Dipolar spectral density

The symbol Re(K ((o)) denotes the real part of the complex spectral density, corresponding to the autocorrelation of the dipolar interactions, while Re(i (co)) is its counterpart for the scalar interaction. The symbol Re(K (a>)) denotes the spectral density describing the cross-correlation of the two parts of the hyperfine interaction. The cross-correlation vanishes at the MSB level of the theory, but in the more complicated case of the lattice containing the electron spin, the cross term may be non-zero. A general expression for the dipolar spectral density is ... 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

Similarly,27 the dipolar-csa cross-correlation spectral density can be expressed as follows... 

The measured spin relaxation parameters (longitudinal and transverse relaxation rates, Ri and P2> and heteronuclear steady-state NOE) are directly related to power spectral densities (SD). These spectral densities, J(w), are related via Fourier transformation with the corresponding correlation functions of reorientional motion. In the case of the backbone amide 15N nucleus, where the major sources of relaxation are dipolar interaction with directly bonded H and 15N CSA, the standard equations read [21] ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

When the nuclear motion is a random proceas, one obtains, for (i), from the theory of random processes (61) the normalized spectral density of the mean square dipolar magnetic field versus frequency (3). The spectral density near a NMR frequency is proportional to the probability of a... 

Fig. 3.14. Plot of the spectral density functions for dipolar relaxation in the presence of an axially symmetric g tensor. Conditions gn = 2.3, gj = 2.0, xc = 2 x 10-9 s, 6 — 0° (upper curve) and 0 = 90° (middle curve) compared with the Solomon behavior (lower curve).

The transition probabilities depend on the mean squared interaction energy relative to the mechanism which causes the transition, times the value of the spectral density at the required frequencies (Eq. (3.14)). The square of the dipolar interaction energy is, as usual (see Eq. (1.4) and Appendix V), proportional to (p, 1 1x2/r3)2, where p and p2 are the magnetic moments of the two spins. The actual equations are... 

This result suggests, if it is assumed that a C-H heteronuclear dipolar relaxation mechanism is operative, that methyl protons dominate the relaxation behavior of these carbons over much of the temperature range studied despite the 1/r dependence of the mechanism. The shorter T] for the CH as compared to the CH2 then arises from the shorter C-H distances. Apparently, the contributions to spectral density in the MHz region of the frequency spectrum due to backbone motions is minor relative to the sidegroup motion. The T p data for the CH and CH2 carbons also give an indication of methyl group rotational frequencies. 

Some systems cannot be well described by translational motion, so instead they require a model based on rotational diffusion. The commonly used model is one where the nuclei and radical form a bound complex, then this complex rotates to modulate dipolar coupling.74 Here, the overall correlation time consists of the rotational correlation time of the solvent complex, xT, and the exchange rate of molecules in and out of the complex, tm, where 1 /tc 1 /tr I 1 /tm. The form of this spectral density function is simpler4,25 ... 

It has been known for a long time that the kind of simplistic distance calibration suggested by Eq. [8] may be subject to systematic errors. First, the intensity of an NOE depends on the spectral density function for the reorientation of the vector between relaxing nuclei. This means that Eq. [8] is valid only if the reference distance and unknown distance are undergoing the same motions. As this is not likely to be the case, distance calibrations have attempted to allow for the possibility of systematic errors.23 Equation [8] also assumes that the dipolar relaxation can be considered in terms of isolated spins relaxing each other. In the presence of spin diffusion, this will lead to a systematic underestimation of distances.41 57 58... 

The size of the dipolar interaction depends primarily on the distance and orientation between the two dipoles, not on the correlation time. By contrast, the rate of change of the dipolar interactions depends on and hence is relevant to the efficiency of relaxation. The total amount of fluctuating fields is independent of Tc, although Tc determines the upper limit of the frequencies of the fields. The three curves in Figure A5-1 must enclose the same area, but their upper limits vary. In curve (a), molecular tumbling is very rapid and the spectral density is low. In curves (b) and (c), the upper limit of frequencies decreases with the lengthening of Tc, so the spectral density increases proportionately to maintain a constant area. 

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