Nuclear Spin Relaxation Theory

Chapter 2 focused on the evolution of a nuclear spin system without examining how it achieves thermal equilibrium with the lattice by energy exchange. The lattice consists of all degrees of freedom, except those of the nuclear spins, associated with molecular rotations and translations in physical systems such as liquid crystals. Spin-lattice relaxation describes how the system of nuclear spins evolves towards thermal equilibrium with the large heat reservoir, the lattice. The spin relaxation rates with which the nuclei arrive at their equilibrium magnetization may be experimentally determined. There is a well-defined connection between the relaxation rates and the dynamics of the lattice provided that the coupling interactions between the nuclear spin system and the lattice are known. Thus, nuclear spin relaxation may be used to study motional processes in molecular systems. 

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In-depth treatments of the topic are available in several books [1-6] and in a large number of review articles. The monograph by Dong [6], for example, focuses on aspects like the dynamics of nuclear spins, orientational order, molecular field theories, nuclear spin relaxation theory, director fluctuation and spin relaxation, rotational and translational dynamics, internal dynamics of flexible mesogens, and multiple-quantum and two-dimensional NMR, topics that will be touched upon very briefly here. Re-... 

Dayie K T, Wagner G and Lefeevre J F 1996 Theory and practice of nuclear spin relaxation in proteins Anna. Rev. Phys. Chem. 47 243-82... 

Gordon R. G. Kinetic theory of nuclear spin relaxation in gases, J. Chem. Phys. 44, 228-34 (1966). 

J. Kowalewski and L. Maler, Nuclear Spin Relaxation in Liquids Theory, Experiments and Applications. Taylor Francis, New York, 2006. 

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. 7. NMRD profiles calculated for slightly asymmetric, weakly deformable complexes with different electron spin quantum numbers (a) cylindrically-symmetric ZFS, E = 0 (b) maximum rhombicity E = DjS. Reprinted from J. Magn. Reson. vol. 146, Nilsson, T. Kowalewski, J., Slow-motion theory of nuclear spin relaxation in paramagnetic low-symmetry complexes A generalization to high electron spin , pp. 345-358, Copyright 2000, with permission from Elsevier.

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

The theory of nuclear spin relaxation (see monographs by Slichter [4], Abragam [5] and McConnell [6] for comprehensive presentations) is usually formulated in terms of the evolution of the density operator, cr, for the spin system under consideration from some kind of a non-equilibrium state, created normally by one or more radio-frequency pulses, to thermal equilibrium, described by Using the Bloch-Wangsness-Redfield (BWR) theory, usually appropriate for the liquid state, we can write [7, 8] ... 

The correlation functions G are the same functions which Oppenheimer and Bloom introduced in their theory of nuclear spin relaxation [304], The function has been called the time-dependent pair distribution function it should not, however, be confused with the van Hove correlation function which is often referred to by the same name. The name time-dependent intermolecular correlation function (TDICF) has, therefore, been proposed for G [71]. 

Redfield limit, and the values for the CH2 protons of his- N,N-diethyldithiocarbamato)iron(iii) iodide, Fe(dtc)2l, a compound for which Te r- When z, rotational reorientation dominates the nuclear relaxation and the Redfield theory can account for the experimental results. When Te Ti values do not increase with Bq as current theory predicts, and non-Redfield relaxation theory (33) has to be employed. By assuming that the spacings of the electron-nuclear spin energy levels are not dominated by Bq but depend on the value of the zero-field splitting parameter, the frequency dependence of the Tj values can be explained. Doddrell et al. (35) have examined the variable temperature and variable field nuclear spin-lattice relaxation times for the protons in Cu(acac)2 and Ru(acac)3. These complexes were chosen since, in the former complex, rotational reorientation appears to be the dominant time-dependent process (36) whereas in the latter complex other time-dependent effects, possibly dynamic Jahn-Teller effects, may be operative. Again current theory will account for the observed Ty values when rotational reorientation dominates the electron and nuclear spin relaxation processes but is inadequate in other situations. More recent studies (37) on the temperature dependence of Ty values of protons of metal acetylacetonate complexes have led to somewhat different conclusions. If rotational reorientation dominates the nuclear and/or electron spin relaxation processes, then a plot of ln( Ty ) against T should be linear with slope Er/R, where r is the activation energy for rotational reorientation. This was found to be the case for Cu, Cr, and Fe complexes with Er 9-2kJ mol" However, for V, Mn, and... 

