Bunch Lengthening

The Haissinski equation solution for bunch lengthening (Potential well region) is given by (V. Serriere, thesis, p. 58)

(1)
\begin{align} \Psi(\phi,\Delta E) = K_1 \Psi_0 e^{ -\left(\frac{1}{2}\left(\frac{\nu_s}{\alpha h \sigma_\delta}\right)^2 \times h|\frac{Z_l \omega_0}{\omega}|(I(\phi)-I(0))\right)} \end{align}

Approximating the bunch distribution as a Gaussian, the dependence of the bunch length on current is given by

(2)
\begin{align} \left(\frac{\sigma_\tau}{\sigma_{\tau_0}}\right)^3 -\left(\frac{\sigma_\tau}{\sigma_{\tau_0}}\right) = \frac{\Delta}{4\sqrt{\pi}} \end{align}

where the parameter $\Delta$ is given by (Serriere)

(3)
\begin{align} \Delta = \frac{2\pi I_b Z_n}{V_m h \cos\phi_s\left(\frac{\alpha\sigma_\delta}{\nu_s}\right)^3} \end{align}

It may also be written (Zotter)

(4)
\begin{align} \Delta = \frac{\alpha_c e I_b Z_n}{E_0\nu_s^2}\left(\frac{c}{\omega_0 \sigma_{z0}}\right)^3 \end{align}

Here, the synchrotron tune $\nu_s$ is given by

(5)
\begin{align} \nu_s = \sqrt{-\frac{V_m h \alpha_c}{2\pi E_0}\cos\phi} \end{align}

It may also be written

(6)
\begin{align} \nu_s = \sqrt{\frac{\alpha_c}{2\pi h}}\left(\left(\frac{V_{rf}}{E_0}\right)^2-\left(\frac{U_0}{E_0}\right)^2\right)^{\frac{1}{4}} \end{align}

and the synchronous phase is given by

(7)
\begin{align} \phi = \pi - \sin^{-1}\frac{U_0}{V_m} \end{align}

This cubic equation has one real solution given by

(8)
\begin{align} \left(\frac{\sigma_\tau}{\sigma_{\tau_0}}\right)=\frac{\sqrt[3]{\sqrt{3} \sqrt{27 Q^2-4}+9 Q}}{\sqrt[3]{2} 3^{2/3}}+\frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{\sqrt{3} \sqrt{27 Q^2-4}+9 Q}} \end{align}

with $Q=\frac{\Delta}{4\sqrt{\pi}}$
For small Q, (certainly, $Q<\frac{1}{2}$), we can use a power series expansion in Q to approximate the bunch length growth. The first few terms are

(9)
\begin{align} \left(\frac{\sigma_\tau}{\sigma_{\tau_0}}\right)\approx 1 + \frac{Q}{2} - \frac{3 Q^2}{8} \end{align}

In Excel Optics, we write the parameter Q as
2*3.1415*D258*K255/(T_Vrf*COS(E238)*(D155*D167/E235)^3)

Longitudinal dynamics

Wolski writes the longitudinal equations of motion:

(10)
\begin{align} \frac{dz}{ds} = -\alpha \delta \end{align}
(11)
\begin{align} \frac{d\delta}{ds} = \frac{q V_{RF}}{C_0 P_0 c} (\sin(\phi_s -kz) -\sin\phi_s) \end{align}

The RF acceptance is given by

(12)
\begin{align} \delta_{RF} = \frac{2\nu_s}{h\alpha}\sqrt{1-\left(\frac{\pi}{2}-\phi\right)\tan\phi_s} \end{align}

It may also be written

(13)
\begin{align} \delta_{RF}=\sqrt{\frac{2 U_0}{\pi \alpha h E_0}}\sqrt{\sqrt{\left(\frac{V_{rf}}{U_0}\right)^2-1}-\arccos(\frac{U_0}{V_{rf}})} \end{align}