KAHIBARO
Table of Contents
Energy Levels in the Nucleus
A nucleus does not always stay in its lowest possible energy state. Just as atoms can exist in excited states, nuclei can also have higher energy configurations. These are called nuclear excited states. In an excited state, the nucleus has more internal energy than it has in its ground state, which is the lowest energy state.
An excited nucleus is usually unstable and tends to move to a lower energy state. When it does, it often releases the extra energy as gamma radiation, which belongs to the next chapter. Here, the focus is on the excited states themselves, what they mean, and how they arise.
Ground State and Excited States
The ground state of a nucleus is its state of minimum energy. Any state with greater energy is an excited state. If the energy of the ground state is written as $E_0$, then an excited state has energy
$$
E_n > E_0
$$
The excitation energy is the difference between the energy of the excited state and the ground state:
$$
E_{\text{exc}} = E_n - E_0
$$
This excitation energy is often measured in electronvolts, usually kilo electronvolts, keV, or mega electronvolts, MeV.
A nuclear excited state is not a different nucleus. The number of protons and neutrons stays the same. Only the internal energy arrangement changes.
Why Excited States Exist
Protons and neutrons inside the nucleus do not have arbitrary energies. They occupy quantized energy levels. This means that only certain discrete energies are allowed. A nucleus can move from one allowed arrangement to another, but it cannot take any energy value in between.
This quantization is a result of quantum physics and of the forces acting inside the nucleus. The strong nuclear force binds the nucleons together, and the possible collective and individual motions of these nucleons produce a set of allowed nuclear states.
A simple way to think about it is that the nucleus has an internal structure, and that structure can be arranged in several specific energy patterns.
How Nuclei Become Excited
A nucleus can be excited in several ways. It may absorb energy during a collision with another particle, during a nuclear reaction, or after a radioactive decay that leaves the daughter nucleus in a higher energy state. It can also be excited when one of its nucleons is promoted to a higher allowed level.
For example, if energy $Q$ is transferred to a nucleus, and this energy matches an allowed excitation energy, the nucleus can move to an excited state:
$$
E_{\text{final}} = E_0 + E_{\text{exc}}
$$
Not every energy transfer produces the same excited state. The final state depends on how much energy is transferred and on quantum rules involving angular momentum and parity.
Discrete Level Structure
Nuclear excited states form a level scheme. Each state has a definite energy. These levels are often shown as horizontal lines, with the ground state at the bottom and excited states above it.
The spacing between levels is not usually regular. Unlike a simple harmonic oscillator, real nuclei often have complicated level patterns. Some levels are close together, while others are far apart.
Excitation Energy and Transitions
If a nucleus moves from a higher state $E_i$ to a lower state $E_f$, the energy difference is
$$
\Delta E = E_i - E_f
$$
This difference is the amount of energy that must be carried away. In many cases, this energy later appears as a gamma photon, though the mechanism of emission belongs to the next chapter.
The key point here is that the transition energy is determined by the difference between two discrete nuclear levels.
Because nuclear energy levels are quantized, the energy released in a transition is also quantized:
$$
\Delta E = E_i - E_f
$$
This is why nuclei produce specific, well defined transition energies.
Spin and Parity of Nuclear States
Each nuclear state is characterized not only by its energy, but also by quantum numbers. Two especially important ones are spin and parity.
The nuclear spin, usually written as $J$, is the total angular momentum of the nucleus. Parity, written as $\pi$, describes how the nuclear wave function behaves under spatial inversion. A state is often labeled as
$$
J^\pi
$$
For example, a state may be written as $0^+$, $2^+$, or $3^-$.
These labels help distinguish different excited states even if their energies are similar. They also determine which transitions between states are allowed or more likely.
Types of Nuclear Excitation
Nuclear excited states can arise from different kinds of internal motion. For beginners, it is useful to separate them into a few broad categories.
| Type of excitation | Basic idea |
|---|---|
| Single-particle excitation | One nucleon moves to a higher allowed level |
| Collective vibrational excitation | The whole nucleus changes shape slightly and oscillates |
| Collective rotational excitation | A non spherical nucleus rotates as a whole |
| More complex excitation | Several nucleons and motions are involved together |
In lighter or closed shell nuclei, single-particle behavior may be especially important. In other nuclei, collective effects can dominate.
