Nuclear spin systems

NMR, nuclear magnetic resonance, is an analytical technique based on the energy differences of nuclear spin systems in a strong magnetic field. It is a powerful technique for structural elucidation of complex molecules. 

Free induction decay A decay time-domain beat pattern obtained when the nuclear spin system is subjected to a radiofrequency pulse and then allowed to precess in the absence of Rf fields. 

A transformation into the rotating frame of reference of the nuclear spin system and integrating over all positions then allows us to rewrite eqn (5) as... 

First order ENDOR frequencies of nonequivalent nuclei or of pairs of magnetically equivalent nuclei are given by Eq. (3.3) which is derived from the direct product spin base. To obtain correct second order shifts and splittings, however, adequate base functions have to be used. We start the discussion of second order contributions with the most simple case of a single nucleus and will then proceed to more complex nuclear spin systems. 

We now come back to the simplest possible nuclear spin system, containing only one kind of nuclei 7, hyperfine-coupled to electron spin S. In the Solomon-Bloembergen-Morgan theory, both spins constitute the spin system with the unperturbed Hamiltonian containing the two Zeeman interactions. The dipole-dipole interaction and the interactions leading to the electron spin relaxation constitute the perturbation, treated by means of the Redfield theory. In this section, we deal with a situation where the electron spin is allowed to be so strongly coupled to the other degrees of freedom that the Redfield treatment of the combined IS spin system is not possible. In Section V, we will be faced with a situation where the electron spin is in... 

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

There is, in addition, another generally different time required to specify the radiofrequency behavior of the nuclear spin system. This time is called the transverse relaxation time and is the time constant for the exponential decay of the transverse (x and y) components of nuclear magnetization M- and M ... 

Optimization of the HMQC technique for the IS nuclear spin system was examined, to avoid the appearance of unwanted peaks or reduce their intensity. In such spin systems I represents a metal atom of spin 1/2 or 1 and S represents n atoms of spin 1/2 coordinated to I. The optimization method can be applied to the analysis of the Li HMQC or HSQC spectra. ... 

The magnetogyric ratio 7 is a constant for a given nucleus. When a nucleus is subjected to a magnetic field H(), the energy of the nuclear spin system is... 

As already mentioned (Sections 1.7.4 and 3.13), when a nuclear spin system experiences fluctuating magnetic fields, relaxation occurs. If these fields are... 

We have already defined the equilibrium magnetization of a spin / in a given magnetic field Bq as Mz(oo), where the (oo) refers to the fact that the sample must have been exposed to the field for time sufficiently long for equilibrium magnetization to be virtually achieved. After any perturbation from equilibrium of the nuclear spin system such that, at time zero after the perturbation, Mz(0) Mz(oo), the system will tend to return to equilibrium with a simple rate law of the type... 

Relaxation theory of nuclear spin systems is well documented in several books18-24 and review articles.4-25-26 Therefore, the theory presented in this chapter is limited to a summary of some of the basic concepts crucial for understanding the material in the following sections. Furthermore, the discussion will be focused on dipolar relaxation, which is known to be the dominant relaxation mechanism in most molecules of chemical interest. For a detailed treatment of other mechanisms, the reader is referred to appropriate review articles.4-18"26... 

A systematic investigation of the application of Lie algebra to NMR was presented.29 The symmetry properties of the nuclear spin systems were naturally included in selection of the sets of the basis operators. With this theoretical framework, the existing sets of basis operators used for various specific purposes can be treated in a unified manner and their respective advantages and disadvantages can be evaluated. A number of 2H MAS spectra calculated on the basis of that theoretical framework are shown in Fig. 2. The... 

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