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Electron diffraction (ED)
Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov
in Electron Crystallography: Electron Microscopy and Electron Diffraction
Physics, Crystallography: Physics
Electron diffraction comes from the combined scattering from many atoms arranged in a crystal. Bragg’s law and the Ewald sphere explain where the diffraction spots will fall on a detector and when ... More
Electron diffraction comes from the combined scattering from many atoms arranged in a crystal. Bragg’s law and the Ewald sphere explain where the diffraction spots will fall on a detector and when the different reflections will be excited. Phase identification and zone-axis orientation is done from one or a few ED patterns. Determination of the unit-cell dimensions come from three main zone axes, high-order Laue zones (HOLZ) or a tilt series. Precession geometry is explained with ray paths and snap-shots from the collection of precession patterns in the TEM. Digital (by software) and analogue (by hardware) precession are compared. The geometrical factors that influence intensities, i.e. the Lorentz factor are given. The very latest techniques for collecting complete 3D ED data is shown, with the two existing geometries, automated diffraction tomography and the rotation method. Convergent-beam electron diffraction (CBED) is briefly introduced.Less
Xiaodong ZouSven HovmöllerPeter Oleynikov
Electron diffraction comes from the combined scattering from many atoms arranged in a crystal. Bragg’s law and the Ewald sphere explain where the diffraction spots will fall on a detector and when the different reflections will be excited. Phase identification and zone-axis orientation is done from one or a few ED patterns. Determination of the unit-cell dimensions come from three main zone axes, high-order Laue zones (HOLZ) or a tilt series. Precession geometry is explained with ray paths and snap-shots from the collection of precession patterns in the TEM. Digital (by software) and analogue (by hardware) precession are compared. The geometrical factors that influence intensities, i.e. the Lorentz factor are given. The very latest techniques for collecting complete 3D ED data is shown, with the two existing geometries, automated diffraction tomography and the rotation method. Convergent-beam electron diffraction (CBED) is briefly introduced.
Keywords: electron diffraction, Bragg’s law, Ewald sphere, phase identification, Laue zones, indexation, unit-cell determination, precession electron diffraction, 3D electron diffraction, convergent-beam electron diffraction
Electron Crystallography: Electron Microscopy and Electron Diffraction
This is the first textbook describing crystal structure determination (especially inorganic) from high-resolution transmission electron microscopy (HRTEM) and electron diffraction (ED). The ... More
This is the first textbook describing crystal structure determination (especially inorganic) from high-resolution transmission electron microscopy (HRTEM) and electron diffraction (ED). The theoretical background and practical procedures are explained with hundreds of figures. The fundamental fact that the crystallographic structure factor phase information is present in EM images is explained by formulas, and HRTEM images and their Fourier transforms.The basics of crystallography are presented in an innovative way compared to standard crystallographic texts, exploiting direct imaging of structures at atomic resolution – impossible with X-ray crystallography. Moving back and forth between real and reciprocal space facilitates a deep understanding of the subject. EM images and ED patterns illustrate the symmetries in crystals.Electron optics and distortion (and how they can be quantified and corrected for) are explained with drawings, formulas and experimental EM images and ED patterns.New developments including precession electron diffraction and 3D reconstruction from HRTEM images and ways of collecting complete 3D ED data are thoroughly covered.We hope this textbook will become the standard for those who investigate crystal structures by EM but want to improve their skills, including going from 2D projections to full 3D structures. It introduces electron crystallography for all crystallographers and materials scientists, who are struggling with crystals, too small or imperfect for single-crystal X-ray diffraction and too complex for X-ray powder diffraction.We believe the easiest way to learn X-ray crystallography is to go via electron crystallography, since you can see both the structure image and the diffraction pattern. We have written this book also with these readers in mind, making reference to X-ray crystallography at every stage. Extensive appendices with worked examples makes the book ideal for courses and self-study.Less
Electron Crystallography : Electron Microscopy and Electron Diffraction
This is the first textbook describing crystal structure determination (especially inorganic) from high-resolution transmission electron microscopy (HRTEM) and electron diffraction (ED). The theoretical background and practical procedures are explained with hundreds of figures. The fundamental fact that the crystallographic structure factor phase information is present in EM images is explained by formulas, and HRTEM images and their Fourier transforms.The basics of crystallography are presented in an innovative way compared to standard crystallographic texts, exploiting direct imaging of structures at atomic resolution – impossible with X-ray crystallography. Moving back and forth between real and reciprocal space facilitates a deep understanding of the subject. EM images and ED patterns illustrate the symmetries in crystals.Electron optics and distortion (and how they can be quantified and corrected for) are explained with drawings, formulas and experimental EM images and ED patterns.New developments including precession electron diffraction and 3D reconstruction from HRTEM images and ways of collecting complete 3D ED data are thoroughly covered.We hope this textbook will become the standard for those who investigate crystal structures by EM but want to improve their skills, including going from 2D projections to full 3D structures. It introduces electron crystallography for all crystallographers and materials scientists, who are struggling with crystals, too small or imperfect for single-crystal X-ray diffraction and too complex for X-ray powder diffraction.We believe the easiest way to learn X-ray crystallography is to go via electron crystallography, since you can see both the structure image and the diffraction pattern. We have written this book also with these readers in mind, making reference to X-ray crystallography at every stage. Extensive appendices with worked examples makes the book ideal for courses and self-study.
