Lattice distortions and

A brief review is given on electronic properties of carbon nanotubes, in particular those in magnetic fields, mainly from a theoretical point of view. The topics include a giant Aharonov-Bohm effect on the band gap and optical absorption spectra, a magnetic-field induced lattice distortion and a magnetisation and susceptibility of ensembles, calculated based on a k p scheme. 

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. 

In the Hamiltonian Eq. (3.39) the first term is the harmonic lattice energy given by Eq. (3.12). It depends only on A iU, i.e., the part of the order parameter that describes the lattice distortions. On the other hand, the electron Hamiltonian Hcl depends on A(.v), which includes the changes of the hopping amplitudes due to both the lattice distortion and the disorder. The free electron part of Hel is given by Eq. (3.10), to which we also add a term Hc 1-1-1 that describes the Coulomb interne-... 

Figure 14. Tunneling to the alternative state at energy can be accompanied by a distortion of the domain boundary and thus populating the ripplon states. All transitions exemplified by solid lines involve tunneling between the intrinsic states and are coupled linearly to the lattice distortion and contribute the strongest to the phonon scattering. The vertical transitions, denoted by the dashed lines, are coupled to the higher order strain (see Appendix A) and contribute only to Rayleigh-type scattering, which is much lower in strength than that due to the resonant transitions.

After each peak has been described by the parameters of a model function, the convolution in Eq. (8.13) can be carried out analytically. As a result, equations are obtained that describe the effects of crystal size, lattice distortion, and instrumental broadening38 on the breadth of the observed peak. Impossible is in this case the separation of different kinds of lattice distortions. 

The Phenomenon. In existing materials the electron density is not even constant inside a single phase. This is obvious for the liquid structure of amorphous regions. Nevertheless, even in crystalline phases lattice distortions and grain boundaries result in variations of the electron density about its mean value. In analogy to the sunlight scattered from the fluctuations of air density, X-rays are scattered from the fluctuations of electron density. 

A mismatch in the lattice parameters of materials on both sides of an interface results in strain that is released by lattice distortions and dislocations next to the interface. Understanding the release mechanisms of the strain is crucial in order to tune the properties of the interface. 

X-ray diffraction patterns from fibres generally contain a few closely overlapping peaks, each broadened by the contributions of crystallite size, crystallite-size distribution, and lattice distortion. In order to achieve complete characterisation of a fibre by X-ray methods, it is first necessary to separate the individual peaks, and then to separate the various profile-broadening contributions. Subsequently, we can obtain measures of crystallite size, lattice distortion and peak area crystallinity, to add to estimates of other characteristics obtained in complementary experiments. 

Figure 7.6 X-ray diffraction patterns for two La-doped PZT ceramics at 40°C and 160°C. (a) PZT (53/47) shows a change in lattice distortion and (b) PZT (54/46) a change in phase composition.

A contribution caused by spin-orbit coupling and closely related to magnetocrystalline anisotropy is magnetoelastic anisotropy. Mechanical stress creates a strain which amounts to a lattice distortion and yields a correction to the magnetocrystalline anisotropy. Surface anisotropy is a manifestation of magnetocrystalline anisotropy, too (sections below and Ch. 3). 

In the very early studies on catalysis little attention was paid to the influence of factors (c), (d), and (e). It is notable, however, that Fricke and co-workers (118,119) did connect the pyrophoric nature of copper and of nickel preparations with their lattice distortions and dispersion. 

It should be stressed that the diffraction methods do not provide complete characterization of lattice distortions and ionic shifts in relaxors due to the compositional disorder of these materials and nanometric scale of polar order. Thus, local methods such as magnetic resonance and, in particular, NMR can be extremely useful in this case. In NMR experiments, the nuclei are sensitive to their local environment at a distance less than 1-2 nm. In addition, NMR operates at a much longer time scale (105-108 s) in comparison with the neutron or X-ray... 

Commensurability. Incommensurate lattice distortions and commensurate-incommensurate phase transitions are often observed in these materials. The incommensurability comes either from an incommensurate Fermi wave vector (2A F, 4kF scattering in charge-transfer salts) or from the counterion stacks (e.g., triiodide-containing materials). 

It is not the purpose here to develop the theory of structural instabilities of quasi-one-dimensional conductors, which has already been treated extensively in a number of papers (e.g., Refs. 34, 46, and 113 to 116). We simply show how lattice fluctuations, lattice distortions, and phase transitions in organic conductors are detected and studied. 

In 5-TiNo,5o, the random distribution of nitrogen atoms undergoes an ordering process below about 880 °C with a concurrent tetragonal distortion of the metal lattice. The so-formed S -TiNo.s phase is only formed from 5-TiNi by lattice distortion and is not an equilibrium phase (e g. it is not formed in isothermal diffusion couples in the form of a phase band). 

Depending on the relative energies for the lattice distortion and the electron-energy redistribution the Peierls transition will occur. The critical temperature, Tp, can be calculated and is given by... 

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