Rigid lattice second-moment

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). 

Fig. 23. Plot of the dipolar broadening parameter AG (left scale) and its relation to the square root of rigid lattice second moment, AM2 = 15 kHz (right scale), for polystyrene networks crosslinked with DVB (solid symbols) and EDM (open symbols) swollen to equilibrium, vs l/n, the reciprocal nominal number of C—C bonds between crosslinks points. Solvents CC14 ( ), CDC13 (A),...

Since crosslinked polymer networks exhibit anisotropic internal motions, the effect of MAS on the line narrowing is explained assuming that the rigid lattice second moment AM2 consists of two terms, analogous to Eq. (34). Then, the residual line width in the NMR MAS experiment, Acomas, can be described by the following relation ... 

Table 7.1 Composition of glasses in the various systems investigated and experimentally determined values of the rigid lattice second moment and of the temperatures where motional narrowing occurs (After Gdbel et al., 1979).

The minimum value of Ti, occurs for Tc= 1. For < Tc< 1, In Ti decreases linearly with increasing wtc, and for wTc>1, In Tj increases linearly with increasing wtc- This theory applies in the motionally-narrowed regime in which the dipolar correlation frequency, Vc = Tcis much faster than the characteristic frequency of the rigid lattice second moment, [see eq. 

Hydrogen motion has also been studied in the lutetium-hydrogen system by Barrere and Tran (1971). At the composition LuHo.17 the hydrogen is in solid solution in the close-packed-hexagonal phase of the lutetium metal. The observed rigid lattice second moment was compatible with random occupation of... 

As Figure 2 shows, the two models differ only by a vertical displacement, which is a measure of the difference in the mean square interparticle dipolar fields (second moments) operative in the two models. The values of these mean square local fields can be calculated easily from the minimum value of Ti or from the rigid lattice values of T2. Because of the 1000-fold ratio between electronic and nuclear magnetic moments, even... 

Figure 7.7 Fast but anisotropic segmental motion results in a solid-like contribution to the NMR signal. This contribution is expressed in terms of a fractional contribution q of the second moment M2 of the rigid lattice line of a single chain or residual dipolar interactions between protons. The line splitting caused by the dipole-dipole interaction depends on the orientation angle q of the internuclear vector of the coupling protons in the magnetic field B(). The distribution of orientation angles changes with the...

Naphthalene, in contrast to benzene, did not show any NMR-spectra line-width narrowing up to its melting temperature of 353 K. The mean experimental second moment was 9.1 compared to 10.1 G, estimated for the rigid crystal. Measurement of spin-lattice relaxation times indicated, however, also a slow reorientational jump motion about an axis normal to the molecular axis An activation energy of 105 kJ/mol was derived. Molecular dynamics simulations suggest that this reorientation about the axis of greatest inertia occurs with a frequency of 100 MHz within 20 K of fusion (353.6 K) Still, no plastic crystal behavior as found in cyclohexane and related compounds (see Sect. 3.1.1) is indicated for benzene or naphthalane, even close to the melting temperature. 

At T < 80 K, the linewidth is quantitatively attributable to the magnetic dipole-dipole coupling of the protons in the rigid benzene lattice. It follows directly from the crystal structure (see Sect. 5.8.1) that each pair of protons ij gives a contribution to the second moment proportional to All together, Andrew and Fades obtained for the second moment in polycrystalline, non-deuterated hydrocarbons a value of 7.159 Gauss. N is here the number of proton pairs whose inter-... 

Anisotropy of the rigid lattice response (such that the frequency of molecular motion is much less than the NMR line width in Hz, which often occurs at low temperature), usually determined from line width or second moment measurements. 

It is convenient to discuss the effect of motion on the intramolecular interactions (where r is constant) separately from the intermolecular interactions where both P and r vary with time. Similar transformations of(3 cos Pjh—l) to those employed in the rigid lattice case, enable us to relate the second moment to the angle 7 between the draw direction and Ho, and cos A and cos A which define the orientation functions for the transversely isotropic situation which has been analysed. We find... 

The second moment values Sj of PMR spectra of V(C,N),Hy phases registered at the temperatures corresponding to the rigid lattice considerably exceed S calculated by the Van Fleck method based on the assumption of two possible (tetra and octa) positions of H. This is possible only if H is located in a shifted position relative to the vacancy. The values of the jump activation energy were found to be 0.31 eV (VQ 84H0 04) and 0.33 eV (V0.79H0.04). 

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