Paul Trap Mathieu Stability

The Paul trap confines charged particles using a combination of static (DC) and oscillating (RF) electric fields. The idealized 1D motion of a single coordinate inside this trap is described by the Mathieu equation: x'' + (a - 2q cos(2τ))x = 0. By studying the monodromy matrix over one full RF cycle, we can map out the stability regimes of the trap.

worked-example

Let's trace the monodromy matrix when a = 0, q = 0.2 starting from a "Position" initial state (x=1, x'=0) over τ = 0 to π.

Initial State: x(0)=1, x'(0)=0 Equation: x'' - 0.4 cos(2τ) x = 0 By numerical integration (RK4, N=4096): x(π) = m11 ≈ 0.9013853059532956 x'(π) = m21 ≈ -0.07509419151026551 Next, starting from Velocity initial state (x=0, x'=1): x(π) = m12 ≈ 2.4969245482301887 x'(π) = m22 ≈ 0.901385305953292 Monodromy Matrix M: [[ 0.9013, 2.4969 ] [-0.0751, 0.9013 ]] Trace = 1.8027 (since |Trace| < 2, Bounded elliptic) Determinant = 1

experiment

In the Interactive Paul Trap module, you can investigate different parameters:

Set DC parameter a to 0.
Set RF parameter q to 0.2.
Observe the Bounded elliptic classification and compare it to the worked example trace of ~1.80.
Set RF parameter q to 0, and observe the Unbounded linear drift where the trace becomes exactly 2.

limitations

This model studies ONE normalized coordinate of an ideal RF Paul trap. It does not claim that this 1D sample proves all 3D trap coordinates are stable. In a real trap, stability must be achieved simultaneously in all radial and axial directions. Furthermore, this simulation ignores space charge effects, collisions, higher-order multipole fields, and non-ideal electrode geometries.

references

Paul, W. (1990). Electromagnetic traps for charged and neutral particles. Reviews of Modern Physics, 62(3), 531-540.
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.62.531

Paul Trap Mathieu Stability