Plane cut

Current and voltage distribution remain completely the same if a plane cuts the sphere through its center and the upper half is removed (half space). Since only half the current flows from the hemisphere, its grounding resistance is obtained by substituting 1/2 for 7 in Eq. (24-8) (see Table 24-1, column 1) ... 

Figure 1.7 The cylindrical symmetry of the H-H ct bond in an H2 molecule. The intersection of a plane cutting through the (T bond is a circle.

We know from Section 1.5 that cr bonds are cylindrically symmetrical. In other words, the intersection of a plane cutting through a carbon-carbon singlebond orbital looks like a circle. Because of this cjdindrical symmetry rotation is possible around carbon-carbon bonds in open-chain molecules. In ethane, for instance, rotation around the C-C bond occurs freely, constantly changing the spatial relationships between the hydrogens on one carbon and those on the other (Figure 3.5),... 

For a homonuclear diatomic molecule such as Cl2 the interatomic surface is clearly a plane passing through the midpoint between the two nuclei—in other words, the point of minimum density. The plane cuts the surface of the electron density relief map in a line that follows the two valleys leading up to the saddle at the midpoint of the ridge between the two peaks of density at the nuclei. This is a line of steepest ascent in the density on the two-dimensional contour map for the Cl2 molecule (Fig. 9). 

Figure 5.48 Optimized-structure views, (a) and (b), and leading 7i

The Ru single crystal was oriented by Laue x<-ray back-scattering to within 1° of the Ru(001) plane, cut by a diamond saw and mechanically polished. After being etched in hot aqua regia for about 15 min, the crystal was spot welded to two tantalum heating wires which were connected to two stainless steel electrodes on a sample manipulator. The temperature was monitored by a Pt/Pt-10% Rh thermocouple which was spot welded to the back of the crystal. 

The minimum of ssq is near the true values slope= 6 and intercepts20 that were used to generate the data (see Data mxb. m). ssq is continuously increasing for parameters moving away from their optimal values. Analysing that behaviour more closely, we can observe that the valley is parabolic in all directions. In other words, any vertical plane cutting through the surface results in a parabola. In particular, this is also the case for vertical planes parallel to the axes, i.e. ssq versus only one parameter is also a parabola. This is a property of so-called linear parameters. 

D plane into two pieces, (c) A 2-D plane cuts a 3-D space into two pieces. By analogy, our 3-D space would cut a 4-D hyperspace into two pieces. 

FIGURE 5.26 The Koch-Cohen cluster illustrated with the back and front planes cut away for clarity. The central section with four tetrahedrally coordinated Fe ions is picked out in bold. 

In molecular structures, planes of symmetry can cut through atoms or through bonds between atoms, (n) CHj has no center but has two planes, as shown in Fig. 5-I(a). Each plane cuts the C and a pair of H s. while bisecting the other H—C—H bond angle, (b) CHCI, has no center and has one plane that cuts the C. H. and one of the Cl s while bisecting the Cl—C—Cl bond angle, as shown in Fig. 5-l(ft). (c) In ethene all six atoms lie in one plane, which, in Fig. 5-l(c). is the plane of the paper. It has a center of symmetry between the C s. It also has three symmetry planes, one of which is the aforementioned plane of the paper. Another plane bisects the C=C bond and is perpendicular to the plane of the paper. The third plane cuts each C while bisecting each H—C—H bond. 

Alternatively, one could say that these planes cut the a axis at intervals ofa/3 (a being the repeat distance in this direction), the b axis at intervals of 6 1, and the c axis at intervals of cf2, the indices being defined as the reciprocals of these intercepts. This comes to the same... 

If, as in Fig. 108 a, the c axis of the crystal is displaced from the axis of rotation in the plane normal to the beam (for the mean position of the crystal), the zero layer (hk0) of the reciprocal lattice is tilted in this same direction, and its plane cuts the sphere of reflection in the circle AD. During the 15° oscillation a number of hkO points pass through the surface of the sphere, and thus X-rays reflected by these hkO planes of the crystal strike the film at corresponding points on the flattened-out film (Fig. 108 b) the spots fall on a curve BAD, whose distance from the equator is a maximum at a Bragg angle 0 = 45° and zero at 6 = 90°. If, on the other hand, the displacement of the c axis is in the plane containing the beam (Fig. 108 c), the spots on the film fall on a curve whose maximum distance from the equator is at 6 = 90° (Fig. 108 d). When the displacement of the c axis has components in both directions,... 

