KAHIBARO
Table of Contents
How a Compound Nucleus Forms
In many nuclear reactions, the incoming particle does not simply bounce off the target nucleus and leave immediately. Instead, it can be captured for a short time. When this happens, the projectile and the target merge into a single excited nuclear system. This temporary system is called a compound nucleus.
The idea of formation is the key point of this chapter. We focus on how the compound nucleus comes into existence, not yet on how it later breaks apart.
The Basic Picture
Suppose a projectile nucleus or particle $a$ approaches a target nucleus $A$. If capture occurs, they combine into an intermediate excited nucleus, often written as
$$
a + A \rightarrow C^*
$$
Here, $C^*$ is the compound nucleus, and the star means that it is in an excited state.
The compound nucleus contains all the nucleons of the target and the projectile. Its atomic number and mass number are the sums of those of the two initial participants:
$$
Z_C = Z_a + Z_A
$$
$$
A_C = A_a + A_A
$$
The energy brought in by the projectile does not stay as simple forward motion. Instead, after capture, that energy is shared among many nucleons inside the nucleus. This produces a highly excited state.
A compound nucleus is formed when the projectile is absorbed by the target and a temporary excited nucleus results:
$$
a + A \rightarrow C^*
$$
The compound nucleus has
$$
Z_C = Z_a + Z_A, \qquad A_C = A_a + A_A
$$
Capture of the Projectile
For formation to happen, the projectile must get close enough to the target nucleus for the nuclear force to act strongly. The nuclear force is short range, so the projectile must approach to a very small distance.
If the projectile is electrically charged, it is repelled by the positive charge of the nucleus. This electrostatic repulsion is called the Coulomb barrier. The projectile must usually have enough energy to overcome this barrier, or in some cases tunnel through it quantum mechanically.
If the projectile is neutral, such as a neutron, there is no Coulomb repulsion. Because of this, neutrons are often captured more easily than charged particles.
Energy in Formation
When the projectile is captured, the total energy of the system becomes excitation energy of the compound nucleus. This excitation energy depends on the incoming kinetic energy and on the reaction energetics.
A simple way to write the excitation energy is
$$
E^* = E_{\text{cm}} + Q_{\text{capture}}
$$
where $E_{\text{cm}}$ is the kinetic energy in the center of mass frame, and $Q_{\text{capture}}$ is the energy released or absorbed in forming the combined nucleus.
A larger excitation energy means that the compound nucleus is formed in a more highly excited state. This does not mean it is stable. It only means the internal nuclear system has more energy to distribute among its nucleons.
In compound nucleus formation, the projectile's incident energy is converted into excitation energy of the temporary nucleus.
The compound nucleus is not usually in its ground state, but in an excited state $C^*$.
Loss of Memory of the Entrance Motion
An important feature of compound nucleus formation is that, once the projectile is absorbed, its energy is rapidly shared among many nucleons. The system reaches a state that no longer remembers the detailed way it was formed.
This means that after formation, the nucleus behaves like a mixed, highly excited system. It does remember conserved quantities such as total energy, total angular momentum, parity, electric charge, and nucleon number. But it does not retain a simple picture of one incoming particle moving through it.
This is what makes the compound nucleus different from a direct reaction, where the interaction is fast and only a few nucleons are involved.
Time Scale of Formation
The projectile is captured very quickly on a nuclear time scale. Once the short range force acts strongly enough, the incoming particle becomes part of the nuclear system. The compound nucleus then exists only for a short lifetime before it decays.
The important idea here is that the formation stage and the decay stage can often be treated separately. First the nucleus forms, then later it decays through one of several possible channels.
Role of Angular Momentum
The projectile usually brings orbital angular momentum, and possibly intrinsic spin. During formation, these combine with the target's angular momentum to give the total angular momentum of the compound nucleus.
So the compound nucleus is not formed in just any state. It must form in a state that satisfies conservation laws. Only certain values of total angular momentum and parity are allowed.
This can be written schematically as
$$
\vec{J} = \vec{I}_A + \vec{I}_a + \vec{L}
$$
where $I_A$ is the target spin, $I_a$ is the projectile spin, and $L$ is the orbital angular momentum of the incoming projectile relative to the target.
Formation of a compound nucleus must obey conservation laws, especially conservation of
$$
\text{energy, charge, nucleon number, angular momentum, and parity}
$$
Why Neutrons Often Form Compound Nuclei Easily
Neutrons are especially effective at forming compound nuclei because they carry no electric charge. They do not face Coulomb repulsion from the positively charged nucleus, so even low energy neutrons can be captured.
This is one reason neutron induced reactions are very important in nuclear physics and nuclear reactors. A slow neutron can be absorbed, forming a compound nucleus in an excited state.
For example,
$$
n + {}^{235}\mathrm{U} \rightarrow {}^{236}\mathrm{U}^*
$$
Here the neutron is captured by uranium-235, forming excited uranium-236.
Resonant Formation
Formation is often especially likely when the incident energy matches an allowed excited state of the combined nucleus. In that case, the probability of forming the compound nucleus becomes much larger. This is called resonance.
You can think of resonance as the incoming system having just the right energy to create a particular excited state of the compound nucleus.
Although the detailed study of resonances belongs to a later topic, the main idea is simple. The compound nucleus forms most strongly when the entrance energy fits one of its quantized excited states.
Comparison with Other Reaction Mechanisms
Not every nuclear reaction forms a compound nucleus. Sometimes the projectile interacts only briefly and directly with the target, causing a fast reaction. In compound nucleus formation, by contrast, the projectile is first absorbed into the whole nuclear system.
The distinction is useful:
| Reaction type | Main feature |
|---|---|
| Direct reaction | Fast interaction, few nucleons involved |
| Compound nucleus formation | Projectile captured, energy shared among many nucleons |
| Compound nucleus decay | Temporary excited nucleus later emits particles or radiation |
This chapter is about the middle line of the table, the creation of the intermediate excited system.
Visual Picture
The process can be pictured as an incoming projectile approaching the target, crossing the interaction region, and being absorbed into a single excited nucleus.
Formation Conditions at a Glance
The main physical requirements for formation can be summarized clearly.
| Requirement | Meaning |
|---|---|
| Sufficient approach | The projectile must come within nuclear force range |
| Barrier penetration or overcoming | Needed for charged projectiles |
| Conservation laws | Energy, angular momentum, parity, charge, and nucleon number must match |
| Available excited state | Formation is enhanced if the energy matches a compound nuclear level |
Final Idea
Formation of a compound nucleus is the step in which the target captures the incoming projectile and creates a temporary, excited combined nucleus. The incoming energy becomes internal excitation energy, and the system rapidly shares this energy among many nucleons. After that, the nucleus behaves as a single excited object whose later decay can proceed in several ways.
The essential idea of compound nucleus formation is:
$$
\text{capture of the projectile} \rightarrow \text{excited combined nucleus} \rightarrow \text{temporary intermediate state}
$$
This intermediate state is the compound nucleus.