Quantum effective

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

We have so far ignored quantum corrections to the virial coefficients by assuming classical statistical mechanics in our discussion of the confignrational PF. Quantum effects, when they are relatively small, can be treated as a perturbation (Friedman 1995) when the leading correction to the PF can be written as... 

When quantum effects are large, the PF can be evaluated by path integral methods [H], Our exposition follows a review article by Gillan [12], Starting with the canonical PF for a system of particles... 

Schweizer E K and Rettner C T 1989 Quantum effects in the scattering of argon from 2H-W(100) Phys. Rev. Lett. 62 3085... 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

Schaaff T G efa/1997 Isolation of smaller nanocrystal Au molecules robust quantum effects in the optical spectra J. Phys. Chem. B 101 7885... 

AIMS) [88]. The inclusion of quantum effects directly in the nuclear motion may be a significant step, as the motion near a conical intersection is known to be very quantum mechanical. 

The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

Quasiclassical calculations are similar to classical trajectory calculations with the addition of terms to account for quantum effects. The inclusion of tunneling and quantized energy levels improves the accuracy of results for light atoms, such as hydrogen transfer, and lower-temperature reactions. 

The electromagnetic spectrum is a quantum effect and the width of a spectral feature is traceable to the Heisenberg uncertainty principle. The mechanical spectrum is a classical resonance effect and the width of a feature indicates a range of closely related r values for the model elements. 

Another reason for interest in microwaves in chemical technology involves the fields of dielectric spectrometry, electron spin resonance (esr), or nuclear magnetic resonance (nmr) (see Magnetic spin resonance). AppHcations in chemical technology relating to microwave quantum effects are of a diagnostic nature and are not reviewed herein. 

Quantum Fluids. Light compounds which exhibit behavior resulting from quantum effects, eg, hydrogen, helium, and neon, are called quantum fluids. 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

Judging from our present knowledge, such a description is far from the whole story. The article of Benderskii and Goldanskii [1992] addressed mostly the vast amount of experimental data accumulated thus far. On the other hand, the major applications of QTST involved gas-phase chemical reactions, where quantum effects were not dominant. All this implies that there is a gap between the possibilities offered by modern quantum theory and the problems of low-temperature chemistry, which apparently are the natural arena for testing this theory. This prompted us to propose a new look at this field, and to consistently describe the theoretical approaches which are adequate even at T = 0. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

As the nanotube diameter increases, more wave vectors become allowed for the circumferential direction, the nanotubes become more two-dimensional and the semiconducting band gap disappears, as is illustrated in Fig. 19 which shows the semiconducting band gap to be proportional to the reciprocal diameter l/dt. At a nanotube diameter of dt 3 nm (Fig. 19), the bandgap becomes comparable to thermal energies at room temperature, showing that small diameter nanotubes are needed to observe these quantum effects. Calculation of the electronic structure for two concentric nanotubes shows that pairs of concentric metal-semiconductor or semiconductor-metal nanotubes are stable [178]. 

Early transport measurements on individual multi-wall nanotubes [187] were carried out on nanotubes with too large an outer diameter to be sensitive to ID quantum effects. Furthermore, contributions from the inner constituent shells which may not make electrical contact with the current source complicate the interpretation of the transport results, and in some cases the measurements were not made at low enough temperatures to be sensitive to 1D effects. Early transport measurements on multiple ropes (arrays) of single-wall armchair carbon nanotubes [188], addressed general issues such as the temperature dependence of the resistivity of nanotube bundles, each containing many single-wall nanotubes with a distribution of diameters d/ and chiral angles 6. Their results confirmed the theoretical prediction that many of the individual nanotubes are metallic. 

Many of the carbon nanotube applications presently under consideration relate to multi-wall carbon nanotubes, partly because of their greater availability, and because the applications do not explicitly depend on the ID quantum effects associated with the small diameter single-wall carbon nanotubes. 

These properties are illustrative of the unique behavior of ID systems on a rolled surface and result from the group symmetry outlined in this paper. Observation of ID quantum effects in carbon nanotubes requires study of tubules of sufficiently small diameter to exhibit measurable quantum effects and, ideally, the measurements should be made on single nanotubes, characterized for their diameter and chirality. Interesting effects can be observed in carbon nanotubes for diameters in the range 1-20 nm, depending... 

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