JUAS 2025
Course 1: The science of particle accelerators
A. Lasheen
$$ \begin{aligned} & \frac{d}{dt}\left(\frac{\Delta E}{\omega_{\mathrm{rev},s}}\right) \\ & = \frac{q V_{\mathrm{rf}}}{2\pi}\left[\sin{\left(\phi\right)} - \sin{\left(\phi_s\right)}\right] \end{aligned}$$
$$ \begin{aligned} \frac{d \phi}{dt} = \frac{h \eta \omega_{\mathrm{rev},s}^2}{\beta_s^2 E_s} \left(\frac{\Delta E}{\omega_{\mathrm{rev},s}}\right) \end{aligned}$$
$$ \begin{aligned} & \frac{d}{dt}\left(\frac{\Delta E}{\omega_\mathrm{rf}}\right) = \frac{q V_{\mathrm{rf}}}{2\pi h}\left[\sin{\left(\phi\right)} - \sin{\left(\phi_s\right)}\right] && \text{(1)} \\ & \frac{d \phi}{dt} = \frac{\eta \omega_\mathrm{rf}^2}{\beta_s^2 E_s} \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right) && \text{(2)} \end{aligned}$$
$$ \begin{aligned} \frac{d}{dt}\left( \frac{d \phi}{dt} \frac{\beta_s^2 E_s}{\eta \omega_\mathrm{rf}^2} \right) = \frac{q V_{\mathrm{rf}}}{2\pi h}\left[\sin{\left(\phi\right)} - \sin{\left(\phi_s\right)}\right] \end{aligned}$$
$$ \begin{aligned} \sin{\left(\phi\right)} - \sin{\left(\phi_s\right)} & = \sin{\left(\phi_s+\Delta\phi\right)} - \sin{\left(\phi_s\right)} \\ & = \sin\phi_s \cos\Delta\phi + \cos\phi_s\sin\Delta\phi - \sin\phi_s \\ & \approx \cos\phi_s\Delta\phi \end{aligned}$$
$$ \begin{aligned} \frac{d^2\Delta\phi}{dt^2} = \frac{q V_{\mathrm{rf}} \eta \omega_\mathrm{rf}^2}{2\pi h\beta_s^2 E_s} \cos\phi_s \Delta \phi \\ \implies \frac{d^2\Delta\phi}{dt^2} + \omega_{s0}^2 \Delta\phi = 0 \end{aligned}$$
$$ \begin{aligned} \omega_{s0} = 2\pi f_{s_0} = \sqrt{- \frac{q V_{\mathrm{rf}} \omega_\mathrm{rf}^2 \eta \cos\phi_s}{2\pi h \beta_s^2 E_s}} \end{aligned}$$
$$ \begin{aligned} \eta \cos\phi_s < 0 \end{aligned}$$
imposes that the synchronous phase is
$$ \begin{aligned} \eta < 0 \rightarrow \phi_s \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \end{aligned}$$
$$ \begin{aligned} \eta > 0 \rightarrow \phi_s \in \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] \end{aligned}$$
$$ \begin{aligned} Q_{s0}=\frac{\omega_{s0}}{\omega_{\mathrm{rev},s}} = \sqrt{- \frac{q V_{\mathrm{rf}} h \eta \cos\phi_s}{2\pi \beta_s^2 E_s}} \end{aligned}$$
$$ \begin{aligned} \Delta\phi\left(t\right) & = \Delta\phi_m\sin\left(\omega_{s0} t\right) \\ \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right)\left(t\right) & = \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right)_m\cos\left(\omega_{s0} t\right) \end{aligned}$$
$$ \begin{aligned} \frac{\left(\Delta E / \omega_\mathrm{rf}\right)_m}{\Delta \phi_m} = \frac{\beta_s^2 E_s}{\left|\eta\right| \omega_\mathrm{rf}^2}\omega_{s0} = \frac{\beta_s^2 E_s}{\left|\eta\right| h^2\omega_{\mathrm{rev},s}} Q_{s0} \end{aligned}$$
