 Nuclear level reconstruction from energy differences

To leading order neglecting nuclear recoil, a measured gamma ray has energy equal to the difference between the upper and lower nuclear energy levels. For a freely recoiling atomic nucleus initially at rest, four-momentum conservation instead giveswhere $M_u$ and $M_l$ are the rest masses of the upper and lower nuclear states. Some transition energy becomes daughter kinetic energy, so high-precision level reconstruction must account for nuclear recoil.

Reconstructing a nuclear gamma ray radiation spectrum amounts to assigning measured photon energies to differences between ordered nuclear energy levels with such corrections where needed. Cascade energy sums constrain the graph, while population-flow balances constrain its orientation and feeding. Experimental central energies need not close exactly; a weighted least squares energy-network fit uses their uncertainties consistently. Multiple graphs can satisfy the same radiation spectrum when coincidence or lifetime information is absent.