larger relative shift of the A-site cation towards the perovskite layer than for the halides. The uranyls, like the halides, tend to have compressed octahedra, with B-Xax / B-Xeq ratio near 0.9. As a consequence the Oeq anions in uranyls do not tend to be extremely over-bonded as with the other oxides. Only two rare earth oxides have been reported to have the aristotype structure. Because of the higher oxidation state of the rare earths, the B-site has an oxidation state of two rather than four. This makes for longer B-X bonds, and thus lower values of τRP.

In fact, for most of the rare earth oxides, τRP < 0.9. Rather than adopting a competing structure or decomposing into other oxides, these structures have a tendency to distort.

− − 0 At room temperature, the preferred tilt system is a p a p c . As explained in Chapter 2, the parallel tilt is preferred because it maximizes the inter-layer anion-anion distance. 3.2.3.4 Temperature and Pressure Effects There is a tendency among crystals to adopt structures of higher symmetry at higher temperatures. As the temperature rises, there is more energy for the librations, or octahedral tilting rotations, to overcome a potential energy barrier. While the octahedral tilts continue to exist, the octahedra swing back and forth around the axes. These librations cause an averaging of the distortions such that the overall structure adopts higher symmetry. At lower temperatures, the distortions “freeze out”, no longer able to overcome the energy barrier. This would suggest that there should be almost no distorted RP structures at high temperatures, and the possibility of more at lower temperatures.

Table 3.4 shows a summary of reported structures at various temperatures. Nd2NiO4 is the only compound reported to have a distorted structure above room temperature, and this is only at 330K. Nd2NiO4 also has a very low tolerance factor. The aristotype is still the preferred structure at low temperature for the alkaline earth oxides. However, the

69 Low Temperature Data Room Temp Data High Temperature Data Compound τRP T (K) Tilt a ref Tilt a c T Tilt a c c (K) 0 0 0 0 0 0 Ba2PbO4 0.904 5 a a c 4.297 a a a c 4.302 13.261 13.299 0 0 0 0 0 0 Ba2SnO4 0.964 12 a a c 4.131 b a a c 4.141 13.283 13.252 0 0 0 Ba2ZrO4 0.948 773 a a c 4.205 c 13.587 0 0 0 0 0 0 0 0 0 Sr2RuO4 0.984 10 a a c 3.864 d a a c 3.873 12.732 973 a a c 3.918 f a0a0c0 12.715 12.79 100 3.860 e 12.729 0 0 0 0 0 0 Sr2MnO4 0.994 4.2 a a c 3.787 g a a c 3.787 12.496 12.496 0 0 0 0 0 0 Sr2FeO4 0.997 4.2 a a c 3.861 h a a c 3.864 12.387 a0a0c0 12.399 100 3.862 i 12.400 0 0 Sr2SnO4 0.891 12 apa0c 4.049 b apapc 4.049 12.583 12.529 a0a0c0 4.053 12.585 0 0 La2NiO4 0.899 10 apapc 3.891 j apapc 3.872 12.626 0 apapc 12.504 70 3.891 12.504 0 0 Pr2NiO4 0.877 15 apa0c 3.882 k apapc 3.882 12.22 12.170 0 0 0 0 0 Nd2NiO4 0.871 1.5 apa0c 3.875 l a a c 3.854 12.214 330 apapc 3.8099 l 12.057 12.135

Table 3.4: High and low temperature Ruddlesden-Popper structures. References for structural data: a. Rosseinsky and Prassides, 1991; b.Green et al., 1996; c. Schpanchenko et al., 1993; d. Huang et al., 1994; e. Walz and Lichtenberg, 1993; f. Gardner et al., 1996; g. Bouloux et al., 1981; h. Dann et al., 1993; i. Dann et al., 1994; j. Lander et al., 1989; k. Fernandez-Diaz et al., 1993; l. Rodriguez-Carvajal et al., 1990.

70 − − 0 lanthanide oxides tend to adopt the tilt system a p a0 c . This is striking because this tilt

− − 0 system is of higher symmetry (tetragonal) than the orthorhombic a p a p c tilt system.

− − 0 However, the a p a0 c tilt system creates two chemically different A-sites which may be why it is not preferred at room temperature. Increasing pressure has the tendency to force a crystal to adopt a more dense structure. The perovskite structure, based on a close-packed anion model, is one of the densest structures. The RP structure is also quite dense, with only the inter-layer spacing providing separation of the ions. Ca2SiO4 and Ca2GeO4, which have the olivine structure under ambient conditions, adopt the RP structure at high pressures.11, 12 3.2.4 Modeling of RP n = 1 structures using lattice energies

To obtain further understanding of the tilting in RP structures, Ca2TiO4 was modeled in the program GULP (General Utility Lattice Program).13 This program can calculate the lattice energy for a given structure and then optimize that structure. The lattice energy is primarily composed of electrostatic forces of attraction and shell repulsion. Also taken into consideration is the “breathing” of the ions. This is represented by a spring constant between the core of each ion and its shell. The core and the shell need not be centered on exactly the same coordinates, allowing for a certain degrees of polarizability. Optimization was achieved under constant pressure at room temperature.

Ca2TiO4 was selected, in part, because this structure is not known to exist. The strontium analog, Sr2TiO4, is undistorted, as is SrTiO3. However CaTiO3, the mineral perovskite, is distorted as the Ca2+ cation is too small for its environment. A similar size- mismatch exists for the theoretical RP structure, so mathematical modeling should indicate what tilt-system it should take on if it did exist. Modeling of Sr2TiO4 in GULP predicts an undistorted structure. As can be seen from table 3.5, two tilt systems distinguish themselves. Most of the tilt systems have identical lattice parameters and essentially no tilt. That is, tilting in

− − 0 these systems is no more stable than no tilt at all. However, the tilt systems a p a0 c and

− − 0 a p a p c show considerable tilt, and lower lattice energy. It is interesting to note that

71 Tilt Space Lattice System Group energy X value tilt angle Lattice constants a 0 a 0c 0 I4/mmm -393.437 0 0 a = 3.783 (139) c = 11.683 − − 0 a a c P42/ncm -393.473 0.025337 5.8704 a = 5.375 p 0 (138) c= 11.613 − − 0 a a c P42/nnm -393.436 0.499984 0.0018 a = 5.350 a 0 (134) c = 11.683 a − a −c 0 Cmca -393.465 0.96808 5.2294 a = 5.360 p p (64) b = 5.379 c = 11.630 a − a −c 0 Cccm -393.437 0.250015 -0.0034 a = 5.350 a a (66) b = 5.350 c = 11.683 0 0 Cmca -393.437 0.249998 0.0005 a = 5.352 a a c p (64) b = 5.355 c = 11.683 0 0 I41/acd -393.442 0.250006 -0.0014 a = 5.350 a a cs (142) c = 23.370

Table 3.5 Lattice energy for Ca2TiO4 as calculated by GULP.

Tilt Space Lattice tilt System Group energy X value angle Lattice constants a 0 a 0c 0 I4/mmm -391.259 0 0 a = 3.879 (139) c = 11.991 a 0 a 0c Cmca -391.332 0.2130 8.419 a = 5.460 p (64) b = 5.460 c = 12.048 a − a −c 0 P42/ncm -391.631 0.0318 11.168 a = 5.507 p 0 (138) c = 11.875 a − a −c 0 Cmca -391.596 0.0120 4.236 a = 5.474 p p (64) b = 5.508 c = 11.912 a 0 a 0c I4/acd - 0.2130 8.419 a = 5.460 s (142) 391.3365 c = 24.098 a − a −c 0 Cccm -391.295 0.2800 -6.843 a = 5.507 a a (66) b = 5.475 c = 12.006 a − a −c 0 P42/nnm -391.311 0.0197 6.996 a = 5.495 a 0 (134) c = 12.012

Table 3.6: Lattice energy for Ca2RuO4 as calculated by GULP

72 these are also the tilt systems for which there are the most examples of distorted structures known. Similarly, when the structure Ca2RuO4 is modeled in GULP, these two tilt systems show significantly lower lattice energy than the other tilt systems (table 3.6).

3.3 Dion-Jacobson Structures

3.3.1 Lattice Strain In section 3.2.1, the problem of mismatched lattices was discussed. Given the similarities between the DJ and RP structures, it would seem logical that the DJ structures could be treated in a similar manner. That is, the lattice constant for DJ structures ought to be an average between the perovskite lattice constant (twice the typical B-X distance) and the CsCl type distance (2/√3 times the A-X distance). Table 3.7 shows many of the known DJ compounds and their theoretical and empirical lattice constants. The theoretical lattice constants were determined from averaging 2* RBX with 2/√3* RAX, where the distances were found using equation 3.2. The valence sBX was n/6 where n is the oxidation state of the B-cation. The valence sAX was m/8 where m is the oxidation state for the A-cation. The compounds are labeled according to their structures where

“A” indicates the aristotype and the reduced cell parameter ar = a, “T” indicates a tetragonal subgroup where √2ar = a, and “O” indicates an orthorhombic subgroup where

2ar = a. Careful inspection of this table will reveal that rather than averaging the desired repeat distances between the two sublayers, the DJ structures adopt a lattice size close to the perovskite lattice size. This expansion of the CsCl type AX layer can be compensated for by changing the symmetry from a cubic local environment to a D4h environment, as the site symmetry is tetragonal, not cubic. That is, the A-X bonds need not get longer, the perovskite-like layers just move closer together to compensate. The only restriction on this closeness is the unfavorable Xax-Xax interactions. The rock salt A-X layer in RP structures does not have this flexibility. 3.3.2 Known n = 1 Dion Jacobson Structures Since the DJ structure does not have the lattice strain typical of RP structures, it is free to relax in whatever way best idealizes the electrostatic bonding environments for the

73 Compound reference 2 RAX/√3 2 RBX a calc ar exp KAlF4 (A) a 3.188 3.603 3.396 3.57 TlAlF4 (A) b 3.371 3.603 3.487 3.63 RbAlF4 (T) c 3.382 3.603 3.493 3.63 RbFeF4 (T) d 3.382 3.871 3.627 3.81 CsFeF4 (T) e 3.578 3.871 3.725 3.90 CsMnF4 (T) f 3.578 3.833 3.706 3.97 CsTiF4 (T) g 3.578 3.977 3.778 3.95 BaUO4 (O) h 3.526 4.150 3.838 4.09 SrUO4 (O) h 3.334 4.150 3.742 4.02 PbUO4 (O) i 3.246 4.150 3.698 4.04

Table 3.7: Theoretical and empirical lattice constants based on the lattice strain model, whereby the calculated lattice constant is the average of the predicted lattice constants for the perovskite and CsCl sub-layers. References to experimental lattice parameters: a. Schoonman and Huggins, 1976; b. Brosset, 1937; c. Bulou and Nouet, 1982; d. Moron et al., 1990; e. Hidaka et al., 1986; f. Massa and Steiner, 1980; g. Sabatier et al., 1982; h. Loopstra and Rietveld, 1969; i. Cremers et al., 1986.

Formula τDJ a b c Vsum Vsum Vsum Vsum Xax ref A B Xax Xeq Xeq P4/mmm a0a0c0 KAlF4 0.98 3.55 6.138 0.969 3.208 0.954 1.104 1.003 a TlAlF4 1.04 3.61 6.37 1.198 2.944 1.013 1.024 1.006 b − + 0 Pbcm a pbp c BaUO4 0.87 5.7553 8.1411 8.2335 2.148 6.174 2.291 1.984 0.857 c 1.869 PbUO4 0.82 5.536 7.968 8.212 1.799 6.654 2.364 2.169 0.845 d 1.962 SrUO4 0.82 5.4896 7.977 8.1297 1.972 6.214 2.209 1.979 0.858 c 1.878 0 0 P4/mbm a a cp KAlF4 0.98 5.043 6.164 0.909 3.063 0.964 1.021 0.964 e RbAlF4 1.04 5.1227 6.2815 1.190 3.033 1.101 1.011 0.953 f P4/nmm + + 0 a p a p c CsMnF4 1.05 7.944 6.337 1.263 2.888 1.112 0.925 0.901 g 1.108 CsFeF4 1.04 7.7874 6.5402 1.206 3.082 1.091 1.034 0.949 h 1.132 CsTiF4 1.02 7.897 6.506 1.238 3.226 1.112 1.080 0.943 i 1.078

Table 3.8: Room temperature data for known n = 1 DJ structures. References to structural data: a. Schoonman and Huggins, 1976; b. Brosset, 1937; c. Loopstra and Rietveld, 1969 d. Cremers et al., 1986. e. Nouet et al., 1981; f. Bulou and Nouet, 1982; g. Hidaka et al., 1986; h. Massa and Steiner, 1980; i. Sabatier et al., 1982; ; 74 ions. Table 3.8 shows many of the compounds known to posses the DJ structure at room temperature. Lattice constants and tilt systems are shown for each compound.

Additionally, the BVS of the different ions were calculated and the ratio of Xax to Xeq is shown. This ratio clearly indicates that for most compounds, the compression of the octahedra predicted from the terminal bonding environment (see Chapter 2) is observed. For many of these compounds, multiple structures have been reported in the literature. For this table, more recent structures, neutron and single crystal diffraction techniques, and structures determined using Reitveld refinement were favored. It should be noted that KAlF4 has been reported to have both the aristotype and distorted structure in ambient conditions, though recent work supports the distorted structure. For TlAlF4, there are no reported room temperature structures after the 1937 one reported here. 14 However, in 1987 Bulou and Nouet reported structures for TlAlF4 in the space group P2/a (12) for the temperatures 200 and 423K. So it is likely that these two compounds are not in fact found in the aristotype at room temperature. Clearly, the aristotype is not as stable as the distorted structures. Below is a discussion of the factors that can affect the structure a particular compound will take. 3.3.2.1 Tolerance factors For perovskites and RP structures a tolerance factor was determined that was useful in determining if the aristotype would be stable for a given combination of A and B cations. In both cases, the idea behind this tolerance factor is that as the size of the A- cation decreases relative to the perovskite framework the bonding requirements of this cation cannot be met and some distortion, either from displacement of the A-cation or concerted movement of the octahedral network, must occur. So the size of the A-cation is important because of its own bonding environment. In the case of the DJ structures, a relatively small A-site cation not only decreases the total bonding it experiences, but also forces the axial anions of different layers closer together. The electrostatic repulsions of these anions will determine a minimum distance necessary between two layers. As such, the tolerance factor for DJ structures serves as a baseline where something has to give, either the octahedra must rotate so the anions are not coming together, or the perovskite plane layers must slip relative to one another.

