= RF-averaged trapping potential
{c}
{title2=$U_{\rm eff}=Q^2|\mathbf E_{\rm amp}|^2/(4m\Omega^2)$}
Write the RF <electric field> as $\mathbf E_{\rm amp}(\mathbf r)\cos\Omega t$, where $\mathbf E_{\rm amp}$ is a real peak-amplitude vector with a common drive phase. Decompose the trajectory into a slow position $\mathbf R$ and fast displacement $\boldsymbol\xi$. Over one short RF cycle, hold $\mathbf R$ fixed. The leading fast equation and solution are
$$
m\ddot{\boldsymbol\xi}=Q\mathbf E_{\rm amp}(\mathbf R)\cos\Omega t,\qquad
\boldsymbol\xi=-\frac{Q\mathbf E_{\rm amp}(\mathbf R)}{m\Omega^2}\cos\Omega t.
$$
The homogeneous slow part is absorbed into $\mathbf R$. Expand the <electric field> to first order in $\boldsymbol\xi$ and average, using $\langle\cos^2\Omega t\rangle=1/2$:
$$
\langle\mathbf F\rangle=-\frac{Q^2}{2m\Omega^2}(\mathbf E_{\rm amp}\cdot\nabla)\mathbf E_{\rm amp}
=-\nabla\left(\frac{Q^2|\mathbf E_{\rm amp}|^2}{4m\Omega^2}\right).
$$
The second equality uses zero <curl> in the quasistatic RF <electric field>. This derives an effective <potential energy>, not an extra static <electric potential>. Its $Q^2$ dependence confines either sign of <electric charge> towards a field-amplitude minimum. Using an RMS field instead would give a coefficient $1/(2m\Omega^2)$ multiplying $Q^2|\mathbf E_{\rm rms}|^2$.
The approximation needs fast displacement small compared with the field's variation length and slow <angular frequencies> small compared with $\Omega$. In the stated <electric quadrupole> model, small $|q_i|$ and $a_i$ of order $q_i^2$ provide a controlled regime, away from stability boundaries. Add $Q\Phi_{\rm dc}$ when $U\ne0$. For $U=0$ the result is
$$
U_{\rm eff}=\frac{Q^2V^2}{4m\Omega^2d^4}(x^2+y^2+4z^2),\qquad
\omega_i\simeq\frac{\Omega}{2}\sqrt{a_i+\frac{q_i^2}{2}}.
$$
For the pure RF case, $\omega_z\simeq2\omega_x=2\omega_y$. The square-root condition describes leading local restoring curvature; it is not a substitute for full <Mathieu equation> stability. Collisions, departures from the ideal <electric quadrupole>, and a displaced RF null require further analysis.
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