KAHIBARO
Table of Contents
Concept and Basic Idea
A transfer reaction is a nuclear reaction in which one or more nucleons are moved from one nucleus to another during the collision. Unlike simple elastic scattering, where the nuclei only change direction, or capture reactions, where a particle is fully absorbed, transfer reactions redistribute nuclear constituents between the reacting partners.
A common symbolic form is
$$
a + A \rightarrow b + B
$$
where the projectile $a$ strikes the target nucleus $A$, and after the reaction the outgoing particle is $b$ and the residual nucleus is $B$. In a transfer reaction, the difference between $a$ and $b$, or between $A$ and $B$, shows what was transferred.
For example, in
$$
d + A \rightarrow p + (A+n)
$$
a deuteron $d$, which contains a proton and a neutron, hits the target. The outgoing proton $p$ leaves, and the neutron is transferred to the target. This is called a neutron stripping reaction.
In
$$
d + A \rightarrow n + (A+p)
$$
the proton is transferred to the target and the neutron exits.
A transfer reaction is identified by a change in nuclear composition caused by the movement of one or more nucleons from the projectile to the target, or from the target to the projectile.
How Transfer Reactions Differ from Other Reactions
The main feature of a transfer reaction is exchange of nuclear particles between the two nuclei. This makes it different from other reaction types.
| Reaction type | Main feature | Example idea |
|---|---|---|
| Elastic scattering | No internal change | Projectile and target remain the same |
| Inelastic scattering | Internal excitation only | Same nuclei, but one is excited |
| Capture reaction | Projectile absorbed | Target gains the whole projectile or a particle |
| Transfer reaction | One or more nucleons exchanged | Projectile loses or gains nucleons |
| Charge-exchange reaction | Proton and neutron roles change | Nuclear charge changes without large mass transfer |
A transfer reaction often leaves both nuclei still present after the interaction, but with different identities than before.
Common Types of Transfer Reactions
Transfer reactions are often classified by what is transferred.
One-Nucleon Transfer
The most common case is transfer of a single neutron or single proton.
Examples are written using shorthand notation such as:
$$
(d,p)
$$
This means the incoming particle is a deuteron and the outgoing particle is a proton. Since a deuteron contains $p+n$, the missing neutron has been transferred to the target.
Similarly,
$$
(d,n)
$$
means proton transfer.
Other common reactions are
$$
(p,d)
$$
and
$$
(^3\mathrm{He},d)
$$
depending on whether the target loses or gains a nucleon.
Two-Nucleon Transfer
Sometimes a pair of nucleons is transferred together, such as two neutrons or a proton-neutron pair. These reactions are useful because paired nucleons inside nuclei often behave in a correlated way.
An example is
$$
(t,p)
$$
where a triton $t$, containing one proton and two neutrons, transfers two neutrons to the target and emits a proton.
Stripping and Pickup Reactions
Transfer reactions are also described by the direction of transfer.
If the projectile gives a nucleon to the target, the process is called a stripping reaction. The projectile is stripped of one of its constituents.
Example:
$$
A(d,p)B
$$
Here the deuteron gives a neutron to the target.
If the projectile takes a nucleon from the target, the process is called a pickup reaction.
Example:
$$
A(p,d)B
$$
Here the incoming proton picks up a neutron from the target and leaves as a deuteron.
Stripping means the projectile loses a nucleon to the target. Pickup means the projectile gains a nucleon from the target.
Reaction Notation
Transfer reactions are commonly written in compact notation:
$$
A(a,b)B
$$
This means target $A$ is bombarded by projectile $a$, outgoing particle $b$ is detected, and residual nucleus $B$ is left behind.
For a transfer reaction, the transferred particle or cluster is inferred from the difference between $a$ and $b$, and between $A$ and $B$.
For example:
$$
{}^{40}\mathrm{Ca}(d,p){}^{41}\mathrm{Ca}
$$
The target gains one neutron. The deuteron becomes a proton, so the neutron has been transferred.
Another example:
$$
{}^{16}\mathrm{O}(p,d){}^{15}\mathrm{O}
$$
The target loses one neutron, which is picked up by the proton to form the outgoing deuteron.
Conservation in Transfer Reactions
As in all nuclear reactions, conservation laws must hold. In transfer reactions this means conservation of charge, nucleon number, energy, and momentum.
