Boltzmann excess

Figure 1.1 Representation of the precession of the magnetic moment about the axis of the applied magnetic field, Bo- The magnitude /r, of the vector corresponds to the Boltzmann excess in the lower energy (a) state.

Any absorption signal observed in a spectroscopic experiment must originate from excess of the population in the lower energy state, the so called Boltzmann excess, which is equal to where TVp and A are the populations in the lower (P) and... 

Even at the highest fields, the NMR experiment would not be practicable if mechanisms did not exist to restore the Boltzmann equilibrium that is perturbed as the result of the absorption of electromagnetic radiation in making an NMR measurement. These mechanisms are known by the general term of relaxation and are not confined to NMR spectroscopy. Because of the small magnitude of the Boltzmann excess in the NMR experiment, relaxation is more critical and more important in NMR than in other forms of spectroscopy. 

A consequence of the small Boltzmann excess in NMR experiments arises from the general spectroscopic principle that absorption cannot occur unless some mechanism exists for a radiationless transition that can restore the excess population in the lower energy state. This is related to the fact that upward and downward transitions are equally probable on collision with an appropriate energy quantum. Therefore excess absorption, that is, observable absorption signals, can only originate from unequal populations, as stated above. 

The mechanisms of radiationless transitions from the upper to the lower energy states are particularly critical in NMR spectroscopy because of the small Boltzmann excess. These mechanisms are termed relaxation and are characterized by their relaxation times T, which are equal to half the time necessary to restore equilibrium by the mechanism considered. Clearly, large values of T indicate inefficient relaxation. Two relaxation mechanisms are important ... 

Since the populations of Hx in the a and j8 states are almost completely equal (recall the vanishingly small Boltzmann excess discussed previously), the two transitions are of equal probability and hence Ha will give rise to a symmetrical doublet. 

The excess of unexcited nuclei over excited nuclei is called the Boltzmann excess. When no radiation falls on the sample, the Boltzmann excess is maximum, Ax. However, when radiation falls on the sample, an increased number of ground-state nuclei become excited and a reduced number remain in the ground state. If the RF field is kept constant, a new equilibrium is reached and the Boltzmann excess decreases to As. When As = Ax, absorption is maximum. When As = 0, absorption is zero. The ratio As/Ax is called Zq, the saturation factor. 

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