KAHIBARO
Table of Contents
Meaning of a Threshold Reaction
In nuclear reactions, some reactions can happen with particles of very low incoming energy, while others require a minimum incoming energy before they can occur at all. This minimum required energy is called the reaction threshold energy.
A threshold appears most clearly in endothermic reactions, where the reaction has a negative Q value. In such reactions, energy must be supplied from the outside to make up for the energy deficit. If the incoming particle does not bring enough kinetic energy, the reaction cannot take place.
Suppose a reaction has the form
$$
a + A \rightarrow b + B
$$
where $a$ is the incident particle and $A$ is the target nucleus, initially at rest in the laboratory frame. If the Q value is negative, then the projectile must have at least some minimum kinetic energy for the reaction to be possible.
A reaction threshold is the minimum projectile kinetic energy required for a reaction to occur when the target is initially at rest.
For endothermic reactions, $Q < 0$, so a threshold energy is required.
Why the Threshold Is Not Just $|Q|$
A beginner might think that if a reaction needs energy $|Q|$, then the threshold should simply be
$$
K_{\text{threshold}} = |Q|
$$
but this is usually not correct in the laboratory frame.
The reason is momentum conservation. Even at the minimum energy needed for the reaction, the final products cannot usually both remain at rest. They must move in such a way that total momentum is conserved. Because of this, some of the projectile energy must go into the kinetic energy of the products, not only into overcoming the negative Q value.
So the threshold energy is generally larger than $|Q|$.
Threshold Condition in the Laboratory Frame
Consider again the reaction
$$
a + A \rightarrow b + B
$$
with target $A$ at rest. The threshold occurs when the final particles move together with the smallest possible total kinetic energy consistent with momentum conservation. In the laboratory frame, the threshold kinetic energy of the projectile is
$$
K_{\text{th}} = -Q \left(1 + \frac{m_a}{m_A}\right)
$$
where $m_a$ is the mass of the incident particle and $m_A$ is the mass of the target nucleus.
This formula is a very useful approximation in nonrelativistic nuclear physics.
For a reaction
$$
a + A \rightarrow b + B
$$
with target $A$ initially at rest and $Q < 0$, the threshold energy is
$$
K_{\text{th}} = -Q \left(1 + \frac{m_a}{m_A}\right)
$$
This is greater than $|Q|$ because momentum must also be conserved.
If the target nucleus is much heavier than the projectile, then $\frac{m_a}{m_A}$ is small, and the threshold energy is close to $|Q|$. If the target is not much heavier, the correction becomes more important.
Physical Interpretation
The threshold energy depends on both energy conservation and momentum conservation. Energy conservation alone tells us that the incident particle must at least supply the missing reaction energy. Momentum conservation adds an extra requirement, because the final nuclei must carry momentum.
At threshold, the products are formed with the smallest possible relative motion. They move together in the same direction in the laboratory frame. This gives the minimum projectile energy that still allows both conservation laws to be satisfied.
You can think of the incoming particle as doing two jobs. First, it supplies the energy needed because $Q < 0$. Second, it provides the momentum that the final products must carry.
Simple Example
Suppose a nuclear reaction has
$$
Q = -2.0 \,\text{MeV}
$$
and the projectile mass is one fourth of the target mass:
$$
\frac{m_a}{m_A} = \frac{1}{4}
$$
Then the threshold energy is
$$
K_{\text{th}} = -(-2.0)\left(1 + \frac{1}{4}\right)
$$
$$
K_{\text{th}} = 2.0 \times 1.25 = 2.5 \,\text{MeV}
$$
Although the energy deficit is only $2.0\,\text{MeV}$, the projectile must actually have at least $2.5\,\text{MeV}$.
Special Cases
There are a few important situations worth noticing.
Exothermic Reactions
If $Q > 0$, the reaction releases energy. In principle, there is no energy threshold from the Q value alone. Even a very slow projectile may cause the reaction, although other effects such as Coulomb repulsion can still make the reaction difficult in practice.
If $Q > 0$, there is no threshold caused by the Q value itself.
If $Q < 0$, a threshold energy is required.
Very Heavy Targets
If the target mass is much larger than the projectile mass, then
$$
\frac{m_a}{m_A} \ll 1
$$
and so
$$
K_{\text{th}} \approx -Q
$$
This is why in many rough estimates, the threshold is taken to be about the magnitude of the negative Q value.
Equal Masses
If the projectile and target have equal mass, then
$$
\frac{m_a}{m_A} = 1
$$
so
$$
K_{\text{th}} = -Q(1+1) = -2Q
$$
For a negative Q value, the threshold is twice the energy deficit.
Summary Table
| Reaction type | Q value | Threshold behavior |
|---|---|---|
| Exothermic | $Q > 0$ | No threshold from Q value alone |
| Thermoneutral | $Q = 0$ | No threshold from Q value alone |
| Endothermic | $Q < 0$ | Requires minimum projectile energy |
| Condition | Threshold formula | |
| Target at rest, nonrelativistic | $K_{\text{th}} = -Q\left(1+\frac{m_a}{m_A}\right)$ | |
| Very heavy target, $m_A \gg m_a$ | $K_{\text{th}} \approx -Q$ |
Visual Picture
The threshold idea can be pictured as an incoming projectile striking a target at rest. Below threshold, the reaction cannot proceed. At threshold, the outgoing products just barely can be created while still satisfying both conservation laws.
At the exact threshold, the final particles have the smallest possible motion relative to each other.
Final Remarks
Reaction thresholds are important because they tell us whether a given projectile beam has enough energy to produce a desired nuclear reaction. They are especially useful in accelerator experiments and in understanding why some reactions occur easily while others require energetic incoming particles.
Key idea:
The threshold energy is not determined by energy conservation alone. It comes from energy conservation together with momentum conservation.
For endothermic reactions with target at rest,
$$
K_{\text{th}} = -Q\left(1+\frac{m_a}{m_A}\right)
$$