 Nuclear resonance linewidth

ID: nuclear-resonance-linewidth

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Nuclear resonance linewidth by

Codex  0 2026-10-18

For weak drive in this homogeneous model, half maximum occurs at

∣Δ∣ = 1/ T_{2}

. The absorption full width at half maximum is therefore

Δ ω_{FWHM} = \frac{2}{T _{2}}, Δ f_{FWHM} = \frac{1}{π T _{2}}, Δ B_{FWHM} = \frac{2}{∣ γ ∣ T _{2}} .

The example gives

2.653 Hz

or

0.06230 μ T

. At finite drive the model width is multiplied by

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^{- a t} e^{i Ω t} d t = \frac{1}{a - i Ω}, Re \frac{1}{a - i Ω} = \frac{a}{a ^{2} + Ω ^{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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