Spin system, nuclear perturbation

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... 

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... 

For molecules that are tumbling rapidly so that magnetic dipole interactions can be neglected, the sum of and W1 is adequate as the complete Hamiltonian to determine energy levels for the nuclear spin system. However, as we noted in Section 5.5, additional terms must be added to account for external perturbations, such as strong rf fields. In this chapter we take the steady-state Hamiltonian as... 

In Chapter 2 we found that a perturbed nuclear spin system relaxes to its equilibrium state or steady state by first-order processes characterized by two relaxation times Ti, the spin-lattice, or longitudinal, relaxation time and T2, the spin-spin, or transverse, relaxation time. Thus far in our treatment of NMR we have not made explicit use of relaxation phenomena, but an understanding of the limitations of many NMR methods requires some knowledge of the processes by which nuclei relax. In addition, as we shall see, there is a great deal of information of chemical value, both structural and dynamic, that can be obtained from relaxation phenomena. 

The goal of all NMR experiments is to determine the change in the separation of the energy levels for different environments. In its simplest form, an NMR experiment consists of three parts, the preparation of the nuclear spin system by placing it in an external magnetic field, its perturbation by applying a pulse of rf radiation, and the detection of phenomena accompanying its return to the initial state when the perturbation is removed. 

Bearing in mind that the nuclear spin system is an ensranble perturbed by the rf-field, we have to consider the possibility that energy from the rf-field is transferred to the nuclear spins and dissipated further to the crystal lattice. These effects can be described by relaxation times that characterize the rates with which the system returns to thermal equilibrium after the perturbation has been switched off. There are the longitudinal or spin-lattice relaxation time Tj and the transverse or spin-spin relaxation time Tj. Including the relaxation effects the equations of motion in the rotating frame (cf. Eq. (19)) are... 

To proceed, it is necessary to consider the straightforward perturbation treatment of two weakly interacting subsystems, in our case the nuclear moment on the one hand and the electrons around fixed nuclei of the molecule on the other. Whereas our treatment is focused on a nuclear spin system and an electronic system, the approach is general. Suppose the Hamiltonian of the combined system is given by... 

Here, S is the pseudo-spin of the system, and F is the KS Fock operator up to first order in the external field or the nuclear spin magnetic perturbation. LWA is suitable for Kramers doublets. Assuming no spatial degeneracy, a pair of degenerate Kramers orbitals is initially calculated from SO DFT by assigning equal occupations of 0.5 to two frontier orbitals. One then chooses one of them, (p, and constructs from its real and imaginary parts of the spin a and l3 components the Kramers pair d>i, 2 as... 

Such reasoning may be extended to more complicated systems. If, however, the magnitude of the spin-spin splitting is comparable to vqB, the chemical shift, this first-order treatment is no longer applicable since the nuclear spin energy levels become perturbed and the spectra become more complex. For the general analysis of NMR spectra, the reader is referred to Pople et al. (109), Roberts (119) and Corio (17a). 

This would imply a very simple linear Zeeman effect but, as we show in chapter 8, additional terms describing the nuclear spin rotation interaction and the spin-spin interaction make the system much more interesting. The nuclear spin transitions are induced by an oscillating magnetic field applied perpendicular to the static magnetic field, the perturbation being represented, for example, by the term... 

The rotational and Zeeman perturbation Hamiltonian (X) to the electronic eigenstates was given in equation (8.105). It did not, however, contain terms which describe the interaction effects arising from nuclear spin. These are of primary importance in molecular beam magnetic resonance studies, so we must now extend our treatment and, in particular, demonstrate the origin of the terms in the effective Hamiltonian already employed to analyse the spectra. Again the treatment will apply to any molecule, but we shall subsequently restrict attention to diatomic systems. 

The partition of the total system into a spin part and a lattice part is, in principle, fuzzy. Other nuclear or electronic spins can either be included in the spin space or into the lattice. In a perturbation treatment, it depends on the interaction strength and on the time-scales of different processes. If nuclear spins are interacting over long periods (e.g. within a small molecule), the different spins cannot be considered separately. However, electron spins can in many cases be put in the lattice space since the difference in time scales for the electron spin and nuclear spin dynamics is prohibitive an effective coupling. 

NMR experiments are an ideal tool for studying the structure and dynamics of liquids, because the measurement itself does not perturb the molecular properties of the system. Furthermore, by selectively studying the nuclear spin of different elements, one can probe different part of the system. In combination with advanced pulse sequence experiments different interactions and processes can be studied. 

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