Nuclear spin Hamiltonian

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 ... 

Ease of manipulation of nuclear spin Hamiltonians (spectral simplification)... 

Average or effective Hamiltonian theory, as introduced to NMR spectroscopy by Waugh and coworkers [55] in the late 1960s, has in all respects been the most important design tool for development of dipolar recoupling experiments (and many other important experiments). In a very simple and transparent manner, this method facilitates delineation of the impact of advanced rf irradiation schemes on the internal nuclear spin Hamiltonians. This impact is evaluated in an ordered fashion, enabling direct focus on the most important terms and, in the refinement process, the less dominant albeit still important terms in a prioritized manner. 

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 third and fourth terms in (1.15) represent the potential energy contributions (in SI units, see General Appendix E) arising from the electron-electron and electron-nuclear interactions, whilst the second term in (1.16) describes the nuclear repulsion term between nuclei with charges Zae and Zf>e. The electron and nuclear spin Hamiltonians introduced into (1.15) are described in detail later. 

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. 

In summary the complete nuclear spin Hamiltonian is given by... 

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. 

Here a is the paramagnetic shift tensor, y is the effective gyromagnetic ratio, and H is the enhanced magnetic field. The quadrupolar energy in eq. (134) includes a pseudoquadrupolar term due to the hyperfine interaction Hhf. A nuclear spin Hamiltonian in the main axes of y and q tensors is written in the form... 

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... 

The effective nuclear spin Hamiltonian of a ligand v has a form analogous to eqs. (135) and (136) ... 

In general, calculating the above effects is like deducing the effective nuclear spin Hamiltonian, described in sect. 3.1. The Hamiltonian of interaction of an impurity ion with surrounding VV ions is written in the general form... 

At higher temperatures the excited states also contribute to the average values, so the corrections become temperature-dependent, similar to the case of nuclear spin Hamiltonians (see sect. 3.3.1). 

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 ... 

All internal nuclear spin Hamiltonians described above present common features that can be explored for a theoretical description of their effects onto the appearance of a NMR... 

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... 

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