We examine discrete models with hexagonal symmetry to compare the sequence of transitions with the alpha-inc-beta phase transition of quartz. We examine a model by Parlinski which employs interactions of nearest and next-nearest neighbor atoms. We numerically determine the configurations which lead to minimum energy for a range of parameters. We then use Golubitsky's results on systems with hexagonal symmetry to derive the bifurcation diagram for Parlinski's model. Finally, we study a large class of modifications to Parlinski's model and show that all such modifications have the same bifurcation picture as the original model. / Ph. D.

Nanomaterials (NMs) are materials in which the size of at least one dimension is less than 100 nm. Examples include quantum dots, nanoparticles, “Buckminsterfullerene (C60)”, carbon nanotubes, graphene and TiO2 thin films. Many research groups have investigated the properties of NMs, and they have reported that some of them are clearly different to those of the bulk materials, and depend on the size of the NMs. Examples include melting temperatures, phase transition pressures, fluorescence spectra, catalytic properties and magnetic properties. Recently, a high-pressure study of Te nano-cylinders revealed compressibility effects that are different to those observed in bulk-Te. Although this study reported an elevation of phase transition pressure compared to the bulk, the authors did not investigate the structures of the high-pressure phases, and it is unclear whether the incommensurate phase found at high pressures in bulk-Te was observed or not. Indeed, it is completely unknown whether the incommensurate phases observed in a number of elements at high pressure also exist in nanoparticle samples of the same materials. The search for, and study of, such phases forms the subject of this thesis. Initial studies of commercial selenium nanoparticles (nano-Se) revealed that the incommensurate phase of bulk selenium (Se-IV) is also found in nano-Se. The transition pressures in nano-Se are slightly higher than those of bulk-Se. However, the nano-Se samples were subsequently found not to have the sizes, shapes, and properties claimed by the vendor, which was confirmed by transmission and scanning electron microscopy. Further commercial samples of nano-Se and nano-Bi were also found to be of extremely poor quality. It was clear, therefore, that a detailed study of incommensurate phases in NMs would require us to make our own samples. Bismuth nanoparticles (nano-Bi) with dimensions 51(6), 52(15), 92(13), 128(45), and 138(27) nm have been successfully synthesised by the author in collaboration with the Hybrid Nano Collods group at the University of St. Andrews. On compression, the nano-Bi samples were found to have the same order of phases Bi-I, Bi-II, Bi-III, and Bi-V and phase transitions as found in bulk-Bi, but were found to exhibit larger phase coexistence. The phase transition pressures on pressure increase were higher than those of the bulk materials, and the smaller the diameter of nano-Bi, the higher the phase-transition pressure. This behaviour is similar to, but more extreme than, that found in CdSe nanoparticles. The incommensurate Bi-III structure has been found in nano-Bi under increases in pressure. However, the di↵raction patterns from Bi-III contain additional unaccounted-for peaks, and this phase is referred to as complex Bi-III. The Debye- Scherrer rings from complex Bi-III are smooth, and do not exhibit the spottiness observed in the diffraction patterns of Bi-III obtained from bulk-Bi. This enables full Rietveld refinement of Bi-III in the nano-samples. Complex Bi-III exists from 3 GPa up to 30 GPa, compared to the stable range of only 2.7 to 7.7 GPa of Bi-III in the bulk material. While such a large range of pressure enables the structure of nano-Bi-III to be studied over a much wider pressure range than bulk-Bi-III, such studies were hampered by the existence of the unaccounted-for peaks. In order to get clean, single-phase patterns of Bi-III, samples of this phase were first prepared on pressure decrease from the higher-pressure Bi-V phase, before recompressing them. Single-phase samples of Bi-III were obtained and were found to be stable up to 14-18 GPa. However, because of phase coexistence, diffraction peaks from Bi-III were still visible at pressures as high as ~30 GPa, which is ~3 times larger than the upper limit pressure of existence of bulk-Bi-III. On pressure re-increase, nano-Bi-III has a higher bulk modulus than bulk-Bi-III. The bulk modulus was found to be size-dependent as it is higher when size decreases. Moreover, nano-Bi has a smaller value of the incommensurate wave vector, which is almost pressure independent, but is found to be particles size dependent. The incommensurate wave vector thus becomes another of the structural and physical properties of nanomaterials that is found to be sample-size dependent.

