Reinhard B. Neder and Thomas Proffen
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199233694
- eISBN:
- 9780191715563
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233694.003.0003
- Subject:
- Physics, Crystallography: Physics
Although it might sound simple, creating a model of a cluster of atoms or an extended crystal has its difficulties. The examples range from creating simple assemblies of atoms to creating the ...
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Although it might sound simple, creating a model of a cluster of atoms or an extended crystal has its difficulties. The examples range from creating simple assemblies of atoms to creating the contents of a unit cell from an asymmetric unit to the building of extended model crystals. This chapter discusses unit cell transformations and general symmetry operations. A disordered perovskite structure is created as an example.Less
Although it might sound simple, creating a model of a cluster of atoms or an extended crystal has its difficulties. The examples range from creating simple assemblies of atoms to creating the contents of a unit cell from an asymmetric unit to the building of extended model crystals. This chapter discusses unit cell transformations and general symmetry operations. A disordered perovskite structure is created as an example.
Naomi E. Chayen, John R. Helliwell, and Edward H. Snell
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199213252
- eISBN:
- 9780191707575
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213252.001.0001
- Subject:
- Physics, Crystallography: Physics
Structural crystallography provides key information to understand the mechanism involved for biological processes. The technique requires high‐quality crystals. The book Macromolecular ...
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Structural crystallography provides key information to understand the mechanism involved for biological processes. The technique requires high‐quality crystals. The book Macromolecular crystallization and crystal perfection covers the techniques to get these high quality crystals and then obtain the best structural data from them. We focus on two areas, the crystal and the diffraction experiment. We briefly address crystallization theory and then focus on practical crystallization strategies discussing screening and optimization. Where high quality crystals are not initially obtained, remediation strategies and alternative approaches are discussed. Diffraction is covered from both the X‐ray and neutron viewpoint. A physical analysis of long and short‐range order is used to explain features seen in the diffraction pattern and the causes of those features. Diffraction disorders are discussed. Factors that cause degradation to the diffraction and strategies to mitigate those factors are addressed. We then address beamline and detector optimization as a means to improve the data quality. Crystallization is still a largely empirical process and our final chapters focus on the use of powder methods, where crystals are small, complementary techniques where we have no crystals at all and what the future holds with the advent of fourth generation X‐ray sources. Overall the book is aimed at both more experienced researchers and graduate students. We aim for it to become a reference work for all researchers in these interdisciplinary subjects on these topics.Less
Structural crystallography provides key information to understand the mechanism involved for biological processes. The technique requires high‐quality crystals. The book Macromolecular crystallization and crystal perfection covers the techniques to get these high quality crystals and then obtain the best structural data from them. We focus on two areas, the crystal and the diffraction experiment. We briefly address crystallization theory and then focus on practical crystallization strategies discussing screening and optimization. Where high quality crystals are not initially obtained, remediation strategies and alternative approaches are discussed. Diffraction is covered from both the X‐ray and neutron viewpoint. A physical analysis of long and short‐range order is used to explain features seen in the diffraction pattern and the causes of those features. Diffraction disorders are discussed. Factors that cause degradation to the diffraction and strategies to mitigate those factors are addressed. We then address beamline and detector optimization as a means to improve the data quality. Crystallization is still a largely empirical process and our final chapters focus on the use of powder methods, where crystals are small, complementary techniques where we have no crystals at all and what the future holds with the advent of fourth generation X‐ray sources. Overall the book is aimed at both more experienced researchers and graduate students. We aim for it to become a reference work for all researchers in these interdisciplinary subjects on these topics.
Sander van Smaalen
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198570820
- eISBN:
- 9780191718762
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570820.001.0001
- Subject:
- Physics, Crystallography: Physics
Aperiodic crystals are crystalline materials with atomic structures that lack translational symmetry. This book gives a comprehensive account of the superspace theory for the description of the ...
