The Spin-Echo Family

Why refocus at all?

After a single 90 degree excitation, the transverse magnetization does not simply decay at the rate set by spin-spin interactions. It decays much faster, at the rate $1/T_2^{*}$, because spins at different positions precess at slightly different frequencies. Some of this spread is random and irreversible (true $T_2$ from fluctuating local fields and diffusion), but much of it is static and reversible: fixed magnetic-field inhomogeneity from an imperfect magnet, susceptibility gradients at tissue interfaces, and chemical-shift offsets. The genius of the spin echo, introduced by Erwin Hahn in 1950, is that a single $180^{\circ}$ refocusing pulse can undo all of the static dephasing and recover a signal that reflects true $T_2$.

This single idea — turn an irreversible-looking decay into a refocusable one — is the foundation of an entire family of sequences: the basic spin echo (SE), fast or turbo spin echo (FSE/TSE), and inversion-recovery variants including STIR and FLAIR. This lesson derives the echo, then builds the family on top of it.

Deriving the Hahn echo

Consider an isochromat (a small group of spins) at position experiencing a constant frequency offset $\Delta\omega$ relative to the rotating-frame reference. After the $90^{\circ}$ pulse places magnetization in the transverse plane, the accumulated phase at time $t$ is simply $\phi(t) = \Delta\omega \, t$. Different isochromats have different $\Delta\omega$, so the phases fan out and the vector sum (the measurable signal) collapses on the timescale $T_2^{*}$.

Now apply a $180^{\circ}$ pulse at time $\tau$. A refocusing pulse about the $y$-axis maps phase $\phi \rightarrow \pi - \phi$, which is equivalent to negating the accumulated phase. Just after the pulse the isochromat phase is $-\Delta\omega\,\tau$. Precession continues at the same $\Delta\omega$, so at a later time $t$ the phase is:

The phase becomes exactly zero when $t = 2\tau$, independent of $\Delta\omega$. Every isochromat, no matter how far off-resonance, returns to zero phase at that instant. The fan of vectors collapses back to alignment and the signal rephases into an echo. The echo time is therefore $TE = 2\tau$, and the static dephasing has been completely undone.

What the refocusing cannot undo is the random component. Molecular tumbling causes the local field — and hence $\Delta\omega$ — to fluctuate during the experiment, so each spin's phase history is not perfectly retraced. This irreversible loss is true $T_2$ decay. The amplitude of the echo is therefore attenuated only by $T_2$, not $T_2^{*}$:

Multiple echoes and the CPMG train

Because the $180^{\circ}$ pulse only inverts phase, you can apply a series of them after one excitation and collect an echo after each. With refocusing pulses spaced evenly, echoes form at $TE = 2\tau, 4\tau, 6\tau, \dots$, and their peak amplitudes trace out the $T_2$ decay curve. Fitting $\ln S$ versus $TE$ to a straight line yields $T_2$ directly — the classic multi-echo measurement.

Two practical problems arise. First, an imperfect $180^{\circ}$ pulse (a true flip angle of, say, $170^{\circ}$) leaves residual longitudinal magnetization and accumulates error over many echoes. Second, diffusion through residual gradients adds spurious decay. The Carr-Purcell-Meiboom-Gill (CPMG) scheme solves both: the $180^{\circ}$ pulses are phase-shifted by $90^{\circ}$ relative to the excitation (applied about $y$ while excitation is about $x$), so flip-angle errors on even echoes cancel those on odd echoes. CPMG is the workhorse refocusing scheme used in essentially every modern spin-echo train.

Fast (turbo) spin echo: encoding many lines per TR

Conventional SE acquires one phase-encoding line per TR, so a $256$-line scan with $TR = 4000$ ms takes over $17$ minutes. Fast spin echo (FSE on GE, TSE on Siemens and Philips) collapses this by phase-encoding each echo of a CPMG train separately. Each $180^{\circ}$ pulse produces an echo that gets its own phase-encode gradient (applied before readout and rewound after), so a single TR fills several lines of k-space at once.

The number of echoes used per TR is the echo-train length (ETL), also called the turbo factor. The acceleration is exactly the turbo factor: a turbo factor of $16$ on a $256$-line acquisition needs only $256/16 = 16$ TRs, cutting a $17$-minute scan to about a minute. The price is that the different lines of k-space are acquired at different real echo times along the decay curve.

Effective TE and k-space ordering

Because contrast in an image is dominated by the center of k-space (low spatial frequencies carry the bulk of the signal energy and gross contrast), the apparent contrast of an FSE image is set by whichever echoes are mapped to the central lines. That echo time is the effective TE ($TE_{eff}$). To produce a T2-weighted image, the sequence orders the train so that late echoes (long real TE) land in the center of k-space; for a proton-density image, early echoes fill the center.

| Property | Conventional SE | Fast / turbo SE | | --- | --- | --- | | Lines per TR | 1 | Equal to the turbo factor (e.g. 4 to 32) | | Scan time for 256 lines | 256 times TR | 256 / turbo factor times TR | | Contrast set by | Single chosen TE | Effective TE from central k-space echo | | Edge sharpness | Uniform | Slight blur from T2 decay across the train | | Fat signal on T2 | Moderate | Brighter (J-coupling decoupled) | | SAR | Lower | Higher (many 180 degree pulses) |

FSE is not free of artifact. Because outer k-space lines are filled by echoes that have decayed further, the high-spatial-frequency data are attenuated, which blurs edges along the phase-encode direction. Longer turbo factors and tissues with short $T_2$ worsen this. A second hallmark is bright fat: in conventional SE, J-coupling among the protons of fatty-acid chains causes additional dephasing that darkens fat on T2 images, but the rapid train of $180^{\circ}$ pulses in FSE disrupts (decouples) this J-modulation, so fat stays conspicuously bright. This is why a separate fat-suppression technique is usually paired with T2 FSE.

