Lorentzian linewidth

Nevertheless, Leggett et al. [1987] have argued, with some provisos, that [with the initial condition (O) = 1] and C(t) may be practically taken the same. If C(t) then obeys the damped oscillator equation (2.41), then the inelastic part of the structure factor has the Lorentzian form with the peaks at [Pg.25]

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. 

This is a Gauss profile with a lorentzian hole, the width of which is determined by the homogeneous linewidth parameters 7ab> T with 7ab " rb depending on the lifetimes 7a, Tb of... 

In many of these descriptions of lineshapes, chemical exchange line-shapes are treated as a unique phenomenon, rather than simply another example of relaxation effects on lineshapes. This is especially true for line-shapes in the intermediate time scale, where severe broadening or overlapping of lines may occur. The complete picture of exchange lineshapes can be somewhat simplified, following Reeves and Shaw [13], who showed that for two sites, the lineshape at coalescence can always be described by two NMR lines. This fact can be exploited to produce a clarified picture of exchange effects on lineshapes and to formulate a new method for the calculation of exchange lineshapes [16, 23]. This method makes use of the fact that lineshapes, even near coalescence, retain Lorentzian characteristics [13] (fig. 3). These lines, or coherences, are each defined by an intensity, phase, position, and linewidth, and for each line in the spectrum, the contribution of that particular line to the overall free induction decay (FID) or spectrum can be calculated. 

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... 

Let us return to the definition of equivalent linewidth in Eq. (H). Since lines at most pressures of interest here can generally be described as Lorentzian in shape, the shape factor in Eq. (H) is given by... 

The NMR spectral shape is a Lorentzian curve, and the linewidth, Avi/2, at half-height of the absorption peak, gives a measure of the spin relaxation rates through Avi/2 = 1 /ttT2. 

The frequency function is a Lorentzian with linewidth at half height of (jt72)-i(= Ri/n). The same, of course, holds in the continuous wave experiment. R2 is a measure of the uncertainty of the energy levels, which gives the linewidth in every spectroscopy. The uncertainty principle, according to which the uncertainty in energy of a level is inversely proportional to the lifetime, tells us that T2 is a measure of the lifetime of the energy levels. 

Figure 111 1 1. EPR signal of 160 and lsO samples of Lai SroflsCuossMnoroCb measured at T=125 K under identical experimental conditions. The solid and dashed lines represent the best fits using a sum of two Lorentzian components with different linewidths a narrow and a broad one...

Results of calculation of second moment line for analyzed systems were obtained by summation over 106 cells in lattice, they are presented in Table 2. The Lorentzian linewidth calculated from these M2 values... 

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] ... 

The Fourier transform converts the FID into a Lorentzian peak with absorptive lineshape (after phase correction). The full width of this peak at one half of the peak s height (the linewidth ) is inversely related to the decay time constant of the FID, ... 

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