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Free induction decay is a receiving signal from freely precessing transverse magnetization after excitation ends. Faraday's law of induction gives coil voltage proportional to the time derivative of linked magnetic flux, with sign set by the winding. At ${\omega }_0{T}_2\gg 1$ its envelope follows the transverse magnetization; a fixed receiver phase can absorb the derivative's near-quadrature shift. We define the calibrated complex signal below to represent magnetization, not raw coil voltage.

For an original teaching model, take positive $\gamma$, a circular drive $B_1(\cos \omega t\hat{\mathbf{x}}-\sin \omega t\hat{\mathbf{y}})$, and axes rotating at $-\omega$ about $z$. Define $u+iv=e^{i\omega t}(M_x+i{M}_y)$, $w=M_z$, $\Delta ={\omega }_0-\omega$ and ${\omega }_1=\gamma B_1$. The Bloch equations becomeA linearly oscillating field of peak amplitude $2B_1$ has a co-rotating component of amplitude $B_1$; neglecting its counter-rotating component requires ${\omega }_1\ll {\omega }_0$ near resonance.

At resonance, a short pulse starting from $(0,0,M_0)$ gives $(u,v,w)=(0,M_0\sin \theta ,M_0\cos \theta )$ with $\theta =\gamma B_1\tau$, neglecting relaxation during duration $\tau$. A receiver phase rotation can make a $\pi /2$ pulse's initial transverse signal real. After the pulse, direct integration givesThese are the two components extracted by quadrature demodulation. Differentiating verifies $\dot{S}=-(T_2^{-1}+i\Delta )S$; taking its magnitude separates decay from detuning.

Choose rounded illustrative $\gamma /(2\pi )=42.58MHz/T$, $B_0=0.200\mathrm{T}$, $B_1=10\mu \mathrm{T}$, $T_1=0.800\mathrm{s}$ and $T_2=0.120\mathrm{s}$. Then $f_0=8.516\mathrm{MHz}$ and the $\pi /2$ duration is $\tau =1/(4(\gamma /(2\pi ))B_1)=0.587\mathrm{ms}$, short relative to both relaxation times. For receiver detuning $\Delta /(2\pi )=5\mathrm{Hz}$ and $S_0=1$, $S(0.050\mathrm{s})=-0.65924i$ and $S(0.100\mathrm{s})=-0.43460$. These are calculated examples, not 1946 measurements.

Erwin Louis Hahn's 1950 apparatus enclosed liquid samples in the coil of a tuned LC circuit, coupled to a gated radio transmitter and radio receiver. One transmitter quadrupled a crystal oscillator's 7.5 MHz to 30 MHz. A separate proton resonance regulator stabilized the static magnetic field.^[ref]

Waiting for thermal equilibrium prepared nuclear polarization. Oscilloscope photographs compared glycerol proton echoes at different field homogeneity: echo width changed; echo time followed twice the pulse separation. Receiver saturation obscured early free induction decay; transmitter leakage required care. The delayed signal tested refocusing; diffusion could still attenuate it.^[ref]

Hahn analyzed equal pulses as well as noting improved recovery with a second pulse twice as long; the ideal sequence above isolates that latter case.^[ref]

A spin echo is a reappearance of transverse magnetization after a pulse sequence reverses an earlier spread of phases. An ensemble can have almost no net transverse magnetization while its components retain enough individual phase memory to reassemble. In the ideal Hahn echo below, a short 90° pulse prepares transverse magnetization, a 180° pulse follows after time $\tau$, and the signal refocuses at $t=2\tau$. The receiver observes freely precessing magnetization, as in free induction decay; no third driving pulse is needed at the echo time.

The useful distinction is between cancellation in an ensemble average and loss of recoverable phase memory. A static spread of Larmor precession frequencies can make free induction decay much faster than transverse nuclear magnetization decay. A spin echo can remove that static spread from the echo amplitude, allowing a more selective measurement of transverse relaxation. It does not in general reverse noise, population relaxation, or every interaction between spins.

Use the rotating frame and positive gyromagnetic ratio convention of free induction decay. Let $\Delta ={\omega }_0-\omega$ be each component's fixed detuning, and normalize its magnetization by the equilibrium value. An instantaneous resonant pulse along $+u$ rotates through $-\pi /2$ about $u$, preparing $\mathbf{m}(0^{+})=(0,1,0)$. A second pulse along $+v$ rotates through $-\pi$ about $v$. These ideal pulses require a drive much stronger than the relevant detunings and negligible relaxation during either pulse. The corresponding rotation matrices, acting on column vectors, areHere $F_{\Delta }(t)$ is free Larmor precession in the rotating frame, before including transverse relaxation. Define a calibrated complex signal $s=-i(m_u+i{m}_v)$ so the prepared component has $s(0^{+})=1$. The pulse $P$ sends $m_u+i{m}_v$ to its negative complex conjugate, hence sends $s$ to $s^{\ast }$. Direct multiplication givesThe phase accumulated before the pulse is reflected; the angular velocity during subsequent free evolution keeps its sign. At $2\tau$ every fixed detuning has zero residual phase. Equivalently, $P{F}_{\Delta }(t)P=F_{\Delta }(-t)$, so $F_{\Delta }(\tau )P{F}_{\Delta }(\tau )=P$. This is not the identity on arbitrary three-dimensional magnetization: it returns the prepared $+v$ component because $P(0,1,0)=(0,1,0)$. The two transverse pulse axes specified here also fix the receiver sign of the echo.

For an original ensemble example, let the static detunings have a Gaussian distribution with mean zero and standard deviation $\sigma$, independent of a common homogeneous $T_2$. Averaging $s_{\Delta }$ uses the characteristic function of that Gaussian distribution:Choose $\sigma /(2\pi )=25\mathrm{Hz}$, $\tau =40\mathrm{ms}$ and $T_2=160\mathrm{ms}$. Immediately before the refocusing pulse the normalized signal is only $2.08\times 10^{-9}$, yet $S(80\mathrm{ms})=e^{-1/2}=0.60653$. A one-pulse free induction decay at 80 ms would instead be about $3.11\times 10^{-35}$. These chosen numbers describe a calculation, not a measured trace. Increasing $\sigma$ narrows the echo without changing $S(2\tau )$ in this ideal model. Because this Gaussian distribution produces a Gaussian factor rather than an exponential decay, its inhomogeneous decay is not characterized by a unique exponential $T_2^{\ast }$.

The exact phase-refocusing time is $2\tau$. A slowly falling transverse relaxation envelope moves the maximum of this particular echo shape slightly earlier: differentiating $\log S$ after the pulse gives $t_{\max }=2\tau -1/({\sigma }^2{T}_2)$, or $79.747\mathrm{ms}$ here. Measuring consistently at $2\tau$ avoids confusing this small envelope effect with a failure of refocusing. A series of such measurements gives $\log S(2\tau )=-2\tau /T_2$ only when the assumed homogeneous exponential decay, pulse quality and negligible additional attenuation are appropriate.