75 The tolerance factor used in table 3.8 assumes that the minimum interlayer spacing is twice the radius of the anion in an ABX4 structure (2RX). The radii used were based on Shannon’s crystal radii.15 Without symmetry changes, the A-cation is still in a tetragonal environment for which the a lattice parameter is defined by the perovskite like layer (2RBX). In the minimally ideal situation, the A-cation would be just touching the anions at each corner of its tetragonal environment, so the length of the body diagonal of this sub-layer would equal 2*RAX. This length, DXax should equal the distances between the axial anions at opposite corners of sub-layer as defined by the perovskite layer and the anion radius.

2 2 2 2 DXax = 4RX + ( 2 * 2RBX ) (equation 3.5) So, the minimal tolerance for the aristotype structure is (see figure 3.4):

2RAX τ DJ = (equation 3.6) 2 2 4RX + 8RBX R = AX (equation 3.7) 2 2 RX + 2RBX It is evident from table 3.8 that there is a clear connection between this tolerance factor and the types of distortions observed. It should be noted that anti-parallel tilts in general will cause the Xax–anions within the A-X sub-layer to rotate in the same direction. As such, without further types of distortions such as plane slippages, these rotations are not favored for compounds with a tolerance less than unity. It is also interesting to observe that when the A-cation is larger than is minimally necessary (τDJ >1), octahedral tilting distortions are also observed. In this case the cause of the distortion cannot be attributed to either separation of the axial anions or compensation for under-bonding at the A-cation site. Rather, the rotations are more likely a compensation for over-bonding at the A-site. In table 3.7, the cesium compounds have reduced lattice parameters that are larger than those prescribed by the size of the

+ + 0 perovskite-like layer. As described in Chapter 2, the tilt system a p a p c will allow some of the A-cations to shift along the c-axis to idealize BVS at this site. This shifting in

76

Figure 3.4: The n = 1 DJ structure with the dimensions of used for the tolterance factor indicated.

77 combination with adjustments to the a lattice parameter and the free positional parameter on Xax allows for a more stable structure. 3.3.2.2 Plane Slippage An alternative to the octahedral rotation for minimizing the electrostatic interactions between the Xax anions in the A-X sub-layer is to allow slippage of the perovskite-like planes with respect to one another. If one plane slips by half the unit cell along both the x and y directions, tetragonal symmetry is preserved and the orientations of the perovskite like slabs is identical to that in the RP structures, and the new aristotype space group is I4/mmm (139). However, rather than the A-cations occupying all of the 4e Wyckoff positions, they occupy half of the 4d positions. This type of slippage, shown in figure 3.5a, is not uncommon when the A-site cation is small, such as Li+ or Na+.16, 17 Another possibility is for the planes to slip only along the x-axis. This will destroy the tetragonal symmetry, with the new aristotype of the orthogonal space group Cmcm (63). The symmetry of this system will in fact allow for tilting along one axis,

+ 0 0 18 ( a p b c ). KFeF4, shown in figure 3.5b, is known to adopt this symmetry above 368 K.

The α-SnWO4 structure is related to these compounds, but allows for out of phase

− − 0 rotations along the x and y axes, aa ba c . A number of tantalates and niobates adopt this structure.19 Examples of compounds exhibiting plane slippages and octahedral tilting distortions are shown in table 3.9. 3.3.2.3 Alternate Structures There are a variety of possible structures for compounds with the stoichiometry

ABX4, and most oxides will adopt an alternate to the n = 1 DJ structure. A related type of layered perovskite can form in which rather than the perovskite planes being sliced along the [100] axis, they are sliced along the [110] axis. In this case, rather than presenting an octahedral corner to the interlayer space, the perovskite slab presents an edge. Examples 20, 21 include CeNbO4 and GdNbO4. Many of the other possible structures have tetrahedral environments for the B-cation. This coordination environment is preferred for small electronegative cations, 6+ such as Cr . Some B-cations prefer a square planar environment, so many of the AAgO4 compounds have this type of environment. Examples of structures with four coordinate 78 Structures with x slippage

BiTaO4 BiNbO4 YNbO4

KFeF4 Structures with xy slippage

LiInF4 LiCoF4 LiMnF4 NaMnF4 [110] layered perovskites

BaMnF4 BiReO4 CeNbO4 GdNbO4 KScF4

Table 3.9: Compounds exhibiting plane slipages

79

Figure 3.5: ABX4 DJ like structures with plane slippages. a).slippage on both x and y b) KFeF4

80 B-sites include the Sheelite and its distorted kin Fergusonite, Zircon, Gaspirite, and β–

SnWO4 structures. 3.3.3 Structures with n > 1 The Dion-Jacobson structures with n > 1 have two or more layers of octahedral building block in each slab. The A-site between the slabs and the A'-site within the perovskite slab are chemically different, which allows for more than one type of A-cation in the aristotype structure. As with the RP structures, there are symmetry differences between odd and even layered structures that affect the space groups of the hettotypes. One notable difference, as described in Chapter 2, is that for even layered systems, it is the anti-parallel tilts that allow the axial anions to separate, rather than the parallel tilts. It would be expected that anti-parallel tilting would predominate in the even layered structures. Table 3.10 lists the known DJ structures with n > 1. Clearly the aristotype is much more stable for these structures. In this case, the A-cation within the perovskite slabs can help to stabilize the structures, as octahedral tilting rotations will change the bonding at this site. For most of the shown structures, the A and A'-cations are different, with the monovalent cation occupying the A-site between the perovskite slabs. This reduces the electrostatic interactions between the slabs, which makes the structure more prone to slippage. It is noteworthy that there are no known examples of fluorides that have the n = 2 DJ structure. There are no known examples of n = 2 structures in the aristotype in which the A and A'-cations are the same. The stoichiometry for the n = 2 phase is the same as that for the pyrochlore structure, A2B2X7, and many compounds with this composition adopt the competing pyrochlore structure.

3.4 Conclusions

The predictive power of the ionic model of layered perovskites is clear. For Ruddlesden-Popper structures, the inherent over-bonding of the equatorial anion makes distortions of the structure unfavorable. Most of these distortions increase bonding at this site. This also destabilizes the structure with respect to other combinations of oxides for which there may be no inherent bond strain. In contrast, the Dion-Jacobson structure is 81 Formula Reference Space Group Tilt System Unshifted 0 0 0 CsLaNb2O7 a P4/mmm (123) a a c 0 0 0 RbLaTa2O7 b P4/mmm (123) a a c 0 0 0 Rb0.86La1.09Ta2O7 b P4/mmm (123) a a c 0 0 0 RbLaTa2O7 c P4/mmm (123) a a c - - 0 RbLaNb2O7 d Imma (74) a ab ac Plane slippage along x

KLaNb2O7 e Cmmm (65)

K1.09l0.968Nb2O7 f Cmmm (65)

KLaTa2O7 g Cmmm (65) Plane slippage along x and y 0 0 0 NaLaNb2O7 e I4/mmm (139) a a c 0 0 0 LiLaNb2O7 e I4/mmm (139) a a c 0 0 0 AgLaNb2O7 e I4/mmm (139) a a c 0 0 0 NaLaTa2O7 h I4/mmm (139) a a c 0 0 0 AgLaTa2O7 i I4/mmm (139) a a c 0 0 0 LiLaTa2O7 B I4/mmm (139) a a c b a 0 a 0c AgLaNb2O7 I41/acd (142) s

Table 3.10: DJ Structures with n = 2. References to structures: a. Kumada et al., 1996; b. Toda and Sato, 1996; c. Honma et al., 1998; d. Armstrong and Anderson, 1994; e. Sato et al., 1993; f. Sato et al., 1992; g. Toda and Honma, 1997; h. Toda et al., 1997; i. Toda et al., 1996.

82 quite open to both octahedral tilting distortions and plane slippages. As predicted, anti- parallel tilts in the n = 1 type structures are not observed unless accompanied by plane slippages. These slippages are more favored for DJ structures than for RP structures due to the anion-anion interaction between layers. In both cases, increasing the number of perovskite-like layers in each slab serves to stabilize the aristotype. When n > 1, the and A'-sites are chemically different and known structures frequently involve a different type of cation at each site.

3.5 References

1. Pauling, L., The principles determining the structure of complex ionic crystals. Journal of the American Chemical Society 1929, 51, 1010-26.

2. Brown, I. D., The Chemical Bond in Inorganic Chemistry: the bond valence model. Oxford University Press: Oxford; New York, 2002; Vol. 12, p 278.

3. Chen, B.-H., Introduction of a Tolerance factor for the Nd2CuO4 (T')-type structure. Journal of Solid State Chemistry 1996, 125, (1), 63-66.

4. Poix, P., Study of the structure [type] potassium nickel fluoride (K2NiF4) by the method of invariants. I. Case of the oxides A2BO4. Journal of Solid State Chemistry 1980, 31, (1), 95-102.

5. Ganguli, D., Cationic radius ratio and formation of dipotassium tetrafluoronickelate- type compounds. Journal of Solid State Chemistry 1979, 30, (3), 353-6.

6. Altermatt, D.; Brown, I. D., The automatic searching for chemical bonds in inorganic crystal structures. Acta Crystallographica, Section B: Structural Science 1985, B41, (4), 240-4.

7. Brese, N. E.; O'Keeffe, M., Bond-valence parameters for solids. Acta Crystallographica, Section B: Structural Science 1991, B47, (2), 192-7.

8. Goldschmidt, V. M., Naturwissenschafsen 1926, 14, 477-485.

9. Lufaso, M. W.; Woodward, P. M., Prediction of the cyrstal structures of perovskites using the software program SPuDS. Acta Crystallographica 2001, B57, 725-738.

10. Troemel, M., Crystal structure of compounds of the Sr2PbO4 type. Zeitschrift fuer Anorganische und Allgemeine Chemie 1969, 371, (5-6), 237-47.

83 11. Liu, L.-G., High pressure calcium silicate, the silicate potassium tetrafluoronickelate(II)-isotype with crystal-chemical and geophysical implications. Physics and Chemistry of Minerals 1978, 3, (3), 291-9.

12. Ringwood, A. E.; Reid, A. F., High pressure polymorphs of olivines: K2NiF4 type. Earth and Planetary Science Letters 1968, 5, (2), 67-70.

13. Gale, J. D.; Rohl, A. L., The General Utility Lattice Program. Mol. Simul. 2003, 29, 291.

14. Bulou, A.; Nouet, J., Structural phase transitions in ferroelastic thallium(I) tetrafluoroaluminate: DSC investigations and structure determinations by neutron powder profile refinement. Journal of Physics C: Solid State Physics 1987, 20, (19), 2885-900.

15. Shannon, R. D., Revised effective ionic radii and systematic studies of interatomic distances in halies and chalcogenides. Acta Cryst. A32, 751-767.

16. Molinier, M.; Massa, W.; Khairoun, S.; Tressaud, A.; Soubeyroux, J. L., Crystal and magnetic structures of NaMnF4. Zeitschrift fuer Naturforschung, B: Chemical Sciences 1991, 46, (12), 1669-73.

17. Wandner, K. H.; Hoppe, R., The first di-rutile: lithium manganese tetrafluoride (with a remark on lithium cobalt tetrafluoride). Zeitschrift fuer Anorganische und Allgemeine Chemie 1987, 546, 113-21.

18. Tomaszewski, P. E., Structural phase transitions in crystals. I. Database. Phase Transitions 1992, 38, (3), 127-220.

19. Choong-Young, L.; Marquart, R.; Qing-Di, Z.; Kennedy, B. J., Structural and spectroscopic studies of Bi Ta1-x Nbx O4. Journal of Solid State Chemistry 2003, 174, 310-318.

20. Santoro, A.; Marezio, M.; Roth, R. S.; Minor, D., Neutron powder diffraction study of the structures of cerium tantalate, cerium niobate, and neodymium tantalate (CeTaO4, CeNbO4, and NdTaO4). Journal of Solid State Chemistry 1980, 35, (2), 167-75.

21. Trunov, V. K.; Kinzhibalo, L. N., Change in lanthanide niobate (LnNbO4) structures along the lanthanide series. Doklady Akademii Nauk SSSR 1982, 263, (2), 348-51 [Crystallogr].

84

CHAPTER 4

STRUCTURAL STUDIES OF NaLaMM'O6 PEROVSKITES AND THE INFLUENCE OF CATION SUBSTITUTION ON THE LAYERED ORDERING OF SODIUM AND LANTHANUM

4.1 Introduction

A described in Chapter 1, the perovskite structure has been studied extensively over the last eighty years. Ordering of the M-site cation has been commonly observed and the implications of this ordering for both structure and properties have been explored in depth. Ordering of the A-site cation has been less readily observed. Most of the examples involve vacancies either at the A-cation or the anion site. Examples of anion deficient perovskites exhibiting A-site layered ordering include the double-perovskites 1-3 1 LnBaFe2O5+x and the triple perovskite YBa2Fe3O8+x. The defect perovskite

BaLaMn2O6 in particular shows the relationship between anion vacancies and the ordering of the A-site cations.4, 5 When prepared directly, this compound is disordered. However, if prepared under appropriate conditions, the La3+ and Ba2+ cations will order to form the oxygen deficient BaLaMn2O5. In this compound the oxygen vacancies are found exclusively in the lanthanum layer. Oxidation at low temperatures will allow a meta-stable ordered form of BaLaMn2O6 to form. Ordering of the A-cations and A-site vacancies has been reported for several 6-8 A1-xMO3 structures, as well as compounds with a mixture of two different cations and vacancies on the A-site (A1-xA'yMO3). For example, the compound (Li3xLa2/3-x)TiO3 has been studied extensively for its lithium ion conductivity.9-11 It has been found that ordering is absent for the stoichiometric LaLiTi2O6, but for higher vacancy concentrations, (and lower values of x) ordering of the A-site is observed. Annealing

85 temperatures also play a role in the ordering of these structures.12 Recently, the structure of La4Mg3W3O18 was reported in which A-cation vacancies concentrated in layers alternating with layers fully occupied by La3+.13 Though it is exceedingly uncommon to see layered ordering of the A-site cations in stoichiometric AA'M2X6 compounds, there are a few examples of A-site ordering in 14, 15 AA'MM'O6 compounds, where there are multiple cations on both the A- and B-sites. 15 In 1984, Sekiya et al. reported making NaLaMgWO6 with a structure that exhibited both rock salt ordering of the M-site cations and layered ordering of the A-site cations.

Interestingly, the analogous NaLaTi2O6 and NaLaZr2O6 show no signs of long-range A- site cation ordering.16 This is a strong indication that the ordering of the A- and M-sites may be cooperatively linked, though the reason for this connection is not clearly understood.

