Linewidth Gaussian

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

FIGURE 9.1 Linewidth versus standard deviation. A gaussian distribution of unit amplitude is plotted on an x-axis scale in units of the standard deviation (or sigma). At 3.15x0 the unit intensity has dropped to 0.1%. The linewidth in simulations is usually expressed as the halfwidth at half height (HWHH), which is equal to circa 1.17x0, or as twice this value that is, the full width at half height (FWHH). 

The inverse Fourier transform of G again yields a Gaussian function g(t) = linewidth Tg of which can be numerically calculated from... 

For a 1 the scattered light spectrum is gaussian with a width determined by the electron temperature, because it is due to the incoherent sum of Thomson scattering from individual, thermally moving electrons. The intensity and spectral linewidth of scattered light therefore yield electron density and temperature. 

Fig. 2. (a) Raw 300 MHz proton spectrum of a mixture of acetone and ethanol in deuteri-ochloroform (b) after reference deconvolution using the acetone signal as reference and an ideal lineshape of a 1 Hz wide Lorentzian and (c) after reference deconvolution with an ideal lineshape characterized by a negative Lorentzian width of 0.1 Hz and a Gaussian width of 0.4 Hz. The 0.1 Hz Lorentzian term represents the approximate difference in natural linewidth between the ethanol and acetone signals, and is responsible for the wings on... 

We have carried out simulations using polynomial least-squares filters of the type described by Savitzky and Golay (1964) to determine the impact of such smoothing on apparent resolution. For quadratic filters, a filter length of one-fourth of the linewidth (at FWHM) does not seriously degrade the apparent resolution of two Gaussian lines in very close proximity. 

Fig. 24 Tunable-diode-laser spectrum of RQ0 of v9 of ethane. Trace (a) is the average of 250,000 scans and exhibits linewidths of 0.0022 cm-1 (the Doppler width is 0.0018 cm-1). Trace (b) results from the deconvolution of the data in trace (a) using a gaussian with a FWHM of 0.0022 cm-1 as a response function. Trace (c) is the Q branch calculated using a model that includes torsional splitting effects Av = 1.95 mk. Trace (c) is calculated for Av = 0.00075 cm-1, which is less than one-half the 300 K Doppler width.

In general, it may be expected that the sites for Fe2+ and Fe3+ in these noncrystalline ion exchange resins will have a large distribution of chemical environments. This expectation should be reflected as a significant broadening of the Mbssbauer resonance, as experimentally observed by Johansson 188). In addition, this broadening should result in a non-Lorenzian spectral line shape. Indeed, a computer analysis of the spectra showed that Gaussian peaks provided a better fit of the data than did Lorenzian peaks. In this case then, the linewidth and peak shape provide information about the distribution of support interactions for the various resonant atoms in the sample. 

Lorentzian or Gaussian, the lineshape (18) is characterised by a linewidth-dependent apparent resonance field Bapparent (corresponding to the absorption maximum), namely, B, decreases as AB increases [33] ... 

At low density (< 1012 cm-3) and temperatures > 100 /jK the two-photon lineshape is a double exponential, exp(- p /<5p0) [3], as expected for Doppler-free two-photon excitation by a Gaussian laser beam of a thermal gas [29]. Here v is the laser detuning from resonance and 8v0 is the linewidth due to the finite interaction time of the atom with the laser beam. At low temperature, lines as narrow as 3 kHz (FWHM at 243 nm) have been observed. A detailed discussion of this lineshape in the trap and the appearance of sidebands due to coherence effects for repeated crossing of the laser beam can be found in [30]. 

A numerical calculation of the line profiles due to the combined effect of the natural lifetime, the light-shift and the saturation has been performed taking into account all possible trajectories of atoms inside the metastable beam. Actually, the study of experimental linewidths shows there are some other stray effects responsible for the broadening of the lines. He have considered their contribution by making a convolution of the line profile with a gaussian curve. 

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