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Aperiodic crystals are crystalline materials with atomic structures that lack translational symmetry. This book gives a comprehensive account of the superspace theory for the description of the crystal structures, and symmetries of incommensurately modulated crystals and composite crystals. It also gives a brief introduction to quasicrystals, thus providing the necessary background for understanding the distinctive features of aperiodic crystals, and it provides the tools for the application of quantitative methods from the realms of crystallography, solid state chemistry, and solid state physics to aperiodic crystal structures. The second half of the book is devoted to crystallographic methods of structural analysis of incommensurate crystals. Thorough accounts are given of the diffraction by incommensurate crystals, the choice of parameters in structure refinements, and the use of superspace in analysing crystal structures. The presentation of methods of structure determination includes direct methods, Fourier methods, Patterson function methods, the maximum entropy method (MEM), and charge flipping. So-called t-plots are introduced as a versatile method for the crystal chemical analysis of incommensurately modulated structures and composite crystals.Less
Aperiodic crystals are crystalline materials with atomic structures that lack translational symmetry. This book gives a comprehensive account of the superspace theory for the description of the crystal structures, and symmetries of incommensurately modulated crystals and composite crystals. It also gives a brief introduction to quasicrystals, thus providing the necessary background for understanding the distinctive features of aperiodic crystals, and it provides the tools for the application of quantitative methods from the realms of crystallography, solid state chemistry, and solid state physics to aperiodic crystal structures. The second half of the book is devoted to crystallographic methods of structural analysis of incommensurate crystals. Thorough accounts are given of the diffraction by incommensurate crystals, the choice of parameters in structure refinements, and the use of superspace in analysing crystal structures. The presentation of methods of structure determination includes direct methods, Fourier methods, Patterson function methods, the maximum entropy method (MEM), and charge flipping. So-called t-plots are introduced as a versatile method for the crystal chemical analysis of incommensurately modulated structures and composite crystals.
Naomi E. Chayen, John R. Helliwell, and Edward H. Snell
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199213252
- eISBN:
- 9780191707575
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213252.003.0020
- Subject:
- Physics, Crystallography: Physics
This chapter summarizes the state‐of‐the‐art in the field and discusses upcoming techniques to solve and to improve the current problems.
This chapter summarizes the state‐of‐the‐art in the field and discusses upcoming techniques to solve and to improve the current problems.
Naomi E. Chayen, John R. Helliwell, and Edward H. Snell
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199213252
- eISBN:
- 9780191707575
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213252.003.0015
- Subject:
- Physics, Crystallography: Physics
Unusual diffraction geometries may seem a curiosity but may stimulate novel avenues of application. Not least they illustrate a diversity of diffraction‐measuring possibilities. Laue diffraction ...
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Unusual diffraction geometries may seem a curiosity but may stimulate novel avenues of application. Not least they illustrate a diversity of diffraction‐measuring possibilities. Laue diffraction including 3‐dimensional detector arrangements is described. The particular congestion of neutron Laue diffraction patterns with big crystals is highlighted. The large‐angle oscillation technique is discussed including the principle with the Ewald sphere construction and practical examples of ‘LOT’ diffraction patterns. Ultra‐fine‐phi‐slicing with perfect or near‐perfect crystals is described. Particular success has been obtained with Laue diffraction where applications to time‐resolved structural intermediates using synchrotron radiation as well as hydrogen and hydration in macromolecular structure are described.Less
Unusual diffraction geometries may seem a curiosity but may stimulate novel avenues of application. Not least they illustrate a diversity of diffraction‐measuring possibilities. Laue diffraction including 3‐dimensional detector arrangements is described. The particular congestion of neutron Laue diffraction patterns with big crystals is highlighted. The large‐angle oscillation technique is discussed including the principle with the Ewald sphere construction and practical examples of ‘LOT’ diffraction patterns. Ultra‐fine‐phi‐slicing with perfect or near‐perfect crystals is described. Particular success has been obtained with Laue diffraction where applications to time‐resolved structural intermediates using synchrotron radiation as well as hydrogen and hydration in macromolecular structure are described.
R. E. Peierls
- Published in print:
- 2001
- Published Online:
- September 2007
- ISBN:
- 9780198507819
- eISBN:
- 9780191709913
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507819.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book develops the quantum theory of solids from the basic principles of quantum mechanics. The emphasis is on a single statement of the ideas underlying the various approximations that have to ...
