The Crystal Clock Synchronizer · AI Generator #GEN-6F77D1E3

Physics · modern-physics

Problem

A space station uses a novel time synchronization system based on X-ray diffraction from a rotating crystal wheel. The wheel contains identical quartz crystals with lattice spacing d=0.31  nmd = 0.31\;\text{nm}, and rotates at ω=120  rad/s\omega = 120\;\text{rad/s}. A stationary X-ray source emits photons of wavelength λ0=0.15  nm\lambda_0 = 0.15\;\text{nm} toward the crystals. Due to the wheel's rotation, the crystals have a velocity component v=18.6  m/sv = 18.6\;\text{m/s} toward the X-ray source when they are positioned for diffraction measurements. The system is designed so that first-order Bragg diffraction (n=1n = 1) occurs at exactly θ=30°\theta = 30° when the crystals are moving toward the source. Find the original rest wavelength λrest\lambda_{\text{rest}} that the X-ray source was designed to emit before accounting for the relativistic Doppler effect caused by the crystal motion.

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