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A graph theoretical approach to the analysis, comparison, and enumeration of crystal structures
(2008)
As an alternative approach to lattices and space groups, this work explores graph theory as a means to model crystal structures. The approach uses quotient graphs and nets - the graph theoretical equivalent of cells and lattices - to represent crystal structures. After a short review of related work, new classes of cycles in nets are introduced and their ability to distinguish between non-isomorphic nets and their computational complexity are evaluated. Then, two methods to estimate a structure’s density from the corresponding net are proposed. The first uses coordination sequences to estimate the number of nodes in a sphere, whereas the second method determines the maximal volume of a unit cell. Based on the quotient graph only, methods are proposed to determine whether nets consist of islands, chains, planes, or penetrating, disconnected sub-nets. An algorithm for the enumeration of crystal structures is revised and extended to a search for structures possessing certain properties. Particular attention is given to the exclusion of redundant nets and those, which, by the nature of their connectivity, cannot correspond to a crystal structure. Nets with four four-coordinated nodes, corresponding to sp3 hybridised carbon polymorphs with four atoms per unit cell, are completely enumerated in order to demonstrate the approach. In order to render quotient graphs and nets independent from crystal structures, they are reintroduced in a purely graph-theoretical way. Based on this, the issue of iso- and automorphism of nets is reexamined. It is shown that the topology of a net (that is the bonds in a crystal) constrains severely the symmetry of the embedding (that is the crystal), and in the case of connected nets the space group except for the setting. Several examples are studied and conclusions on phases are drawn (pseudo-cubic FeS2 versus pyrite; α- versus β- quartz; marcasite- versus rutile-like phases). As the automorphisms of certain quotient graphs stipulate a translational symmetry higher than an arbitrary embedding of the corresponding net would show, they are examined in more detail and a method to reduce the size of such quotient graphs is proposed. Besides two instructional examples with 2-dimensional graphs, the halite, calcite, magnesite, barytocalcite, and a strontium feldspar structures are discussed. For some of the structures it is shown that the quotient graph which is equivalent to a centred cell is reduced to a quotient graph equivalent to the primitive cell. For the partially disordered strontium feldspar, it is shown that even if it could be annealed to an ordered structure, the unit cell would likely remain unchanged. For the calcite and barytocalcite structures it is shown that the equivalent nets are not isomorphic.
In dieser Arbeit wurden zwei Idealkristallsysteme und drei Systeme, die im weitesten Sinne als Domänenkristalle zu bezeichnen sind, mit quantenmechanischen Methoden untersucht, die auf Dichte-Funktional-Theorie basieren. Dabei wurden je nach System unterschiedliche Eigenschaften der jeweiligen Kristalle betrachtet. Zusätzlich wurden die berechneten Domänenkristalle jeweils mit entsprechenden Idealkristallen bezüglich ihrer Stabilität verglichen. Insgesamt konnte gezeigt werden, dass sich die hier verwendeten Rechenmethoden sehr gut zur Untersuchung von Grundzuständen und Strukturen unter hydrostatischem Druck sowie von Bindungseigenschaften eignen. Desweiteren lieferten die Ergebnisse starke Hinweise darauf, dass Kristalle mit Strukturgradienten nur dann existieren können, wenn sie sich vom Idealkristall um sehr geringe Energien unterscheiden, die unter der Fehlergrenze der hier angewendeten Methode (2-3 kJ/Mol) liegen.
Recently, carbonates have attracted a lot of attention, due to the recognition of their importance in the global carbon cycle. This was enabled by improvement of the experimental techniques that allow for investigating the stability, structure, and physical properties of materials and high-pressures and high-temperatures, that is, they allow for investigating minerals and geochemical processes at the conditions occurring deep inside Earth. Although a lot of research has been focused on carbonates, there are still some open questions regarding their structure and physical properties at such extreme conditions. The aim of this thesis is to establish a deeper understanding of the nature of the phase transitions in carbonates by studying how do the atoms building up the crystal structure vibrate, that is lattice dynamics. The methodology adapted in this study is a combination of experimental and computational methods which allows for a very thorough examination of the problem. The computational approach allows to determine parameters that are elusive or tedious to measure, and the experimental results provide a solid benchmark for the calculations. This tandem of methods has been widely used for investigating lattice dynamics of various materials. In this study it was used to elucidate the structure and properties of carbonates in the deep Earth conditions
Many natural minerals exist in the form of a solid solution. The systematic changes in structural and physical properties of oxide solid solutions are of geological importance and allow for wide applications. In order to understand the composition-structureproperty relations, substitutional solid solutions of CuxZn2−xTiO4, ZnxMg1−xTi2O5 and CuxMg1−xTi2O5 have been synthesised by mechanochemical activation assisted solid state synthesis. Self-propagating high-temperature synthesis has been employed to achieve the interstitial solid solutions of Ti5Si3Zx (Z refers to the element boron or oxygen).
