It is useful to model nuclei as spinning bar magnets with an intrinsic magnetic dipole moment, $\boldsymbol{\mu}$, and an angular momentum, $\bf L$. In free space, a magnetic moment, $\boldsymbol{\mu}$, is free to point in any direction. However, if an external magnetic field $\bf B$ is present, $\boldsymbol{\mu}$ will try to align itself with the external field. When we consider the effect of conservation of angular momentum, we find that $\boldsymbol{\mu}$ will not fully align with $\bf B$, but will instead precess about the axis defined by $\bf B$. This behavior is analogous to a spinning top precessing in a gravitational field. In a material containing large numbers of such nuclei, the sum of all the aligned nuclear dipole moments results in a net (bulk) magnetization in the sample. It is the magnitude of these bulk magnetizations, and their behavior over time which is measured in PNMR.

Consider the behavior of a bar magnet with dipole moment $\boldsymbol{\mu}$ in a magnetic field $\bf B$ as shown in Fig. 1. When placed in an external magnetic field $\bf B$, the dipole will feel a torque given by

that will act to align $\boldsymbol{\mu}$ with $\bf B$ and minimize the energy

Figure 1: Spinning bar magnet in an external magnetic field.

However, if we add angular momentum $\bf L$ to the bar magnet, the torque will not align the moment with the field, but will instead causes it to precess about the axis defined by $\bf B$. This is analogous to a spinning top; gravity acts to pull the top down to the table, but the angular momentum keeps the top spinning at a constant angle with respect the vertical. The frequency at which the bar magnet precesses is given by

where we introduce the gyromagnetic ratio,

For a bar magnet, the magnetic moment and angular momentum can be separately adjusted, but in the case of elementary particles, the ratio $\gamma$ is intrinsic and fixed. Note that there is a component of $\boldsymbol{\mu}$ which projects onto the $z$-axis, $\mu_z$, and which does not vary as the dipole precesses about the z-axis. Additionally, there is a component projecting into the $xy$-plane,$\mu_{xy}$, which does precess about the $z$-axis with an angular velocity $\omega$. The vector components of $\boldsymbol{\mu}$ can thus be written as

Like the spinning bar magnet described above, protons have a magnetic dipole moment and carry angular momentum. In the proton's case, the moment $\boldsymbol{\mu}$ is parallel and points in the same direction as the angular momentum,$\bf I$. (In the nuclear case, the angular momentum is due to intrinsic nuclear spin and is designated by $\bf I$ rather than $\bf L$ to distinguish it from other types of angular momentum, such as orbital angular momentum). When placed in an external magnetic field, ${\bf B} = B_0 \hat{\bf z}$, the proton will precess about that field.

The most significant difference from the bar magnet, though, is that the angular momentum $\bf I$ is quantized along the $z$-axis (which is defined to be the axis of the external magnetic field) such that  ${\bf{I}}_Z = m_l\hbar$, where $m_l$ is the so-called magnetic quantum number. This number obeys $-s < m_l < s$ (in integer steps) where $s$ is the intrinsic spin quantum number. For protons with $s = 1/2$, we have $m_l = \pm 1/2$. Physically, this means that a proton can have the component of its angular momentum either aligned preferentially in the direction of $\bf B$ ($m_l = + 1/2$) or in the opposite direction ($m_l = - 1/2$). Using Eqs. (2) and (4) above, and substituting the nuclear spin $\bf I$ for $\bf L$, we can write the energy of the proton's spin state as

where again ${\bf{I}} _Z = m_l\hbar$. For $m_l = \pm 1/2$ , the two possible energy states for a proton in an external magnetic field are shown in Fig. 2. Note that because of the negative sign, the orientation where the magnetic moment is preferentially aligned with the field $m_l = + 1/2$) is a lower energy state than the case where the moment is anti-aligned ($m_l = - 1/2$).

Figure 2: The two allowed energy states of a proton (spin 1/2 particle) in a magnetic field.

The energy separation$\Delta E$ between the two states can be written in terms of an angular frequency,

where we use the same relation between angular frequency, gyromagnetic ratio, and field as the classical bar magnet, Eq. (3):

For the proton, the resonant frequency, $f_0 = \omega_0 / 2\pi$, is

and its gyromagnetic ratio $\gamma$ is

NOTE: Gauss (symbol: G) is the traditional unit for magnetic fields in nuclear magnetic resonance, but Tesla (symbol: T) is the SI unit, where 1 T = $10^4$ G.