Effect of the Boltzmann Distribution

Several important points are contained in Eq. 2.12. First, the transition probability increases quadratically with both y and Bl. [Whereas the energy absorbed varies as B j2, the observed NMR signal, which is proportional to an induced voltage in a coil (see Chapter 3), varies linearly with f. ] Second, the matrix element furnishes the selection rule Am = 1, so that transitions are permitted only between adjacent energy levels and thus give only a single line at a frequency 

the resonance condition is expressed in the delta function. Actually, the delta function would predict an infinitely sharp line, which is unrealistic therefore, it is replaced by a line shape function g(v), which has the property that 

For nuclei with / = l/2 there is only one transition, so Eq. 2.16 becomes 

The tendency of nuclei to align with the magnetic field and thus to drop into the lowest energy level is opposed by thermal motions, which tend to equalize the populations in the 21 + 1 levels. The resultant equilibrium distribution is the usual compromise predicted by the Boltzmann equation. For simplicity we consider only nuclei with I = V2, so that we need include only two energy levels, the lower corresponding to m = +V2 (state a) and the upper to m = — /2 (state / ). For the I = /2 system the Boltzmann equation is 

By substitution of the values of E from Eq. 2.5, and by introducing Eqs. 2.1 and 2.2, we find that this becomes 

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