Symmetrized multipole analysis of orientational distributions

Abstract

A rigorous treatment of the constraints imposed by site and body symmetry on the orientational distribution of librating molecules is presented in terms of multipole expansions. Radial densities of the site-symmetric dynamic multipole expansion are linear combinations of the radial densities of the body-symmetric static multipole expansion of the same multipole order. The transformation from static to dynamic radial densities is transmitted by libration matrices, which can be included as parameters in a structure factor model. The matrix elements are connected to the rotational dynamics of the molecule. They are expansion coefficients of the orientational distribution in a basis obtained by site and body symmetrization of the real Wigner functions. Both symmetrizations are reduced to selecting terms from a general basis according to simple index rules. Independence of the kind of density function and Fourier invariance make the formalism useful in combined neutron and X-ray diffraction studies.

Photoluminescence of BaZrO3 explained by a order/disorded transformation

“…Recently, the use of microwave heating was proven to be effective for ceramic compounds in the field of powder preparation with both expected and unexpected merits: e.g., kinetic enhancement, low reaction temperature, time reduction, control of the overall particle size and aggregation process as well as new and interesting properties. [22][23][24][25][26][27][28][29][30][31] On the other hand, further research by our group has significantly expanded our efforts to demonstrate that the MWHT method is one of the most efficient, versatile, and highly cost-effective approaches to obtain crystalline, single-phase micro and/or nanoscale materials at lower temperatures and shorter reaction times but with little residual impurities (normally \1 ppm) [29][30][31][32]. The low cost and convenience of the process, good reproducibility, high yield and clean reactions of the present synthetic method provide an incentive to scale it up for industrial production.…”

Explicit Orthonormal Fixed Bases for Spaces of Functions that are Totally Symmetric Under the Rotational Symmetries of a Platonic Solid

“…Work on the construction of basis functions for various representations of the icosahedral group, including the T~ (O, ~o) functions for the identity representation, has been done previously by Altmann (1957), Cohan (1958), Elcoro, Perez-Mato & Madariaga (1994), Heuser-Hofmann & Weyrich (1985), Jack & Harrison (1975), Kara & Kurki-Suonio (1981), Laporte (1948), Liu, Ping &Chen (1990), McLellan (1961) and Meyer (1954), among others. The approaches used include group-theoretical approaches and approaches through combining primitive invariant polynomials in the Cartesian coordinates.…”

Three-Dimensional Reciprocal Form Factors and Momentum Densities of Electrons from Compton Experiments I. Symmetry-Adapted Series Expansion of the Electron Momentum Density

“…It was first considered in connection with solutions to the Laplace and the Helmholtz equation with polyhedral boundary conditions (Klein [46], Goursat [47], Pockels [48], Poole [49], Hodgkinson [50], Laporte [51]), in connection with the problem of term splitting when passing from spherical symmetry to another point symmetry (Bethe [52], see also the whole literature on ligand-field Unauthenticated Download Date | 5/11/18 1:52 PM theory), and in connection with the rotational-vibrational spectrum of the methane molecule (Ehlert [53], Jahn [54,55], Hecht [56], Moret-Bailly [57], Fox and Ozier [58]). Systematic group-theoretical studies for a larger number of point groups are due to Laporte [51], Meyer [59], Melvin [60], Altmann [61,62], Döring [63], Bradley and Cracknell [64], and Kurki-Suonio et al [65,66,43]. Owing to the long period over which the literature is spread, the focus of interest and the formalisms are very heterogeneous.…”

“…Rose [40]. Messiah [41], Spain and Smith [42], Kara and Kurki-Suonio [43] as well as Korn and Korn [44] omit the factor (-1)'" in (11); only [40][41][42][43] introduce it in (7) instead, while neither of [43,44] compensates it in (8). Sneddon [45] hold.…”

“…The electron momentum density, n (p), in the repeated zone scheme of the reciprocal lattice, [43,64,66] and are based on z as the polar axis. The integer number k is S 0, while the index / is non-negative and even in all cases because of the inversion centre.…”