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Square Lattice Bragg Scattering

Seeing Atoms with Waves​

When waves scatter from a periodic structure like a crystal, they create diffraction patterns that reveal the atomic arrangement. This is how we determine the structure of everything from simple salts to complex proteins.

Square Lattice Structure​

The square lattice is the simplest 2D crystal structure:

    ●───●───●───●───●
    │   │   │   │   │
    ●───●───●───●───●
    │   │   │   │   │
    ●───●───●───●───●
    │   │   │   │   │
    ●───●───●───●───●

    Lattice constant: a
    Basis vectors: a₁ = (a,0), a₂ = (0,a)

Real materials with square-like arrangements:

  • Surface of cubic crystals (NaCl 100 face)
  • 2D electron systems in semiconductors
  • Optical lattices for ultracold atoms

Bragg's Law​

Constructive interference occurs when: 2d sin(θ) = nλ

Where:

  • d is the spacing between lattice planes
  • θ is the scattering angle
  • λ is the wavelength
  • n is the diffraction order

For a square lattice, the allowed scattering vectors form a reciprocal lattice - also square!

The Simulation​

Grid: 512 × 512 points
Lattice spacing: 10.0 units
Potential depth: 25.0 units
Well radius: 2.0 units
Incoming momentum: kx = 5.0

What to Watch For​

  1. Initial approach: Gaussian wavepacket traveling toward the crystal
  2. Scattering: Part reflects, part transmits, part diffracts into Bragg peaks
  3. Diffraction pattern: Spots appear at specific angles in k-space
  4. Fourfold symmetry: Pattern reflects the square symmetry of the lattice

Animation​

The video above shows the wavepacket scattering in real-time, revealing the characteristic Bragg diffraction pattern.

The Reciprocal Lattice​

The diffraction pattern reveals the reciprocal lattice - the Fourier transform of the real-space lattice:

Real Space (atoms)         Reciprocal Space (diffraction)

    ● ● ● ●                      ★   ★   ★
    ● ● ● ●        FFT           ★   ★   ★
    ● ● ● ●       ───→           ★   ★   ★
    ● ● ● ●                      ★   ★   ★

Square lattice             Square pattern of spots
spacing = a                spacing = 2π/a

Run It Yourself​

claude -p "Demonstrate Bragg scattering: Create a square 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_square.gif" \
  --allowedTools "mcp__quantum-mcp__*"

Physics Insights​

Why Specific Angles?​

The crystal acts as a perfect diffraction grating in 2D. Only waves scattered at certain angles add constructively - all others cancel out by destructive interference.

What Sets the Intensity?​

The brightness of each diffraction spot depends on:

  1. Structure factor: How atoms are arranged in the unit cell
  2. Form factor: The scattering strength of each atom
  3. Temperature: Thermal motion reduces peak intensity (Debye-Waller factor)

Comparison with Other Lattices​

See related demos for hexagonal lattice animations.

Different lattices produce different diffraction patterns:

  • Square: Fourfold symmetric spots
  • Hexagonal: Sixfold symmetric spots