4.2 Effects of distortions cation ordering on X-ray powder diffraction

X-Ray and Neutron diffraction of polycrystalline samples are common structural characterization techniques that rely on the translational symmetry of the crystal. Diffraction of X-rays or neutrons off of planes of atoms within the crystal produces reflections at particular angles dependant upon the distance between the planes of atoms, known as the d-spacing. In polycrystalline, or powder, samples, all possible orientations are presented to the incident radiation and there is an averaging of the pattern as peaks with common d-spacings overlap. This makes the patterns for symmetric structures much simpler than those for lower symmetry structures. The ideal ternary perovskite has a cubic structure, as described in chapter 1. The powder diffraction pattern of this structure is easily indexed. Any alteration to the structure typically lowers the symmetry of the cell and gives rise to additional reflections, or peaks, which can indicate atomic displacements, octahedral tilting, or ion ordering. Complete analyses of the affects of M-cation ordering and octahedral tilting have been described by Glazer,17 Woodward,18 and Barnes,19 and are summarized below along with the additional affects of A-cation ordering.

86 4.2.1 M-site Cation Ordering The ordering of cations is determined in large part by differences between those cations. Cations of similar size, charge, and electronic structure do not have strong differences in their preferred environment and will tend to distribute themselves randomly. However, when there is a large difference in size and/or charge, ordering can stabilize a structure by allowing each cation to have its preferred environment rather than an average environment.20 In the case of Mg2+ and W6+, it is not electrostatically favorable for the hexavalent tungsten ions to be near each other, nor for an anion to be in direct contact with two such highly charged cations, which could lead to over-bonding. Additionally, the Mg2+ is somewhat larger than the W6+, and an ordered arrangement will allow for the O-Mg bonds on average to be longer than the O-W bonds. A rock salt ordering of the two M-cations will ensure that each oxygen is in contact with exactly one Mg2+ and one W6+, allowing the anion between them to shift towards the tungsten cation and ensuring the most favorable electrostatic arrangement. In fact this is the most commonly observed type of ordering for M-cations in a 1:1 ratio in perovskites.20 This

+ type of arrangement is a basis for the R1 irreducible representation (irrep), and causes a supercell to form with face centered cubic symmetry (space group 225, Fm3m ). (Irreducible representations are discussed in more detail in Chapter 2). Most perovskites are not distorted much from cubic symmetry, and their XPD patterns can be indexed using a pseudo-cubic cell of lattice parameter approximately twice that of a typical perovskite. A strong peak in the region of 30°-32° 2θ (for Cu Kα excitation) is characteristic of perovskites and is the 220 peak for a perovskite of doubled unit cell parameters. The d-spacing of this peak can be used to determine the lattice constant and other peaks assigned from there. When there is no ordering or distortion, the peaks will only have even-even-even (eee) indices. Ordering of the M-cations in a rock salt fashion will cause the cell to adopt face centered cubic symmetry and a doubled unit cell length. As such, peaks will appear with odd-odd-odd (ooo) indices. Characteristic is the (111) peak near 19° 2θ. When the long rang order is nearly 100%, and there is a large difference in the scattering power of the two M-cations, this peak can

87 be quite strong. Face centering does not allow peaks whose indexing is not all odd or all even, so additional reflections are an indication of further distortion.

4.2.2 A-site Cation Ordering The A-cations in perovskites tend to order in layers rather than in the rock-salt fashion observed for M-cations. Electrostatically, this rock-salt ordering should be observed for the A-cations, so the driving force for the layering is not intuitive. The anion environment is the critical determining factor. Each anion is surrounded by two M-cations and four A-cations, with the M-cations being directly opposite one another. The rock-salt ordering of the M-cations moves the anion from the fixed 3d position (½, 0, 0) in space group Pm3m to the 24e position in Fm3m with a free positional parameter (x, 0, 0). (Note that doubling the a lattice parameter causes Z to increase from 1 to 8). This allows the anion to shift toward the smaller or more highly charged cation to relieve bonding strain. In the case of rock-salt ordering for the A-cations, a basis for

− the irrep R2 , the new space group is again Fm3m , but the anion is on the fixed 24d site (0, ¼, ¼). The like A-cations are directly opposite one another, as shown in figure 4.1a, so there is no mechanism for the anion to displace towards the cation with the smaller

− radius. However, if the cations order in layers, irrep X 3 , the perovskite adopts the space group P4 / mmm (123) and there are three crystallographically different positions for the anions. One third of the anions sit on either the 1b or 1a Wyckoff position, in which they are surrounded by only one type of A-cation. The remaining 2/3 of the anions sit on the 4i position, surrounded by 2 M-cations, two A-cations, and two A´-cations, as shown in figure 4.1b. This enables the anion to shift toward the layer with the smaller and/or more highly charged A-site cations. The layered ordering of the A-site cation destroys the cubic symmetry, as well as the face centering. A tetragonal cell is formed and even-even-odd (eeo) peaks appear. A strong reflection near 11° 2θ (the 001 peak) is a clear indicator of A-site ordering in perovskites. Ordering of the A-cations also removes the mirror plane perpendicular to the c-axis that runs through the M-cations. In the space group P4 / mmm , in which ordering of the A-cations, but not the M-cations occurs, a shift of M-cations does not give rise to 88 B B (a) (b)

A2 A1 A1 A2

A2 A1 A2 A1

B B

Figure 4.1 The anion environment in a perovskite with (a) rock salt type A-cation ordering, and (b) layered A-cation ordering.

89 new difference reflections. However, when ordering of the M-cations is imposed, a displacement of these cations from their pseudo mirror planes gives rise to odd-odd-even

+ − (ooe) peaks, l ≠ 0. A combination of the A-site and M-site ordering (both R1 and X 3 ) causes the symmetry of the space group P4 / nmm (129). 4.2.3 Distortions and Group Theoretical Analysis As described in chapter 1, the cubic geometry of the perovskite is stabilized when the tolerance factor, τ, is nearly unity, indicating a good match for the A and M cationic radii. When the A-cation is too small, the tolerance factor falls below 1.0 and distorted geometries are preferred. The same can be said for perovskites with multiple cations on the A and/or M-sites, with the tolerance factor typically being determined using the average of the cationic radii at each site. In ternary perovskites, it is the tilt system a −b + a − that is most commonly observed.21 However, ordering of cations with different sizes and charges adds new driving forces to the types of distortions that are preferred. Also, the symmetry of the tilt systems will be influenced by the ordering of the cations. Therefore, a group theoretical analysis of the tilting in perovskites with both A- and M-site ordering was performed to determine the possible tilt systems and resulting symmetries. Analysis of symmetry reduction was performed using the online version of ISOTROPY.22 The parent space group for ternary perovskites, Pm3m , was used along with the

+ − + − irreps R1 and X 3 for the cation ordering. R1 is a one dimensional irrep, while X 3 is three dimensional. Its three parameters may be generically written as (a, b, c), with the separate letters representing the same distortion in different directions. If the parameters are all 0, there is no distortion. Layered ordering of the A-cations occurs when the parameters are (a, 0, 0). However, if the symmetry is lowered slightly, the parameter sets (a, b, 0), (a, a, b), and (a, b, c) will allow layered ordering, with multiple positions for at least one of the A-cations. Therefore, the fully ordered arrangement is possible, but not

+ − required with these distortions. The combination of the R1 and X 3 irreps results in the space group P4 / nmm , which is an isotropy subgroup of Pm3m . Use of the alternative parameter sets for the layered ordering does not affect the selection of P4 / nmm as the

90 aristotype space group for combined ordering, but their inclusion in the analysis is necessary for completeness when considering octahedral tilting.

+ Further distortions from octahedral tilting can be either in-phase, irrep M 3 , or

+ out-of-phase, irrep R4 , as described in chapter 1. These distortions may be analyzed either from the parent space group, Pm3m , in combination with the ordering irreps or starting with the ordered aristotype ( P4 / nmm ). The analysis for figure 4.2 was completed using the simultaneous combination of all distortion irreps, and thus is an analysis from the parent space group. This results in ten parameters, as, with the

+ exception of R1 , the irreps are all three dimensional. Only simple tilts were considered, as defined by Howard and Stokes,23 in which tilting along a particular axis, if it exists, is either in-phase, or out-of-phase. As described above, parameter sets allowing but not requiring layered ordering of the A-cations were included for completeness. When this type of A-site order along with rock-salt ordering of the M-site cations imposed, there are 12 unique combinations with octahedral tilting distortions. As expected, tilt systems that involve only rotations about the c-axis, a 0 a 0c − and a 0 a 0c + , retain tetragonal symmetry, and thus add subtle changes in intensity to the eeo peaks. In-phase tilts of equal magnitude about the a and b-axes, a + a + c 0 , also retain the tetragonal symmetry, though they increase the size of the a lattice parameter by a factor of √2. In contrast, out-of- phase tilts of equal magnitude, a − a −c 0 , do not expand the unit cell, but lower the symmetry to monoclinic. Further distortions will lower the symmetry further. As can be seen from figure 4.2, the tilt systems can be combined in a variety of ways. There are a number of conceivable combinations which do not appear. This is because certain combinations destroy enough symmetry elements such that tilts about two or more axes are not required to be of equal magnitude. For example, the removal of cubic symmetry upon ordering of the A-cations removes the possibility of equal tilting about all three axes. Also, combining tilting about the c-axis with tilting about the a- and b-axes will destroy the symmetry elements that require tilting about the latter two axes to be equivalent. Therefore, by symmetry a–a–c– (or a+a+c+) is not distinct from a–b–c– (or

91

A and M cation ordering, no tilting P4/nmm (129) √2 x √2 x 2

Pmaa (49) P 4 2 m (111) P4212 (90) C2/m (12) P21/m (11) P4/n (85) a+b0c0 a+a+c0 a0a0c+ a0b-c0 a-a-c0 a0a0c- 2 x 2 x 2 2 x 2 x 2 √2 x √2 x 2 2 x 2 x 2 √2 x √2 x 2 √2 x √2 x 2

P222 (16) P112/a (13) P 4 (81) C2 (5) P21 (4) P1 (2) a+b+c+ a+b-c0 a+a+c- a-b0c+ a-a-c+ a-b-c- 2 x 2 x 2 2 x 2 x 2 2 x 2 x 2 2 x 2 x 2 √2 x √2 x 2 √2 x √2 x 2

Figure 4.2 Phase diagram for octahedral tilting distortions in perovskites with both A- and M-site cation ordering.

92 a+b+c+). This criterion for identifying unique tilt systems has been used previously by Howard and Stokes.23 Evidence of octahedral tilting in ternary perovskites is given by the appearance of difference reflections. Just as the ordering of the M-site cations will give rise to pseudocubic (ooo) peaks, the various tilting distortions will also give rise to pseudocubic peaks. For example, the out-of-phase tilting generally will also cause (ooo) peaks to appear. However, peaks due to octahedral tilting will be weak in intensity compared to those due to cation ordering, provided the ordering is extensive and there is a great deal of contrast between the scattering powers of the cations. In-phase rotations will cause difference reflections to occur with (ooe) indices. These are the same peaks that appear for the M-cation shift that is allowed in the P4 / nmm aristotype, so appearance of these peaks is not necessarily a clear indication of tilting. The displacement of the A-cation will cause (eeo) peaks to appear. These peaks are the same peaks that derive their intensity from the A-cation ordering. Again, with sufficient contrast and long range order, these peaks can be quite strong. As such, the indicators of octahedral tilting in ternary perovskites are obscured by the ordering of the A- and M-site cations in the compounds discussed below.

4.3 Experimental

4.3.1 Synthesis Perovskites were prepared using standard solid state techniques from oxides and carbonates of the metals. It was found that making MgWO4 as a precursor for the

ALaMgWO6 compounds yielded purer samples with better crystallinity. This was prepared from MgO (B&A) and WO3 (Cerac) with a 15% excess of MgO added due to its hygroscopic nature. The sample was heated at temperatures up to 900°C until impurity peak disappeared and the X-ray diffraction pattern did not change. La2O3 is also hygroscopic. For some of the samples, an excess of 17% (determined by thermogravimetric analysis) was added. For others, La2O3 (B&A) was heated to 900°C for several hours to drive off water and stored in a dessicator until it was weighed out. The method used is indicated in table 4.1. For the Alkali metals, a 10% excess of the

93

Compound Sample Maximum heating temperature Impurities NaLaMgWO6 115 900 (6) 1050 (18) NaLaMgTeO6 145 600(2.5) 900(6) 1050(12) 1100(24) 1200 (6) 185b 900(6) 1100(12) NaLaScNbO6 127b 900(4) 1100(24) 180a* 900(6) 1100(8) 1200(20) 1250(4) sc NaScO2 NaLaScSbO6 123a 900(6) 1050(6) 1100(12) 1150(16) 180b* 900(6) 1100(8) 1200(20) 1250(4) sc NaScO2 NaLaTi2O6 065c 187a 850(6) 1100(18) NaLaZr2O6 127a 900(4) 1100(24) KLaMgWO6 083a KWO4 LiLaMgWO6 093b

Table 4.1: Synthetic conditions for each of the samples. * These samples were prepared with dried La2O3. (sc) Samples were slow cooled at a rate less than or equal to 1°/min.

94 metal carbonate was added at the initial grinding stage, and then after each heating cycle above 1000°C. Metal oxides used include: TeO2 (Cerac), Sc2O3 (Boulder), Nb2O5

(Aldrich), Sb2O3 (Cerac), TiO2 (GFS), and ZrO2 (Johnson Matthey). Aside from MgWO4, no precursors were used. Stoichiometric amounts of the appropriate metal oxides and alkali carbonate were ground intimately with acetone in an agate mortar for several minutes. The mixtures were then placed in high form alumina crucibles and heated as described in table 4.1. 4.3.2 Structural Characterization (XPD, NPD, SAED) X-ray diffraction relies on the ability of electron clouds surrounding an atom to scatter X-rays. The scattering length of an atom is approximately proportional to the square of the atomic number, so that heavier elements have much larger scattering lengths than light elements. As a consequence X-ray diffraction can be very sensitive, particularly for compounds containing metals. The positions of heavy elements can be determined with great accuracy. Because of the large range of scattering lengths, there is also a high contrast between elements of different atomic number occupying similar sites. This allows for detection of ordering or supercells. However, because oxygen is a relatively light element, it can be difficult to determine the location of oxygen anions in an extended crystal structure, and indications of lowered symmetry due to oxygen displacement in the form of weak reflections of may be overlooked as impurity peaks. X- ray powder diffraction (XPD) data were collected in Bragg-Brentano geometry from 10- 120° 2θ and a step size ~0.0144° using a Bruker D8 X-ray powder diffractometer (40 kV- 50 mA, sealed Cu X-ray tube, λ=1.54056 Å) equipped with an incident beam Ge 111 monochromator and a Braun linear position sensitive detector. Powder diffraction takes advantage of the averaging of orientations for polycrystalline (powder) samples. All possible orientations of the crystallites are assumed to be presented to the X-rays and used in the refinement of the data collected. In these samples, synchronous rotation of the flat plate sample holder was used in an attempt to minimize the effects of preferred orientation.