Vibrational and Rotational States
Some nuclei behave approximately like vibrating drops of nuclear matter. In such nuclei, excited states can correspond to shape oscillations. Other nuclei are not perfectly spherical and can have rotational excited states, where the nucleus rotates with quantized angular momentum.
These patterns are important because they create recognizable sequences of energy levels. A rotational band, for example, often contains states with increasing spin. A vibrational nucleus may show low lying states associated with one quantum of vibration.
The detailed theory belongs to nuclear models, but the main idea is that excited states are often signatures of the internal shape and motion of the nucleus.
Metastable Excited States
Most excited nuclear states decay very quickly. However, some excited states have unusually long lifetimes. These are called metastable states, or nuclear isomers.
A metastable state is still an excited state, but it remains excited long enough to be identified separately. Such a state is often marked with an "m" after the nuclide symbol, such as technetium 99m, written as ${}^{99m}\mathrm{Tc}$.
These long lifetimes occur because the transition to a lower state may be strongly hindered by quantum rules, such as a large change in angular momentum.
A metastable state is an excited nucleus with an unusually long lifetime. It is not a different chemical element, and it has the same numbers of protons and neutrons as the ground state nucleus.
Lifetime of Excited States
An excited state does not have a fixed decay time for each individual nucleus. Nuclear decay is probabilistic. Instead, one speaks of an average lifetime $\tau$ or a half life if the state is long lived enough.
Very short lived excited states may exist for times such as $10^{-12}\,\text{s}$ or less. Metastable states can last much longer, from microseconds to years in special cases.
A shorter lifetime usually means the state decays more readily. A longer lifetime means the transition is less probable.
Energy Scale of Nuclear Excited States
Nuclear excitation energies are much larger than many atomic excitation energies. Atomic excited states are often in the electronvolt range, while nuclear excited states are commonly in the keV to MeV range.
| System | Typical excitation energy |
|---|---|
| Atomic electrons | eV |
| Nucleus | keV to MeV |
This large energy scale reflects the much stronger binding and much smaller size of the nucleus compared with the atom.
Example of Level Differences
Suppose a nucleus has a ground state at $0\,\text{keV}$ and an excited state at $250\,\text{keV}$. The excitation energy is
$$
E_{\text{exc}} = 250\,\text{keV}
$$
If there is another state at $100\,\text{keV}$, then a transition from $250\,\text{keV}$ to $100\,\text{keV}$ has energy
$$
\Delta E = 250\,\text{keV} - 100\,\text{keV} = 150\,\text{keV}
$$
A later transition from $100\,\text{keV}$ to the ground state would have energy
$$
\Delta E = 100\,\text{keV}
$$
This shows that an excited nucleus can decay in steps through intermediate levels.
Relation to Nuclear Identity
It is important to distinguish an excited state from a change in nuclear composition. If a nucleus changes its number of protons or neutrons, it becomes a different nuclide. In an excited state, the nuclide remains the same, but its internal energy is higher.
For example, a daughter nucleus formed after alpha or beta decay may be created in an excited state. The identity of the daughter nucleus is already fixed by the decay. The excited state simply means that the daughter is not yet in its ground state.
Spectroscopic View
Because each nucleus has its own set of allowed excited states, the pattern of its level energies acts like a fingerprint. By measuring the energies associated with transitions between states, physicists can infer the internal level structure of the nucleus.
This is one of the foundations of nuclear spectroscopy. The observed transition energies reveal which excited states exist and how they are connected.
Main Ideas to Remember
Nuclear excited states are higher energy configurations of a nucleus with the same numbers of protons and neutrons as the ground state nucleus. These states are quantized, so only certain excitation energies are allowed. Each state is characterized by energy and quantum numbers such as spin and parity. Excited nuclei usually move to lower energy states, often through one or more transitions, and some special excited states are metastable and can live much longer than ordinary excited states.
Key facts:
$$
E_{\text{exc}} = E_n - E_0
$$
$$
\Delta E = E_i - E_f
$$
Nuclear excited states are discrete, quantized, and belong to the same nuclide as the ground state.