Let G be a group. A weak Cayley table isomorphism $\phi$: G $\rightarrow$ G is a bijection satisfying two conditions: (i) $phi$ sends conjugacy classes to conjugacy classes; and (ii) $\phi$(g1)$\phi$(g2) is conjugate to $\phi$(g1g2) for all g1, g2 in G. The set of all such mappings forms a group W(G) under composition. We study W(G) for fifty-six of the two hundred nineteen three-dimensional crystallographic groups G as well as some other groups. These fifty-six groups are related to our previous work on wallpaper groups.

The Himalayan orogen provides a natural laboratory to test models of orogenic development due to large-scale continental collision. The Greater Himalayan Series (GHS), a lithotectonic unit continuous along the entire length of the belt, comprises the metamorphic core of the Himalayan orogen and underlies the highest topography. GHS rocks are exposed as a moderately north-dipping slab bounded below by the Main Central Thrust (MCT) and above by the South Tibetan Detachment System (STDS) of normal faults. Coeval reverse- and normal-sense motion on the crustal-scale MCT and STDS ductile shear zones allows the GHS to be modeled as an extruded wedge or channel of mid-crustal material. Due to this unique tectonic setting, the deformation path of rocks within the bounding shear zones and throughout the core of the GHS profoundly influences the efficiency of extrusion and exhumation processes. Attempts to quantify GHS deformation and metamorphic evolution have provided significant insight into Himalayan orogenic development, but these structural and petrologic studies are often conducted in isolation. Penetrative deformation fabrics developed under mid-upper amphibolite facies conditions within the GHS argue that deformation and metamorphism were coupled, and this should be considered in studies aimed at quantifying GHS teconometamorphic evolution. This work focuses on two projects related to the coupled deformation, thermal and metamorphic evolution during extrusion and exhumation of the GHS, focused on the lower and upper margins of the slab. A detailed examination of the P--T history of a schist collected from within the MCT zone of the Sutlej River, NW India, provides insight into the path experienced by these rocks as they traveled through the crust in response to the extreme shortening related to India-Asia collision. Combined forward thermodynamic and diffusion modeling indicates compositional zoning preserved in garnet has remained unmodified since growth and can be related directly to the P--T--X evolution of rocks from this zone. Classic porphyroblast--matrix relationships coupled with the above models provide a structural framework within which to interpret the microstructures and provide additional constraints on the relative timing of metamorphic and deformation events. A combined microstructural and quartz petrofabric study of rocks from the highest structural levels of the GHS in the Zanskar region was completed. This work provides the first quantitative estimate of temperatures attending normal-sense shearing along the Zanskar Shear Zone, the westernmost strand of the STDS. Results indicate penetrative top-N (extensional) deformation occurred at elevated temperatures and resulted in the telescoping of isothermal surfaces present during shearing and extrusion of GHS rocks. Simple geometric models invoking heterogeneous simple shear parallel to the overlying detachment require dip-slip displacement magnitudes on the order of 15--40 km, identical to estimates derived from nearby barometric analyses. Finally, focus is given to the rotational behavior of rigid inclusions suspended in a flowing viscous matrix from a theoretical perspective. Predictions of clast rotational behavior have been used to construct several kinematic vorticity estimation techniques that have become widely adopted for quantitative studies of naturally deformed rocks. Despite the popularity of the techniques, however, basic questions regarding clast-based analyses remain open. Therefore a numerical model was constructed and a systematic investigation of 2- and 3D clasts suspended in steady and non-steady plane-strain flows was undertaken to determine likely sources of error and the intrinsic strengths and limitations of the techniques. / Ph. D.

