Excess population of nuclei

There is no net loss of energy, but the spread of energy among the contiguous nuclei concerned results in broadening of band. This relaxation does not contribute to the maintenance of the excess population of nuclei in a lower energy state. 

FIGURE 3.9 The excess population of nuclei in the lower spin state at 60 MHz. 

Figure 1 (A) The precession of an individual magnetic moment n about the external magnetic field Bq. (B) The precession of magnetic moments in the a (mi = +5) and (m, = -5) spin states. (C) The resultant magnetic moment Af of a large number of nuclei of the same kind, reflecting the small excess population of nuclei in the a spin state.

The transition probability for the upward transition (absorption) is equal to that for the downward transition (stimulated emission). The contribution of spontaneous emission is neglible at radiofrequencies. Thus, if there were equal populations of nuclei in the a and f spin states, there would be zero net absorption by a macroscopic sample. The possibility of observable NMR absorption depends on the lower state having at least a slight excess in population. At thermal equlibrium, the ratio of populations follows a Boltzmann distribution... 

Second, since the magnetogyric ratio of a nucleus is smaller than that of hydrogen (Table 5.2), nuclei always have resonance at a frequency lower than protons. Recall that at lower frequencies, the excess spin population of nuclei is reduced (Table 5.3) this, in turn, reduces the sensitivity of NMR detection procedures. 

When a high-power decoupling field is applied to one of the two nuclei, it results in an equalization of the populations of levels 1 and 3, as well as of levels 2 and 4. What this means, in effect, is that half the excess d) of nuclei in levels 3 and 4 are promoted to levels 1 and 2. The populations of the levels at equilibrium and immediately after application of the decoupling field are shown in Table 3.3. 

Some nuclei commonly observed by nuclear magnetic resonance are listed with their important physical constants in Table VI. The difference in the population of nuclei between energy states is usually very small, with lower states being occupied by only a few excess nuclei per million. If the states were equally populated, net absorption... 

As mentioned in the text, there is only a slight excess of nuclei in the ground state (about 13 in a million protons at 100 MHz). Would you expect in the case of a C-NMR experiment for the same population difference to prevail ... 

Is there any reason to expect that there will be an excess of nuclei in the lower spin state The answer is a qualified yes. For any system of energy levels at thermal equilibrium, there will always be more particles in the lower state(s) than in the upper state(s). However, there will always be some particles in the upper state(s). What we really need is an equation relating the energy gap (AE) between the states to the relative populations of (numbers of particles in) each of those states. This time, quantum mechanics comes to our rescue in the form of the Boltzmann distribution ... 

Our NMR theory is almost complete, but there is one more thing to consider before we set about designing a spectrometer. We indicated previously that at equilibrium in the absence of an external magnetic field, all nuclear spin states are degenerate and, therefore, of equal probability and population. Then, when immersed in a magnetic field, the spin states establish a new (Boltzmann) equilibrium distribution with a slight excess of nuclei in the lower energy state. 

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