KAHIBARO
Table of Contents
Magnetic moment as a property of a nucleus
A nucleus can behave like a tiny magnet. This magnetic behavior is called the nuclear magnetic moment. It is connected to the motion and intrinsic angular momentum of the charged particles inside the nucleus, mainly protons, and also to the spin structure of both protons and neutrons.
The idea is similar to a small current loop. A moving electric charge creates a magnetic effect. Inside a nucleus, protons carry charge and move in complicated quantum states. In addition, nucleons have intrinsic spin, and spin also contributes to magnetism. Even though the neutron has no net electric charge, it still has a magnetic moment because it is not a simple pointlike neutral object, it has internal charged constituents.
The nuclear magnetic moment is important because it tells us about nuclear structure. By measuring it, physicists learn how protons and neutrons are arranged and how their angular momenta combine.
Relation to angular momentum
In quantum physics, the magnetic moment of a nucleus is related to its total angular momentum, usually denoted by $\vec I$. The magnetic moment is a vector, usually written as $\vec \mu$.
A simple proportional relation is
$$
\vec \mu = g\, \mu_N \, \frac{\vec I}{\hbar}
$$
where $g$ is the nuclear $g$ factor, $\mu_N$ is the nuclear magneton, and $\hbar$ is the reduced Planck constant.
This equation says that the magnetic moment points along, or sometimes opposite to, the angular momentum direction, depending on the sign of $g$.
The nuclear magnetic moment is proportional to nuclear angular momentum, but the proportionality is not universal. It depends on the nuclear $g$ factor:
$$
\vec \mu = g\, \mu_N \, \frac{\vec I}{\hbar}
$$
Different nuclei have different $g$ factors.
The nuclear magneton
To express nuclear magnetic moments, physicists use a special unit called the nuclear magneton. It is defined as
$$
\mu_N = \frac{e\hbar}{2m_p}
$$
where $e$ is the elementary charge and $m_p$ is the proton mass.
This plays a role for nuclei similar to the Bohr magneton for electrons, but it is much smaller because the proton mass is much larger than the electron mass.
A useful comparison is shown below.
| Quantity | Definition | Typical use |
|---|---|---|
| Bohr magneton | $\mu_B = \dfrac{e\hbar}{2m_e}$ | Atomic and electron magnetism |
| Nuclear magneton | $\mu_N = \dfrac{e\hbar}{2m_p}$ | Nuclear magnetism |
Because $m_p \gg m_e$, we have $\mu_N \ll \mu_B$. This is why nuclear magnetic effects are generally much weaker than electronic magnetic effects.
Sources of the nuclear magnetic moment
The nuclear magnetic moment comes from two main contributions. One is orbital motion of charged nucleons, mainly protons. The other is intrinsic spin of the nucleons.
If a proton moves in an orbit inside the nucleus, its motion contributes to the magnetic moment because it is a charged particle. Both protons and neutrons also have intrinsic spin magnetic moments. The proton has a positive magnetic moment, while the neutron has a nonzero magnetic moment with opposite sign.
So the total nuclear magnetic moment is the quantum sum of many contributions from individual nucleons. Because nuclei are quantum systems, these contributions do not simply add like ordinary arrows. They combine according to angular momentum rules.
Why neutrons contribute
At first this may seem surprising. A neutron has zero net charge, so why should it have a magnetic moment?
The reason is that the neutron is made of charged quarks. Its internal charge distribution and internal motion produce a magnetic moment. Therefore, even a neutral particle can have a nonzero magnetic moment if it has internal structure.
This is one of the reasons nuclear magnetic moments are so useful. They reveal that nucleons are not simple classical particles.
Sign and magnitude
The magnetic moment can be positive or negative. The sign tells us about the direction of the magnetic moment relative to the angular momentum.
If the $g$ factor is positive, the magnetic moment tends to align with the angular momentum. If the $g$ factor is negative, it tends to point in the opposite direction.
The magnitude depends on how the nucleons are arranged. Nuclei with different numbers of protons and neutrons can have very different magnetic moments, even if their total spin is similar.
A nucleus with zero total spin usually has zero magnetic moment.
If
$$
I = 0
$$
then typically
$$
\mu = 0
$$
because there is no net angular momentum to generate a permanent nuclear magnetic moment.
Magnetic moment in an external magnetic field
When a nucleus is placed in an external magnetic field $\vec B$, its magnetic moment interacts with that field. The energy of interaction is
$$
U = -\vec \mu \cdot \vec B
$$
This means that different orientations of the nuclear magnetic moment can have different energies. In quantum mechanics, only certain orientations are allowed. So a magnetic field can split a nuclear energy level into several closely spaced levels.