Inversion recovery, STIR, and FLAIR

Inversion recovery (IR) prepends a $180^{\circ}$ inversion pulse before the spin-echo excitation. The inversion flips $M_z$ from $+M_0$ to $-M_0$, and during the inversion time ($TI$) the longitudinal magnetization recovers along its $T_1$ curve, passing through zero on its way back to equilibrium:

A tissue is nulled when $M_z(TI) = 0$, which occurs when $1 - 2e^{-TI/T_1} = 0$, giving the famous null condition:

Choosing $TI$ to null a specific tissue is the entire point of STIR and FLAIR. STIR (Short TI Inversion Recovery) nulls fat: at $1.5$ T fat has $T_1 \approx 250$ ms, so $TI_{null} \approx 0.693 \times 250 \approx 170$ ms (typically set around $150$ to $170$ ms). Because STIR suppresses fat by its short $T_1$ rather than by chemical shift, it is robust to field inhomogeneity and works well over large fields of view and near metal. FLAIR (Fluid-Attenuated Inversion Recovery) nulls CSF: CSF has $T_1 \approx 4000$ ms, so $TI_{null} \approx 0.693 \times 4000 \approx 2500$ to $2800$ ms. FLAIR is the standard for seeing periventricular and cortical lesions that would otherwise be hidden by bright CSF on T2.

| Sequence | Tissue nulled | Approx T1 at 1.5 T | Approx TI | | --- | --- | --- | --- | | STIR | Fat | 250 ms | 150 to 170 ms | | FLAIR | CSF (free water) | 4000 ms | 2200 to 2800 ms | | IR (T1 weighting) | None (used for high T1 contrast) | varies | 300 to 700 ms |

How TR and TE set the weighting

Spin-echo image contrast is governed by two operator-chosen parameters. TR controls how much $T_1$ recovery occurs between excitations and therefore the $T_1$ weighting; TE controls how much $T_2$ decay occurs before sampling and therefore the $T_2$ weighting. Combining the recovery and decay terms gives the working signal equation for spin echo:

• T1-weighted: short TR (around 400 to 700 ms) maximizes T1 differences, short TE (around 10 to 20 ms) minimizes T2 influence. Fat bright, fluid dark.
• T2-weighted: long TR (2500 ms or more) removes T1 dependence, long TE (around 80 to 120 ms) emphasizes T2 differences. Fluid and many lesions bright.
• Proton-density (PD): long TR removes T1 dependence, short TE removes T2 dependence, leaving contrast driven by spin density.
• Avoid: short TR with long TE produces mixed weighting and low signal — generally not diagnostically useful.

The interactive diagram below lets you place the $90^{\circ}$ excitation, the $180^{\circ}$ refocusing pulse, and the readout, and watch where the echo forms relative to the pulse spacing. Set up a single-echo SE first, then add refocusing pulses to build a CPMG/FSE train and observe how the echoes line up at multiples of $2\tau$.

Now use the contrast playground to sweep $TR$ and $TE$ across tissues with different $T_1$ and $T_2$. Try to reproduce the three canonical weightings from Eq. 5.5: drive TR short for T1 weighting, push TE long for T2 weighting, and find the long-TR short-TE corner that isolates proton density.

SAR: the cost of all those 180s

Every RF pulse deposits energy in tissue, quantified as the specific absorption rate (SAR) in watts per kilogram. The energy of a pulse scales roughly with the square of the flip angle and, critically, with the square of the main field strength $B_0$, because the Larmor frequency — and thus the RF frequency — rises with $B_0$. A $180^{\circ}$ refocusing pulse deposits about four times the energy of a $90^{\circ}$ pulse of the same shape.

FSE multiplies the number of $180^{\circ}$ pulses per TR by the turbo factor, so a long echo train is an RF-heavy sequence. Moving from $1.5$ T to $3$ T roughly quadruples SAR for the same pulses. This is the dominant constraint on high-field FSE: scans can hit the regulatory SAR limit and the scanner must lengthen TR, drop slices, or reduce the refocusing flip angle. The standard mitigation is variable-flip-angle (hyperecho or TRAPS-style) refocusing, where the $180^{\circ}$ pulses are replaced by lower flip angles (often $120^{\circ}$ or less); this dramatically cuts SAR while a carefully designed flip-angle sweep maintains usable signal across a very long train (the basis of single-slab 3D FSE sequences such as SPACE, CUBE, and VISTA).