Crystal Structures

Interactive 3D crystal structure visualization - Explore cubic crystal systems (simple cubic, body-centered cubic, face-centered cubic), Bragg equation, Miller indices, d-spacing, atomic density, and X-ray diffraction patterns

Crystal Type: Simple Cubic
Lattice Constant: 1.00 Å
Coordination Number: 6
Packing Factor: 0.52

Miller Indices (hkl)

d-spacing: d100 = 1.00 Å

Bragg Equation

2d·sinθ = nλ
Bragg Condition: 2 × 1.00 × sin(30.0°) = 1.00
nλ: 1.54
Condition NOT met

Atomic Density

ρ = n / V = n / a³
Atoms per cell: 1
Cell volume: 1.00 ų
Density: 1.00 atoms/ų

X-ray Diffraction Pattern

Crystal Parameters

Crystal Structure Equations

Bragg Equation 2d·sinθ = nλ (n = 1,2,3...)
Cubic System a = b = c, α = β = γ = 90°
d-spacing d_(hkl) = a/√(h²+k²+l²)
Atomic Density ρ = n/V (n = atoms per cell, V = a³)
Coordination Numbers: sc = 6, bcc = 8, fcc = 12
Packing Factors: sc = 0.52, bcc = 0.68, fcc = 0.74
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What are Crystal Structures?

Crystal structures are ordered arrangements of atoms in three-dimensional space. The unit cell is the smallest repeating unit that shows the full symmetry of the crystal structure. In cubic crystal systems, the unit cell is a cube with atoms at specific positions.

Cubic Crystal Structures

Simple Cubic (sc): Atoms are located only at the corners of the cube. Each corner atom is shared by 8 unit cells, giving 1 atom per unit cell. The coordination number is 6, and the packing efficiency is 52%.

Body-Centered Cubic (bcc): Atoms are at the corners and one atom at the body center. This gives 2 atoms per unit cell. The coordination number is 8, and the packing efficiency is 68%.

Face-Centered Cubic (fcc): Atoms are at the corners and the centers of all faces. This gives 4 atoms per unit cell. The coordination number is 12, and the packing efficiency is 74% (highest possible for equal spheres).

Miller Indices

Miller indices (hkl) are a notation system for crystal planes and directions in crystal lattices. They represent the reciprocal of the fractional intercepts that the plane makes with the crystallographic axes. For example, the (100) plane is parallel to the y and z axes and intercepts the x-axis at 1 unit cell length.

Bragg's Equation and X-ray Diffraction

Bragg's equation (2d·sinθ = nλ) describes the condition for constructive interference of X-rays scattered by crystal lattice planes. When X-rays of wavelength λ strike crystal planes with spacing d at angle θ, they will be strongly reflected only when the path difference between waves reflected from adjacent planes equals an integer multiple of the wavelength (nλ). This principle is the foundation of X-ray diffraction analysis, which is used to determine crystal structures.

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