Why does nuclear recoil usually prevent resonance absorption of gamma rays?

Nuclear recoil prevents resonance absorption of gamma rays because of the laws of conservation of momentum and energy. When a nucleus emits a gamma-ray photon, it must recoil to conserve momentum, which carries away a portion of the transition energy as kinetic recoil energy ($E_R$). Consequently, the emitted photon has slightly less energy than the difference between the nuclear energy states.12

When this photon strikes an absorbing nucleus, the absorber must also recoil to conserve momentum, requiring an additional amount of energy ($E_R$) to facilitate the transition. Because the emitted photon is missing a total of $2E_R$—the energy lost by the source during emission and the energy required by the absorber during capture—it lacks the precise energy necessary for resonance absorption. While this effect is negligible for visible light, the recoil energy for gamma rays is significantly larger, and their energy transitions are much more sharply defined, making the mismatch large enough to prevent resonance in free nuclei. The Mössbauer effect overcomes this by embedding nuclei in a solid lattice, allowing the entire crystal to absorb the recoil momentum and reducing the energy loss to nearly zero.23