Nuclear Magnetic Energy Levels

In contrast to bar magnets, the magnetic moments of spinning nuclei do not align in the direction of B0, no matter how strong this field is. Instead, the Larmor precession is accelerated by increasing the strength B0 of the field vector B0  

The energy of a magnetic moment ft in a field B0 is given by the product of B0 and /r0. Thus from eq. (1.3), 

Following quantum mechanical rules, a nucleus with total spin quantum number / may occupy (21 + 1) different energy levels when placed in a magnetic field. For nuclei with 7 = 4, e.g. 1H, 13C, 15N, 19F, 31P, two spin alignments relative to B0 arise (Fig. 1.3) these are symbolized by + and — 

Precession of the nuclear spins about B0 is energetically favored (Fig. 1.3 (b)) since the component of the nuclear magnetic moment vector fi in the direction of B0 reinforces the magnetic field (Fig. 1.3 (a)). 

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At equilibrium, the nuclear magnetic energy levels are populated according to a Boltzmann distribution which favors the lower state. For the two orientations relative to B0 of nuclei with 1 = the spin populations may be symbolized by N+ and N (Fig. 1.3 (b)). The distribution 7V+/7V can be expressed by the Boltzmann factor, recalling that... 

Relaxation can be described in terms of the magnetization vector components. At resonance, the equilibrium magnetization M0 parallel to B0 decreases to Mz, due to the transitions between the nuclear magnetic energy levels caused by the alternating field B,. Following resonance, the equilibrium of the nuclear spins with their lattice and with each other is restored by relaxation. 

Clearly, nuclear screening decreases the spacing of nuclear magnetic energy levels. An increase in magnetic shielding... 

NMR is simply described as a coherent radiofrequency (rf) spectroscopy involving the nuclear magnetic energy levels. Hence the technique requires that a nucleus possesses a magnetic moment fi) which is related to its quantum mechanical spin (/) via /r = yhl, where y is the gyromagnetic ratio of the nucleus. The spin can have various z components (nh) where —Ispin states is lifted by an external... 

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. ...

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

We saw in Chapter 7 that the resonance frequency of a quadrupolar nucleus is dependent on the orientation of the molecule in which it resides. Molecular tumbling now causes fluctuating electric fields, which induce transitions among the nuclear quadrupole energy levels. The resulting nuclear relaxation is observed in the NMR just as though the relaxation had occurred by a magnetic mechanism. 

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... 

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