Intralayer interactions

Of particular importance to carbon nanotube physics are the many possible symmetries or geometries that can be realized on a cylindrical surface in carbon nanotubes without the introduction of strain. For ID systems on a cylindrical surface, translational symmetry with a screw axis could affect the electronic structure and related properties. The exotic electronic properties of ID carbon nanotubes are seen to arise predominately from intralayer interactions, rather than from interlayer interactions between multilayers within a single carbon nanotube or between two different nanotubes. Since the symmetry of a single nanotube is essential for understanding the basic physics of carbon nanotubes, most of this article focuses on the symmetry properties of single layer nanotubes, with a brief discussion also provided for two-layer nanotubes and an ordered array of similar nanotubes. 

Kaolinite is easily cleaved perpendicular to the c direction, since the interactions between the aluminosilicate layers are much weaker than the intralayer interactions. Therefore as a part of this work the structure and surface energy, of the resulting 001 surface are considered. The surface energy may be evaluated from the energy of a single layer surface block of clay in a vacuum, U, and the energy of a portion of the bulk clay, U, containing the same number of atoms as the surface block. 

Some multilayered HTSC, for example Bi2Sr2CaCu20x, show an anisotropy of the elastic moduli inherent for layered crystals, and negative thermal expansion in a direction within the layer [16], which can be described by formula (2). At the same time for multilayered structures such as HTSC 1-2-3, where the interlayer interaction between all layers is of the same order, the intralayer interaction essentially varies from one layer to another layer. Local anisotropy of chain type is characteristic for layers with weak intralayer interaction (a layer of the rare earth and a layer of chains Cu-O). In these layers the root-mean-square displacement of atoms in a direction within the plane is beyond the classical limit at lower temperatures, and is appreciably higher than the root-mean-square... 

To first order, we consider the molecular structure of the surface layers to be identical to that of the bulk layers. Consequently, all the characteristics corresponding to short-range intralayer interactions (e.g. Davydov splitting, vibrational frequencies, excitonic band structure, vibronic relaxations are similar for bulk and surface layers). In fact, we shall see that even slight changes may be detected. They will be analyzed in Section III.C, devoted to surface reconstruction. Therefore, our crystal model consists of (a,b) monolayers translated in energy relative to the bulk excitation by 206, 10, and 2cm-1 for the first three layers, as indicated in Fig. 3.5. No other changes are considered in this first-order crystal model. 

Figure 4-13. Evolution with the size of the sexithienyl cluster of the excitation energies from the ground state to the lowest excited state (open circles), to the high-lying excited state strongly coupled to the ground state (open squares), and to the lowest charge transfer-excited state (open triangles). In aU cases, only intralayer interactions have been considered.

All consequences of the fiexoelectric effect considered so far are indirect and are not directly observable. Fiexoelectric interactions change the elastic constant (Eq. 5.16) in the SmC phase, which influences the layer polarization (Eq. 5.42). The layer polarization influences intralayer interactions and interactions to nearest layers and is consequently a source of interactions to more distant layers. The fiexoelectric interactions stabilize different structures with longer periodicities. But these are all indirect effects, which often cannot be separated from other effects. As the fiexoelectric effect is of achiral origin, there is no simple way to isolate the influence of flexoelectricity on the structure or the macroscopic properties. 

Next we examine what happens when we stack these unit cells. Consider first the point r. Here A3 = 0, so we stack the unit cells with no change. If we start with jTanti of Fig. 15-34a, we obtain Fig. 15-35a, which shows that the unit cells interact in an antibonding manner. Here, then, is a Bloch sum that is antibonding within and also between unit cells. The symbol tt indicates two destabilizing interactions. Each arrow represents the same amount of energy because the interlayer distance within a unit cell is the same as that between unit cells and also because there are equal numbers of inter- and intralayer interactions in the crystal. Similar sketches easily demonstrate how the other three unit-cell function sets of Fig. 15-34 behave at F Tfbond, E tt tond tt- Therefore, at F, the it and tt bands are both... 

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