Magnetic Resonance — Question 7

Question 7

Exponential transverse decay sets a Lorentzian absorption width

In a rotating frame, transverse coherence decays as $C(t)=e^{-t/T_2}$ for $t\ge0$. Define the absorptive response $A(\delta)=\mathrm{Re}\int_0^\infty C(t)e^{i\delta t}dt$. Let $T_2=2.00\,\mathrm{ms}$.

1. Derive the absorptive line shape.

2. Find its full width at half maximum in angular frequency and ordinary frequency.

Question 7 - Solution

Exponential transverse decay sets a Lorentzian absorption width

(a) Method and calculation

$$\int_0^\infty e^{-(1/T_2-i\delta)t}dt=\frac{1}{1/T_2-i\delta}.$$$$\boxed{A(\delta)=\frac{T_2}{1+\delta^2T_2^2}}.$$

(b) Interpretation and further calculation

$$A=A(0)/2\Rightarrow|\delta|=1/T_2\Rightarrow\boxed{\Delta\omega_{\mathrm{FWHM}}=2/T_2=1000\,\mathrm{s^{-1}}}.$$$$\boxed{\Delta f_{\mathrm{FWHM}}=\Delta\omega/(2\pi)=1/(\pi T_2)=159.2\,\mathrm{Hz}}.$$

Physical check

This is the real absorptive transform of a causal exponential. Angular frequency and ordinary frequency differ by 2 pi; confusing them changes the inferred relaxation time.