Key Crystal Structures

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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 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. Exams will ask you to connect structure to properties.

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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 directly 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, combined with 12 independent slip systems (on {111} planes in ⟨110⟩ directions), enables excellent ductility.
• 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. These materials deform plastically rather than fracture because dislocations can move easily along multiple slip planes.
• The conventional cubic cell contains 4 atoms (8 corner atoms × 1/8 + 6 face atoms × 1/2).

Hexagonal Close-Packed (HCP)

• Same 74% packing efficiency as FCC but with hexagonal symmetry and ABAB stacking sequence. The unit cell contains 2 atoms (basis of two atoms per lattice point).
• Coordination number of 12 matches FCC, but only 3 independent basal slip systems (on the {0001} plane) make HCP metals generally less ductile.
• Found in magnesium, titanium, and zinc. These materials often show anisotropic mechanical properties because deformation depends strongly on crystal orientation relative to the applied stress.
• The ideal $c/a$ ratio is $\sqrt{8/3} \approx 1.633$, though real HCP metals deviate from this. Zinc, for instance, has $c/a \approx 1.856$, which further limits its slip behavior.

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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, phase stability, and solubility of interstitial atoms.

Body-Centered Cubic (BCC)

• Coordination number of 8 with atoms at cube corners plus one center atom. The conventional cell contains 2 atoms (8 × 1/8 + 1 center).
• 68% packing efficiency leaves more interstitial space, which affects diffusion rates and the solubility of small atoms like carbon. This is directly relevant to the metallurgy of steel: carbon dissolves interstitially in BCC iron (ferrite) but with limited solubility compared to FCC iron (austenite).
• Found in iron (below 912°C), chromium, tungsten, and the alkali metals. BCC metals often have high melting points and tend to show a ductile-to-brittle transition at low temperatures, unlike FCC metals.
• Slip in BCC is more complex than in FCC. Slip occurs on {110}, {112}, and {123} planes in ⟨111⟩ directions, and the lack of a true close-packed plane means dislocation behavior is temperature-sensitive.

Simple Cubic

• Coordination number of only 6 with atoms at corners only. The unit cell contains just 1 atom (8 × 1/8).
• 52% packing efficiency is extremely low, making this structure thermodynamically unfavorable for most elements.
• Polonium is the only element that adopts this structure at standard conditions. This is a classic exam fact, and the reason is straightforward: atoms almost always rearrange into higher-coordination structures to lower their energy.

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Covalent Network Structures

When directional covalent bonding dominates over metallic bonding, geometry follows orbital hybridization rather than packing efficiency. Tetrahedral coordination from $sp^3$ 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. The structure is an FCC Bravais lattice with a two-atom basis (atoms at 0,0,0 and 1/4,1/4,1/4 in fractional coordinates), giving 8 atoms per conventional cell.
• Low packing efficiency (~34%) but extreme hardness due to strong, directional $sp^3$ covalent bonds throughout the structure. The large band gap in diamond (5.5 eV) also follows from the strength of these bonds.
• Found in carbon (diamond), silicon, and germanium. This structure is responsible for both the hardest natural material and the foundation of semiconductor technology. Silicon's moderate band gap (1.1 eV) in this structure is what makes it ideal for transistors.

Zinc Blende (Sphalerite)

• Diamond cubic structure with two atom types occupying the two basis sites. One FCC sublattice holds cations, the other holds anions, both in tetrahedral coordination.
• Coordination number of 4 for both species: each Zn is surrounded by four S atoms and vice versa. The bonding is partially ionic, partially covalent, which tunes the band gap.
• Critical semiconductor structure found in ZnS, GaAs, and InP. This is the basis for most III-V compound semiconductors used in optoelectronics, high-speed electronics, and laser diodes.

Wurtzite

• Hexagonal analog of zinc blende with the same tetrahedral coordination but ABAB stacking instead of ABCABC. Think of it as the HCP version of the zinc blende structure.
• Coordination number of 4 maintained, but the hexagonal symmetry breaks inversion symmetry more strongly, creating different electronic and piezoelectric properties.
• Found in ZnO, GaN, and AlN. These materials are essential for LED technology (GaN-based blue LEDs won the 2014 Nobel Prize) and high-power electronics.

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Ionic Crystal Structures

Ionic structures balance electrostatic attraction with geometric constraints set by ion size ratios. The radius ratio $r_+/r_-$ predicts coordination: larger ratios allow higher coordination numbers because the cation is big enough to "prop open" space for more anion neighbors.

Sodium Chloride (Rock Salt)

• Two interpenetrating FCC sublattices, one of Na$^+$ and one of Cl$^-$. Each Na$^+$ sits in an octahedral hole of the Cl$^-$ sublattice, and vice versa.
• Coordination number of 6 for both ions, forming octahedral geometry. This is consistent with radius ratios in the range 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 same-charge repulsive contacts. This is why ionic crystals cleave along specific planes rather than deforming plastically.
• Many ionic compounds adopt this structure: NaCl, MgO, FeO, and most alkali halides (except the cesium halides).

Cesium Chloride

• Simple cubic arrangement with Cs$^+$ at the body center and Cl$^-$ at the corners (or vice versa). This is not BCC, because the two sites are occupied by different ions and are therefore not equivalent.
• Coordination number of 8 for both ions, reflecting the large radius ratio when cation and anion sizes are similar ($r_+/r_- > 0.732$).
• Found in CsCl, CsBr, and CsI. The larger cesium cation allows this higher coordination compared to smaller alkali metals like sodium.

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Complex Functional Structures

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

Perovskite

• $ABX_3$ formula with cubic symmetry in the ideal case. A common description: A cations at corners, B cation at body center, X anions at face centers. Equivalently, the B cation sits at the center of an octahedron of X anions, with A cations filling the spaces between octahedra.
• Highly tunable properties through cation substitution. The same structure hosts ferroelectrics (BaTiO$_3$), high-temperature superconductors (related layered perovskites), colossal magnetoresistance materials (La$_{1-x}$Sr$_x$MnO$_3$), and hybrid organic-inorganic solar cell absorbers (CH$_3$NH$_3$PbI$_3$).
• The Goldschmidt tolerance factor $t = \frac{r_A + r_X}{\sqrt{2}(r_B + r_X)}$ predicts structural stability. When $t \approx 1$, the ideal cubic perovskite is stable. Deviations from $t = 1$ lead to octahedral tilting and lower-symmetry distortions (orthorhombic, tetragonal), which often drive the ferroelectric and magnetic transitions that make perovskites so useful.

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Quick Reference Table

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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. Explain why copper (FCC) is more ductile than titanium (HCP) despite both being metals with close-packed structures. Which structural concepts should you discuss?

6. Iron transforms from BCC (ferrite) to FCC (austenite) at 912°C. How does this change in structure affect the solubility of carbon, and why?