Unit Cells

Definition and Concept

Unit Cell Explained

Unit cell: smallest repeating structural unit of a crystal lattice. Repeats in 3D to form entire lattice. Defines symmetry, structure, and properties.

Crystal Lattice Relationship

Crystal lattice: infinite periodic arrangement of points in space. Unit cell: finite volume enclosing lattice points and atomic basis.

Importance

Determines macroscopic physical and chemical properties. Basis for X-ray diffraction, materials design, and solid state analysis.

"The unit cell is the fundamental descriptor of crystalline order and symmetry." -- C. Hammond, Solid State Chemistry

Lattice Parameters

Definition

Lattice parameters: lengths (a, b, c) and interaxial angles (α, β, γ) defining unit cell geometry. Units: Ångströms and degrees.

Measurement

Determined experimentally via X-ray diffraction and electron microscopy. Essential for identifying crystal system.

Role in Structure

Parameters influence atomic packing, density, and symmetry. Variations affect material properties like conductivity and hardness.

a ≠ b ≠ cα ≠ β ≠ γ(General triclinic cell)

Bravais Lattices

Concept

Bravais lattices: 14 distinct 3D lattice types classifying all possible crystal lattice symmetries.

Classification

Grouped into 7 crystal systems: cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, trigonal.

Examples

Cubic system: simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC). Hexagonal system: hexagonal close packed (HCP).

Types of Unit Cells

Simple (Primitive) Unit Cell

Contains lattice points only at corners. Volume minimal. Example: simple cubic.

Body-Centered Unit Cell

Lattice points at corners and one at center. Increased packing density. Example: BCC structure.

Face-Centered Unit Cell

Lattice points at corners and center of each face. Highest packing efficiency among cubic cells. Example: FCC structure.

Packing Efficiency

Definition

Packing efficiency: fraction of unit cell volume occupied by atoms. Expressed as percentage.

Calculation Method

Calculate total atomic volume within unit cell divided by cell volume. Atoms treated as hard spheres.

Typical Values

Simple cubic: 52.4%. BCC: 68%. FCC and HCP: 74% (maximum packing in metals).

Packing Efficiency (%) = (Volume of atoms in unit cell / Volume of unit cell) × 100

Coordination Number

Definition

Number of nearest neighbor atoms surrounding a central atom in the unit cell.

Relation to Unit Cell Type

Simple cubic: CN = 6. BCC: CN = 8. FCC and HCP: CN = 12.

Implications

Higher CN: stronger metallic bonding, higher density, increased stability.

Primitive and Centered Unit Cells

Primitive Unit Cell

Contains one lattice point per cell. Smallest possible volume. Basis for defining lattice.

Centered Unit Cell

Additional lattice points at face centers, body center, or base centers. Increases symmetry.

Comparison

Centered cells often larger and more symmetric than primitive cells. Both describe same lattice by different choices.

Symmetry in Unit Cells

Symmetry Elements

Rotation axes, mirror planes, inversion centers, glide planes define symmetry. Affect physical properties.

Space Groups

Combination of symmetry operations and lattice translations. 230 distinct 3D space groups categorize crystals.

Impact on Unit Cell

Symmetry reduces independent atomic positions, simplifies structural description.

Calculation of Unit Cell Contents

Number of Atoms per Unit Cell

Sum of fractional contributions of atoms at corners, edges, faces, and inside cell.

Fractional Contributions

Corner atom: 1/8, edge atom: 1/4, face atom: 1/2, inside atom: 1.

Example Calculation

FCC unit cell: 8 corners × 1/8 + 6 faces × 1/2 = 4 atoms per cell.

Atoms per unit cell = Σ (Number of atoms × fractional contribution)

Applications in Solid State Chemistry

Material Design

Unit cell manipulation tailors electronic, optical, mechanical properties of materials.

Phase Identification

Unit cell parameters used to identify phases and polymorphs in crystalline materials.

Defect Analysis

Unit cell defects impact conductivity, magnetism, and catalytic activity.

Experimental Determination

X-ray Diffraction (XRD)

Primary method. Measures diffraction angles to calculate lattice constants and symmetry.

Electron and Neutron Diffraction

Complementary techniques for light atoms and magnetic structures.

Microscopy Techniques

High-resolution TEM and STM visualize unit cell dimensions directly in some cases.

Common Crystal Structures

Simple Cubic

Rare in metals. CN = 6, packing efficiency 52.4%. Example: Polonium.

Body-Centered Cubic (BCC)

Moderate packing, CN = 8, packing efficiency 68%. Examples: Fe (α-Fe), Cr, W.

Face-Centered Cubic (FCC)

High packing, CN = 12, packing efficiency 74%. Examples: Cu, Al, Au, Ag.

Hexagonal Close Packed (HCP)

Packing efficiency 74%, CN = 12. Examples: Mg, Ti, Zn.

References

• West, A.R., Solid State Chemistry and Its Applications, Wiley, 2nd Ed., 2014, pp. 123-178.
• Kittel, C., Introduction to Solid State Physics, Wiley, 8th Ed., 2004, pp. 45-90.
• Cullity, B.D., Stock, S.R., Elements of X-ray Diffraction, Prentice Hall, 3rd Ed., 2001, pp. 67-115.
• Callister, W.D., Rethwisch, D.G., Materials Science and Engineering: An Introduction, Wiley, 9th Ed., 2013, pp. 210-255.
• Burns, R.G., Mineralogical Applications of Crystal Field Theory, Cambridge University Press, 1993, pp. 33-70.