KAHIBARO
Table of Contents
Angular Momentum Inside the Nucleus
Nuclear angular momentum describes the total angular momentum carried by a nucleus. In nuclear physics, this quantity is very important because it helps classify nuclear states, predict transitions between states, and understand how protons and neutrons arrange themselves inside the nucleus.
A nucleus is made of protons and neutrons, which are collectively called nucleons. Each nucleon can contribute angular momentum in two ways. One part comes from its motion inside the nucleus, called orbital angular momentum. Another part comes from its intrinsic spin, which is a built-in quantum property. The total nuclear angular momentum is formed by combining these contributions from all nucleons.
This chapter focuses on the idea of angular momentum as a property of the whole nucleus. The specific meanings of nuclear spin and nuclear magnetic moment are treated in their own chapters.
Orbital and Intrinsic Contributions
In classical mechanics, angular momentum is related to rotational motion and is written as
$$
\vec{L} = \vec{r} \times \vec{p}
$$
where $\vec{r}$ is position and $\vec{p}$ is linear momentum. In nuclei, however, angular momentum must be described using quantum mechanics. The motion of a nucleon around the center of the nucleus contributes orbital angular momentum, usually denoted by $\vec{L}$. In addition, each nucleon has intrinsic spin, denoted by $\vec{S}$.
The total angular momentum of a nucleon is
$$
\vec{j} = \vec{l} + \vec{s}
$$
where $l$ is the orbital angular momentum quantum number and $s$ is the spin quantum number. For protons and neutrons,
$$
s = \frac{1}{2}
$$
The total angular momentum of the entire nucleus is then obtained by combining the angular momenta of all nucleons.
Important quantum rule:
For a nucleon,
$$
\vec{j} = \vec{l} + \vec{s}, \qquad s = \frac{1}{2}
$$
For the whole nucleus, the total angular momentum is the vector sum of all individual nucleon angular momenta.
Quantization of Nuclear Angular Momentum
Unlike classical angular momentum, nuclear angular momentum is quantized. This means it can only take certain allowed values.
If a nucleus has total angular momentum quantum number $J$, then the magnitude of its angular momentum is
$$
|\vec{J}| = \sqrt{J(J+1)}\,\hbar
$$
and one component, usually taken along the $z$ axis, is
$$
J_z = m_J \hbar
$$
where
$$
m_J = -J, -J+1, \dots, J-1, J
$$
So a state with angular momentum $J$ has $2J+1$ possible orientations.
This is one of the key features of nuclear states. A nucleus is not free to have any angular momentum it wants. Only specific values are allowed.
For a nuclear state with angular momentum quantum number $J$,
$$
|\vec{J}| = \sqrt{J(J+1)}\,\hbar
$$
and
$$
J_z = m_J \hbar, \qquad m_J = -J, \dots, J
$$
The number of allowed magnetic substates is
$$
2J+1
$$
Integer and Half-Integer Values
The total nuclear angular momentum quantum number $J$ may be integer or half-integer. Which kind appears depends on the total number of nucleons.
A nucleus contains protons and neutrons, and each nucleon has spin $\frac{1}{2}$. When many such angular momenta are combined, the result depends on whether the nucleus has an even or odd number of nucleons.
In general, nuclei with even mass number $A$ often have integer total angular momentum, while nuclei with odd mass number often have half-integer total angular momentum. In particular, many even-even nuclei, meaning nuclei with even numbers of protons and even numbers of neutrons, have ground state angular momentum
$$
J = 0
$$
This happens because nucleons tend to pair off in such a way that their angular momenta cancel.
The following table shows common patterns.
| Type of nucleus | Typical ground state angular momentum |
|---|---|
| Even $Z$, even $N$ | Often $J = 0$ |
| Odd $A$ | Often half-integer $J$ |
| Odd $Z$, odd $N$ | Often nonzero integer $J$ |
These are common trends, not absolute rules for every excited state.
Coupling of Angular Momenta
When two angular momenta are combined in quantum mechanics, the possible total values follow a definite rule. If two angular momenta $j_1$ and $j_2$ are added, then the total angular momentum $J$ can take values
$$
J = |j_1 - j_2|, |j_1 - j_2| + 1, \dots, j_1 + j_2
$$
This rule is used repeatedly in nuclear physics. A nucleus is built from many nucleons, so angular momentum coupling becomes a central tool.
For example, if one nucleon has $j_1 = \frac{3}{2}$ and another has $j_2 = \frac{1}{2}$, then the total can be
$$
J = 1 \text{ or } 2
$$
This does not mean the nucleus always has both values at once. Rather, these are the allowed total angular momentum states that can result from that coupling.
