Atomic packing factor

The atomic packing factor (APF) is the ratio of the hard sphere volume inside the unit cell to the volume of the unit cell. For an fee system with foiu- atoms inside the unit cell, the hard sphere volume is 4 x 4R /3 and the unit cell volume is Therefore, the APF is 

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The atomic packing factor is the fraction of the volume of a crystal that is filled up by its atoms. In other words, the higher the atomic packing factor, the less empty space there is in the material. 

A close-packed structure is one that has the maximum volume of the unit cell occupied by atoms. The occupied fraction of the unit cell can be determined by calculating the atomic packing factor (APF) ... 

This structure is rather open the atomic packing factor (APF) for GaAs is only 0.41. In the GaAs structure each atom has only four nearest neighbors the coordination... 

Polonium (Po) is a radioactive element that was discovered in 1898 by Marie Sklodowska-Curie and Pierre Curie. Po is used in brushes to remove dust from photographic films and to avoid charge static accumulation produced by several processes, such as the rolling of paper, wire, and sheet metal. In addition, Po has been alloyed with beryllium to be used as a neutron source. All these and other applications depend on Po s structural properties. Po is the only element of the periodic table that adopts the simple cubic (sc) structure at ambient pressure (a few other elements such as Ca-III and As-II present the sc, but only at high pressure [1]), and this structure has a low atomic packing factor and is rare in nature. The first experimental studies of Po s crystal structure, by using electron diffraction, were reported in 1936 by Rollier et al. [2]. Several years later, Beamer and Maxwell [3,4] and Sando and Lange [5] reported on their X-ray diffraction experiments on metallic Po. From these reports, we know that Po exhibits two structural phases the a phase (a-Po), which has the sc structure p Pmim)], a = 3.345(2) A [4], and the /3 phase (/3-Po), stable above 77(9)°C, which has the rhombohedral (r) structure [Df (/ 3m)], a = 3.359(1) A, and a = 98.22(5)°. 

For the FCC structure, the atomic packing factor is 0.74, which is the maximum packing possible for spheres all having the same diameter. Computation of this APF is also included as an example problem. Metals typically have relatively large atomic packing factors to maximize the shielding provided by the free electron cloud. 

The coordination number for the BCC crystal structure is 8 each center atom has as nearest neighbors its eight corner atoms. Because the coordination number is less for BCC than for FCC, the atomic packing factor is also lower for BCC—0.68 versus 0.74. 

It is also possible to have a unit cell that consists of atoms situated only at the corners of a cube. This is called the simple cubic (SC) crystal structure-, hard-sphere and reduced-sphere models are shown, respectively, in Figures 33a and 33b. None of the metallic elements have this crystal structure because of its relatively low atomic packing factor (see Concept Check 3.1). The only simple-cubic element is polonium, which is considered to be a metalloid (or semi-metal). 

The coordination number and the atomic packing factor for the HCP crystal structure are the same as for FCC 12 and 0.74, respectively. The HCP metals include cadmium, magnesium, titanium, and zinc some of these are listed in Table 3.1. 

Show that the atomic packing factor for the FCC crystal structure is 0.74. 

It is important to note that many of the principles and concepts addressed in previous discussions in this chapter also apply to crystalline ceramic and polymeric systems (Chapters 12 and 14). For example, crystal structures are most often described in terms of unit cells, which are normally more complex than those for FCC, BCC, and HCP. In addition, for these other systems, we are often interested in determining atomic packing factors and densities, using modified forms of Equations 3.3 and 3.8. Furthermore, according to unit cell geometry, crystal structures of these other material types are grouped within the seven crystal systems. 

Coordination number—the number of nearest-neighbor atoms, and Atomic packing factor—the fraction of solid sphere volume in the unit cell. 

Metallic materials have relatively high atomic packing factors, which means that these interstitial positions are relatively small. Consequently, the atomic diameter of an interstitial impurity must be substantially smaller than that of the host atoms. Normally, the maximum allowable concentration of interstitial impurity atoms is low (less than 10%). Even very small impurity atoms are ordinarily larger than the interstitial sites, and as a consequence, they introduce some lattice strains on the adjacent host atoms. Problems 4.8 and 4.9 call for determination of the radii of impurity atoms r (in terms of R, the host atom radius) that just fit into tetrahedral and octahedral interstitial positions of both BCC and FCC without introducing any lattice strains. 

Compute the atomic packing factor for cesium O chloride using the ionic radii in Table 12.3 and assuming that the ions touch along the cube diagonals. 

If the density of this material is 5.24 g/cm, compute its atomic packing factor. For this computation, you will need to use the ionic radii listed in Table 12.3. 

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