Linear chain of atoms

This bismuth-III structure is also observed for antimony from 10 to 28 GPa and for bismuth from 2.8 to 8 GPa. At even higher pressures antimony and bismuth adopt the body-centered cubic packing of spheres which is typical for metals. Bi-III has a peculiar incommensurate composite crystal structure. It can be described by two intergrown partial structures that are not compatible metrically with one another (Fig. 11.11). The partial structure 1 consists of square antiprisms which share faces along c and which are connected by tetrahedral building blocks. The partial structure 2 forms linear chains of atoms that run along c in the midst of the square antiprisms. In addition, to compensate for the... 

Figure 1.28. (a) Infinite periodic linear chain of atoms separated by the distance a, (b) band structure for 1 and p atomic orbitals pointing perpendicularly to the chain direction, (c) band structure for p atomic orbitals aligned along the chain direction and (d) DOS of a ID infinite periodic linear chain. 

Long flexible pol3nners have a large number of Internal degrees of freedom. The typical primary structure of such molecules Is a linear chain of atoms (often carbon) connected by chemical bonds ("backbone"). Usually, every other backbone atom carries a side group. By rotation about the bonds In the backbone the molecule changes Its shape, and since there are many of these bonds In a polymer. a wide spectrum of conformations Is available. The rotation is hindered by the side groups, so that some of these conformations may be rather unfavourable. In some macromolecules (for instance proteins) sequences of preferred bond orientations show up as helical or folded sections In the molecules (secondary structure). 

For the linear chain of atoms (see fig. 4.7), the total energy per atom within the embedded atom context is given by... 

The standard textbook exercise for illuminating the ideas discussed above in their full generality is that of the linear chain of atoms. The simplest variant of this exercise assumes that all the masses are the same as are all of the springs joining them. In this case, the dynamical matrix itself is one-dimensional and hence its... 

The properties of the linear chain of atoms fall into three sets those for the intramolecular, the intermolecular and the closed-shell Ne-H interactions. The intramolecular bond lengths vary from Re = 1.39 au, the equilibrium value to 1.04 au obtained for the central molecule in the Ne H6 Ne vise at a pressure of 158 GPa. Only at the highest pressures do the intermolecular separations approach Re. Unlike the LDA calculations on solid hep hydrogen, no increase in separation over Re is found for high pressures. The Ne-H separations decrease at a slower rate. 

Foreman and Lomer were the first to report that if the vibrations propagate in the cubic crystal lattice along the high symmetrical directions (111), (110), (100), the mathematical description of the process can be reduced to a linear chain of atoms (48). The problem of vibrations of parallel atomic planes is reduced to the problem of interacting oscillating points of the equal mass. Every oscillatory mode can be treated as related to a separate linear chain. 

Representation of (a) a linear chain of atoms and (b) the approximation of a linear chain of atoms as the enlarged area from a ring of atoms having a very large radius. 

Consider an example of a polymer composed of a linear chain of atoms, with N atoms, each separated by a distance d. Therefore, the total length of the chain is (N - )d K Nd for large number of atoms. We can treat this problem as electrons confined to a one-dimensional infinite potential well. The eigenenergies are given by... 

The atoms of the metal oxides, as in all solids, partieipate in thermal vibrations around their equilibrium positions. These vibrations are determined by the structure of the solid and the interactions between the atom or ions in the lattice and, in general, their detailed description is very complicated. However, there are differetn simplified models for simulation of the collective vibrations of the atoms. Let us consider a linear chain of atoms in a soUd at a distance a between neighbors (Scheme la). In this case the shorter wavelength will be... 

Consider a monoatomic linear chain of atoms with mass m and nearest-neighbour interactions of the Lennard-Jones type with (p(r) given by (4.2). Prove that the zero-point energy... 

A well-known concept to introduce collective vibrations of atoms is based on the dynamics of a linear chain of atoms, which are coupled via harmonic springs to each other. Let us consider the longitudinal vibrations for such an infinite chain where we consider the atomic displacements u around the atomic equilibrium positions, according to Figure 8.2.1a. The force F acting on atom n is the sum... 

The linear chain model has also been used to study the role of energy dissipation in atom surface collisions. It is of course too simplistic to approximate the surface by a linear chain of atoms but the model is easily solvable and can give some qualitative insight into the problem. Thus the problem can be formulated by adding the following equation to the set of equations for the spring or chain atoms... 

TABLE 2.5 Energy Transfer Ei t to a Linear Chain of Atoms with Mass 63.5 amu for Collisions with an Atom with Mass 4 amu and 20.2 amu... 

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