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This book develops the quantum theory of solids from the basic principles of quantum mechanics. The emphasis is on a single statement of the ideas underlying the various approximations that have to be used in the study of this subject. Care is taken to separate sound arguments from conjecture. The treatment covers the electron theory of metals as well as the dynamics of crystals, including the author's work on the thermal conductivity of crystals.Less
This book develops the quantum theory of solids from the basic principles of quantum mechanics. The emphasis is on a single statement of the ideas underlying the various approximations that have to be used in the study of this subject. Care is taken to separate sound arguments from conjecture. The treatment covers the electron theory of metals as well as the dynamics of crystals, including the author's work on the thermal conductivity of crystals.
Erich H. Kisi and Christopher J. Howard
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780198515944
- eISBN:
- 9780191705663
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198515944.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book covers the theory, practicalities, and the extensive applications of neutron powder diffraction in materials science, physics, chemistry, mineralogy, and engineering. Various highlight ...
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This book covers the theory, practicalities, and the extensive applications of neutron powder diffraction in materials science, physics, chemistry, mineralogy, and engineering. Various highlight applications of neutron powder diffraction are outlined in the introduction, then the theory is developed and instrumentation described sufficient for a return to the applications. The book covers the use of neutron powder diffraction in the solution (hard) and refinement (more straightforward) of crystal and magnetic structures, applications of powder diffraction in quantitative phase analysis, extraction of microstructural information from powder diffraction patterns, and the applications of neutron diffraction in studies of elastic properties and for the measurement of residual stress. Additional theory to underpin these various applications is developed as required.Less
This book covers the theory, practicalities, and the extensive applications of neutron powder diffraction in materials science, physics, chemistry, mineralogy, and engineering. Various highlight applications of neutron powder diffraction are outlined in the introduction, then the theory is developed and instrumentation described sufficient for a return to the applications. The book covers the use of neutron powder diffraction in the solution (hard) and refinement (more straightforward) of crystal and magnetic structures, applications of powder diffraction in quantitative phase analysis, extraction of microstructural information from powder diffraction patterns, and the applications of neutron diffraction in studies of elastic properties and for the measurement of residual stress. Additional theory to underpin these various applications is developed as required.
Douglas L. Dorset
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529088
- eISBN:
- 9780191712838
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529088.001.0001
- Subject:
- Physics, Crystallography: Physics
This book describes the solid state behaviour of organic materials based the polymethylene chain, i.e., the functional molecular component of polyethylenes, soaps, detergents, edible fats, lipids, ...
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This book describes the solid state behaviour of organic materials based the polymethylene chain, i.e., the functional molecular component of polyethylenes, soaps, detergents, edible fats, lipids, oils, greases, and waxes. Along with chain unsaturation and branching, polydispersity, i.e., the aggregation of several polymethylene chain lengths, is shown to control various physical properties, including the preservation of metastable phases (polymorphic as well as ‘rotator’ forms). Using linear chain waxes as model materials, this book explores how solid solutions are stabilized and what structures are possible. Strictly linear molecules are compared to those functionalized with ‘head-groups’. The onset of fractionation, followed by formation of eutectic phases, is discussed, again describing the structures of favoured molecular assemblies. The rationale for polydisperse aggregation derives from the early work of A. I. Kitaigorodskii, demonstrating how certain homeomorphic parameters such as relative molecular shape and volume, as well as favoured crystalline polymorphs, lead to stable solid solutions. Relevant to high-molecular weight polymers, the influence of chain-folding is also discussed. A comprehensive review of known linear chain single crystal structures, including the alkanes, cycloalkanes, perfluoroalkanes, fatty alcohols, fatty acids, fatty acid esters, and cholesteryl esters, is presented to show how molecular shape, including chain branching, influences layer packing and co-solubility. Finally, a critique of previously suggested models for petroleum and natural wax assemblies is given, based on current crystallographic and spectroscopic information. This includes single crystal structures based on electron diffraction data. Although constrained to single chain molecules in the examples discussed, cited behaviour can be generalized to multiple chain-containing fats and lipids.Less
This book describes the solid state behaviour of organic materials based the polymethylene chain, i.e., the functional molecular component of polyethylenes, soaps, detergents, edible fats, lipids, oils, greases, and waxes. Along with chain unsaturation and branching, polydispersity, i.e., the aggregation of several polymethylene chain lengths, is shown to control various physical properties, including the preservation of metastable phases (polymorphic as well as ‘rotator’ forms). Using linear chain waxes as model materials, this book explores how solid solutions are stabilized and what structures are possible. Strictly linear molecules are compared to those functionalized with ‘head-groups’. The onset of fractionation, followed by formation of eutectic phases, is discussed, again describing the structures of favoured molecular assemblies. The rationale for polydisperse aggregation derives from the early work of A. I. Kitaigorodskii, demonstrating how certain homeomorphic parameters such as relative molecular shape and volume, as well as favoured crystalline polymorphs, lead to stable solid solutions. Relevant to high-molecular weight polymers, the influence of chain-folding is also discussed. A comprehensive review of known linear chain single crystal structures, including the alkanes, cycloalkanes, perfluoroalkanes, fatty alcohols, fatty acids, fatty acid esters, and cholesteryl esters, is presented to show how molecular shape, including chain branching, influences layer packing and co-solubility. Finally, a critique of previously suggested models for petroleum and natural wax assemblies is given, based on current crystallographic and spectroscopic information. This includes single crystal structures based on electron diffraction data. Although constrained to single chain molecules in the examples discussed, cited behaviour can be generalized to multiple chain-containing fats and lipids.
