Magnetic Resonance — Question 8

Question 8

A static frequency offset is refocused by a spin echo

After an ideal transverse preparation, a spin has rotating-frame coherence $c(0)=1$ and constant offset $\delta$. Free evolution gives $c(t)=e^{-i\delta t}c(0)$. At $t=\tau$, an ideal pulse conjugates c. Irreversible decay contributes $e^{-t/T_2}$.

1. Find the offset-dependent phase for $t>\tau$ and the echo time.

2. If the echo amplitude at $2\tau=.120\,\mathrm s$ is half its initial value, infer $T_2$.

Question 8 - Solution

A static frequency offset is refocused by a spin echo

(a) Method and calculation

$$c(\tau^-)=e^{-i\delta\tau},\quad c(\tau^+)=e^{+i\delta\tau}.$$$$c(t)\propto e^{-i\delta(t-\tau)}e^{+i\delta\tau}=e^{-i\delta(t-2\tau)}\Rightarrow\boxed{t_{\mathrm{echo}}=2\tau}.$$

(b) Interpretation and further calculation

$$M_{\mathrm{echo}}/M(0)=e^{-2\tau/T_2}=1/2.$$$$\boxed{T_2=.120/\ln2=.1731\,\mathrm s}.$$

Physical check

An ideal echo refocuses static offsets, but not the irreversible transverse decay represented by T2. Time-dependent noise may not be fully refocused.