Single layered perovskites with the chemical formula La2−xSrxTO4 (T = transition metal) exhibit a variety of intriguing ordering phenomena. The most outstanding is the occurrence of high temperature superconductivity in La2−xSrxCuO4, which can be considered as the prototype system for the more complex cuprates. Some cuprates show incommensurate static charge order at low temperatures [38–40]. For others it is believed that charges are dynamically correlated [39, 147, 259]. Such effects are difficult to measure if the charges fluctuate. In contrast to the cuprate La2−xSrxCuO4 the isostructural nickelates and cobaltates remain insulating over a wide doping range [112, 134, 135, 138]. While incommensurate charge stripe order is long known for the nickelates, recently also evidence for charge stripes in cobaltates has been published [174]. Single crystal rods, with ≈10cm length and ≈0.8cm diameter, have been grown by the traveling solvent floating zone technique using an optical four mirror furnace. We investigated strontium doped nickelates in the range 0.15 ≤ x ≤ 0.22. In addition, also co-doped nickelates have been investigated. A large number of samples with different doping concentrations enabled us to systematically characterize the sample properties. Powder X-ray diffraction measurements were used to determine the lattice parameters. For the nickelates we could confirm the doping dependence of the lattice constants reported in literature [202]. The main interest for the cobaltate system was in the strontium doping range 1/3 ≤ x ≤ 1/2. It was previously reported that the ab-lattice parameter exhibits an anomalous peak around a Sr doping x ≈ 1/3 [140]. We could not confirm such an anomaly for our samples and, instead, we observe a strictly monotonic doping dependence of the lattice parameters which we attribute to the close to perfect stoichiometry of our samples. Samples with the 214-layered perovskite structure can be synthesized over a wide range of oxygen off-stoichiometry. However, the oxygen content can have similarly strong influence on the sample properties as strontium doping. It is therefore essential for data interpretation to determine the oxygen off stoichiometry. EDX and WDX measurements were used to confirm the oxygen content in our nickelates to be nearly stoichiometric. The oxygen content determination of the cobaltates is somewhat more difficult. Thermogravimetry measurements in a flow of Ar/H2 confirmed a nearly stoichiometric oxygen content δ in La2−xSrxCoO4+δ for all samples. We used neutron diffraction measurements to determine the magnetic order in our nickelate samples. In stripe ordered nickelates a small titanium co-doping of the order of 5% is suficcient to supress the incommensurate magnetism and restore antiferromagnetic order. Within the series of zinc co-doped nickelates three samples exhibit an incommensurability epsilon ≈ 1/8, indicating the stabilization of an intermediate stripe pattern with an eightfold unit cell. Compared to the epsilon ≈ 1/3 regime the correlation length is greatly reduced. The magnon dispersion of two samples within the intermediate stripe phases with epsilon ≈ 1/8 and epsilon ≈ 1/4 has been measured with neutron spectroscopy. The observed dispersion neither resembles the one in the undoped nor the 1/3 strontium doped samples. Despite the amount of disorder in our co-doped nickelate materials there are no clear signs for the emergence of hourglass spectra which is most likely caused by a strong exchange interaction across the holes. We investigated the charge and magnetic order in the incommensurate regime of La2−xSrxCoO4 with doping 0.33 ≤ x ≤ 0.5 by elastic neutron scattering and hard X-ray synchrotron measurements. In contrast to the established opinion that this phase is characterized by charge stripe order we were able to show that no charge stripes are present. Instead we found that checkerboard charge order, which is most stable at x = 1/2, persists to a much lower doping than previously thought. The absence of charge stripes is also in agreement with the dispersion of the top most Co-O bond stretching phonon mode. Charge order can induce an anomaly in this branch according to the modulation vector ~q. We observed a softening at ~q = (1/2 1/2 0), which is consistent with our expectations for a checkerboard charge ordered phase. Inelastic neutron measurements revealed an additional high energy part of the hourglass dispersion which has not been reported so far. The entire lowenergy spin excitations that belong to the classical hour-glass dispersion are mostly in-plane excitations, the newly discovered high-energy magnon mode arises from out-of-plane excitations. The resemblance between the low energy excitations below the neck of the hourglass with the excitations in La1.5Sr0.5CoO4 and similarly between the high energy excitations with those observed in La2CoO4 suggests that the observed dispersion is not a single dispersion, but instead consists of two dispersions with distinct origin. In this model the low-energy dispersion arises mainly from magnetic excitations of hole doped regions and the high-energy part would be connected to magnetic excitations within the undoped islands. The absence of charge stripe order in the insulating cobaltates in combination with an unmagnetic low spin state for Co+3 requires a different explanation for the presence of incommensurate magnetic order. We propose a picture on the basis of the ideal checkerboard charge order of the half doped reference system. Decreasing the strontium concentration requires the replacement of Co+3 by Co+2, effectively resulting in the competition between the antiferromagnetic order of the undoped and the antiferromagnetic order of the half doped compound. The induced frustration can be released by a twisting of magnetic moments away from their antiferromagnetic orientation, ultimately leading to the observed incommensurate magnetic order.

