The Nuclear Spin Hamiltonian

The evolution of the density matrix is governed by Eq. (2.10) in which the Hamiltonian for the spin system must be specified. It is noted here that the relaxation effects arising from dissipative interactions between the spin system and the lattice have not been included in the equation. The nuclear spin Hamiltonian contains only nuclear spin operators and a few phenomenological parameters that originate from averaging the full Hamiltonian for a molecular system over the lattice coordinates. These magnetic resonance parameters can, at least in principle, be deduced by quantum chemical calculations [2.3]. The terms that will be needed for discussion in this monograph will be summarized here. 

The Zeeman interaction between the magnetic moment of a nucleus and the static magnetic field Bq is linear in the spin operators  

This interaction is modified by the chemical shielding on the nucleus of the fields produced by the surrounding electrons. The chemical shift interaction can be incorporated into Eq. (2.22) to give 

It is noted that Eq. (2.24) is true only when the Zeeman term is larger than any other interactions that contribute to the spin Hamiltonian. The interaction with the radiofrequency (r.f.) field has the same form as the Zeeman interaction 

The Hrf t) can be made time-independent by a transformation into a coordinate frame that rotates with the radiofrequency uji about the Z axis  

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If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

Except for coi (transition frequencies of the nuclear spin Hamiltonian) all values are temperature-dependent. From the previous subsection the behaviour of a>s is known. From the anomalous contribution to the birefringence which is proportional to (Sp ) we get the information concerning Ai. If we assume that the damping of the soft mode is non-critical (which is generally accepted), Eq. 10 describes a transition from an under-damped mode to an over-damped one as Tc is approached from either side. 

The nuclear spin hamiltonian (H) for the Zeeman (Hz) and the quadrupole (Hq) interactions may be written... 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

The development of the effective Hamiltonian has been due to many authors. In condensed phase electron spin magnetic resonance the so-called spin Hamiltonian [20,21] is an example of an effective Hamiltonian, as is the nuclear spin Hamiltonian [22] used in liquid phase nuclear magnetic resonance. In gas phase studies, the first investigation of a free radical by microwave spectroscopy [23] introduced the ideas of the effective Hamiltonian, as also did the first microwave magnetic resonance study [24], Miller [25] was one of the first to develop the more formal aspects of the subject, particularly so far as gas phase studies are concerned, and Carrington, Levy and Miller [26] have reviewed the theory of microwave magnetic resonance, and the use of the effective Hamiltonian. 

Nuclear relaxation in paramagnetic complexes occurs due to the time dependent terms in the nuclear spin Hamiltonian. The amount of relaxation effect is dependent on the intensity of electron-nuclear interaction and the rate at which this interaction is interrupted. Thus the relaxation rates of ligand nuclei are determined by the two factors, namely, molecular structure and molecular dynamics in solution. Thus the relaxation rates of ligand nuclei shed light on molecular structure and dynamics in solution. 

Direct observation of very small dipolar couplings between distant protons in the overcrowded spectra of biological molecules represents a challenging task. Now Bax and co-workers have presented a method which, by the appropriate manipulation of the nuclear spin Hamiltonian, allows effective decoupling of all protons outside a spectral region of interest and facilitates observation of interactions between remote protons separated by distances of up to 12 A. The method has been applied to measure remote dipolar couplings in an unlabelled nucleic acid. 

At axial symmetry of the R-ion position in a crystal, the main axes of those tensors coincide, Yx Yy" Yx> y Yl denotes a principal crystal axis), so the nuclear spin Hamiltonian can be written as... 

Under conditions at which NMR experiments are usually performed, the interactions between the nucleus and the electromagnetic fields present in its environment (including the interactions with electrons, other nuclei, other ions, and so on) are well described using the concept of the nuclear spin Hamiltonian CHmiciear)- This Hamiltonian contains only terms that depend on the orientation of the nuclear spin and, therefore, its matrix representation is usually given in the m) basis, which corresponds to eigenstates of the Zeeman Hamiltonian (Hz). It is convenient to write the nuclear spin Hamiltonian in the form ... 

The random molecular motional processes, which make the nuclear spin Hamiltonian H (f) time dependent, are responsible for the nuclear spin relaxation. In order to... 

We next consider the effect of finite nuclear size on the nuclear spin Hamiltonian. The electric moments were derived by considering the Coulomb interaction of the nuclear charge density, expanded in a multipole series, with the electrons. By analogy, the magnetic moments are derived by considering the Gaunt interaction of the nucleus with the electrons. It is at this point that we must consider, at least as a formal entity, the nuclear wave function, and from it obtain a nuclear spin density that interacts with the electron spin density. 

As noted earlier, we will confine our discussion to the case when the interaction of the spins with the magnetic field, the Zeeman interaction, is dominant so that the direction of the applied, static field determines the axis of quantization and the experiments yield the component of the various interactions in this direction. It is now convenient to define this as the Z direction. A very detailed discussion of the spin interactions, and their contribution to the nuclear spin Hamiltonian has been given by Smith, Palke and Gerig. 

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