Lorentzian lineshape spinning

Fig. 9. The solid curve depicts the super-Lorentzian lineshape g(u>) for a single spin species in vesicles according to Eq. (57) for Q -= 100. The frequency scale is given in units of I/T2. The wide skirts on which the sharp line is superimposed are illustrated by the plot of 10g(aj) versus u> (dashed curve). A Lorentzian line (x 10) of width 100/72 is shown for comparison (small-dashed curve). Reproduced with permission from M. Bloom, Chem. Phys. Lipids, 1975, 14, 107. 1975 American Chemical Society.

The definition of the convolution product is quite clear like the one of the Fourier transforms, it has a given mathematical expression. An important property of convolution is that the product of two functions corresponds to the Fourier transform of the convolution product of their Fourier transforms. In the context of high-resolution FT-NMR, a typical example is the signal of a given spin coupled to a spin one half. In the time domain, the relaxation gives rise to an exponential decay multiplied by a cosine function under the influence of the coupling. In the frequency domain, the first corresponds to a Lorentzian lineshape while the second corresponds to a doublet of delta functions. The spectrum of such a spin has a lineshape which is the result of the convolution product of the Lorentzian with the doublet of delta functions. In contrast, the word deconvolution is not always used with equal clarity. Sometimes it is meant as the strict reverse process of convolution, in which case it corresponds to a division in the reciprocal domain, but it is often used more loosely to mean simplification. This lack of clarity is due to the diversity of solutions offered to the problem of deconvolution, depending on the function to be deconvoluted, the quality one wishes to obtain, and other parameters. 

Fig. 6. Paramagnetic critical scattering in La2Cu04. Energy-integrating scans across the two-dimensional rod of scattering measure the Fourier transform of the instantaneous spin-spin correlation function in La2Cu04 above T. The peak width (corrected for finite instrumental resolution) is the inverse correlation length for antiferromagnetic spin fluctuations. The dashed lines show the experimental resolution function, and the solid lines are the results of fits to a Lorentzian lineshape convolved with the resolution for three different temperatures. From Kcimer et al. (1992).

An ESR line is not infinitely sharp it has a shape and width due to spin relaxation. The equations of motion for Mx, My, and M in the presence of an applied field Ho and including the spin relaxation processes discussed above are called the Bloch equations. The solution to these equations predicts a Lorentzian line with a halfwidth at halfheight of Lorentzian lineshapes are indeed often found for free radicals in liquids. In this case T2 can be determined from the linewidth. The Bloch equations also predict how the ESR signal intensity will vary with increasing microwave power. The ESR signal increases, reaches a maximum, and then decreases with increasing microwave power this behavior is called power saturation. From an analysis of the power saturation curve of ESR intensity versus microwave power, it is possible to determine Ti. 

In principle we could deconvolute the experimental spectrum with the instrumental lineshape, if that were known, to recover the true spectrum. In our example we have some good experimental evidence as to the form of the instrumental lineshape since the acetone signal is (apart from small carbon-13 satellites) a singlet, its experimental shape is just the instrumental lineshape convoluted by a Lorentzian of width l/(7rr2 ), where is the spin-spin relaxation time of the acetone protons. How can we use this experimental evidence to correct the imperfect experimental spectrum The simplest way to deconvolute one function fi uj) by another f2 ( ) is to Fourier transform the ratio of their inverse Fourier transforms ... 

The basic HMQC pulse sequence (Figure 13) is the shortest, simplest HMQC experiment and gives the best sensitivity, ft is used both to determine the chemical shifts of an insensitive nuclide (spin S) coupled to a nuclide of higher sensitivity (spin I) and to correlate the coupled pairs of I and S spins. The experiment is not phase sensitive hence lineshapes are not Lorentzian and coupling constants cannot... 

First, either a Lorentzian or Gaussian filter is applied to the FID to reduce the amount of noise. The choice of lineshape will depend on the shape of the frequency domain spectrum, the lineshape is related to how the fluorine spins interact with their environment. The filter linewidth is generally similar to or slightly less than the T2 value (T2 can be estimated from the spectral linewidth). After application of the time domain filter, a fast Fourier transform (FFT) is performed. The resultant frequency domain spectrum will then need to undergo phase adjustment to obtain a pure absorption spectrum. The amount of receiver dead time (time lost between the end of the excitation pulse and the first useful detection time point) will determine the presence and extent of baseline artifact present as well as how difficult phase adjustment will be to accomplish. 

It has been established that the mobility of protons can be estimated by the detaUed analysis of NMR lineshape. Fen e et al. theoretically treated the influence of isotropic thermal motion upon MAS NMR for spin I = 1/2. When the broadening results from an inhomogeneous magnetic dipolar interaction, the Uneshape is expressed by Lorentzian line with Une width given by the following equation. 

Measurements by photographic photometry require careful calibration due to the nonlinear response of photographic plates saturation effects can lead to erroneous values. Line profiles can be recorded photoelectrically, if the stability of the source intensity and the wavelength scanning mechanism are adequate. Often individual rotational lines are composed of incompletely resolved spin or hyperfine multiplet components. The contribution to the linewidth from such unresolved components can vary with J (or TV). In order to obtain the FWHM of an individual component, it is necessary to construct a model for the observed lineshape that takes into account calculated level splitttings and transition intensities. An average of the widths for two lines corresponding to predissociated levels of the same parity and J -value (for example the P and R lines of a 1II — 1E+ transition) can minimize experimental uncertainties. A theoretical Lorentzian shape is assumed here for simplicity, but in some cases, as explained in Section 7.9, interference effects with the continuum can result in asymmetric Fano-type lineshapes. 

A nuclear magnetic resonance line is usually found to have one of two ideal lineshapes - Gaussian, or more often, Lorentzian. A Gaussian line is found when there is a random distribution of static fields within the sample. A Lorentzian line by contrast arises because the spin lifetime follows a first-order decay law. Weighting functions can be applied to a free-induction decay to generate... 

Figure 5. ESR lineshapes of a spin probe gx = 2.0093, gy = 2.0064, gj = 2.00215, A = 0) undergoing reorientation with jump angle 6 = 80° and rotational correlation times r = 900 (top), 45 (middle), and 9 ns (bottom). The lineshapes are convoluted by a Lorentzian with width 1/7J = 5.26 MHz to account for the changes of the Larmor frequency, occurring each on average, due to the...

Calculations of (pickup) and 5), (resonator) were made for a spin packet for the same values of 7), T2, Bi, /w and Mq as those for set (a) listed above, off-resonance by Acd=20 MHz. The resulting calculated signals are included in Figures 5 and 6, respectively. The required initial value of magnetization. Mo, off resonanee is expected to be smaller than that at resonanee, as it depends on deviation Aco and the EPR lineshape (e.g. Gaussian, Lorentzian) depending on the properties of the sample. But in order to demonstrate the dependence of the signals on T, T2 and facilitate easy comparison with the situation when the spin packet is at resonance. 

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