Keywords: electron crystallography, crystal structure, electron microscopy, electron diffraction, crystallography, high-resolution transmission electron microscopy, crystal structure factor, symmetry, precession electron diffraction, 3D reconstruction
Electron crystallography ‐ an introduction
The historical introduction covers the parallel developments in EM on inorganic and biological samples. Vainshtein, Pinsker and Zvyagin in Moscow solved structures from electron-diffraction patterns ... More
The historical introduction covers the parallel developments in EM on inorganic and biological samples. Vainshtein, Pinsker and Zvyagin in Moscow solved structures from electron-diffraction patterns in the 1950s. Aaron Klug in 1968 pioneered computer image processing of EM images and realised that the crystallographic structure factor phase information can be read out directly in numbers from the Fourier transform. A few years later the EM optics was powerful enough to allow metals to be seen in oxides. The first determination of atomic coordinates in a crystal from EM images was done in 1984. Compared to X-ray crystallography, electron microscopy has some advantages; crystals a million times smaller (even defects) can be studied and the crystallographic structure factor phases can be read out numerically. Drawbacks are radiation damage (especially for organics) and multiple diffraction.Less
The historical introduction covers the parallel developments in EM on inorganic and biological samples. Vainshtein, Pinsker and Zvyagin in Moscow solved structures from electron-diffraction patterns in the 1950s. Aaron Klug in 1968 pioneered computer image processing of EM images and realised that the crystallographic structure factor phase information can be read out directly in numbers from the Fourier transform. A few years later the EM optics was powerful enough to allow metals to be seen in oxides. The first determination of atomic coordinates in a crystal from EM images was done in 1984. Compared to X-ray crystallography, electron microscopy has some advantages; crystals a million times smaller (even defects) can be studied and the crystallographic structure factor phases can be read out numerically. Drawbacks are radiation damage (especially for organics) and multiple diffraction.
Keywords: electron crystallography, history of electron crystallography, crystal structure, disordered compounds, crystallographic structure factor phase, electron microscopy, electron diffraction, crystallography, electron microscopy image, crystal structure factor, image processing
Diffraction of Electrons and Neutrons
Kannan M. Krishnan
in Principles of Materials Characterization and Metrology
Physics, Condensed Matter Physics / Materials
Electron scattering, significantly stronger than that for X-rays, is sensitive to surfaces and small volumes of materials. Low-energy electron diffraction (LEED) provides information on surface ... More
Electron scattering, significantly stronger than that for X-rays, is sensitive to surfaces and small volumes of materials. Low-energy electron diffraction (LEED) provides information on surface reconstruction and the arrangement of adsorbed atoms. Reflection high energy electron diffraction (RHEED) probes surface crystallography, and monitors, in situ, mechanisms of thin film growth. Transmission electron diffraction reveals a planar cross-section of the reciprocal lattice, where intensities are products of the structure and lattice amplitude factors determined by the overall shape of the specimen. The amplitude of any diffracted beam at the exit surface oscillates with thickness (fringes) and the excitation error (bend contours). Selected area diffraction produce spot or ring patterns, where low-index zone-axis orientations reflect the symmetry of the crystal, and double-diffraction shows positive intensities even for reflections forbidden by extinction rules. Kikuchi lines appear as pairs of dark and bright lines, and help in tilting the specimen. A focused probe produces convergent beam electron diffraction (CBED), useful for symmetry analysis at nanoscale resolution. Neutrons interact with the nucleus and the magnetic moment of the atom via the spin of the neutron; the latter finds particular use in studies of magnetic order. The atomic scattering factor for neutrons shows negligible angular dependence.Less
Electron scattering, significantly stronger than that for X-rays, is sensitive to surfaces and small volumes of materials. Low-energy electron diffraction (LEED) provides information on surface reconstruction and the arrangement of adsorbed atoms. Reflection high energy electron diffraction (RHEED) probes surface crystallography, and monitors, in situ, mechanisms of thin film growth. Transmission electron diffraction reveals a planar cross-section of the reciprocal lattice, where intensities are products of the structure and lattice amplitude factors determined by the overall shape of the specimen. The amplitude of any diffracted beam at the exit surface oscillates with thickness (fringes) and the excitation error (bend contours). Selected area diffraction produce spot or ring patterns, where low-index zone-axis orientations reflect the symmetry of the crystal, and double-diffraction shows positive intensities even for reflections forbidden by extinction rules. Kikuchi lines appear as pairs of dark and bright lines, and help in tilting the specimen. A focused probe produces convergent beam electron diffraction (CBED), useful for symmetry analysis at nanoscale resolution. Neutrons interact with the nucleus and the magnetic moment of the atom via the spin of the neutron; the latter finds particular use in studies of magnetic order. The atomic scattering factor for neutrons shows negligible angular dependence.
