Relaxation theory

Redfleld A G 1996 Relaxation theory density matrix formulation Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chichester Wiley) pp 4085-92... 

A good introductory textbook, includes a nice and detailed presentation of relaxation theory at the level of Solomon equations. 

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

Comparison between Different Relaxation Theories Available in the Literature... 

Even when they have a partial crystallinity, conducting polymers swell and shrink, changing their volume in a reverse way during redox processes a relaxation of the polymeric structure has to occur, decreasing the crystallinity to zero percent after a new cycle. In the literature, different relaxation theories (Table 7) have been developed that include structural aspects at the molecular level magnetic or mechanical properties of the constituent materials at the macroscopic level or the depolarization currents of the materials. 

Storozhev A. V. Nonresonance effects in the binary relaxation theory, Chem. Phys. 138, 81-8 (1989). 

The first example of chemically induced multiplet polarization was observed on treatment of a solution of n-butyl bromide and n-butyl lithium in hexane with a little ether to initiate reaction by depolymerizing the organometallic compound (Ward and Lawler, 1967). Polarization (E/A) of the protons on carbon atoms 1 and 2 in the 1-butene produced was observed and taken as evidence of the correctness of an earlier suggestion (Bryce-Smith, 1956) that radical intermediates are involved in this elimination. Similar observations were made in the reaction of t-butyl lithium with n-butyl bromide when both 1-butene and isobutene were found to be polarized. The observations were particularly significant because multiplet polarization could not be explained by the electron-nuclear cross-relaxation theory of CIDNP then being advanced to explain net polarization (Lawler, 1967 Bargon and Fischer, 1967). 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

When other relaxation mechanisms are involved, such as chemical-shift anisotropy or spin-rotation interactions, they cannot be separated by application of the foregoing relaxation theory. Then, the full density-matrix formalism should be employed. 

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

Rather sophisticated applications of Mossbauer spectroscopy have been developed for measurements of lifetimes. Adler et al. [37] determined the relaxation times for LS -HS fluctuation in a SCO compound by analysing the line shape of the Mossbauer spectra using a relaxation theory proposed by Blume [38]. A delayed coincidence technique was used to construct a special Mossbauer spectrometer for time-differential measurements as discussed in Chap. 19. 

Recent solid state NMR studies of liquid crystalline materials are surveyed. The review deals first with some background information in order to facilitate discussions on various NMR (13C, ll, 21 , I9F etc.) works to be followed. This includes the following spin Hamiltonians, spin relaxation theory, and a survey of recent solid state NMR methods (mainly 13C) for liquid crystals on the one hand, while on the other hand molecular ordering of mesogens and motional models for liquid crystals. NMR studies done since 1997 on both solutes and solvent molecules are discussed. For the latter, thermotropic and lyotropic liquid crystals are included with an emphasis on newly discovered liquid crystalline materials. For the solute studies, both small molecules and weakly ordered biomolecules are briefly surveyed. 

E. The Brownian-Dynamic model Microscopic Formulation of Onsager Relaxation Theory... 

The relaxation theory used in the Appendix to describe the principle of TROSY clearly tells us what to expect, but it is always a little more satisfying if one can obtain a simple physical picture of what is happening. We consider a system of two isolated scalar coupled spins of magnitude %, 1H (I) and 15N (S), with a scalar coupling constant JHN. Transverse relaxation of this spin system is dominated by the DD coupling between spins XH and 15N and by the CSA of each individual spin. The relaxation rates of the individual multiplet components of spin 15N are now discussed assuming an axially symmetric 15N CSA tensor with the axial principal component parallel to the 15N-XH vector as shown in Fig. 10.2. 

On the basic of relaxation theory the concept of TROSY is described. We consider a system of two scalar coupled spins A, I and S, with a scalar coupling constant JIS, which is located in a protein molecule. Usually, I represents H and S represents 15N in a 15N-1H moiety. Transverse relaxation of this spin system is dominated by the DD coupling between I and S and by CSA of each individual spin. An additional relaxation mechanism is the DD coupling with a small number of remote protons, / <. The relaxation rates of the individual multiplet components in a single quantum spectrum may then be widely different (Fig. 10.3) [2, 9]. They can be described using the single-transition basis opera-... 

Besides the benefits of the increasing correlation time of the virus, the quantitative interpretation of the saturation transfer might be limited by the breakdown of the traditional relaxation theory based on the motional nar-... 

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 ... 

A more general formulation of relaxation theory, suitable for systems with scalar spin-spin couplings (J-couplings) or for systems with spin quantum numbers higher than 1/2, is known as the Wangsness, Bloch and Redfield (WBR) theory or the Redfield theory 17). In analogy with the Solomon-Bloembergen formulation, the Redfield theory is also based on the second-order perturbation approach, which in certain situations (not uncommon in... 

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc <5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1> the expan-... 

A general trend which could be noticed over the last few years and which may be expected to develop further in the near future involves a closer coupling between the use of general tools of computational chemistry (ab initio and semi-empirical quantum chemistry, statistical-mechanical simulations) and relaxation theory. When applied to model systems, the computational chemistry methods have the potential of providing new insights on how to develop theoretical models, as well as of yielding estimates of the parameters occurring in the models. 

The first step in the characterization of a new super-paramagnetic colloid is obviously the evaluation of its relaxometric properties, which determine its potential efficiency for MRI (27,28). Relating these valuable relaxometric data to morphological and physical properties of the particles may be carried out thanks to a proton relaxivity theory. 

Finally, we should mention the sample temperature control. It is a direct consequence of all relaxation theories that, in any FFC NMRD application. 

---