# When do you use the Boltzmann Distribution in NMR?

Jan 16, 2016

You use the Boltzmann distribution to calculate the relative number of nuclei in each spin state.

#### Explanation:

The magnetic moments of the nuclei in a magnetic field can align either parallel to the field (lower energy) or antiparallel (higher energy).

The energy separation ΔE between these states is relatively small, and the energy from thermal collisions is sufficient to place many nuclei into higher energy spin states.

The Boltzmann distribution describes the number of nuclei in each spin state.

N_+/N_(-) = e^-((ΔE)/(kT))

At room temperature, the number of spins in the lower energy level, ${N}_{-}$, slightly outnumbers the number in the upper level, ${N}_{+}$.

In a 60 MHz machine at 298 K,

${N}_{+} / {N}_{-} = \text{1 000 000"/"1 000 009}$

In other words, the lower energy spin state has a population excess of about
9 in 2 million molecules.

Using a higher frequency (increasing ΔE) increases the population excess.