General curvature radii

Another point of view is to consider the space-time curvature induced by the rest mass mo of the electron outside a volume of radius rq. According to general relativity theory, the curvature radius Rg around the electron would be given by [18,19] ... 

We study here the meniscus of a liquid facing a vertical plate, shown in Figure 2.13. We now return to the general equation (2.15). Here one of the curvature radius is infinite. The curvature C = 1/R can be expressed as... 

A confocal F.P,I, often called a spherical interferometer [4,33], consists of two spherical mirrors M, with equal curvatures (radius r) which oppose each other at a distance d = r (Fig,4,52a), These interferometers have gained great importance in laser-physics firstly, as high-resolution spectrum analyzers for detecting the mode structure and linewidth of lasers [4,34] and secondly, in the nearly confocal form, as laser resonators. We discuss the former application in this section, while laser resonators will be treated in a more general way in Sect,6,3,... 

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... 

A particular important property of silicon electrodes (semiconductors in general) is the sensitivity of the rate of electrochemical reactions to the radius of curvature of the surface. Since an electric field is present in the space charge layer near the surface of a semiconductor, the vector of the field varies with the radius of surface curvature. The surface concentration of charge carriers and the rate of carrier supply, which are determined by the field vector, are thus affected by surface curvature. The situation is different on a metal surface. There exists no such a field inside the metal near the surface and all sites on a metal surface, whether it is curved not, is identical in this aspect. 

Similar analysis can be made for other types of materials. Thus, as a generalization, the curvature of a surface causes field intensification, which results in a higher current than that on a flat surface. Although the detailed current flow mechanism can be different for different types of materials under different potentials and illumination conditions, the effect of surface curvature on the field intensification at local areas is the same. The important point is that the order of magnitude for the radius of curvature that can cause a significant effect on field intensification is different for the substrates of different widths of the space charge layer. This is a principle factor that determines the dimensions of the pores. 

The so-called Derjaguin equation relates in a general way the force F h) between curved surfaces to the interaction energy per unit area E(h), provided the radius of curvature R is larger than the range of the interactions [17]. Adopting the Derjaguin approximation, one obtains ... 

For intermediate values of rja, or for tubes of intermediate size, no general formula has been given. Bashforth and Adams have however published tables from which the form of any capillary surface may be calculated, and with the aid of these Sugden has further calculated a table of values of rjh for all values of rja, between 0 and 6. h is here the radius of curvature at the crown... 

Schematically, the steps in this process are shown in Fig. 14.4. At the beginning, the local radius of the tip end is small. Field emission can be easily established. A high current though the tip end then causes local melting. The local curvature at the end of the tip suddenly decreases. The field emission current is then reduced dramatically. The tip end recrystallizes to have a relatively large radius. Feenstra et al. (1987a) observed that the tips prepared in this way always provide reproducible tunneling spectra, although atomic-resolution topographic images are generally not observed.

Detonation Front and Shock Front. Detonation Zone and Shock Zone. The shape of the detonation wave and density-distance particle velocity-distance relations behind the wave front are of considerable practical theoretical importance. The deton wave emerging from the end of an unconfined cylindrical chge of a condensed expl is in general spherical in shape. The curvature of this front has a marked effect on both rate pressure of deton. It has been found that there is a minimum radius of convex curvature for each expl, below which deton will not propagate. The min radius of curvature is primarily that at which the divergence is so great that the energy released from the chem reaction of the very small vol of expl involved is insufficient to compensate for the rapid increase in area in the deton front. 

Equation (10.4) is a special case of a more general concept represented by the Young and Laplace equation. A sphere possesses a constant radius of curvature. For an area element belonging to a nonspherical curved surface there can exist two radii of curvature (rj and 2)- If the two radii of curvature are maintained constant while an element of the surface is stretched along the x-axis from x to x + dx and along the y-axis from y to y + dy the work performed will be... 

The Laplace equation in this form is general and applies equally well to geometrical bodies whose radii of curvature are constant over the entire surface to more intricate shapes for which the Rs, are a function of surface position. In the instance of constant radii of curvature across the surface, Eq. (2.68) reduces for several common cases. For spherical surfaces, R = R2 = R, where R is the radius of the sphere, and Eq. (2.68) becomes ... 

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