KAHIBARO
Table of Contents
Formation and Decay Through Resonant States
In many nuclear reactions, the projectile and target do not always react in a smooth, gradual way as the projectile energy changes. Instead, at certain special energies, the probability of reaction can rise sharply. These sharp increases are called resonances.
A resonance occurs when the incoming particles temporarily form a compound nucleus in an excited state whose energy matches the total energy available in the collision. This intermediate state is not stable. It exists only for a very short time, then decays into one of the possible exit channels.
The basic picture is
$$
a + A \rightarrow C^* \rightarrow b + B
$$
where $a$ is the projectile, $A$ is the target nucleus, $C^*$ is the excited compound nucleus, and $b + B$ are the final products.
At resonance, the reaction becomes much more likely because the system is able to occupy a real nuclear energy level of the compound nucleus.
Energy Matching
Nuclei have discrete energy levels. If the combined energy of the projectile and target corresponds closely to one of these levels, the compound nucleus can be formed very efficiently.
Suppose the compound nucleus has an excited level at energy $E_R$. If the center of mass energy of the incoming particles is near $E_R$, the reaction cross section becomes large. This is the essential condition for resonance.
A resonance appears when the incident energy matches, or nearly matches, an excited energy level of the compound nucleus.
Mathematically, resonance occurs near
$$
E \approx E_R
$$
where $E$ is the collision energy and $E_R$ is the resonance energy.
This does not mean the reaction only happens at one exact energy. Because the compound state has a finite lifetime, the resonance has a finite width in energy.
Resonance Width and Lifetime
A resonant state is short lived. The shorter its lifetime, the less precisely its energy is defined. This produces an energy spread called the resonance width, usually written as $\Gamma$.
The relation between lifetime $\tau$ and width $\Gamma$ is
$$
\Gamma \approx \frac{\hbar}{\tau}
$$
A very short lifetime means a large width. A longer lifetime means a narrower resonance.
The lifetime width relation is
$$
\Gamma \approx \frac{\hbar}{\tau}
$$
Narrow resonance, long lifetime.
Broad resonance, short lifetime.
This is an important idea because experimental reaction curves often show either narrow sharp peaks or broad smooth humps, depending on the lifetime of the intermediate compound state.
Resonance Shape
The cross section near a resonance usually has a peaked shape. A common ideal form is the Breit-Wigner formula. For a single isolated resonance, the energy dependence is approximately
$$
\sigma(E) \propto \frac{\Gamma_{\text{in}} \Gamma_{\text{out}}}{(E - E_R)^2 + (\Gamma/2)^2}
$$
Here, $\Gamma_{\text{in}}$ describes coupling to the entrance channel, $\Gamma_{\text{out}}$ describes coupling to the exit channel, and $\Gamma$ is the total width.
The total width is the sum of the partial widths for all allowed decay channels,
$$
\Gamma = \Gamma_1 + \Gamma_2 + \Gamma_3 + \cdots
$$
Each partial width measures how strongly the resonant state decays into one specific channel.
Near an isolated resonance, the cross section has a peak centered at $E_R$.
A simplified resonance form is
$$
\sigma(E) \propto \frac{\Gamma_{\text{in}} \Gamma_{\text{out}}}{(E - E_R)^2 + (\Gamma/2)^2}
$$
The peak is highest near $E = E_R$.
Partial Widths and Decay Channels
A compound nucleus can decay in several ways. It may emit a neutron, proton, alpha particle, or gamma ray, depending on what channels are energetically allowed and consistent with conservation laws.
Each possible decay mode has its own partial width. If a state can decay by neutron emission and gamma emission, then
$$
\Gamma = \Gamma_n + \Gamma_\gamma
$$
If more channels are open, more terms are added.
The branching tendency of the resonance into a given channel depends on the ratio of that channel's partial width to the total width. For example, the fraction of decays into channel $i$ is roughly
$$
\frac{\Gamma_i}{\Gamma}
$$
So a resonance may be formed strongly but still decay only weakly into a channel of interest if that channel has a small partial width.
Resonances in Cross Section Graphs
When experimentalists measure reaction cross section as a function of projectile energy, resonances appear as peaks. These peaks give information about nuclear structure.
From the location of the peak, one finds the resonant energy. From the width of the peak, one estimates the lifetime. From the height and shape, one learns how strongly the entrance and exit channels are coupled to the resonant state.
| Quantity seen in experiment | Physical meaning |
|---|---|
| Peak position | Resonance energy $E_R$ |
| Peak width | Total width $\Gamma$ |
| Narrow peak | Longer-lived state |
| Broad peak | Shorter-lived state |
| Peak strength | Coupling to reaction channels |
Resonant Versus Nonresonant Reactions
Not every nuclear reaction proceeds through a clear resonance. Some reactions vary smoothly with energy and are called nonresonant. In resonant reactions, the intermediate compound nucleus plays a dominant role through a specific excited state.
In practice, measured cross sections may contain both resonant and nonresonant contributions. The resonant part produces strong energy dependent structure, while the nonresonant background changes more slowly.
Angular Momentum and Resonance Conditions
A resonance is not determined by energy alone. The compound state must also have the correct angular momentum and parity to be formed from the entrance channel and to decay into the exit channel.
This means that even if the energy is close to an excited level, the resonance may be weak or absent if the quantum numbers do not match properly. Thus resonances provide detailed information about nuclear states, not just their energies.
Example of the Resonance Idea
Imagine a projectile approaches a target nucleus. If its energy is far from any excited level of the compound nucleus, the reaction probability may be modest. But if the energy is tuned close to an allowed excited state, the system can temporarily form that state, and the reaction probability rises sharply.
This is similar to driving an oscillating system at one of its natural frequencies, where the response becomes much larger. In nuclear physics, the matching is in energy, and the response is a larger reaction cross section.
Visualizing a Resonance Peak
The center of the peak marks the resonance energy $E_R$. The width of the peak indicates $\Gamma$.
Importance of Resonances
Resonances are extremely important in nuclear physics because they reveal the internal energy level structure of nuclei. They also strongly affect reaction rates. In some cases, especially in astrophysical environments, a single narrow resonance can dominate the probability of a nuclear reaction.
By studying resonances, physicists learn about excited nuclear states, lifetimes, decay modes, and the mechanisms by which nuclear reactions proceed.
Resonances are signatures of temporary compound nucleus states.
They are identified by peaks in the reaction cross section, and they provide direct information about
$$
E_R, \quad \Gamma, \quad \tau, \quad \text{and decay channels}
$$
Summary
A resonance is a sharp increase in reaction probability that occurs when the projectile and target form a compound nucleus in an excited state with matching energy. The resonance energy gives the position of the state, and the resonance width gives information about its lifetime through
$$
\Gamma \approx \frac{\hbar}{\tau}
$$
The cross section near resonance has a peaked form, often described by the Breit-Wigner expression. Different exit channels contribute partial widths, and the total width is their sum. Resonances are therefore one of the clearest experimental tools for studying nuclear excited states and compound nucleus behavior.