Key Concepts of Crystal Lattice Structures

Why This Matters

Crystal lattice structures aren't just abstract geometry—they're the foundation for understanding why materials behave the way they do. When you're tested on solid state physics, you're being evaluated on your ability to connect atomic arrangement to macroscopic properties like conductivity, hardness, and mechanical strength. The way atoms pack together determines everything from why copper conducts electricity so well to why diamond can scratch virtually any other material.

The concepts here span multiple interconnected ideas: symmetry and periodicity, packing efficiency, reciprocal space, and crystallographic notation. You'll need to understand how real-space lattices translate into reciprocal space for diffraction analysis, and how unit cell geometry constrains material properties. Don't just memorize the 14 Bravais lattices or the formula for Miller indices—know why different structures exist and what physical consequences each arrangement produces.

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Common Crystal Structures

These are the specific atomic arrangements you'll encounter most frequently. The key principle here is that atoms arrange themselves to minimize energy, which typically means maximizing packing efficiency while satisfying bonding requirements.

Simple Cubic (SC)

• One atom per unit cell—atoms sit only at cube corners, each shared among 8 adjacent cells
• Packing efficiency of 52%—the lowest of the cubic structures, leaving significant void space
• Coordination number of 6—rarely found in nature due to inefficient packing; polonium is the only element with this structure

Body-Centered Cubic (BCC)

• Two atoms per unit cell—corner atoms plus one atom at the cube's center
• Packing efficiency of 68%—intermediate density found in alkali metals (Na, K) and transition metals (Fe, W, Cr)
• Coordination number of 8—the central atom touches all 8 corner atoms, influencing ductility and slip behavior

Face-Centered Cubic (FCC)

• Four atoms per unit cell—corner atoms plus atoms at the center of each face
• Packing efficiency of 74%—maximum possible for spheres, shared with HCP; found in Cu, Al, Au, Ag
• 12-fold coordination—high symmetry enables multiple slip systems, making FCC metals highly ductile

Hexagonal Close-Packed (HCP)

• ABAB stacking sequence—two alternating hexagonal layers with atoms nested in the gaps
• Packing efficiency of 74%—identical to FCC but with different stacking (ABCABC for FCC vs. ABAB for HCP)
• Fewer slip systems than FCC—found in Mg, Ti, Zn; generally less ductile due to limited deformation pathways

Diamond Structure

• FCC lattice with tetrahedral basis—two interpenetrating FCC lattices offset by $\frac{1}{4}$ of the body diagonal
• Each atom bonded to 4 neighbors—$sp^3$ hybridization creates strong covalent bonds in all directions
• Low packing efficiency (~34%)—the tetrahedral geometry sacrifices density for bond strength, yielding extreme hardness

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Mathematical Frameworks for Lattices

These tools let you describe, classify, and analyze crystal structures systematically. The underlying principle is that periodic structures can be completely characterized by their symmetry operations and repeating units.

Bravais Lattices

• 14 distinct lattice types—the complete set of unique ways to arrange points periodically in 3D space
• Classified into 7 crystal systems—based on unit cell geometry: cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, triclinic
• Foundation for all crystal structures—any crystal can be described as a Bravais lattice plus a basis of atoms

Crystal Systems (7 Types)

• Defined by lattice parameters—relationships between edge lengths ($a, b, c$) and angles ($\alpha, \beta, \gamma$)
• Cubic is most symmetric—$a = b = c$ and $\alpha = \beta = \gamma = 90°$; triclinic is least symmetric with no constraints
• Symmetry determines properties—higher symmetry generally means more isotropic physical behavior

Miller Indices

• Notation for planes and directions—$(hkl)$ for planes, $[uvw]$ for directions; derived from reciprocals of axis intercepts
• Negative indices written with bars—e.g., $(\bar{1}10)$ indicates a negative intercept on the first axis
• Essential for diffraction analysis—Bragg's law relates $d_{hkl}$ spacing to diffraction angles; for cubic systems, $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

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Unit Cells and Space-Filling

Different ways to define the repeating unit of a crystal serve different purposes. The key insight is that there's no single "correct" unit cell—you choose based on what property you're calculating.

Primitive Unit Cell

• Contains exactly one lattice point—the minimum repeating unit that generates the full lattice by translation
• Not always the most convenient choice—may have awkward angles that obscure symmetry (e.g., primitive cell of BCC is rhombohedral)
• Volume relates to conventional cell—for BCC, primitive cell volume is $\frac{1}{2}$ the conventional cubic cell

Wigner-Seitz Cell

• Constructed by perpendicular bisectors—draw lines to all nearest neighbors and bisect them; the enclosed region is the cell
• Preserves full point symmetry—unlike arbitrary primitive cells, the Wigner-Seitz cell reflects the lattice's rotational symmetry
• Connects to Brillouin zones—the first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice

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Reciprocal Space Concepts

These concepts transform real-space periodicity into momentum-space analysis. The fundamental principle is that periodic structures in real space create discrete allowed states in reciprocal (momentum) space.

Reciprocal Lattice

• Fourier transform of the direct lattice—reciprocal lattice vectors $\vec{G}$ satisfy $e^{i\vec{G} \cdot \vec{R}} = 1$ for all lattice vectors $\vec{R}$
• Basis vectors defined by: $\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$ and cyclic permutations
• Diffraction condition—constructive interference occurs when scattering vector equals a reciprocal lattice vector (Laue condition)

Brillouin Zones

• Wigner-Seitz cells in reciprocal space—the first Brillouin zone contains all unique $\vec{k}$ values before periodicity repeats
• Zone boundaries mark band gaps—electrons with wavevectors at zone edges experience Bragg reflection, opening energy gaps
• Higher zones exist—the $n$th zone contains points reached by crossing $n-1$ zone boundaries; all zones have equal volume

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Quick Reference Table

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Self-Check Questions

1. Both FCC and HCP achieve 74% packing efficiency—what structural difference explains why FCC metals are typically more ductile than HCP metals?

2. If given lattice parameters $a = b \neq c$ and $\alpha = \beta = \gamma = 90°$, which crystal system does this describe, and how many Bravais lattices exist within it?

3. Compare the Wigner-Seitz cell and the first Brillouin zone: how are they constructed, and what is their mathematical relationship?

4. The reciprocal lattice of an FCC direct lattice has what structure? Explain why this relationship matters for interpreting X-ray diffraction data.

5. A problem gives you Miller indices $(111)$ for a cubic crystal with lattice constant $a$. Calculate the interplanar spacing $d_{111}$ and explain what physical measurement would reveal this spacing.