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 dhkl 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
Recommended Textbooks
- "Introduction to Solid State Physics" by Charles Kittel
- "The Structure of Materials" by De Graef & McHenry
- "Elements of X-ray Diffraction" by Cullity & Stock
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
