Slow neutron-capture paths and branching points

A neutron has no electric charge, so there is no entrance-channel Coulomb penetration factor of the charged-particle type. Its capture cross section still depends on energy, resonances, level structure, and available gamma transitions. For thermal neutrons with a Maxwell relative distribution, the capture hazard per target is neutron density times the thermal reactivity. A Maxwellian-averaged cross section is commonly defined by dividing that reactivity by a stated thermal speed; its normalization should be checked before multiplying a tabulated value by a velocity.

In dilute targets with a maintained neutron reservoir, this hazard gives a target's exponential survival against capture alone. Beta decay supplies another hazard, obtained from its mean lifetime or half-life conversion. Stellar weak rates can differ from terrestrial values because of ionization, thermal excitation, and electron environment. A branch prediction therefore needs rates appropriate to the relevant temperature and density. Merely labeling an isotope unstable does not decide which outgoing path dominates under all astrophysical conditions.

Suppose a single unstable nucleus has only capture and beta-decay exits and the two hazards are constant while it waits. Survival against either is exponential with their sum. Integrating the capture hazard times that survival gives the eventual capture probability. This is not the probability of capture during an arbitrarily short exposure, because a finite exposure can leave an undecayed and uncaptured population. Counting that remaining population is essential when comparing a laboratory irradiation with an ideal eventual branch fraction.

When the hazards are comparable, changing neutron density moves the branch fraction strongly. Increasing the capture rate far above the decay rate drives the capture fraction toward one; taking neutron density toward zero drives it toward zero. These limiting checks help catch inverted ratios. More elaborate branch points can include internal isomeric states, thermally coupled levels, and several destruction routes, in which case a two-hazard formula is a reduced model whose omitted channels must be shown to be negligible.

Away from branch points, a simple capture chain can approach a steady local flow when the neutron exposure changes slowly and source and sink boundary effects are remote. Production into one isotope then balances its capture destruction. Under a common neutron distribution this yields an approximate inverse relation between abundance and capture cross section. Nuclei near closed neutron shells often have smaller capture rates and can accumulate, but the detailed pattern still depends on the exposure history and nuclear data.

The local relation is not valid for every isotope from the start of irradiation. A finite neutron pulse, a magic-number bottleneck, a strong branch, or a changing seed inventory can prevent stationarity. Neutrons are supplied by separate reactions and can be absorbed by poisons as well as the heavy-element path. A branch ratio can constrain a model of the environment, but one observed isotope ratio does not uniquely fix all stellar mixing, neutron-source, and temperature histories without additional evidence.