Angular momentum addition rule:
If two angular momenta $j_1$ and $j_2$ combine, then
$$
J = |j_1 - j_2|, |j_1 - j_2| + 1, \dots, j_1 + j_2
$$
Pairing and Cancellation
One of the most important ideas in nuclei is pairing. Nucleons often form pairs in which their angular momenta point in opposite ways, giving nearly zero net contribution.
This is why even-even nuclei often have $J = 0$ in their ground states. If all nucleons are paired, the total angular momentum cancels out. In contrast, if there is one unpaired nucleon, that nucleon often determines the angular momentum of the whole nucleus.
This idea is especially useful in the shell model. Although the shell model itself belongs to another chapter, one simple consequence is very important here. Closed shells and paired nucleons tend to give low total angular momentum, while unpaired nucleons give the dominant contribution.
Ground States and Excited States
A nucleus can exist in different energy states. Each state has its own angular momentum quantum number $J$. The ground state is the lowest-energy state, and excited states are higher-energy states.
It is common to label nuclear states by their angular momentum. For example, a nucleus may have a ground state with $J = 0$ and excited states with $J = 2$, $J = 4$, and so on. In another nucleus, the ground state may be $J = \frac{1}{2}$ or $J = \frac{3}{2}$.
Thus, angular momentum is not just a property of the nucleus as a permanent object. It is a property of a particular nuclear state.
Spectroscopic Notation
Nuclear states are often labeled by $J$, and frequently by parity as well, though parity belongs more fully to other discussions. A notation such as
$$
J^\pi = 0^+, \quad \frac{1}{2}^-, \quad 2^+
$$
is often used to identify nuclear states. In this notation, $J$ gives the total angular momentum quantum number.
For this chapter, the important point is that $J$ is the standard label for nuclear angular momentum.
Physical Interpretation
It is tempting to imagine the nucleus as a tiny rigid ball spinning like a planet or a top. That picture is not correct in a literal classical sense. Nuclear angular momentum is a quantum quantity. It is connected to internal motion and intrinsic particle properties, not simply to visible rotation of a solid object.
Still, the idea of angular momentum remains useful because it controls many observable effects. It influences the structure of energy levels and the allowed changes between one nuclear state and another.
Simple Example of Angular Momentum Coupling
Suppose a nucleus has all nucleons paired except one. Let that unpaired nucleon have orbital angular momentum
$$
l = 1
$$
and spin
$$
s = \frac{1}{2}
$$
Then its total angular momentum can be
$$
j = l \pm s = 1 \pm \frac{1}{2}
$$
So the allowed values are
$$
j = \frac{1}{2}, \frac{3}{2}
$$
If this nucleon dominates the nuclear state, then the nucleus may have total angular momentum $J = \frac{1}{2}$ or $J = \frac{3}{2}$, depending on which state is occupied.
This simple example shows how orbital motion and intrinsic spin combine to give nuclear angular momentum.
Visualizing Quantized Angular Momentum
The magnitude of angular momentum is fixed by $J$, but its projection along a chosen axis can take only certain values. This is often shown as a set of discrete allowed projections.
For $J = 2$, there are five possible projections, corresponding to $m_J = -2, -1, 0, 1, 2$.
Summary Relations
The essential mathematical relations for nuclear angular momentum are gathered here.
| Quantity | Formula | ||
|---|---|---|---|
| Total nucleon angular momentum | $\vec{j} = \vec{l} + \vec{s}$ | ||
| Nucleon spin | $s = \frac{1}{2}$ | ||
| Nuclear angular momentum magnitude | $ | \vec{J} | = \sqrt{J(J+1)}\,\hbar$ |
| Projection on chosen axis | $J_z = m_J \hbar$ | ||
| Allowed $m_J$ values | $m_J = -J, \dots, J$ | ||
| Number of substates | $2J+1$ | ||
| Addition of two angular momenta | $J = | j_1-j_2 | , \dots, j_1+j_2$ |
Key ideas to remember:
Nuclear angular momentum is a quantum quantity.
It comes from orbital motion and intrinsic spin of nucleons.
It is quantized, and a nuclear state is labeled by $J$.
Paired nucleons often cancel each other's contributions, so unpaired nucleons often determine the total angular momentum.
Final Perspective
Nuclear angular momentum is one of the main labels used to describe nuclear states. It arises from combining the orbital and spin angular momenta of the nucleons. Because nuclei are quantum systems, angular momentum is quantized and follows strict addition rules. In many nuclei, pairing causes large cancellations, so the total angular momentum may be small even though many nucleons are present.
Understanding this quantity prepares the way for studying nuclear spin, magnetic moments, and the structure of nuclear energy levels.