Population of Energy levels:

When placed in an external magnetic field, all the spin magnetic moments in the sample, do not occupy the lowest available energy state. You know that the population of different energy levels is governed by Boltzmann distribution law. If n₁ and n₂ are number of spins in the lower and upper energy states in that order, their ratio at a given temperature, is given through

 n₂ (upper)/ n₁ (lower) = e ^{( - ?E/ kT )}                                                                            

where, ?E is the energy difference between the two energy levels and k is the Boltzmann constant (k = 1.380658 ×10^-23 J/K). As the energy difference between the two spin states is extremely small, an upper level will always be appreciably populated at all the temperatures above absolute zero (0 K).  Let us calculate this ratio for protons in a magnetic field of 1.5 T at room temperature (300 K); ?E is computed using the Eq. 12.4.

?E = g_Nµ_NB₀

= 5.585× 5.05 × 10^-27 J T^-1 × 1.5 T

= 4.23 × 10^-26 J

Since ?E/kT is very small, e ^(-?E/kT) may be approximated to (1- ?E/kT). Thus, the ratio of nuclear spin populations in two energy states will be:

=1-(4.23×10-²⁶/1.380658×10^-23×300)=1-1.02×10^-5

= 0.9999898

Since transition from lower state to upper state involves absorption of radiation and that from the higher state to lower state the emission of radiation, no absorption can occur unless lower energy level has excess of protons. In the example given above, it is found that the lower energy level has about 2 excess spins for every 10⁵ spins in the upper level. Though, it is a very small number but it is finite and hence absorption is observed. In the absence of this small excess population, no NMR could be observed.