Diffusion attenuation of a spin echo ($S(2\tau )=\exp [-2\tau /T_2-2D{\gamma }^2{G}^2{\tau }^3/3]$)

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 $2\tau$ isStatic 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 $G=d{B}_z/dx$ 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 $\mathbb{E}[W_s{W}_t]=\min (s,t)$. Writing $Q(r)={\int }_r^{2\tau }y(t)dt$ givesThe characteristic function of a zero-mean Gaussian distribution now supplies the extra factor $\mathbb{E}[e^{i\Phi }]=e^{-\mathrm{Var}\Phi /2}$, soHere $\gamma$ 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 $\gamma /(2\pi )=42.58MHz/T$, $G=0.010T/m$ and $D=2.0\times 10^{-9}{\mathrm{m}}^2/s$. 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.