= Ion trap
{wiki}
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.
\Image[/ion-trap-functional-comparison.png]
{title=Paul and Penning traps use different confinement mechanisms}
{description=Original comparison of the ideal fields derived below. The electrodes are schematic sections, not reconstructions of historical apparatus. For the Paul panel the arrows show phase $\Omega t=\pi$ for positive charge; half a cycle reverses them. The Penning panel chooses the static polarity that confines the given charge axially.}
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