Magnetic Resonance — Question 6

Question 6

Longitudinal recovery reveals the relaxation time from a null crossing

After a perfect inversion pulse, a sample has $M_z(0)=-M_0$. It obeys $dM_z/dt=(M_0-M_z)/T_1$. The signal crosses zero at $t_0=.420\,\mathrm s$.

1. Solve the recovery equation.

2. Infer $T_1$ and find when the signal first reaches $0.90M_0$.

Question 6 - Solution

Longitudinal recovery reveals the relaxation time from a null crossing

(a) Method and calculation

$$M_z-M_0=[M_z(0)-M_0]e^{-t/T_1}.$$$$\boxed{M_z(t)=M_0(1-2e^{-t/T_1})}.$$

(b) Interpretation and further calculation

$$0=1-2e^{-t_0/T_1}\Rightarrow\boxed{T_1=t_0/\ln2=.6059\,\mathrm s}.$$$$.90=1-2e^{-t/T_1}\Rightarrow\boxed{t=T_1\ln20=1.815\,\mathrm s}.$$

Physical check

The zero crossing is not one relaxation time. The specified inversion initial condition accounts for the factor two in the recovery curve.