Empty-core radius

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

Explain the concept of a pseudopotential. Aluminium is fee with a lattice constant of a = 7.7 au. It is well described by an Ashcroft empty core pseudopotential of core radius 1.1 au. Show that the lattice must be expanded by 14% for the 2n/a(200) Fourier component of the pseudopotential to vanish. 

Figure 5.7 The Ashkroft pseudopotential. This is an empty core model potential, in which the potential is zero inside a radius / c(0 which is different for each /. / is the azimuthal quantum number.

The different location of polar and amphiphilic molecules within water-containing reversed micelles is depicted in Figure 6. Polar solutes, by increasing the micellar core matter of spherical micelles, induce an increase in the micellar radius, while amphiphilic molecules, being preferentially solubihzed in the water/surfactant interface and consequently increasing the interfacial surface, lead to a decrease in the miceUar radius [49,136,137], These effects can easily be embodied in Eqs. (3) and (4), aUowing a quantitative evaluation of the mean micellar radius and number density of reversed miceUes in the presence of polar and amphiphilic solubilizates. Moreover it must be pointed out that, as a function of the specific distribution law of the solubihzate molecules and on a time scale shorter than that of the material exchange process, the system appears polydisperse and composed of empty and differently occupied reversed miceUes [136],... 

The possibility that the core of a dislocation could be empty was first recognised by Frank [1], If the strain energy density arising from a dislocation is sufficiently large, it may become energetically favourable to remove the material near the core and place it in an unstrained environment far from any dislocations. This process creates additional surface area around the core of the dislocation. The equilibrium radius of the hollow core of a screw dislocation is given by... 

The concept behind this theory is illustrated in Fig. 17. The vibrating molecule is approximated as a spherical cavity within a continuum solvent, and the vibrational motion is approximated as a spherical breathing of the cavity. The radius of the cavity is determined by a balancing of forces the tendency of the solvent to collapse an empty cavity, the intermolecular van der Waals attraction of the vibrator for the solvent molecules, and the intermolecular repulsion between the solvent molecules and the core of the vibrator. When the vibrator is in v = 1, the mean bond length of the vibrating bonds is longer due to anharmonicity. The increased bond length... 

For the number of shells in both structures, each lattice is related to the radius (R) of the nanoparticle [27-29]. Therefore, the value of R contains a number of shells and the size of a nanoparticle increases as the number of shells increases. The shells (R) and their numbers are only bounded to the nearest-neighbour pair exchange interactions (J) between spins. To provide the magnetization of the whole particle, each of the spin sites, which stand for the atomistic moments in the nanoparticle, are described by Ising spin variables that take on the values S1-= l, 0. For a core/surface (C/S) morphology, all spins in the nanoparticle are organized in three components that are core (C, filled circles), interface (or core-surface) (CS) and surface (S, empty circles) parts. The number of spins in these parts within the C/S-type nanoparticle are denoted byNc, Ncs and Ns, respectively. But, the total number of spins (N) in a C/S nanoparticle covers only C and S spin numbers, i.e. N =NC + Ns. On the other hand, the numbers of spin pairs for C, CS and S regions in 2D are defined by N [,=(N (.y(. /2)-Ncs,... 

Brunauer, Mikhail and Bodor (13) developed a method of pore structure analysis which employs a hydraulic radius, rj, as a measure of pore size. This radius is defined as r[ = V/S where V is the volume of a group of pore "cores" with the wall surface area S. The "core" refers to the empty... 

The active micro-reactors described above cannot be recycled because the SiH moieties cannot be renewed. Recyelable micro-networks may be realized in the form of passive miero-reactors which do not actively take part in the reaction but merely provide the confined reaction space. For this purpose hollow micro-networks are synthesized first, a micro-emulsion of linear poly(dimethyl-siloxane) (PDMS) of low molar mass (M = 2000-3000 g/mol) is prepared and the endgroups are deactivated by reaction with methoxytrimethylsilane. Subsequent addition of trimethoxymethyl-silane leads to core-shell particles with the core formed by linear PDMS surrounded by a crosslinked network shell. Due to the extremely small mesh size of the outer network shell the PDMS ehains become topologically trapped and do not diffuse out of the micro-network over periods of several months (Fig. 3). However, if the mesh size of the outer shell is increased by condensation of trimethoxymethylsilane and dimethoxydimethylsilane the linear PDMS chains readily diffuse out of the network core and are removed by ultrafiltration. The remaining empty or hollow micro-network collapses upon drying (Fig. 4). So far, shape-persistent, hollow particles are prepared of approximately 20 nm radius, which may be viewed as structures similar to crosslinked vesicles. At this stage the reactants cannot be concentrated within the micro-network in respect to the continuous phase. 

The Kihara potential function [12] is used as described in McKoy and Sinanoglu [13]. The Kihara potential parameters, a (the radius of the spherical molecular core), a (the collision diameter), and e (the characteristic energy) are taken from Tohidi-Kalorazi [14], The fugacity of water in the empty hydrate lattice, // in Equation , can be calculated by ... 

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