Title: Boltzmann distribution of nuclear spins
Long Title: IUPAC Gold Book - Boltzmann distribution of nuclear spins
DOI: 10.1351/goldbook.08316
Status: current

Definition
Distribution of nuclear spins among their possible energy levels at thermal equilibrium.

Note
For a positive magnetogyric ratio, produces population excess (polarization) in the direction of \(\boldsymbol{B_{0}}\). The population difference between the spin state can be expressed as \[\frac{N_i}{N} = \frac{e^{-{\varepsilon}_{i}/kT}}{\sum_{j=1}^{M}{e}^{-{\varepsilon}_{j}/kT}}\] where \(T\) is the thermodynamic temperature, \(k\) is the Boltzmann constant, \(N_{\rm{i}}\) is the number of spins in excited state \(i\), \(N\) is the total number of spins and \(\varepsilon_{i}\) and \(\varepsilon_{j}\) are the energies of states \(i\) and \(j\) respectively, and \(M\) is the number of possible energy levels.

Related Term
- energy levels: https://goldbook.iupac.org/terms/view/08246

Source
- PAC, 2021, 93, 647. 'Glossary of methods and terms used in analytical spectroscopy (IUPAC Recommendations 2019)' on page 682 (https://doi.org/10.1515/pac-2019-0203)

Other Outputs
- html: https://goldbook.iupac.org/terms/view/08316/html
- json: https://goldbook.iupac.org/terms/view/08316/json
- xml: https://goldbook.iupac.org/terms/view/08316/xml

Citation: Citation: 'Boltzmann distribution of nuclear spins' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.08316
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Accessed: 2026-06-28T11:10:51+00:00