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

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

We define

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

(3)with

(4)with $\omega_\xi = \frac{\xi \omega_0}{\alpha_c}$

We can write this as a function of $\alpha$ and $\xi$.

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)This is a coupled bunch instability.

The impedance is also given by

(7)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)