Hexagonal point lattice, planes

It is known that a metallic ID system is unstable against lattice distortion and turns into an insulator. In CNTs instabilities associated two kinds of distortions are possible, in-plane and out-of-plane distortions as shown in Fig. 8. The inplane or Kekuld distortion has the form that the hexagon network has alternating short and long bonds (-u and 2u, respectively) like in the classical benzene molecule [8,9,10]. Due to the distortion the first Brillouin zone reduees to one-third of the original one and both K and K points are folded onto the F point in a new Brillouin zone. For an out-of-plane distortion the sites A and B are displaced up and down ( 2) with respect to the cylindrical surface [11]. Because of a finite curvature of a CNT the mirror symmetry about its surface are broken and thus the energy of sites A and B shift in the opposite direction. 

Fig. 9 (a) Sketch of the structure of the hexagonal columnar phase of DNA, showing parallel molecules hexagonally packed in the plane perpendicular to their axis, a and 4 are the lattice parameters, (b) COL developable domains observed in polarized microscopy, w indicates defect walls between differently oriented domains, while 7t stands for point defect around which DNA molecules continuously bend (size bar is 10 pm). Adapted with permission from [27]... 

Figure 16.5. A hexagonal planar net is generated by the fundamental translations a1 a2 (each of length a) and a12 — 2n/3. To generate a space lattice with three-fold rotational symmetry, the second and third layers must be translated so that Pi lies over the points marked P2 and P3, respectively, that is at (1/3 2/3 1/3) and (2/3 1/3 2/3). If using hexagonal coordinates a3 is normal to the plane of a1 a2 and lies along e3, so that this unit cell (3R) contains three lattice points (Figure 16.4).

In order to specify a crystal direction, a vector is drawn from the origin to some point P. This vector will have projections u on the a axis, d on the b axis, and W on the c axis. The three numbers are divided by the highest common denominator to give the set of smallest integers, u, v, and w. The direction is then denoted in brackets as [uvw]. Sets of equivalent directions are labeled u v w). For cubic systems, the [h k /] direction is always orthogonal to the (hkl) plane of the same indices. With the other crystal systems, this simple relationship does not hold. For example, in the hexagonal lattice, the normal to the (1 0 0) plane is in the [2 1 0] direction, the [100] direction being 120° to the (1 0 0) plane (see Practice Problem 4). 

I shall introduce here a concept that is common to all methods. The Brillouin zone (BZ) is described in most solid state physics textbooks. It is defined as a reciprocal lattice cell bounded by the planes that are perpendicular bisectors of the vectors from the origin to the reciprocal lattice points. Fig. 2 illustrates the first BZ cell for a hexagonal lattice. 

Figure 5.14. The relationship between the lattice parameters of the hexagonal unit cell (solid and dotted lines) and the related orthorhombic unit cell (dashed lines). The unit cell parameter perpendicular to the plane of the projection is identical in both crystal systems. The smaller orthorhombic unit cell found using the DICVOL91 indexing program is indicated by the thick solid vectors (a o and c o). Open circles show lattice points and the dash-dotted vector illustrates the C-translation in the conforming orthorhombic lattice.

For specificity, let us consider the application of these ideas to a-CraOa. Its crystal structure can be represented as a hexagonal close-packed lattice of oxide ions (the closed-packed layers of oxide ions alternate ababab. ..) in which two-thirds of the octahedral holes are filled with Cr3+ ions in a systematic fashion. Suppose the crystal to be cleaved in a close-packed plane in the presence of water. To preserve electrical neutrality, the oxide ions in this plane must be equally divided between the surfaces of the faces being formed. As a result, each Cr3+ in the layer below these oxide ions would be five-coordinate and in a square pyramidal configuration. Each ion would react with a molecule of water following which a proton would move from each adsorbed water molecule to an adjacent oxide ion. Thus, the outer face would consist of a close-packed layer of hydroxide ions. This is shown in Fig. 2. The basic point is that electrical neutrality and six-coordination can both be preserved by replacing what would be a plane of oxide ions in bulk by an equivalent plane of hydroxide ions at the surface. Similar ideas obtain on alumina. 

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