The energy of a straight single-wall carbon nanotube (SWNT) with different helicity is proposed by making continuum limit of the curvature elastic energy of a single-layer curved graphite carbon. We find that the zigzag is the only stable state with the lowest energy for the fixed diameter tubule. Our helicity energy formula agrees well with the energy of the zigzag configuration and armchair configuration SWNT's calculated by quantum molecular dynamics numerical simulation.

A continuum model of the configurations of multiwalled carbon nanotubes (MWNT's) is studied in this paper. The conformation energy of the MWNT is proposed, and the shape equations for a MWNT are derived. From the shape equations, the solutions corresponding to the straight carbon nanotube and coiled carbon nanotube are given respectively. The configurations of the ring MWNT and the plane curve MWNT are also determined theoretically. It is revealed that there are only a very few valid values of the ring radius of the ring MWNT in the case of the fixed parameters of its cross section. The mismatched effect between the interlayer lattices enriches the variation of ring MWNT's with different size. It is found that the conformation energy per unit arc length of the ring is double of that of straight MWNT for same tube cross section. Typical configurations of the plane curve MWNT obtained from the shape equations agree well with the TEM images of the nanotubes. The total conformation energy of the plane curve MWNT is in the same order of the ring MWNT one for enough long arc length.

We investigate ring formation of single-walled carbon nanotubes theoretically in this paper. The conformation energy of the nanotubes is identified as the curvature elastic energy of a curved graphite and the intertube van der Waals interaction. The entropy of ring nanotubes depends on the total number of carbon atoms. Assuming continuity, we find that the ring single-walled carbon nanotube is stable only when the radius of the tube is valued 6.958

In this paper, we study the morphological transition of bacterial colonies exposed to ultraviolet radiation by modifying the bacteria model proposed by Delprato et al. Our model considers four factors: the lubricant fluid generated by bacterial colonies, a chemotaxis initiated by the ultraviolet radiation, the intensity of the ultraviolet radiation, and the bacteria's two-stage destruction rate with given radiation intensities. Using this modified model, we simulate the ringlike pattern formation of the bacterial colony exposed to uniform ultraviolet radiation. The following is shown. 1 Without the UV radiation the colony forms a disklike pattern and reaches a constant front velocity. 2 After the radiation is switched on, the bacterial population migrates to the edge of the colony and forms a ringlike pattern. As the intensity of the UV radiation is increased the ring forms faster and the outer velocity of the colony decreases. 3 For higher radiation intensities the total population decreases, while for lower intensities the total population increases initially at a small rate and then decreases. 4 After the UV radiation is switched off, the bacterial population grows both outward as well as into the inner region, and the colony's outer front velocity recovers to a constant value. All these results agree well with the experimental observations Phys. Rev. Lett. 87, 158102 (2001). Along with the chemotaxis, we find that lubricant fluid and the two-stage destruction rate are critical to the dynamics of the growth of the bacterial colony when exposed to UV radiation, and these were not previously considered.

The specific heat of single-walled carbon nanotubes SWNT's is studied theoretically in this paper. Making use of the continuum model, phonon dispersion relations are derived numerically and analytically. Finally, expressions for specific heat versus temperature are derived theoretically, and we find that the lattice wave propagating along the length of the SWNT plays the principle role in deciding the value of the specific heat. Our theoretical results agree well with experimental data Science 289, 1730 2000.

The vibrational spectra of double-wall carbon nanotubes (DWNTs) are studied theoretically within a continuum model in this paper. The phonon dispersion relations are derived numerically and analytically, and two radial breathing modes (RBMs) are calculated analytically, which agree with experimental data [Phys. Rev. B 66, 075416 (2002)]. By comparison of the RBM frequencies with the ones of isolated single-wall carbon nanotubes, we find that there is a systematic upward shift for DWNTs RBM frequencies due to the interlayer van der Waals interactions. In case of counterphase modes, the upshift magnitude of RBM frequencies increases with an increase of the outer-layer radius R. However, for inphase RBMs, the upshift magnitude of RBM frequencies may increase or decrease with an increase of R. We then discuss four acoustic phonons and find that there is no transverse acoustic phonon in DWNTs. Finally, the general phonon dispersion relations are calculated.

In this paper, the optimum diameter of stable single-walled carbon nanotubes (SWNT's) in a rope is calculated theoretically using its conformation energy, which consists of curvature elastic energy and intertube van derWaals energy. An equlibrium shape equation for single layer fullerenes is derived by means of variation of the conformation energy. For straight SWNT's in an SWNT rope, the solution to the equlibrium shape equation gives the optimum diameter to be 1.36nm for stable SWNT's. This agrees well with the experimental observation [Nature 388, 756 (1997); Science 273, 483 (1998)].

The chiralities of double-walled carbon nanotubes (DWNTs) are assigned using two radial breathing modes (RBMs) of the DWNTs unambiguously, which are calculated theoretically in this paper by use of the modified continuum model. According to experimental data, we assign the specific chiralities to the inner and outer layers by use of two RBM frequencies of the DWNTs, and then calculate the diameters of the two layers and even the interlayer distance. The interlayer spacing between two layers b is found to be not a constant, but is in the range 0.321 to 0.402nm, which agrees well with experimental data [W. Ren et al., Chem. Phys. Lett. 359, 196 (2002); R. Pfeiffer et al., Phys. Rev. Lett. 90, 225501 (2003)] and calculated result [R. Saito et al., Chem. Phys. Lett. 348, 187 (2001)]. The values of both the upshift and downshift of DWNTs' RBM frequencies are not constant, but vary with the interlayer spacings b and their chiralities. By calculating and comparing the case of the chiralities being taken into account with that of the chiralities being neglected in the van der Waals forces, we find that the effect of chiralities should not be neglected in order to assign the chiralities of DWNTs though the difference between the two RBM frequencies caused by the different chiralities of inner and outer tubes is small. In addition, it is found that there can be several values of B1 B2 corresponding to the same value of B2 B1, arising because one inner (outer) tube can be paired with several outer (inner) tubes, which can be explained by two RBM frequencies of the DWNTs.

The energy band structure of quantum wires with both Rashba and Dresselhaus spin-orbit couplings in a magnetic field is studied theoretically. We find that the magnetic field strongly affects the subband structure of the wire. With both couplings we obtain a beating pattern in the magnetoresistance. Compared with the quantum wire with only Rashba spin-orbit coupling, the beating pattern can be suppressed even if the coupling strengths are large. This is due to the interplay of the two spin-orbit couplings. The disappearing threshold of the nodes depends on both coupling strengths. When the coupling strengths change, the number of nodes of the beating pattern also varies.

In this paper, we investigate the motion of spiral waves in the complex Ginzburg-Landau equation CGLE analytically and numerically. We find that the tip of the spiral wave drifts primarily in the direction of the electric field and there is a smaller component of the drift that is perpendicular to the field when a uniform field is applied to the system. The velocity of the tip is uniform and its component along the electric field is equal to the strength of the field. When the CGLE system is driven by white noise, a diffusion law for the vortex core of the spiral wave is derived at long time explicitly. The diffusion constant is found to be D T/C2, in which T is the noise strength and C is the core asymptotic factor of the spiral wave. When the external force is a simple oscillation we find that the tip of the spiral wave drifts if the frequency of the external force is the same as that of the system. Our analytical results are verified using numerical simulations.