Particle topology

Keywords. Nanocrystals, Core-shell particles. Topology, Opals, Capsules, Surface plasmon absorption, Bragg diffraction. Arrays, Colloidal crystals... 

The parameter C increases with the molar mass and is related to the branching probability p. For self-similar structures, one expects that the particle topology does not vary with molar mass, that is, the form factor should be independent of molar mass. These nonrandomly branched... 

The detailed consideration of these equations is due largely to Kozeny [50] the reader is also referred to Collins [51]. However, it is apparent that, subject to assumptions concerning the topology of the porous system, the determination of K provides an estimate of Ao- It should be remembered that Ao will be the external area of the particles and will not include internal area due to pores (note Ref. 52). Somewhat similar equations apply in the case of gas flow the reader is referred to Barrer [53] and Kraus and co-workers [54]. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

This picture of a fresh particle coating shows the characteristic topology of the PSA. 

Fig. 19. Topology and peel removal of discrete PSA particles from a substrate.

Suzuki, Y Nishio, I, Quasielastic-Light-Scattering Study of the Movement of Particles in Gels Topological Structure of Pores in Gels, Physical Review B 45, 4614, 1992. 

Figure 7.5 Two topologically distinct types of mesoporous gold sponge, each with 50 volume % gold, (a) Swiss-cheese morphology produced by de-alloying, (b) aggregated particle morphology produced by sintering of nanoparticles.

Thus for an ideal two-phase system the total calibrated intensity that is scattered into the reciprocal space is the product of the square of the contrast between the phases and the product of the volume fractions of the phases, Vi (1 — Vi) = V1V2. V1V2 is the composition parameter66 of a two-phase system which is accessible in SAXS experiments. The total intensity of the photons scattered into space is thus independent from the arrangement and the shapes of the particles in the material (i.e., the topology). Moreover, Eq. (8.54) shows that in the raw data the intensity is as well proportional to the irradiated volume. From this fact a technical procedure to adjust the intensity that falls on the detector is readily established. If, for example, we do not receive a number of counts that is sufficient for good counting statistics, we may open the slits or increase the thickness of a thin sample. 

The quantitative analysis of a multiphase topology comprises the formulation of structure models and the fitting of measured data. Fitting is discussed in Chap. 11. In this section the setup of topological models is discussed. The problem arises from the fact that most structural models of particle correlation are anisotropic and the visualization of structure in anisotropic materials by means of the CDF shows that suitable models must be rather complex. Thus a direct fit of anisotropic data would require fitting of a measured 3D or 2D function by a complex model. Both the effort to setup such models, and the computational effort to fit the data are very high. 

The analytical structural model for the topology of the nanostructure is defined in Isr (5). For many imaginable topologies such models can be derived by application of scattering theory. Several publications consider layer topologies [9,84,231] and structural entities built from cylindrical particles [240,241], In the following sections let us demonstrate the principle procedure by means of a typical study [84],... 

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