Fapp. The plane cuts the body normal to the applied force. There are two contributions from the body itself. One is the projection of the surface capillary force per unit length (7s) onto the normal direction and integrated over the bounding curve. The second is the normal stress crnn integrated over the cross-sectional area—in the case of fluids bounded by a surface of uniform curvature s, ann = -ysns [4]. 

In addition to forming cr- and tr-bonds similar to those formed by p-orbitals, d-orbitals may overlap in 8-bonds with two nodal planes cutting through the intemuclear axis. Draw overlap diagrams showing how d-orbitals can combine in these three ways. 

Figure 10.21 Forms of the n molecular orbitals for chains of from one to seven atoms. Symmetry labels (S, symmetric A, antisymmetric) refer to symmetry with respect to a perpendicular mirror plane cutting the chain at its midpoint. Reprinted with permission from M. J. Goldstein and R. Hoffman, J. Amer. Chem. Soc., 93, 6193 (1971). Copyright by the American Chemical Society.

Why is it that some objects are chiral but others aren t In general, an object is not chiral if, like the coffee mug, it has a symmetry plane cutting through its middle so that one half of the object is a mirror image of the other half. If you were to cut the mug in half, one half of the mug would be the mirror image of the other half. A hand, however, has no symmetry plane and is therefore chiral. If you were to cut a hand in two, one "half" of the hand would not be a mirror image of the other half (Figure 20.20). 

In Fig. 4.3, an additional set of planes, and thus an additional source of diffraction, is indicated. The lattice (dark lines) is shown in section parallel to the ab faces or the xy plane. The dashed lines represent the intersection of a set of equivalent, parallel planes that are perpendicular to the xy plane of the paper. Note that the planes cut each a edge into two parts and each b edge into one part, so these planes have indices 210. Because all (210) planes are parallel to the z axis (which is perpendicular to the plane of the paper), the / index is zero. [Or equivalently, because the planes are infinite in extent, and are coincident with c edges, and thus do not cut edges parallel to the z axis, there are zero (210) planes per unit cell in the z direction.] As another example, for any plane in the set shown in Fig. 4.4, the first plane encountered from any lattice point cuts that unit cell at a/2 and b 3, so the indices are 230. 

All planes perpendicular to the xy plane have indices hkO. Planes perpendicular to the xz plane have indices hOk, and so forth. Many additional sets of planes are not perpendicular to x,y, or z. For example, the (234) planes cut the unit cell edges a into two parts, b into three parts and c into four parts. (See Fig. 4.5.)... 

The methods discussed in this section can be extended to systems that have more than three components. The problem is to convert each system to a pseudobinary system. For a quaternary system, the properties of an equilateral tetrahedron may be used to depict the composition of the system. The composition axes would be four lines drawn from the four apexes perpendicular to the opposite faces. Planes cutting the tetrahedron parallel to the bases would represent pseudoternary systems for which one composition variable would be constant. Pseudobinary systems would be depicted by the intersections of two of the pseudoternary planes. Indeed, the experimental measurements and calculations would be extensive. 

The multiplicity factor, m, specifies the number of equivalent lattice planes that may all cause reflection at the same Bragg angle position, that is, the number of equally spaced planes cutting a unit cell in a particular, Qikl), crystalline plane family. In the case of low symmetry systems, the multiplicity factor will be low every time. On the other hand, for high symmetry systems, a single family of... 

The points at which the s-lines of the particles are cut by this moving plane represent simultaneous events. Thus the s-diagram suggests that an observer is merely watching the points at which his time plane cuts the s-lines as it moves along. 

The surface tension is conveniently regarded as the deficiency in hydrostatic pressure across a plane cutting the surface perpendicularly, compared with a plane of similar area wholly in one of the bulk phases this is the aspect of surface tension used by Bakker (see Chap. I, 4, p. 4). The work done in increasing the volume in one of the bulk phases by dV is P dV that done in increasing the volume of the surface phase by dVa, and its area by dA, is PdVa—ydA, where y is the surface tension. 

What is the essential difference between these two compounds that means one is super imp osable on its mirror image and one is not The answer is symmetry. Acetone cyanohydrin has a plane of symmetry running through the molecule. This plane cuts the central carbon and the Of 1 and CN groups in half and has one methyl group on each side. 

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