Single particle form factor

The form factor term, P(q), contains information on the distribution of segments within a single dendrimer. Models can be used to fit the scattering from various types of particles, common ones being a Zimm function which describes scattering from a collection of units with a Gaussian distribution (equation (3a)), a... 

FIG. 12 Pattern of light scattered from a single layer of colloidal particles in the disordered phase. The particles are polystyrene spheres, of diameter 2 /glass plates. Except for the contribution of the form factor P(k), which depends on the scattering angle, and normalization and geometrical factors, this picture shows directly the static structure factor of the system. 

The above derivation can also be applied to colloidal or polymer-based liquids and is then used to calculate the so-called form factors of soft matter samples. The major difference between a monoatomic liquid and a polymer chain in the melt or in solution is that the total structure factor consists of two parts. The first is the inter-particle structure factor and the second the intra-particle structure factor. This second part is also often called the particle form factor P(q). Using Eq. (2.38) it is straightforward to calculate P(q) for a given soft matter sample. A good example is the form factor of a single polymer coil in a melt [88, 92]. The pair correlation function of such a coil is given by... 

In the case of hybrid functionals, still another mode of implementation has become popular. This alternative, which also avoids solution of Eq. (91), is to calculate the derivative of the hybrid functional with respect to the singleparticle orbitals, and not with respect to the density as in (91). The resulting single-particle equation is of Hartree-Fock form, with a nonlocal potential, and with a weight factor in front of the Fock term. Strictly speaking, the orbital derivative is not what the HK theorem demands, but rather a Hartree-Fock like procedure, but in practice it is a convenient and successful approach. This scheme, in which self-consistency is obtained with respect to the singleparticle orbitals, can be considered an evolution of the Hartree-Fock Kohn-Sham method [6], and is how hybrids are commonly implemented. Recently, it has also been used for Meta-GGAs [2]. For occupied orbitals, results obtained from orbital selfconsistency differ little from those obtained from the OEP. 

Here we consider how to obtain the single-particle form factor even when labeled molecules are present at a high concentration. For this purpose let us first consider a single-component, bulk polymer, consisting of N molecules in volume Vy each molecule with Z = v/vu segments. The amplitude of scattering from such a polymer is... 

We define the single-particle (or intramolecular) form factor P(q) by... 

The solid-phase diffusion of monoatomic metal (or its oxide) particles to give multiatomic cluster particles. It is assumed that diffusion processes in the solid phase are activated. The rate of diffusion = DnC = C Di where C = Cla is the current amount of reaction centers per one cell of the a size, N is the size of A-atomic cluster, Di = Doexp[- aE/( T)] is the diffusion coefficient for a monoatomic particle, EaX) and Do = vexp(A57/ ) are the varying parameters, the energy of activation, and the entropy factor ((v 10 sec" ). Such a model assumes that the coexistence of two or more separate particles in one cell is not possible because they immediately form a single cluster. In the isotropic medium the diffusing particle with a corresponding probability can move in one of the 26 directions. 

In our recent smdies, we focused on several complicating factors arising in studies of nanoparticles of a non-negligible size (e.g., polymeric micelles, vesicles) that can carry several fluorescent labels. When the dimensions of such particles become comparable to the typical dimensions of the effective volume (coi, (O2), the correlated motion of the fluorophores located on a single particle affects the shape of the autocorrelation function. Recently, an approximate expression for the FCS autocorrelation function of diffusing particles of finite size has been derived by Wu et al. [85]. They have shown that the autocorrelation function of uniformly labeled spherical particles can be expressed in a form similar to (12) where the diffusion time, concentration, and dimensions of the active volume are replaced by corresponding apparent quantities that depend on the particle size. Qualitatively, the same results were obtained in our computer simulations, which are discussed later (see Sect. 4.3). 

The inner diffraction effect is produced when the individual particles of the atom capable of vibration, i.e. the electrons contained in the atoms, are dispersed and give rise to secondary radiations which interfere with one another. In a liquid built up of single atoms— A, Kr, Xe, Hg, Ga— the result is that, on account of intra-atomic interference, the dependence of intensity distribution on the angle of diffraction is already affected. This influence is generally expressed by a factor, which, because of its origin, is called the atomic form factor its action is that more intensity is scattered in the directions near the primary beam than if the interaction of the individual electrons is not taken into account. 

In interacting systems the optical and orientation factors in a are no longer separable quantities. The induced optical effect is determined both by the single particle scattering [form factor P q)] and by the pair distribution function [structure factor 5( )], the latter being direction-dependent [42]. Since in addition to orientation the electric field causes particle translation, even for spherical particles the radial distribution function g q, /) and the "static" structure factor attain time-dependent induced anisotropy. The deformed surface potential also contributes to this effect. 

This effect originates from the anisotropic dipole-dipole correlations not accounted for by the Maier-Meier theory operating with a single particle distribution function. When, with decreasing temperature, the smectic density wave p(z) develops (even at the short-range scale) the longitudinal dipole moments prefer to form antiparallel pairs and the apparent molecular dipole moment becomes smaller. This would reduce positive s. Theoretically, dipole-dipole correlations may be taken into account by introducing the so-called Kirkwood factors. 

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