Monopole structures

For 1 = 0 the monopole structure can be determined completely if Boo(h) is known. In fact, the phase problem of reduces to the determination of the sign of F(,(,. This is usually not too difficult a task for Bqo as plausible arguments can be made concerning the corresponding radial mass distribution v fr) of the resonant label atoms. Once the signs of the sinusoidal function BQQ(h) are known for each peak, the phases of Ao(,(h) can be determined directly by using the cross term. 

Figure 5.16. Configuration of layers in monopole structures with (a) one k — lor (b) two k= disclinations (c) boojum obtains from the monopole when the line defect...

The boojum has been introduced by Mermin [62] for superfluid 3He-A as a way of reducing the energy of the monopole. In the 3He-A spherical volume, the energy of a monopole decreases when the line shrinks into a point at the surface this point is the boojum. However, in the cholesteric phase, such a transformation violates the equidistance between the layers. As a result, the monopole structure remains stable, at least when R/p 1. Only when R/p I, can the isolated point defect with An rotations of the director field... 

As already discussed, there are no isolated point defects in the cholesteric phase, 7r2(9 = SO 3)/D2) = 0. However, singular point defects can serve as the ends of linear solitons, as in the case of the monopole structure, in which the nonsingular disclinations can be considered as linear solitons. 

The most rigid cholesteric biopolymers have other types of layer textures, the monopole structures or Robinson spherulites discussed in the previous section the layers are approximately along concentric spheres (positive Gaussian curvature). An extensive study and review of cholesteric spherulites in materials of biological interest can be foimd in Bouligand and Livolant [57]. 

We assume that the double bonds in 1,3-butadiene would be the same as in ethylene if they did not interact with one another. Introduction of the known geometry of 1,3-butadiene in the s-trans conformation and the monopole charge of 0.49 e on each carbon yields an interaction energy <5 — 0.48 ev between the two double bonds. Simpson found the empirical value <5 = 1.91 ev from his assumption that only a London interaction was present. Hence it appears that only a small part of the interaction between double bonds in 1,3-butadiene is a London type of second-order electrical effect and the larger part is a conjugation or resonance associated with the structure with a double bond in the central position. 

In practice, the choice of parameters to be refined in the structural models requires a delicate balance between the risk of overfitting and the imposition of unnecessary bias from a rigidly constrained model. When the amount of experimental data is limited, and the model too flexible, high correlations between parameters arise during the least-squares fit, as is often the case with monopole populations and atomic displacement parameters [6], or with exponents for the various radial deformation functions [7]. 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

BUSTER has been run against the L-alanine noisy data the structure factor phases and amplitudes for this acentric structure were no longer fitted exactly but only within the limits imposed by the noise. As in the calculations against noise-free data, a fragment of atomic core monopoles was used the non-uniform prior prejudice was obtained from a superposition of spherical valence monopoles. For each reflexion, the likelihood function was non-zero for a set of structure factor values around this procrystal structure factor the latter acted therefore as a soft target for the MaxEnt structure factor amplitude and phase. 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

The calculation of the thermally-smeared core fragment and the valence monopoles densities was carried out by a Fourier transform of a set of aliased structure factors computed with the program VALRAY [46] details of this calculation have been published elsewhere [49],... 

The first approximation to the description of Rydberg levels treats the benzene ion-core as a monopole. This description is known not to be quantitatively accurate. Calculations which include the symmetry of the molecular ion, and the charge delocalization, lead to an energy level spectrum in much better agreement with experiment. Thus, it seems unlikely that the geometric structure of the molecular ion can be completely neglected in the study of photoionization. 

Fig. 12a-e. Non-monopole (quadrupole) relaxation and shape distortion of a core hole in open shell type of situations (a) Lowest order self-energy diagram (b) one-electron picture of the pulling down of empty levels below the Fermi level (c) shape distortion of one-electron orbitals due to quadrupole relaxation (d, e) schematic core level spectra in the case of (d) closed and (e) open ground state shell structure (d)... 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

One finds that solid inclusions tend to define the low frequency structure in k, the attenuation edge and the resonance in c, by means of the dipole resonance, whereas cavities tend to define it by means of the monopole resonance (35). Since the dipole resonance of a solid inclusion involves center of mass motion of the inclusion, this resonance tends to be at lower frequencies than that due to the monopole of a cavity which merely involves cavity wall motion. The heavier the inclusions in the case of solid inclusion and the softer the matrix material in the case of cavities, the lower in frequency in which significant attenuation is achieved (attenuation edge) and the larger the corresponding scattering cross sections. 

In a rigid matrix material, the low frequency resonance structure of cavities tend to be smeared out and overlapping. Monopole scattering is then relatively weak, but dipole and quadrupole scattering, with accompanying mode conversion, enhance attenuation throughout the low frequency region. 

---