In a theoretical treatment, it is necessary to make approximations in the derivation of the spectral densities (Appendix A.2 - equation (A7)), that is, the Fourier transforms of time correlation functions of perturbations used to express the nuclear spin relaxation times. These theories have been tested against experiments and their limitations have been examined under varying conditions. The advantage of MD simulations to evaluate the theoretical models is the realism of the description and that many approximations in the theoretical model can be tested separately. Because of the conceptual differences between theories and the arbitrariness in their parameterization, it is often not possible discriminate between... 

MD simulations of nuclear spin relaxation in liquids were initiated at a time when the development of theoretical models for many mechanisms was more or less stagnant. Since then simulations have been used in combination with both theory and experiment to develop new ideas, and MD simulations is becoming recognized as a vital tool for the understanding of the relaxation processes. 

J. Kowalewski, L. Nordenskidld, N. Benetis, and P.-O. Westlund, Theory of Nuclear Spin Relaxation in Paramagnetic Systems in Solution, Progr. NMR Spectr., 17 (1985), 141-185. 

There has been a growing recognition of the significance of the symmetrization postulate for nuclear spin relaxation of quantum rotors in the solid state. However, even the conventional theories of the latter phenomenon, based on the classical jump model, are specialized to such an extent that for a proper presentation of the problem a separate review should be provided. Therefore, only a brief reference will be made here to a recent paper where a consistently quantum description of the relaxation behaviour of weakly hindered CD3 rotors is reported.The relevance of the latter work to the content of the present review stems from the fact that the relaxation processes are described therein in terms of essentially the same quantum coherences as those entering the DQR theory of NMR line shapes addressed in Section 4.1. This points to a relative generality of the DQR theory. 

By the same experimental technique, the temperature dependence of the nuclear spin relaxation rates was investigated for the radical cations of dimethoxy- and trimethoxybenzenes [89], The rates of these processes do not appear to be accessible by other methods. As was shown, l/Tfd of an aromatic proton in these radicals is proportional to the square of its hyperfine coupling constant. This result could be explained qualitatively by a simple MO model. Relaxation predominantly occurs by the dipolar interaction between the proton and the unpaired spin density in the pz orbital of the carbon atom the proton is attached to. Calculations on the basis of this model were performed with the density matrix formalism of MO theory and gave an agreement of experimental and predicted relaxation rates within a factor of 2. 

The first microscopic theory for the phenomenon of nuclear spin relaxation was presented by Bloembergen, Purcell and Pound (BPP) in 1948 [2]. They related the spin-lattice relaxation rate to the transition probabilities between the nuclear spin energy levels. The BPP paper constitutes the foundation on which most of the subsequent theory has been built, but contains some faults which were corrected by Solomon in 1955... 

G. Soda and H. Chihara, "Note on the theory of nuclear spin relaxation exact formulae in the weak collision limit," J. Phys. Soc. Japan 36, 954-958 (1974). 

A surprising amount of insight concerning nuclear spin relaxation can be obtained simply by treating the various available spin states as analogous to chemical states linked by kinetics. Although the spin transition probabilities such as W+ remain to be determined either by experiment or by quantum mechanical theory, as do kinetic rate constants, nevertheless the flow of spins obeys essentially the same kinetic laws as does any other equilibrating system. 

There is a relationship between l/Tj and the Knight shift in metals that is very sensitive to electron correlation and exchange. To explain this and to derive its implications for expanded alkali metals, we digress briefly to summarize some results of the general theory of nuclear spin relaxation in metals. For noninteracting electrons, the theory of nearly free electrons gives an explicit expression for the integral in Eq. (3.10)... 

T.C. Stringfellow, Density Matrix Theory and Nuclear Spin Relaxation (University of Wisconsin—Madison School of Pharmacy, Madison, 2003). 

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