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A cyclotron repeatedly passes ions through an alternating electric field between two hollow electrodes called cyclotron dees. A nearly uniform magnetic field perpendicular to their plane bends the charged particles back toward the gap. Inside each cyclotron dee, ideal shielding suppresses the accelerating electric field, so each segment is a constant-speed circular motion. The next gap crossing increases the speed and the radius of curvature. The complete path consists of connected, progressively larger arcs, rather than one circle or a smooth spiral with continuous acceleration everywhere.

The striking feature is a timing cancellation: the longer arc traveled by a faster particle takes the same time. This cyclotron resonance condition lets one fixed radio frequency accelerate many successive gap crossings. The following construction and cyclotron energy and orbit radius example describe an ideal model; the Lawrence-Livingston cyclotron experiment explains what the builders actually measured.

Polykarp Kusch and Henry Michael Foley compared hyperfine structure resonance frequencies for different electronic states of gallium, indium and sodium. Their 1948 paper reports mutually consistent departures from the ratios expected with orbital factor $1$ and spin factor $2$. Comparing several states constrained the alternative explanation of perturbed atomic levels.^[ref]

The Rabi molecular beam magnetic resonance experiment explains the readout principle: a resonant transition changes an atom's magnetic response, and hence its transmitted beam signal. Kusch describes rapidly interchangeable ovens, several radio frequency oscillators and careful field homogenization. Measurements taken during monotonic field drift were reduced to a common field.^[ref]

Real hyperfine structure lines are not simply bare electronic $g_J$ times field. Nuclear magnetic moments, known interaction constants and nonlinear field dependence enter the inference.^[ref] Field homogeneity controls line broadening; alternating species addresses drift. Neither control removes an error in the atomic model. The beauty of the comparison is the separation of two problems: cancelling a shared calibration scale and testing whether several physical systems imply the same electron spin g-factor.

Nuclear magnetic resonance detects the response of atomic nuclei with a magnetic moment to a radio frequency magnetic field in a steady magnetic field. Its striking possibility is to measure a nuclear property through an ordinary electrical signal from a bulk sample: an enormous number of tiny magnetic moments can produce a measurable change in a receiving circuit.

For a nucleus with spin angular momentum $\mathbf{J}$, define its gyromagnetic ratio by $\mathbf{\mu }=\gamma \mathbf{J}$. Throughout this article $\gamma$ is angular, measured in radians per second per tesla; $\gamma /(2\pi )$ is the corresponding value in cycles per second per tesla. With ${\mathbf{B}}_0=B_0\hat{\mathbf{z}}$, the Hamiltonian is $H=-\gamma B_0{J}_z$. Adjacent energy levels differ byThus a measured resonance frequency and a calibrated magnetic field determine $|\gamma |$. Converting that into a nuclear magnetic moment also requires the spin quantum number: the maximum projection has magnitude $|\gamma |\hslash I$. This simple experiment does not by itself determine the sign of $\gamma$.

The Rabi molecular beam magnetic resonance experiment detects transitions through changes in a selected molecular beam. The bulk experiments below instead detect electrical consequences of nuclear magnetization in matter at ordinary density. A small Boltzmann distribution population imbalance is multiplied by a very large number of nuclei. For noninteracting spin one-half nuclei of number density $n$ and positive $\gamma$, their equilibrium magnetization isThis follows by weighting the two projections $\pm \gamma \hslash /2$ with their Boltzmann distribution probabilities. It explains both the smallness of the bulk polarization at ordinary temperature and the value of many nuclei contributing to one receiving circuit.

NMR spectroscopy measures resonant responses of atomic nuclei to applied electromagnetic radiation in a magnetic field. In residual-group monitoring of polyurethane formation, distinguishable signal integrals supply a relative residual-group readout; overlap or very weak signals can limit that inference.

The rubidium experiment used laser cooling, a magneto-optical trap and magnetic trapping of neutral atoms with a rotating bias field. Radio frequency transitions removed energetic atoms; elastic collisions restored thermal equilibrium. Before absorption imaging, the trap was decompressed, then nearly released with a gradient supporting against gravity. Varying the final evaporation cut revealed a narrow anisotropic core within an isotropic thermal cloud. The broad component supplied a simultaneous comparison through the same optics. Resolution and size calibration limited inference; interactions broadened the core relative to an ideal gas.^[ref]

The 1995 sodium experiment combined laser cooling with a magnetic quadrupole trap and a repulsive optical plug protecting its field zero. Radio frequency evaporation preceded release and absorption imaging after time of flight. Lowering the evaporation cut produced broad thermal wings and a narrow core. Moving the plug changed the core's expansion orientation while the thermal distribution remained spherical: a control connecting the feature to confinement. Overlap of the two trapped pockets and residual switch-off acceleration limited quantitative shape comparisons; repulsive interactions affected expansion.^[ref]