Frequency domain

These components fit the real-valued Fourier spectrum displayed in the Frequency workspace, as a function of frequency \(\nu\) in MHz (see Frequency-domain fitting for the workflow). They are only offered when fitting in the frequency domain, where the picker presents them as a flat list.

The physical motivation mirrors the time domain through the Fourier-transform pairs of the standard relaxation envelopes: an exponentially relaxing (dynamically broadened) signal \(e^{-\lambda t}\) gives a Lorentzian line of width \(\mathrm{FWHM} = \lambda/\pi\), while a Gaussian envelope \(e^{-(\sigma t)^2}\) (static, dense field distribution) gives a Gaussian line of width \(\mathrm{FWHM} = 2\sigma\sqrt{\ln 2}/\pi\). Both peaks are parameterised directly by their full width at half maximum — the fwhm parameter is the literal width of the line on the plot, which is why the factors \(4\ln 2\) and \(4\) appear in the expressions below. A fitted width therefore converts back to a time-domain rate via \(\lambda = \pi\,\mathrm{FWHM}\) or \(\sigma = \pi\,\mathrm{FWHM}/(2\sqrt{\ln 2})\), and a peak centre to a local field via \(B_0 = \nu_0/(\gamma_\mu/2\pi)\). Peak heights are in the arbitrary units of the displayed spectrum (they depend on apodisation and normalisation), so physical conclusions should rest on positions and widths rather than absolute heights.

GaussianPeak

\[S(\nu) = h\,\exp\!\left[-4\ln 2\, \frac{(\nu-\nu_0)^2}{w^2}\right], \qquad w \equiv \mathrm{FWHM}\]

A Gaussian spectral line — appropriate when the underlying time-domain envelope is Gaussian (static, dense field distribution): the spectral-domain counterpart of Gaussian. The width \(w\) is the literal full width at half maximum, \(S(\nu_0 \pm w/2) = h/2\) exactly.

Name	Symbol	Unit	Description
height	\(h\)	a.u.	Peak height.
nu0	\(\nu_0\)	MHz	Peak centre.
fwhm	\(w\)	MHz	Full width at half maximum.

LorentzianPeak

\[S(\nu) = \frac{h}{1 + 4\,(\nu-\nu_0)^2/w^2}, \qquad w \equiv \mathrm{FWHM}\]

A Lorentzian spectral line — appropriate when the time-domain envelope is exponential (dynamic broadening, or a dilute static field distribution): the spectral-domain counterpart of Exponential. Parameters as for GaussianPeak, with the same half-height convention. Note the heavy Lorentzian tails: fit windows should extend several FWHM beyond the peak or the width will be underestimated; a line that is neither Gaussian nor Lorentzian usually signals overlapping sites, which are better fitted as two peaks.

ConstantBackground

\[S(\nu) = b_g\]

A flat spectral baseline, absorbing white noise and the flat part of any apodisation pedestal. Parameter: bg (a.u.).

LinearBackground

\[S(\nu) = b_g + m\,\nu\]

A linearly sloped baseline for spectra with a gentle trend across the fit window, e.g. the shoulder of a distant intense line or low-frequency 1/f-like leakage. Parameters: bg (a.u.), slope (a.u./MHz). Prefer ConstantBackground unless the slope is clearly visible — the slope is strongly correlated with the peak parameters in narrow windows.