<p>This thesis is based on a study of five distinct intermetallic systems with the aim of expanding the general knowledge of aperiodically modulated crystal structures. Families of compounds that contain a variety of superstructures together with incommensurately modulated structures have been investigated mainly by means of single crystal X-ray diffraction and higher dimensional structure models.</p><p>A uniform (3+1)-dimensional structure for Bi-Se phases was developed with the composition as a single variable. The structure description is based on a cubic NaCl type structure with homoatomic layer stackings. It is shown by computational modelling that the formation energies of bismuth selenides with more than 40 at. % Bi are close to zero, a result that supports the idea of a continuous series of stackings corresponding to an ordered solid solution of Bi in Bi₂Se₃.</p><p>The Nowotny chimney-ladder structures are described with a (3+1)-dimensional composite structure, valid for all such compounds regardless of the included elements, the composition or the valence electron concentration. A new member is added to this family by the ZrBi_1.62 compound. The modulation is believed to arise as a secondary effect of the criteria of a fixed electron count.</p><p>A symmetry analysis is presented for the <i>RE</i>_1+ε(MB)₄ (<i>RE</i> = rare earth elements, M = iron metal elements) family of compounds and a uniform (3+1)-dimensional composite structure description has been developed. The modulation may be due to the presence of unusually short contacts between the <i>RE</i> channel atoms, giving rise to a rotational modulation of the (MB)₄ tetraederstern chains.</p><p>A (3+1)-dimensional incommensurate structure has been determined for the novel δ₁ – CoZn compound. The structure displays a unique assembly of fused icosahedra and the modulation is induced by geometric strain.</p><p>The structure of the K(PtSi)₄ compound was re-determined. Despite a close kinship with the <i>RE</i>_1+ε(MB)₄ compounds, this structure is not modulated.</p>

This thesis is based on a study of five distinct intermetallic systems with the aim of expanding the general knowledge of aperiodically modulated crystal structures. Families of compounds that contain a variety of superstructures together with incommensurately modulated structures have been investigated mainly by means of single crystal X-ray diffraction and higher dimensional structure models. A uniform (3+1)-dimensional structure for Bi-Se phases was developed with the composition as a single variable. The structure description is based on a cubic NaCl type structure with homoatomic layer stackings. It is shown by computational modelling that the formation energies of bismuth selenides with more than 40 at. % Bi are close to zero, a result that supports the idea of a continuous series of stackings corresponding to an ordered solid solution of Bi in Bi2Se3. The Nowotny chimney-ladder structures are described with a (3+1)-dimensional composite structure, valid for all such compounds regardless of the included elements, the composition or the valence electron concentration. A new member is added to this family by the ZrBi1.62 compound. The modulation is believed to arise as a secondary effect of the criteria of a fixed electron count. A symmetry analysis is presented for the RE1+ε(MB)4 (RE = rare earth elements, M = iron metal elements) family of compounds and a uniform (3+1)-dimensional composite structure description has been developed. The modulation may be due to the presence of unusually short contacts between the RE channel atoms, giving rise to a rotational modulation of the (MB)4 tetraederstern chains. A (3+1)-dimensional incommensurate structure has been determined for the novel δ1 – CoZn compound. The structure displays a unique assembly of fused icosahedra and the modulation is induced by geometric strain. The structure of the K(PtSi)4 compound was re-determined. Despite a close kinship with the RE1+ε(MB)4 compounds, this structure is not modulated.

