Spectral spin relaxation

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 major limitation in using protein NMR spectroscopy in drug discovery has been the molecular weight limitation imposed by nuclear spin relaxation (line broadening) and increased spectral complexity associated with macromolecules larger than 35 kDa [5]. The most recent developments in NMR spectroscopy aimed at overcoming these problems will be briefly reviewed in Sect. 21.2. 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

Fig. 3. Variation of the completely reduced dipole-dipole spectral density (see text) for the model of a low-symmetry complex for S = 3/2. Reprinted from J. Magn. Reson., vol. 59,Westlund, RO. Wennerstrom, H. Nordenskiold, L. Kowalewski, J. Benetis, N., Nuclear Spin-Lattice and Spin-Spin Relaxation in Paramagnetic Systems in the Slow-Motion Regime for Electron Spin. III. Dipole-Dipole and Scalar Spin-Spin Interaction for S = 3/2 and 5/2 , pp. 91-109, Copyright 1984, with permission from Elsevier.

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

The carbon nuclei are to be identified with I, the proton nuclei with S, and the carbon-proton internuclear distance with r. The spectral density is the Fourier transform of a correlation function which is usually based on a probabilistic description of the motion modulating the dipole-dipole interactions. The spin-spin relaxation time, T2, is usually written directly as a function of spectral densities ( ). 

The lattice models provide useful interpretations of spin relaxation in dissolved polymers and rubbery or amorphous bulk polymers. Very large data bases are required to distinguish the interpretive ability of lattice models from other models, but as yet no important distinction between the lattice models is apparent. In solution, the spectral density at several frequencies can be determined by observing both carbon-13 and proton relaxation processes. However, all the frequencies are rather hl unless T2 data are also included which then involves the prospect of systematic errors. It should be mentioned that only effective rotational motions of either very local or very long range nature are required to account for solution observations. The local... 

Another class of toroidal NMR probes is formed by toroid cavity detectors (TCD) [31]. These TCD have been mounted into cylindrical metallic autoclaves to study spin relaxation effects of gases under pressure [31]. If the toroid cavity detector is the metallic pressure vessel itself it is called a toroid cavity autoclave (TCA) probe [32, 33]. These probes can be tuned to higher resonance frequencies than TCDs and also show better spectral resolution [29]. Figure 2.12 shows the design of a TCA where the autoclave body is built from phosphorus bronze [33] keeping the field distortions of the magnetic field Bq, induced by susceptibility mismatches... 

---