95 While neutron diffraction is a much lower resolution technique, and less convenient in the sense that it is only available at a few facilities around the country, it has the advantage of relying on a scattering that is not based on atomic number. Neutrons scatter off of the nucleus rather than the electron cloud, and the scattering length of the atoms does not follow a pattern based on the atomic number. As such, the scattering length of oxygen (5.803 fm) is comparable to that of the many metals (W = 4.86 fm, La = 8.24 fm, Na = 3.63 fm). Reflections due to changes in oxygen anion position, such as by octahedral tilting, will be much more readily observed in neutron powder diffraction (NPD) data. NPD data were collected at beamline BT-1 at the National Institute of Standards and Technology, Gaithersburg, Maryland. The beamline is equipped with 32 detectors, a Cu (311) monochromator and a 15´ collimator. The ~8g samples were packed in neutron transparent vanadium canisters and patterns were collected from 3 to 165° 2 theta for approximately 6hrs. 4.3.3 Data Analysis Structure refinements of XPD were performed using the Rietveld method24 as implemented in the TOPAS software package25. Using the results of the group theoretical analysis described in section 4.2.3, the peak splitting was analyzed to assign the appropriate space group and starting positions were determined. A Pawley fit was used to determine the lattice parameters. This information was used for a full structural fit, and atomic positions for non-fixed atoms were allowed to refine, starting with the heaviest atoms. Thermal parameters were allowed to refine, with the parameters of the A-cations being fixed to be equal, as were the thermal parameters of the M-cations. Axial oxygen ion displacement parameters were refined individually, while the displacement parameters of the equatorial oxygen ions (those in the M-cation plane) were constrained to be equal. This was done to prevent unwanted correlations between individual displacement parameters. Finally, the cation mixing was refined for both the A and B cation sites. Long range ordering was determined by the LRO = (2s-1)*100%, where s is the fractional occupancy of the dominant cation on one of the sites. When s was nearly 1, complete ordering was assumed.

96 Structural refinements of NPD were performed using GSAS.26, 27 When possible, the lattice parameters and Wyckoff positions from the XPD were used as starting models. For some structures, additional peaks due to displacements of the oxygen anions indicated that other space groups must be considered. Starting models for structures without A cation ordering were generated using SPuDS.21 Atomic positions for non-fixed atoms were allowed to refine, starting with the atoms with the greatest scattering factors. Again thermal parameters of the A-site cations were constrained to be equal, as were the thermal parameters for the B-site cations. When necessary for stable refinement, the thermal parameters of certain oxygen anion positions were allowed to refine anisotropically. Because of the greater contrast in XPD, the cation ordering was left as determined through refinement of those data. Finally, the structure and positions determined through GSAS were then used as a starting model for TOPAS refinement of the XPD to ensure that the new positions did not conflict. Structural analysis of some compounds (NaLaMgWO6 and NaLaTi2O6) was performed using Transmision Electron Microscopy (TEM) and Scattered Angle Electron Diffraction (SAED). Tilt angles and cation displacements were determined using the program IVTON. Refined atomic positions are used to determine a polyhedron around the M- cations. The centroid of this polyhedron is determined as the center of gravity of the anions that determine the vertices of the polyhedron. The displacement of the M-site cation is calculated as the distance in Ångstroms between the cation and the polyhedron centroid. Because this displacement can be quite large for some M-site cations, the tilt angle is determined from the centroid position rather than the cation position, where the degree of tilt is one half of 180 minus the M-O-M' angle.

4.4 Results

4.4.1 NaLaMgWO6

The first compound prepared was NaLaMgWO6, as this had previously been 15 prepared by Sekiya et al. The XPD pattern of NaLaMgWO6 is shown in figure 4.3. Clear evidence of M-cation ordering is seen in the strong (111) peak (around 19° 2θ) and the weaker (311) peak (around 33° 2θ). Similarly, the strong (001) peak near 11° 2θ is an

97

Figure 4.3 XPD pattern for NaLaMgWO6 (open circles), fit in C2/m (solid line), and difference pattern. The inset shows the details of the diffraction pattern for the pseudocubic <440> peaks, along with the fit in C2/m (solid line) and P21/m (dashed line).

98 indicator of the A-cation ordering. While it is possible to get a fit of the pattern in the parent space group, P4/nmm, peak profiles, particularly of the pseudocubic <440> peaks, suggest that the structure has lower symmetry. Original fitting was done using the space 15 28 group P21 / m , as assigned by Sekiya et al. and subsequently Arillo et al. This is an isotropy subgroup of P4/nmm and is consistent with octahedral tilting, a − a −c 0 , as described in figure 4.2. However, the reflection splitting, particularly of the psuedocubic <440> peaks, does not support this assignment. The profile of the <440> set of peaks along with the refinement of the XPD data in both C2/m (solid line) and P21/m (dashed line) and are shown in the inset of figure 4.3. The space group C2/m clearly gives a better fit of these peaks, as well as a better fit overall for the pattern. XPD refinement Systematic absences and peak splitting were used to determine a more appropriate space group and it was found that the XRD pattern does support the assignment in a monoclinic space group. The group theoretical analysis of these perovskites, shown in figure 4.2 presents three likely candidates for the monoclinic space

0 − 0 − + 0 groups, not including P21 / m . The other three are C2/m (a b c ), P2 / c ( a b c ) and C2 ( a −b0c + ). The difference between the a 0b −c 0 and a −b0c + tilt systems is the removal of a mirror plane due to the addition of a rotation about the c-axis. Based on the analysis, the appropriate space group is C2/m. The space group P2 / c gives an inferior fit to the other two, and was not considered further. Tilting about the c-axis, which distinguishes C2/m from C2, causes subtle changes in the intensity of the eeo peaks, which in the case of layered ordering can be quite intense due to the contrast of the A-site cations. As such, it is difficult to deconvolute the contribution of the two distortions in X-ray powder data. The fit for C2/m and C2 were similar so the higher symmetry selection was preferred.

The Rietveld refinement yielded a goodness-of-fit parameter, Rwp = 0.145. This compares favorably to the goodness-of-fit parameter that was obtained for a Pawley fit to the pattern, Rwp = 0.140, strongly supporting the validity of the refined structure. Structural data for this compound are given in table 4.2.

99 NaLaMgWO6 NaLaScNbO6 NaLaMgTeO6 Space C2/m Rwp = C2/m Rwp P21/m Rwp = Group a0b-c0 0.128 a0b-c0 =0.0842 a-a-c0 0.0738 2 2 2 Tilt System a = 7.8118(4) Χ = 5.72 a = 7.9820(6) Χ = 5.55 a = 5.5679(3) Χ = 2.29 Lattice b = 7.8165(4) R(F2) = b = 7.9816(7) R(F2) = b = 5.5564(2) R(F2) = Parameters c = 7.9010(3) 0.155 c = 8.0253(4) 0.986 c = 7.8669(4) 0.0538 β = 90.137(5) β = 90.01(1) β = 90.036(6) A (occ) Na U(iso) Na [0.81(2)] U(iso) Na [0.61(1)] U(iso) 4g: 0, y , 0 0.0061(5) La [0.19(2)] 0.0050(8) La [0.39(1)] 0.0110(6) y= 0.221(1) 4g 0, y , 0 2e x, ¼, z y= 0.277(2) x=0.756(2) z=0.007(2)

A' (occ) La U(iso) La: [0.81(2)] U (iso) Na [0.61(1)] U(iso) 4h: 0, y , ½ 0.0061(5) Na: [0.19(2)] 0.0050(8) La [0.39(1)] 0.0110(6) y= 0.2439(5) 4h 0, y , ½ 2e x, ¼, z y= 0.2601(8) x=0.744(2) z=0.493(2)

M Mg U(iso) Sc U(iso) Mg U(iso) 4i x, 0, z 0.0061(5) 4i x, 0, z 0.0099(6) 2e x, ¼, z 0.0071(3) x=0.742(1) x=0.7489(9) x=0.255(2) z=0.747(1) z=0.7540(9) z=0.251(1)

M' W U(iso) Nb U(iso) Te U(iso) 4i x, 0, z 0.0061(5) 4i x, 0, z 0.0099(6) 2e x, ¼, z 0.0071(3) x=0.252(1) x=0.751(1) x=0.253(2) z=0.733(1) z=0.248(1) z=0.744(1)

O(Na) 4i x, 0, z U(iso) 4i x, 0, z U(iso) 2e x, ¼, z U(iso) x=0.787(1) 0.036(2) x=0.8051(6) -0.015(1) x=0.314(1) -0.003(1) z=0.005(2) z=0.015(1) z=0.0103(8)

O(La) 4i x, 0, z U(iso) 4i x, 0, z U(iso) 2e x, ¼, z U(iso) x=0.7050(8) 0.017(1) x=0.706(2) 0.042(3) x=0.195(1) 0.011(2) z=0.498(1) z=0.496(3) z=0.4997(8)

Oeq1 4i x, 0, z U(iso) 4i x, 0, z U(aniso) 4f x, y, z U(iso) x=0.014(2) 0.013(1) x=0.006(2) -0.04(2) x=0.521(2) 0.031(2) z=0.721(1) z=0.696(1) 0.19(1) y=0.019(2) -0.017(2) z=0.729(1)

Oeq2 4i x, 0, z U(iso) 4i x, 0, z U(iso) 4f x, y, z U(aniso) x=0.503(1) 0.005(1) x=0.481(3) 0.057(5) x=-0.026(2) 0.013(4) z=0.8002(7) z=0.773(3) y=0.014(2) 0.039(4) z=0.780(1) 0.018(3)

Oeq3 4j x, y, z U(iso) 4j x, y, z U(aniso) x=0.741(2) 0.043(2) x=0.771(3) 0.12(1) y=0.257(2) y=0.243(1) 0.008(3) z=0.7712(8) z=0.742(2) 0.11(1)

Table 4.2: Crystallographic Data for NPD data refined using GSAS (continued) 100 Table 4.2 Continued NaLaScSbO6 NaLaZr2O6 NaLaTi2O6 Space Group P21/n Rwp = Imma Rwp = Pnma Rwp = - + - - - 0 - + - Tilt System a b a 0.092 a a b 0.081 a b a 0.0571 2 2 2 Lattice a=5.5635(2) Χ = 4.86 a=5.7742(2) Χ = 2.17 a=5.4789 Χ = 1.93 Parameters b= 5.6405(2) R(F2) = b=8.1426(2) R(F2) = b=7.7469 R(F2) = c=7.9701(4) 0.115 c=5.7246(1) 0.0865 c=5.4872 0.0.0365 A Na/La U(iso) Na/La U(iso) Na/La U(iso) (occupancy) 2e x,y,z 0.0147(6) 4e o, ¼, z 0.0104(3) 4c x, ¼, z 0.0157(4) x=0.500(3) z= 0.4924(9) x=0.0239(4) y=0.504(1) z=0.5044(8) z=0.250(2)

M 2a 0, ½, 0 U(iso) 4a 0,0,0 U(iso) 4a 0,0,0 U(iso) 0.018(1) 0.0064(4) 0.0049(3)

M 2b ½, 0, 0 U(iso) -0.006(1)

O 2e x, y, z U(iso) 4e 0, ¼, z U(iso) 4c x, ¼, z U(iso) x=0.259(2) 0.020(2) z= 0.9551(6) 0.0070(5) x=0.4807(5) 0.0124(6) y=0.245(2) z=0.4187(5) z=0.042(1)

O 2e x, y, z U(iso) 8f ¼, y. ¼ U(aniso)* 8d x, y, z U(iso) x=0.231(2) 0.014(2) y= 0.4762(3) 0.014(1) x=0.2911(4) 0.0183(5) y=0.245(2) -0.0041(9) y=0.0421(3) z=0.037(1) 0.0086(7) z=0.7898(4)

O 2e x, y, z U(iso) x=0.568(1) 0.008(2) y=0.000(2) z=0.247(1)

101 NPD refinement

The neutron powder diffraction data show the structure of NaLaMgWO6 to be more complex than the XPD data indicates. As discussed above, NPD data is more sensitive to oxygen position than XPD data. The data collected at NIST were analyzed in both P21 / m and C2/m. The Reitveld refinements yielded similar goodness of fit parameters for the two space groups: wRp = 0.127 and 0.128 respectively. However, close inspection of the observed and calculated patterns indicated that neither space group could completely account for all of the observed reflections. Figure 4.4 shows a close up of three sections of the neutron powder pattern. In particular, the <311> peaks near 38° 2θ have satellites on either side for which there should be no reflections in either space group. Similar extraneous peaks are visible near 62 and 70° 2θ. These peaks can be accounted for if a larger unit cell is used in which either or both of the a and b axes is ~

11Å (2√2 ap). A cell this large produces so many peaks that above 60° almost any reflection could be indexed, so the peaks near 38° are more telling. The supercell with the larger a or b axis is necessary to account for these satellite peaks, and base centering with such a supercell will remove the reflections. The possibility of impurities was considered to account for these reflections. However, similar satellites were visible in patterns of NaLaScNbO6 and KLaMgWO6. The only common metal is the lanthanum, and there are no known metal oxides of the consitituent materials, with or without lanthanum, that account for the additional peaks. However, the supercell predicts additional reflections at precisely the correct d-spacing. There are no simple tilt systems that create such a supercell. Because these reflections are apparent only in the neutron data, it is reasonable to attribute their presence to a shift in the oxygen anion positions. A le Bail fit was done in the space group Pm for cells of size 11 x 5.5 x 8 and 5.5 x 11 x 8. While the le Bail fits were similar to those for the smaller unit cells (wRp = 0.125), the Reitveld refinement for a cell with so many variables proved extremely unstable. The best fit was achieved for the 11 x 5.5 x 8 cell (wRp = 0.152). Further analysis of the data will be necessary to determine more clearly what the complex tilting pattern is that leads to this expanded cell.

102

Figure 4.4 Neutron diffraction data for selected regions of NaLaMgWO6 (open circles). Overlaid are calculated patterns for refinements in C2/m (solid) and P21/m (dashed). Below are difference patterns for each.

103

Figure 4.5: XPD patterns for six perovskites. M-site ordering peaks are found near 19°, while A-site ordering peaks are found near 11° and 25°. Patterns are labeled from back to front: (a) NaLaMgWO6, (b) NaLaScNbO6, (c) NaLaMgTeWO6 (d) NaLaScSbO6, (e) NaLaZr2O6, and (f) NaLaTi2O6.

