Anisotropic oscillator model

There exist a variety of extensions of the basic shell model. One variation for molecular systems uses an anisotropic oscillator to couple the core and shell charges,thus allowing for anisotropic polarizability in nonspherical systems. Other modifications of the basic shell model that account for explicit environment dependence include a deformable or breathing shelF ° and shell models allowing for charge transfer between neighboring sites. 

Figure 14.12a demonstrates the Rietveld refinement pattern for time-of-flight (TOF) neutron diffraction data measured at room temperature for LiFeP04. Fitting was satisfactory (/ p = 2.66%, Rf - 0.46%, 5=1.34) with accurately refined atomic positions as well as anisotropic atomic displacement parameters for all atoms under the classical harmonic oscillation model. 

In their initial paper, Condon, Altar, and Eyring chose to examine the effect of a perturbing potential of the form A xyz, where is a constant, on a three-dimensional anisotropic harmonic oscillator. Their purpose was to show that an electron moving in a properly dissymmetric potential field could give rise to optical activity. The harmonic oscillator model was not essential to their main thesis, but was chosen in order to have basis functions with which to carry through the perturbation calculations in detail. Nevertheless, they did also show how the harmonic potential field could be adapted to describe chromophoric electrons in actual molecules. 

The Drude oscillators are typically treated as isotropic on the atomic level. However, it is possible to extend the model to include atom-based anisotropic polarizability. When anisotropy is included, the harmonic self-energy of the Drude oscillators becomes... 

Silicon is a model for the fundamental electronic and mechanical properties of Group IV crystals and the basic material for electronic device technology. Coherent optical phonons in Si revealed the ultrafast formation of renormalized quasiparticles in time-frequency space [47]. The anisotropic transient reflectivity of n-doped Si(001) featured the coherent optical phonon oscillation with a frequency of 15.3 THz, when the [110] crystalline axis was parallel to the pump polarization (Fig. 2.11). Rotation of the sample by 45° led to disappearance of the coherent oscillation, which confirmed the ISRS generation,... 

The polarizability tensor, a, introduced in section 4.1.2, is a measure of the facility of the electron distribution to distortion by an imposed electric field. The structure of the electron distribution will generally be anisotropic, giving rise to intrinsic birefringence. This optical anisotropy reflects the average electron distribution whereas vibrational and rotational modes of the molecules making up a sample will cause the polarizability to fluctuate in time. These modes are discrete, and considering a particular vibrational frequency, vk, the oscillating polarizability can be modeled as... 

Figure 33. Herringbone orientational correlation functions F (3.15) in the inset and the logarithmic derivatives 62 In (3.18) as a function of distance I in units of the lattice constant a - 4.26 A in the disordered phase of the anisotropic-planar-rotor model (2.5) from Monte Carlo simulations at 7" = 25.5 K and a linear system size of L = 180. The different symbols distinguish the three symmetty axes a, and the dashed line marks the plateau 2/. In the inset all three F fall on top of each other and the different symbols denote here the two oscillating parts of the antiferromagnetic-like ordering pattern. (Adapted from Fig. 1 of Ref. 273.)...

The ab initio potentials used in solid nitrogen are from Refs. [31] and [32]. They have been respresented by a spherical expansion, Eq. (3), with coefficients up to = 6 and Lg = 6 inclusive, which describe the anisotropic short-range repulsion, the multipole-multipole interactions and the anisotropic dispersion interactions. They have also been fitted by a site-site model potential, Eq. (5), with force centers shifted away from the atoms, optimized for each interaction contribution. In the most advanced lattice dynamics model used, the TDH or RPA model, the libra-tions are expanded in spherical harmonics up to / = 12 and the translational vibrations in harmonic oscillator functions up to = 4, inclusive. 

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