Final answer:
The atomic packing factor for HCP can be calculated by determining the volume occupied by atoms in the unit cell and dividing it by the volume of the unit cell itself. The ideal APF for HCP is approximately 0.74.
Step-by-step explanation:
Calculating Atomic Packing Factor for HCP
To calculate the atomic packing factor (APF) for hexagonal close-packed (HCP) crystal structures, we need to consider the number of atoms in the unit cell and their volume as well as the volume of the unit cell itself. In HCP, each atom has 12 nearest neighbors and the atoms adopt an ABABAB... stacking sequence. The total number of atoms in one HCP unit cell is 6 (2 from the top and bottom faces, and 1/3 of each of 12 in the corners).
To obtain the APF, we use these steps:
- Calculate the volume of a single atom using the atomic radius (r), which is typically provided. For an atom treated as a sphere: V_atom = (4/3)πr^3.
- Multiply V_atom by the number of atoms in the unit cell to get the total volume occupied by atoms: V_total_atoms = 6 * V_atom.
- Find the volume of the HCP unit cell using the unit cell's dimensions which can be obtained through geometry and knowing that the c/a ratio is about 1.633 for ideal HCP.
- Calculate the APF as the ratio of the total volume occupied by atoms (V_total_atoms) to the volume of the unit cell (V_cell): APF = V_total_atoms / V_cell.
Since both HCP and CCP structures fill 74 percent of the available space, we expect an ideal HCP APF to be approximately 0.74.