Bonds Brillouin zones

These surprising results can be understood on the basis of the electronic structure of a graphene sheet which is found to be a zero gap semiconductor [177] with bonding and antibonding tt bands that are degenerate at the TsT-point (zone corner) of the hexagonal 2D Brillouin zone. The periodic boundary... 

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. 

The resonating-valence-bond theory of metals discussed in this paper differs from the older theory in making use of all nine stable outer orbitals of the transition metals, for occupancy by unshared electrons and for use in bond formation the number of valency electrons is consequently considered to be much larger for these metals than has been hitherto accepted. The metallic orbital, an extra orbital necessary for unsynchronized resonance of valence bonds, is considered to be the characteristic structural feature of a metal. It has been found possible to develop a system of metallic radii that permits a detailed discussion to be given of the observed interatomic distances of a metal in terms of its electronic structure. Some peculiar metallic structures can be understood by use of the postulate that the most simple fractional bond orders correspond to the most stable modes of resonance of bonds. The existence of Brillouin zones is compatible with the resonating-valence-bond theory, and the new metallic valencies for metals and alloys with filled-zone properties can be correlated with the electron numbers for important Brillouin polyhedra. 

Brillouin zones and the resonating-valence-bond theory... 

The pictures in Fig. 10.12 give an impression of how s orbitals interact with each other in a square lattice. Depending on the k values, i.e. for different points in the Brillouin zone, different kinds of interactions result. Between adjacent atoms there are only bonding... 

The theory of band structures belongs to the world of solid state physicists, who like to think in terms of collective properties, band dispersions, Brillouin zones and reciprocal space [9,10]. This is not the favorite language of a chemist, who prefers to think in terms of molecular orbitals and bonds. Hoffmann gives an excellent and highly instructive comparison of the physical and chemical pictures of bonding [6], In this appendix we try to use as much as possible the chemical language of molecular orbitals. Before talking about metals we recall a few concepts from molecular orbital theory. 

In white tin, the nearest neighbor interatomic distance is increased to about 3 A. As expected, this lowers the energy of the 5s anti-bonding functions, and in a considerable portion of the Brillouin zone the 5 antibonding functions lie below the 5p like functions in energy hence, there are substantially more than one 5s electron and substantially less than three 5p electrons per atom. Judging from the Mdssbauer data (one could get this presumably from the band calculations with enough effort). 

Therefore, as shown in Fig. 7.1(a), the bottom of the band is at the centre of the Brillouin zone (0,0,0), whereas the top of the band is at the zone boundary ( / )( 1,1,1), since ssa < 0. It follows from eqn (7.1) that the bottom and top of the band correspond to perfect bonding and antibonding states, respectively, between all six neighbouring atoms, so that the width of the s band is 2 6ssoj, as expected. The corresponding density of states is shown in Fig. 7.1(b). The van Hove singularities, arising from the flat bands at the Brillouin zone boundaries, are clearly visible. 

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