Rapid neutron-capture flow and freezeout

Compare the capture hazard with the relevant beta rate for the nuclei actually populated. Fast capture relative to beta decay allows several neutron additions between charge-changing events, but temperature also controls the reverse photodisintegration rate. When both capture and its reverse are faster than beta transfer and environmental change, an isotopic chain can approach equilibrium. Its abundance ratios depend on neutron density, nuclear separation energies, and partition functions. The beta links between chains can remain slower and carry directed flow rather than satisfying weak detailed balance.

A waiting-point approximation concentrates each chain near its favored isotopes and follows their beta transfers. If those transfers also become stationary over a suitable interval, the chain population times its effective beta rate can be approximately constant across charge. This steady beta flow is an additional approximation, not beta equilibrium and not a necessary feature of every r-process trajectory. In a cold rapid-capture environment, neglecting photodisintegration removes the hot equilibrium argument and makes capture-rate competition directly important.

Choose a parcel after seed formation and define a seed count and its average mass number. If each remaining free neutron is eventually incorporated into one of those seeds, no new seeds form, no fission occurs, and there are no external losses, the mean final mass number increases by the initial number of free neutrons per seed. Beta-minus decay changes charge while keeping mass number fixed in the absence of delayed particle emission. This gives a useful conservation bound before a large network is solved.

The bound is conditional. Some neutrons may escape, remain free until they decay, or be captured by particles not counted in the chosen seed population. Beta-delayed neutron emission returns neutrons and reduces the emitting nucleus's mass number; subsequent recapture can redistribute them. Fission changes the number and distribution of heavy nuclei as well as releasing neutrons. Once these effects matter, dividing the initial neutron inventory by an assumed fixed seed count is no longer a complete prediction of any particular abundance peak.

As neutron density falls, capture slows and may become comparable to beta decay or expansion. A hot isotopic equilibrium can fail before all neutron captures stop, and residual captures can still move the abundance pattern. Freezeout therefore needs a named balance: loss of capture–photodisintegration equilibrium, departure from rapid-capture flow, or exhaustion of free neutrons are related but different events. A single temperature threshold cannot replace the timescale comparisons for an arbitrary ejecta trajectory.

After the main capture phase, beta decays, delayed neutrons, and possible fission reshape the products before they become long-lived material. Nuclear masses and rates far from stability are often uncertain, so the same qualitative flow mechanism can produce different quantitative patterns under different inputs and histories. The examples below use explicit budgets and hazards to expose robust accounting. They neither assign a universal abundance pattern to all astrophysical sites nor infer a site's total ejecta yield from a local mean-mass estimate.