Charged particles can be trapped in two main ways: Paul traps and Penning traps. This post dives into Penning traps, their physics and how they keep ions in place. Want to learn about Paul traps? See my earlier posts on Paul traps and ion crystallization.

Charged particles in a magnetic field

A charged particle of charge $q$ and mass $m$ moving in a magnetic field $B$ experiences the Lorentz force,

Since the force is always perpendicular to the velocity, it does no work: the kinetic energy is conserved and the speed stays constant. The motion is a uniform circular orbit in the plane orthogonal to $B$, with radius $r_c$ and frequency $\omega_c$ given by,

This is the cyclotron motion. If the particle has a velocity component $v_{\parallel}$ along $B$, its path is helical: circular motion in the plane perpendicular to $B$ plus free propagation along the field lines.

Below is an interactive simulation of $N$ interacting particles in a magnetic field $B$. Each particle starts with a random velocity in $x$, $y$, and $z$. The magnetic field affects motion in the $x–y$ plane but leaves motion along $z$ unchanged. Particles with smaller $z$-momentum move more slowly in that direction and can appear confined within the region.

Use the slider below the animation to adjust the number of particles and the field strength $B$. As expected from Eq.2, larger $B$ produces tighter, faster radial orbits.

The animation shows that the magnetic field provides confinement only in the radial plane. A particle with $v_z = 0$ may appear stationary along $z$, but in a real trap tiny electric-field imperfections give it a small axial kick. With no restoring force along that axis, the particle simply keeps going and eventually escapes the trap. To achieve full 3D confinement, we need a restoring force along z. As for the case of Paul traps, this can be done by adding a static DC electric field in the form of a quadrupole potential,

we can make sure that the particle is trapped in the $z$-direction. Let's now look at the Lagrangian of the whole system,

For a uniform magnetic field $\mathbf{B} = B\,\hat{z}$, is convenient to use a symmetric gauge,

Plugging these into the minimal-coupling Lagrangian gives,

Along z, the magnetic term does nothing, so the the equation of motion is the same as for a harmonic oscillator,

with $\omega_z = \sqrt{q\frac{2 V_0}{m d^2}}$.

On the $x-y$ plane, the equation of motion are slightly more complicated and the two coordinates mix,

These can be solved by introducing the complex coordinate $u(t) = x(t) + i\,y(t)$ so that $\dot{u} = \dot{x} + i\dot{y}$ and $\ddot{u} = \ddot{x} + i\ddot{y}$. In this coordinate system the radial dynamics reduce to a single complex equation,

which, by assuming $u(t) = u_0 e^{-i\omega t}$, can be reduced to a quadratic equation $\omega^2 - \omega_c\omega + \frac{\omega_z^2}{2} = 0$ with solutions,

The larger root, $\omega_+$, is known as the modified cyclotron frequency. It corresponds to a rapid circular motion with a relatively small radius. The smaller root, $\omega_-$, is the magnetron frequency, which corresponds to a much slower circular drift with a larger radius. These two motions together describe the full trajectory of a charged particle in a Penning trap. For both frequencies to be real, the condition $\omega_c^2 > 2\omega_z^2$ must be satisfied. The plot below illustrates the motional frequencies as function of the axial frequency for a given magnetic field.

Perhaps more interactive, the animation below show the motion of a charged particle in a Penning trap with variable magnetic and electric fields. In the inset you can see the various cyclotron, magnetron and axial frequencies and how they vary with magnetic and electric field strength.

Cooling and quantum

In order to quantize the motion of a charged particle in a Penning trap we have to move to an Hamiltonian description after a Legendre transformation of Eq.6.

Defining $\omega_c = \frac{qB}{m}$, $\omega_z^2 = \frac{2 q V_0}{m d^2}$ and $\Omega^2 = \frac{\omega_z^2}{2}$ the Hamiltonian can be written as,

Introducing a new set of complex coordinates $u = x + i y$ and $v = x - i y$ with corresponding momenta $p_u = \frac{1} {2}(p_x - i p_y)$ and $p_v = \frac{1} {2}(p_x + i p_y)$ the Hamiltonian Eq. 12 becomes diagonal in terms of u, v and their conjugate momenta.

And where we defined

and,

The frequency $\omega_+$ is also called the modified cyclotron frequency, this is larger than the usual cyclotron frequency and largely positive. In contrast, $\omega_-$, known as the magnetron frequency, corresponds to a negative-energy mode in the Hamiltonian. This negative-energy character arises when we define the complex coordinates which allow the Hamiltonian to separate into two independent modes. The “negative” does not imply a negative angular velocity; it reflects the structure of the Hamiltonian in this complex (rotating) basis. Finally, to ensure stable trapping (i.e., real frequencies), we have to satisfy $\omega_c^2 > 2\,\omega_z^2$.

Perhaps one of the most confusing dynamics of the motion of a charged particle in a penning trap is that, in order to be trapped the particle should always be in motion (as the Lorentz force is needed to provide confinement). But in the quantum ground state, there’s no motion at all! To get a feel for this, we can imagine the quantum motion as the average behavior of many classical particles, each starting in slightly different positions. The animation below shows exactly that, alongside the classical motion of a single particle, with and without cooling.