La famille des composés halogènes et inter-halogènes représentent des solides moléculaires adoptant des phases denses communes avec des solides moléculaires diatomiques comme l’azote et l’hydrogène. Parmi les transformations structurales et électroniques induites sous haute pression et observées dans ces solides, on note, entre autres, la dissociation moléculaire et la métallisation. De plus, l’étude des phases denses de l’iode a permis récemment l’observation d’une structure cristalline possédant une modulation dite incommensurable, c’est-à-dire une modulation possédant une périodicité différente de celle de la structure cristalline, jetant ainsi une lumière nouvelle sur le processus de dissociation moléculaire dans les solides halogènes. Dans ce mémoire, on propose d’étudier les changements structuraux dans monobromure d’iode (IBr), un composé inter-halogène possédant des propriétés structurales semblables à celles de deux composés halogènes, soit l’iode (I2) et le brome (Br2) sous leur forme solide. Des expériences de diffraction des rayons X de poudres en utilisant un rayonnement synchrotron ont été réalisées à température ambiante sur l’IBr en variant la pression jusqu’aux environs de 60 GPa. La nature chimique particulière du composé IBr a nécessité la mise au point de techniques de chargement d’échantillon destinées à préserver l’intégrité chimique de la substance utilisée. On rapporte également l’observation d’une phase de l’IBr présentant une modulation incommensurable. Les phases observées dans l’IBr permettent d’établir des parallèles avec les phases denses rapportées dans I2 et Br2 par le biais d’un modèle phénoménologique décrivant la séquence structurale des solides halogènes sous forme condensée.

La famille des composés halogènes et inter-halogènes représentent des solides moléculaires adoptant des phases denses communes avec des solides moléculaires diatomiques comme l’azote et l’hydrogène. Parmi les transformations structurales et électroniques induites sous haute pression et observées dans ces solides, on note, entre autres, la dissociation moléculaire et la métallisation. De plus, l’étude des phases denses de l’iode a permis récemment l’observation d’une structure cristalline possédant une modulation dite incommensurable, c’est-à-dire une modulation possédant une périodicité différente de celle de la structure cristalline, jetant ainsi une lumière nouvelle sur le processus de dissociation moléculaire dans les solides halogènes. Dans ce mémoire, on propose d’étudier les changements structuraux dans monobromure d’iode (IBr), un composé inter-halogène possédant des propriétés structurales semblables à celles de deux composés halogènes, soit l’iode (I2) et le brome (Br2) sous leur forme solide. Des expériences de diffraction des rayons X de poudres en utilisant un rayonnement synchrotron ont été réalisées à température ambiante sur l’IBr en variant la pression jusqu’aux environs de 60 GPa. La nature chimique particulière du composé IBr a nécessité la mise au point de techniques de chargement d’échantillon destinées à préserver l’intégrité chimique de la substance utilisée. On rapporte également l’observation d’une phase de l’IBr présentant une modulation incommensurable. Les phases observées dans l’IBr permettent d’établir des parallèles avec les phases denses rapportées dans I2 et Br2 par le biais d’un modèle phénoménologique décrivant la séquence structurale des solides halogènes sous forme condensée.

Im Rahmen dieser Arbeit wurden thermodynamische Modellationen, Synthesen und Strukturuntersuchungen zu Lanthanoidpolysulfiden und den selenärmeren Lanthanoid-polyseleniden durchgeführt. Bereits bekannte thermodynamische Daten der Lanthanoidpolysulfide, welche durch Gesamtdruckmessungen ermittelt wurden, konnten genutzt werden, um neue Synthesewege zur Darstellung der Lanthanoidpolysulfide thermodynamisch zu modellieren. Im Rahmen dieser Modellierungen wurde für Praseodym exemplarisch gezeigt, dass ein chemischer Gasphasentransport der Polysulfide unter Zugabe von Br2(l) und unter der Ausbildung des Transportmittels SBr2(g) möglich ist. Des Weiteren konnte die Synthese der Verbindungen LnS1.9 (Ln = La – Nd, Gd) unter Verwendung von HgS als Sulfiddonor durchgeführt und anhand der entwickelten elektrochemischen Spannungsreihe anorganischer sulfidischer Festkörper sowie weiterführender thermodynamischer Rechnungen rationalisiert werden. Die durch thermodynamische Rechnungen optimierten Synthesen lieferten Kristalle guter bis hervorragender Qualität, welche durch anschließende Röntgen¬beugungsexperimente charakterisiert wurden. Anhand dieser Daten konnten die Kristallstrukturen einer Vielzahl von Lanthanoidpolychalkogeniden LnQ2–δ (Ln = La – Ho, Y; Q = S, Se; 0 £  £ 0.15) gelöst und verfeinert werden. Die Strukturtypen, welche im Rahmen dieser Arbeit für die Lanthanoidpolychalkogenide gefunden wurden, sind der CeSe2-Typ, der CeSe1.9-Typ, eine niedersymmetrische Form des GdSe1.875-Typs und eine inkommensurabel modulierte Variante des PrSe1.85-Typs. Da es sich bei allen beschriebenen Verbindungen um Überstrukturen des ZrSSi-Typs handelt, wurden zur Veranschaulichung dieser Tatsache Bärnighausen-Stammbäume für die gefundenen Strukturtypen aufgestellt. Anhand dieser Stammbäume wurde illustriert, wie sich die Atomlagen der niedersymmetrischen Überstrukturen aus denen des hochsymmetrischen Aristotyps ableiten lassen.

