= Paul trap Mathieu parameters
{c}
{title2=$a_x=4QU/(md^2\Omega^2),\ q_x=2QV/(md^2\Omega^2)$}
Specify the convention before quoting a stability parameter. Choose the ideal, rotationally symmetric <electric potential>
$$
\Phi(\mathbf r,t)=\frac{U-V\cos\Omega t}{2d^2}(x^2+y^2-2z^2),\qquad m>0,\quad d>0,\quad\Omega>0.
$$
Here $U$ is a static voltage coefficient, $V$ is a peak RF voltage coefficient, $d$ is the length entering this displayed normalization, $m$ is <mass>, $Q$ is signed <electric charge>, and $\Omega$ is the drive <angular frequency>. The coefficient $V$ is neither a peak-to-peak voltage nor an RMS voltage. An apparatus's ring-to-cap voltage requires matching its electrode surfaces to this <electric potential>; electrode radius alone does not define our $d$. The minus sign fixes the RF phase at $t=0$.
With $\tau=\Omega t/2$, <Newton's second law> gives a <Mathieu equation> for each Cartesian coordinate $u_i$:
$$
\frac{d^2u_i}{d\tau^2}+(a_i-2q_i\cos2\tau)u_i=0,\qquad
(a_y,q_y)=(a_x,q_x),\qquad
(a_z,q_z)=(-2a_x,-2q_x).
$$
Thus a three-dimensional <Paul trap> needs simultaneous bounded solutions in all three directions. Changing the sign of $V$ reverses $q_i$ and shifts the RF phase by half a cycle; it does not change the ideal infinite-time stability regions. Rescaling the voltage or length convention changes the numerical parameters. <Floquet theory> determines stability of the full <Mathieu equation>; the small-parameter approximation below does not establish stability for arbitrary $q_i$. Bounded mathematical motion can still hit an electrode if its amplitude exceeds the available space.
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