Space Charge
Wakefields
Charged particles traversing in a slowly varying long beam will induce a wakefiled voltage given by
\[V_{W} = -\frac{i}{\omega_s}\frac{Z}{n}\frac{d\lambda}{d\tau},\]where the longitudinal charge density is given by $\lambda(\tau)$ and the space charge impedance $Z/n$ is given by
\[\frac{Z}{n}=-j\frac{Z_o}{\beta_s\gamma_s^2}{g}\]where $g$ is the space charge geometry factor, $Z_0$ is the impedance of free space, and $n=f/f_s$.
Induced Voltage
The induced voltage gradient due to space charge can therfore be written as
\[V'_{SC}(\tau) = \frac{g}{\omega_s}\frac{Z_0}{\beta_s\gamma_s^2}\frac{d^2\lambda}{d\tau^2}\]Synchrotron Frequency Shift
The synchrotron frequency is generalized by
\[\Omega^2=-\frac{\eta}{\beta_s^2E_s}\frac{q}{T_s}V'(\tau)\]where
\[V'(\tau) = V'_{RF}(\tau)+V'_{SC}(\tau)\]and
\[V'_{RF}=V_g\frac{d}{d\tau}g(\phi)\approx h\omega_s V_g\cos\varphi\]The frequency tune shift is given by
\[\mu=\Omega/\Omega_s=\sqrt{1+\frac{V'_{SC}}{V'_{RF}}}\]For a parabolic distribution given by
\[\lambda(\tau) = \frac{3Q}{2L_\tau}(1-4\frac{\tau^2}{L_\tau^2})\]and so
\[\lambda''=-\frac{12Q}{L_\tau^3}\]therefore the gradient is given by
\[V'_{SC}(\tau) = -\frac{12Q}{L_\tau^3}\frac{g}{\omega_s}\frac{Z_0}{\beta_s\gamma_s^2}\]and so the frequency spread is shifted by a constant given vy $V’_{SC}(\tau)$
Effective Voltage
Given a distribution of particles with varying synchrotron frequency, the effective synchrotron frequency is given by
\[<\mu> = \oint\lambda(\phi)\mu(\phi)d\phi\]For a parabolic distribution we have
\[<\mu> = \int_\infty \frac{3}{2L}(1-4\frac{\tau^2}{L^2})(1-\frac{(h\omega_s\tau)^2}{16})d\tau\] \[<\mu> = 1-\frac{\sigma_{\hat{\phi}}^2}{16}\]from
\[\mu^2=\Omega^2/\Omega_s^2\propto V'(\tau)\propto V_g\]and that
\[\mu = \Omega/\Omega_s\]The implied gap voltage $V_g$ given by the interpreted effective voltage $
This needs to include space charge tho