The unit cell for Al2 O3 has hexagonal symmetry with lattice...

The unit cell for $\mathrm{Al}_{2} \mathrm{O}_{3}$ has hexagonal symmetry with lattice parameters $a=0.4759 \mathrm{nm}$ and $c=1.2989 \mathrm{nm} .$ If the density of this material is $3.99 \mathrm{g} / \mathrm{cm}^{3},$ calculate its atomic packing factor. For this computation use ionic radii listed in Table 12.3.

Key Concepts

Atomic Packing Factor

The atomic packing factor (APF) is the fraction of space in a crystal structure that is actually occupied by the atoms, assuming they are rigid spheres. It is calculated by dividing the total volume of the atoms present in the unit cell by the volume of the unit cell itself. APF is an important measure in solid state physics and materials science as it quantifies how densely the atoms are packed in a given structure.

Unit Cell and Lattice Parameters

The unit cell is the smallest repeating entity in a crystal that, when repeated in all directions, forms the entire crystal lattice. Lattice parameters, such as edge lengths and lattice constants, define the geometry and size of the unit cell. These parameters are critical for determining the unit cell volume and, consequently, for calculating properties like density and atomic packing factor.

Crystal Structure (Hexagonal Symmetry)

Crystal structures are classified based on the symmetry of the unit cell. In a hexagonal system, the crystal is defined by two equal lattice parameters in the basal plane and one different lattice parameter along the perpendicular axis. Hexagonal symmetry influences how atoms are arranged and how efficiently they pack together, which directly affects the calculation of the atomic packing factor.

Ionic Radii

Ionic radii are the effective radii of ions within a crystal, and they serve as a model for the size of the atoms or ions in the structure. By assuming atoms as spheres with these radii, one can estimate the volume occupied by each ion. This estimation is crucial for computing the atomic packing factor, especially in crystals where ionic bonding predominates and ions have well-defined sizes.

Material Density and Its Relation to Crystal Structure

The density of a material is determined by the mass of the atoms within a unit cell and the volume of that unit cell. It serves as an important parameter in assessing how closely atoms are packed within a crystal lattice. By relating density to the number of atoms and their arrangement, one can validate the computed atomic packing factor and gain insight into the material's structural efficiency.

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