Ted Janssen, Gervais Chapuis, and Marc de Boissieu
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198567776
- eISBN:
- 9780191718335
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567776.001.0001
- Subject:
- Physics, Crystallography: Physics
Until the 1970s, all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the last decades a new class of solid state matter, called aperiodic crystals, has been found. ...
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Until the 1970s, all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the last decades a new class of solid state matter, called aperiodic crystals, has been found. It is a long range ordered structure, but without lattice periodicity. It is found in a wide range of materials: organic and anorganic compounds, minerals (including a substantial portion of the earths crust), and metallic alloys, under various pressures and temperatures. Because of the lack of periodicity, the usual techniques for the study of structure and physical properties no longer work, and new techniques have to be developed. This book deals with the characterization of the structure, the structure determination, and the study of the physical properties, especially dynamical and electronic properties of aperiodic crystals. The treatment is based on a description in a space with more dimensions than three, the so-called superspace. This allows us to generalise the standard crystallography and to look differently at the dynamics. The three main classes of aperiodic crystals, modulated phases, incommensurate composites, and quasicrystals are treated from a unified point of view, which stresses the similarities of the various systems.Less
Until the 1970s, all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the last decades a new class of solid state matter, called aperiodic crystals, has been found. It is a long range ordered structure, but without lattice periodicity. It is found in a wide range of materials: organic and anorganic compounds, minerals (including a substantial portion of the earths crust), and metallic alloys, under various pressures and temperatures. Because of the lack of periodicity, the usual techniques for the study of structure and physical properties no longer work, and new techniques have to be developed. This book deals with the characterization of the structure, the structure determination, and the study of the physical properties, especially dynamical and electronic properties of aperiodic crystals. The treatment is based on a description in a space with more dimensions than three, the so-called superspace. This allows us to generalise the standard crystallography and to look differently at the dynamics. The three main classes of aperiodic crystals, modulated phases, incommensurate composites, and quasicrystals are treated from a unified point of view, which stresses the similarities of the various systems.
Sander Van Smaalen
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198570820
- eISBN:
- 9780191718762
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570820.003.0009
- Subject:
- Physics, Crystallography: Physics
This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal ...
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This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal lattice in superspace, the point symmetry of the diffraction pattern, and the reflection conditions.Less
This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal lattice in superspace, the point symmetry of the diffraction pattern, and the reflection conditions.
Peter Müller, Regine Herbst-Irmer, Anthony L. Spek, Thomas R. Schneider, and Michael R. Sawaya
- Published in print:
- 2006
- Published Online:
- September 2010
- ISBN:
- 9780198570769
- eISBN:
- 9780191718809
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570769.001.0001
- Subject:
- Physics, Crystallography: Physics
This book covers advanced aspects of practical crystal structure refinement, focusing on practical problems in the everyday life of a crystallographer. After an introduction to SHELXL in the first ...
