Rotators restricted rotator spectral function

We have obtained the expression given in GT, p. 225 for the spectral function of free rotors moving in a homogeneous potential in the interval between strong collisions see also VIG, Eqs. (7.12) and (7.13). So, the subscript F means free. The subscript R in Eq. (74c) is used as an initial letter of restriction. Indeed, as it follows from the comparison of Eq. (77) with Eq. (74a), the second term of the last equation expresses the steric-restriction effect arising for free rotation due to a potential wall. If we set, for example, p = 7t, what corresponds to a complete rotation (without restriction) of a dipole-moment vector p, then we find from Eqs. (74a)-(74c) that LR z) = 0 and L(z) = Lj,(z). This result confirms our statement about restriction. ... 

We account for only the torque proportional to the string s expansion AL, which produces the main effect considered in this work. For calculation we employ the spectral function (SF) Lstr(Z), which is linearly connected with the spectrum of the dipolar ACF (see Section II), with Z x Y being the reduced complex frequency. Its imaginary part Y is in inverse proportion to the lifetime tstr of the dipoles exerting restricted rotation. The dimensionless absorption Astr is related to the SF Lstr as... 

Finally, restricted 7-motion within an apex angle 0o [Eq. (7.80)] has been employed to interpret [7.11] spectral densities of aromatic deuterons in the nematic and smectic A phases of 50.7-d4. Assuming that a-, and 7-motion are completely uncorrelated, and neglecting the -motion [i.e., set / (0) = /3 t)], the spectral densities were evaluated by describing the a-motion by a simplified model of uniaxial free rotational diffusion about the director and the 7-motion by the reduced correlated functions given in Eq. (7.80). It remains to be examined whether the anisotropic viscosity model in conjunction with restricted 7-motion [i.e., Eq. (7.82)] would be better in interpreting spectral densities of motion in various smectic phases. Also, there is still no convincing NMR evidence for biased 7-motion in nematic or smectic A phases. 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

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