104 4.4.2 NaLaMgTeO6

A comparison of the XPD patterns for NaLaMgWO6, NaLaMgTeO6,

NaLaScNbO6, NaLaScSbO6, NaLaTi2O6, and NaLaZr2O6 is shown in figure 4.5. The peaks for NaLaMgTeO6 (figure 4.5c) were significantly broader than those for

NaLaMgWO6, especially the (eeo) peaks indicating A-site ordering. As such, it was difficult to resolve the peak splitting in order to accurately determine a space group for proper refinement. Some features of note in this pattern are the small (001) peak and the absence of (ooe) peaks. The (001) peak, an indicator of A-site ordering is quite small and broad. This indicates only a small degree of long range order. Comparing the full width at half maximum for the eeo peaks verses the eee peaks will provide insight into the degree of ordering as compared to the crystallite size. Using the Sherrer formula (equation 4.1), the size of the ordered domains can be determined. 0.9λ d = (equation 4.1) β cosθ

In this formula, the size of the domain (d) is related to the Cu Kα wavelength (λ), the full width at half maximum for the diffraction peak (β), and the diffraction angle (θ). Peaks with (eeo) indices, related to the A-site ordering, give an estimated domain size of around 40Å, while the average crystallite size from the remaining peaks is at least 1300Å. The (ooe) peaks, most notably those near 28° 2θ and 43° 2θ, are an indication of M-cation displacement. As will be discussed below, this displacement is often coupled with the ordering of the A-site cations. In the case of NaLaMgTeO6, there is no evidence of peaks at these angles. The NPD pattern indicates lower symmetry than the XPD pattern does. Intensity at the ooo peaks is normally an indication of either B-site cation ordering or out of-phase octahedral tilting. Because the ordering peaks are so strong in the XPD, it is difficult to determine if there is an octahedral tilting. However, in NPD, magnesium and tellurium have very similar scattering factors (5.4 fm and 5.8 fm respectively), and their contrast is small. Intensity in the (ooo) reflections is due primarily to out-of-phase octahedral tilting. Furthermore, this type of tilting distortion does not give intensity to the (ooo) peaks in which h = k = l. The NPD pattern has noticeably less intensity at the 105 <111> pseudo-cubic reflection than the other (ooo) reflections, a strong indication that out-of-phase tilting is a factor in this structure. However, this compound has a high degree of pseudo symmetry in that the lattice parameters in each direction are very similar, and β only marginally deviates from 90°. There is very little peak splitting in either the XPD or NPD patterns, and peak broadening due to short range ordering and anti-phase boundaries make stable refinements difficult.

In refining the data for NaLaMgTeO6, initially no A-site ordering was considered. Ordering at the A-site reduces the symmetry greatly and makes stable refinement difficult. In the NPD data, only the initial 001 peak seems to indicate A-site order, so it is assumed that other parameters should fit well based on the rest of the pattern. The observations noted above indicated out-of-phase tilting along at least one of the axes. Initially refinements were attempted in space groups I4/m (a0a0c-) and I2/m (a0b-b-). While the latter produced a better fit, the placement of the oxygen anions and the anisotropic thermal factors for the equatorial oxygens indicated an additional tilt - about the third axis. Further refinements were completed using space groups P21/m (a b+a-), R3(a-a-a-), and P1(a-b-c-). While the fit improved slightly as the symmetry was lowered, this is to be expected, as there are more parameters to adjust. However, the pseudo-symmetry made the refinements quite unstable and the affect observed for the tilt system a0b-b- was not diminished. As such, this latter tilt system was accepted. GSAS will allow for constraints to be placed on the lengths of bonds. These soft constraints usually do not improve the fit of the pattern, but may enable a more reasonable structure to be determined. In this case, the O-Te and O-Mg bond lengths were constrained to be near their expected values (1.92 and 2.1 Å respectively). The overall fit of the pattern was not affected to a great extent (wRp =). Finally the thermal parameters for the B-cations were allowed to refine anisotropically. The space groups selected for this system has the B-cations on fixed positions. If there were a displacement of the B-cations, as observed with W in the NaLaMgWO6 compound, there would be a greater thermal displacement along one axis than the others. This type of anisotropy was not observed.

106 To determine the extent to A-cation ordering, the tilt system a-a-c0 for perovskites with A-site ordering (space group P21/m) was used. However, the atoms were constrained to mimic the body centering of the a0b-b- tilt system. That is, in the space group P21/m, there are two sites for A-cations. During the refinement, these sites, equally occupied by Na+ and La3+, were constrained to move synergistically to maintain the body centering. Also, each of the oxygen positions split into two positions, and these oxygen anions were constrained to move in tandem as well. Once the refinement was stable, the constraints on the oxygen anions were removed. Only the removal of the constraints on the equatorial oxygens improved the refinement. The thermal displacement parameters on the oxygen anions were allowed to refine anisotropically. Only anisotropy on one site improved the fit, and this produced negative values for 2 parameters. The occupancies of the A-sites were allowed to refine such that the total occupancy sum of Na+ and La3+ was equal to unity. The long range ordering was determined to be 22%. This refinement was comparable to that of the a0b-b- tilt system with no ordering, with rWp = 0.0738.

4.4.3 NaLaScNbO6

The XPD pattern of NaLaScNbO6 indicates that that both A- and M-site ordering are present, due to the peaks at 11° and 19°. The XPD pattern for this compound is shown in figure 4.5b. Clearly, there is a much higher degree of Na/La ordering for this compound than for NaLaMgTeO6. However, the ordering peaks show similar broadening, though to a lesser degree. Like NaLaMgWO6, NaLaScNbO6 appears to be monoclinic, possibly having the tilt system a +b −c 0 (space group 13, P112/a).

However, the peak splitting is much less in NaLaScNbO6 than in NaLaMgWO6, so refinement of the XPD data in monoclinic space groups does not significantly improve the fit over the aristotype space group.

The NPD pattern of NaLaScNbO6 revealed a small impurity of NaScO2 (also present in the NaLaScSbO6 sample). This impurity is made of the lightest elements with the smallest X-ray scattering factors, so it was not observed in the XPD data. Impurity peaks in the neutron data are visible at 16.2°, 33.1° and 39.5° 2θ. The NPD pattern was 0 - 0 refined in space group C2/m, tilt system a b c , with NaScO2 added as a second phase

107 making up 5% of the composition after the phase fractions were refined. Similar to

NaLaMgWO6, the reflection at 38° 2θ contained small satellite peaks. Because the peak splitting was much less these satellite peaks were closer to the main peak, and less intense. Refinement in this space group was not able to account for some of the intensity at certain peaks, which may be due to additional impurities. The final wRp value was 0.0842, and the LRO of the A-site was refined as 62%. The M-site ordering was assumed to be 100% based on the XPD pattern, for which the Sc/Nb contrast was great and the oxygen anion position contribution to intensity of the (ooo) peaks was minimal.

4.4.4 NaLaScSbO6 Neither the XPD nor the NPD data indicate there is A-site ordering in

NaLaScSbO6, as there is no evidence of eeo peaks in either pattern. XPD data indicated that the ordering of the B-site cations was nearly 100% (see figure 4.5d). This is to be expected of antimony compounds. Without bonding d orbitals on the metal, oxygen has difficulty forming 180° bond angles between two antimony ions. The same oxygen 2p orbital would be required to overlap with antimony 5p orbitals on either side. If the M-O-

M bond angle is less than 180°, both the pz and px orbitals can be involved with the metal- oxygen bonding. As such, main group metals like antimony often prefer structural arrangements that allow for lower bond angles. However, if each oxygen is bonded only to one antimony and one transition metal, the pz orbital can interact with the antimony while the px or py orbital can π-bond with the transition metal. This means that rock-salt ordering on the M-sites is highly favored when there is one main group and one transition metal cation on the M-site. There were no observable difference reflections or peak splitting in the XPD data. However, in the NPD data, the ooo peaks were quite strong, with less intensity on the <111> reflection than the other ooo reflections. Tilting rotations do not give intensity to peaks where h=k=l, so this is an indication of out-of- phase tilting. - + - The best fit was given for the tilt system a b a , space group P21/n (14). This is also the most common tilt system for ternary perovskites. The wRp for this refinement was 7.7% with an Rf2 value of 0.072. The A-site cations were assumed to distribute randomly, and the thermal parameters for these cations were set to be equal. The thermal

108 parameters for the B-site cations were also constrained to be equal. This was done to reduce the effect of correlations between these parameters. Because the M-site cations sit on fixed positions in this tilt system, they are not allowed to displace. To check for displacement, the thermal parameters of the M-cations were allowed to refine anisotropically. This caused the refinement to diverge, so it is concluded there is no displacement of the M-cations. The structural parameters for NaLaScSbO6 are shown in table 4.2. M-O-M' bond angles of 160° indicate a rotation of approximately 10°.

4.4.5 NaLaTi2O6 and NaLaZr2O6

In the cases of NaLaTi2O6 and NaLaZr2O6, there is only one B-site cation, so there is no possibility of ordering on this site. As such, the XPD patterns show no indicative intensity at the (ooo) reflections (see figures 4.5e and 4.5f). Most notably, the ordering peak near 19° 2θ is missing. Additionally, there is no strong evidence in either the XPD or the NPD patterns indicating long range ordering of the A-site cations. After prolonged heating, a small broad peak appears in the NaLaZr2O6 XPD pattern, and SAED patterns of NaLaTi2O6 indicate the possibility of short range ordering. Without any degree of intensity at the eeo reflections, refinements were necessarily completed assuming a random distribution of the A-site cations. A small amount of intensity at the <311> reflection site indicates out-of-phase tilting.

NaLaZr2O6 was easily refined for both XPD and NPD in the space group Pnma, using the tilt system a −b + a − . The crystallographic information for this structure is found in table 4.2. NaLaTi2O6 was highly pseudo-cubic, and determination of the appropriate tilt system and space group were more challenging. Analysis of the neutron data indicated that the tilt system a − a −b0 , in the space group Imma (74) was the most appropriate. The Le Bail fit of the neutron pattern was better for Imma than for Pnma (wRp= 0.058 and 0.063 respectively). In both of these space groups, the M-cation sits on a fixed position of mutually perpendicular mirror planes, so no displacement of the M-site cation is allowed. To test for M-cation displacement, the thermal parameters of Ti and Zr were allowed to refine anisotropically. For NaLaTi2O6, this gave no improvement to the fit. For NaLaZr2O6, the fit improved slightly (wRp = 0.0810), however, the several of the parameters were negative, which is meaningless, and/or had errors of the same magnitude

109 as the parameter itself. Thus it is concluded that on average, there is no displacement of the M-cations from the centroid of the octahedra. The degree of octahedral tilting for the structures was determined by the M-O-M bond angle. For NaLaTi2O6, a small tilt angle of around 7° is found, while for

NaLaZr2O6, the tilt angle was much larger, near 13°. See table 4.3.

4.4.6 KLaMgWO6 and LiLaMgWO6

KLaMgWO6 and LiLaMgWO6 were prepared as described in section 4.3, and analyzed by XPD. These samples were also analyzed at Argonne National Laboratory (Ref information) by NPD. This analysis revealed that the samples were both hygroscopic and susceptible to decomposition. The data could be treated qualitatively and the lattice parameters could be determined quantitatively, but impurity peaks in the neutron data made full pattern fitting difficult. XPD data was collected soon after preparation and does appear to have impurity peaks. These data were suitable for refinement of the metal positions, but the oxygen and lithium positions, as discussed above, were difficult to determine without supporting NPD data. Subsequent attempts to synthesize these compounds without impurities has thus far proved unsuccessful. XPD data for both samples clearly indicate A-site ordering, as each pattern has a strong (001) ordering peak near 11°. Greater size differences between La3+ and A+ strongly favor ordering, so in the absence of other differences between these compounds and

NaLaMgWO6, ordering of the A-site cations is to be expected. The peak splitting in

KLaMgWO6 was very distinct, and so refinement of the XPD data was done in space group C2/m (12). The results of this refinement are shown in table 4.4. The lithium compound maintained a high degree of pseudo-symmetry, such that peak splitting and difference reflections gave no indication of octahedral tilting distortions. As was shown with NaLaMgTeO6 and NaLaScSbO6, these observations in XPD data can be misleading. So the refinement of the data in the aristotype space group P4/nmm should be taken as preliminary, and only indicative of the cell size and heavy metal positions.

110 Compound Tilt Angle Displacement (Å) M M'

NaLaMgWO6 15° 0.016 0.074

NaLaScNbO6 9° 0.013 0.012

NaLaMgTeO6 8° 0.019 0.018

NaLaScSbO6 10° 0 0

NaLaZr2O6 13° 0

NaLaTi2O6 6° 0

Table 4.3: Measurement of tilt and M-cation displacement from octahedral center.

111 KLaMgWO6 LiLaMgWO6 C2/m P4/nmm Rwp =0.146 Rwp =0.0637 a=7.9110(2) a= 5.523 b=7.8726(2) c= 7.892 c=7.9795(2) β=90.00(1)

Tablel 4.4: Pawley refinement of data for KLaMgWO6, LiLaMgWO6

112 4.5 Discussion

The ordering of the A-site cations in NaLaMgWO6 and NaLaScNbO6 is unusual in that it does not involve vacancies of either the cations or the anions. They are among the few examples of stoichiometric double perovskites that exhibit this type of ordering. It is unique too, because of the similar size of the Na+ and La3+ cations. One would expect ordering to be favored when both the charge and the size of the cations differ greatly, as with La3+ and vacancies. In fact, the ordering of the A-sites cations in

KLaMgWO6 and LiLaMgWO6 is more easily understood because of the greater difference in size of the cations. A clue to the stabilization of this ordering is in the displacement of the M-site cations. Transition metals with a d0 electron count have empty low lying d orbitals, which are close in energy to the oxygen 2p non-bonding orbitals. When the symmetry of the perovskite is reduced from cubic to tetragonal, as with the ordering of the A-sites, mixing of these orbitals can occur such that the oxygen 2p orbitals are stabilized and the empty metal nd orbitals are destabilized. This effect is known as the second order Jahn- Teller effect (SOJT). The greater the electronegativity of the metal, the lower the energy and the more easily it can mix with the Oxygen 2p orbitals. While the main group metals may be more electronegative than the transition metals, the nd orbitals are filled and lie below the Fermi level. Distortions that involve rehybridization of the d orbitals are not favored for these compounds.

Displacement of the M-cations is observed in WO3, which has a distorted perovskite structure without the A-site cations. This displacement is cooperative as the short bonds created by the displacement of the W6+ cause the relaxation of the W-O bond on the other side of the oxygen anion, which forces the next octahedra to have the W6+ to displace in the same direction. However, when the M-site is ordered, as in Ba2MgWO6, the cooperative displacement mechanism is disrupted and on average there is no displacement of the W6+ cation. So ordering of the A-site cations is another mechanism for inducing this type of displacement.

