Key Crystal Structures

---

Why This Matters

Crystal structures are the foundation of everything you'll study in condensed matter physics—they determine how electrons move through materials, why some metals bend while others shatter, and what makes semiconductors work in your devices. When you're being tested on concepts like band structure, mechanical properties, ionic bonding, and semiconductor behavior, the underlying crystal geometry is almost always the key to understanding why materials behave the way they do.

Don't just memorize that copper is FCC and iron is BCC. Know why close-packed structures conduct heat efficiently, how coordination number relates to bonding type, and what makes certain structures ideal for semiconductors. The exam will ask you to connect structure to properties—that's where the points are.

---

Close-Packed Metallic Structures

These structures maximize atomic packing efficiency at 74%, achieved through different stacking sequences of close-packed planes. The distinction between FCC and HCP comes down to ABCABC vs. ABABAB stacking, which affects slip systems and mechanical behavior.

Face-Centered Cubic (FCC)

• Coordination number of 12 with atoms at cube corners and face centers—this high coordination enables excellent ductility through multiple slip systems
• 74% packing efficiency represents the theoretical maximum for identical spheres, contributing to high density and thermal conductivity
• Common in ductile metals like aluminum, copper, gold, and nickel—materials that deform plastically rather than fracture

Hexagonal Close-Packed (HCP)

• Same 74% packing efficiency as FCC but with hexagonal symmetry and ABAB stacking sequence—two atoms per unit cell
• Coordination number of 12 matches FCC, but fewer slip systems make HCP metals generally less ductile
• Found in magnesium, titanium, and zinc—materials often showing anisotropic mechanical properties due to limited slip planes

---

Open Metallic Structures

Lower packing efficiency doesn't mean inferior—BCC and simple cubic structures offer distinct advantages in certain applications. The "open" nature of these structures affects diffusion rates and phase stability at different temperatures.

Body-Centered Cubic (BCC)

• Coordination number of 8 with atoms at cube corners plus one center atom—lower than close-packed structures but still metallic bonding dominant
• 68% packing efficiency leaves more interstitial space, affecting diffusion and solubility of small atoms like carbon
• Found in iron (at room temperature), chromium, and tungsten—often in metals requiring high strength or high melting points

Simple Cubic

• Coordination number of only 6 with atoms at corners only—the lowest coordination among common structures
• 52% packing efficiency is extremely low, making this structure thermodynamically unfavorable for most elements
• Polonium is the only element with this structure at standard conditions—a classic exam fact demonstrating why simple cubic is rare

---

Covalent Network Structures

When directional covalent bonding dominates over metallic bonding, geometry follows orbital hybridization rather than packing efficiency. Tetrahedral coordination ($sp^3$ hybridization) is the signature of these structures.

Diamond Cubic

• Tetrahedral bonding with coordination number 4—each atom forms four covalent bonds at 109.5° angles, creating an FCC lattice with a two-atom basis
• Low packing efficiency (~34%) but extreme hardness due to strong, directional $sp^3$ covalent bonds throughout the structure
• Found in diamond and silicon—the structure responsible for both the hardest natural material and the foundation of semiconductor technology

Zinc Blende (Sphalerite)

• FCC-based structure with two atom types in tetrahedral coordination—essentially diamond cubic with alternating zinc and sulfur atoms
• Coordination number of 4 for both species, with each Zn surrounded by four S atoms and vice versa
• Critical semiconductor structure found in ZnS, GaAs, and InP—the basis for most III-V compound semiconductors

Wurtzite

• Hexagonal analog of zinc blende with the same tetrahedral coordination but ABAB stacking instead of ABCABC
• Coordination number of 4 maintained, but hexagonal symmetry creates different electronic and piezoelectric properties
• Found in ZnO, GaN, and AlN—important for LED technology and high-power electronics

---

Ionic Crystal Structures

Ionic structures balance electrostatic attraction with ion size ratios. The radius ratio rule predicts coordination: larger ratios allow higher coordination numbers.

Sodium Chloride (Rock Salt)

• FCC arrangement of both ion types interpenetrating each other—each Na$^+$ sits in an octahedral hole of the Cl$^-$ sublattice
• Coordination number of 6 for both ions, forming octahedral geometry consistent with intermediate radius ratios (0.414–0.732)
• High melting point and brittleness result from strong ionic bonding and the inability of ions to slip past each other without creating repulsive contacts

Cesium Chloride

• Simple cubic arrangement with Cs$^+$ at body center and Cl$^-$ at corners (or vice versa)—not actually BCC since the two sites aren't equivalent
• Coordination number of 8 for both ions, reflecting the large radius ratio when cation and anion sizes are similar
• Found in CsCl, CsBr, and CsI—the larger cesium cation allows this higher coordination compared to smaller alkali metals

---

Complex Functional Structures

Some structures derive their importance from compositional flexibility rather than geometric simplicity. These structures often exhibit emergent properties like ferroelectricity and superconductivity.

Perovskite

• ABX$_3$ formula with cubic symmetry—A cations at corners, B cation at body center, and X anions at face centers (or equivalent descriptions)
• Highly tunable properties through cation substitution—the same structure hosts ferroelectrics (BaTiO$_3$), superconductors, and solar cell absorbers
• Tolerance factor $t = \frac{r_A + r_X}{\sqrt{2}(r_B + r_X)}$ predicts structural stability—when $t \approx 1$, cubic perovskite is stable

---

Quick Reference Table

---

Self-Check Questions

1. Which two structures share 74% packing efficiency, and what stacking difference explains their distinct mechanical properties?

2. A material has coordination number 4 and tetrahedral bonding. Name three possible crystal structures it could adopt, and explain what distinguishes them.

3. Compare and contrast the rock salt and cesium chloride structures: what physical factor determines which structure an ionic compound will adopt?

4. Why is simple cubic structure so rare in nature, and what single element famously adopts it?

5. An FRQ asks you to explain why copper is more ductile than titanium despite both being metals. Which structural concepts should you discuss in your response?