Consider
$$
{}^{A}_{Z}X + d \rightarrow {}^{A+1}_{Z}X + p
$$
The neutron number increases by one in the target nucleus, but the total number of nucleons and total charge remain unchanged.
The reaction energy is described by the $Q$ value:
$$
Q = \left(m_a + m_A - m_b - m_B\right)c^2
$$
A positive $Q$ value means energy is released, and a negative $Q$ value means energy must be supplied.
For any transfer reaction, always check conservation of mass number and atomic number:
$$
A_a + A_A = A_b + A_B
$$
and
$$
Z_a + Z_A = Z_b + Z_B
$$
Why Transfer Reactions Are Important
Transfer reactions are especially valuable in nuclear structure studies. Because only one or a few nucleons are exchanged, they can probe specific nuclear states.
When a nucleon is transferred into or out of a nucleus, the final nucleus may be left in a particular energy level. By measuring the energy and angle of the outgoing particle, physicists learn about those levels and about how nucleons are arranged inside the nucleus.
Transfer reactions help study:
| What is studied | Why transfer reactions help |
|---|---|
| Single-particle energy levels | A transferred nucleon can populate specific nuclear orbitals |
| Spin and parity of nuclear states | Angular patterns give structural clues |
| Shell structure | Reactions reveal which orbitals are occupied |
| Pairing effects | Two-nucleon transfer is sensitive to nucleon pairing |
Angular Distributions
One of the most useful observables in transfer reactions is the angular distribution of the outgoing particle. The number of detected particles often depends strongly on the emission angle.
This pattern carries information about the orbital angular momentum transferred in the reaction. Different transferred angular momenta produce different angular shapes. Because of this, transfer reactions are a powerful spectroscopic tool.
A beginner does not need the full theory here, but the key idea is simple. The way particles emerge from the reaction is not random. Their angular pattern reflects the internal quantum state involved in the transfer.
Energy and Threshold Considerations
A transfer reaction can occur only if the projectile has enough energy to overcome the interaction requirements. Even if the reaction is energetically allowed, its probability may depend strongly on incident energy.
At low to moderate energies, transfer reactions are often quite selective. This is useful in experiments because specific states can be populated more clearly than in more violent reactions.
The measured outgoing-particle energy helps identify which final nuclear state was produced. If the residual nucleus is left excited, some of the available energy goes into excitation energy, so the outgoing particle has less kinetic energy.
Example of a Neutron Transfer Reaction
Consider the reaction
$$
{}^{12}\mathrm{C}(d,p){}^{13}\mathrm{C}
$$
The incoming deuteron contains one proton and one neutron. After the collision, the proton leaves and the neutron remains with the carbon nucleus. So the target has gained a neutron.
This can be visualized as:
$$
{}^{12}\mathrm{C} + (p+n) \rightarrow {}^{13}\mathrm{C} + p
$$
The final nucleus may be produced in its ground state or in an excited state. By measuring the proton energy, one can tell which state of ${}^{13}\mathrm{C}$ was formed.
Simple Reaction Sketch
Interpreting the Symbols
It is often helpful to compare the projectile and ejectile directly.
| Reaction notation | Interpretation |
|---|---|
| $(d,p)$ | Deuteron enters, proton exits, neutron transferred to target |
| $(d,n)$ | Deuteron enters, neutron exits, proton transferred to target |
| $(p,d)$ | Proton enters, deuteron exits, neutron picked up from target |
| $(^3\mathrm{He},d)$ | Helium-3 enters, deuteron exits, proton transferred |
| $(t,p)$ | Triton enters, proton exits, two neutrons transferred |
This shorthand is widely used because it immediately tells the experimenter what kind of transfer has occurred.
Physical Picture
A useful mental picture is that the projectile comes close enough to the target for nuclear forces to act strongly. During this close encounter, one constituent may move from one nucleus to the other. After that exchange, the products separate.
This is not like a chemical transfer. It is a quantum process governed by nuclear wave behavior, conservation laws, and interaction probabilities. Still, the exchange picture is a good starting intuition.
Key Ideas to Remember
Transfer reactions are nuclear reactions in which nucleons are exchanged between projectile and target.
Typical notation is $A(a,b)B$.
Examples such as $(d,p)$ and $(p,d)$ are especially important for studying nuclear structure.
These reactions are powerful because they can populate specific nuclear states and reveal information about nuclear shells and energy levels.