Lineshape density matrix formalism

To conclude this section on 2D NMR of liquid crystals, some studies of more exotic liquid crystalline systems are pointed out. Polymer dispersed nematic liquid crystals have attracted much attention because of their applications as optical display panels. Deuteron 2D quadrupole echo experiments have been reported [9.28] in the isotropic and nematic phases of / -deuterated 5CB dispersed in polymers. A similar technique was used [9.29] to study two model bilayer membranes. Both studies allow determination of the lineshape F(u ) due to quadrupolar interactions and the homogeneous linewidth L(u ) of the individual lines [9.28]. The 2D quadrupole echo experiment has also been used [9.30] to separate chemical shift and quadrupolar splitting information of a perdeuterated solute dissolved in a lyotropic liquid crystal. The method was compared with the multiple-quantum spectroscopy that is based on the observation of double-quantum coherence whose evolution depends on the chemical shift but not on the quadrupolar splitting. The multiple-quantum method was found to give a substantial chemical shift resolution. The pulse sequences for these methods and their treatment using density matrix formalism were summarized [9.30] for a spin 1=1 system with non-zero chemical shift. Finally, 2D deuteron exchange NMR was used [9.31] to study ring inversion of solutes in liquid crystalline solvents. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

The correlation functions in Eq. (5) are then expanded in the usual way2 in terms of spectral densities (see Section 3 for further details). As Eq. (5) shows, the relaxation operator involves products of Hamiltonian matrix elements and thus has the effect of redistributing coherence between the various matrix elements or coherences/populations of the density matrix through its involvement in Eq. (2). Kristensen and Farnan40 use their formalism to calculate the central transition lineshapes for lvO (/= 5/2) for both fully relaxed and partially relaxed conditions under different motional models. Some examples are shown in Fig. 27. 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

A relatively formal derivation in Ref. [11], which is based on the NMR lineshape analysis ideas of Alexander and Binsch [95, 96] shows that the whole dynamics is determined by the following Liouville von Neumann equation for the density matrix p ... 

We further wish to emphasize the formal relationship that exists between the optical lineshape and the elements of the molecular density matrix. The general expression for the absorption lineshape is [44] ... 

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