Crystal Structures

Introduction

Crystal structures define the ordered arrangement of atoms, ions, or molecules in a solid. They determine physical properties: hardness, conductivity, optical behavior. Solid state chemistry relies on understanding these structures for material design and analysis.

"The structure is the key to understanding the properties of solids." -- Linus Pauling

Crystal Lattice

Definition

Crystal lattice: 3D array of points representing atom positions repeated periodically in space. Defines translational symmetry. Basis atoms attached to each lattice point form the crystal structure.

Lattice Parameters

Parameters: three edge lengths (a, b, c) and three interaxial angles (α, β, γ). Determine unit cell geometry. Units: angstroms (Å) for lengths, degrees for angles.

Symmetry Elements

Include rotation axes, mirror planes, inversion centers. Symmetry restricts possible lattice types. Essential for classifying crystals into systems and lattices.

Unit Cell

Definition

Smallest repeating unit that fully describes crystal structure by translation. Contains lattice points and basis atoms. Replicates in 3D to form entire lattice.

Types of Unit Cells

Primitive (P): lattice points only at corners. Centered: body-centered (I), face-centered (F), base-centered (C). Affect packing and density.

Volume Calculation

Volume V given by:

V = abc √(1 + 2cosαcosβcosγ - cos²α - cos²β - cos²γ)

Bravais Lattices

Definition

Fourteen unique 3D lattices that describe all possible translational symmetries in crystals. Classified by lattice system and centering type.

Classification

Seven crystal systems combined with centering: Primitive (P), Body-centered (I), Face-centered (F), Base-centered (C). Only 14 combinations are unique lattices.

List of Bravais Lattices

Includes cubic P, I, F; tetragonal P, I; orthorhombic P, C, I, F; monoclinic P, C; triclinic P; hexagonal P; rhombohedral (trigonal) R.

Crystal Systems

Overview

Seven crystal systems defined by unit cell parameters: cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, trigonal (rhombohedral).

Parameter Constraints

Cubic: a=b=c, α=β=γ=90°. Tetragonal: a=b≠c, α=β=γ=90°. Hexagonal: a=b≠c, α=β=90°, γ=120°, etc.

Symmetry and Properties

Higher symmetry (cubic) results in isotropic properties. Lower symmetry (triclinic) causes anisotropy in physical properties.

Packing Efficiency

Definition

Fraction of volume occupied by atoms assuming hard spheres within unit cell. Key for density and stability of metals and ionic crystals.

Common Packing Types

Simple cubic (52%), body-centered cubic (68%), face-centered cubic (74%), hexagonal close-packed (74%).

Calculation Example

Packing Efficiency = (Volume of atoms in cell / Volume of unit cell) × 100%For FCC:Atoms per unit cell = 4Atomic radius = rUnit cell edge a = 2√2 rPacking efficiency = (4 × (4/3)πr³) / a³ × 100% ≈ 74% 

Coordination Number

Definition

Number of nearest neighbor atoms or ions surrounding a central atom in crystal. Indicates bonding environment and stability.

Typical Values

Simple cubic: 6, BCC: 8, FCC & HCP: 12, ionic crystals vary depending on ion size ratios.

Importance

Determines density, mechanical properties, electronic structure. High coordination: dense packing, metallic bonding. Low coordination: open structures, covalent bonding.

Types of Crystals

Ionic Crystals

Composed of cations and anions. Bonding: strong electrostatic forces. Properties: high melting points, brittleness, electrical insulators in solid state.

Metallic Crystals

Atoms arranged in close-packed lattices. Bonding: delocalized electron sea. Properties: conductivity, malleability, ductility.

Covalent Crystals

Atoms connected by covalent bonds in extended networks. Examples: diamond, quartz. Properties: hardness, high melting points, poor electrical conductivity.

Molecular Crystals

Discrete molecules held by van der Waals or hydrogen bonds. Properties: low melting points, soft, electrical insulators.

Polymorphism

Definition

Ability of a substance to exist in more than one crystal structure. Influences physical and chemical properties.

Examples

Carbon: diamond (tetrahedral) vs graphite (layered hexagonal). TiO2: rutile vs anatase vs brookite phases.

Factors Affecting Polymorphism

Temperature, pressure, impurities, synthesis method. Control over polymorph essential in pharmaceuticals, materials science.

Defects in Crystals

Point Defects

Vacancies: missing atoms. Interstitials: atoms in non-lattice sites. Substitutional: foreign atoms replacing host atoms.

Line Defects (Dislocations)

Edge and screw dislocations. Affect mechanical strength, ductility. Crucial in plastic deformation.

Planar Defects

Grain boundaries, stacking faults. Influence grain size, corrosion resistance, electrical properties.

X-ray Crystallography

Principle

X-rays diffract by electron clouds in crystal lattice. Constructive interference at specific angles given by Bragg's Law yields diffraction pattern.

Bragg's Law

nλ = 2d sin θwhere,n = order of reflection (integer),λ = wavelength of incident X-rays,d = interplanar spacing,θ = angle of incidence/reflection.

Applications

Determination of atomic coordinates, lattice parameters, symmetry. Essential for structure elucidation in inorganic and organic materials.

Applications

Materials Design

Crystal engineering for catalysts, semiconductors, superconductors. Control over structure affects performance.

Pharmaceuticals

Polymorph screening to optimize solubility, stability, bioavailability of drugs.

Nanotechnology

Crystal growth control for nanomaterials with tailored properties.

Geology and Mineralogy

Identification and classification of minerals based on crystal structure.

References

• West, A.R., Solid State Chemistry and Its Applications, Wiley, 2014, pp. 45-120.
• Kittel, C., Introduction to Solid State Physics, 8th ed., Wiley, 2005, pp. 1-80.
• Cullity, B.D., Stock, S.R., Elements of X-ray Diffraction, 3rd ed., Prentice Hall, 2001, pp. 75-150.
• Shannon, R.D., "Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides," Acta Crystallographica A32, 1976, pp. 751-767.
• Hahn, T. (Ed.), International Tables for Crystallography, Vol. A: Space-Group Symmetry, Springer, 2016, pp. 1-250.