Crystallography: An Introduction

Introduction to Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. The word "crystallography" derives from the Greek words crystallon, meaning "cold drop" or "frozen drop", and graphein, meaning "to write" ¹ .

Key Concept

Crystals are solids whose constituent atoms, molecules, or ions are arranged in an ordered, repeating pattern extending in all three spatial dimensions ¹ .

Historical Development

Crystallography as a science originated in the 17th century with the work of Nicolaus Steno, who discovered the law of constancy of interfacial angles ¹ . This was followed by René-Just Haüy's theory of crystal structure in the 18th century.

Interactive 3D crystal lattice representation

Crystal Structures and Lattices

The fundamental concept in crystallography is the crystal lattice, which describes the periodic arrangement of atoms in a crystal ¹ .

Crystal Systems

There are seven crystal systems, which categorize crystals based on their symmetry elements and lattice parameters ¹ :

Crystal System	Axial Lengths	Axial Angles	Examples
Cubic	a = b = c	α = β = γ = 90°	Diamond, NaCl
Tetragonal	a = b ≠ c	α = β = γ = 90°	Zircon, TiO₂
Orthorhombic	a ≠ b ≠ c	α = β = γ = 90°	Barite, Sulfur
Hexagonal	a = b ≠ c	α = β = 90°, γ = 120°	Graphite, Quartz
Trigonal	a = b = c	α = β = γ ≠ 90°	Calcite, Corundum
Monoclinic	a ≠ b ≠ c	α = γ = 90° ≠ β	Gypsum, Sucrose
Triclinic	a ≠ b ≠ c	α ≠ β ≠ γ ≠ 90°	Kyanite, Turquoise

Distribution of Crystal Systems

Bravais Lattices

In three dimensions, there are 14 possible Bravais lattices that describe the arrangement of lattice points in space ¹ . These are derived from the seven crystal systems by considering different lattice centering.

Crystal Growth Techniques

Various methods are used to grow single crystals for crystallographic studies ¹ :

Slow Evaporation

Solvent is allowed to slowly evaporate from a solution, leading to supersaturation and crystal formation.

Temperature Gradient

A temperature difference is maintained across the growth solution to control crystallization.

Vapor Diffusion

A precipitant diffuses slowly into the protein solution, gradually reducing solubility.

Melt Growth

Crystals are grown from molten material by controlled cooling (e.g., Czochralski method).

Symmetry in Crystals

Symmetry is a fundamental concept in crystallography that describes the repetitive patterns in crystal structures ¹ .

Point Symmetry Operations

Point symmetry operations leave at least one point in space unchanged ¹ . These include:

Rotation

Rotation of the crystal by a specific angle around an axis. Crystals can only have 1-, 2-, 3-, 4-, or 6-fold rotation axes due to the translational symmetry of the lattice ¹ .

Reflection

Mirror reflection across a plane, which divides the crystal into two mirror-image halves.

Inversion

Transformation of every point (x,y,z) to (-x,-y,-z) through a center of symmetry.

Rotoinversion

Combination of rotation and inversion, including the important case of 4-fold rotoinversion.

Crystal Classes

There are 32 crystal classes (point groups) that describe the possible combinations of symmetry elements in crystals ¹ .

Crystal Classes by System

Space Groups

When translational symmetry is combined with point symmetry, we obtain 230 space groups that describe all possible symmetric arrangements in three-dimensional crystals ¹ .

Triclinic: 2 space groups

Monoclinic: 13 space groups

Orthorhombic: 59 space groups

X-Ray Diffraction

X-ray diffraction is the primary experimental technique used to determine crystal structures ¹ .

Bragg's Law

The fundamental equation of X-ray diffraction is Bragg's Law ¹ :

nλ = 2d sinθ

Where:
n = order of reflection
λ = wavelength of X-rays
d = interplanar spacing
θ = angle of incidence

When X-rays interact with a crystal, they are scattered by the electrons in the atoms. If the scattered waves are in phase, they constructively interfere and produce a diffraction pattern ¹ .

Diffraction Methods

Laue Method

Uses white radiation and a stationary single crystal. Mainly used for orientation determination.

Rotating Crystal Method

Uses monochromatic radiation and a rotating single crystal. Provides complete diffraction data.

Powder Method

Uses monochromatic radiation and a powdered sample. Useful for phase identification.

Four-Circle Diffractometer

Allows precise orientation of the crystal to measure all reflections systematically.

X-Ray Diffraction Process

Crystal Preparation
Grow a high-quality single crystal
Data Collection
Expose crystal to X-rays and record diffraction pattern
Data Processing
Index reflections and determine unit cell parameters
Structure Solution
Determine phases and calculate electron density
Refinement
Improve the atomic model to fit the experimental data

Structure Factors

The structure factor F(hkl) describes how the incident X-ray beam is scattered by the crystal in the direction corresponding to the diffraction spot with indices h,k,l ¹ . It contains information about both the amplitude and phase of the diffracted wave.

Applications of Crystallography

Crystallography has numerous applications across various scientific and technological fields ¹ .

Pharmaceuticals

Determining the crystal structures of drugs and their targets enables rational drug design and optimization of pharmaceutical properties.

Polymorph screening
Protein-ligand interactions
Salt selection

Materials Science

Understanding structure-property relationships guides the development of new materials with tailored characteristics.

Semiconductors
Superconductors
Catalysts
Battery materials

Structural Biology

Revealing the 3D structures of biological macromolecules provides insights into their function and mechanisms.

Enzyme mechanisms
Membrane proteins
Virus structures
Drug targets

Crystal Gallery