Structure of crystalline solids

X-rays Electromagnetic radiation with wavelengths ranging between 10"10 and lO cm. X-rays diffraction A physical method for determining the structure of crystalline solids by exposing the solids to X-rays and then studying the varying intensity of the difracted rays due to interference effects. 

A general presentation and discussion of the origin of structure of crystalline solids and of the structural stability of compounds and solid solutions was given by Villars (1995) and Pettifor (1995). For an introduction to the electronic structure of extended systems, see Hoffmann (1987, 1988). In this chapter a brief sampling of some useful semi-empirical correlations and, respectively, of methods of classifying (predicting) phase and structure formation will be summarized. 

When studying the electronic structure of crystalline solids, physicists tend to think in terms of the mathematical concept of electronic bands, the so-called dispersion... 

All of the other chapters in this book deal with the symmetries of finite (discrete) objects. We now turn to the symmetry properties of infinite arrays. The end use for the concepts to be developed here is in understanding the rules governing the structures of crystalline solids. While an individual crystal is obviously not infinite, the atoms, ions, or molecules within it arrange themselves as though they were part of an infinite array. Only at, or very close to, the surface is this not the case this surface effect does not, in practice, diminish the utility of the theory to be developed. 

In this chapter we mainly deal with the microscopic structure of crystalline solid surfaces and with the methods used to analyze this structure and the chemical composition (introductions are Refs. [309-311]). 

Ice (H20) is a molecular crystalline solid. Six water molecules bond to each other to form a hexagonal pattern. This pattern is reflected in the hexagonal geometry exhibited in snowflakes. In Activity 4.1, you will construct models of basic crystalline solids and grow molecular-solid crystals. Then you will consider the basic structures of crystalline solids and look upon these structures as three-dimensional works of art. 

Research three-dimensional works of art that resemble the atomic packing arrangements of crystalline solids and explain the artwork in terms of the structures of crystalline solids. 

Many works of art, two-dimensional as well as three-dimensional, depict subject matter resembling (or resemble, in the case of sculpture) structures of crystalline solids, such as the following ... 

Students will discuss the differences and similarities in the four bonding structures of crystalline solids using the three-dimensional clay artwork. 

Raman spectroscopy has seldom been used for elucidating structures of crystalline solids. However, with IR, it has proved valuable for observing isostructural rare-earth borates (110) and for measurement of phonon coupling in the zinc borate Zn40(B02)8 (304). 

Bravais lattice — used to describe atomic structure of crystalline -> solid materials [i,ii], is an infinite array of points generated by a set of discrete translation operations, providing the same arrangement and orientation when viewed from any lattice point. A three-dimensional Bravais lattice consists of all points with position vectors R ... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

Solids are classified as crystalline solids or amorphous solids. Crystalline solids, such as an ice cube or a sodium chloride crystal, have a definite melting point. Amorphous solids, such as a chocolate bar or glass, get softer and softer as the temperature is raised. The structures of crystalline solids feature regularly repeating arrangements of the constituent particles. The structure of amorphous solids is not regular, but something like that of liquids sometimes, amorphous solids are called supercooled liquids. ... 

The structures of crystalline solids are most commonly determined by X-ray diffraction. Diffraction occurs when beams of light are scattered from a regular... 

X-ray diffraction a technique for establishing the structures of crystalline solids by directing X rays of a single wavelength at a crystal and obtaining a diffraction pattern from which interatomic spaces can be determined. (16.3)... 

Both X-ray and neutron diffraction methods are applied to determine the structure of crystalline solid hydrates. Because of the very small scattering cross-section of hydrogen atoms for X-rays it is much desirable to solve the crystal structure by means of neutron diffraction techniques. 

The concepts of crystalline state and symmetry are just about synonymous today, although the general sense of symmetry is much older than the idea of symmetrical arrangement of atoms in the structures of crystalline solids. Following Webster s dictionary, symmetry is the beauty of form arising from balanced proportions , and to be symmetrical is to have the correspondence in size, shape and relative position or parts on opposite sides of a dividing line or median plane or about a center or axis . 

X ray Electromagnetic radiation of shorter wavelength X-ray diffraction An analytical technique used to than ultraviolet radiation (10-11 m to 10-9 m or 0.01 nm determine the structures of crystalline solids. 

An immediate and most important consequence of the band structure of crystalline solids is the distinction between metals and nonmetals that reflects the position of the Fermi energy vis-a-vis the band energy. Before addressing this issue it is important to consider the energy scales involved. The following points are relevant for this consideration ... 

The band structure of crystalline solids is usually obtained by solving the Schrodinger equation of an approximate one-electron problem. In the case of non-metallic materials, such as semiconductors and insulators, there are essentially no free electrons. This problem is taken care of by the Bloch theorem. This important theorem assumes a potential energy profile V(r) being periodic with the periodicity of the lattice. In this case the Schrodinger equation is given by... 

The X-ray diffraction technique is the most commonly used experimental method for investigating the crystal structures of crystalline solids. As the underlying theory and methods are detailed in several specific textbooks (e.g.. Ref. [1]), only a brief description of the essential features will be provided at this point. 

Electronic structures of crystalline solids are mostly calculated on the basis of DFT. In this approach an open-shell system is described by spin polarized electronic band structures, in which the up-spin and down-spin bands are allowed to have different orbital... 

This is one of the few areas of Mn" chemistry for which a significant amount of structural data is available. Yet, the only pattern to emerge is one of variety of structure, mainly polymeric, and the absence of any hard evidence for a dimeric tetra-bridged species of the copper(II) acetate structure. There is a distinct impression, when first following through the literature, of encountering with almost each new structure some unexpected, novel feature, and the data certainly underline the sheer futility of attempting to define structure of crystalline solids by other than X-ray methods. 

The structures of crystalline solids are most commonly determined by X-ray diffraction. Diffraction occurs when beams of light are scattered from a regular array of points in which the spacings between the components are comparable with the wavelength of the light. Diffraction is due to constructive interference when the waves of parallel beams are in phase and to destructive interference when the waves are out of phase. 

Henderson, Eds., Academic Press, New York, 1975, p. 147. Hartree-Fock Studies of Electronic Structures of Crystalline Solids. 

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