Nuclear magnetic resonance and spin relaxation

For a spin-one-half nucleus in a strong uniform field, the leading resonance frequency is the angular gyromagnetic ratio times field divided by two pi. Define the fractional shift relative to a suitable reference at the same field. Its spin contribution is proportional to the uniform static electronic susceptibility, with a hyperfine coupling whose units depend on the susceptibility convention. Orbital or chemical shifts, demagnetizing fields, and macroscopic susceptibility corrections may also contribute. A measured total shift cannot be substituted for its spin part without separating these terms.

Nuclei with spin greater than one half additionally possess an electric quadrupole moment. Coupling to an electric-field gradient can split or shift lines, and second-order effects may affect even a central transition. Structural changes can therefore move a line without changing spin susceptibility. Multiple sites, orientation-dependent tensors, and disorder broaden the interpretation further. Field and orientation dependence, known reference compounds, and comparisons across nuclear species help disentangle magnetic and quadrupolar contributions before a microscopic conclusion is assigned.

Longitudinal relaxation restores the nuclear population imbalance after a pulse perturbs it. Transverse local fields drive transitions between nuclear Zeeman levels, so their noise near the Larmor angular frequency controls a weak-coupling T1 rate. The electronic response enters through a momentum sum weighted by the hyperfine form factor. In contrast, the spin shift samples the uniform static response. Antiferromagnetic fluctuations can therefore enhance relaxation strongly while changing the uniform shift relatively little, or can be filtered at a particular nuclear site.

A useful illustrative classical-noise model takes each transverse field component to have equal variance and an exponentially decaying correlation. In the high-temperature, stationary, weak-coupling and motional-narrowing regime its Lorentzian spectral density gives the rate below. Slowing the noise first increases the rate, then decreases it once the correlation time becomes long compared with the inverse Larmor frequency. Thus a relaxation maximum does not by itself show that fluctuation amplitude peaks; the spectral weight may simply move through the nuclear detection window.

For fully saturated spin-one-half nuclei with a single uniform relaxation time, longitudinal magnetization recovers exponentially. Inversion recovery has a different amplitude, while quadrupolar level populations, distributions of environments, or incomplete saturation can require multiple exponentials. T2 characterizes transverse relaxation, while the free-induction time T2* can additionally include static frequency inhomogeneity; an echo refocuses part of that static phase spread. A broad line is therefore not automatically a short T1, and a fitted stretched exponential needs a physical interpretation rather than automatic identification with one microscopic rate.

A simple noninteracting metal with contact hyperfine coupling gives the Korringa relation between spin shift and relaxation. It uses a dimensionless spin shift, not a percentage number or a shift contaminated by orbital contributions. Correlations, multiband weights, momentum-dependent couplings, and spin-orbit effects can alter the comparison. Likewise, a decrease below a superconducting transition must be interpreted with field orientation, orbital subtraction, vortex contributions, and the pairing model specified; a single relaxation curve cannot uniquely identify a gap symmetry.