7.5 : Atomic Nuclei: Nuclear Spin State Population Distribution

Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N₋) and spin +½ (N₊) states.

Here, ΔE is the energy difference between the states, k is the Boltzmann constant (1.38 × 10⁻²³ J·K⁻¹), and T is the absolute temperature measured in kelvins. The energy difference can be expressed as hν, where h is Planck’s constant (6.626 × 10⁻³⁴ J·s) and ν is the operating frequency of the NMR instrument.

For example, in an instrument operating at 60 MHz at 298 K, the ratio is slightly less than 1 (0.999991), implying that the lower energy state has approximately 9 to 10 excess nuclei in a total population of about 2,000,000 nuclei. The excess population is small but significant, as these spins are responsible for the net magnetization that produces the NMR signal. Using a higher operating frequency increases the energy gap between the spin states and the excess population.