Second moment Gaussian line

Let (AHy)A denote the limiting second moment for zero fluorine concentration and (AHy) the second moment of F due to other F nuclei when the lattice under consideration is full. Then for any intermediate concentration of F the line width for a Gaussian shape is... 

G Using the expression for the second moment of Gaussian lines, we can write ... 

However, the Lorentzian form of the dipolar broadening function, which has the advantage of mathematical simplicity, is not suitable for an interpretation in terms of second moments it is replaced with a Gaussian dipolar function S(oa, AG), where the parameters AG correspond to the appropriate fractions of the square root of the intra-group rigid lattice second moments. With appropriate values for AG, calculated and experimental line shapes I(oo) are found to be in a good agreement for cross-linked polyethylene oxide) swollen in chloroform 1U). 

In general, if the NMR line has any Lorentzian character to it, the second moment is not an ideal parameter for study because so much of the crucial information is lost in the wings of the spectrum. A line can be considered to be a Gaussian when S /CSg) =3. When such difficulties exist, it may be easier to get the true second and fourth moments from the FID as will be described shortly. For discussions of moments of Lorentzian-like lines, the relationship between the true moment and the measurable moment, and which component of the second moment is affected by molecular motion, see Abragam (Appendix A), Sections IV.II.B, IV.IV.A, X.V.A. 

Gaussian carbon-13 line whose second moment is known... 

In a rigid polycrystalline system, a nucleus does not experience an isolated dipole interaction but many such interactions arising from neighboring nuclei having (in most instances) different R and 0 relationships with the nucleus. The result is a Gaussian shaped resonance line centered at 0)q and characterized by a second moment given by7... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

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