Quadrupole density

Considering an effective polarization that combines the polarization, magnetization and quadrupole density [9, 10],... 

It should be noted, however, that the flexoelectric effect is not necessarily related to the ordering of molecular dipoles. Frost and Marcerou proposed another microscopic mechanism of the flexoelectric effect, which requires neither the asymmetry of the molecular shape nor the permanent molecular dipole. The macroscopic polarization may simply appear in the direction of the gradient of average density of the molecular quadrupole moments. The quadrupole mechanism of flexoelectricity is more general because, in principle, it should manifest itself in any anisotropic material with a non-zero quadrupole density including solid crystals d and elastomers. 

According to Eq. (1.1) the quadrupole contribution to the flexoelec-tric polarization is determined by the gradient of the average quadrupole density, which can be written in the following form using Eq. (1.5) ... 

The orientationally ordered nematic has a non-zero quadrupole density, irrespective of the shape or polar nature of the constituent molecules. Prost and Marcerou realized that, as a spatial gradient in the quadrupole density is equivalent to polarization, all nematics have non-zero flexocoefficients. In other words, flexoelectricity is a universal property of all nematic liquid crystals. As the nematic is a prime example of soft condensed matter, the magnitude of the flexocoefficient can be simply estimated as... 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

The reduction of the C— Br and C—1 group moments from 1.10 and 0.90 in bromo- and iodo-benzene to about 0.80 and 0.50 in 2-bromo- and 2-iodo-thiophene has been ascribed to the larger weight of resonance forms such as (8) and (9) in the thiophene series. The chlorine, nuclear, quadrupole, resonance frequencies of chloro-substituted thiophenes are much higher than those of the corresponding benzene derivatives. This has been ascribed to a relayed inductive effect originating in the polarity of the C—S o-bond in thiophenes. The refractive indices, densities, and surface tension of thiophene, alkyl- and halo-thiophenes, and of some other derivatives have been... 

The factors Kn are elastic constants for the nematic phase and Icb is the Boltzmann constant. Therefore a combination of molecular electronic structure, orientational order and continuum elasticity are all involved in determining the flexoelectric polarisation. Polarisation can also be produced in the presence of an average gradient in the density of quadrupoles. This is... 

Schwerdtfeger, P., Bast, R., Gerry, M.C.L., Jacob, C.R., Jansen, M., Kelld, V., Mudring, A.V., Sadlej, A.J., Saue, T, Sdhnel, T. and Wagner, F.E. (2005) The quadrupole moment of the 3 /2 nuclear groimd state of Au from electric field gradient relativistic coupled cluster and density functional theory of small molecules and the solid slide. Journal of Chemical Physics, 122,124317-1-124317-9. 

Thierfelder, C., Schwerdtfeger, P. and Saue, T. (2007) Cu and Au Nuclear Quadrupole Moments from Four-Component Relativistic Density Functional Calculations using Exact Long-Range Exchange. Physical Review A, 76, 034502-1-034502-4. 

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