113 Why should the ordering of the A-site have an effect on the symmetry of the M- site? It is the bonding environment at the anions that plays a critical role here. As discussed above, there are 3 chemically different oxygen positions in perovskites with A- + 3+ site layering: within the Na layer (ONa), within the La layer (OLa), and between the A- 29 cation layers, equatorial to the M-sites (OM). Idealized bond valences can be calculated by assuming that each type of metal oxygen bond has only one bond valence. The bond valences are then calculated to give a BVS equal to the oxidation state for each metal. n+ That is, for M , sMO= n/N, where N is the coordination number of the metal: s(W-O) = 1, s(Mg-O) = 1/3, s(La-O) = ¼ and s(Na-O) = 1/12. The BVS for each oxygen anion is then calculated as follows:

BVS (ONa) = 4sNaO + sMgO + sWO = 4(1/12) +1/3 + 1 = 1 2/3

BVS (OLa) = 4sLaO + sMgO + sWO = 4(1/4) +1/3 + 1 = 2 1/3

BVS (OM) = 2sNaO + 2sLaO + sMgO + sWO = 2(1/12) +2(1/4) + 1/3 + 1 = 2 According to Pauling’s second rule,29 the BVS of oxygen should equal its oxidation state, 2. While this is the case for the oxygen between the A-cation layers, it is not for the oxygen ions within the A-cation layers. ONa is under-bonded while OLa is over-bonded. This is an inherent bond strain that exists even if quantitative bond valences, and no octahedral tilting system alone can relieve this bond strain.

If the OLa is over-bonded, this strain could be reduced by the M-cations shifting away from this oxygen, which would in turn cause a shift toward the ONa, which is under- bonded. The effect is quite cooperative and serves to reinforce the ordering of the A-site cations. As discussed above, this type of displacement is favored for d0 transition metals, such at W6+. So the ordering of the A-site cations and the SOJT displacement of the transition metal M-cations is synergistic. Of course, the effect could also be attributed to electrostatics as the more highly charged La3+ has a stronger repulsion for the M-site cations than Na+. If this were the primary cause of the cation displacement, we would expect to see this type of shift for the main group M’-cations as well. However, as discussed in the results section, there is little or no indication of a displacement for Te6+ or Sb5+ as compared to their transition metal counterparts, W6+ and Nb5+. In fact the M- cations, Mg2+ and Sc3+, show minimal displacement as well. Furthermore, though both

114 Ti4+ and Zr4+ are d0 transition metals, without the ordering of the A-site cations these cations do not displace either. It is unclear why the complete occupancy of the M-site is less favorable for stabilizing the layered ordering of the A-site cations, and further study on this topic is needed.

4.6 Conclusion

The structures of several NaLaMM’O6 compounds have been determined through a combination of XPD and NPD. The ordering of the A-cations in layers for stoichiometric perovskites has been shown to be dependant upon the presence of d0 transition metals at the M’-cation site. Such metals are susceptible to second order Jahn Teller displacements that compensate for the over- or under-bonding of the anions in the A-site layers. An ordered arrangement of these d0 transition metals with lower valent metals appears to be necessary to reinforce the ordering at the A-site. It is also concluded that the bonding environment at the anion site plays a critical role in the structure determination, and serves as an electrostatic driving force for the SOJT. As such, the electrostatics and the SOJT effect work cooperatively to produce both layered ordering of the A-site ions and displacements of the d0 transition metals.

4.7 References

1. Karen, P.; Kjekshus, A.; Huang, Q.; Karen, V. L.; Lynn, J. W.; Rosov, N.; Sora, I. N.; Santoro, A., Neutron powder diffraction study of nuclear and magnetic structures of oxidized and reduced YBa2Fe3O8+w. Journal of Solid State Chemistry 2003, 174, (1), 87-95.

2. Karen, P.; Suard, E.; Fauth, F., Crystal Structure of Stoichiometric YBa2Fe3O8. Inorganic Chemistry 2005, 44, (23), 8170-8172.

3. Woodward, P. M.; Karen, P., Mixed Valence in YBaFe2O5. Inorganic Chemistry 2003, 42, (4), 1121-1129.

4. Caignaert, V.; Millange, F.; Domenges, B.; Raveau, B.; Suard, E., A New Ordered Oxygen-Deficient Manganite Perovskite: LaBaMn2O5.5. Crystal and Magnetic Structure. Chemistry of Materials 1999, 11, (4), 930-938. 115 5. Millange, F.; Caignaert, V.; Domenges, B.; Raveau, B.; Suard, E., Order-Disorder Phenomena in New LaBaMn2O6-x CMR Perovskites. Crystal and Magnetic Structure. Chemistry of Materials 1998, 10, (7), 1974-1983.

6. Chakhmouradian, A. R.; Mitchell, R. H.; Burns, P. C., The A-site deficient ordered perovskite Th0.25.box.0.75NbO3: a re-investigation. Journal of Alloys and Compounds 2000, 307, (1-2), 149-156.

7. Kennedy, B. J.; Howard, C. J.; Kubota, Y.; Kato, K., Phase transition behavior in the A-site deficient perovskite oxide La1/3NbO3. Journal of Solid State Chemistry 2004, 177, (12), 4552-4556.

8. Sefat, A. S.; Amow, G.; Wu, M.-Y.; Botton, G. A.; Greedan, J. E., High-resolution EELS study of the vacancy-doped metal/insulator system, Nd1- xTiO3, x = 0 to 0.33. Journal of Solid State Chemistry 2005, 178, (4), 1008-1016.

9. Belous, A. G., Lithium ion conductors based on the perovskite La2/3-xLi3xTiO3. Journal of the European Ceramic Society 2001, 21, 1797-1800.

10. Harada, Y.; Ishigaki, T.; Kawai, H.; Kuwano, J., Lithium ion conductivity of polycrystalline perovskite La0.67-xLi3xTiO3 with ordered and disordered arrangements of the A-site ions. Solid State Ionics 1998, 108, (1-4), 407-413.

11. Rivera, A.; Leon, C.; Santamaria, J.; Varex, A.; Paris, M. A.; Sanz, J., Li3xLa(2/3)- xTiO3 fast ionic conductors. Correlation between lithium mobility and structure. Journal of Non-Crystalline Solids 2002, 307-310, 992-998.

12. Garcia-Martin, S.; Alario-Franco, M. A.; Ehrenberg, H.; Rodriguez-Carvajal, J.; Amador, U., Crystal Structure and Microstructure of Some La2/3-xLi3xTiO3 Oxides: An Example of the Complementary Use of Electron Diffraction and Microscopy and Synchrotron X-ray Diffraction To Study Complex Materials. Journal of the American Chemical Society 2004, 126, (11), 3587-3596.

13. Khalyavin, D. D.; Senos, A. M. R.; Mantas, P. Q., Crystal structure of La4Mg3W3O18 layered oxide. Journal of Physics: Condensed Matter 2005, 17, (17), 2585-2595.

14. Dupont, L.; Chai, L.; Davies, P. K., A- and B-site order in (Na1/2La1/2)(Mg1/3Ta2/3)O3 perovskites. Materials Research Society Symposium Proceedings 1999, 547, (Solid-State Chemistry of Inorganic Materials II), 93-98.

15. Sekiya, T.; Yamamoto, T.; Torii, Y., Cation ordering in sodium lanthanum magnesium tungstate (NaLa)(MgW)O6) with the perovskite structure. Bulletin of the Chemical Society of Japan 1984, 57, (7), 1859-62.

116 16. Belous, A. G.; Novitskaya, G. N.; Gavrilova, L. G.; Polyanetskaya, S. V.; Makarova, Z. Y., Lanthanum Titanate-Zirconates with the Perovskite Structure. Soviet Progress in Chemistry 1985, 51, (1), 13.

17. Glazer, A. M., Simple ways of determining perovskite structures. Acta Crystallographica, Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography 1975, A31, (6), 756-62.

18. Woodward, P. M. Structural distortions, phase transitions, and cation ordering in the perovskite and tungsten trioxide structures. Oregon State University, Corvallis, Oregon, 1997.

19. Barnes, P. W. Exploring structural changes and distortions in quaternary perovskites and defect pyrochlores using powder diffraction techniques. The Ohio State University, Columbus, Ohio, 2003.

20. Davies, P. K., Cation ordering in complex oxides. Current Opinion in Solid State & Materials Science 1999, 4, (5), 467-471.

21. Lufaso, M. W.; Woodward, P. M., Prediction of the cyrstal structures of perovskites using the software program SPuDS. Acta Crystallographica 2001, B57, 725-738.

22. Stokes, H. T.; Hatch, D. M. ISOTROPY, 2002.

23. Howard, C. J.; Stokes, H. T., Group-Theoretical Analysis of Octahedral Tilting in Perovskites. Acta Crystallographica 1998, B54, 782-789.

24. Young, R. A.; Editor, The Rietveld Method. [In: Int. Union Crystallogr. Monogr. Crystallogr., 1993; 5]. Oxford: London, 1993; p 298 pp.

25. Cheary, R. W.; Coelho, A., A fundamental parameters approach to x-ray line-profile fitting. Journal of Applied Crystallography 1992, 25, (2), 109-21.

26. Larson, A. C.; Dreele, R. B. V. General Structure Analysis System (GSAS); 2000; pp 86-748.

27. Toby, B. H., EXPGUI, a graphical user interface for GSAS. Journal of Applied Crystallography 2001, 34, (2), 210-213.

28. Arillo, M. A.; Gomez, J.; Lopez, M. L.; Pico, C.; Veiga, M. L., Structural and electrical characterization of new materials with perovskite structure. Solid State Ionics 1997, 95, 241-248.

29. Pauling, L., The principles determining the structure of complex ionic crystals. Journal of the American Chemical Society 1929, 51, 1010-26.

117

CHAPTER 5

STRUCTURAL STUDIES OF NaBiMM'O6 STRUCTURES AND THE INFLUENCE OF THE STEREO-ACTIVE LONE PAIR ON ORDERING AND THE STABILITY OF THE PEROVSKITE STRUCTURE

5.1 Introduction In this study, an attempt was made to prepare perovskites with analogous composition to those in Chapter 4, with the lanthanum replaced by bismuth. Lanthanum and bismuth both have +3 oxidation states, and are of approximately the same radius (1.3Å in 8 fold coordination). However, bismuth has two valence electrons, making the outer shell more polarizable than that of lanthanum. As the A-cation in perovskites is not thought to bond very covalently, and serves primarily as charge balance, such a substitution, in theory, should not alter the structure greatly. As such, if layered ordering of the A-cations could be achieved with a highly polarizable cation, it is possible that very high dielectric constants could be reached. Ferroelectricity, however, would depend on the atomic shifts occurring in such a way as to remove any real or pseudo inversion centers. When the symmetry of a crystal contains an inversion center, displacement of an ion in one direction is balanced by displacement of the same type of ion in the opposite direction elsewhere in the cell. As such, net polarization is not possible. A pseudo inversion center would mimic this affect and allow only a very small polarization. In this final chapter, some of the dielectric properties of the various materials studied here are examined. Beyond the knowledge of chemical trends that enhances our understanding of how we can predict the structures of materials, the dielectric properties are what make these materials interesting on a practical level. Recently, bismuth-zinc- niobium metal oxides have been studied as possible microwave dielectrics.1, 2 118 Bi1.5ZnNb1.5O7 (BZN) has the pyrochlore structure in which bismuth and half the zinc atoms occupy the A-site and niobium and the remainder of the zinc atoms occupy the octahedral B-sites. It has been shown that the true composition of BZN may be a defect structure with 92% occupancy of the zinc atoms. The dielectric constant of this material has been reported to be 143 in the bulk3 and 190 for thin films.2

5.2 Introduction to dielectrics

Impedance measurements are used to determine the dielectric constant of a material. A single frequency voltage is applied to the material of interest and the phase shift and amplitude of the resulting current are measured. The complex permittivity is expressed as ε = (ε' + iε''), where ε' and ε'' are the real and imaginary components, respectively. The imaginary part of the permittivity is a measure of the AC signal energy loss as it passes through the material. The dielectric constant of a material is a measure of its relative capacitance:

κ = C p Cv , where Cp is the capacitance of the material and Cv is the capacitance of a

-12 vacuum of the same dimensions. The permittivity of free space, ε 0 , is 8.854 x 10 F/m, and is related to the capacitance of a volume by its area and thickness: ε A C = 0 (equation 5.1) v d where A is the area, and d is the separation of the electrodes (thickness). Note the role these dimensions play in determining the capacitance of a material. Smaller dielectric constants can still be useful in microwave dielectric materials. The loss of a material, also known as the dissipation factor, results both from intrinsic structural features of a material,4 as well as extrinsic properties such as porosity, grain boundaries, defects and vacancies, and crystallite orientation. For sintered pellets, as those studied here, these factors dominate, so the sintering conditions play an important roll in the measured loss of a material. A correction for porosity (P = 1-D, where D is the percent of the theoretical density achieved in the sample) to the measured value of ε can be made for samples near or greater than 80% density by the Bruggman effective medium.5, 6 119 The quality factor, Q, is a measure of the dielectric loss. In the sub-microwave frequency region, Q is equal to 1/tan δ where δ is the angle between the real and imaginary components. In the microwave region, there is typically a peak in the signal amplitude, and the quality factor is a measure of the width of this peak at the resonant frequency. A narrow peak (high Q), leads to high selectivity in microwave dielectric materials. The dielectric constant of a material may vary with temperature. It is preferable that this variation be minimal for materials used in practical applications. The variability is measured through the temperature coefficients of resonant frequency (τf), the dielectric constant (τε), or capacitance (τc). The coefficient of capacitance, τc, can be measured by: 1 δε τ = (equation 5.2) c ε δT The linear slope of the graph of the dielectric constant versus the temperature in kelvin was used for this equation. This coefficient is related to the coefficient of the dielectric 7 constant by the coefficient of linear expansion of a material, αL. For perovskites, this was typically found to be between 8 and 20 ppm/K. High temperature studies of some of the materials studied here (NaLaMgWO6 and NaBiScSbO6) found that 10 ppm/K was a reasonable estimate of this factor.

τ ε = τ c −α L (equation 5.3) The temperature coefficient of frequency is then related by:

⎛τ ε ⎞ τ f = −⎜ + α L ⎟ (equation 5.4) ⎝ 2 ⎠

For microwave dielectrics, τf needs to be minimal, typically less than 50 ppm/K.

