Обертання об’єму: коливання за лад магнітного поля

Spin Precession

A nucleus with spin behaves like a tiny bar magnet with angular momentum attached.

Drop an ordinary bar magnet into a field and it swings to align with it; a spinning nuclear magnetic moment does something different, because it also has angular momentum resisting that swing — exactly like a spinning top under gravity, it precesses: the spin axis traces a cone around the field direction rather than collapsing straight onto it.

That precession, at a frequency fixed only by the nucleus and the field strength, is the entire physical basis of nuclear magnetic resonance and, by extension, MRI.

The influence of external magnetic fields on nuclear spins

A fundamental concept in Magnetic Resonance Imaging (MRI) is the interaction between an externally applied magnetic field and the spin angular momentum of atomic nuclei, primarily hydrogen-1 protons. When a sample containing these nuclei is placed within a strong static magnetic field, denoted as B₀, the nuclei align themselves with respect to this field, though not perfectly. This alignment isn’t static; instead, the nuclei constantly precess—that is, they rotate around the direction of the applied field – at a rate determined by their gyromagnetic ratio (γ).

This precession frequency is known as the Larmor frequency (ω), and it's directly proportional to both the strength of the magnetic field (B₀) and the gyromagnetic ratio. The equation describing this relationship is ω = γB₀, where ω represents the precession rate in radians per second, γ is a constant specific to each nucleus (typically 42.58 MHz/T for hydrogen), and B₀ is the strength of the static magnetic field, usually measured in Tesla.

The Larmor frequency is crucial because it dictates the resonant condition for energy absorption. When radiofrequency pulses are applied at this precise frequency, nuclei absorb energy and transition to higher energy states. As these nuclei return to their equilibrium state, they emit a signal that can be detected and used to construct an image.

Larmor equation:        omega = gamma * B0
proton gyromagnetic ratio: gamma / 2*pi ~ 42.58 MHz/T
at clinical field B0 = 1.5 T:  f = 42.58 * 1.5 ~ 63.9 MHz  (the RF pulse frequency)

Перекочування спінів за допомогою резонансного RF імпульсу

При рівновазі популяція протонів, що осідають, здійснює обертання з чистою магнітною дипольною моментом, паралельним до вектора B₀ (осі z), яка невидима для приймального контуру через відсутність змін у трансверзальному полі, необхідних для індукції сигналу. Надсилайте короткий RF імпульс точно на частоту Ломберса, щоб резонансно змістити цей чистий магнітний момент на вибраний кут повороту (зазвичай 90°), що є єдиною причиною, чому резонанс має значення тут: RF поза резонансом просто не ефективно взаємодіє з обертаючимися протонами. Після повороту магнітний момент має компоненти, що обертаються в трансверзальному плані, на частоті Ломберса, і цей обертається компонент є тим, що приймає антена та перетворює на необроблений сигнал MRI.

Relaxation: T1 and T2 are two separate clocks

Once the RF pulse ends, the tipped magnetisation doesn’t stay tipped — it relaxes back toward equilibrium along two independent exponential timescales. T1 relaxation (spin-lattice) is the z-component regrowing back to its full equilibrium value as the spins dump energy into their molecular surroundings; T2 relaxation (spin-spin) is the transverse component decaying as individual spins gradually dephase from each other due to local field variations between neighbouring nuclei, with no net energy loss required. T2 is always faster than or equal to T1, and different tissues have distinctly different T1 and T2 values, which is precisely why an MRI scanner can generate contrast between, say, grey matter, white matter and cerebrospinal fluid just by choosing when in the T1/T2 recovery curves it samples the signal.

dMz/dt = -(Mz - M0) / T1        // longitudinal recovery toward equilibrium M0
dMxy/dt = -Mxy / T2              // transverse decay, dephasing between spins

solutions:  Mz(t)  = M0 + (Mz(0) - M0) * exp(-t/T1)
            Mxy(t) = Mxy(0) * exp(-t/T2)

The Bloch equations tie it all together

Felix Bloch combined precession and relaxation into one compact set of equations in 1946 describing the full motion of the magnetisation vector M under a field B, plus the two decay terms — the Bloch equations — and they remain the standard classical description of NMR and MRI signal behaviour. Everything a clinical scanner does — spin-echo sequences, gradient-echo sequences, fat suppression, diffusion weighting — is ultimately a specific pattern of RF pulses and field gradients designed to exploit differences in Larmor frequency, T1 or T2 between tissues, all built on this single precessing-vector picture.

Frequently asked questions

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