$$ \begin{aligned} & \Delta\phi\left(t\right) = \Delta\phi_m\sin\left(\omega_{s0} t\right) \\ \implies & \Delta\dot{\phi} = \Delta\phi_m\omega_{s0}\cos\left(\omega_{s0} t\right) \\ \implies & \frac{\eta \omega_\mathrm{rf}^2}{\beta_s^2 E_s} \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right)_m \cos\left(\omega_{s0} t\right) = \Delta\phi_m\omega_{s0}\cos\left(\omega_{s0} t\right) \end{aligned}$$
$$ \begin{aligned} \left(\frac{\Delta\phi}{\Delta\phi_m}\right)^2 + \left(\frac{\Delta E / \omega_\mathrm{rf}}{\left[\Delta E / \omega_\mathrm{rf}\right]_m}\right)^2 = 1 \end{aligned}$$
$$ \begin{aligned} \varepsilon_{l,0} = \frac{\pi}{\omega_\mathrm{rf}} \Delta E_m \Delta\phi_m = \pi \Delta E_m \Delta\tau_m \end{aligned}$$
$$ \begin{aligned} \varepsilon_{l,0} = \pi \Delta E_m \frac{\tau_l}{2} & = \frac{\pi\beta_s^2 E_s}{4\left|\eta\right|}\omega_{s0} \tau_l^2 \\ & = \frac{\pi\left|\eta\right|}{\beta_s^2 E_s} \frac{1}{\omega_{s0}} \Delta E_m^2 \end{aligned}$$
$$ \begin{aligned} \varepsilon_{l,0} = \pi \Delta E_m \frac{\tau_l}{2} & = \pi \omega_\mathrm{rf} \Delta \phi_m \frac{\beta_s^2 E_s}{\left|\eta\right| \omega_\mathrm{rf}^2}\omega_{s0} \frac{\tau_l}{2} = \pi \frac{\omega_\mathrm{rf}\tau_l}{2} \frac{\beta_s^2 E_s}{\left|\eta\right| \omega_\mathrm{rf}}\omega_{s0} \frac{\tau_l}{2} \\ \varepsilon_{l,0} & = \frac{\pi\beta_s^2 E_s}{4\left|\eta\right|}\omega_{s0} \tau_l^2 \\ & = \frac{\pi\beta_s^2 E_s}{4\left|\eta\right|} \sqrt{- \frac{q V_{\mathrm{rf}} \omega_\mathrm{rf}^2 \eta \cos\phi_s}{2\pi h \beta_s^2 E_s}} \tau_l^2 \\ & = \tau_l^2 \sqrt{- \frac{\pi}{32} \frac{\omega_{\mathrm{rev},s}^2 \beta_s^2 E_s}{\eta} q V_{\mathrm{rf}} h \cos\phi_s} \end{aligned}$$
$$ \begin{aligned} \varepsilon_{l,0} = \frac{\pi}{\omega_\mathrm{rf}} \Delta E_m \Delta\phi_m & = \frac{\pi}{\omega_\mathrm{rf}} \Delta E_m \frac{1}{\omega_{s0}}\frac{\left|\eta\right| \omega_\mathrm{rf}^2}{\beta_s^2 E_s} \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right)_m \\ \varepsilon_{l,0} & = \frac{\pi\left|\eta\right|}{\beta_s^2 E_s} \frac{1}{\omega_{s0}} \Delta E_m^2 \\ & = \frac{\pi\left|\eta\right|}{\beta_s^2 E_s} \sqrt{- \frac{2\pi h \beta_s^2 E_s}{q V_{\mathrm{rf}} \omega_\mathrm{rf}^2 \eta \cos\phi_s}} \Delta E_m^2 \\ & = \Delta E_m^2 \sqrt{- 2\pi^3 \frac{ \eta }{\omega_{\mathrm{rev},s}^2 \beta_s^2 E_s} \frac{1}{q V_{\mathrm{rf}} h \cos\phi_s}} \end{aligned}$$
$$ \begin{aligned} \left| \frac{\dot{\omega}_{s0}}{\omega_{s0}} \right| \ll \omega_{s0} \end{aligned}$$
$$ \begin{aligned} \alpha_\mathrm{ad} = \left| \frac{1}{\omega_{s0}^2}\frac{d\omega_{s0}}{dt} \right| \ll 1 \end{aligned}$$
