KAHIBARO
Table of Contents
Angular momentum inside the nucleus
Nuclear spin is the intrinsic angular momentum of a nucleus. It is a quantum property, not a picture of the nucleus as a tiny hard ball literally spinning like a planet or a top. The nucleus is made of protons and neutrons, and each of these particles already has its own intrinsic spin. The total nuclear spin comes from combining the intrinsic spins of the nucleons with their orbital angular motion inside the nucleus.
The symbol usually used for nuclear spin is $I$. Like all quantum angular momentum, it comes in specific allowed values, such as
$$
I = 0,\ \frac{1}{2},\ 1,\ \frac{3}{2},\ 2,\dots
$$
The magnitude of the nuclear angular momentum is
$$
|\mathbf{I}| = \sqrt{I(I+1)}\,\hbar
$$
where $\hbar$ is the reduced Planck constant.
Nuclear spin is a quantum angular momentum. It is not a classical spinning sphere.
Its magnitude is
$$
|\mathbf{I}| = \sqrt{I(I+1)}\,\hbar
$$
and its component along a chosen axis can take only the values
$$
I_z = m_I \hbar
$$
with
$$
m_I = -I, -I+1, \dots, I-1, I
$$
Quantization of nuclear spin
A nucleus does not have any arbitrary orientation of angular momentum when measured along a chosen axis. If we choose the $z$ axis, the measurable component is
$$
I_z = m_I \hbar
$$
where $m_I$ is the magnetic quantum number of the nucleus. For a given spin $I$, there are
$$
2I+1
$$
possible values of $m_I$.
For example, if $I = 1$, then
$$
m_I = -1, 0, +1
$$
so there are three allowed orientations relative to the chosen axis. If $I = \frac{1}{2}$, then there are two possible values,
$$
m_I = -\frac{1}{2}, +\frac{1}{2}
$$
This quantization becomes very important when the nucleus is placed in a magnetic field, because the different orientations can have different energies.
Where nuclear spin comes from
The total spin of a nucleus comes from adding several angular momenta. Each proton and neutron contributes intrinsic spin, and each may also have orbital angular momentum because it moves within the nucleus. The nuclear spin is the total result of these contributions.
In simple terms, nuclear spin depends on how the nucleons pair up. Nucleons often prefer to form pairs with opposite angular momenta, so their contributions cancel. Because of this, many nuclei have small total spin, and some have zero spin.
A useful pattern is this:
| Type of nucleus | Typical spin behavior |
|---|---|
| Even number of protons, even number of neutrons | Often $I = 0$ |
| Odd mass number nuclei | Often half integer spin |
| Odd number of protons and odd number of neutrons | Often integer spin |
This pattern is not a complete rule for predicting every nucleus, but it is a very useful guide.
A common rule of thumb is:
Even-even nuclei often have
$$
I = 0
$$
because nucleon angular momenta tend to pair and cancel.
Integer and half integer spin
Nuclear spin can be either integer or half integer. Which one occurs depends on the total number of nucleons, that is, protons plus neutrons.
If the mass number $A$ is even, the nucleus has an even total number of fermions, and the nuclear spin is usually an integer. If $A$ is odd, the nucleus has an odd total number of fermions, and the nuclear spin is half integer.
Examples help make this clear.
| Nucleus | Protons | Neutrons | Mass number $A$ | Typical spin type |
|---|---|---|---|---|
| ${}^{12}\mathrm{C}$ | 6 | 6 | 12 | integer, in fact $I=0$ |
| ${}^{14}\mathrm{N}$ | 7 | 7 | 14 | integer |
| ${}^{1}\mathrm{H}$ | 1 | 0 | 1 | half integer, $I=\frac{1}{2}$ |
| ${}^{13}\mathrm{C}$ | 6 | 7 | 13 | half integer |
Examples of nuclear spin values
Different nuclei have different spin values. A few important examples are listed below.
| Nucleus | Nuclear spin $I$ |
|---|---|
| ${}^{1}\mathrm{H}$ | $\frac{1}{2}$ |
| ${}^{2}\mathrm{H}$, deuterium | $1$ |
| ${}^{4}\mathrm{He}$ | $0$ |
| ${}^{12}\mathrm{C}$ | $0$ |
| ${}^{13}\mathrm{C}$ | $\frac{1}{2}$ |
| ${}^{14}\mathrm{N}$ | $1$ |
These values are measured experimentally and are essential in nuclear physics, spectroscopy, and magnetic resonance techniques.
Spin states and degeneracy
When no external magnetic field is present, all the different $m_I$ states for a given $I$ usually have the same energy. This is called degeneracy. For example, a nucleus with $I = \frac{3}{2}$ has four allowed values of $m_I$,
$$
m_I = -\frac{3}{2}, -\frac{1}{2}, +\frac{1}{2}, +\frac{3}{2}
$$
and in the absence of a field these four spin orientations are generally equal in energy.
The number of possible spin states is
$$
2I+1
$$
so a spin zero nucleus has only one state, while a spin one nucleus has three states.
For a nucleus with spin $I$, the number of allowed spin projections is
$$
2I+1
$$
This is one of the most important counting rules in nuclear spin physics.
Why nuclear spin matters
Nuclear spin affects how nuclei behave in magnetic fields and how they interact with radiation and with other particles. Nuclei with nonzero spin can have multiple orientation states, and this makes them observable in methods such as nuclear magnetic resonance. Spin also helps determine the angular momentum of nuclear energy levels and influences the selection rules for nuclear transitions.
A nucleus with $I=0$ is much simpler in this respect because it has no different spin projections and no intrinsic angular momentum direction to orient in space.
Simple picture of spin projections
The figure below shows the allowed projected values along a chosen axis for several spin values.
Summary
Nuclear spin is the total intrinsic angular momentum of a nucleus. It is described by the quantum number $I$, which may be integer or half integer. Its magnitude is $\sqrt{I(I+1)}\,\hbar$, and its projection on a chosen axis is $m_I\hbar$, where $m_I$ takes $2I+1$ allowed values. Nuclear spin arises from the combination of proton and neutron spins and orbital angular momenta, with pairing often causing cancellation. This is why many even-even nuclei have spin zero, while odd nuclei often have nonzero spin.