Compound nuclei and Breit–Wigner resonances

A resonance state can decay through several allowed channels, each with a partial energy width. Their sum is the total width, and the fraction assigned to a given exit is its branching probability in the exponential-decay approximation. Time reversal relates coupling into a specified entrance to coupling out through its inverse channel. It does not equate that entrance partial width with the sum of all decay widths. The total mean lifetime is set by the total width.

The compound hypothesis suppresses detailed memory of the entrance mechanism after equilibration, but retains energy, angular momentum, parity, and any other relevant conserved quantities. It therefore does not justify identical decay patterns for arbitrarily different resonance spins or excitation energies. Direct reactions can coexist and interfere, especially in the elastic channel. A formation-times-branching description is an approximation whose usefulness depends on the degree of internal mixing and the energy resolution of the experiment.

For a single isolated resonance, the reaction cross section contains the product of entrance and exit partial widths divided by detuning squared plus one quarter of total width squared. The prefactor is an entrance-channel wavelength area times the spin factor. Near a narrow resonance, treating that area and the widths as constant gives a Lorentzian whose full width at half maximum equals the total width. At the peak, the entrance-to-exit coupling balance matters as much as the lifetime.

This constant-width approximation requires a resonance well separated from neighbors, a background that can be neglected or explicitly modeled, and sufficiently slow energy variation of penetrabilities and wave number across the width. Threshold-adjacent or broad resonances can violate these assumptions. If many levels overlap, adding isolated probabilities can miss interference and statistical correlations. Experimental beam spread, Doppler broadening, and resolution convolutions can also make a measured line wider than the natural width.

For time-reversal-invariant dynamics, the same resonance couplings appear in the inverse process at the same total energy. However, cross sections divide a transition rate by incident flux and average over initial spin states. Different channel momenta and spin degeneracies therefore produce different forward and reverse cross sections. Detailed balance compares their spin-degeneracy times momentum-squared weighted values, rather than asserting that the raw cross sections are equal.

This comparison must specify individual target and residual internal states. Thermally populated excited nuclei require an additional ensemble average; a laboratory ground-state cross section cannot simply be used as that stellar average. Identical particles and polarization also require appropriate counting conventions beyond the simple distinguishable spin-average formula. Before applying reciprocity, check that both channels are open at the chosen total energy and that the inverse calculation uses its own centre-of-mass wave number.