Keywords: Low energy electron diffraction (LEED), Reflection high energy electron diffraction (RHEED), Surface crystallography and reconstruction, Rheed oscillations in thin film growth, Lattice amplitude factors, Column approximation, Kinematic and dynamical scattering, Selected area diffraction, Convergent beam electron diffraction (CBED), Neutron scattering
Experimental methods in X‐ray and neutron crystallography
Monaco Hugo L. and Gilberto Artioli
in Fundamentals of Crystallography
Physics, Crystallography: Physics, Condensed Matter Physics / Materials
This chapter discusses the experimental methods used to study the diffraction of X-rays and neutrons by crystalline materials. Although electrons are also diffracted by crystals, electron diffraction ... More
This chapter discusses the experimental methods used to study the diffraction of X-rays and neutrons by crystalline materials. Although electrons are also diffracted by crystals, electron diffraction is not discussed in any detail here. The chapter begins by describing how X-rays and neutrons are produced and how one can define the beam of radiation that will interact with the crystalline sample. The specimens that receive attention are single crystals and polycrystalline materials, the latter being ensembles of a large number of small single crystals or aggregates of small coherently scattering crystalline domains. The chapter proceeds by discussing in separate sections methods used to record diffraction patterns and to measure quantitatively the intensity of radiation scattered by these two types of specimen.Less
Monaco Hugo L.Gilberto Artioli
This chapter discusses the experimental methods used to study the diffraction of X-rays and neutrons by crystalline materials. Although electrons are also diffracted by crystals, electron diffraction is not discussed in any detail here. The chapter begins by describing how X-rays and neutrons are produced and how one can define the beam of radiation that will interact with the crystalline sample. The specimens that receive attention are single crystals and polycrystalline materials, the latter being ensembles of a large number of small single crystals or aggregates of small coherently scattering crystalline domains. The chapter proceeds by discussing in separate sections methods used to record diffraction patterns and to measure quantitatively the intensity of radiation scattered by these two types of specimen.
Keywords: experimental methods, diffraction, X-rays, neutrons, crystalline materials, electron diffraction
Structure refinement
Structure refinement is a way to get more accurate atomic coordinates using ED intensities. A preliminary structure obtained from HRTEM or ED data can be refined with ED or X-ray powder diffraction ... More
Structure refinement is a way to get more accurate atomic coordinates using ED intensities. A preliminary structure obtained from HRTEM or ED data can be refined with ED or X-ray powder diffraction data. R-values are used to follow how the refinement progresses towards a better structure. The importance of high-quality data and data completeness are stressed. After refinement, the atomic coordinates can reach an accuracy of 0.02 Å.Less
Structure refinement is a way to get more accurate atomic coordinates using ED intensities. A preliminary structure obtained from HRTEM or ED data can be refined with ED or X-ray powder diffraction data. R-values are used to follow how the refinement progresses towards a better structure. The importance of high-quality data and data completeness are stressed. After refinement, the atomic coordinates can reach an accuracy of 0.02 Å.
Keywords: structure refinement, electron diffraction, R-values, data quality, data completeness, X-ray powder diffraction, accurate atomic coordinates
Structure and bonding in the gaseous dihalides and dialkyl derivatives of the Group 2 and Group 12 metals
Arne Haaland
in Molecules and Models: The molecular structures of main group element compounds
This chapter opens with a discussion of the difficulties of establishing unequivocally whether a monomeric, gaseous Group 2 metal dihalide has a linear or angular equilibrium structure. The ... More
This chapter opens with a discussion of the difficulties of establishing unequivocally whether a monomeric, gaseous Group 2 metal dihalide has a linear or angular equilibrium structure. The accumulated experimental evidence shows that most of these triatomic molecules are linear, but gaseous CaF2, SrF2, SrCl2, and all the barium dihalides are angular. These angular structures are not in accord with the Valence Shell Electron Pair Repulsion (VSEPR) model, but may be rationalized in terms of the polarizable ion model or in terms of sd hybridization on the metal atom. Bond distances and mean bond energies are listed. The bonding may be described in terms of two delocalized (i.e., three-center) or two localized (i.e., two-center) bonding molecular orbitals. The gaseous dihalides of the Group 12 metal atoms are all linear. The bonding radius of Hg is found to be two or three pm smaller than that of Cd, an anomaly which may be due to a combination of the f-block contraction and relativistic effects.Less
This chapter opens with a discussion of the difficulties of establishing unequivocally whether a monomeric, gaseous Group 2 metal dihalide has a linear or angular equilibrium structure. The accumulated experimental evidence shows that most of these triatomic molecules are linear, but gaseous CaF2, SrF2, SrCl2, and all the barium dihalides are angular. These angular structures are not in accord with the Valence Shell Electron Pair Repulsion (VSEPR) model, but may be rationalized in terms of the polarizable ion model or in terms of sd hybridization on the metal atom. Bond distances and mean bond energies are listed. The bonding may be described in terms of two delocalized (i.e., three-center) or two localized (i.e., two-center) bonding molecular orbitals. The gaseous dihalides of the Group 12 metal atoms are all linear. The bonding radius of Hg is found to be two or three pm smaller than that of Cd, an anomaly which may be due to a combination of the f-block contraction and relativistic effects.
