KAHIBARO
Table of Contents
Energy Absorption in Nuclear Reactions
An endothermic nuclear reaction is a reaction that requires an input of energy in order to occur. In this type of reaction, the total mass-energy of the products is greater than that of the initial particles, so the reaction cannot proceed unless some extra kinetic energy is supplied by the incoming particles.
The energy balance of a nuclear reaction is described by the $Q$ value. For a general reaction,
$$
a + A \rightarrow b + B
$$
the $Q$ value is
$$
Q = \left(m_a + m_A - m_b - m_B\right)c^2
$$
If $Q < 0$, the reaction is endothermic. This means the reaction absorbs energy.
For an endothermic reaction, the defining condition is
$$
Q < 0
$$
A negative $Q$ value means that energy must be supplied from outside, usually as kinetic energy of the incoming projectile.
Physical Meaning of Negative Q
In nuclear physics, mass and energy are interchangeable through Einstein’s relation $E = mc^2$. If the final rest mass is larger than the initial rest mass, then that extra rest energy must come from somewhere. It comes from the kinetic energy of the particles before the reaction.
So, in an endothermic reaction, part of the initial kinetic energy is converted into additional rest mass of the reaction products. As a result, the incoming particle must have enough energy for the reaction to be possible at all.
This is different from an exothermic reaction, where energy is released because the products have less rest mass than the reactants.
Threshold Energy
A negative $Q$ value does not just mean that energy is needed. It also means there is usually a minimum projectile energy, called the threshold energy, below which the reaction cannot occur.
If the target nucleus $A$ is initially at rest in the laboratory frame, then the threshold energy is greater than just $|Q|$. This happens because momentum must also be conserved, and some kinetic energy must remain in the final particles.
For the reaction
$$
a + A \rightarrow b + B
$$
with target $A$ at rest, the threshold energy is approximately
$$
E_{\mathrm{th}} \approx -Q \left(1 + \frac{m_a}{m_A}\right)
$$
for nonrelativistic cases.
In an endothermic reaction, the minimum projectile energy is not usually just $|Q|$.
Because both energy and momentum must be conserved,
$$
E_{\mathrm{th}} > |Q|
$$
for a target initially at rest.
Why the Threshold Is Larger Than |Q|
It is tempting to think that supplying exactly $|Q|$ should be enough. But that would ignore momentum conservation. In the laboratory frame, the projectile brings momentum into the reaction. After the reaction, the products must carry that momentum. Therefore, the products cannot both be completely at rest. Some kinetic energy must remain in the final state.
This extra kinetic energy raises the required incident energy above $|Q|$.
A heavier target nucleus usually lowers the correction factor $\frac{m_a}{m_A}$. That is why reactions on heavy nuclei often have threshold energies closer to $|Q|$ than reactions on light targets.
Example of an Endothermic Reaction
Consider a reaction with
$$
Q = -2.0 \,\text{MeV}
$$
If the projectile mass is much smaller than the target mass, then the threshold energy is only slightly greater than $2.0 \,\text{MeV}$. But if the projectile and target have similar masses, the threshold can be noticeably larger.
For example, if
$$
\frac{m_a}{m_A} = 1
$$
then
$$
E_{\mathrm{th}} \approx -Q(1+1) = 4.0 \,\text{MeV}
$$
So even though the reaction absorbs only $2.0\,\text{MeV}$ in mass-energy terms, the projectile may need about $4.0\,\text{MeV}$ of kinetic energy in the lab frame.
Comparison with Exothermic Reactions
The main contrast between exothermic and endothermic reactions is shown below.
| Reaction type | Sign of $Q$ | Energy behavior | Minimum incident energy |
|---|---|---|---|
| Exothermic | $Q > 0$ | Releases energy | May occur even with very small incident energy, if other conditions allow |
| Endothermic | $Q < 0$ | Absorbs energy | Requires a threshold energy |
This distinction is very important in experiments, because not every energetically allowed reaction occurs at all projectile energies.
Energy Diagram
A simple way to visualize an endothermic reaction is to think of the products as lying at a higher energy level than the reactants.
The products are shown at a higher energy than the reactants, so energy must be supplied to move from the initial state to the final state.
Practical Importance
Endothermic reactions are especially important when choosing beam energies in nuclear experiments. If the projectile energy is below threshold, the reaction will not occur, no matter how long one waits. This makes threshold energy a key design parameter in accelerators, detectors, and reaction studies.
In astrophysics and reactor physics, endothermic reactions can also matter because they may be suppressed unless temperatures or particle energies are high enough.
A reaction with negative $Q$ is energetically forbidden unless the incoming particles provide enough kinetic energy to reach at least the threshold energy.
Summary
An endothermic nuclear reaction absorbs energy and has a negative $Q$ value. The needed energy comes from the kinetic energy of the incoming particles. Because momentum must also be conserved, the required projectile energy in the laboratory frame is greater than $|Q|$. This minimum required energy is called the threshold energy, and it determines whether the reaction can occur at all.