The Structure of Solids

This brings us to a discussion of the structure of solids and how structure is determined. 

it is easy to see that in Bragg s x-ray diffraction equation, the angle, 9, is actually the angle between a given plrnif. nf atoms in the structure and the path of the x-ray beam. The unit, d , is defined as the distance between planes of the lattice and X. is the wavelength of the radiation. 

As we said, we usually define planes, not points, in the lattice. The reason that we do this is that the waves of electromagnetic radiation are constructively diffracted by planes of atoms in the solid rather than points in the lattice. One system that has come into general use is that of MnXER INDICES which is represented by h, k, 1.  

In this system, we use a, b, c (not unit-cell vectors) to represent the lengths of intercepts which define the planes within the unit cell. Miller Indices are the reciprocals of the intercepts, a, b c, of the chosen plane on the x, y, z - directions in the lattice. 

To illustrate this concept, examine the symmetry elements of our cube, shown as follows  

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It can be readily confirmed thaf by decreases as the number of bonds N increases and/or llieir length (r ) decreases. This relationship between the bond strength and the number of neighbours provides a useful way to rationalise the structure of solids. Thus the high coordination of metals suggests that it is more effective for them to form more bonds, even though each individual bond is weakened as a consequence. Materials such as silicon achieve the balance for an infermediate number of neighbours and molecular solids have the smallest atomic coordination numbers. 

In the last chapter we examined data for the yield strengths exhibited by materials. But what would we expect From our understanding of the structure of solids and the stiffness of the bonds between the atoms, can we estimate what the yield strength should be A simple calculation (given in the next section) overestimates it grossly. This is because real crystals contain defects, dislocations, which move easily. When they move, the crystal deforms the stress needed to move them is the yield strength. Dislocations are the carriers of deformation, much as electrons are the carriers of charge. 

Below a temperature of Toi 260 K, the Ceo molecules completely lose two of their three degrees of rotational freedom, and the residual degree of freedom is a ratcheting rotational motion for each of the four molecules within the unit cell about a different (111) axis [43, 45, 46, 47]. The structure of solid Ceo below Tqi becomes simple cubic (space group Tji or PaS) with a lattice constant ao = 14.17A and four Ceo molecules per unit cell, as the four oriented molecules within the fee structure become inequivalent [see Fig. 2(a)] [43, 45]. Supporting evidence for the phase transition at Tqi 260 K is... 

The arrangement of atoms, ions, and molecules within a crystal is determined by x-ray diffraction (Major Technique 3, which follows this chapter), one of the most useful techniques for determining the structures of solids. 

This chapter will present more advanced topics than those of the first chapter in terms of determining the structure of solids. Consequently, you will gain some knowledge of how one goes about determining the structure of a solid, even if you never have to do it. 

In this section, we will present the basis developed to explain the structure of solids. That is, the concepts that were perfected in order to accurately describe how atoms or ions fit together to form a solid phase. This work was accomplished by many prior workers who established the rationale used to define the structure of a symmetrical solid. As you will recall, we said that the basic difference between a gas, liquid and that of a solid lay in the orderliness of the solid, compared to the other phases of the same material. 

We have already indicated that solids can have several forms or symmetries. To elucidate the structure of solids in more detail, at least three postulates apply ... 

This completes our discussion of the beisis and factors developed by past investigators to describe and conceptulize the structure of solids. You will note that we have not yet fully described the s)unmetry factor of solids. The reason for this is that we use symmetry factors to characterize solid structure without resorting to the theoretical basis of structure determination. That is, we have a standard method for categorizing solid structures. We say that salt, NaCl, is cubic. That is, the Na" ion and the Cl ion are alternately arranged in a close-packed cubic structure. The next section now investigates these structure protocols. 

We have Investigated the structure of solids In the second chapter and the nature of point defects of the solid in the third chapter. We are now ready to describe how solids react. This will Include the mechanisms Involved when solids form by reaction from constituent compounds. We will also describe some methods of measurement and how one determines extent and rate of the soUd state reaction actually taking place. We will also show how the presence and/or formation of point defects affect reactivity In solid state reactions. They do so, but not In the memner that you might suspect. We will also show how solid state reactions progress, particularly those involving silicates where several different phases appear as a function of both time and relative ratios of reacting components. 

With this imaging system it is possible to study virtually all metals and alloys, many semiconductors and some ceramic materials. The image contrast from alloys and two-phase materials is difficult to predict quantitatively, as the effects of variations in chemistry on local field ion emission characteristics are not fully understood. However, in general, more refractory phases image more brightly in the FIM. Information regarding the structure of solid solutions, ordered alloys, and precipitates in alloys has been obtained by FIM. 

Adsorption is a physicochemical process whereby ionic and nonionic solutes become concentrated from solution at solid-liquid interfaces.3132 Adsorption and desorption are caused by interactions between and among molecules in solution and those in the structure of solid surfaces. Adsorption is a major mechanism affecting the mobility of heavy metals and toxic organic substances and is thus a major consideration when assessing transport. Because adsorption is usually fully or partly reversible (desorption), only rarely can it be considered a detoxification process for fate-assessment purposes. Although adsorption does not directly affect the toxicity of a substance, the substance may be rendered nontoxic by concurrent transformation processes such as hydrolysis and biodegradation. Many chemical and physical properties of both aqueous and solid phases affect adsorption, and the physical chemistry of the process itself is complex. For example, adsorption of one ion may result in desorption of another ion (known as ion exchange). 

Hamilton, W. C., and Ibers, J. A. (1968). Hydrogen Bonding in Solids. W.A. Benjamin, New York. This book shows how hydrogen bonding is an important factor in the structure of solids. 

The clusters 59 and 60 (Figure 2.3-12) form an Al cluster framework, which, amongst others, can be described as a distorted section from the structure of solid aluminum, as is shown in addition in Figure 2.3-12 ( molecular nanostructured element modifications ). The alternative description of cluster 59 as a sandwich compound, wherein an Al3+ ion is coordinated by two aromatic AI3R32 rings (cf. Ga3R32, Section 2.3.2), is not confirmed by quantum chemical calculations [89]. 

The results of a comprehensive 13 C CP/MAS NMR study of the structure of solid polypeptides, prepared by polymerization of amino acid N-carboxyanhydrides under various conditions, have been reported105,107. In the case of poly(L-alanine) it was found that... 

This is by no means a complete catalog, but, hopefully some introduction to the possibilities of studying the structure of solids with the Mossbauer eflEect. 

Besides the spectroscopic investigations of solids by laser-excited spontaneous Raman or Brillouin scattering already discussed in Sections III.6 and 7, much new insight into the optical properties and the structure of solids has been gained by studying nonlinear optical effects. (Surveys and more detailed information about nonlinear optics can be found in refs. 306-308))... 

Modern instrumental methods of analysis have provided scientists with a wealth of information regarding the nature of the solid state and the reactivity of solids. Knowledge of the structure of solids and an ability to study thermal behavior are essential to an understanding of the behavior of high-energy materials. 

Owing to its single composition and pure covalent bonding, diamond is a standard solid material, and takes the role of the most appropriate sample in explaining the effects upon structure-sensitive properties when the structure of solid material deviates from the ideal state. 

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