Planar Density

The (111) high planar density face of the fee structure showing how the atomic planar density is calculated. 

---

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... 

Combine your answers appropriately to arrive at the planar density, PD. If r = 1.36 A, what is the number density (number of atoms per unit area) for the (111) plane in FCC ... 

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. 

Table 1. Summary of XSW studies of aqueous ion adsorption at the calcite (1014) surface the adsorbate ions coherent coverage, coherent fractions and coherent positions. c =yjoT40-lML corresponds to the (1014) planar density of Ca2+ ions.

The slip direction is usually the direction having the smallest spacing between atoms or ions of the same type (the highest linear density). In metals, the slip plane is often the closest packed plane (the highest planar density). In ceramics, we consider planar density, but there is often... 

In order now to relate the average jump frequency A of a particle i to bi or to let us analyze the particle flux passing through an imagined unit interface lying between two lattice planes of an ideal isotropic solution. The planar density (number of atoms per cm ) of particles of type i is A,x lattice plane in front of the unit interface, and lattice plane... 

Writing the corresponding a as planar density of the (Jongitudinal) effective charge, O = /2), Fig. 6.1.5 yielded... 

Fig. 2b Ni (100) total density of states and planar density of states for central plane, second plane from the surface, and the surface plane.

In fact, thorough analyses of X-ray diffraction patterns of SPS films exhibiting different crystalline and co-crystalline phases, and related evaluations of degrees of orientation, have allowed the conclusion that the three observed uniplanar orientations correspond to the three simplest orientations of the high planar-density ac layers (i.e., of close-packed alternated enantiomorphous SPS helices, Rg. 10.2a,b) with respect to the film plane. In particular, it has been proposed that the three uniplanar orientations of SPS should be named Ou cu, Uu Cj, and C//, indicating crystalline phase orientations presenting the a and c axes parallel (//) or perpendicular ( ) to the film plane (Hg. 10.2c-e) [75]. 

Figure 2.7 Transverse planar density (c/J of clay (filled) and solvent (open) particles versus x at T= 1-10 with clay-solvent interaction e s= 1, —1, -2 plates are in the...

The planar density (number of atoms per unit area) of the (111) face of an fee structure can be found from Figure 4.16. The altitude of the equilateral triangle inscribed by the six circles is /V2R. Its area is therefore iVSRp-. The triangle contains three 1 /2 circles and three 1 /6 circles, so the number of atoms included is 4. The planar density becomes... 

The bcc structure has no close-packed planes like the (111) plane in the fee structure. The planes with the highest planar densities are the 101, 211, and 321 families. The (101) face shown in Figure 4.17 contains two atoms and has an area given by or 16V2/3R2. The planar density is given by... 

The planar fraction is the area covered by the atoms whose centers lie in the plane divided by the area of the plane. Unlike the planar density, which contains the radius of the atoms, the planar fraction is a pure number and is perhaps more meaningful. Planes with the highest planar fraction are the slip planes along which metals deform plastically. The importance of this parameter is discussed in more detail in Chapter 9. 

The planar fraction is just the planar density multiplied by ttR. From Equation 4.17, the planar fraction of the (111) plane in the fee system is... 

Relations between the atomic radius R and the lattice dimension a in the cubic systems are important for determining properties such as mass density, APF, planar densities, and... 

The two previous sections discussed the equivalency of nonparallel crystallographic directions and planes. Directional equivalency is related to linear density in the sense that, for a particular material, equivalent directions have identical linear densities. The corresponding parameter for crystallographic planes is planar density, and planes having the same planar density values are also equivalent. 

In an analogous manner, planar density (PD) is taken as the number of atoms per unit area that are centered on a particular crystallographic plane, or... 

Linear and planar densities are important considerations relative to the process of slip— that is, the mechanism by which metals plastically deform (Section 7.4). Slip occurs on the most densely packed crystallographic planes and, in those planes, along directions having the greatest atomic packing. 

---