Atomic space patterns

The dimensionality of the diffraction problem will have a strong effect on how the diffraction pattern appears. For example in a ID problem, e.g., diffraction from a single Une of atoms spaced apart, only the component ofS in the direction along the line is constrained. For a 2D problem, e.g., the one encountered in RHEED, two components of S in the plane of the surface are constrained. For a 3D problem, e.g., X-ray scattering from a bulk crystal, three components of S are constrained. 

Anyone who has seen the well-formed crystals of minerals in our museums must have been impressed by the great variety of shapes exhibited cubes and octahedra, prisms of various kinds, pyramids and double pyramids, flat plates of various shapes, rhombohedra and other less symmetrical parallelepipeda, and many other shapes less easy to describe in a word or two. These crystal shapes are extremely fascinating in themselves artists (notably Durer) have used crystal shapes for formal or symbolic purposes, while many a natural philosopher has been drawn to the attempt to understand first of all the geometry of crystal shapes considered simply as solid figures, and then the manner in which these shapes are formed by the anisotropic growth of atomic and molecular space-patterns. 

But this book has a practical object, as its title proclaims. Our purpose in this chapter is to inquire to what extent crystal shapes can be criteria for identification, and how much they tell us about the atomic and molecular space-patterns within them. 

This is one reason for studying crystal shapes. Another and more weighty reason is that crystal shapes tell us a great deal about the relative dimensions and the symmetries of the atomic and molecular space-patterns constituting the crystalline material. 

A crystal is not, in actual fact, a simple array of points, each of which is a pattern-unit. In the first place, an atom is not a point its electrons, which scatter the X-rays, are distributed over distances commensurable with the interatomic distances. Secondly, each pattern-unit in a crystal often consists, not of one atom, but a group of atoms. The pattern-unit is thus not a point, but has a diffuse and often irregular form. However, for the purpose of diffraction theory, as far as it is carried in this chapter, the diffuse pattern-unit may be mentally replaced by a point. It will be shown in Chapter VII that the form of the pattern-unit affects the intensities of the diffracted beams but it does not affect their positions, which depend only on the space-lattice, the fundamental arrangement of identical pattern-units. 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

The occurrence of discrete energy states of a bound electron in an atom and the co-existence of wave and particle properties in a free electron, can be accounted for on the basis of wave mechanics (Schrodinger, 1926). Observation of the diffraction patterns produced when electrons of known energy encounter crystals of known atomic spacing shows that the wave length, A, associated with an electron of velocity, v, is given by... 

Radiation incident upon a crystal is scattered in a variety of ways. When the wavelength of the radiation is similar to that of the atom spacing in a crystal the scattering, which is termed diffraction, gives rise to a set of well defined beams arranged with a characteristic geometry, to form a diffraction pattern. The positions and intensities of the diffracted beams are a function of the arrangements of the atoms in space and some other atomic properties, especially, in the case of X-rays, the atomic... 

Fig. 13 Real-space model (a, b) of the commensurate c 2 x. 2)R45° and incommensurate c(. 2 x p)R45° bromide structures observed on Au(OOl). The open circles correspond to bromide atoms, (c, d) show the corresponding in-plane reciprocal space patterns in which the squares are scattering from the Au substrate and the circles are Br reflections. In (d), the peaks move outward along K with increasing potential. (Taken from Ref [57].)...

Atoms in a molecule are bound by chemical bonds residing in a space between the bounded atoms. The pattern of such bonds has to be assumed. Each chemical bond represents an electron bonding pair (see the present chapter). 

Crystallinity is a state of molecular structure referring to a long-range periodic geometric pattern of atomic spacings. In semi-crystalline polymers, such as polyethylene (PE), the degree of crystallinity (% crystallinity) influences the degree of stiffness, hardness, and... 

Electron diffraction is an important technique for the study of crystalline materials [64, 65]. It is regularly used to identify crystal structures and local orientation. The directions in which electrons are diffracted from a specimen relate to the atomic spacings and orientation of the material (Section 3.2). A crystal has a regular arrangement of atoms and so in the TEM it will produce a diffraction pattern consisting of sharp spots. 

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