<p>The morphologies and growth behavior of Cu₆Sn₅ intermetallic compound (IMC) formed between Sn-based solder and large-grain polycrystalline Cu substrate were systematically investigated. Hexagonal Cu₆Sn₅ grains were observed to form at certain reflow condition, which matches well with the literature results for IMC growing on single crystal substrate. The kinetics of IMC growth was also investigated and different mechanisms were proposed for different evolution stages. It was observed that facet formation should be a growth shape rather than an equilibrium shape, and the orientation relationship between Cu and Cu₆Sn₅ was studied using scanning electron microscope (SEM) and Electron backscatter diffraction (EBSD), and were visualized on inverse pole figure. </p>

<p> The crystal structures of VOP2SiO8, VO(PO3)2, V(PO3)3, (VO)2P2O7 and VAsO5 have been determined with the help of x-rays, and are compared with the known structures in the V-P-As-O system. All the vanadium atoms in the mixed oxides are octahedrally coordinated and the V^+4 O6 and V^+5 O6 octahedra are all characterized by one short vanadyl bond. V^+3 O6 groups are nearly regular. The tetrahedral phosphorus is found in structural elements ranging from infinite metaphosphate chains (V(PO3)3, VO(PO3)2), to pyrophosphate groups ((VO)2P2O7) to isolated tetrahedra (VAsO5 and VOP2SiO8). Both structural and substitutional disordering is evident in the V-P-As-O system, and is discussed together with a detailed model for stacking faults in VAsO5.</p> <p> Some of the phases in the V-P-As-O system are known to catalyze the oxidation of butene to maleic anhydride, and certain structural features of the compounds are related to this catalytic activity. α-VPO5 can be related to (VO)2P2O7 through the formation of shears in a manner similar to shear formation in V2O5, and such a mechanism is proposed as a means whereby an α-VPO5 catalyst can change into (VO)2P2O7, the known composition of the spent catalyst.</p> / Thesis / Doctor of Philosophy (PhD)

<p> The alkali metal dichromates show extensive polymorphism. The crystal structures of the polymorphs α-Na2Cr2O7, β-Na2Cr2O7, β1-Rb2Cr2O7 and P21/c NaRbCr2O7 have been determined with x-ray methods. Crystal data were determined for β2-Rb2Cr2O7, P1 Cs2Cr2O7 and the P21/c NaCsCr2O7.</p> <p> The dichromate ions found in this work have been compared with the dichromate ions found in other crystal structure determinations. The anions are described in terms of the bridging oxygen angles b and the torsion angles α1 and α2. Many of the dichromate ions are close to having C2v symmetry with values for α1 and α2 close to zero and bridging angles of around 124°. But there is a number of dichromates with α1 = -α2 and 0°<|α|<60° for which the bridging angle varies between 131° to 141°.</p> <p> The structures determined in this work are discussed as part of a unified description of thortveitite like and dichromate like structures in terms of layers of Y2O7 anions. In terms of this description and Brown and Calvo's classification a structure is proposed for the β2Rb2Cr2O7, while for the structure of NaCsCr2O7 it is suggested that it is isostructural to that of P21/c NaRbCr2O7. The phase transition of α-Na2Cr2O7 to β-Na2Cr2O7 is considered and it is suggested that a twisting thermal mode plays an important role in this as well as in other transitions.</p> / Thesis / Doctor of Philosophy (PhD)

Yes / For dental human enamel, what is the precise mineralization progression spatially and the precise timings of mineralization? This is an important question in the fundamental understanding of matrix-mediated biomineralization events, but in particular because we can use our understanding of this natural tissue growth in humans to develop biomimetic approaches to repair and replace lost enamel tissue. It is important to understand human tissues in particular since different species have quite distinct spatial and temporal progression of mineralization. In this study, five human central incisors at different stages of enamel maturation/mineralization were spatially mapped using synchrotron X-ray diffraction and X-ray microtomography techniques. From the earliest developmental stage, two crystallite-orientation populations coexist with angular separations between the crystallite populations averaging approximately 40o and varying as a function of position with the tooth crown. In general, population one had significantly lower texture magnitude and contributed a higher percentage to the overall crystalline structure, compared to population two which only contributed 20-30% but had significantly higher texture magnitude. This quantitative analysis allows us to understand the complex and co-operative structure-function relationship between two populations of crystallites within human enamel. There was an increase in the mineral concentration from the enamel-dentin junction peripherally and from the incisal tip cervically as a function of maturation time. Quantitative backscattered-electron analyses revealed that mineralization of prism cores precedes that of prism boundaries. These results provide new insights into the precise understanding of the natural growth of human enamel. / Partly funded by NERC grant ”Timelines in Teeth” NE/F018096/2.

