Spin, Magnetization & the Larmor Equation

Why hydrogen, and why a magnet

Every clinical MR image is, at bottom, a map of hydrogen nuclei — single protons, overwhelmingly in water and fat. Three facts make the proton the ideal reporter: it is abundant in tissue, it possesses the largest magnetic moment of the common biological nuclei, and it carries the quantum property we will exploit throughout this course — spin.

Spin is an intrinsic form of angular momentum. It is not a literal rotation, but it behaves like one: a spinning charge produces a tiny magnetic dipole moment $\vec{\mu}$. For a nucleus, the moment is proportional to its spin angular momentum $\vec{S}$:

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

The constant of proportionality $\gamma$ is the gyromagnetic ratio, and it is the single most important number in MRI. For hydrogen, $\gamma / 2\pi \approx 42.58\ \text{MHz/T}$.

From one spin to net magnetization

Left alone, the magnetic moments of the protons in a voxel point in every direction at random, and their vector sum is zero — there is nothing to detect. Place the tissue in the scanner's strong static field $\vec{B}_0$ (conventionally along $+z$), and two things happen.

First, each moment precesses about $\vec{B}_0$ like a gyroscope in gravity, rather than simply aligning. Second, quantum mechanically the spins distribute between a slightly lower-energy state (aligned with $\vec{B}_0$) and a slightly higher-energy state (anti-aligned). The split is tiny, so the populations are governed by the Boltzmann distribution, and the aligned state wins by only a few parts per million at body temperature.

That minuscule excess is everything. Summed over the enormous number of protons in a voxel, it yields a measurable net magnetization vector $\vec{M}_0$ pointing along $\vec{B}_0$:

Two consequences of Eq. 1.1 will recur for the rest of the course. The signal grows with proton density $N$ — the basis of PD weighting — and it grows with field strength $B_0$. The near-linear gain in available magnetization is the central reason 3 T scanners can deliver more signal (and why ultra-high-field research pushes to 7 T and beyond).

Precession and the Larmor equation

A magnetic moment in a field experiences a torque, and the equation of motion for the net magnetization (ignoring relaxation for now) is

$\frac{d\vec{M}}{dt} = \gamma\, \vec{M} \times \vec{B}_0 .$

A cross product means $\vec{M}$ does not move toward $\vec{B}_0$; it moves perpendicular to both — it precesses around $\vec{B}_0$ at a fixed angle, tracing a cone. The angular frequency of that precession is the Larmor frequency:

The Larmor equation is the hinge on which all of MRI turns. Read it three ways:

• Resonance. To tip the magnetization, we must irradiate the tissue at exactly $f_0$. At 1.5 T that is about 63.9 MHz; at 3 T, about 127.7 MHz — radiofrequencies, not the dangerous ionizing energies of X-rays.
• Spatial encoding (preview). If we deliberately make $B_0$ vary with position using gradients, then by Eq. 1.2 frequency varies with position. That is precisely how we will later turn one signal into an image.
• Chemical shift. Electrons partially shield the nucleus, so fat and water protons see slightly different local fields and precess at slightly different frequencies — the origin of fat-water separation and of chemical-shift artifact.

The rotating frame — the trick that makes everything simple

Watching $\vec{M}$ precess at 64 million times a second is hopeless. So we adopt a coordinate system — the rotating frame — that spins about $z$ at the Larmor frequency. In that frame, on-resonance magnetization appears stationary, because we are turning with it. The fast precession is "subtracted out," leaving only the slow, interesting dynamics: excitation and relaxation.

In the rotating frame, the radiofrequency excitation field $\vec{B}_1$ — which in the lab frame is itself oscillating at $f_0$ — appears as a small static field lying in the transverse plane (say, along $x'$). And $\vec{M}$ does to $\vec{B}_1$ exactly what it did to $\vec{B}_0$: it precesses about it. A $\vec{B}_1$ pulse of amplitude $B_1$ applied for a duration $\tau$ therefore tips $\vec{M}$ away from $z$ by a flip angle

A 90° pulse rotates all of $\vec{M}_0$ into the transverse plane, where it can induce a signal in the receive coil. A 180° pulse inverts it. Only the transverse component $M_{xy}$ is detectable — the receive coil senses the rotating transverse magnetization by Faraday induction; the longitudinal component $M_z$ is invisible to it.

Try it: the Bloch simulator

The widget below integrates the magnetization in the rotating frame. Apply a 90° pulse and watch the vector swing into the transverse plane; the readout shows $|M_{xy}|$ — the signal you could actually detect. Then leave it alone and watch relaxation (the subject of the next lesson) pull it back to equilibrium. Try a small flip angle, change the frame to "lab" to see the precession reappear (slowed for visualization), and introduce a little off-resonance.

Putting it together

We now have the complete vocabulary of an MR experiment's first instant. A static field $\vec{B}_0$ creates a small equilibrium magnetization $\vec{M}_0$ (Eq. 1.1) that precesses at the Larmor frequency (Eq. 1.2). A resonant $\vec{B}_1$ pulse, viewed in the rotating frame, tips that magnetization by a flip angle (Eq. 1.3) into the transverse plane, where it becomes detectable signal. What happens next — how that signal decays and how the longitudinal magnetization recovers — is relaxation, and it is what makes one tissue look different from another.