A nucleus bound in an iron crystal emits its gamma ray recoil-free with probability…
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Nuclear Physics · Momentum, energy and conservation laws
The Nucleus Resolved, Ch Gamma decay: multipoles, isomers, internal conversion and the Mossbauer line
Problem
A 57Fe nucleus bound in an iron crystal emits its 14.4keV gamma ray recoil-free with probability (the Lamb-Mössbauer factor) f=0.75. The natural linewidth of the transition is Γ=4.64×10−9eV. Someone argues: "For a recoil-free event, no momentum at all is transferred to the crystal: the photon simply carries away all of the momentum released in the decay, and momentum conservation between the nucleus and the photon no longer applies to these events." Take the emitting crystal to have an effective mass M=1.0mg that recoils as a single rigid body in a recoil-free event.
((a)) Compute the photon momentum, and the recoil energy and recoil speed this mass actually acquires.
((b)) Compare the recoil energy with Γ and explain what is wrong with the claim.