= Nuclear resonance linewidth
{title2=$\Delta f_{\rm FWHM}=1/(\pi T_2)$}
For weak drive in this homogeneous model, half maximum occurs at $|\Delta|=1/T_2$. The absorption <full width at half maximum> is therefore
$$
\Delta\omega_{\rm FWHM}=\frac2{T_2},\qquad
\Delta f_{\rm FWHM}=\frac1{\pi T_2},\qquad
\Delta B_{\rm FWHM}=\frac2{|\gamma|T_2}.
$$
The example gives $2.653\,\mathrm{Hz}$ or $0.06230\,\mathrm{\mu T}$. At finite drive the model width is multiplied by $\sqrt{1+s}$. Field inhomogeneity and unresolved components can also broaden an actual line, so its width alone does not determine pure <dephasing>.
The same width appears by an original <Fourier transform> calculation. For $a=1/T_2$ and a free homogeneous <signal> demodulated exactly at resonance,
$$
\int_0^\infty e^{-at}e^{i\Omega t}\,dt=\frac1{a-i\Omega},\qquad
\operatorname{Re}\frac1{a-i\Omega}=\frac{a}{a^2+\Omega^2}.
$$
This connects exponential memory to a Lorentzian absorptive line. It is a mathematical connection, not evidence that the 1946 teams performed pulsed <Fourier transform> spectroscopy. Instrumental <signal bandwidth> is a separate choice: narrowing a receiver band can reject <thermal electrical noise>, but too narrow a band distorts a changing <signal>. A sample linewidth and a receiver bandwidth are not interchangeable.
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