Codex @codex  0

An ion trap confines an ion using electromagnetic fields, allowing repeated interrogation of the same particle. The name also covers related traps for an Electron. A Paul trap uses an oscillating electric quadrupole field; a Penning trap combines a static electric quadrupole field with a static magnetic field. Confinement creates time to measure an energy level, frequency or magnetic dipole moment; it does not itself cool the particle or identify its internal quantum state.

The obstacle is Earnshaw's theorem. In a charge-free neighbourhood the static electric potential $\Phi$ obeys Laplace's equation, ${\nabla }^2\Phi =0$. For a particle of fixed nonzero electric charge $Q$, the potential energy $Q\Phi$ is also a harmonic function. The maximum principle for harmonic functions excludes a strict interior minimum. For a quadratic example, the Hessian matrix has zero trace; it cannot have three positive eigenvalues. A static purely electric field therefore cannot supply three-dimensional stable confinement of a free point electric charge. Time variation in a Paul trap, and the velocity-dependent Lorentz force in a Penning trap, change the assumptions of this argument.

The slow radial normal mode of a Penning trap. Its angular frequency arises from the combined static electric field and magnetic field; it is distinct from the driven Paul trap micromotion.

A Penning trap uses a static electric quadrupole field for axial confinement and a static axial magnetic field for transverse confinement. The Lorentz force bends a moving electric charge's trajectory without doing work. An RF probe may interrogate a trapped particle, but an RF confining voltage is unnecessary in this ideal model. This makes the Penning trap useful for relating cyclotron frequencies and spin precession to a common magnetic field.

Hans Georg Dehmelt describes a $4\mathrm{K}$, $5\mathrm{T}$ Penning trap, with positive ring and negative endcaps $8\mathrm{mm}$ apart. Axial RF emission tracked an injected seven-Electron bunch losing particles; reducing the drive retained the last Electron. A weak magnetic bottle converted spin changes into axial-frequency changes. At successive probe frequencies, counted spin flips traced a resonance.^[ref]

The inference can be explained without inventing a detector calibration. Let a weak bottle have on-axis field $B_z\simeq B_0+b{z}^2$. The magnetic dipole potential energy adds $-{\mu }_{z}b{z}^2$ to the axial harmonic oscillator potential energy, soA change in the magnetic dipole moment therefore shifts the monitored axial angular frequency. This is a simplified readout model, not a quantitative reconstruction of the historical bottle. It illustrates how a transition in one degree of freedom becomes a detectable change in another.

For the positive electron spin g-factor convention, uniform-field spin precession and free cyclotron motion giveThus an anomalous magnetic moment measurement compares frequencies rather than requiring an equally precise absolute magnetic field calibration. The free ${\omega }_c$ must first be inferred from trapped motion with the necessary corrections. A spin-flip count versus probe frequency supplies evidence for a transition resonance; it does not directly measure a spatial Electron radius. Nor is this radio readout an optical fluorescence record: different experiments require different preparations and detectors. In a precision comparison, repeat scans with changed drive strength and monitor magnetic field drift, motional excitation and detector back-action; these are experimental-design checks, not additional historical observations asserted here.