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This book covers advanced aspects of practical crystal structure refinement, focusing on practical problems in the everyday life of a crystallographer. After an introduction to SHELXL in the first chapter, the second chapter provides a brief survey of crystal structure refinement. The next few chapters address the various aspects of structure refinement, from the treatment of hydrogen atoms to the assignment of atom types, to disorder, to non-crystallographic symmetry and twinning. One chapter is dedicated to the refinement of macromolecular structures and two shorter chapters deal with structure validation. In most chapters, the book gives refinement examples, based on the program SHELXL, describing every problem in detail.Less
This book covers advanced aspects of practical crystal structure refinement, focusing on practical problems in the everyday life of a crystallographer. After an introduction to SHELXL in the first chapter, the second chapter provides a brief survey of crystal structure refinement. The next few chapters address the various aspects of structure refinement, from the treatment of hydrogen atoms to the assignment of atom types, to disorder, to non-crystallographic symmetry and twinning. One chapter is dedicated to the refinement of macromolecular structures and two shorter chapters deal with structure validation. In most chapters, the book gives refinement examples, based on the program SHELXL, describing every problem in detail.
Sergey V. Krivovichev
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199213207
- eISBN:
- 9780191707117
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213207.001.1
- Subject:
- Physics, Crystallography: Physics
This book deals with the structural crystallography of inorganic oxysalts in general. A special emphasis is placed upon structural topology and the methods of its description. The latter include ...
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This book deals with the structural crystallography of inorganic oxysalts in general. A special emphasis is placed upon structural topology and the methods of its description. The latter include graph theory, nets, 2-D and 3-D tilings, polyhedra, etc. The structures considered range from minerals to organically templated oxysalts, for all of which this book provides a unified approach to structure interpretation and classification. Most of the structures are analysed and it is shown that they possess the same topological genealogy and relationships, sometimes despite their obvious chemical differences. In order to expand the range of oxysalts considered, the book offers traditional schemes and also alternative approaches such as anion topologis, anion-centered polyhedra and cation arrays. It also looks into the amazingly complex and diverse world of inorganic oxysalts.Less
This book deals with the structural crystallography of inorganic oxysalts in general. A special emphasis is placed upon structural topology and the methods of its description. The latter include graph theory, nets, 2-D and 3-D tilings, polyhedra, etc. The structures considered range from minerals to organically templated oxysalts, for all of which this book provides a unified approach to structure interpretation and classification. Most of the structures are analysed and it is shown that they possess the same topological genealogy and relationships, sometimes despite their obvious chemical differences. In order to expand the range of oxysalts considered, the book offers traditional schemes and also alternative approaches such as anion topologis, anion-centered polyhedra and cation arrays. It also looks into the amazingly complex and diverse world of inorganic oxysalts.
William Clegg, Alexander J Blake, Jacqueline M Cole, John S O Evans, Peter Main, Simon Parsons, and David J Watkin (eds)
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.001.0001
- Subject:
- Physics, Crystallography: Physics
This book presents a less mathematical approach to X-ray crystal structure determination than is given in some detailed texts and concentrates on practical aspects. The book provides the necessary ...
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This book presents a less mathematical approach to X-ray crystal structure determination than is given in some detailed texts and concentrates on practical aspects. The book provides the necessary conceptual framework for understanding and applying the techniques described, but also gives practical advice on topics such as growing crystals, solving and refining structures, and understanding and using the results. There are also plenty of worked examples and problems provided (with answers), to reinforce the material presented. The book is based on the intensive course run by the Chemical Crystallography Group of the British Crystallographic Association every two years, and the material is drawn from the 2007 and 2009 courses. Much of the material of the first edition in 2001 has been significantly updated and expanded, and some new topics have been added. The approach to several of the topics is somewhat different as a result of changes in the authorship and the course teaching team. These changes reflect developments in the subject.Less
This book presents a less mathematical approach to X-ray crystal structure determination than is given in some detailed texts and concentrates on practical aspects. The book provides the necessary conceptual framework for understanding and applying the techniques described, but also gives practical advice on topics such as growing crystals, solving and refining structures, and understanding and using the results. There are also plenty of worked examples and problems provided (with answers), to reinforce the material presented. The book is based on the intensive course run by the Chemical Crystallography Group of the British Crystallographic Association every two years, and the material is drawn from the 2007 and 2009 courses. Much of the material of the first edition in 2001 has been significantly updated and expanded, and some new topics have been added. The approach to several of the topics is somewhat different as a result of changes in the authorship and the course teaching team. These changes reflect developments in the subject.
