Solids lattice periodicity

In general, it is expected that AHS will differ from AHL, since the difference in volume between the liquid and solid alloy and the existence of a lattice periodicity will produce slight changes in the distribution of charge between the components, a small change in... 

Solid contacts are incommensurate in most cases, except for two crystals with the same lattice constant in perfect alignment. That is to say, a commensurate contact will become incommensurate if one of the objects is turned by a certain angle. This is illustrated in Fig. 30, where open and solid circles represent the top-layer atoms at the upper and lower solids, respectively. The left sector shows two surfaces in commensurate contact while the right one shows the same solids in contact but with the upper surface turned by 90 degrees. Since the lattice period on the two surfaces, when measured in the x direction, are 5 3 A and 5 A, respectively, which gives a ratio of irrational value, the contact becomes incommensurate. 

One of the most powerful methods of direct structural analysis of solids is provided by HRTEM, whereby two or more Bragg reflections are used for imaging. Following Menter s first images of crystal lattice periodicity (26) and... 

Clearly, the most prominent imperfection in a crystalline solid is its surface, since it represents a cutoff of the lattice periodicity. The surface can be defined as constituting one atomic-molecular layer. This definition is sometimes not particularly useful, however. lu certaiu cases the system or property of iuterest requires that additioual layers be cousidered as the surface. ... 

In Chapter 8, the simple case of totally immiscible solids, exhibiting a minimum melting eutectic, was discussed. There are a variety of other behaviors that can be demonstrated in solid-liquid equilibria. For example, a solid solution may be formed. In a solid solution, the arrangement of atoms shows some degree of randomness on the molecular level. This occurs in a substitutional solid solution, where the components are very similar and can substitute for each other in the solid lattice. Although the lattice is regular, which atoms in the lattice are substituted is random. (If the substitution were periodic, the system would be a compound.) Copper and nickel illustrate this behavior and form a substitutional solid solution at all concentrations. Another type of solid solution is an interstitial... 

The diffraction condition for electrons in a solid, with periodicity a, is satisfied when the wave vectors lie on the bisector plane of a reciprocal lattice vector, that is, at the BZ boundaries. This is given by ... 

In earlier chapters, it was seen how a qualitative energy-level diagram for the smallest repeating chemical point group, or lattice point (known to crystallographers as the basis, or asymmetric unit), can be used to approximate the relative placement of the energy bands in a solid at the center of the BZ. This is so because the LCAO-MO theory is equivalent to the LCAO band scheme, minus consideration of the lattice periodicity. The present chapter will investigate how the orbital interactions vary for different values of the wave vector over the BZ. 

This subsection is devoted to the study of the influence of the various radiation-induced imperfections, on the properties of nonmetallic solids, taking into account the lifetime of these imperfections. From this particular point of view, and in a general way, one may distinguish between structural and electronic imperfections. To the first group, belong all the imperfections which modify the lattice periodicity they comprise lattice defects (Section III,C,1) and dislocations, which are only mentioned as a reminder in this paper. Electronic imperfections are described in Section III,C,2. 

The discovery of five-fold symmetry prompted the arf-interim Commission on Aperiodic Crystals of the International Union of Crystallography to change the definition of a crystal as a periodic three-dimensional arrangement of identical unit cells to the following ...by crystal we mean any solid having an essentially discrete diffraction diagram, and by aperiodic crystal we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent . International Union of Crystallography. Report of the Executive Committee for 1991, Acta Cryst. A48,922 - 946 (1992). 

This linear problem is thus exactly soluble. On the practical level, however, one cannot carry out the diagonalization (4.11) for macroscopic systems without additional considerations, for example, by invoking the lattice periodicity as shown below. The important physical message at this point is that atomic motions in solids can be described, in the harmonic approximation, as motion of independent harmonic oscillators. It is important to note that even though we used a classical mechanics language above, what was actually done is to replace the interatomic potential by its expansion to quadratic order. Therefore, an identical independent harmonic oscillator picture holds also in the quantum regime. 

Fig. 2.14. Energy as a function of shear for a generic crystalline solid. The periodicity of the energy profile reflects the existence of certain lattice invariant shears which correspond to the stress-free strains considered in the method of eigenstrains (adapted from Tadmor et al. (1996)).

In directing their study mainly to complexes or molecules (in particular those containing transition metal ions) that can be considered largely in isolation from the host matrix and for which a localized electron approach is appropriate, chemists have paid less attention to the study of many interesting aspects of the study of concentrated systems. In particular the lattice periodicity of such compounds encourages a band theoretical approach to bonding and structure, and experimental and theoretical study by solid-state physicists has been intense, but the spread of knowledge between dis-... 

In solids the periodicity of the crystal lattice causes the appearance of a new set of quantum numbers, usually denoted by the wavevector k (units of 1/cm), which ranges from k = - kj ax to k = kjjiax one-dimensional solid of lattice parameter, or repeat distance a, kmax= 7c/a). In the free-electron approximation the energy as a function of ky, k2 in one, two, and three dimensions is given by ... 

The main alternative approach to force field methods that has emerged in the past few years is the application of solid-state (periodic) DFT calculations for the lattice energy minimization and energetic evaluation of predicted crystal structures. The main weakness of DFT is its failure to account for the attractive dispersion interactions between... 

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