The changes in the crystal structure and physical properties due to the formation of solid solutions are investigated by employing X-ray diffraction, neutron diffraction, Raman spectroscopy, low-temperature heat capacity, thermal expansion, scanning electron microscopy, UV-vis spectroscopy, plane-wave ultrasound spectroscopy and density functional theory calculations.
Development of the flash-heating method for measuring melting temperatures in the diamond anvil cell
(2016)
A new ‘laser flash-heating’ method has been developed for measuring melting temperatures above 2000 K in a diamond anvil cell at gigapascals of pressure. It overcomes the general difficulties in detecting an onset of melting in a diamond anvil cell. It also circumvents the notorious experimental difficulties associated with the long heating durations of the CW laser-heating and the short timescales in the pulsed laser-heating and shock-compression experiments.
In this method, the duration of heating a sample is tuned to avoid chemical reactions of the sample with the diamond anvils and the surrounding pressure medium, while maintaining the accuracy of the temperature measurements. The absence of chemical reactions is confirmed by the EDS technique. Melt detection is now unambiguous from the analysis of textures on the surface and in depth of the recovered samples using the SEM and FIBM techniques, respectively. Using this method, the following has been achieved.
1. The melting curve of hcp-Re has been measured to 48 GPa, 4200 K for the first time. It has a significantly steeper slope than those observed for other transition metals like W and Mo with bcc structures. Above 20 GPa, Re becomes the most refractory metal surpassing W.
2. The melting curve of bcc-Mo has been measured to 45 GPa, 3100 K. It agrees with previous melt-slopes approaching zero value with pressure as reported in the LHDAC experiments using ADXRD and visual observation techniques for inferring the onset of melting. Flash-heating experiments at pressures higher than 50 GPa are required to further corroborate the flat melt-slope and resolve the long standing controversy about melting of Mo.
3. The melting curve of bcc-Ta has been measured to 85 GPa, 4300 K. Unlike in previous experiments using ADXRD and visual observation as probes, it has been tightly bracketed with an unambiguous detection of the onset of melting, without any chemical reaction. The present melting curve cannot be reconciled with shock measurements and theoretical predictions, and the precision of measurements calls for a reevaluation of theoretical, shock compression, and other DAC approaches to determine melting at high pressures. A further analysis with TEM technique for investigating the structure of the heated portion below and above melting temperatures of Ta may benefit in resolving various phase transitions predicted to explain the vast discrepancies in the reported melt-slopes.
When extrapolated to one atmosphere pressure, all the measured flashmelting curves agree with the known melting points.
Die nachfolgende Dissertation wurde an der Goethe-Universität Frankfurt am Institut für Geowissenschaften (FB 11) in der Arbeitsgruppe Kristallographie und Mineralogie (AG Winkler) verfasst. Die Betreuung der hier durchgeführten Arbeiten erfolgte hauptsächlich durch Prof. B. Winkler in Zusammenarbeit mit Dr. L. Bayarjargal, PD Dr. E. Haussühl und PD Dr. V. Vinograd. Bei dem vorliegenden Manuskript handelt es sich um eine kumulative bzw. publikationsbasierte Dissertation, welche die Forschungsergebnisse verschiedener bereits veröffentlichter wissenschaftlicher Fachartikel zusammenfasst.
Die Arbeit beschreibt verschiedene Synthesen und Untersuchungen an Carbonaten und teilt sich im Wesentlichen in zwei Abschnitte. Zum einen wurden Experimente mit Carbonaten bei Extrembedingungen bzw. unter hohen Drücken und hohen Temperaturen durchgeführt, wie sie auch im Inneren der Erde zu finden sind. Im zweiten Teil wurden Carbonate bei Raumbedingungen synthetisiert und der Einbau von Seltenerdelementen untersucht. Grundsätzlich werden jedoch in beiden Teilen dieser Arbeit die Strukturen und Eigenschaften verschiedener Carbonate und eine mögliche Kationensubstitution bzw. die Synthese isostruktureller Verbindungen erforscht.