Plastic deformation in crystalline materials largely depends on the properties of dislocations, in particular their mobility. While continuum description of deformation of a crystalline metal can be made reasonably well by considering the elastic properties of dislocations and neglecting the core, crystallographic aspects of dislocation motion require precise understanding of the core effects. The concept of the generalized stacking fault (GSF) energy was proposed as means to describe this. GSF energy, a fundamental property of a given material, can be determined using first principles total energy calculations. In this thesis, we use GSF to understand some of the intriguing mechanical responses recently observed in some metallic systems. First, we examine the structures and stacking fault energies in Mg-Zn-Y alloy system. This system is unique in the sense that trace additions of Zn and or Y result in long period stacking sequences such as 6l and 14l, as reported in recent literature. Further, these alloys exhibit extraordinary mechanical properties. We attempt to rationalize these experimental observations through first principles calculations of energies of periodic structures with different stacking sequences and stacking faults. For pure Mg, we find that the 6-layer structure with the ABACAB stacking is most stable after the lowest energy hcp structure with ABAB stacking. Charge density analysis shows that the 2l and 6l structures are electronically similar, which might be a cause for better stability of 6l structure over a 4l sequence or other periodic structures. Addition of 2 atomic% Y leads to stabilization of the structure to 6l sequence whereas the addition of 2 atomic% Zn makes the 6l energetically comparable to that of the hcp. Stacking fault (SF) on the basal plane of 6l structure is higher in energy than that of the hcp 2l Mg, which further increases upon Y doping and decreases significantly with Zn doping. SF energy surface for the prismatic slip indicates dissociation of dislocations in alloys with a 6l structure. Thus, in an Mg-Zn-Y alloy, Y stabilizes the long periodicity, while Zn doping provides a synergistic effect in improving the mechanical properties alongwith strengthening due to long periodic phases. Our investigation of surface properties and magnetism in Ni revealed that, the universal binding energy relation (UBER) derived earlier to describe the cohesion between two rigid atomic planes, does not accurately capture the cohesive properties when the cleavage cracked surfaces are allowed to relax through atomic displacements. We find that two characteristic length-scales are involved in the cleavage of a crystal accompanied by structural relaxation at the cleaved surface. Based on that, we suggest a modified functional form of UBER that is analytical and at the same time accurately models the properties of relaxed surfaces upon cleavage. We demonstrate the generality as well as the validity of this modified UBER through first-principles density functional theory calculations of cleavage in fcc, bcc, and hcp metals, as well as covalently bonded materials. We also found that the cohesive law (stress-displacement relation) differs significantly in the case where cracked surfaces are allowed to relax, with lower peak stresses occuring at higher displacements. We have attempted understanding these ideas through images obtained from electronic densities and eigen states. Our work should be useful in providing inputs to multi-scale simulations of fracture in materials. The third phase of the work reports the stacking fault energy and twinning in Ni with a particular emphasis on the size effect. Experimental and computational research on Nan crystalline metals (mostly on Ni) indicates unique facets of dislocation activity (extended partial dislocations) and modes of deformation (twinning). In order to capture the intrinsic scaling eject in the nano-regime, it is imperative to account for the complex electronic structure of the metal in question. The stacking fault (SF) and twinning fault (TF) energies in nano-thin elm of Ni with 7, 13, 19, and 25 layers of (111) planes were determined using rest-principles density functional theory (DFT) total energy calculations. Generalized planar fault (GPF) energy curves of the nano-thin alms show higher extreme vis-a-vis the bulk, indicating that creation of SFs in nano-Ni is relatively difficult. In contrast, the ratios of energy barriers relevant to nucleation of dislocations and twinning support the observed enhanced tendency for extended partial dislocation formation and twinning in the nano-thin films in comparison with bulk. Our results should be useful in understanding deformation behavior of nano-structured Ni-based alloys used as advanced structural materials.

