Centrosymmetric particles

Wang H, Yan E C Y, Borguet E and Eisenthal K B 1996 Second harmonic generation from the surface of centrosymmetric particles in bulk solution Chem. Phys. Lett. 259 15-20... 

In agreement with the considerations developed above for spherical particles, the first purely local contribution vanishes for perfectly centrosymmetrical particles, and in particular for spheres. In that case, the model developed above coincides with that proposed by J.I. Dadap et al. [39, 40]. In particular, the two non local contributions correspond to the effective dipole p ff and effective quadrupole contributions introduced in that work. The effective dipole arises from retardation effects taken into account at the fundamental frequency and the effective quadrupole to retardation effects at the harmonic frequency only. Contributions with retardation effects at both the fundamental and the harmonic frequencies would be of higher orders of the parameter x. As we shall see later, the different contributions may be distinguished through their angular and polarization patterns. 

The intensity of scattering given by eqn. (36) is then for centrosymmetrical particles ... 

Estimate of the HETP intradiffusion coefficients, when based on HETP dependence on concentration, has been shown to be close in value to the quantity calculated with the theoretical formula obtained by using the model of centrosymmetrical diffusion in an ion-exchanger bead [89,90]. The values in this formula of r /D, where r is the effective particle radius, were determined in thin-layer kinetic experiments with the same ion-exchanger fraction. 

Radiation and particles, i.e. x-rays, neutrons and electrons, interact with a crystal in a way that the resulting diffraction pattern is always centrosymmetric, regardless of whether an inversion center is present in the crystal or not. This leads to another classification of crystallographic point groups, called Laue classes. The Laue class defines the symmetry of the diffraction pattern produced by a single crystal, and can be easily inferred from a point group by adding the center of inversion (see Table 1.10). 

Wave functions (13) apply to any pair of coupled centrosymmetric protons, irrespective of the distance and of the magnitude of the coupling term. Quantum correlation arises exclusively from normal coordinates representing dynamics of indistinguishable particles. It has been proposed [Keen 2003] that quantum entanglement could be due to the overlap integral of the one-particle wave functions >Pi, (a o , ) for protons ... 

To treat more realistic models for the kinetics of biomolecular encounters, a simulation approach has been developed and applied to proteins.370,371,3718 This approach merges stochastic dynamics methodology (Chapt. IV.D) with the analytic result for the Debye rate constant for a pair of particles, moving diffusively through solvent with a centrosymmetric interaction potential.365 The analytical expression for the Debye rate constant, ko(,R), to first achieve a separation, R, is given by... 

Rg is the radius of gyration of the shape function whereas the quantity a is related to the internal distribution of the electron density. The quantity 3 is the square of the distance between the centers of gravity of T(f) and of the internal distribution T(f )Ap(f) and vanishes for particles with centrosymmetric structure. 

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