Calculations Involving Unit-Cell Dimensions

You can determine the structure and dimensions of a unit cell by diffraction methods, which we will describe briefly in the next section. Once you know the unit-cell dimensions and the structure of a crystal, however, some interesting calculations are possible. For instance, snppose you find that a metallic solid has a face-centered cubic unit cell with all atoms at lattice points, and yon determine the edge length of the unit cell. From this unit-cell dimension, you can calculate the volume of the unit cell. Then, knowing the density of the metal, you can calculate the mass of the atoms in the unit cell. Because you know that the unit cell is face-centered cubic with aU atoms at lattice points and that such a unit cell has four atoms, you can obtain the mass of an individual atom. This determination of the mass of a single atom gave one of the first accurate determinations of Avogadro s number. The calculations are shown in the next example. 

Calculating Atomic Mass from Unit-Cell Dimension and Density 

X-ray diffraction from crystals provides one of the most accurate ways of determining Avogadro s number. Silver crystallizes in a face-centered cubic lattice with all atoms at the lattice points. The length of an edge of the unit cell was determined by x-ray diffraction to be 408.6 pm (4.086 A). The density of silver is 10.50 g/cm. Calculate the mass of a silver atom. Then, using the known value of the atomic weight, calculate Avogadro s number. 

Knowing the edge length of a unit cell, you can calculate the unit-cell volume. Then, from the density, you can find the mass of the unit cell and hence the mass of a silver atom. 

SOLUTION You obtain the volume, V, of the unit cell by cubing the length of an edge. 

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The impressive development in molecular simulation stems from important improvement in both codes and computers. It made available accurate atomistic simulation of fluorine derivatives. Depiction of the SPF is most commonly achieved by quantum calculations. The major difficulty in describing interactions involving fluorine atoms actually lies in a correct description of the electrostatic effects [38]. Crystal unit cell dimensions [39,40] and the thermal behavior of fluoropolymers [41 3] were thus originally difficult to reproduce. Dihedral potential energy (Equation 6.3) that plays a central role in the backbone dynamics was also incorrectly depicted. In this section, illustrative examples of force fields specifically dedicated to fluoropolymers and more transferable force fields are reviewed. 

As molecular packing calculations involve just simple lattice energy minimizations another set of tests have focused on the finite temperature effects. For this purpose, Sorescu et al. [112] have performed isothermal-isobaric Monte Carlo and molecular dynamics simulations in the temperature range 4.2-325 K, at ambient pressure. It was found that the calculated crystal structures at 300 K were in outstanding agreement with experiment within 2% for lattice dimensions and almost no rotational and translational disorder of the molecules in the unit cell. Moreover, the space group symmetry was maintained throughout the simulations. Finally, the calculated expansion coefficients were determined to be in reasonable accord with experiment. 

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