Nuclear spin

Nuclear spin is a quantum mechanical observable that obeys the same angular momentum algebra as electron spin. The total spin operator I has eigenvalues I(I+1)ħ², and its projection along a quantization axis takes values m_I = -I, -I+1, ..., I, giving 2I+1 possible states.¹ The spin quantum number I is determined by the coupling of proton and neutron spins and orbital angular momenta within the nucleus, following the shell model. For even-even nuclei (even numbers of both protons and neutrons), I = 0 in the ground state, while odd-A nuclei typically have half-integer spins, and odd-odd nuclei can have integer spins.²

Nuclei with non-zero spin possess a magnetic moment proportional to their spin, with the gyromagnetic ratio γ determining the Larmor frequency in an external magnetic field. This is the basis of nuclear magnetic resonance (NMR), where transitions between spin states are induced by radiofrequency pulses. In medical imaging, MRI exploits the nuclear spin of hydrogen-1 (proton) in water and fat to generate detailed anatomical images.³ The sensitivity of NMR depends on the natural abundance and gyromagnetic ratio of the isotope; for example, ¹³C and ¹⁵N have lower sensitivity but are useful for studying organic molecules.⁴

Nuclear spin couples to electron angular momentum, producing hyperfine splitting in atomic and molecular spectra. This interaction, described by the Fermi contact term and dipolar coupling, is exploited in atomic clocks and precision measurements of fundamental constants.⁵ Nuclear spin also underpins techniques such as nuclear quadrupole resonance (NQR) for nuclei with I ≥ 1, which is used in explosive detection and materials science. In quantum computing, nuclear spins are promising qubits due to their long coherence times, as demonstrated in nitrogen-vacancy centers in diamond.⁶

Beyond the familiar applications, nuclear spin has subtle implications. For instance, the spin of the deuteron (I=1) leads to a small electric quadrupole moment, which affects the binding energy of heavy hydrogen. The phenomenon of spin isomers, such as para- and ortho-hydrogen, arises from the coupling of nuclear spins with rotational states, influencing thermal conductivity and chemical reactivity.⁷ In astrophysics, the 21-cm line of neutral hydrogen, resulting from a hyperfine transition, is a crucial probe of the early universe. Additionally, the nuclear spin of ³He is used in neutron spin filters and in polarized targets for particle physics experiments.⁸

Spin quantum number

A quantum number that quantifies the intrinsic angular momentum of a particle or nucleus.

Gyromagnetic ratio

The ratio of the magnetic moment to the angular momentum of a particle or nucleus.

Hyperfine structure

The splitting of spectral lines due to the interaction between nuclear spin and electron angular momentum.