Keywords: alkaline earth metal dihalides, potential energy curves, dipole moment, gas electron diffraction, molecular beam deflection, angular structures, polarizable ion model, VSEPR model, Group 12 metal dihalides, Group 12 metal dimethyl derivatives
3D electron crystallography
All structures are 3-dimensional, but each EM image is just a 2D projection of the 3D structure. Until recently, almost all work in the EM field has been 2D. 3D electron crystallography can be made ... More
All structures are 3-dimensional, but each EM image is just a 2D projection of the 3D structure. Until recently, almost all work in the EM field has been 2D. 3D electron crystallography can be made by combining several HRTEM images and/or ED patterns taken from different orientations (zone axes). A crystal with all unit-cell dimensions over 5 Å will never give rise to projections that can be interpreted in terms of atomic coordinates. A 3D structure is needed. The procedure from image collection and interpretation, in terms of zone axis and projected symmetry, to combination of data into a 3D map, is demonstrated. All the steps are followed in great detail, using a very complex intermetallic structure as an example. HRTEM images (before and after image processing) and ED patterns from several zone axes are show together with a table of amplitudes and phases extracted from the HRTEM images and ED patterns. The remarkably clear 3D potential map is interpreted in terms of atomic structure.Less
All structures are 3-dimensional, but each EM image is just a 2D projection of the 3D structure. Until recently, almost all work in the EM field has been 2D. 3D electron crystallography can be made by combining several HRTEM images and/or ED patterns taken from different orientations (zone axes). A crystal with all unit-cell dimensions over 5 Å will never give rise to projections that can be interpreted in terms of atomic coordinates. A 3D structure is needed. The procedure from image collection and interpretation, in terms of zone axis and projected symmetry, to combination of data into a 3D map, is demonstrated. All the steps are followed in great detail, using a very complex intermetallic structure as an example. HRTEM images (before and after image processing) and ED patterns from several zone axes are show together with a table of amplitudes and phases extracted from the HRTEM images and ED patterns. The remarkably clear 3D potential map is interpreted in terms of atomic structure.
Keywords: 3D electron crystallography, 3D reconstruction, high-resolution electron microscopy, electron diffraction, combining 2D projections, zone axes, merging projections, intermetallic compounds, 3D potential map, 3D atomic coordinates, contrast transfer function, symmetry, origin determination
Jenny Pickworth Glusker and Kenneth N. Trueblood
in Crystal Structure Analysis: A Primer
Chemistry, Crystallography: Chemistry
A common approach to crystal structure analysis by X-ray diffraction presented in texts that have been written for nonspecialists involves the Bragg equation, and a discussion in terms of ... More
A common approach to crystal structure analysis by X-ray diffraction presented in texts that have been written for nonspecialists involves the Bragg equation, and a discussion in terms of “reflection” of X rays from crystal lattice planes (Bragg, 1913). While the Bragg equation, which implies this “reflection,” has proved extremely useful, it does not really help in understanding the process of X-ray diffraction. Therefore we will proceed instead by way of an elementary consideration of diffraction phenomena generally, and then diffraction from periodic structures (such as crystals), making use of optical analogies (Jenkins and White, 1957; Taylor and Lipson, 1964; Harburn et al., 1975). The eyes of most animals, including humans, comprise efficient optical systems for forming images of objects by the recombination of visible radiation scattered by these objects. Many things are, of course, too small to be detected by the unaided human eye, but an enlarged image of some of them can be formed with a microscope—using visible light for objects with dimensions comparable to or larger than the wavelength of this light (about 6 × 10−7 m), or using electrons of high energy (and thus short wavelength) in an electron microscope. In order to “see” the fine details of molecular structure (with dimensions 10−8 to 10−10 m), it is necessary to use radiation of a wavelength comparable to, or smaller than, the dimensions of the distances between atoms. Such radiation is readily available (1) in the X rays produced by bombarding a target composed of an element of intermediate atomic number (for example, between Cr and Mo in the Periodic Table) with fast electrons, or from a synchrotron source, (2) in neutrons from a nuclear reactor or spallation source, or (3) in electrons with energies of 10–50 keV. Each of these kinds of radiation is scattered by the atoms of the sample, just as is ordinary light, and if we could recombine this scattered radiation, as a microscope can, we could form an image of the scattering matter. Less
Jenny Pickworth GluskerKenneth N. Trueblood
A common approach to crystal structure analysis by X-ray diffraction presented in texts that have been written for nonspecialists involves the Bragg equation, and a discussion in terms of “reflection” of X rays from crystal lattice planes (Bragg, 1913). While the Bragg equation, which implies this “reflection,” has proved extremely useful, it does not really help in understanding the process of X-ray diffraction. Therefore we will proceed instead by way of an elementary consideration of diffraction phenomena generally, and then diffraction from periodic structures (such as crystals), making use of optical analogies (Jenkins and White, 1957; Taylor and Lipson, 1964; Harburn et al., 1975). The eyes of most animals, including humans, comprise efficient optical systems for forming images of objects by the recombination of visible radiation scattered by these objects. Many things are, of course, too small to be detected by the unaided human eye, but an enlarged image of some of them can be formed with a microscope—using visible light for objects with dimensions comparable to or larger than the wavelength of this light (about 6 × 10−7 m), or using electrons of high energy (and thus short wavelength) in an electron microscope. In order to “see” the fine details of molecular structure (with dimensions 10−8 to 10−10 m), it is necessary to use radiation of a wavelength comparable to, or smaller than, the dimensions of the distances between atoms. Such radiation is readily available (1) in the X rays produced by bombarding a target composed of an element of intermediate atomic number (for example, between Cr and Mo in the Periodic Table) with fast electrons, or from a synchrotron source, (2) in neutrons from a nuclear reactor or spallation source, or (3) in electrons with energies of 10–50 keV. Each of these kinds of radiation is scattered by the atoms of the sample, just as is ordinary light, and if we could recombine this scattered radiation, as a microscope can, we could form an image of the scattering matter.
