Atomic and Planar Densities

To understand the description of ion implantation and ion doses, one must know the atomic density interplanar distance between planes and the number of atoms cm-2 on a given plane. In cubic systems with an atomic density of N atoms cm-3, the crystal lattice parameter, ac, is given by 

The atomic volume can be calculated without the use of crystallography. The atomic density N of atoms cm-3 is given by 

The average areal density of a monolayer, Ns atoms cm-2, also can be estimated without the use of crystallography by taking the atomic density N to the 2/3 power. 

Equation (1.5) gives the average areal density of one monolayer for a material with an atomic density N. 

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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. 

Crystallographic directional and planar eqnivalencies are related to atomic linear and planar densities, respectively. 

The Lewis dot formulas for the three resonance structures (one is shown) predicts 3 regions of high electron density around the central N atom and a trigonal planar electronic and ionic geometry. The N atom has sp2 hybridization (Section 28-16). The three-dimensional structure is shown on the next page. 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

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... 

Embedded atom potentials have been extensively used for performing atomistic simulations of point, line and planar defects in metals and alloys (e.g. Vitek and Srolovitz 1989). The pair potential ( ), atomic charge density pBtom(r), and embedding function F(p) are usually fitted to reproduce the known equilibrium atomic volume, elastic moduli, and ground state structure of the perfect defect-free lattice. However, the prediction of ground state structure, especially the competition between the common metallic structure types fee, bcc, and hep, requires a more careful treatment of the pair potential contribution ( ) than that provided by the semiempirical embedded atom potential. This is considered in the next chapter. 

A number of structural questions will be addressed. What is the range of typical P=C bond lengths and how does this vary with substituent Is the C planar or pyramidal, and does this depend on substituents What is the geometry about P The answers to these questions will determine the hybridization of the P and C atoms. We shall also examine the density distribution to determine the charges on the atoms and the degree of d orbital participation. 

Historically, bis(aminotroponeiminato) nickel(II) complexes have been veiy instructive. The compounds are either pseudotetrahedral or display a tetrahedral-planar equilibrium. The ligands contain seven-membered rings showing alternation of proton shifts and spin densities (Table 2.5). The interest lies in the variety of R derivatives which show how spin density can be transmitted through it bonds, whereas it cannot be transmitted through sp3 carbons or through ethereal oxygen atoms [48,49]. 

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