$$ \begin{aligned} \tau_l \propto \varepsilon_{l,0}^{1/2}\ V_\mathrm{rf}^{-1/4}\ h^{-1/4} \ E_s^{-1/4}\ \eta^{1/4} \end{aligned}$$
$$ \begin{aligned} \Delta E_m \propto \varepsilon_{l,0}^{1/2}\ V_\mathrm{rf}^{1/4}\ h^{1/4} \ E_s^{1/4}\ \eta^{-1/4} \end{aligned}$$
$$ \begin{aligned} f_{s0}&=\frac{1}{2\cdot 3.14}\sqrt{\frac{1\cdot 4.5\cdot 10^6\cdot (4620 \cdot 2\cdot3.14 /23.11\cdot 10^{6} )^2 \cdot 1.385 \cdot 10^{-3} }{2\cdot 3.14 \cdot 4620 \cdot (14/14.03)^2 \cdot 14.03\cdot 10^{9}}} \\ &\approx 784\ \text{Hz} \\ Q_{s0} & = 784\cdot 23.11\cdot 10^{-6} \approx 1.81\cdot 10^{-2} \end{aligned}$$
$$ \begin{aligned} f_{s0}&=\frac{1}{2\cdot 3.14}\sqrt{\frac{1\cdot 4.5\cdot 10^6\cdot (4620 \cdot 2\cdot3.14 /23.05\cdot 10^{6} )^2 \cdot 3.082 \cdot 10^{-3} }{2\cdot 3.14 \cdot 4620 \cdot 1 \cdot 450\cdot 10^{9}}} \\ &\approx 206\ \text{Hz} \\ Q_{s0} & = 206\cdot 23.05\cdot 10^{-6} \approx 4.76\cdot 10^{-3} \end{aligned}$$
$$ \begin{aligned} \varepsilon_{l,0} = \frac{3.14\cdot (14/14.03)^2 \cdot 14.03\cdot 10^{9}}{4 \cdot 1.385 \cdot 10^{-3}} 2\cdot 3.14 \cdot 784 \cdot (3\cdot 10^{-9})^2 \approx 0.35 \ \text{eVs} \end{aligned}$$
$$ \begin{aligned} \delta_p & = 2\frac{\Delta E_m}{\beta_s^2 E_s} = 4\frac{\varepsilon_{l,0}}{\pi \tau_l \beta_s^2 E_s} = \frac{4\cdot 0.35}{3.14\cdot 3\cdot 10^{-9} (14/14.03)^2 \cdot 14.03\cdot 10^{9}} \\ & \approx 1.06 \times 10^{-2} \end{aligned}$$
$$ \begin{aligned} \tau_{l,\mathrm{high}} = \tau_{l,\mathrm{low}} \left(\frac{E_\mathrm{high}}{E_\mathrm{low}}\right)^{-1/4} = 3 \cdot \left(\frac{450}{14.03}\right)^{-1/4} \approx 1.26 \ \text{ns} \end{aligned}$$
$$ \begin{aligned} & \tau_{l,\mathrm{high}} = \tau_{l,\mathrm{low}} \left(\frac{V_\mathrm{high}}{V_\mathrm{low}}\right)^{-1/4} \\ \implies & V_\mathrm{high} = V_\mathrm{low} \left(\frac{\tau_{l,\mathrm{high}} }{\tau_{l,\mathrm{low}}}\right)^{-4} = V_\mathrm{low} \times 16 \end{aligned} $$
$$ \begin{aligned} & \frac{d}{dt}\left(\frac{\Delta E}{\omega_\mathrm{rf}}\right) \\ & = \frac{q V_{\mathrm{rf}}}{2\pi h}\left[\sin{\left(\phi\right)} - \sin{\left(\phi_s\right)}\right] \end{aligned}$$
$$ \begin{aligned} \frac{d \phi}{dt} = \frac{\eta \omega_\mathrm{rf}^2}{\beta_s^2 E_s} \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right) \end{aligned}$$
$$ \begin{aligned} \frac{d}{dt}\left( \frac{d \phi}{dt} \frac{\beta_s^2 E_s}{\eta \omega_\mathrm{rf}^2} \right) = \frac{q V_{\mathrm{rf}}}{2\pi h}\left(\sin{\phi} - \sin{\phi_s}\right) \end{aligned}$$
$$ \begin{aligned} \frac{d^2 \phi}{dt^2} + \frac{\omega_{s0}^2}{\cos\phi_s} \left(\sin{\phi} - \sin{\phi_s}\right) = 0 \end{aligned}$$
