Nuclear reactions and threshold energy

A nuclear reaction that absorbs energy (a negative Q-value) cannot proceed at just any projectile energy — momentum conservation means some of the projectile's kinetic energy is unavoidably tied up in moving the products forward, so the projectile needs strictly more kinetic energy than |Q| to make the reaction happen at all.

Practise this concept

Every nuclear reaction has a Q-value: the total kinetic energy released, equal to the rest-mass energy of the reactants minus the rest-mass energy of the products. An exothermic reaction (Q positive) releases energy and can happen even with a projectile at rest. An endothermic reaction (Q negative) needs to be fed energy from somewhere, and that somewhere is the projectile's kinetic energy.

The naive expectation is that a projectile with kinetic energy equal to |Q| should just barely make an endothermic reaction happen. That is wrong, and the reason is momentum conservation. In the lab frame the target nucleus starts at rest, so the whole system carries whatever momentum the incoming projectile brought with it — momentum that has to still be there after the reaction, carried forward by the products. Building up that shared forward motion costs kinetic energy that never becomes available to pay for the reaction itself. The projectile therefore needs somewhat more than |Q|, and the exact excess depends on how the momentum has to be split: a light projectile hitting a much heavier target wastes very little of its energy on recoil, while a projectile close in mass to the target wastes a lot.

This is exactly why particle accelerators built to produce or study threshold reactions are specified well above the reaction's raw Q-value — the accelerator has to supply the recoil tax as well as the reaction's own energy deficit.

Key ideas

Requires: Nuclear binding energy, Conservation laws in particle interactions

Formulas

E_{th} = |Q| \cdot \frac{m_a + m_X}{m_X}

Symbol	Name	Unit
$Q$	reaction energy	MeV (megaelectronvolt)
$m_a$	projectile mass	u (atomic mass unit)
$m_X$	target mass	u (atomic mass unit)
$E_th$	threshold energy	MeV (megaelectronvolt)

Common misconceptions

Believing the threshold kinetic energy simply equals the magnitude of the reaction's energy deficit, |Q| — ignoring that momentum conservation forces some of that kinetic energy into the products' recoil rather than into the reaction itself.
Mixing up which mass belongs in the correction factor's denominator — using the projectile's own mass instead of the (usually much heavier) stationary target's mass.
Carrying the reaction energy's negative sign straight through the threshold-energy formula, arriving at a nonsensical negative kinetic energy instead of taking its magnitude.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Physics of the microworldLOFY.01.009