No description available.

Texture, microstructure and formability were studied in Direct Chill Cast (DC) and Strip Cast (SC) aluminum alloys with regard to crystallographic anisotropy, the Portevin-Le Chatelier effect and aging softening behavior. It was found that material properties change greatly with manufacturing processes (DC vs. SC) and chemical composition (3xxx vs. 5 xxx alloys). DC cast hot band materials are usually fully recrystallized and have strong softening textures while SC hot band materials have a rolling structure with strong deformation textures. Softening textures cause 90 earing while deformation textures result in 45 earing after deep drawing. During cold rolling, 90 earing in DC cast hot band materials decreases and eventually changes to 45 earing after certain degrees of cold reduction. Correspondingly, the intensity of the softening texture components in DC cast hot band materials decreases while the intensity of deformation texture components increases with increasing degrees of cold reduction. These two kinds of textures interact and attempt to balance each other during cold rolling which produces resultant earing. However, this is not true for SC hot band materials since it's hard to obtain strong softening textures and thus 90 earing in these materials. 5 xxx Al-Mg alloys are more difficult to work than 3 xxx aluminum alloys. Elevated temperature annealing which greatly reduces the strength (hardness) improves significantly the workability of Al-Mg alloys. On the other hand, the Portevin-Le Chatelier effect and aging softening behavior are stronger in Al-Mg alloys than in 3xxx aluminum alloys and both increase with increasing cold reduction and with increasing Mg content. An apparent tensile anisotropy exists in as received SC hot band materials. The tensile yield strength (YS) is smaller in the QD (45 to the rolling direction), and larger in the RD (rolling direction) and the TD (transverse direction). There is no obvious difference in YS between these RD and TD directions. The average stress drop of serrations in the PLC effect, D s , is strongest in the TD, smallest in the RD with QD in between but closer to TD. However, no tensile anisotropy was observed in a fully recrystallized DC hot band or in solution treated SC hot band materials. It was found that a rolling structure favors mechanical anisotropy while a recrystallized structure prevents it. The tensile anisotropy is due to anisotropic distributions of microstructures, i.e., dislocations, precipitates and solute atoms. A random microstructure is associated with material that shows little or no mechanical anisotropy. An elongated or preferably orientated microstructure is associated with material with high mechanical anisotropy. Recovery thermal treatments at sufficiently high temperatures so that dislocation annihilation and microstructure rearrangement occurs when applied to the final gauge material also lowers mechanical anisotropy because of the reduction in intensity of the elongated (preferably orientated) microstructure. In addition, plastically deforming the material in a more homogenous manner (such as cross rolling as compared to straight rolling) produces a more uniform microstructure with an accompanying lower mechanical anisotropy.

Magnesium (Mg) alloy sheets possess several key properties that make them attractive as lightweight replacements for heavier ferrous and non-ferrous alloy sheets. However, Mg alloys need to be formed at elevated temperatures to overcome their limited room-temperature formabilities. For example, commercial forming is presently conducted at 450°C. Deformation behavior of the most commonly used wrought Mg alloy, AZ31B-H24, and two potentially competitive materials, AZ31B-HR and ZEK100 alloy sheets, with weaker crystallographic textures, are studied in uniaxial tension at 450°C and lower temperatures. The underlying physics of deformation including the operating deformation mechanisms, grain growth, normal and planar anisotropy, and strain hardening are used to construct material constitutive models capable of predicting forming for all three Mg alloy sheets at 450°C and 350°C. The material models constructed are implemented in finite-element-method (FEM) simulations and validated using biaxial bulge forming, an independent testing method. Forming limit diagrams are presented for the AZ31B-H24 and ZEK100 alloy sheets at temperatures from 450°C down to 250°C. The results suggest that forming processes at temperatures lower than 450°C are potentially viable for manufacturing complex Mg components. / text