Honeycomb lattice

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

Fig. 1. Two-dimensional honeycomb lattice of graphene primitive lattice vectors R and R2 are depicted outlining primitive unit cell.

So, despite the very small diameter of the MWCNT with respeet to the de Broglie wavelengths of the charge carriers, the cylindrical structure of the honeycomb lattice gives rise to a 2D electron gas for both weak localisation and UCF effects. Indeed, both the amplitude and the temperature dependence of the conductance fluctuations were found to be consistent with the universal conductance fluctuations models for mesoscopic 2D systems applied to the particular cylindrical structure of MWCNTs [10]. 

Fig. 16. (a) The chiral vector OA or Ch = nhi + md2 is defined on the honeycomb lattice of carbon atoms by unit vectors ai and a of a graphene layer and the chiral angle with respect to the zigzag axis (9 = 0°). Also shown are the lattice vector... 

Lattice-sublattice relations and their application to orientational structures on square and honeycomb lattices... 

As an example, we now apply the relation (3.2.2) to the case of a complex honeycomb lattice,124 with hexagonal sides equal to a. The basis vectors... 

Fig. 3.4. The ground-state configuration of dipoles on a complex honeycomb lattice and the corresponding triangular basic and dense lattices (a) the first Brillouin zones for the reciprocal basic and dense lattices (b).

Let us analyse the effect caused by the thermodynamic fluctuations of order parameter on the ground state of a dipole honeycomb lattice. Two factors determine the situation just as they do in the case of a square lattice.54,56,81 These are the... 

Figure 9.38 The simplest types of regular 2D lattices (A) the Bethe lattice (Z=3) (B) the honeycomb lattice (Z=3), The simple square lattice (Z=4), (C) the simple square lattice (Z=4), (D) Kagome lattice (Z=4), (E) the triangular lattice (Z=6).

Carbon nanotubes (CNTs) constitute a nanostructured carbon material that consists of rolled up layers of sp2 hybridized carbon atoms forming a honeycomb lattice. After diamond, graphite and fullerenes, the one-dimensional tubular structure of CNTs is considered the 4th allotrope of carbon (graphene is the 5th). 

Although a honeycomb lattice theoretically consists of sp2 atoms, the carbon s ability to represent intermediate states of hybridization leads to another kind of defect to counterbalance the strain energy induced by high curvature. This so-called rehybridization results in a higher n-character of the C-C bonds [24]. Furthermore, local sp3 hybridization can be induced though chemical treatment, such as after thermal elimination of functional groups. 

Single crystals of /S-A1203 are essentially two dimensional conductors. The conducting plane has hexagonal symmetry (honeycomb lattice). This characteristic feature made -alumina a useful model substance for testing atomistic transport theory, for example with the aid of computer simulations. Low dimensionality and high symmetry reduce the computing time of the simulations considerably (e.g., for the calculation of correlation factors of solid solutions). 

Ewert KK, Evans HM, Zidovska A, Bouxsein NF, Ahmad A, Safinya CR (2006) A columnar phase of dendritic lipid-based cationic liposome-DNA complexes for gene delivery hexago-nally ordered cylindrical micelles embedded in a DNA honeycomb lattice. J Am Chem Soc 128 3998-4006... 

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