Pendellösung Effect

Definition

The Pendellösung effect (German for “pendulum solution”) describes the periodic transfer of energy between an incident beam and diffracted beam during dynamical diffraction in perfect crystals. It manifests as oscillations in diffracted intensity with varying crystal thickness or orientation.

Key Mechanism

1. Two-Wave Coupling:
   When X-rays/neutrons satisfy Bragg’s condition (\(2d\sinθ=nλ\)), the incident wave (\(ψ_0\)) and diffracted wave (\(ψ_h\)) become coupled via the crystal’s periodic potential.

2. Interference Pattern:
   The waves form a standing wave pattern whose nodes/antinodes periodically shift between atomic planes, causing energy exchange. The Pendellösung distance (\(Λ\)) is the characteristic thickness for complete energy transfer:
   \(Λ = \frac{πV_c \cosθ}{λ |F_h|}\)
   where \(V_c\) is unit cell volume, \(F_h\) is structure factor.

Observations

• Thickness Fringes: In wedge-shaped crystals, intensity oscillates with thickness (e.g., silicon wafers).
• Rocking Curves: For monochromatic beams, oscillations appear during angular scans near Bragg angle.

Applications

• Precision measurement of structure factors (\(F_h\)).
• Phase contrast imaging in neutron interferometry.
• Testing crystal perfection in semiconductor materials.

Quantum Analogue

Analogous to Rabi oscillations in two-level quantum systems, where the diffraction potential acts like a coupling field.