5.3 Experimental

5.3.1 Synthesis

The compositions NaBiMM’O6 (M= Mg, Sc, Ga, M’ = W, Te, Nb, Sb) were attempted. In the case of NaBiMgWO6, this composition could not be prepared by standard solid state synthesis, and resulted in a mixture of oxides. Compounds were prepared from Na2CO3 and the relevant metal oxides. In the case of NaBiScSbO6,

120 ScSbO4 was prepared as a precursor due to the difficulty of getting scandium oxide to react without volatilizing the antimony. This was prepared from ScO3 (Boulder) and

SbO3 (Cerac) with excess antimony oxide being added and heating until the XPD pattern was unchanged and showed no signs of impurity. Other metal oxides used were Ga2O3

(Alpha Aesar), Nb2O5 (Aldrich), MgO (Allied), and WO3 (Cerac). All of the samples appeared to be prone to decomposition, either when exposed more than briefly to ambient conditions or after prolonged heating. As such, it was difficult to maintain pure samples for analysis. Stoichiometric amounts of the metal oxides and a 10% excess of Na2CO3 were ground intimately with acetone in an agate mortar with pestle, and the contents placed in a covered high form alumina crucible.

Due to the low melting point of Bi2O3, the samples were preheated between 500 and 700°C for 3-6 hours for initial reaction. Samples were then heated at higher temperatures until the XPD diffraction patterns did not change, with the final annealing temperature between 950 and 1100°C. In the case of NaBiScSbO6, the pattern was quite complex, and high temperature XPD patterns were taken to understand the phases produced. This revealed that the composition was of two dominant phases, a perovskite phase and a pyrochlore phase. Different heating cycles were needed to produce either phase. 5.3.2 Structural Characterization Structural characterization of the bismuth compounds was done using powder diffraction techniques. Due to the difficulty in producing pure samples, samples of sizeable enough quantities for NPD were not obtained and XPD was the sole means of structural characterization. A detailed description of XPD is given in chapter 4. XPD data for these compounds were collected in Bragg-Brentano geometry from 10-120° 2θ and a step size ~0.0144° using a Bruker D8 X-ray powder diffractometer (40 kV-50 mA, sealed Cu X-ray tube, λ=1.54056 Å) equipped with an incident beam Ge 111 monochromator and a Braun linear position sensitive detector. Pawley refinements were completed using the TOPAS software package. 5.3.3 Dielectric Measurements After verification of purity by XPD, samples were ground and pelletized into half inch pellets with a stainless steel uniaxial press (Carver Laboratory Press, Model C) using

121 approximately 0.5-1 gram of material for each pellet. Three metric tons of pressure were used for 20-60 seconds. Pellets were returned to the alumina crucible and covered with muffling powder of the same material. The pellets were then heated at a temperature 50- 100° higher than the final annealing temperature. XPD patterns of the muffling powder and one pellet were used to verify consistent structure, and the remaining pellets were measured for mass, diameter, and thickness. The thickness, radius, and mass of the pellets were used to determine the density of the samples. Theoretical density was calculated using the cell parameters from the Pawley fits and the following equation, where Z is the number of formula units per cell,

FW is the formula weight, NA is Avagadro’s number and Vu is the unit cell volume. Z ⋅ FW ρ = (equation 5.6) N A ⋅Vu Dielectric measurements were taken of at least one pellet for each sample, with a preference for the thinnest and/or most dense pellet. Pellets were coated with a circle of InGa paint of diameter 0.7 mm, centered on both flat surfaces of the pellets. A Solartron SI 1260 impedance/gain-phase analyzer was used for measurements in the sub- microwave frequency range of 10 to 107 Hz. A parallel plate capacitor arrangement with platinum contacts on opposing sides of the pellet was used. A sample holder was fashioned from parallel plates held together by spring tension. Alligator clips were soldered to the BNC connectors of the Solartron and clipped to wire leads from the electrodes. The pellets secured in the sample holder were placed, in turn, in a large test tube along with a standard mercury thermometer in an oil bath. The oil bath was used to heat the pellets above 100°C. The temperature was recorded at several intervals and the impedance measurements made at these temperatures.

5.4 Results

5.4.1 Comparison of NaLaScNbO6 and NaBiScNbO6

Two preparations of NaBiScNbO6 (NBSNO) were used in this study. In the first preparation, 135b, the sample was annealed at 950°C, pelletized, and then sintered at

122

Figure 5.1: XPD pattern for NaBiScNbO6. The inset shows a close-up of the 10-40° range. Note the lack of ordering peaks at 11° and 19°. Peak splitting indicates symmetry reduction due to octahedral tilting.

123 1050°C. In the second, the sample was annealed at 800°C, washed with concentrated nitric acid, and annealed again at 900°C. The sample was then pelletized and heated to

1000°C. Before prilling, a 10% excess of Na2CO3 was added to the sample to aid in the formation of larger crystallites and sintering of the pellets. Like NaLaScNbO6,

NaBiScNbO6 clearly formed a perovskite structure. However, the ordering of the A-and B-cations clearly present in the lanthanum structure was absent (See figure 5.1). With no ordering of the cations, NBSNO forms a disordered perovskite, with cell edge 7.963Å. The structure was refined in space group Pnma, tilt system a −b + a − . Details of the structure are shown in table 5.1. Dielectric measurements were taken of both NLSNO and NBSNO. The NLSNO pellet had a density 56% of the theoretical density, while the NBSNO pellet was 73% dense. Low density of the pellets leads to much greater loss and makes it difficult to determine if the lower dielectric constant is inherent to the material or a result of the porosity of the pellet. For reliable measurements of the dielectric constants of materials, pellets of at least 90% density are required. As such these results can only be compared qualitatively. The dielectric constants of these materials are reported in table 5.2.

5.4.2 Comparison of NaLaScSbO6 and NaBiScSbO6

When prepared as the other compounds in this chapter were, NaBiScSbO6 (NBSSO) formed both a perovskite and a pyrochlore phase. XPD patterns of the sample were taken from 500°C to 1200°C. It was seen that at 1000°C the pyrochlore phase was almost completely absent. However, when the compound was heated further, the perovskite phase disappeared and the pyrochlore phase was dominant with a small impurity believed to be a scandium rich ternary oxide (Sc5.6Sb1.4O11.9). As such, an attempt was made to make each phase in isolation in order to determne the properties of the individual phases and to compare the two. It was not possible to prepare the pyrochlore phase without the impurity. Under normal conditions, the lanthanum analog of this compound only forms the perovskite phase. However, when heated to extreme conditions (in excess of 1500°C) there is evidence of both a perovskite and pyrochlore phase of the lanthanum composition as well. Structures for both phases of NaBiScSbO6 are shown in Table 5.1. As with

124

Compound Space Group Lattice Parameters (Å)

NaBiScNbO6 Pnma 5.631 7.963 5.615 NaBiScSbO6 Fm3m 7.9504 NaBiScSbO6 Fd3m 10.45 NaBiGaSbO6 Fd3m 10.33

Table 5.1: Pawley fit lattice parameters for bismuth compounds

Compound Sample % κ D κ κ Τcf density norm Brugg man NaBiScNbO6 171a 73 130 0.1 178 213 -1300 NaLaScNbO6 171b 59 33 0.03 56 58 NaBiScSbO6 175a 80 94 0.1 117 186 60 NBSSO4 82 109 0.2 132 156 -5000 NaLaScSbO6 123a 89 47 0.07 53 80 -260 NaBiGaSbO6 135a 98 45 0.04 46 80 -120

Table 5.2: Dielectric data for analogous lanthanum and bismuth perovskites, as well as NaBiGaSbO6. κ normalized was calculated by dividing κ by the % density.

125

Figure 5.2: XPD Pattern for NaBiScSbO6 in the perovskite phase. The inset shows a close-up of the 10-40° range. The strong peak at 19° is an indication of B-cation ordering.

126

Figure 5.3: XPD patterns of NaBiScSbO6 taken while heating. The final pattern at 1200°C was consistent with the pattern when the sample was returned to room temperature.

127 NaLaScSbO6, the perovskite phase does not exhibit any ordering of the A-cations (see figure 5.2). The dielectric constant of the perovskite phase of NBSSO was measured for two different samples. These are reported in Table 5.2. It was not possible to remove the scandium rich impurity from the pyrochlore phase.

5.4.3 Other NaBiMM’O6 structures

An attempt was made to prepare NaBiMgWO6 (NBMWO), using the methods described in the experimental section. However, no pure phases were formed, only a mixture of unidentifiable metal oxides. As such, no dielectric measurements were made.

NaBiGaSbO6 (NBGSO) formed only a pyrochlore phase. The XPD pattern is shown in figure 5.4. The sample was refined in space group Fd3m , and the a lattice constant found to be 10.33Å. The pellets were extremely dense, and a pellet of 98% density was used for the dielectric measurements. While the dielectric constant for this sample was quite low in comparison to the other bismuth oxides, the temperature coefficient was also quite small. 5.4.4 Dielectric measurements A summary of the dielectric measurements made on the materials studied here is shown in Table 5.2. The relative permittivity at 106 Hz is shown, along with the value of

τf. Also listed is the porosity of the material and the dielectric constant corrected for the porosity, both by the crude estimation of dividing by the theoretical density and by the Bruggeman effective medium formulation.

5.5 Discussion

5.5.1 Relative Stability of Pyrochlore and Perovskite Structures

The stoichiometries of the perovskite and pyrochlore structures (AMX3 and

A2B2X7 respectively), do not seem to allow them to be competing structures. However, as described in chapter 1, one of the anions in the pyrochlore composition is chemically distinct from the other six. There are numerous examples of defect pyrochlores in which this X' anion is absent, giving both structures the same stoichiometry. A comparison of the two structures has been made in detail.8 Theoretical calculations of the electrostatic energy of pyrochlores were made based on the free positional parameter, x, of the anion

128

Figure 5.4: XPD pattern for NaBiGaSbO6.

129 site. For a value of x = 0.3125, the B-site is perfectly octahedral. An increase in this value leads to a compressed trigonal antiprism. The value of the x parameter can be influenced by the nature of the A- and B-cations. For example, when the A-cation is a d10 transition metal, the value of x tends to be low, near or below 0.3125. Also, for a given A-cation, the value of x increases with the radius of the B-cation. As the x-parameter changes, the stability of the pyrochlore structure changes. For pyrochlores in which the A and B-cations are of different charges (2+/4+ or 1+/5+), the pyrochlore structure is destabilized. The stability of the perovskite structure is similar to that of the pyrochlore structure over these ranges. When the charge difference is small, the pyrochlore structure

1+ 5+ is more stable at higher values of x. However, for A2 B2 O6 pyrochlores, the perovskite structure is always more stable. The analysis done by Subramanian et al.8 does not include calculations of pyrochlores with multiple A- and B-cations. The average charges for the A- and B- cations for the compounds discussed here would be 2+ and 4+ respectively. Using this information, stabilization of the perovskite structure would be predicted for low values of x, while the pyrochlore structure would be predicted for higher values of x. Thus the nature of the A-cation and the size of the B-cations are important factors in the stability of the phases considered here. It would appear from the compounds studied here, that the electronic structure of the B-cations is of critical importance. It is clear that when both B-cations are transition metals, as with NBSNO and NBMWO, the pyrochlore phase is not preferred. In the pyrochlore structure, the M-O-M' bond angle is 120-140° (check this), whereas in the perovskite structure, the bond angle is 180° for an undistorted structure, and near this for distorted ones. The oxygen anion has no d-orbitals available for bonding. The 2p orbitals are oriented orthogonally to one another. This means that it is not possible for the oxygen to form strong sigma bonds with two metals at 180°, because only one p- orbital would need to be involved in two different bonds. The main group metals, such as Ga and Sb, also only have p-orbitals for bonding, so it is preferred for the M-O-M bond angle to be significantly less than 180°, as this will allow two different Op orbitals to be involved in σ bonding. In contrast, transition metals with empty d-orbitals can

130 π-bond to the oxygen anions, allowing the anion to form both a σ and a π-bond with the two neighboring metal atoms. As such, M-O-M' bond angles close to 180° are favored for maximum overlap of the Op and Md-orbitals. This is why NaBiScNbO6 is found only in the perovskite structure, while NaBiScSbO6 and NaBiGaSbO6 both have the pyrochlore structure.

Why then does NaLaScSbO6 have primarily the perovskite structure while

NaBiScSbO6 forms both stable perovskite and pyrochlore structures? As discussed in chapter 4, the perovskite structure can be stabilized by ordering of the M and M'-cations, such that each oxygen anion is bonded to at least one transition metal. This is the case with the perovskite form of NaBiScSbO6. It is also likely that the pyrochlore phase of this compound is deficient in scandium, as evidenced by the scandium rich impurity. But this pyrochlore phase forms much more readily than in the lanthanum compound. It is likely here that the lone pair on the bismuth ion is stabilizing the pyrochlore structure. In the defect pyrochlore structure, the O' anions at the 8b site are missing. In a stoichiometric pyrochlore, these anions have a relatively short bond length to the A- cations and sit between these cations. Their removal exposes the highly charged cations to one another, which is electronically unfavored and destabilizes the structure, as in the case of NaLaScSbO6. In the case of a lone pair cation, however, the lone pairs on neighboring A-cations can interact forming a band state. Electron density from the 6s orbitals is found at the vacant 8b site.8, 9 5.5.2 Comparison of Dielectric constants The dielectric constant of a material can be extremely dependent on the sample preparation. Porosity of the material, grain boundaries, and ionic conductivity can all factor into the decrease of the dielectric constant or the increase of dielectric loss in the material. It is for this reason that samples of high density, greater than 90%, are needed for reliable quantitative measurements. Such materials will have low porosity with the grains grown together. In this study, it was not possible for pellets to be prepared at this density in most cases. However, qualitative comparison of the materials can lend some insight into how different aspects of structure and composition can affect the dielectric constants of materials.

131 5.5.2.1 Effect of ns2 cation on dielectric constant Looking at table 5.2 it is clear that substituting Bi3+ for La3+ has a significant effect on the dielectric constant of perovskites. The lanthanum structures have values of κ that are barely above a background reading. Meanwhile, the bismuth compounds have κ values near or above 100, before any corrections for porosity. The lone pair ion, Bi3+, has a polarizable valence shell that clearly contributes to the overall polarizability of this structure. Similar effects have been observed for compounds containing lead, which also has 6s2 electrons. Lead, however, is not desirable as a component in commercially made materials due to its toxicity. Bi3+ has a higher charge which makes it difficult to incorporate into the A-site of perovskites without compensating with a complimentary lower-valent A-cation like Na+. It should also be noted that the bismuth compounds have a much greater frequency dependence than do the lanthanum compounds (see Figure 5.5). At lower frequency, the ions have a longer amount of time to shift positions before the electric field is reversed. An increase in polarization at lower frequencies can be an indication of ionic conductivity. 5.5.2.2 Effect of d0 cation on dielectric constant As discussed in chapter 4, the d0 transition metal cations are susceptible to second order Jahn-Teller distortions, allowing them to displace from high symmetry positions in order to lower the energy of oxygen 2p bonding orbitals. Main group cations have no such driving force for displacement. It should come as no surprise, then, that the compounds containing the transition metal niobium had greater dielectric constants than those calculated for the antimony containing compounds. In fact the greatest value of κ was for the compound containing both bismuth and niobium: NaBiScNbO6. 5.5.2.3 Effect of structure on dielectric constant The pyrochlore structure is cubic and has the symmetry of the space group Fd3m (#227). The B-cations are octahedrally coordinated, and the octahedra form corner-sharing tetrahedral clusters. This extended network has channels through which the A-O network weaves. Due to the diamond glide, the pyrochlore structure does not inherently lend itself to ferroelectric behavior. Simple ordering patterns of either the

132

Figure 5.5: Frequency dependence of the dielectric constant κ for four compounds.