Keywords: Bragg reflection, Bragg’s Law (Bragg equation) Laue method, X rays, X-ray diffraction, amplitude angles of incidence and reflection, constructive interference, crystal lattice, cubic unit cell destructive interference, diffraction, diffraction photograph, elastic scattering, electron diffraction, electron microscopy, electrons (X-ray scatterers in atoms), envelope (diffraction pattern), grating, diffraction, inelastic scattering, intensity, interference lattice, (crystal), lens microscope, molecular transform, monoclinic unit cell, myoglobin neutrons and neutron diffraction, optical diffraction pattern, order of diffraction phase, phase difference, phthalocyanine molecule polarized X rays, potassium chloride, precession camera and method, reciprocal space, relative phase angle, sampling regions (diffraction pattern), slit diffraction, synchrotron radiation translation (lattice), unit cell
Methods of study
Falko P. Netzer and Claudine Noguera
in Oxide Thin Films and Nanostructures
Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology
The experimental and theoretical characterization of oxide nanostructures is addressed. The experimental techniques are classified according to their information content, revealing atomic geometry, ... More
The experimental and theoretical characterization of oxide nanostructures is addressed. The experimental techniques are classified according to their information content, revealing atomic geometry, chemical composition, electronic structure as well as magnetic, vibrational and chemical properties. Due to the nanometer scale dimensions of oxide nanosystems, many experimental techniques are derived fom the field of surface science and involve ultrahigh vacuum technology. The quantum-theoretical simulations for the description of oxide materials are presented by progressing from simple to increasingly sophisticated methods; the latter become necessary to accurately treat electron correlation effects, which are significant in many oxide materials, in particular at low dimension. Electronic structure methods, total energy methods and atomic structure simulation methods are introduced and discussed.Less
Falko P. NetzerClaudine Noguera
The experimental and theoretical characterization of oxide nanostructures is addressed. The experimental techniques are classified according to their information content, revealing atomic geometry, chemical composition, electronic structure as well as magnetic, vibrational and chemical properties. Due to the nanometer scale dimensions of oxide nanosystems, many experimental techniques are derived fom the field of surface science and involve ultrahigh vacuum technology. The quantum-theoretical simulations for the description of oxide materials are presented by progressing from simple to increasingly sophisticated methods; the latter become necessary to accurately treat electron correlation effects, which are significant in many oxide materials, in particular at low dimension. Electronic structure methods, total energy methods and atomic structure simulation methods are introduced and discussed.
Keywords: Electron spectroscopy, electron microscopy, scanning probe microscopy, electron diffraction, X-ray diffraction, photon absorption, electronic structure methods, structural optimization
On Comparing Experimental and Calculated Structural Parameters
Lothar Schäfer and John D. Ewbank
in Molecular Orbital Calculations for Biological Systems
Chemistry, Quantum and Theoretical Chemistry
The tacit assumption underlying all science is that, of two competing theories, the one in closer agreement with experiment is the better one. In structural chemistry the same principle applies ... More
The tacit assumption underlying all science is that, of two competing theories, the one in closer agreement with experiment is the better one. In structural chemistry the same principle applies but, when calculated and experimental structures are compared, closer is not necessarily better. Structures from ab initio calculations, specifically, must not be the same as the experimental counterparts the way they are observed. This is so because ab initio geometries refer to nonexistent, vibrationless states at the minimum of potential energy, whereas structural observables represent specifically defined averages over distributions of vibrational states. In general, if one wants to make meaningful comparisons between calculated and experimental molecular structures, one must take recourse of statistical formalisms to describe the effects of vibration on the observed parameters. Among the parameters of interest to structural chemists, internuclear distances are especially important because other variables, such as bond angles, dihedral angles, and even crystal spacings, can be readily derived from them. However, how a rigid torsional angle derived from an ab initio calculation compares with the corresponding experimental value in a molecule subject to vibrational anharmonicity, is not so easy to determine. The same holds for the lattice parameters of a molecule in a dynamical crystal, and their temperature dependence as a function of the molecular potential energy surface. In contrast, vibrational effects are readily defined and best described for internuclear distances, bonded and non-bonded ones. In general, all observed internuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. To illustrate how the operations of an experimental technique affect the nature of its observables, gas electron diffraction shall be used as an example. Less
Lothar SchäferJohn D. Ewbank
The tacit assumption underlying all science is that, of two competing theories, the one in closer agreement with experiment is the better one. In structural chemistry the same principle applies but, when calculated and experimental structures are compared, closer is not necessarily better. Structures from ab initio calculations, specifically, must not be the same as the experimental counterparts the way they are observed. This is so because ab initio geometries refer to nonexistent, vibrationless states at the minimum of potential energy, whereas structural observables represent specifically defined averages over distributions of vibrational states. In general, if one wants to make meaningful comparisons between calculated and experimental molecular structures, one must take recourse of statistical formalisms to describe the effects of vibration on the observed parameters. Among the parameters of interest to structural chemists, internuclear distances are especially important because other variables, such as bond angles, dihedral angles, and even crystal spacings, can be readily derived from them. However, how a rigid torsional angle derived from an ab initio calculation compares with the corresponding experimental value in a molecule subject to vibrational anharmonicity, is not so easy to determine. The same holds for the lattice parameters of a molecule in a dynamical crystal, and their temperature dependence as a function of the molecular potential energy surface. In contrast, vibrational effects are readily defined and best described for internuclear distances, bonded and non-bonded ones. In general, all observed internuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. To illustrate how the operations of an experimental technique affect the nature of its observables, gas electron diffraction shall be used as an example.
