Perturbation, nuclear spin

We have demonstrated that the principle of multiple selective excitation with subsequent data processing appropriate to disentangle the superimposed responses may successfully be incorporated into selective ID and 2D pulse sequences. This allows to improve the overall efficiency especially of experiments with an inherent low sensitivity. In most of the presented examples a series of single selective pulses has been applied, giving rise to unwanted relaxation losses and setting an upper limit to the number of perturbed nuclear spins. With the application of single multiple selec-... 

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. 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

The perturbation may also be an internal magnetic moment I, arising from a nuclear spin (I is here taken to include the proportionality constants between spin and moment, i.e. it includes the gAfJ-N factor from Section 8.2). 

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

Magnetic dipole interaction Hm (4.47) and electric quadmpole interaction //q (4.29) both depend on the magnetic quantum numbers of the nuclear spin. Therefore, their combined Hamiltonian may be difficult to evaluate. There are closed-form solutions of the problem [64], but relatively simple expressions exist only for a few special cases [65]. In Sect. 4.5.1 it will be shown which kind of information can be obtained from a perturbation treatment if one interaction of the two is much weaker than the other and will be shown below. In general, however, if the interactions are of the same order of magnitude, eQV Jl, and... 

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 ... 

B) How many lines are expected from this model The total number of nuclear spin states is (2 f + 1) x (2I2 + 1) x (2/3 + 1). Thus, if the model structure has six protons (I = 1/2), there should be (2 x 1/2 + l)6 = 26 = 64 nuclear spin states. If some of the nuclei are expected to be equivalent, then the number of lines will be less than the number of spin states, i.e., some of the spin states will be degenerate (to first-order in perturbation theory). Thus, if the six protons are in three groups of two, it is as if you had three spin-1 nuclei and you expect (2 x 1 + l)3 = 33 = 27 distinct lines. If there is one group of four equivalent protons and another group of two, then it is as if you had one spin-2 nucleus and one spin-1 nucleus and you expect (2x2+ 1)(2 x 1+1) =15 lines. 

This is a simplified Hamiltonian that ignores the direct interaction of any nuclear spins with the applied field, B. Because of the larger coupling, Ah to most transition metal nuclei, however, it is often necessary to use second-order perturbation theory to accurately determine the isotropic parameters g and A. Consider, for example, the ESR spectrum of vanadium(iv) in acidic aqueous solution (Figure 3.1), where the species is [V0(H20)5]2+. 

Fig. 4 Relaxation pathways between quadrupole-perturbed Zeeman levels of / = 3/2 nuclear spin. Reprinted from [50]...

Gv( f) covering symmetry67. For orientations of B0 in the mirror plane S, the symmetry group of the spin Hamiltonian is < 9f = C2h(e2f). The direct product base of the nuclear spin functions of two geometrically equivalent nuclei reduces to two classes, containing six A-type and three B-type functions, respectively. Second order perturbation theory applied to H = UtHU, where U symmetrizes the base functions of the Hamil-... 

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... 

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