= Diffusion attenuation of a spin echo
{title2=$S(2\tau)=\exp[-2\tau/T_2-2D\gamma^2G^2\tau^3/3]$}
A <spin echo> refocuses a nucleus's fixed detuning. A nucleus that moves through a <magnetic field gradient> can have different detunings during the two intervals. To see the distinction without assuming that all <noise> is reversible, let $\Delta(t)$ vary during free evolution. With the same pulse convention, its residual <phase> at $2\tau$ is
$$
\Phi=\int_0^\tau\Delta(t)\,dt-\int_\tau^{2\tau}\Delta(t)\,dt
=\int_0^{2\tau}y(t)\Delta(t)\,dt,\qquad
y(t)=\begin{cases}+1,&t<\tau,\\-1,&t>\tau.\end{cases}
$$
Static detuning cancels because $\int y(t)\,dt=0$. A detuning $\Delta_a$ during the first interval and $\Delta_b$ during the second leaves $\Phi=\tau(\Delta_a-\Delta_b)$, which need not vanish. The final component <signal> is $e^{i\Phi}$ before relaxation; changing the overall receiver convention changes its conjugation but not its magnitude.
For an original quantitative model, take a constant <magnetic field gradient> $G=dB_z/dx$ and unrestricted one-dimensional <Brownian motion> $x(t)=x_0+\sqrt{2D}W_t$, where $D$ is the <diffusion coefficient> and $W_t$ is standard <Brownian motion>. Then $\Delta(t)=\Delta_0+\gamma Gx(t)$. The constant terms cancel, and the <Gaussian distribution> of the remaining <phase> follows from the <Brownian covariance kernel> $\mathbb E[W_sW_t]=\min(s,t)$. Writing $Q(r)=\int_r^{2\tau}y(t)\,dt$ gives
$$
Q(r)=\begin{cases}-r,&0\leq r\leq\tau,\\r-2\tau,&\tau\leq r\leq2\tau,\end{cases}\qquad
\operatorname{Var}\Phi=2D\gamma^2G^2\int_0^{2\tau}Q(r)^2\,dr
=\frac43D\gamma^2G^2\tau^3.
$$
The <characteristic function> of a zero-mean <Gaussian distribution> now supplies the extra factor $\mathbb E[e^{i\Phi}]=e^{-\operatorname{Var}\Phi/2}$, so
$$
S(2\tau)=\exp\!\left[-\frac{2\tau}{T_2}-\frac23D\gamma^2G^2\tau^3\right].
$$
Here $\gamma$ is in radians per second per tesla, $G$ in tesla per metre, and $D$ in square metres per second: the extra exponent is <dimensionless>. The figure chooses $\gamma/(2\pi)=42.58\,\mathrm{MHz/T}$, $G=0.010\,\mathrm{T/m}$ and $D=2.0\times10^{-9}\,\mathrm{m^2/s}$. With the earlier $\tau$ and $T_2$, diffusion reduces the echo from $0.60653$ to about $0.32930$. These are illustrative model parameters, not values extracted from Hahn's experiment.
This result assumes instantaneous perfect pulses, a constant <magnetic field gradient>, unrestricted <diffusion> with constant <diffusion coefficient>, and independent homogeneous <transverse relaxation>. Restricted motion, changing gradients, finite pulses and interactions between <spins> can require different attenuation laws. Even a fixed <Hamiltonian> is refocused only for the terms whose sign the pulse reverses: a simultaneous 180° rotation of two <spins> leaves their $J_{z,1}J_{z,2}$ product unchanged, because both factors change sign. A recovered echo therefore demonstrates a particular recoverable phase history, rather than reversal of every microscopic process.
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