Keywords: anharmonicity, crystalline state, electron diffraction operator, gas electron diffraction, internuclear distances, least squares diffraction, methyl hydrazinocarboxylate, protein crystallography, radial distribution function, vibration
Layer packing of polymethylene chains
DOUGLAS L. DORSET
in Crystallography of the Polymethylene Chain: An Inquiry into the Structure of Waxes
This chapter discusses energetic principles important for the close packing of linear polymethylene chains that lead to favoured layer packing arrays, revealing that a hard sphere repulsive model is ... More
This chapter discusses energetic principles important for the close packing of linear polymethylene chains that lead to favoured layer packing arrays, revealing that a hard sphere repulsive model is a good first approximation. The methylene subcell concept is introduced, both from a theoretical basis as well as from observation, based on known crystal structures of representative materials. Convenient methods for identifying methylene subcells, based on powder diffraction, electron diffraction, and infrared spectroscopy, are presented.Less
This chapter discusses energetic principles important for the close packing of linear polymethylene chains that lead to favoured layer packing arrays, revealing that a hard sphere repulsive model is a good first approximation. The methylene subcell concept is introduced, both from a theoretical basis as well as from observation, based on known crystal structures of representative materials. Convenient methods for identifying methylene subcells, based on powder diffraction, electron diffraction, and infrared spectroscopy, are presented.
Keywords: methylene subcells, crystal structures, powder diffraction, electron diffraction, infrared spectroscopy, subcell packing
Polydispersity and the paraffin chain—statement of the problem
This chapter introduces the problem of polydispersity for understanding properties of materials containing polymethylene chains. A model has been proposed for the assembly of molecular chains in a ... More
This chapter introduces the problem of polydispersity for understanding properties of materials containing polymethylene chains. A model has been proposed for the assembly of molecular chains in a petroleum waxes that will remain a focal point for later discussions in the book. The energetics of molecular crystalline polymorphism is discussed as are the methods used for the construction and interpretation of binary phase diagrams. Variances to symmetry rules for chain co-solubility proposed by A. I. Kitaigorodskii are presented for further consideration in later chapters.Less
This chapter introduces the problem of polydispersity for understanding properties of materials containing polymethylene chains. A model has been proposed for the assembly of molecular chains in a petroleum waxes that will remain a focal point for later discussions in the book. The energetics of molecular crystalline polymorphism is discussed as are the methods used for the construction and interpretation of binary phase diagrams. Variances to symmetry rules for chain co-solubility proposed by A. I. Kitaigorodskii are presented for further consideration in later chapters.
Keywords: chain polymorphism, co-solubility, binary phase diagrams, powder diffraction, electron diffraction, calorimetry
A scanning electron microscope (SEM) focuses an electron beam to a sharp probe, with its diameter, which depends on the acceleration voltage and the aberration coefficients of the probe-forming lens, ... More
A scanning electron microscope (SEM) focuses an electron beam to a sharp probe, with its diameter, which depends on the acceleration voltage and the aberration coefficients of the probe-forming lens, determining SEM resolution. This electron beam is scanned over the specimen and signals arising from a variety of beam-specimen interactions are recorded to form images using different detectors positioned in the specimen chamber. Secondary electrons, detected with the Everton-Thornley detector, reveal the topography and electrical properties; back-scattered electrons provide information about the average atomic number and local crystallography of the specimen. Ferromagnetic materials alter the trajectory of secondary (Type I) and back-scattered (Type II) electrons to provide magnetic contrast. The magnetic polarization of the secondary electrons can also be analyzed directly (SEMPA) to image domains. The electron beam also excites characteristic X-rays for chemical microanalysis. Luminescent specimens produce light (Cathodoluminescence); these photons provide information on the electronic structure, particularly the defect states, of the specimen. Environmental SEMs, with differential pumping, image the specimen in a gaseous environment and/or under hydration for biological materials. A SEM combined with a focused ion beam (FIB) column is used for nano-fabrication, including preparation of electron-transparent TEM specimens.Less
A scanning electron microscope (SEM) focuses an electron beam to a sharp probe, with its diameter, which depends on the acceleration voltage and the aberration coefficients of the probe-forming lens, determining SEM resolution. This electron beam is scanned over the specimen and signals arising from a variety of beam-specimen interactions are recorded to form images using different detectors positioned in the specimen chamber. Secondary electrons, detected with the Everton-Thornley detector, reveal the topography and electrical properties; back-scattered electrons provide information about the average atomic number and local crystallography of the specimen. Ferromagnetic materials alter the trajectory of secondary (Type I) and back-scattered (Type II) electrons to provide magnetic contrast. The magnetic polarization of the secondary electrons can also be analyzed directly (SEMPA) to image domains. The electron beam also excites characteristic X-rays for chemical microanalysis. Luminescent specimens produce light (Cathodoluminescence); these photons provide information on the electronic structure, particularly the defect states, of the specimen. Environmental SEMs, with differential pumping, image the specimen in a gaseous environment and/or under hydration for biological materials. A SEM combined with a focused ion beam (FIB) column is used for nano-fabrication, including preparation of electron-transparent TEM specimens.
