BOND LENGTHS AND BOND VALENCES OF IONS BONDED TO OXYGEN: THEIR VARIABILITY IN INORGANIC CRYSTALS by Olivier Gagné A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of DOCTOR OF PHILOSOPHY Department of Geological Sciences University of Manitoba Winnipeg Copyright © 2016 by Olivier Gagné ABSTRACT A large amount of information concerning interatomic distances in the solid state is available, but little has been done in recent times to comprehensively filter, summarize and analyze this information. Here, I examine the distribution of bond lengths for 135 ions bonded to oxygen, using 180,331 bond lengths extracted from 9367 refined crystal structures collected from the Inorganic Crystal Structure Database (ICSD). The data are used to evaluate the parameterization of the bond-length—bond-valence relation of the bond-valence model. Published bond-valence parameters for 135 cations bonded to oxygen, and the various methods used in their derivation, are evaluated. New equations to model the relation are tested and the common form of the equation is found to be satisfactory. A new method (the Generalized Reduced Gradient Method, GRG method) is used to derive new bond-valence parameters for 135 cations bonded to oxygen, leading to significant improvements in fit for many of the ions. The improved parameterization is used to gain crystal-chemical insight into the milarite structure. A literature review of 350+ published compositions is done to review the end- members of the milarite group and to identify compositions that should have been described as distinct minerals species. The a priori bond-valences are calculated for minerals of this structure, and are used to examine the controls of bond topology on site occupancy, notably by localizing the major source of strain of the structure (the B site). Examination of the compositions of all known milarite-group minerals shows that compositions with a fully occupied B site are less common than those with a vacant B site, in accord with the idea that the B site is a local region of high strain in the structure. ii The bond-length distributions for the ions of the alkali and alkaline-earth metal families are examined. Variations in mean bond-lengths are only partly explained by the distortion theorem of the bond-valence model. I have found that bond length also correlates with the amount of vibrational displacement of the constituent ions. The validity of some uncommon coordination numbers, e.g., [3]-coordinated Li+, [3]- coordinated Be2+, is confirmed. iii ACKNOWLEDGEMENTS First and foremost, I thank my thesis advisor Frank. C. Hawthorne for his help, support, encouragement and dedication over the course of my degree, and for providing me with an environment in which I was able to flourish. I thank members of my Ph.D. thesis committee, Elena Sokolova, Norman Halden, Scott Kroeker and Peter Budzelaar for their support, positivity, and helpful commentary, and the support staff of the department of Geological Sciences at the University of Manitoba. I thank the Natural Sciences and Engineering Research Council (NSERC) for a PGS-D3 scholarship, the University of Manitoba’s Faculty of Graduate Studies for a Duff Roblin and GETS Fellowships, the Mineralogical Association of Canada for awarding me with a MAC Foundation scholarship, the C.H.R. Faculty of Environment, Earth and Resources for an Entrance and Graduate Student Initiative Scholarships, and the Department of Geological Sciences for Rita Wadien scholarships. I thank Patrick Mercier and Yvon Lepage of the National Research Council for providing help in shaping research themes early on in the project, and Éric Martineault and Yulia Uvarova whom I could always count on for advice and entertaining discussions. Last but not least, I thank my family members, who were always present and supportive of my (often questionable) decisions. Merci Maman, Mathieu et Amélie. Finally, I thank my friends, especially the ones that tend to jump out of airplanes and things, whom I could always count on to clear my mind in time of need. Shout out to my 4-way skydiving team, team Alphaghetti. iv “This is for the kids who have nowhere to turn Who have nothing to live for You think you haven't the will to persist You have to search within yourself” -Jamey Jasta v TABLE OF CONTENTS Abstract.............................................................................................................................ii Acknowledgements..........................................................................................................iv Dedication.........................................................................................................................v List of Tables..................................................................................................................xvi List of Figures.................................................................................................................xix Contributions of Authors...............................................................................................xxiv List of Copyrighted Material for which Permission was Obtained................................xxvii Chapter 1. Introduction..................................................................................................1 1.1 General introduction……………………………………………………...……..….2 1.1.1 Concepts of crystal chemistry............................................................3 1.1.1.1 The internal structure of crystals..........................................4 1.1.1.2 Interatomic distances and ionic radii....................................8 1.1.1.3 The chemical bond and chemical bonding...........................9 1.1.2 Interatomic distances in the solid state............................................11 1.1.2.1 Bond lengths for cations bonded to O2-..............................12 1.1.3 The types of chemical bonds...........................................................13 vi 1.1.3.1 Ionic bonding……………………………………………..……14 1.1.3.2 Covalent bonding…………………………........................…17 1.1.3.3 Metallic bonding………………………………........…………19 1.1.3.4 Van der Waals bonding…………………….........................19 1.1.3.5 Hydrogen bonding…………...............………………………20 1.1.4 Theories of chemical bonding..........................................................20 1.1.4.1 Valence-bond theory..........................................................21 1.1.4.2 Molecular orbital (MO) theory.............................................21 1.1.4.3 Bond-valence theory..........................................................22 1.1.4.3.1 Theoretical basis of bond-valence theory.............23 1.1.4.3.2 The bond-valence model......................................27 1.2 Objectives and Significance of Research......................................................29 1.3 Structure of the thesis…………………............................................…...…….31 1.4 References……………………................................................................…….33 Chapter 2. Comprehensive derivation of bond-valence parameters for ion pairs involving oxygen……...................................................................................................41 2.1 Synopsis........................................................................................................42 2.2 Abstract..........................................................................................................42 2.3 Introduction....................................................................................................43 vii 2.4 Experimental bond-lengths used in this work................................................45 2.5 Method of evaluation of bond-valence parameters........................................46 2.6 Evaluation of published oxide bond-valence parameters..............................47 2.7 The hydrogen atom........................................................................................49 2.8 Use of bond-valence parameters for hydrogen-oxygen bonds......................50 2.9 Comments on fixing the B parameter............................................................51 2.10 Comments on the level of fit........................................................................52 2.11 Parameterization..........................................................................................53 2.11.1 Methods of derivation of the bond-valence parameters.................53 2.11.1.1 Least-squares fitting.........................................................53 2.11.1.2 Fixing the B parameter.....................................................54 2.11.1.3 Graphical method: cation and anion sums.......................55 2.11.1.2 Graphical method: cation sums........................................56 2.11.1.5 RMSD minimization..........................................................58 2.11.2 Generalized Reduced Gradient (GRG) method of RMSD minimization...................................................................................59 2.11.2.1 The Generalized Reduced Gradient method....................60 2.11.2.2 Weighting scheme............................................................63 2.11.3 Comparison of the most common methods of derivation...............63 viii 2.11.4 General considerations..................................................................70 2.11.4.1 Optimizing both cation and anion bond-valence sums.....70 2.11.4.2 On the universality of the bond-valence