NMR MRI Relation Cross Scans

recovery-fraction-three-row-scan

Changing the recovered fraction changes the final longitudinal row. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Recovery fraction multiplies the remaining gap

The start and equilibrium stay fixed; the scan changes only the recovered fraction of the remaining gap.

Mfinal=Mstart+f(M∞−Mstart)M_{\text{final}}=M_{\text{start}}+f(M_{\infty}-M_{\text{start}})
Recovery fraction rowThe final value is start plus a checked fraction of the gap.start 4gap 12fraction 1/2final 10

Three fractions land at three exact final values

Every row recomputes the same gap before applying the pinned fraction.

fgapincrementMf14123712126103412913\begin{array}{c|c|c|c}f&\text{gap}&\text{increment}&M_f\\\frac{1}{4}&12&3&7\\\frac{1}{2}&12&6&10\\\frac{3}{4}&12&9&13\\\end{array}

The middle row is a half-gap recovery

The final display cites the same checked recovery row as the table.

4+12⋅12=104+\frac{1}{2}\cdot12=10
Recovery fraction rowThe final value is start plus a checked fraction of the gap.start 4gap 12fraction 1/2final 10