Reaction Q-values and threshold energies

For a two-body reaction, define Q as initial rest energy minus final rest energy for the specified internal states. Positive Q releases rest energy into kinetic energy; negative Q requires kinetic input. Exciting a product lowers the available Q by that excitation energy. Rest masses already contain nuclear binding energy, so adding a binding-energy release again would double count it. State whether a quoted mass corresponds to a ground state or an isomer before using the difference.

Neutral-atom masses contain electron rest masses and electron binding energies. They can be used directly for many reactions when the summed nuclear charge, and hence the number of neutral-atom electrons, balances on both sides, provided electronic binding corrections are negligible at the desired precision. Bare projectiles, ionized products, and beta processes need explicit care. For neutral-atom beta-minus masses the electron rest contribution cancels in the usual Q formula; positron emission instead requires subtracting two electron rest energies, up to the relevant atomic corrections.

A projectile and stationary target carry nonzero total momentum. At an endothermic two-body threshold, the final particles have zero relative momentum in their centre-of-mass frame, but move together in the laboratory. The invariant total energy reaches the sum of their rest energies at this point. Solving that condition for the laboratory projectile kinetic energy gives an exact relativistic threshold, assuming the channel is otherwise allowed and no extra particles are present.

At nuclear beam energies small compared with rest energies, the leading expression is the magnitude of Q multiplied by one plus the projectile-to-target mass ratio. The extra term is the energy tied up in centre-of-mass translation. It becomes small for a light projectile on a very heavy target, but is substantial for comparable masses. The nonrelativistic approximation discards a calculable correction quadratic in Q; quoting an exact threshold and then using inconsistent approximate masses would defeat its precision.

A positive Q-value removes an endothermic kinematic threshold, but it does not remove charged-particle Coulomb repulsion, centrifugal barriers, or restrictive quantum numbers. Tunneling permits sub-barrier reactions, so a Coulomb barrier is a suppression scale rather than an absolute energy-conservation threshold. Similarly, a channel with the correct energy may have a tiny matrix element or negligible phase space near opening. Rates require dynamics in addition to the mass balance.

A useful calculation separates three questions: whether the listed particles conserve charge and nucleon number, whether total energy and momentum can realize the channel, and how likely the transition is. These checks also clarify inverse reactions, whose Q-values have opposite signs for the same specified states. Experimental thresholds can appear broadened by target thermal motion, beam spread, energy loss, and detector sensitivity; an observed onset is not automatically the ideal threshold of a target at rest.