$$ \begin{aligned} \frac{\dot{\phi}^2}{2 \omega_{s0}^2} - \frac{ \cos\phi + \phi\sin\phi_s }{\cos\phi_s} = \mathcal{H} \end{aligned}$$
$$ \begin{aligned} d\left(x^2\right) = 2\ x\ dx \quad \rightarrow \quad \dot{\phi} \ddot{\phi} = \frac{1}{2}\frac{1}{dt}\frac{d\left(\phi^2\right)}{dt^2} \end{aligned}$$
$$ \begin{aligned} & \frac{\omega_{s0}^2}{\cos\phi_s} \int{\left(\sin{\phi} - \sin{\phi_s}\right)\frac{d\phi}{dt} dt} \\ = & \frac{\omega_{s0}^2}{\cos\phi_s} \left[ \int{\sin{\phi}\ d\phi} - \int{\sin{\phi_s}d\phi} \right] \\ = & - \frac{\omega_{s0}^2}{\cos\phi_s} \left( \cos\phi + \phi\sin\phi_s \right) \end{aligned}$$
$$ \begin{aligned} \mathcal{H} = \frac{\dot{\phi}^2}{2 \omega_{s0}^2} - \frac{\cos\phi - \cos\phi_s + \left(\phi-\phi_s\right)\sin\phi_s}{\cos\phi_s} \end{aligned}$$
$$ \begin{aligned} \mathcal{H} = \frac{\eta \omega_\mathrm{rf}^2}{2 \beta_s^2 E_s} \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right)^2 + \frac{q V_{\mathrm{rf}}}{2\pi h} \left[\cos\phi - \cos\phi_s + \left(\phi-\phi_s\right)\sin\phi_s\right] \end{aligned}$$
$$ \begin{aligned} \frac{d\phi}{dt} = \frac{\partial \mathcal{H}}{\partial \left(\Delta E / \omega_\mathrm{rf}\right)} \quad \text{and} \quad \frac{d\left(\Delta E / \omega_\mathrm{rf}\right)}{dt} = - \frac{\partial \mathcal{H}}{\partial \phi} \end{aligned}$$
$$ \begin{aligned} \mathcal{H} = \mathcal{T}\left(\frac{\Delta E}{\omega_\mathrm{rf}}\right) + \mathcal{U}\left(\phi\right) \end{aligned}$$
$$ \begin{aligned} \frac{dq}{dt} = \frac{\partial \mathcal{H}}{\partial p} \quad \text{and} \quad \frac{dp}{dt} = - \frac{\partial \mathcal{H}}{\partial q} \end{aligned}$$
$$ \begin{aligned} \mathcal{H} = \int{\frac{\partial \mathcal{H}}{\partial p} dp} + \int{\frac{\partial \mathcal{H}}{\partial q} dq} \end{aligned}$$
$$ \begin{aligned} \mathcal{H} = & \frac{\eta \omega_\mathrm{rf}^2}{2 \beta_s^2 E_s} \left(\frac{\Delta E}{\omega_\mathrm{rf}}\right)^2 + \mathcal{U}\left(\phi\right) \\ \mathcal{H}_u & = \mathcal{U}\left(\pi-\phi_s\right) \end{aligned}$$
$$ \begin{aligned} \Delta E_{\mathrm{sep}} = \pm \sqrt{\frac{2 \beta_s^2 E_s}{\left|\eta\right|}} \sqrt{\mathcal{U}\left(\pi-\phi_s\right) - \mathcal{U}\left(\phi\right)} \end{aligned}$$
$$ \begin{aligned} \Delta E_{\mathrm{sep,m}} = \sqrt{\frac{2 q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|}} Y\left(\phi_s\right) \end{aligned}$$
$$ \begin{aligned} Y\left(\phi_s\right) = \left|- \cos\phi_s + \frac{\left(\pi-2\phi_s\right)}{2}\sin\phi_s \right|^{1/2} \end{aligned}$$
