Hollow cores radius

As mentioned before and assuming the vahdity of the continuum elasticity theory at the dislocation core, F. C. Frank derived the expression for the characteristic radius of a hollow core (Frank, 1951) ... 

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... 

Normalized to unstrained calcite. Dissolution rate calculation as a function of radius r of the hollow core, based on Eq. 11, for p = 10locm-2 (ST = total perfect surface area Sd = total surface area of hollow cores). 

Above Ccrlt (i.e. E or F in Figure 1), there are two real roots to the equation, so there is a minimum and a maximum in the A G function. If a pit is nucleated at the core, the pit should spontaneously open until its radius fulfills the condition that A G is at a minimum ("10 A). There is then an activation barrier Ag (=AGmax m -A Gm. jmul) toward further opening of the pit into a macroscopic etch pit. Monte Carlo simulations of etch pit formation have shown that such hollow tubes should be stable for some materials, including guartz (27). Above Ccr t, the height of the activation barrier (Ag ) will determine the rate of formation of etch pits. If metastable equilibrium is assumed for the pit nuclei size distribution, the rate of formation of pits per unit area, J, for concentrations above critical should have the form ... 

On compression to the ionization radius of an atom the equivalent of one electron becomes decoupled from the atomic core and finds itself in an impenetrable hollow sphere at constant potential, conveniently defined as V = 0. This problem, which is closely related to the problem of an electron confined to a one-dimensional finite line segment, has been studied in great detail. The Hamiltonian... 

While the outer radius of the apoferritin molecule (a = 63.5 A) can be accurately determined, for example by dynamic light scattering,21 the inner radius (of the hollow shell) is more difficult to estimate. The iron cores of the ferritin have a maximum diameter of the order of 80 A, while the six hydrophobic channels have a length of about 12 A.27 In what follows the value 20 A will be employed for the average thickness of the shell. The Hamaker... 

If atom A is represented as a hollow sphere charged with the valence charge q then the potential has the same value at each point within this sphere namely q /r where r is the average radius of the valence shell. Thus any change in q changes this potential by Aq /rA and the model predicts that all core levels will be changed by this amount... 

Of course, such large states are rather fragile as stressed above, Rydberg states can only develop if there is enough free space around the atom for the wavefunction of the electron to extend well outside the atomic core. This aspect seems to have been appreciated most clearly in the early days of quantum theory by Sommerfeld and Welker [33], who considered a H atom enclosed in a hollow sphere,10 and showed that, if the radius of the sphere is less than 1.835 ao, then the energy of the system becomes positive, i.e. the electron attempts to escape by exerting a pressure on the inner surface of the sphere. 

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. 

We made calculations in two cases (1) when all pure metal (core) was consumed by the reaction Cb (y — x ) > 1 and (2) when the internal radius of the compound shell became equal to the radius of the remaining metallic core (void in the hollow shell between the compound shell and the remaining metallic core with connecting bridges shrinks prior to the end of the reaction)... 

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