Planar atomic density

In a manner similar to that nsed to calculate the density of a nnit cell, we can calcn-late the density of atoms on a plane, or planar density. The perpendicular intersection of a plane and sphere is a circle, so the radius of the atoms will be helpful in calcnlat-ing the area they occnpy on the plane. Refer back to Example Problem 1.4 when we calcnlated the lattice parameter for a BCC metal. The section shown along the body diagonal is actually the (110) plane. The body-centered atom is entirely enclosed by this plane, and the comer atoms are located at the confluence of four adjacent planes, so each contributes 1/4 of an atom to the (110) plane. So, there are a total of two atoms on the (110) plane. If we know the lattice parameter or atomic radius, we can calculate the area of the plane, Ap, the area occupied by the atoms, Ac, and the corresponding... 

In equation (4), A is the number density of atoms per unit surface area A is the dispersion constant the subscripts 5 and / refer to the adsorbent and adsorbate, respectively and do = 0.S5 asf is the z-coordinate at which the 10-4 potential for a single planar surface passes through its zero-point value. The 10-4 potential is obtained by integration of the Lennard-Jones 12-6 potential over an infinite planar surface. The dispersion constants A and Aff represent the adsorbate-adsorbent and adsorbate-adsorbate interactions, respectively these coefficients are calculated from the Kirkwood-Muller equations in the original HK paper [6], Combining equations (2-4) yields an equation that relates filling pressure to pore width ... 

Determination of surface atom density on nanocrystals can be difficult, and imprecise, especially for very small particles that cannot be easily characterized microscopically. Nevertheless, reasonable accuracy can be obtained by using theoretical calculations informed by empirical data. In this work, the CdTe nanocrystals that were prepared (2.5-6 nm diameter) were found to be in the zinc blende crystal structure, allowing the use of the bulk density and interplanar distances of zinc blende CdTe in these calculations. It is likely that a variety of crystalline facets are exposed on individual nanocrystals, each with a range of planar densities of atoms. It is also likely that there is a distribution of different facets exposed across an assembly of nanocrystals. Therefore, one may obtain an effective average number of surface atoms per nanocrystal by averaging the surface densities of commonly exposed facets in zinc blende nanocrystals over the calculated surface area of the nanocrystal. In this work we chose to use the commonly observed (Iff), (100), and (110) zinc blende planes, which are representative of the lattice structure, with both polar and nonpolar surfaces. For this calculation, we defined a surface atom as an atom (either Cd or Te ) located on a nanocrystal facet with one or more unpassivated orbitals. Some facets, such as Cd -terminated 111 faces, have closely underlying Te atoms that are less than 1 A beneath the surface plane. These atoms reside in the voids between Cd atoms, and thus are likely to be sterically accessible from the surface, but because they are completely passivated, they were not included in this definition. 

Dislocations do not move with the same degree of ease on all erystallographie planes of atoms and in all crystallographic directions. Typically, there is a preferred plane, and in that plane there are specific directions along which dislocation motion occurs. This plane is called the slip plane it follows that the direction of movement is called the slip direction. This combination of the slip plane and the slip direction is termed the slip system. The slip system depends on the crystal structure of the metal and is such that the atomic distortion that accompanies the motion of a dislocation is a minimum. For a particular crystal structure, the slip plane is the plane that has the densest atomic packing—that is, has the greatest planar density. The slip direction corresponds to the direction in this plane that is most closely packed with atoms—that is, has the highest linear density. Planar and linear atomic densities were discussed in Section 3.11. 

Further evidence for the unusual nature of benzene is that all its carbon-carbon bonds have the same length—139 pm—intermediate between typical single (154 pm) and double (134 pm) bonds. In addition, an electrostatic potential map shows that the electron density in all six carbon-carbon bonds is identical. Thus, benzene is a planar molecule with the shape of a regular hexagon. All C-C—C bond angles are 120°, all six carbon atoms are sp2-hybridized. and each carbon has a p orbital perpendicular to the plane of the six-membered ring. 

The density of 87O has been determined as 2.15 g cm at 25 °C and calculated from the lattice constants as 2.179 g cm at —110 °C, measured by a single-crystal X-ray diffraction analysis [1, 64, 65]. The 87O molecules are of Cl symmetry and consist of chair-hke seven-membered homocycles with the exocyclic oxygen atom in an axial position see Fig. 2. Most remarkably are the two almost planar groups 0-8-8-8 (torsion angle r=2.9°) and 8-S-8-8 (r=6.3- ). 

The special feature of the dithiocarbamato ligand is an additional 7r-electron flow from the nitrogen atom to the sulfur atoms via a planar delocalised rr-orbital system. The net effect is a strong electron donation, resulting in a high electron density on the metal. )... 

So far as steric effects are concerned, the least energy-demanding direction of approach by the nucleophile to the carbonyl carbon atom will be from above, or below, the substantially planar carbonyl compound. It is also likely to be from slightly to the rear of the carbon atom (cf. 12), because of potential coulombic repulsion between the approaching nucleophile and the high electron density at the carbonyl oxygen atom ... 

Perhaps the most important conclusion to be drawn from results for metal atoms in groups such as 7SiL3 or -PL3+ is undetectably small (70,71). Indeed, the R2C- moiety displays hyperfine interaction with H and 13C that suggest normal planarity at carbon with essentially unit spin-density thereon, and coupling to the metal atom (specifically, 31P) is small and probably negative. This implies that spin-density is acquired by spin-polarisation of the C-M o-electrons and not by p -d delocalisation, as is so often... 

---