Sheets graphene

The circumference of any carbon nanotube is expressed in terms of the chiral vector = nai ma2 which connects two crystallographically equivalent sites on a 2D graphene sheet [see Fig. 16(a)] [162]. The construction in... 

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... 

Closely related to the ID dispersion relations for the carbon nanotubes is the ID density of states shown in Fig. 20 for (a) a semiconducting (10,0) zigzag carbon nanotube, and (b) a metallic (9,0) zigzag carbon nanotube. The results show that the metallic nanotubes have a small, but non-vanishing 1D density of states, whereas for a 2D graphene sheet (dashed curve) the density of states... 

Fig. 20. Electronic 1D density of states per unit cell of a 2D graphene sheet for two (n, 0) zigzag nanotubes (a) the (10,0) nanotube which has semiconducting behavior, (b) the (9, 0) nanotube which has metallic behavior. Also shown in the figure is the density of states for the 2D graphene sheet (dotted line) [178].

Fig. 24. Adsorption of lithium on the internal surfaces of micropores formed by single, bi, and trilayers of graphene sheets in hard carbon.

Powder X-ray diffraction and SAXS were employed here to explore the microstructure of hard carbon samples with high capacities. Powder X-ray diffraction measurements were made on all the samples listed in Table 4. We concentrate here on sample BrlOOO, shown in Fig. 27. A weak and broad (002) Bragg peak (near 22°) is observed. Well formed (100) (at about 43.3°) and (110) (near 80°) peaks are also seen. The sample is predominantly made up of graphene sheets with a lateral extension of about 20-30A (referring to Table 2, applying the Scherrer equation to the (100) peaks). These layers are not stacked in a parallel fashion, and therefore, there must be small pores or voids between them. We used SAXS to probe these pores. 

Fig. 32. Reversible capacity of microporous carbon prepared from phenolic resins heated between 940 to 1 I00°C plotted as a function of the X-ray ratio R. R is a parameter which is empirically correlated to the fraction of single-layer graphene sheets in the samples.

Interest has rapidly focused on the single-walled, capped tubes, as shown in Figure 11.7. They can currently be grown up to 100 pm in length, i.e., about 100,000 times their diameter. As the figure shows, there are two ways of folding a graphene sheet in such a way that the resultant tube can be seamlessly closed with a Cfto hemisphere... one way uses a cylinder axis parallel to some of the C—C bonds in... 

Carbon nanotube research was greatly stimulated by the initial report of observation of carbon tubules of nanometer dimensions[l] and the subsequent report on the observation of conditions for the synthesis of large quantities of nanotubes[2,3]. Since these early reports, much work has been done, and the results show basically that carbon nanotubes behave like rolled-up cylinders of graphene sheets of bonded carbon atoms, except that the tubule diameters in some cases are small enough to exhibit the effects of one-dimensional (ID) periodicity. In this article, we review simple aspects of the symmetry of carbon nanotubules (both monolayer and multilayer) and comment on the significance of symmetry for the unique properties predicted for carbon nanotubes because of their ID periodicity. 

In the theoretical carbon nanotube literature, the focus is on single-wall tubules, cylindrical in shape with caps at each end, such that the two caps can be joined together to form a fullerene. The cylindrical portions of the tubules consist of a single graphene sheet that is shaped to form the cylinder. With the recent discovery of methods to prepare single-w alled nanotubes[4,5), it is now possible to test the predictions of the theoretical calculations. 

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. 3. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The pairs of integers ( , ) in the figure specify chiral vectors Cy, (see Table I) for carbon nanotubes, including zigzag, armchair, and chiral tubules. Below each pair of integers (n,m) is listed the number of distinct caps that can be joined continuously to the cylindrical carbon tubule denoted by (n,wi)[6]. The circled dots denote metallic tubules and the small dots are for semiconducting tubules.

Fig. 2. By rolling up a graphene sheet (a single layer of ear-bon atoms from a 3D graphite erystal) as a cylinder and capping each end of the eyiinder with half of a fullerene molecule, a fullerene-derived tubule, one layer in thickness, is formed. Shown here is a schematic theoretical model for a single-wall carbon tubule with the tubule axis OB (see Fig. 1) normal to (a) the 6 = 30° direction (an armchair tubule), (b) the 6 = 0° direction (a zigzag tubule), and (c) a general direction B with 0 < 6 < 30° (a chiral tubule). The actual tubules shown in the figure correspond to (n,m) values of (a) (5,5), (b) (9,0), and (c) (10,5).

Fig. 3. (a) Depiction of central Brillouin zone and allowed graphene sheet states for a [4,3] nanolube conformation. Note Fermi level for graphene occurs at K points at vertices of hexagonal Brillouin zone, (b) Extended Brillouin zone pie-ture of [4,3] nanotube. Note that top left hexagon is equivalent to bottom right hexagon. 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

Fig. 5. Band gap as a function of nanotube radius calculated using empirical tight-binding Hamiltonian. Solid line gives estimate using Taylor expansion of graphene sheet results in eqn. (7).

Carbon nanotubes were first thought of as perfeet seamless eylindrieal graphene sheets —a defeet-free strueture. However, with time and as more studies have been undertaken, it is elear that nanotubes are not neeessarily that perfeet this issue is not simple bc-eause of a variety of seemingly eontradictory observations. The issue is further eomplicated by the faet that the quality of a nanotube sample depends very mueh on the type of maehine used to prepare it[l]. Although nanotubes have been available in large quantities sinee 1992[2], it is only recently that a purification method was found[3]. So, it is now possible to undertake various accurate property measurements of nanotubes. However, for those measurements to be meaningful, the presence and role of defeets must be elearly understood. 

The question whieh then arises is What do we call a defect in a nanotube To answer this question, we need to define what would be a perfeet nanotube. Nanotubes are mieroerystals whose properties are mainly defined by the hexagonal network that forms the eentral cylindrical part of the tube. After all, with an aspect ratio (length over diameter) of 100 to 1000, the tip structure will be a small perturbation except near the ends. This is clear from Raman studies[4] and is also the basis for calculations on nanotube proper-ties[5-7]. So, a perfect nanotube would be a cylindrical graphene sheet composed only of hexagons having a minimum of defects at the tips to form a closed seamless structure. 

From Euler s theorem, one can derive the following simple relation between the number and type of cycles n, (where the subscript / stands for the number of sides to the ring) necessary to close the hexagonal network of a graphene sheet ... 

Fig. 5. Conformations of the 4 types of defect lines that can occur in the graphene sheet 18].

Following a standard notation[ 12,13], a cylindrical tubule can be described by the (L,M) couple of integers, as represented in Fig. 1. When the plane graphene sheet (Fig. 1) is rolled into a cylinder so that the equivalent points 0 and M of the graphene sheet are superimposed, a tubule labeled (L,M) is formed. L and M are the numbers of six membered rings separating 0 from L and L from M, respectively. Without loss of generality, it can be assumed that L>M. 

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