Center of curvature

Foi shts placed at from the center of curvature, the electrons passed by this analyzer foUow the equipotential surface described by R. With an acceptance angle 8a shown in Figure 22 and a sht width w, the energy resolution of the CFIA is given by... 

When the isobars are curved, an additional force, a centrifugal force outward from the center of curvature, enters into the balance of forces. In the case of curvature around low pressure, a balance of forces occurs when the pressure gradient force equals the sum of the coriolis and centrifugal forces (Fig. 17-12) and the wind continues parallel to the isobars. In the... 

The CHA is shown in schematic cross-section in Fig. 2.5 [2.5]. Two hemispheres of radii ri (inner) and T2 (outer) are positioned concentrically. Potentials -Vi and -V2 are applied to the inner and outer hemispheres, respectively, with V2 greater than Vi. The source S and the focus E are in the same plane as the center of curvature, and Tq is the radius of the equipotential surface between the hemispheres. If electrons of energy E = eVo are injected at S along the equipotential surface, they will be focused at Eif ... 

The radius of curvature R of a plane curve at any point P is the distance along the normal (the perpendicular to the tangent to the curve at point P) on the concave side of the curve to the center of curvature (Figure 1-33). If the equation of the curve is y = f(x)... 

For a spherical mirror the conjugate points lie on top of each other at the center of curvature (COC). Light emitted from the COC of the spherical mirror will focus back onto the COC without any additional system aberration. 

The first environmental problem encountered would be air turbulence. The light travels a 200 m path from center of curvature to the mirror and back. To give an example of the potential problem, assume the temperature of the air... 

A nearly perfect diverging wavefront would exit the test plate appearing as though it came from a source 100 m away. A segment would be positioned so that its mean center of curvature was coincident with that virtual source 100 m away. In the worst case in our example, the un-equal air path would be about 4 m rather than 204 m. Interference would take place between the wavefront reflected off the 100 m radius side of the test plate and the segment. The roughly 3 m back to the source and beamsplitter is common path and will not affect the interference pattern. 

Concentric Having a common center of curvature or symmetry, [nih]... 

Some interesting features of the Stoll formula are worth noting. First, only the sum R + d) appears in the formula. Therefore, only the distance of the center of curvature of the tip to the sample surface matters. The radius R becomes irrelevant. Second, the STM corrugation decays exponentially with tip-sample separation. By extrapolating the formula to (/ +1/) = 0, the corrugation coincides with that of the metal surface. These features are also found to be consistent with experimental results and with the results of the Tersoff-Hamann theory, as described below. 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

Fig. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

The STM images of large superstructures on metal surfaces exhibit a very simple form. As shown first time by Tersoff and Hamann (1983, 1985), at the low-bias limit, the STM images of large superstructures on metal surfaces are independent of tip electronic states, and an STM image is simply a contour of an important quantity of the sample surface only the Fermi-level local density of states (LDOS), taken at the center of curvature of the tip. An attempt was also made to interpret the observed atom-resolved images of semiconductors... 

Equation C.23 is the form of the Gibbs-Thomson equation introduced in Eq. C.17. It is a conditionfor mechanical equilibrium in a two-phase system with a curved interface. The phase located on the side of the interface toward its center of curvature (e.g. the (3 phase in Fig. C.5), has the higher pressure. Note also that for a flat interface, Eq. C.23 gives Pa = P0, as expected. 

Real image" means that the image is on the same side of the mirror as the object). This relationship, known as the "thin-lens fomula," is approximate it holds exactly only when u = v = R, that is, when a point source and its image are both at the center of curvature C. Also, if the object is moved to I, then the image will form at O that is, the points O and I are "conjugate," and... 

Centripetal force. The centripetal force is the radial component of the net force acting on a body when the problem is analyzed in an inertial system. The force is inward toward the instantaneous center of curvature of the path of the body. The size of the force is mv2/ where r is the instantaneous radius of curvature. See centrifugal force. 

For refractive surfaces, define the surface radius to be the directed distance from a surface to its center of curvature. Thus a surface convex to the incident light is positive, one concave to the incident light is negative. The surface equation is then n/s + n /s = (n -n)/R where s and s are the... 

One can say that the mirror folds the length axis at the mirror, so that emergent rays to a real image at the left represent a positive value of s. We are forced also to declare that the mirror also flips the sign of the surface radius. For reflective surfaces, the radius of curvature is defined to be the directed distance from a surface to its center of curvature, measured with respect to the axis used for the emergent light. With this qualification the convention for the signs of s and R is the same for mirrors as for refractive surfaces. 

Equations (7-66) and (7-67), or related versions, have been used by Hupes and Jencks and by Castro and co-workers to account for curvature. The quantity p (J) defines the center of curvature of the plot and is expected to occur when the pK of the nucleophile is equal to the pK of the leaving group." For weaker nucleophiles pKa < p/(2) breakdown of the tetrahedral intermediate will be rate determining, because the leaving group X is a stronger nucleophile than is N , so k2< k i if, however, pX > pX, the nucleophilic attack is rate determining. 

To estimate the expected number of tries required to find a point + j such that A(J> < 0 we consider a random walk in which a step is determined by chooang a random point from a uniform distribution on the unit hypersphere centered at and moving in that direction a constant distance AX. The probability that the step so chosen has a component pointing toward the center of curvature of the contour = constant (toward the stationary point) equals the fraction... 

In any point of the crystal tangent to the circle with center C, the normal to the curved crystal planes passes through O, which is the center of curvature of these planes. Let S be a point of the circle such that the angle (SCO) = n- 20>. with 0>. as the corresponding Bragg angle. For any point M of the single crystal, we then have ... 

Then, the radius of curvature, R between the center of curvature and the tangent points (for example Ra in Figure 4.4 b) can be found from the simple distance formula of analytical geometry ... 

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