Crystal Structures and Packing Fractions

Vocabulary-style flashcards covering crystal structure properties, including atom contributions, geometric relationships, and packing fractions for SC, BCC, and FCC systems.

Simple Cubic (SC) structure packing fraction

$$52\%$$

Number of corner atoms in a BCC structure

$8$ atoms at the corners

Number of center atoms in a BCC structure

$1$ atom exactly in the center

Contribution of corner atoms to a BCC unit cell

$8 \times (\frac{1}{8}) = 1$ atom

Contribution of the center atom to a BCC unit cell

$1$ atom (since it is not shared by neighboring cells)

Total number of atoms per unit cell in a BCC structure

$2$ atoms

Formula for the volume of a single spherical atom

$$\frac{4}{3} \times \pi \times R^3$$

Formula for total volume occupied by atoms in a unit cell

$$\text{Number of atoms } (Z) \times (\frac{4}{3} \times \pi \times R^3)$$

Formula for the volume of a cubic unit cell

$$a^3$$

Packing fraction

Ratio of total volume occupied by atoms in a unit cell to the total volume of the unit cell

Relationship between edge length ($a$) and atomic radius ($R$) for BCC

$$a = \frac{4R}{\sqrt{3}}$$

Packing fraction of a Body-Centered Cubic (BCC) structure

$$68.0\%$$

Number of corner atoms in an FCC structure

$8$ atoms at the corners

Number of face atoms in an FCC structure

$6$ atoms (one on each of the $6$ faces)

Contribution of face atoms to an FCC unit cell

$6 \times (\frac{1}{2}) = 3$ atoms (since each face is shared by $2$ cells)

Total number of atoms per unit cell ($Z$) in an FCC structure

$4$ atoms (1 from corners + 3 from faces)

Relationship between edge length ($a$) and atomic radius ($R$) for FCC

$$a = \frac{4R}{\sqrt{2}}$$

Formula for total volume occupied by atoms in an FCC unit cell

$$\frac{16}{3} \times \pi \times R^3$$

Packing fraction of a Face-Centered Cubic (FCC) structure

$$74.0\%$$