Materials Science — Definition of Atomic Packing Factor (APF) What is the term for the ratio of the volume actually occupied by atoms to the total volume of the crystal's unit cell?

Introduction / Context:
This question tests your understanding of a fundamental crystallography metric used in materials science and metallurgy: the atomic packing factor (APF). APF quantifies how efficiently atoms pack within a unit cell, directly influencing density, slip behavior, and many mechanical properties of metals and alloys.

Given Data / Assumptions:

• We consider ideal hard-sphere atoms in a crystalline solid.
• The “unit cell” is the smallest repeating volume that fully describes the crystal structure.
• The ratio is defined as occupied atomic volume divided by total unit-cell volume.

Concept / Approach:
APF is calculated as APF = (total volume of atoms contained in the unit cell) / (volume of the unit cell). Famous results are: APF for face-centered cubic (FCC) = 0.74, for hexagonal close-packed (HCP) = 0.74, and for body-centered cubic (BCC) = 0.68 (using the hard-sphere model). These values help explain relative densities and ease of slip (FCC and HCP are close-packed; BCC is not).

Step-by-Step Solution:
Identify the quantity asked: the ratio of atomic volume to unit-cell volume.Recall definition: this is by convention the atomic packing factor (APF).Differentiate from coordination number (counts nearest neighbors) and space lattice (geometric framework of points).Select the named term that matches the ratio definition: atomic packing factor.

Verification / Alternative check:
Compute APF for FCC as a check: APF = (4 * (4/3) * π * r^3) / a^3 with a = 2√2 r → APF ≈ 0.74, a standard textbook value.

Why Other Options Are Wrong:
Coordination number: counts nearest neighbors (e.g., 12 in FCC/HCP) and is not a volume ratio.Space lattice: an array of points in space; it does not measure filled volume.None of these / packing efficiency: while “packing efficiency” is colloquial, the formal term used in materials curricula is APF.

Common Pitfalls:
Confusing APF with density; APF contributes to density but density also depends on atomic mass and lattice parameter.

Final Answer:
atomic packing factor