Characteristic radius

Colby et al. [35] proposed an interesting experimental approach to measure the static exponents. They noticed that it is hard to accurately measure the chemical extent of reaction, p, and thus eliminated this variable (more precisely the distance from the gel point p — pc ) from the scaling relations. For example combining Eqs. 2-5 and 2-6 yields the following relation between the weight average molecular weight, Mw, and the characteristic radius, Rchar ... 

The experiments discussed in this chapter have shown that a variety of chiral molecules self-assemble into cylindrical tubules and helical ribbons. These are indeed surprising structures because of their high curvature. One would normally expect the lowest energy state of a bilayer membrane to be flat or to have the minimum curvature needed to close off the edges of the membrane. By contrast, these structures have a high curvature, with a characteristic radius that depends on the material but is always fairly small compared with vesicles or other membrane structures. Thus, the key issue in understanding the formation of tubules and helical ribbons is how to explain the morphology with a characteristic 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) ... 

Finite and nonzero values of the integrated quantities (B.31)-(B.32) can now be obtained by shrinking the characteristic radius Tq to very small values, as represented by... 

It has thus been shown that the present theory of charged particle equilibria necessarily leads to point-like configurations with an excessively small characteristic radius r0, permitted, in principle, even to approach the limit r0 = 0. In this way the integrated field quantities can be rendered finite and nonzero. As pointed out in Section V.A.l.b, a strictly vanishing radius would not become physically acceptable, whereas a nonzero but very small radius is reconcilable both with experiments and with the present analysis. It would leave space for some form of internal particle structure. A small but lower limit of the radius would also be supported by considerations based on general relativity [15,20]. 

It should be noted that SASA itself can be defined in many ways (see, for instance, Pascual-Ahuir, Silla, and Tunon 1994). In the simplest approach, one imagines solvent molecules to be spheres having some characteristic radius. The SASA is then generated by the center of... 

Let us consider a slit-like pore of width D along whose walls the ip(x) potential is localized (Fig. 4). We shall regard the interaction of monomers with the walls as a short-range interaction and the characteristic radius of interaction as being of the order of the segment size a. The exact assignment of the form of the potential is immaterial for our purposes, since it describes the effective interaction of units with the pore walls, renormalized by the solvent molecules. Conditions are to be as follows ... 

Fig. 4a and b. Distribution of the segment density at different values of the energy 0 (a) and schematic picture of lattice-like chain of length N in a slit-like pore of width D (b). 0cis the critical energy characteristic of the case when the entropy losses of the macromolecule in the pore are compensated by the energy of interaction with the wall. attractive potential of a depth 0 and with a characteristic radius of interaction r0 of the order of the segment size a... 

Figure 4.40 Deflection of an electron in a magnetic field which is perpendicular to its original path and leads in the plane of drawing to a new path with characteristic radius R. The distance s along the original path is related to the distance d as indicated.

Of course, sharper tips produce sharper images. Commercially available supertips have a diameter in the 0.1 pm range, and a characteristic radius at the tip of about 20 nm. Such tips can be made by focusing an intense electron beam of a scanning electron microscope in the presence of a low pressure of hydrocarbons... 

Figure 6. Calculated results from the similarity solution plotted as a function of time. Here R0 is the characteristic radius for energy deposition, A is the nonlinear amplitude of the temperature and density functions, T (R = 0) is the central temperature, and I is the induction parameter. The indicates the predicted induction time r0 = 1.0 x 10 4 sec E0 — 4.0 X 10 ergs R0 = 0.1 cm.

The problem is avoided by modification of the characteristic atomic radii to match the condition of strain that exists in molecules. As a typical example the variation of calculated dissociation energy with characteristic radius is shown in the table below for the C-C bonds in ethane, ethene and acetylene. The discrepancy approaches experimental uncertainty for all three bonds at the same value of rc = 1.85 A, which also models diatomic C2 correctly. 

Table 5.1 Dissociation energy as a function of characteristic radius...

An unexpected feature of Table 5.1 is the remarkable similarity between the energies calculated from the characteristic radius rc and those calculated from the ionization radius r0, for the same interactions, but with bond orders increased by unity. It means that the steric factor which is responsible for the increase in bond order i.e. screening of the internuclear repulsion) is also correctly described by an adjustment to r o to compensate for modified valence density. Calculating backwards from first-order D0 = 210 kjmol-1, an effective zero-order C-C bond length of 1.72 A is obtained. 

Molecular-mechanics force fields distinguish between general and 1,3 non-bonded interactions. The obvious reason for this distinction is that the distance between ligands is affected when linked to the same central atom. Their final non-bonded separation depends, not only on ligand type, but also on the size of the central atom. In such a three-atom system the relevant parameters are the characteristic radius (rc) of the central atom, together with the... 

The basic relationship between the intensity profile I(s) and the particle given by a number of atoms N and number of electrons per atom n and a characteristic radius r is given by the following ... 

In this equation, r and z are cylindrical coordinates (z = 0 being the axis of symmetry of the field), C is a constant, k is field curvature, and Rm is the characteristic radius. The... 

The effective nuclear charge on a ligand atom was established as Ze = ka 3, where a is a characteristic radius which can now be identified with the ionization radius of the atom, and k — 0.8 was found empirically to produce effective nuclear charges relative to Z H) = 1. This formula follows from the formulation... 

In most cases, Do is about 10 cm s, and it then follows that the characteristic radius of an UME is smaller than 6 [im. However, it is important to notice that Do is sometimes appreciably smaller than 10 cm s if O is a very large molecule, or, more likely, if a viscous or glass-like medium is used planar diffusion would then prevail even at a radius of 6 pm. Note also that a microelectrode with a radius of 100 pm experiences a substantial contribution from spherical diffusion. If the geometry of the electrode differs from the simple disk or microsphere shape, the definition is less clear. In general, the characteristic behavior of a UME is observed if at least one of the dimensions is in the micrometer range. For instance, a band electrode with a width of less than 10 pm behaves as an UME even if it is several mm long. 

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