This effect is the basis of magnetic resonance methods and many spectroscopic techniques.
The interaction energy of a nuclear magnetic moment with a magnetic field is
$$
U = -\vec \mu \cdot \vec B
$$
A magnetic field can therefore split nuclear states into different energy levels.
Quantized orientations
If the nucleus has spin quantum number $I$, then the component of angular momentum along a chosen axis can take the values
$$
I_z = m_I \hbar
$$
where
$$
m_I = -I, -I+1, \dots, I-1, I
$$
Because the magnetic moment is related to angular momentum, the magnetic moment component along the field direction is also quantized. This leads to a set of allowed magnetic energies in a field.
For a field along the $z$ axis, the energy becomes
$$
U = -\mu_z B
$$
and with the proportionality between $\mu_z$ and $I_z$, the allowed energies are discrete.
Example of level splitting
A nucleus with spin $I = \tfrac{1}{2}$ has two allowed values of $m_I$:
$$
m_I = +\frac{1}{2}, \, -\frac{1}{2}
$$
So in an external magnetic field it has two possible magnetic energy states. This is the simplest and most important case in many applications.
A nucleus with spin $I = 1$ has three allowed orientations, so it has three magnetic sublevels in a field.
Simple shell model picture
A basic nuclear shell model gives a first estimate of magnetic moments. In many nuclei, paired nucleons tend to cancel each other's contributions. Then the main contribution often comes from one unpaired nucleon.
If the unpaired particle is a proton, the magnetic moment is often quite different from the case of an unpaired neutron. This is because the proton is charged and the neutron is neutral, and because their intrinsic spin magnetic moments are different.
This simple picture is useful, but real nuclei often deviate from it because nucleons interact strongly and collective effects can appear.
Measured moments and nuclear structure
Measured nuclear magnetic moments are compared with theoretical predictions. Good agreement supports a model of nuclear structure. Disagreement often signals that the nucleus is more complicated than the simplest picture suggests.
For example, measured moments can reveal whether a nucleus behaves like a single unpaired nucleon outside a closed shell, or whether many nucleons contribute collectively.
In this way, the nuclear magnetic moment is a powerful experimental probe of the nucleus.
Typical examples
Some well known nuclei have the following general behavior.
| Nucleus | Spin $I$ | Magnetic moment behavior |
|---|---|---|
| $^1\mathrm{H}$, proton nucleus | $\tfrac{1}{2}$ | Large positive magnetic moment |
| Neutron | $\tfrac{1}{2}$ | Nonzero negative magnetic moment |
| Even-even nuclei | often $0$ | Usually zero magnetic moment |
| Nuclei with one unpaired nucleon | nonzero | Often have measurable magnetic moments |
The exact numerical values are important in advanced nuclear physics, but for a beginner the key point is that the moment depends strongly on nuclear composition and quantum structure.
Connection to resonance methods
A nucleus with a magnetic moment in a magnetic field can absorb electromagnetic radiation of the right frequency and change from one magnetic sublevel to another. This is the basis of nuclear magnetic resonance, often called NMR.
The required energy difference satisfies
$$
\Delta E = h f
$$
where $f$ is the radiation frequency and $h$ is Planck's constant.
Because the magnetic splitting depends on the magnetic moment, resonance measurements allow very precise determination of nuclear magnetic properties.
Magnetic resonance occurs when the radiation frequency matches the energy splitting:
$$
\Delta E = h f
$$
This makes nuclear magnetic moments measurable with high precision.
Visual picture
A simple picture is to imagine the nucleus as a tiny spinning magnet placed in an external field. The field tries to orient the magnetic moment, but quantum mechanics allows only certain orientations.
Main ideas to remember
The nuclear magnetic moment is the magnetic character of a nucleus arising from nucleon spin and proton orbital motion. It is proportional to the nuclear angular momentum through a nucleus-dependent $g$ factor. It is measured in units of the nuclear magneton,
$$
\mu_N = \frac{e\hbar}{2m_p}
$$
and it interacts with a magnetic field through
$$
U = -\vec \mu \cdot \vec B
$$
This interaction causes energy splitting and makes techniques such as nuclear magnetic resonance possible. Most importantly, the value of the nuclear magnetic moment provides direct information about nuclear structure.
Key formulas for nuclear magnetic moment:
$$
\vec \mu = g\, \mu_N \, \frac{\vec I}{\hbar}
$$
$$
\mu_N = \frac{e\hbar}{2m_p}
$$
$$
U = -\vec \mu \cdot \vec B
$$
$$
\Delta E = h f
$$
These relations connect nuclear structure, magnetism, and spectroscopy.