Quantum diffraction

The next field of applications of elementary catastrophe theory are optical and quantum diffraction phenomena. In the description of short wave phenomena, such as propagation of electromagnetic waves, water waves, collisions of atoms and molecules or molecular photodissociation, a number of physical quantities occurring in a theoretical formulation of the phenomenon may be represented, using the principle of superposition, by the integral... 

In Fig. 3 we report typical total differential scattering data measured for He-NO( n) at two collision energies. The quenching of the quantum diffraction oscillations in the angular dependence of the total DCS with respect to what is expected from a spherical interaction contains information on the anisotropy of the repulsive wall of the potential. The analysis of these.data has permitted us to derive a full anisotropic PES, which represents a significant improvement with respect to a previous determination. Second virial coefficients were also included in the analysis. 

Fig. 2.14 Quantum diffraction probabilities versus normal incidence energy, for H2(n = 0, 7 = 0)/ft( 111), for the zeroth and the first diffraction order, Po and Pi, along the incidence direction [lOl]. RotationaUy elastic diffraction probability into the zeroth and first order, Pq and Pf, is also shown. Pj refers to the part of Pq due to diffraction into the (01) and (01) out-of-plane diffraction peaks. These results correspond to E = 0.69 eV...

As stressed earlier, the actual pair interaction potential v(r) in a monatomic fluid depends on the quantum nature of the particles and is a function of their distance. No dependence on the temperature in these functions is included. As a matter of principle, such a thing cannot be. This absence is displayed clearly within the exact PI scheme when formulating the quantum statistical problem (e.g., Eqs. 25-27). However, in the semiclassical cases one finds potentials that are built by using the underlying v(r) as a reference, which is corrected so as to include quantum diffraction information (J, h,m) relevant to the system under study. This extra dependence can influence the calculation of semiclassical properties, as indicated by the thermodynamic derivative procedures [120]. 

In general, in the study of systems at equihbrium the main advantage of the semiclassical approaches is that they can be utilized in simulation work by following essentially the same techniques as in the classical case (i.e., one works with Ng particles). Moreover, whenever they can be applied, semiclassical approaches can work excellently and represent a substantial saving in computational effort as compared to the exact PI xP-calculations. On the negative side, even the best semiclassical approaches cannot cope with very large quantum diffraction effects. Actually, they do not capture all the information needed to characterize the quantum system (e.g., some g (r) structures cannot be computed). In the literature of this field there are different names for these pair potentials (e.g., classical effective potentials, semiclassical effective potentials, quantum effective pair potentials, etc.). The absence of external fields will be assumed in this section, and the system will be ruled by the Hamiltonian... 

For small quantum diffraction effects at low densities the results reported by Gibson are accurate enough for practical purposes [103]. 

Some new numerical results for fluid He, fluid He, and the hard-sphere fluid, under quantum diffraction effects, are given below to illustrate a number of the basic main points discussed in this chapter. The particle masses (amu) have been set to m( He) = 4.0026, m("He) = 3.01603, and m(hard sphere) = 28.0134. PIMC simulations in the canonical ensemble using the necklace normal-mode moves have been employed. The Metropolis algorithm has been apphed with the general acceptance criterion set to 50% of the attempted moves for each normal mode. In the helium simulations the propagator SCVJ (a = 1 / 3) has been utihzed. The quantum hard-sphere fluid results presented in this chapter have been obtained from a further processing of data reported in Ref. 96. Also, for fluid He Monte Carlo classical (CLAS) and effective potential QFH calculations have been performed by following the standard procedures. 

Diffraction is based on wave interference, whether the wave is an electromagnetic wave (optical, x-ray, etc), or a quantum mechanical wave associated with a particle (electron, neutron, atom, etc), or any other kind of wave. To obtain infonnation about atomic positions, one exploits the interference between different scattering trajectories among atoms in a solid or at a surface, since this interference is very sensitive to differences in patii lengths and hence to relative atomic positions (see chapter B1.9). 

Treacy E B 1969 Optical pulse compression with diffraction gratings IEEE J. Quantum. Electron. 5 454-8... 

Section 4.04.1.2.1). The spectroscopic and the diffraction results refer to molecules in different vibrational quantum states. In neither case are the- distances those of the hypothetical minimum of the potential function (the optimized geometry). Nevertheless, the experimental evidence appears to be strong enough to lead to the conclusion that the electron redistribution, which takes place upon transfer of a molecule from the gas phase to the crystalline phase, results in experimentally observable changes in bond lengths. 

There are restrictions on the values of the quantum numbers which elecuons can occupy in tluee-diirrensioiral metal structures which can be determined by application of tire Bragg diffraction equation... 

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