= Free induction decay
{title2=$S(t)=S_0e^{-t/T_2}e^{-i\Delta t}$}
{wiki}
<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 $\omega_0T_2\gg1$ 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 $u+iv=e^{i\omega t}(M_x+iM_y)$, $w=M_z$, $\Delta=\omega_0-\omega$ and $\omega_1=\gamma B_1$. The <Bloch equations> become
$$
\dot u=-u/T_2+\Delta v,\qquad
\dot v=-\Delta u-v/T_2+\omega_1w,\qquad
\dot w=-\omega_1v-(w-M_0)/T_1.
$$
A 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 gives
$$
S(t)=S_0e^{-t/T_2}e^{-i\Delta t},\qquad
I(t)=\operatorname{Re}S(t),\qquad Q(t)=\operatorname{Im}S(t).
$$
These 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 $\gamma/(2\pi)=42.58\,\mathrm{MHz/T}$, $B_0=0.200\,\mathrm T$, $B_1=10\,\mathrm{\mu T}$, $T_1=0.800\,\mathrm s$ 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.
\Image[/nuclear-magnetic-resonance-rotating-frame-decay-and-linewidth.png]
{title=A pulse, rotating-frame quadratures and the corresponding resonance width}
{description=Original calculated <Bloch equations> example with the parameters above. The resonant pulse rotates <magnetization> towards $+v$; a receiver phase makes $S(0)=1$. The later free <signal> has 5 Hz detuning. The final panel shows the normalized weak-drive homogeneous absorption profile, centered on resonance. None of these curves is a historical trace.}
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