$$ \begin{aligned} \mathcal{U}\left(\pi-\phi_s\right) & = \frac{q V_{\mathrm{rf}} }{2\pi h} \left[\cos\left(\pi-\phi_s\right) - \cos\phi_s + \left(\pi-2\phi_s\right)\sin\phi_s\right] \\ & = \frac{q V_{\mathrm{rf}}}{2\pi h} \left[\cos\pi\cos\phi_s + \sin\pi\sin\phi_s - \cos\phi_s + \left(\pi-2\phi_s\right)\sin\phi_s\right] \\ & = \frac{q V_{\mathrm{rf}}}{2\pi h} \left[-2\cos\phi_s + \left(\pi-2\phi_s\right)\sin\phi_s\right] \\ & = \frac{q V_{\mathrm{rf}}}{\pi h} \left[- \cos\phi_s + \frac{\left(\pi-2\phi_s\right)}{2}\sin\phi_s\right] \end{aligned}$$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} = 2 \sqrt{\frac{2 \beta_s^2 E_s}{\left|\eta\right|\omega_\mathrm{rf}^2}} \int_{\phi_u}^{\phi_m}{\sqrt{\mathcal{U}\left(\pi-\phi_s\right) - \mathcal{U}\left(\phi\right)}\ d\phi} \end{aligned}$$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} = \frac{8}{\omega_\mathrm{rf}} \sqrt{\frac{2 q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|}} \Gamma\left(\phi_s\right) \\ \end{aligned}$$
$$ \begin{aligned} \Gamma\left(\phi_s\right) = \frac{1}{4\sqrt{2}} \int_{\phi_u}^{\phi_m}{\sqrt{-\cos\phi_s - \cos\phi + \left(\pi - \phi - \phi_s\right)\sin\phi_s}\ d\phi} \end{aligned}$$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} \approx \frac{8}{\omega_\mathrm{rf}} \sqrt{\frac{2 q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|}} \frac{1-\sin\phi_s}{1+\sin\phi_s} \\ \end{aligned}$$
For the stationary RF bucket $\mathcal{A}_\mathrm{bk} = 8 \Delta E_{\mathrm{sep,m}}/\omega_\mathrm{rf}$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} & = 2 \sqrt{\frac{2 \beta_s^2 E_s}{\left|\eta\right|\omega_\mathrm{rf}^2}} \int_{\phi_u}^{\phi_m}{\sqrt{\mathcal{U}\left(\pi-\phi_s\right) - \mathcal{U}\left(\phi\right)}\ d\phi} \\ & = 2 \sqrt{\frac{q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|\omega_\mathrm{rf}^2}} \int_{\phi_u}^{\phi_m}{\sqrt{...}\ d\phi} \end{aligned}$$
$$ \begin{aligned} ... = & \left[\cos\left(\pi-\phi_s\right) - \cos\phi_s + \left(\pi-2\phi_s\right)\sin\phi_s\right] \\ & - \left[\cos\phi- \cos\phi_s + \left(\phi-\phi_s\right)\sin\phi_s\right] \\ & = \left[- 2\cos\phi_s + \left(\pi-2\phi_s\right)\sin\phi_s\right] \\ & - \left[\cos\phi- \cos\phi_s + \left(\phi-\phi_s\right)\sin\phi_s\right] \\ & = -\cos\phi_s - \cos\phi + \left(\pi - \phi - \phi_s\right)\sin\phi_s \end{aligned}$$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} & = \frac{8}{\omega_\mathrm{rf}} \sqrt{\frac{2 q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|}} \frac{1}{4\sqrt{2}} \int_{\phi_u}^{\phi_m}{\sqrt{...}\ d\phi} \\ & = \frac{8}{\omega_\mathrm{rf}} \sqrt{\frac{2 q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|}} \Gamma\left(\phi_s\right) \\ \end{aligned}$$
