Crystallography Guide

1 Crystal Systems

All crystals in nature can be classified into one of seven crystal systems based on their unit cell geometry and symmetry operations. The unit cell is the smallest repeating unit that defines the crystal structure.

The Seven Crystal Systems

Cubic: a = b = c, α = β = γ = 90°
Examples: Diamond, NaCl, Cu, Fe
Highest symmetry system with three equal axes at right angles.
Tetragonal: a = b ≠ c, α = β = γ = 90°
Examples: TiO₂ (rutile), SnO₂, ZrSiO₄
Two equal axes perpendicular to a unique axis.
Orthorhombic: a ≠ b ≠ c, α = β = γ = 90°
Examples: BaSO₄, CaCO₃ (aragonite), α-S
Three unequal axes all at right angles.
Hexagonal: a = b ≠ c, α = β = 90°, γ = 120°
Examples: Graphite, ZnO, Mg, SiO₂ (quartz)
Two equal axes at 120° with a perpendicular unique axis.
Rhombohedral: a = b = c, α = β = γ ≠ 90°
Examples: CaCO₃ (calcite), Bi, As
Three equal axes equally inclined to each other.
Monoclinic: a ≠ b ≠ c, α = γ = 90° ≠ β
Examples: Gypsum, FeS₂, many minerals
Three unequal axes, one oblique angle.
Triclinic: a ≠ b ≠ c, α ≠ β ≠ γ
Examples: CuSO₄·5H₂O, K₂Cr₂O₇
Lowest symmetry with all axes and angles different.

                Why It Matters
                Crystal system determines mechanical properties, optical behavior, and anisotropic characteristics of materials. For instance, cubic crystals are isotropic, while hexagonal crystals show different properties along different directions.
            

2 Miller Indices

Miller indices are a notation system to describe crystallographic planes and directions. They're fundamental to understanding material behavior, X-ray diffraction, and crystal structure.

Notation for Planes: (hkl)

Miller indices describe a plane by the reciprocals of its intercepts on the crystallographic axes.

A plane intercepts the axes at:

$$\text{a-axis: } \dfrac{a}{h} \qquad \text{b-axis: } \dfrac{b}{k} \qquad \text{c-axis: } \dfrac{c}{l}$$

Algorithm to determine Miller indices:

Identify the plane's intercepts on the a, b, and c axes
Take the reciprocals of these intercept values
Clear any fractions by multiplying by the least common denominator
Reduce to smallest integers and enclose in parentheses: (hkl)

(100) Plane

Perpendicular to the a-axis

Parallel to b and c axes

(110) Plane

Cuts a and b axes equally

Parallel to c axis

(111) Plane

Cuts all three axes

At equal distances

Notation for Directions: [uvw]

Square brackets denote a direction vector in the crystal lattice:

$$\vec{r} = u\vec{a} + v\vec{b} + w\vec{c}$$

                Key Insight
                In cubic crystals, direction [hkl] is perpendicular to plane (hkl). This is NOT true for other crystal systems due to non-orthogonal axes.
            

3 Interplanar Spacing

The interplanar spacing d_hkl is the perpendicular distance between adjacent parallel crystallographic planes with Miller indices (hkl). This is one of the most important parameters in crystallography.

Why It Matters

X-ray Diffraction: Bragg's Law relates interplanar spacing to diffraction angles
Material Identification: Each material has characteristic d-spacings
Strain Analysis: Changes indicate residual stress or plastic deformation
Phase Analysis: Different structures exhibit distinct d-spacing patterns

Bragg's Law

$$n \lambda = 2 d_{hkl} \sin\theta$$

where n is the order of diffraction, λ is the X-ray wavelength, and θ is the Bragg angle

Cubic System Formula

$$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$$

Other Crystal Systems

Tetragonal:

$$\frac{1}{d_{hkl}^2} = \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}$$

Hexagonal:

$$\frac{1}{d_{hkl}^2} = \frac{4}{3}\left(\frac{h^2 + hk + k^2}{a^2}\right) + \frac{l^2}{c^2}$$

                Real-World Application
                Thermal expansion increases dhkl proportionally with temperature. Residual stress analysis via XRD exploits this: compressive stress decreases dhkl, while tensile stress increases it.
            

4 Crystallographic Planes & Directions

Importance in Materials Science

Crystallographic planes and directions control many material properties:

Slip Systems: Plastic deformation occurs on specific slip systems
Example: FCC metals slip on {111}⟨110⟩ (12 independent slip systems)
Cleavage: Materials fracture preferentially along low-index planes
Example: Mica cleaves perfectly on (001) basal planes
Surface Energy: Varies with atomic packing density
Close-packed planes like {111} in FCC have lower surface energy
Anisotropic Properties: Young's modulus, electrical conductivity, and thermal conductivity are direction-dependent

{100} Family

6 equivalent planes

Cleavage planes in NaCl, MgO

{110} Family

12 equivalent planes

Primary slip planes in BCC

{111} Family

8 equivalent planes

Primary slip planes in FCC

                Engineering Design
                Texture engineering exploits crystallographic anisotropy. Single-crystal turbine blades are grown with the [001] direction along the blade axis to maximize creep resistance.
            

5 Using the Visualization Tool

Getting Started

Select a crystal system (start with cubic for simplicity)
Adjust lattice parameters to see how the unit cell changes
Enter Miller indices (try 100, 110, 111)
Rotate the 3D view to understand plane orientation
Enable "Show Plane Family" to see equivalent planes

Tips for Learning

Use the example buttons to see common important planes
Watch how d-spacing changes with different Miller indices
Try hexagonal system to see 4-index Miller-Bravais notation
Hover over info icons for quick explanations

6 Further Learning

Crystallographic Databases

ICDD PDF Database (powder diffraction)
Crystallography Open Database (COD)
American Mineralogist Crystal Structure Database

Related Software

VESTA - 3D visualization of crystal structures
CrystalMaker - professional crystallography toolkit
Mercury - Cambridge Structural Database viewer