= Static inhomogeneity refocusing
{title2=$S(2\tau)=e^{-2\tau/T_2}$}
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>, are
$$
R_u(-\pi/2)=\begin{pmatrix}1&0&0\\0&0&1\\0&-1&0\end{pmatrix},\qquad
P=R_v(-\pi)=\begin{pmatrix}-1&0&0\\0&1&0\\0&0&-1\end{pmatrix},\qquad
F_\Delta(t)=\begin{pmatrix}\cos\Delta t&\sin\Delta t&0\\-\sin\Delta t&\cos\Delta t&0\\0&0&1\end{pmatrix}.
$$
Here $F_\Delta(t)$ is free <Larmor precession> in the <rotating frame>, before including <transverse relaxation>. Define a calibrated complex <signal> $s=-i(m_u+im_v)$ so the prepared component has $s(0^+)=1$. The pulse $P$ sends $m_u+im_v$ to its negative <complex conjugate>, hence sends $s$ to $s^*$. Direct multiplication gives
$$
s_\Delta(t)=\begin{cases}
e^{-t/T_2}e^{-i\Delta t},&0\leq t<\tau,\\
e^{-t/T_2}e^{-i\Delta(t-2\tau)},&t>\tau.
\end{cases}
$$
The <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, $PF_\Delta(t)P=F_\Delta(-t)$, so $F_\Delta(\tau)PF_\Delta(\tau)=P$. 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.
\Image[/hahn-echo-phase-refocusing.png]
{title=How a 180° pulse refocuses a static spread of <Larmor precession> <frequencies>}
{description=Original <Bloch vector> projections and accumulated <phases> for five illustrative detunings, with no relaxation. A $-\pi/2$ rotation about $u$ prepares $+v$; a $-\pi$ rotation about $v$ reflects transverse phases at $t=\tau$. Colors identify the same components throughout. Their subsequent free <angular velocities> are unchanged.}
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>:
$$
p(\Delta)=\frac{e^{-\Delta^2/(2\sigma^2)}}{\sqrt{2\pi}\sigma},\qquad
S(t)=e^{-t/T_2}\begin{cases}
e^{-\sigma^2t^2/2},&0\leq t<\tau,\\
e^{-\sigma^2(t-2\tau)^2/2},&t>\tau.
\end{cases}
$$
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 $2.08\times10^{-9}$, yet $S(80\,\mathrm{ms})=e^{-1/2}=0.60653$. A one-pulse <free induction decay> at 80 ms would instead be about $3.11\times10^{-35}$. 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 $T_2^*$.
\Image[/hahn-echo-gaussian-ensemble.png]
{title=Synthetic <Gaussian distribution> ensemble: a vanished <free induction decay> and a recovered <spin echo>}
{description=Left: the stated instantaneous $90_u^\circ$ and $180_v^\circ$ sequence, ensemble magnitude, one-pulse comparison and homogeneous $T_2$ envelope. Direct sums over weighted detunings agree with the analytic curve. Right: samples at exact refocusing times for a series of separate sequences, with and without the <diffusion> model below. No panel reproduces an experimental trace.}
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 $t_{\rm max}=2\tau-1/(\sigma^2T_2)$, 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.
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