133 A- or the B- site, such as layering or the rock salt type ordering seen in perovskites, are not possible. Almost all methods of displacing the A-cations, as would be observed in a ferroelectric material, maintain the inversion center of the structure.

5.6 Conclusion

While the work in this chapter is preliminary and qualitative, some important observations have been made. Substitution of different types of cations within a material, even when it maintains the same basic structure, can have a profound impact on the material properties of the bulk compound. As was seen in Chapter 4, inclusion of d0 transition metal ions will allow for cationic displacements that can enhance the polarizablity, and thus the dielectric properties of a material. In this case, NaA'ScM'O6 perovskites containing Nb5+ as the M'-cation had higher dielectric constants than those containing Sb5+. Additionally, substituting a cation with 6s2 electrons, which are polarizable, for a cation with an empty valence shell also enhanced the dielectric properties of the material. That is, NaBiScM'O6 compounds had higher dielectric constants than NaLaScM'O6 compounds. Though the preliminary measurements indicating high losses and frequency dependences of these materials make them unsuitable as practical dielectric materials, the principles illustrated here could be applied in the development of other materials.

5.7 Chapter 5 References

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2. Tagantsev, A. K.; Lu, J.; Stemmer, S., Temperature dependence of the dielectric tunability of pyrochlore bismuth zinc niobate thin films. Applied Physics Letters 2005, 86, (3), 032901/1-032901/3.

3. Cann, D. P.; Randall, C. A.; Shrout, T. R., Investigation of the dielectric properties of bismuth pyrochlores. Solid State Communications 1996, 100, (7), 529-534.

4. Penn, S. J.; Alford, N. M.; Templeton, A.; Wang, X.; Xu, M.; Reece, M.; Schrapel, K., Effect of porosity and grain size on the microwave dielectric properties of sintered alumina. Journal of the American Ceramic Society 1997, 80, (7), 1885-1888. 134 5. Geyer, R. G.; Mantese, J.; Baker-Jarvis, J., NIST Tech. Note 1994, 1371.

6. Geyer, R. G.; Baker-Jarvis, J.; Vanderah, T. A.; Mantese, J., Complex permittivity and permeability estimation of composite electroceramics. Ceramic Transactions 1998, 88, (Advances in Dielectric Ceramic Materials), 115-128.

7. Colla, E. L.; Reaney, I. M.; Setter, N., Effect of structural changes in complex perovskites on the temperature coefficient of the relative permittivity. Journal of Applied Physics 1993, 74, (5), 3414-25.

8. Subramanian, M. A.; Aravamudan, G.; Rao, G. V. S., Oxide pyrochlores - a review. Progress in Solid State Chemistry 1983, 15, (2), 55-143.

9. Longo, J. M.; Raccah, P. M.; Kafalas, J. A.; Pierce, J. W., Preparation and structure of a pyrochlore and perovskite in the BiRhO3+x system. Materials Research Bulletin 1972, 7, (2), 137-46.

135

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151

APPENDIX ADDITIONAL TABLES

This Appendix contains tables with information on additional group theoretical analysis.

+ Table A1 contains the sub-groups based on irreps N1 and P4. These irreps cause super-anti-parallel tilting distortions along the a- and/or b-axis. These distortions are

+ in-phase from layer to layer in for subgroups of irrep N1 , and out-of-phase for subgroups of irrep P4. Table A2 shows the sub-groups resulting from combinations of simple tilts of n = 1 DJ structures. Simple tilts are those that occur about only one axis and can be either in-phase or out-of-phase, parallel or anti-parallel. Table A.3 shows the sub-groups resulting from the distortions of DJ structures with n > 1. The table shows even layering as odd layering will resemble the n = 1 case.

152

Tilt System Space Unit-cell Irrep Group size (Direction Vectors) + 0 0 C2/m a≈√2a + as b c p N1 (12) b≈ c0 (a,0,0,a) c≈√2ap β≈90° + + 0 I4/mmm a≈2a + as a0 c p N1 (139) c≈2c0 (a,a,0,0)

+ + 0 I4 /amd a≈2a + asa a0 c 1 p N1 (141) c≈2c0 (a,0,a,0)

+ + 0 A2/m a≈√2a + as as c p N1 (12) b≈2√2ap (a,a,a,a) c≈2c0 β ≈90 − 0 0 Fddd a≈2a P as b c p 4 (70) b≈2ap (a,a,0,0) c≈2c0

− − 0 Ibam a≈2c P as as c 0 4 (72) b≈√2ap (a,a,a,-a) c≈√2ap − − 0 a≈√2a P asa asa c I 42d p 4 (122) b≈√2ap (a,0,0,0) c≈2c0

Table A1: Possible super-anti-parallel octahedral tilting distortions for n = 1 RP structures.

153 Tilt System Space Group Unit-cell Irrep size (Direction Vector) a+ b+ c Pmmn a≈2a + p p p p M 2 (59) b≈2ap (c) c≈c − 0 X 2 (a,b) a- b0c C2/m a≈2a + p p p M 2 (12) b≈2ap (c) c≈c + 0 M γ≈90° 5 (a,a) a- a- c P2 /c a≈c + p p p 1 0 M 2 (14) b≈√2ap (c) c≈√2a + p M γ≈90° 5 (a,0) a- b- c P-1 a≈c + p p p 0 M 2 (2) b≈√2ap (c) c≈√2a + p M α, β, γ≈90° 5 (a,b) a- b+ c0 Pbcm a≈c + p p 0 M 5 (57) b≈2ap (a,a) c≈2a p X − 2 (b,b) a+ b+ c Immm a≈ 2a + a a p p M 2 (71) b≈ 2c0 (c) c≈ 2a + p R 1 (a,b) a- b0c Cmcm a≈ 2a + a p p M 2 (63) b≈ 2ap (c) c≈ 2c − 0 A 5 (a,a) Table A2: Possible tilt systems for n = 1 DJ structures resulting from combinations of irreps with simple tilts. (continued)

154 Table A2 continued (continued)

Tilt System Space Group Unit-cell Irrep size (Direction Vector) a- a- c Pnma a≈ √2a + a a p p M 2 (62) b≈ 2c0 (c) c≈ √2a − p A 5 (a,0) a- b- c P2 /m a≈ √2a + a a p 1 p M 2 (11) b≈ 2c0 (c) c≈ √2a − p A β ≈ 90° 5 (a,b) a+ b0c Cmcm a≈ 2a + a a p R1 (63) b≈ 2c0 (a,a) c≈ 2a − p A 4 (c) a+ a+ c P4 /nmc a≈ 2a + a a a 2 p R1 (137) b≈ 2ap (a,b) c≈ 2c − 0 A 4 (c) a+ b+ c Pmmn a≈ 2a + a a a p R1 (59) b≈ 2ap (a,b) c≈ 2c − 0 A 4 (c) a- b0c C2/m a ≈ 2a − a a p A5 (12) b ≈ 2ap (a,a) c ≈ C 0 A− β ≈ 90 4 (c) a- a- c I2/a a ≈√2a − a a a p A5 (15) b ≈ √2ap (a,0) c ≈ c 0 A− β ≈ 4 (c)

155 Table A2 continued (continued)

Tilt System Space Group Unit-cell Irrep size (Direction Vector) a- b- c P-1 a ≈ * − a a a A5 (2) b ≈ √2ap (a,b) c ≈ √2a p A− β ≈ 4 (c) a+ b0c Cmcm a≈ 2a − p a p X 2 (63) b≈ 2c0 (a,a) c≈ 2a − p A 4 (c) a+ a+ c P4 /nmc a≈ 2a − p p a 2 p X 2 (137) b≈ 2ap (a,0) c≈ 2c − 0 A 4 (c) a+ b+ c Pmmn a≈ 2a − p p a p X 2 (59) b≈ 2ap (a,b) c≈ 2c − 0 A 4 (c) a- b0c Cmca a≈ 2a + p a p M 5 b≈ 2ap (a,a) c≈ 2c 0 A− 4 (c) a- a- c Pbcn a ≈ √2a + p p a p M 5 (60) b ≈ √2ap (a,0) c ≈ 2c 0 A− 4 (c) a- b- c P2 /c a ≈ √2a + p p a 1 p M 5 (14) b ≈ 2c0 (a,b) c ≈ 2a p A− β ≈ 90° 4 (c)

156 Table A2 continued (continued)

Tilt System Space Group Unit-cell Irrep size (Direction Vector) a+ b+ c0 Cmcm a≈ 2a − p a p X 2 (63) b≈ 2c0 (a,a) c≈ 2a + p R 1 (b,b) a+ b+ c Cmcm a≈ 2a − p a a p X 2 (63) b≈ 2c0 (a,a) c≈ 2a + p R 1 (b,b) − A4 (c) a- b- c0 Ccca a≈ 2a + p a p M 5 (68) b≈ 2ap (a,a) c≈ 2c 0 A− 5 (b,b) a- b- c C2/c a≈ 2a + p a p p M 5 (15) b≈ 2ap (a,a) c≈ 2c 0 A− β ≈ 90° 5 (b,b) a- b- c C2/c a≈ 2a + p a a p M 5 (15) b≈ 2ap (a,a) c≈ 2c 0 A− β ≈ 90° 4 (c) a- b+ c0 Cmca a≈ 2a − a p p A5 (64) b≈ 2c0 (a,a) c≈ 2a p X − 2 (b,b) a- b+ c C2/m a≈ 2c − a p a 0 A5 (12) b≈ 2ap (a,a) c≈ 2a p X − β ≈ 90° 2 (b,b) − A4 (c)

157 Table A2 continued

Tilt System Space Group Unit-cell Irrep size (Direction Vector) a- b+ c0 Ibam a≈ 2c + p a 0 M 5 (72) b≈ 2ap (a,a) c≈ 2a p R + 1 (b,b) a- b+ c Pbcm a≈ 2a + p a a p M 5 (57) b≈ 2c0 (a,a) c≈ 2a p R + 1 (b,b) − A4 (c) a- b+ c0 Cmcm a ≈ 2√2a − a a p A5 (63) b ≈ 2√2ap (a,a) c ≈ 2c 0 R + 1 (b,b)

a- b+ c P2 /m a ≈ √2a − a a a 1 p A5 (11) b ≈ 2c0 (a,a) c ≈ √2a p R + β ≈ 90° 1 (b,b) − A4 (c) a- b+ c Pnma a≈ 2a + p p a p M 5 (62) b≈ 2ap (a,a) c≈ 2c 0 X − 2 (b,b) − A4 (c)

158

Tilt Space Unit Irrep System Group Cell size 0 0 0 a a c I4/mmm a = a0 (139) c = c0 a0a0c Cmca a ≈ c + p 0 X 2 (64) b ≈ √2a0 c ≈ √2a0 a0a0c P4/mbm a ≈ √2a + 0 0 X 2 (127) c ≈ c0 a0a0c Ccca a ≈ c − a 0 X 1 (68) b ≈ √2a0 c ≈ √2a0 a0a0c P4/nbm a ≈ √2a − 0 0 X 1 (125) c ≈ c0 a a c0 P4 /mnm a ≈ √2a − p 0 2 0 X 3 (136) c ≈ c0 a a c0 Cmcm a ≈ c − p p 0 X 3 (63) b ≈ √2a0 c ≈ √2a0 a b c0 Pnnm a ≈ √2a − p p 0 X 3 (58) b ≈ √2a0 c ≈ c0 a a c0 P4 /mcm a ≈ √2a − p 0 2 0 X 4 (132) c ≈ c0 a a c0 Cmma a ≈ c − a a 0 X 4 (67) b ≈ √2a0 c ≈ √2a0 a b c0 Pccm a ≈ √2a − a a 0 X 4 (49) b ≈ √2a0 c ≈ c0 a a c Amm2 a ≈ c − p 0 0 0 X 3 (38) b ≈ 2a0 + X 2 c ≈ 2a0 a a c+ Cmc2 a ≈ c − p p p 1 0 X 3 (36) b ≈ √2a0 + X 2 c ≈ √2a0 a a c+ Pnma a ≈ √2a − p p a 0 X 3 (62) b ≈ c0 + X 2 c ≈ √2a0

Table A3: Possible tilting distortions for n > 1 DJ structures (continued)

159 Table A3(cont’d)

Tilt Space Unit Irrep System Group Cell size a b c Pmn2 a ≈ c − p p p 1 0 X 3 (31) b ≈ √2a0 + X 2 c ≈ √2a0 a a c Amm2 a ≈ c − a 0 0 0 X 4 (38) b ≈ 2a0 + X 2 c ≈ 2a0 a a c Abm2 a ≈ √2a − a a a 0 X 4 (39) b ≈ c0 + X 2 c ≈ √2a0 a a c Pbcm a ≈ √2a − a a p 0 X 4 (57) b ≈ √2a0 + X 2 c ≈ c0 a b c Pma2 a ≈ c − a a p 0 X 4 (28) b ≈ √2a0 + X 2 c ≈ √2a0 a a c- C2/m a ≈ c − p 0 0 0 X 3 (12) b ≈ √2a0 − X 1 c ≈ √2a0 γ ≈ 90 a a c- Pbcn a ≈ c − p p a 0 X 3 (60) b ≈ √2a0 − X 1 c ≈ √2a0 a a c- C2/c a ≈ c − p p p 0 X 3 (15) b ≈ √2a0 − X 1 c ≈ √2a0 β ≈ 90 a b c- P2_1/c β ≈ − p p p X 3 (14) − X 1 a a c- C2/m a ≈ 2a − a 0 0 0 X 4 (12) b ≈ 2a0 − X 1 c ≈ c0 β ≈ 90 a a c- Pcca a ≈ c − a a a 0 X 4 (54) b ≈ √2a0 − X 1 c ≈ √2a0

160 Table A3 (cont’d)

Tilt Space Unit Irrep System Group Cell size a a c- P2/c β ≈ − a a p X 4 (13) − X 1 a b c- P2/c a ≈ √2a − a a p 0 X 4 (13) b ≈ √2a − 0 X c ≈ c0 1 β ≈ 90 a a c0 P4_2/m a ≈ √2a − p a 0 X 3 (84) c ≈ c0 − X 4

Table A3: Possible tilting distortions for n > 1 DJ structures

161