Crystal structures and unit cells Which of the following space lattices has exactly fourteen atoms contained within a single conventional unit cell?

Introduction / Context:
This question tests recall of how many atoms are effectively contained inside the conventional unit cell for common metallic crystal structures. Knowing these counts helps with density calculations, packing factors, and interpreting diffraction data in materials science and metallurgy.

Given Data / Assumptions:

• Conventional unit cells are used (cubic for BCC and FCC, hexagonal for HCP).
• Corner atoms contribute 1/8 each, face-centered atoms contribute 1/2 each, body-centered atoms contribute 1 each, and atoms fully inside the cell count as 1.
• No exotic or complex structures are implied beyond the listed common lattices.

Concept / Approach:
The effective number of atoms per conventional unit cell is a standard result: BCC has 2 atoms per cell; FCC has 4 atoms per cell; the conventional HCP cell contains 6 atoms. Diamond-cubic contains 8 atoms per cubic cell. Fourteen atoms per unit cell is not the atom count of these common lattices; therefore, among the options given, the correct choice is “none of these”.

Step-by-Step Solution:
For BCC: 8 corners * 1/8 + 1 body center * 1 = 2 atoms.For FCC: 8 corners * 1/8 + 6 faces * 1/2 = 1 + 3 = 4 atoms.For HCP (conventional hexagonal cell): 12 corners * 1/6 + 2 faces * 1/2 + 3 centered inside = 2 + 1 + 3 = 6 atoms.For diamond-cubic: 8 atoms effectively per conventional cubic cell.No listed lattice yields 14 atoms per unit cell; hence “none of these”.

Verification / Alternative check:
Textbook tables of lattice types consistently list: BCC = 2, FCC = 4, HCP (conventional) = 6, diamond-cubic = 8. Fourteen is a coordination number sometimes encountered in other contexts, but not the atom count for these unit cells.

Why Other Options Are Wrong:
BCC (2), FCC (4), HCP (6), diamond-cubic (8) do not match 14; selecting any of them would contradict standard crystallography results.

Common Pitfalls:
Confusing “coordination number” with “atoms per unit cell”. Coordination number for FCC/HCP is 12; BCC is 8—none equals 14 for these structures.

Final Answer:
none of these