NMR MRI Relation Cross Scans

spin-echo-recoverable-loss-three-row-scan

A spin echo restores recoverable dephasing but excludes irreversible loss. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Echo signal separates recoverable and irreversible buckets

The coherent bucket stays fixed; changing the recoverable bucket changes the echo while the loss remains excluded.

Secho=Scoh+SrecoverableS_{\text{echo}}=S_{\text{coh}}+S_{\text{recoverable}}
Spin-echo refocus rowRecoverable signal returns; irreversible loss stays outside the echo.coherent 6recoverable 4lost 2echo 10

More recoverable dephasing gives a larger echo

Each row still closes the same initial bucket; only the recoverable/lost split changes.

recoverablelostSechoFecho15771224823421056\begin{array}{c|c|c|c}\text{recoverable}&\text{lost}&S_{\text{echo}}&F_{\text{echo}}\\1&5&7&\frac{7}{12}\\2&4&8&\frac{2}{3}\\4&2&10&\frac{5}{6}\\\end{array}

The final row refocuses the recoverable bucket but leaves loss out

The visible echo is not the full initial bucket; irreversible loss is still excluded.

6+4=106+4=10
Spin-echo refocus rowRecoverable signal returns; irreversible loss stays outside the echo.coherent 6recoverable 4lost 2echo 10