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Hexagonal Lattice Bragg Scattering

The Structure of Graphene​

The hexagonal (honeycomb) lattice is the structure of graphene - a single layer of carbon atoms that won the 2010 Nobel Prize in Physics. It's also found in hexagonal boron nitride and the surfaces of many materials.

Honeycomb Structure​

The hexagonal lattice has two atoms per unit cell, creating the distinctive honeycomb pattern:

        ●───●       ●───●       ●───●
       /     \     /     \     /     \
      ●       ●───●       ●───●       ●
       \     /     \     /     \     /
        ●───●       ●───●       ●───●
       /     \     /     \     /     \
      ●       ●───●       ●───●       ●
       \     /     \     /     \     /
        ●───●       ●───●       ●───●

    Two-atom basis creates the honeycomb

Key Features​

  • Two atoms per unit cell (A and B sublattices)
  • Three nearest neighbors per atom
  • Sixfold rotational symmetry
  • Dirac cones in the band structure (for graphene)

Why Graphene is Special​

Graphene's electrons behave as massless Dirac fermions, leading to:

  • Extremely high electron mobility
  • Unusual quantum Hall effect
  • Potential for quantum computing
  • Revolutionary material applications

Understanding its diffraction pattern helps characterize graphene samples.

The Simulation​

Grid: 512 × 512 points
Lattice constant: 10.0 units
Potential depth: 25.0 units
Two atoms per unit cell
Incoming momentum: kx = 5.0

Animation​

The video above shows the wavepacket scattering from the honeycomb lattice, revealing the sixfold symmetric Bragg pattern.

The Diffraction Pattern​

The hexagonal lattice produces a sixfold symmetric diffraction pattern:

                 ★
                / \
              ★     ★
               \   /
            ★───●───★
               /   \
              ★     ★
                \ /
                 ★

    Six primary diffraction spots
    (plus higher-order spots)

The two-atom basis creates additional features:

  • Some spots are systematically absent (structure factor = 0)
  • Remaining spots show specific intensity ratios
  • Pattern uniquely identifies the honeycomb structure

Reciprocal Lattice​

The reciprocal lattice of a hexagonal lattice is also hexagonal, but rotated 30°:

Real SpaceReciprocal Space
Hexagonal, angle = 60°Hexagonal, angle = 60°
Spacing = aSpacing = 4π/(√3 a)
Aligned with xRotated 30°

Run It Yourself​

claude -p "Demonstrate Bragg scattering: Create a hexagonal lattice with Gaussian points \
  (spacing=25, depth=100) starting at x=85, wavepacket at x=40 with momentum=[0.25,0], \
  run 1200 steps, show potential overlay, and save to /tmp/bragg_hexagonal.gif" \
  --allowedTools "mcp__quantum-mcp__*"

X-ray Crystallography​

This same principle is used to determine:

  • Crystal structures of new materials
  • Protein structures (X-ray crystallography)
  • DNA structure (Franklin and Wilkins' famous Photo 51)

The pattern of spots tells us:

  1. Positions: Crystal symmetry
  2. Intensities: Atomic positions within the unit cell
  3. Widths: Crystal quality and domain size

Real-World Applications​

Graphene Characterization​

Low-Energy Electron Diffraction (LEED) on graphene shows the hexagonal pattern, confirming:

  • Single-layer vs multi-layer samples
  • Orientation relative to substrate
  • Defect density and crystal quality

Material Identification​

Different materials produce unique "fingerprint" patterns:

  • Hexagonal: Graphene, hBN, MoS₂
  • Square: Cubic crystals
  • Complex: Proteins, minerals