Keywords: Scanning electron microscope, Everton-Thornley detector, Secondary and back-scattered electrons, Beam-specimen interactions, Electron back-scattered diffraction (EBSD), Type I and Type II magnetic contrats, Cathodoluminescence, Environmental SEM, Focussed ion beam milling, Nanofabrication
The surface of ice
Victor F. Petrenko and Robert W. Whitworth
in Physics of Ice
Especially at temperatures approaching the melting point, the surface of ice has properties significantly different from the bulk, and the surface is sometimes referred to as having a ‘liquid-like ... More
Especially at temperatures approaching the melting point, the surface of ice has properties significantly different from the bulk, and the surface is sometimes referred to as having a ‘liquid-like layer’. This chapter reviews many experiments to study this surface layer including scanning tunnelling microscopy, low energy electron diffraction, X-ray diffraction, optical ellipsometry, surface conductivity, surface potentials, nuclear magnetic resonance, and scanning force microscopy. Different experiments reveal unusual features to different depths and at different temperatures. Attempts at interpretation involve both thermodynamics and molecular modelling. The surface energy and the angle of contact with the liquid have been determined and are relevant to adhesion.Less
Victor F. PetrenkoRobert W. Whitworth
Especially at temperatures approaching the melting point, the surface of ice has properties significantly different from the bulk, and the surface is sometimes referred to as having a ‘liquid-like layer’. This chapter reviews many experiments to study this surface layer including scanning tunnelling microscopy, low energy electron diffraction, X-ray diffraction, optical ellipsometry, surface conductivity, surface potentials, nuclear magnetic resonance, and scanning force microscopy. Different experiments reveal unusual features to different depths and at different temperatures. Attempts at interpretation involve both thermodynamics and molecular modelling. The surface energy and the angle of contact with the liquid have been determined and are relevant to adhesion.
Keywords: ice, surface, liquid-like layer, scanning tunnelling microscopy, low energy electron diffraction, X-ray diffraction, ellipsometry, surface conductivity, scanning force microscopy, surface energy
in Title Pages
Much of our present knowledge of the architecture of molecules has been obtained from studies of the diffraction of X rays or neutrons by crystals. X rays are scattered by the electrons of atoms ... More
Much of our present knowledge of the architecture of molecules has been obtained from studies of the diffraction of X rays or neutrons by crystals. X rays are scattered by the electrons of atoms and ions, and the interference between the X rays scattered by the different atoms or ions. in a crystal can result in a diffraction pattern. Similarly, neutrons are scattered by the nuclei of atoms. Measurements on a crystal diffraction pattern can lead to information on the arrangement of atoms or ions within the crystal. This is the experimental technique to be described in this book. X-ray diffraction was first used to establish the three-dimensional arrangement of atoms in a crystal by William Lawrence Bragg in 1913 (Bragg, 1913), shortly after Wilhelm Conrad Röntgen had discovered X rays and Max von Laue had shown in 1912 that these X rays could be diffracted by crystals (Röntgen, 1895; Friedrich et al., 1912). Later, in 1927 and 1936 respectively, it was also shown that electrons and neutrons could be diffracted by crystals (Davisson and Germer, 1927; von Halban and Preiswerk, 1936; Mitchell and Powers, 1936). Bragg found from X-ray diffraction studies that, in crystals of sodium chloride, each sodium is surrounded by six equidistant chlorines and each chlorine by six equidistant sodiums. No discrete molecules of NaCl were found and therefore Bragg surmised that the crystal consisted of sodium ions and chloride ions rather than individual (noncharged) atoms (Bragg, 1913); this had been predicted earlier by William Barlow and William Jackson Pope (Barlow and Pope, 1907), but had not, prior to the research of the Braggs, been demonstrated experimentally. A decade and a half later, in 1928, Kathleen Lonsdale used X-ray diffraction methods to show that the benzene ring is a flat regular hexagon in which all carbon–carbon bonds are equal in length, rather than a ring structure that contains alternating single and double bonds (Lonsdale, 1928).Her experimental result, later confirmed by spectroscopic studies (Stoicheff, 1954), was of great significance in chemistry. Less
Much of our present knowledge of the architecture of molecules has been obtained from studies of the diffraction of X rays or neutrons by crystals. X rays are scattered by the electrons of atoms and ions, and the interference between the X rays scattered by the different atoms or ions. in a crystal can result in a diffraction pattern. Similarly, neutrons are scattered by the nuclei of atoms. Measurements on a crystal diffraction pattern can lead to information on the arrangement of atoms or ions within the crystal. This is the experimental technique to be described in this book. X-ray diffraction was first used to establish the three-dimensional arrangement of atoms in a crystal by William Lawrence Bragg in 1913 (Bragg, 1913), shortly after Wilhelm Conrad Röntgen had discovered X rays and Max von Laue had shown in 1912 that these X rays could be diffracted by crystals (Röntgen, 1895; Friedrich et al., 1912). Later, in 1927 and 1936 respectively, it was also shown that electrons and neutrons could be diffracted by crystals (Davisson and Germer, 1927; von Halban and Preiswerk, 1936; Mitchell and Powers, 1936). Bragg found from X-ray diffraction studies that, in crystals of sodium chloride, each sodium is surrounded by six equidistant chlorines and each chlorine by six equidistant sodiums. No discrete molecules of NaCl were found and therefore Bragg surmised that the crystal consisted of sodium ions and chloride ions rather than individual (noncharged) atoms (Bragg, 1913); this had been predicted earlier by William Barlow and William Jackson Pope (Barlow and Pope, 1907), but had not, prior to the research of the Braggs, been demonstrated experimentally. A decade and a half later, in 1928, Kathleen Lonsdale used X-ray diffraction methods to show that the benzene ring is a flat regular hexagon in which all carbon–carbon bonds are equal in length, rather than a ring structure that contains alternating single and double bonds (Lonsdale, 1928).Her experimental result, later confirmed by spectroscopic studies (Stoicheff, 1954), was of great significance in chemistry.
