Crystal Structure Types (BCC, FCC, HCP) Cheat Sheet

Key Facts

• In a body-centered cubic unit cell, the number of atoms per unit cell is $n = 2$ because $8$ corner atoms contribute $1$ atom total and the body-center atom contributes $1$ atom.
• In a face-centered cubic unit cell, the number of atoms per unit cell is $n = 4$ because $8$ corners contribute $1$ atom total and $6$ face atoms contribute $3$ atoms total.
• For BCC, atoms touch along the body diagonal, so $4r = \sqrt{3}a$ and $a = \frac{4r}{\sqrt{3}}$.
• For FCC, atoms touch along the face diagonal, so $4r = \sqrt{2}a$ and $a = 2\sqrt{2}r$.
• The atomic packing factor is $APF = \frac{\text{volume of atoms in unit cell}}{\text{volume of unit cell}}$.
• BCC has coordination number $8$ and atomic packing factor $APF \approx 0.68$.
• FCC and HCP both have coordination number $12$ and atomic packing factor $APF \approx 0.74$.
• Crystal density is calculated with $\rho = \frac{nM}{N_A V_{cell}}$, where $M$ is molar mass and $N_A = 6.022 \times 10^{23}\ \text{mol}^{-1}$.

Vocabulary

Unit cell

A unit cell is the smallest repeating three-dimensional block that shows the symmetry and arrangement of atoms in a crystal.

Body-centered cubic

Body-centered cubic, or BCC, is a cubic structure with atoms at the corners and one atom at the center of the cube.

Face-centered cubic

Face-centered cubic, or FCC, is a cubic structure with atoms at the corners and at the centers of all six faces.

Hexagonal close-packed

Hexagonal close-packed, or HCP, is a close-packed structure with layers arranged in an $ABAB$ repeating pattern.

Coordination number

Coordination number is the number of nearest neighboring atoms touching a given atom in a crystal structure.

Atomic packing factor

Atomic packing factor is the fraction of a unit cell's volume occupied by atoms, given by $APF = \frac{V_{atoms}}{V_{cell}}$.