La famille des composés halogènes et inter-halogènes représentent des solides moléculaires adoptant des phases denses communes avec des solides moléculaires diatomiques comme l’azote et l’hydrogène. Parmi les transformations structurales et électroniques induites sous haute pression et observées dans ces solides, on note, entre autres, la dissociation moléculaire et la métallisation. De plus, l’étude des phases denses de l’iode a permis récemment l’observation d’une structure cristalline possédant une modulation dite incommensurable, c’est-à-dire une modulation possédant une périodicité différente de celle de la structure cristalline, jetant ainsi une lumière nouvelle sur le processus de dissociation moléculaire dans les solides halogènes. Dans ce mémoire, on propose d’étudier les changements structuraux dans monobromure d’iode (IBr), un composé inter-halogène possédant des propriétés structurales semblables à celles de deux composés halogènes, soit l’iode (I2) et le brome (Br2) sous leur forme solide. Des expériences de diffraction des rayons X de poudres en utilisant un rayonnement synchrotron ont été réalisées à température ambiante sur l’IBr en variant la pression jusqu’aux environs de 60 GPa. La nature chimique particulière du composé IBr a nécessité la mise au point de techniques de chargement d’échantillon destinées à préserver l’intégrité chimique de la substance utilisée. On rapporte également l’observation d’une phase de l’IBr présentant une modulation incommensurable. Les phases observées dans l’IBr permettent d’établir des parallèles avec les phases denses rapportées dans I2 et Br2 par le biais d’un modèle phénoménologique décrivant la séquence structurale des solides halogènes sous forme condensée.

La famille des composés halogènes et inter-halogènes représentent des solides moléculaires adoptant des phases denses communes avec des solides moléculaires diatomiques comme l’azote et l’hydrogène. Parmi les transformations structurales et électroniques induites sous haute pression et observées dans ces solides, on note, entre autres, la dissociation moléculaire et la métallisation. De plus, l’étude des phases denses de l’iode a permis récemment l’observation d’une structure cristalline possédant une modulation dite incommensurable, c’est-à-dire une modulation possédant une périodicité différente de celle de la structure cristalline, jetant ainsi une lumière nouvelle sur le processus de dissociation moléculaire dans les solides halogènes. Dans ce mémoire, on propose d’étudier les changements structuraux dans monobromure d’iode (IBr), un composé inter-halogène possédant des propriétés structurales semblables à celles de deux composés halogènes, soit l’iode (I2) et le brome (Br2) sous leur forme solide. Des expériences de diffraction des rayons X de poudres en utilisant un rayonnement synchrotron ont été réalisées à température ambiante sur l’IBr en variant la pression jusqu’aux environs de 60 GPa. La nature chimique particulière du composé IBr a nécessité la mise au point de techniques de chargement d’échantillon destinées à préserver l’intégrité chimique de la substance utilisée. On rapporte également l’observation d’une phase de l’IBr présentant une modulation incommensurable. Les phases observées dans l’IBr permettent d’établir des parallèles avec les phases denses rapportées dans I2 et Br2 par le biais d’un modèle phénoménologique décrivant la séquence structurale des solides halogènes sous forme condensée.