Keywords: Fourier synthesis, amplitude, benzene, electron diffraction, hexamethylbenzene, lens, microscope, phase, relative phase angle, sinusoidal wave
Micro- and noncrystalline materials
The crystalline state is characterized by a high degree of internal order. There are two types of order that we will discuss here. One is chemical order, which consists of the connectivity (bond ... More
The crystalline state is characterized by a high degree of internal order. There are two types of order that we will discuss here. One is chemical order, which consists of the connectivity (bond lengths and bond angles) and stoichiometry in organic and many inorganic molecules, or just stoichiometry in minerals, metals, and other such materials. Some degree of chemical ordering exists for any molecule consisting of more than one atom, and the molecular structure of chemically simple gas molecules can be determined by gaseous electron diffraction or by high-resolution infrared spectroscopy. The second type of order to be discussed is geometrical order, which is the regular arrangement of entities in space such as in cubes, cylinders, coiled coils, and many other arrangements. For a compound to be crystalline it is necessary for the geometrical order of the individual entities (which must each have the same overall conformation) to extend indefinitely (that is, apparently infinitely) in three dimensions such that a three-dimensional repeat unit can be defined from diffraction data. Single crystals of quartz, diamond, silicon, or potassium dihydrogen phosphate can be grown to be as large as six or more inches across. Imagine how many atoms or ions must be identically arranged to create such macroscopic perfection! Sometimes, however, this geometrical order does not extend very far, and microarrays of molecules or ions, while themselves ordered, are disordered with respect to each other on a macroscopic scale. In such a case the three-dimensional order does not extend far enough to give a sharp diffraction pattern. The crystal quality is then described as “poor” or the crystal is considered to be microcrystalline, as in the naturally occurring clay minerals. On the other hand, in certain solid materials the spatial extent of geometrical order may be less than three-dimensional, and this reduced order gives rise to interesting properties. For example, the geometrical order may exist only in two dimensions; this is the case for mica and graphite, which consist of planar structures with much weaker forces between the layers so that cleavage and slippage are readily observed. Less
The crystalline state is characterized by a high degree of internal order. There are two types of order that we will discuss here. One is chemical order, which consists of the connectivity (bond lengths and bond angles) and stoichiometry in organic and many inorganic molecules, or just stoichiometry in minerals, metals, and other such materials. Some degree of chemical ordering exists for any molecule consisting of more than one atom, and the molecular structure of chemically simple gas molecules can be determined by gaseous electron diffraction or by high-resolution infrared spectroscopy. The second type of order to be discussed is geometrical order, which is the regular arrangement of entities in space such as in cubes, cylinders, coiled coils, and many other arrangements. For a compound to be crystalline it is necessary for the geometrical order of the individual entities (which must each have the same overall conformation) to extend indefinitely (that is, apparently infinitely) in three dimensions such that a three-dimensional repeat unit can be defined from diffraction data. Single crystals of quartz, diamond, silicon, or potassium dihydrogen phosphate can be grown to be as large as six or more inches across. Imagine how many atoms or ions must be identically arranged to create such macroscopic perfection! Sometimes, however, this geometrical order does not extend very far, and microarrays of molecules or ions, while themselves ordered, are disordered with respect to each other on a macroscopic scale. In such a case the three-dimensional order does not extend far enough to give a sharp diffraction pattern. The crystal quality is then described as “poor” or the crystal is considered to be microcrystalline, as in the naturally occurring clay minerals. On the other hand, in certain solid materials the spatial extent of geometrical order may be less than three-dimensional, and this reduced order gives rise to interesting properties. For example, the geometrical order may exist only in two dimensions; this is the case for mica and graphite, which consist of planar structures with much weaker forces between the layers so that cleavage and slippage are readily observed.
Keywords: amorphous materials, biological macromolecules, diamond, electron diffraction, glass, hydrogen bond, liquid crystals, macromolecules, powder diffraction