Crystal Structures — Counting Atoms in the HCP Unit Cell How many atoms are effectively contained in a conventional close-packed hexagonal (HCP) unit cell, and where are they located?

Introduction / Context:
The hexagonal close-packed (HCP) structure is one of the two classic close-packed metal crystal structures (the other is FCC). Correctly counting the number of atoms per unit cell is essential for deriving density, slip system counts, and understanding anisotropy in HCP metals like magnesium, titanium, and zinc.

Given Data / Assumptions:

• We use the conventional hexagonal unit cell for HCP (not the smaller primitive cell).
• Atoms at corners and faces are shared among neighboring cells; their fractional contributions must be accounted for.
• Atomic hard-sphere model is assumed.

Concept / Approach:
In the conventional HCP cell, the atom accounting is: 12 corner atoms of the hexagon (each shared by 6 cells) contribute 12 * (1/6) = 2 atoms; 2 atoms at the centers of the top and bottom hexagonal faces (each shared by 2 cells) contribute 2 * (1/2) = 1 atom; and there are 3 atoms entirely inside the unit cell. Total = 2 + 1 + 3 = 6 atoms. This is consistent with the close-packed stacking ABAB… in HCP and the known APF of 0.74.

Step-by-Step Solution:
Identify all lattice positions in the conventional HCP cell.Compute contributions: corners (1/6 each) → 2 atoms.Add face-center atoms (1/2 each) → 1 atom.Include interior atoms (fully contained) → 3 atoms.Sum contributions: 2 + 1 + 3 = 6 atoms per unit cell.

Verification / Alternative check:
Cross-check density derivations for Mg or Ti using Z = 6; results align with published densities when lattice parameters and atomic masses are used.

Why Other Options Are Wrong:
Options (a) and (c) describe cubic cells (BCC and FCC features), not HCP.Option (b) ignores interior atoms and miscounts sharing, yielding the wrong total.Option (e) refers to a primitive lattice with one atom; the conventional HCP cell contains six atoms.

Common Pitfalls:
Mixing up “conventional” versus “primitive” cells; HCP conventional cell has six atoms even though the lattice can be described with a smaller primitive cell.

Final Answer:
six atoms: 12 corner atoms of the hexagon contributing 2 total, 2 face-centre atoms contributing 1 total, and 3 atoms wholly inside the cell