$$ \begin{aligned} \Gamma\left(\phi_s\right) & = \frac{1}{4\sqrt{2}} \int_{\phi_u}^{\phi_m}{\sqrt{-\cos\phi_s - \cos\phi + \left(\pi - \phi - \phi_s\right)\sin\phi_s}\ d\phi} \\ & \approx \frac{1-\sin\phi_s}{1+\sin\phi_s} \end{aligned}$$
$$ \begin{aligned} \varepsilon_l = & 2 \sqrt{\frac{2 \beta_s^2 E_s}{\left|\eta\right|\omega_\mathrm{rf}^2}} \quad \cdot \\ & \int_{\phi_{b,l}}^{\phi_{b,r}}{\sqrt{\mathcal{U}\left(\phi_{b,lr}\right) - \mathcal{U}\left(\phi\right)}\ d\phi} \end{aligned}$$
The filling factor is commonly defined in emittance: $\varepsilon_l/\mathcal{A}_\mathrm{bk}$ or in energy: $\Delta E_{b,m}/\Delta E_{\mathrm{sep},m}$
$$ \begin{aligned} T_s = \int_{\phi_{b,l}}^{\phi_{b,r}}{\frac{d\phi}{\dot{\phi}}} \end{aligned}$$
$$ \begin{aligned} \frac{\omega_s}{\omega_{s0}} & = \frac{\pi}{2 K\left(\sin\frac{\phi_{b}}{2}\right)} \\ & \approx 1 - \frac{\phi_b^2}{16} \end{aligned}$$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} = & \frac{8}{4620\cdot 2\cdot 3.14 / 23.11 \cdot 10^6} \cdot \\ & \sqrt{\frac{2\cdot 1 \cdot 4.5\cdot 10^6 \cdot (14/14.03)^2 \cdot 14.03 \cdot 10^9}{3.14\cdot 4620 \cdot 1.385\cdot 10^{-3}}} \\ & \approx 0.50\ \text{eVs} \end{aligned}$$
$$ \begin{aligned} \Delta E_{\mathrm{sep},m} = & \sqrt{\frac{2\cdot 1 \cdot 4.5\cdot 10^6 \cdot (14/14.03)^2 \cdot 14.03 \cdot 10^9}{3.14\cdot 4620 \cdot 1.385\cdot 10^{-3}}} \\ & \approx 79.1\ \text{MeV} \end{aligned}$$
$$ \begin{aligned} \mathcal{A}_\mathrm{bk} = & \frac{8}{4620\cdot 2\cdot 3.14 / 23.05 \cdot 10^6} \cdot \\ & \sqrt{\frac{2\cdot 1 \cdot 4.5\cdot 10^6 \cdot 1 \cdot 450 \cdot 10^9}{3.14\cdot 4620 \cdot 3.082 \cdot 10^{-3}}} \\ & \approx 1.91\ \text{eVs} \end{aligned}$$
$$ \begin{aligned} \Delta E_{\mathrm{sep},m} = & \sqrt{\frac{2\cdot 1 \cdot 4.5\cdot 10^6 \cdot 1 \cdot 450 \cdot 10^9}{3.14\cdot 4620 \cdot 3.082 \cdot 10^{-3}}} \\ & \approx 301\ \text{MeV} \end{aligned}$$
$$ \frac{\Delta E_{\mathrm{sep},m,2}}{\Delta E_{\mathrm{sep},m,1}} = 0.9 = \sqrt{\frac{V_\mathrm{rf,2}}{V_\mathrm{rf,1}} } \quad \rightarrow \quad V_\mathrm{rf,2} = 0.9^2 V_\mathrm{rf,1} \approx 0.81 V_\mathrm{rf,1} $$
$$ \begin{aligned} \omega_{s0} = 2\pi f_{s_0} = \sqrt{- \frac{q V_{\mathrm{rf}} h \omega_{\mathrm{rev},s}^2 \eta \cos\phi_s}{2\pi \beta_s^2 E_s}} \end{aligned}$$
$$ \begin{aligned} \Delta E_{\mathrm{sep,m}} = \sqrt{\frac{2 q V_{\mathrm{rf}} \beta_s^2 E_s}{\pi h \left|\eta\right|}} \left|- \cos\phi_s + \frac{\left(\pi-2\phi_s\right)}{2}\sin\phi_s \right|^{1/2} \end{aligned}$$
$$ \begin{aligned} \frac{\omega_s}{\omega_{s0}} = \frac{\pi}{2 K\left(\sin\frac{\phi_{b}}{2}\right)} \approx 1 - \frac{\phi_b^2}{16} \end{aligned}$$