Antônio M. Figueiredo Neto and Silvio R. A. Salinas
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198525509
- eISBN:
- 9780191711756
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525509.001.0001
- Subject:
- Physics, Crystallography: Physics
Mixtures of surfactant or amphiphilic molecules and solvents are known to display a large number of lyotropic mesophases. Although the physics of thermotropic liquid crystals has been vastly ...
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Mixtures of surfactant or amphiphilic molecules and solvents are known to display a large number of lyotropic mesophases. Although the physics of thermotropic liquid crystals has been vastly discussed in the literature, lyotropic mesophases have been much less explored. This book gives a unifying presentation of the structural and physical properties of lyotropic liquid crystalline systems, including a set of experimental results and a discussion of the characterization of different structures and the corresponding phase transitions. The initial chapters contain a description of the main experimental results and techniques associated with the characterization of lyotropic mixtures, and an overview of some theoretical approaches to account for the phase transitions and phase diagrams of these mesophases. This book includes discussions on periodically organized lyotropic systems, micellar and bicontinuous structures, modulated and cholesteric structures, and a number of related developments (magnetic colloids, microemulsions, films, and lyo-banana mesophases).Less
Mixtures of surfactant or amphiphilic molecules and solvents are known to display a large number of lyotropic mesophases. Although the physics of thermotropic liquid crystals has been vastly discussed in the literature, lyotropic mesophases have been much less explored. This book gives a unifying presentation of the structural and physical properties of lyotropic liquid crystalline systems, including a set of experimental results and a discussion of the characterization of different structures and the corresponding phase transitions. The initial chapters contain a description of the main experimental results and techniques associated with the characterization of lyotropic mixtures, and an overview of some theoretical approaches to account for the phase transitions and phase diagrams of these mesophases. This book includes discussions on periodically organized lyotropic systems, micellar and bicontinuous structures, modulated and cholesteric structures, and a number of related developments (magnetic colloids, microemulsions, films, and lyo-banana mesophases).
Victor F. Petrenko and Robert W. Whitworth
- Published in print:
- 2002
- Published Online:
- February 2010
- ISBN:
- 9780198518945
- eISBN:
- 9780191707247
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198518945.001.0001
- Subject:
- Physics, Crystallography: Physics
Ice is one of the most abundant and environmentally important materials on Earth, and its unique and intriguing physical properties present fascinating areas of study. This book takes as its subject ...
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Ice is one of the most abundant and environmentally important materials on Earth, and its unique and intriguing physical properties present fascinating areas of study. This book takes as its subject the physics of ice: the properties of the material itself and the ways in which these properties are interpreted in terms of water molecules and crystalline structure. Although ice has a simple crystal structure its hydrogen bonding results in unique properties, which continue to be the subject of active research. An understanding of these properties is essential in fields such as glaciology, ice mechanics, ice adhesion, the dating of ice cores in Antarctica and Greenland, meteorology, thunderstorm electricity, and the study of comets or the icy moons of the outer planets in the solar system. In this book the physical principles underlying the properties of ice are carefully developed. Much work on important topics such as the current understanding of the electrical, mechanical, and surface properties of ice, or the occurrence of many different crystalline phases, are developed.Less
Ice is one of the most abundant and environmentally important materials on Earth, and its unique and intriguing physical properties present fascinating areas of study. This book takes as its subject the physics of ice: the properties of the material itself and the ways in which these properties are interpreted in terms of water molecules and crystalline structure. Although ice has a simple crystal structure its hydrogen bonding results in unique properties, which continue to be the subject of active research. An understanding of these properties is essential in fields such as glaciology, ice mechanics, ice adhesion, the dating of ice cores in Antarctica and Greenland, meteorology, thunderstorm electricity, and the study of comets or the icy moons of the outer planets in the solar system. In this book the physical principles underlying the properties of ice are carefully developed. Much work on important topics such as the current understanding of the electrical, mechanical, and surface properties of ice, or the occurrence of many different crystalline phases, are developed.
GAUTAM R. DESIRAJU and THOMAS STEINER
- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198509707
- eISBN:
- 9780191708206
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509707.003.0006
- Subject:
- Physics, Crystallography: Physics
The weak hydrogen bond was first identified in 1935, but it was only in the early 1990s that it really permeated into the consciousness of chemists and biologists. It only seems natural that this ...
