Nuclear spin energy levels

From calculating the mean value of the Hamiltonian (12.82), we obtain the energy of the nuclear spins in the magnetie field 

Note that since HJab has the dimension of the energy, then Jab itself is a frequency and may be expressed in Hz. 

Due to the presence of the rest of the molecule (electron shielding) the Larmor frequency va = o-a) is changed by (ta with respect to the Lar- 

Example 3. The carbon nucleus in an external magnetic field 

Let us consider a single carbon nucleus (spin quantum number /c = j) in a molecule. 

---

Nuclear spin energy levels for a proton or carbon-13, / = 1/2. 

Figure 4 Schematic representation of the populations of the nuclear spin energy levels of a quadrupolar nucleus with spin 5/2 (such as Mg) under a strong magnetic field and a perturbative quadrupole coupling showing (A) populations at thermal equilibrium, (B) populations after complete saturation of the satellite transitions, and (C) populations after complete inversion of the satellite transitions, following the order first, inversion of STl and ST4 and then inversion of ST2 and ST3. The numbers at left of each level (named pj in the text) are proportional to the population of that level, with —hVl/ 2k T= 0. ...

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the magnitude of the spin-spin splitting is comparable to vqB, the chemical shift, this first-order treatment is no longer applicable since the nuclear spin energy levels become perturbed and the spectra become more complex.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). 

Equation (8.23) is for gN positive for gN negative, a minus sign must be added.] Although there are 2/+1 nuclear-spin energy levels, they are equally spaced, and the selection rule allows only transitions between adjacent levels hence we get a single NMR absorption frequency. 

K f. U, + Ix2 tj) 2. (It might be thought necessary to consider also the effect of the relative populations of the levels on the transition intensities, but the separation between nuclear-spin energy levels is much less than kT, so that the very slight differences in population can be ignored here.)... 

In the first-order approximation to NMR spectra when H° is taken as the first two terms in (8.78), the nuclear-spin energy levels are, in general, degenerate. Show for the A2X2 case (where A and X are protons) that the functions (8.79) are the correct zeroth-order wave functions for the perturbation H where H is the last term in (8.78). 

RF radiation (radio waves) matches the spacing between nuclear-spin energy levels artificially split by a strong magnetic field. 

To decide which nuclear spins are responsible for observed effects, it is sometimes useful to perform experiments on samples selectively isotopi-cally enriched. The interaction of the nuclear momenta with an applied magnetic field splits the nuclear spin energy levels, and the transitions between those levels can be observed and studied. For the particular case of / = 3, as in proton NMR, the splitting between the two w7 = nuclear... 

Transitions between nuclear spin energy levels are modulated by ... 

Redfield limit, and the values for the CH2 protons of his- N,N-diethyldithiocarbamato)iron(iii) iodide, Fe(dtc)2l, a compound for which Te r- When z, rotational reorientation dominates the nuclear relaxation and the Redfield theory can account for the experimental results. When Te Ti values do not increase with Bq as current theory predicts, and non-Redfield relaxation theory (33) has to be employed. By assuming that the spacings of the electron-nuclear spin energy levels are not dominated by Bq but depend on the value of the zero-field splitting parameter, the frequency dependence of the Tj values can be explained. Doddrell et al. (35) have examined the variable temperature and variable field nuclear spin-lattice relaxation times for the protons in Cu(acac)2 and Ru(acac)3. These complexes were chosen since, in the former complex, rotational reorientation appears to be the dominant time-dependent process (36) whereas in the latter complex other time-dependent effects, possibly dynamic Jahn-Teller effects, may be operative. Again current theory will account for the observed Ty values when rotational reorientation dominates the electron and nuclear spin relaxation processes but is inadequate in other situations. More recent studies (37) on the temperature dependence of Ty values of protons of metal acetylacetonate complexes have led to somewhat different conclusions. If rotational reorientation dominates the nuclear and/or electron spin relaxation processes, then a plot of ln( Ty ) against T should be linear with slope Er/R, where r is the activation energy for rotational reorientation. This was found to be the case for Cu, Cr, and Fe complexes with Er 9-2kJ mol" However, for V, Mn, and... 

Chapter 2 considers how we can understand the form of the NMR spectrum in terms of the underlying nuclear spin energy levels. Although this approach is more complex than the familiar successive splitting method for constructing multiplets it does help us understand how to think about multi-plets in terms of active and passive spins. This approach also makes it possible to understand the form of multiple quantum spectra, which will be useful to us later on in the course. The chapter closes with a discussion of strongly coupled spectra and how they can be analysed. 

In molecular hydrogen, the existence of nuclear-spin energy levels is responsible for the distinction between ortho and para hydrogen, which correspond to the triplet and singlet (i.e., parallel and antiparallel) orientations, respectively, of the two nuclei in H2. Because of the coupling of the rotational and spin levels, ortho and para hydrogen differ in specific heat and certain other properties. The correlated orientation of the nuclear spins in para H2 has re-... 

The first microscopic theory for the phenomenon of nuclear spin relaxation was presented by Bloembergen, Purcell and Pound (BPP) in 1948 [2]. They related the spin-lattice relaxation rate to the transition probabilities between the nuclear spin energy levels. The BPP paper constitutes the foundation on which most of the subsequent theory has been built, but contains some faults which were corrected by Solomon in 1955... 

For the splitting of the nuclear spin energy levels in the presence of an external magnetic field, we then obtain... 

Signal Averaging Empirical Hamiltonian Nuclear Spin Energy Levels The Ramsey Theory of the NMR Chemical Shift (4(g))... 

The method used to excite the nuclei and achieve resonance must clearly be capable of covering all of the Larmor frequencies in the sample. This is achieved in the Fourier transform (FT) method by simultaneously exciting all the Larmor frequencies by application of a pulse (short burst) of rf signal (Bj) at or near all vqS, which results in the equalization of the populations of the nuclear spin energy levels. Equilibrium spin populations are reestablished in a free-induction-decay (FID) process following the rf pulse. The vector diagram in Fig. 20.5 can be used to visualize the effect of the rf pulse (61) on the nuclear spins and their subsequent FID to equilibrium. 

---