Packing close

What type of two-dimensional lattice describes the structure of a single layer of close-packed atoms  

Spheres sit in depressions marked with yellow dots 

Spheres sit in depressions that lie directly over spheres of first layer, ABAB... stacking. 

Spheres sit in depressions marked with red dots centers of third-layer spheres offset from centers of spheres in first two layers, ABC ABC... stacking. 

It is well known that in two dimensions close-packed spheres have a hexagonal arrangement in which each sphere is tangent to six others in the plane. In three dimensions, there are two possible ways to achieve close packing. They have to do with the way closed-packed planes are stacked together. In both cases, however, each sphere is tangent to, or coordinated by, twelve others. 

The second case corresponds to three close-packed layers staggered relative to each other. It is not until the fourth layer that the sequence is repeated. This is known as cubic close packed, or CCP, and is represented as (... ABCABCABC...). Geometric considerations show that, for equal-sized spheres in both the CCP and HCP arrangements. 

Second layer atoms in B sites Third layer directly above first layer 

05 percent of the total volume is occupied by the spheres. The packing densities of other nonclosed packed stmctures are given in Table 3.1. 

Atom Location Number of Unit Cells Sharing Atom Fraction of Atom Within Unit Cell 

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The wave function T i oo ( = 11 / = 0, w = 0) corresponds to a spherical electronic distribution around the nucleus and is an example of an s orbital. Solutions of other wave functions may be described in terms of p and d orbitals, atomic radii Half the closest distance of approach of atoms in the structure of the elements. This is easily defined for regular structures, e.g. close-packed metals, but is less easy to define in elements with irregular structures, e.g. As. The values may differ between allo-tropes (e.g. C-C 1 -54 A in diamond and 1 -42 A in planes of graphite). Atomic radii are very different from ionic and covalent radii. 

If the surface tension of a liquid is lowered by the addition of a solute, then, by the Gibbs equation, the solute must be adsorbed at the interface. This adsorption may amount to enough to correspond to a monomolecular layer of solute on the surface. For example, the limiting value of in Fig. Ill-12 gives an area per molecule of 52.0 A, which is about that expected for a close-packed... 

Use Eq. Ill-15 and related equations to calculate and the energy of vaporization of argon. Take m to be eo of Problem 6, and assume argon to have a close-packed structure of spheres 3.4 A in diameter. 

An adsorption isotherm known as the Temkin equation [149] has the form tt = ofF /F where a is a constant and F" is the limiting surface excess for a close-packed... 

A LEED pattern is obtained for the (111) surface of an element that crystallizes in the face-centered close-packed system. Show what the pattern should look like in symmetry appearance. Consider only first-order nearest-neighbor diffractions. 

Consider the case of an emulsion of 1 liter of oil in 1 liter of water having oil droplets of 0.6 /rm diameter. If the oil-water interface contains a close-packed monolayer of surfactant of 18 per molecule, calculate how many moles of surfactant are present. 

This structure is called close packed because the number of atoms per unit volume is quite large compared with other simple crystal structures. 

Figure Al.3.23. Phase diagram of silicon in various polymorphs from an ab initio pseudopotential calculation [34], The volume is nonnalized to the experimental volume. The binding energy is the total electronic energy of the valence electrons. The slope of the dashed curve gives the pressure to transfomi silicon in the diamond structure to the p-Sn structure. Otlier polymorphs listed include face-centred cubic (fee), body-centred cubic (bee), simple hexagonal (sh), simple cubic (sc) and hexagonal close-packed (licp) structures.

Relatively strong adsorbate-adsorbate interactions have a different effect the adsorbates attempt to first optimize the bonding between them, before trying to satisfy their bonding to the substrate. This typically results in close-packed overlayers with an internal periodicity that it is not matched, or at least is poorly matched, to the substrate lattice. One thus finds well ordered overlayers whose periodicity is generally not closely related to the substrate lattice tiiis leads... 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Figure Bl.21.1 shows a number of other clean umeconstnicted low-Miller-index surfaces. Most surfaces studied in surface science have low Miller indices, like (111), (110) and (100). These planes correspond to relatively close-packed surfaces that are atomically rather smooth. With fee materials, the (111) surface is the densest and smoothest, followed by the (100) surface the (110) surface is somewhat more open , in the sense that an additional atom with the same or smaller diameter can bond directly to an atom in the second substrate layer. For the hexagonal close-packed (licp) materials, the (0001) surface is very similar to the fee (111) surface the difference only occurs deeper into the surface, namely in the fashion of stacking of the hexagonal close-packed monolayers onto each other (ABABAB.. . versus ABCABC.. ., in the convenient layerstacking notation). The hep (1010) surface resembles the fee (110) surface to some extent, in that it also...

The central idea underlying measurements of the area of powders with high surface areas is relatively simple. Adsorb a close-packed monolayer on the surface and measure the number A of these molecules adsorbed per unit mass of the material (usually per gram). If the specific area occupied by each molecule is A then the... 

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