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The weak hydrogen bond was first identified in 1935, but it was only in the early 1990s that it really permeated into the consciousness of chemists and biologists. It only seems natural that this interaction was explored first using spectroscopy, followed by crystallography. In structural supramolecular chemistry, a crystal structure is often not the result of hierarchic interaction preferences but a convolution of a large number of strong and weak interactions, each of which affect the rest intimately. Methods for codification of crystal structures must take this into account if they are to be accurate and useful. Of course, the goal of a subject like crystal engineering is to design systems where the interaction preferences are hierarchic, or in other words where the interaction interference is at a minimum. However, most crystal structures are not so predictable and the challenge posed by weak hydrogen bonding effects to the dogma of crystal engineering remains a real one.Less
The weak hydrogen bond was first identified in 1935, but it was only in the early 1990s that it really permeated into the consciousness of chemists and biologists. It only seems natural that this interaction was explored first using spectroscopy, followed by crystallography. In structural supramolecular chemistry, a crystal structure is often not the result of hierarchic interaction preferences but a convolution of a large number of strong and weak interactions, each of which affect the rest intimately. Methods for codification of crystal structures must take this into account if they are to be accurate and useful. Of course, the goal of a subject like crystal engineering is to design systems where the interaction preferences are hierarchic, or in other words where the interaction interference is at a minimum. However, most crystal structures are not so predictable and the challenge posed by weak hydrogen bonding effects to the dogma of crystal engineering remains a real one.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0002
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and ...
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This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ-part of the multiple Dirichlet series in terms of crystal graphs.Less
This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ-part of the multiple Dirichlet series in terms of crystal graphs.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0018
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter translates Statements A and B into Statements A′ and B′ in the language of crystal bases, and explains in this language how Statement B′ implies Statement A′. It first introduces the ...
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This chapter translates Statements A and B into Statements A′ and B′ in the language of crystal bases, and explains in this language how Statement B′ implies Statement A′. It first introduces the relevant definition, which is provisional since it assumes that we can give an appropriate definition of boxing and circling for Ω. The crystal graph formulation in Statement A′ is somewhat simpler than its Gelfand-Tsetlin counterpart. In particular, in the formulation of Statement A, there were two different Gelfand-Tsetlin patterns that were related by the Schützenberger involution. In the crystal graph formulation, different decompositions of the long element simply result in different paths from the same vertex v to the lowest weight vector.Less
This chapter translates Statements A and B into Statements A′ and B′ in the language of crystal bases, and explains in this language how Statement B′ implies Statement A′. It first introduces the relevant definition, which is provisional since it assumes that we can give an appropriate definition of boxing and circling for Ω. The crystal graph formulation in Statement A′ is somewhat simpler than its Gelfand-Tsetlin counterpart. In particular, in the formulation of Statement A, there were two different Gelfand-Tsetlin patterns that were related by the Schützenberger involution. In the crystal graph formulation, different decompositions of the long element simply result in different paths from the same vertex v to the lowest weight vector.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, ...
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Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.Less
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.
Apala Majumdar, Jonathan Robbins, and Maxim Zyskin
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0014
- Subject:
- Mathematics, Probability / Statistics, Analysis
This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet ...
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This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet energy of homo-topy classes of such harmonic maps, subject to tangent boundary conditions, and investigate lower and upper bounds for this Dirichlet energy on each homotopy class; local minimisers of this energy correspond to equilibrium and metastable configurations. A lower bound for the infimum Dirichlet energy for a given homotopy class is obtained as a sum of minimal connections between fractional defects at the vertices. In certain cases, this lower bound can be improved. For a rectangular prism, upper bounds are obtained from locally conformal solutions of the Euler-Lagrange equations, with the ratio of the upper and lower bounds bounded independently of homotopy type.Less
This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet energy of homo-topy classes of such harmonic maps, subject to tangent boundary conditions, and investigate lower and upper bounds for this Dirichlet energy on each homotopy class; local minimisers of this energy correspond to equilibrium and metastable configurations. A lower bound for the infimum Dirichlet energy for a given homotopy class is obtained as a sum of minimal connections between fractional defects at the vertices. In certain cases, this lower bound can be improved. For a rectangular prism, upper bounds are obtained from locally conformal solutions of the Euler-Lagrange equations, with the ratio of the upper and lower bounds bounded independently of homotopy type.
