Instabilities

Kernel et. al. gives the instability limit for transverse single bunch as

(1)
\begin{align} I_{th} = \frac{4\frac{E}{e}\alpha \sigma_\delta(I_{th})|\omega_{\xi}-\omega_q|\sigma_{\tau}(I_{th})}{\sqrt{\frac{2}{3}}|\beta Z_\perp|} \end{align}

This assumes that the function $G(I)$ is unchanged.

We define

(2)
\begin{align} G(I) = \frac{\sigma_z(I) \sigma_\delta(I)}{\sigma_{z0}\sigma_{\delta 0}} \end{align}

In terms of G(I), we can write the equation as

(3)
$$I_{th} = I_0 G(I_{th})$$

with

(4)
\begin{align} I_0 = \frac{4\frac{E}{e}\alpha \sigma_{\delta 0}|\omega_{\xi}-\omega_q|\sigma_{\tau 0}}{\sqrt{\frac{2}{3}}|\beta Z_\perp|} \end{align}

with $\omega_\xi = \frac{\xi \omega_0}{\alpha_c}$
We can write this as a function of $\alpha$ and $\xi$.

(5)
\begin{align} I_0 = A\alpha + B\xi \end{align}

Solving this equation yields the threshold.
To solve for the instability limit, we must iterate:

In general, we have different modes with growth rates, and you look for when the mode can grow faster than it is damped. Then you have an instability.
Sacherer? Laclare?
Multi-particle code?

For resistive wall instability, Perron quotes Nagaoka with the following formula:

(6)
\begin{align} \frac{1}{\tau} = \frac{\beta\omega_0 I}{4\pi \frac{E}{e}}\frac{R}{b_{eff}^3}\sqrt{\frac{2\rho}{(1-\Delta Q_\beta)\omega_0\epsilon_0}} \end{align}

This is a coupled bunch instability.

The impedance is also given by

(7)
\begin{align} \frac{\mu R Z_0}{\mu_0 b^3} \delta_s \end{align}

and $\delta_s$ scales as $\frac{1}{\sqrt{\omega}}$
The TMCI threshold is proportional to $\nu_s*\sigma_\tau$

Sacharer's formula which gives the complex frequency shift Δωcm of a mode m depending on current….

(8)
\begin{align} \Delta \omega = i C \frac{I_b \beta}{4\pi \sigma_t \frac{E}{e}} Z_{eff} \end{align}