15.1 Physics of Nuclear Magnetic Resonance

Nuclear Magnetic Resonance (NMR) is the physical phenomenon that makes MRI possible. It describes how atomic nuclei with magnetic properties respond when placed in a strong external magnetic field and then hit with radiofrequency (RF) energy. To understand MRI at any depth, you need a solid grasp of nuclear spin, the Larmor precession, and the relaxation processes that ultimately create image contrast.

Nuclear Spin and Magnetic Moment

Fundamental Properties of Atomic Nuclei

Every atomic nucleus has a property called nuclear spin, an intrinsic angular momentum characterized by the spin quantum number $I$. Nuclei with $I = 0$ (like carbon-12) have no net spin and are invisible to MRI. The nuclei that matter for MRI are those with $I \neq 0$, most importantly hydrogen-1 ($^1H$), which has $I = \frac{1}{2}$ and is abundant in biological tissue.

A nucleus with non-zero spin also possesses a magnetic moment $\vec{\mu}$, related to its spin angular momentum $\vec{J}$ by:

$\vec{\mu} = \gamma \vec{J}$

The constant $\gamma$ is the gyromagnetic ratio, a nucleus-specific value that tells you how strongly the magnetic moment couples to angular momentum. For protons, $\gamma = 42.58 \text{ MHz/T}$. This number shows up constantly in MRI calculations, so it's worth memorizing.

When you place these magnetic nuclei in an external magnetic field $B_0$, the field exerts a torque on each magnetic moment. Rather than simply snapping into alignment (the way a compass needle would), the nuclei precess around the field direction, much like a spinning top wobbles around the gravitational axis.

Larmor Precession and Resonance Frequency

The rate of this precession is called the Larmor frequency:

$\omega_0 = \gamma B_0$

This is the single most important equation in MRI physics. It tells you that the precession frequency scales linearly with field strength. For protons:

• At 1.0 T: $f_0 = 42.58 \text{ MHz}$
• At 1.5 T: $f_0 = 63.87 \text{ MHz}$
• At 3.0 T: $f_0 = 127.74 \text{ MHz}$

Resonance occurs when you apply an oscillating magnetic field (the RF pulse) at exactly the Larmor frequency. Only at this matching frequency can the RF field efficiently exchange energy with the nuclear spins. This frequency-matching requirement is what makes NMR selective: you can target specific nuclei by tuning your RF pulse to their Larmor frequency.

Resonance and Magnetization

The Magnetization Vector

A single proton's magnetic moment is far too small to detect. What you actually measure in MRI is the net magnetization vector $\vec{M}$, which represents the summed magnetic moments of billions of protons in a volume of tissue.

At thermal equilibrium inside $B_0$, a slight majority of proton spins align parallel to the field rather than anti-parallel. This population difference is tiny (on the order of a few per million at clinical field strengths), but it's enough to produce a measurable net longitudinal magnetization $M_z$ pointing along $B_0$.

There is no net transverse magnetization ($M_{xy} = 0$) at equilibrium because the individual spins precess with random phases, canceling each other out in the transverse plane.

Manipulating Magnetization with RF Pulses

Applying an RF pulse at the Larmor frequency tips $\vec{M}$ away from the longitudinal axis. The flip angle depends on the pulse's amplitude and duration:

• A 90° pulse rotates $M_z$ entirely into the transverse plane, maximizing $M_{xy}$ and zeroing out $M_z$.
• A 180° pulse inverts the magnetization from $+M_z$ to $-M_z$ (used in inversion recovery sequences and spin-echo refocusing).

Once tipped into the transverse plane, the precessing $M_{xy}$ component induces a voltage in the receiver coil. This is the actual MRI signal. Controlling flip angles and pulse timing is the basis of all MRI pulse sequences.

Relaxation Processes

After the RF pulse ends, the magnetization doesn't stay tipped forever. It returns to equilibrium through two independent relaxation mechanisms, and the rates of these processes differ between tissues. That difference is what gives MRI its soft-tissue contrast.

Longitudinal Relaxation (T1)

T1 relaxation (spin-lattice relaxation) is the recovery of $M_z$ back to its equilibrium value. The "lattice" here refers to the surrounding molecular environment. During T1 recovery, excited spins release energy to nearby molecules, gradually restoring the longitudinal magnetization.

The recovery follows an exponential curve:

$M_z(t) = M_0 \left(1 - e^{-t/T_1}\right)$

(assuming $M_z$ was zeroed by a 90° pulse at $t = 0$).

The T1 time constant is the time it takes for $M_z$ to recover to about 63% of its equilibrium value. Typical values at 1.5 T:

• Fat: ~260–300 ms (short T1, recovers quickly)
• Gray matter: ~900 ms
• CSF/water: ~2000–4000 ms (long T1, recovers slowly)

Fat has a short T1 because its molecular tumbling rate is close to the Larmor frequency, making energy transfer to the lattice efficient. In T1-weighted images, tissues with short T1 appear bright because they recover more signal before the next excitation pulse.

Transverse Relaxation (T2) and Free Induction Decay

T2 relaxation (spin-spin relaxation) is the decay of $M_{xy}$ after excitation. This happens because individual spins lose phase coherence with each other. Some spins precess slightly faster and some slightly slower due to tiny local field variations from neighboring nuclei. Over time, the spins fan out in the transverse plane and their signals cancel.

The decay follows:

$M_{xy}(t) = M_{xy}(0) \cdot e^{-t/T_2}$

The T2 time constant is the time for $M_{xy}$ to decay to about 37% of its initial value. Typical values at 1.5 T:

• Fat: ~80–100 ms
• Gray matter: ~100 ms
• CSF/water: ~1500–2000 ms (long T2)

Note that the original guide listed water as having a shorter T2 than fat. That's incorrect. Free water (like CSF) actually has a very long T2 because water molecules are small and tumble rapidly, producing fewer slow spin-spin interactions. In T2-weighted images, tissues with long T2 (like fluid) appear bright.

T2 vs. T2*: In practice, transverse magnetization decays faster than the true T2 because of additional dephasing from static magnetic field inhomogeneities (from the magnet itself, tissue interfaces, etc.). This faster observed decay rate is called T2*, where $T_2^* < T_2$. Spin-echo sequences can refocus the static inhomogeneity component and recover the true T2, while gradient-echo sequences are sensitive to T2*.

The decaying transverse magnetization induces a signal in the receiver coil called the free induction decay (FID). The FID is the raw time-domain signal that gets Fourier-transformed to produce frequency information for image reconstruction.

T1 and T2 Are Independent

A common point of confusion: T1 and T2 are separate processes happening simultaneously but driven by different physical mechanisms. T1 involves energy exchange with the lattice. T2 involves loss of phase coherence between spins. T2 is always less than or equal to T1 ($T_2 \leq T_1$), because any process that causes T1 recovery also disrupts phase coherence, but not vice versa.