SCMULT

Tracks through a zero length multipole to simulate space charge effects
Parallel capable? : yes

Parameter Name

Units

Type

Default

Description

This element simulates transverse space charge (SC) kick using K.Y. Ng's formula ( K.Y. Ng, FNAL, private communication).

The linear SC force is given by:

$$\Delta x'=\frac{K_{sc}Le^{-z^2/(2\sigma_z^2)}}{\sqrt{2\pi}\sigma_z} \frac{x}{\sigma_x(\sigma_x+\sigma_y)}$$

$$\Delta y'=\frac{K_{sc}Le^{-z^2/(2\sigma_z^2)}}{\sqrt{2\pi}\sigma_z} \frac{y}{\sigma_y(\sigma_x+\sigma_y)}$$ (30)

where $K_{sc}=\frac{2Nr_e}{\gamma^3\beta^2}$, $L$ is the integrating length, $\sigma_{x,y,z}$ are rms beam size.

The non-linear SC force is given by:

$$\Delta x'=\frac{K_{sc}Le^{-z^2/(2\sigma_z^2)}}{2\sigma_z\sqr... ...x}{\sigma_y}} {\sqrt{2(\sigma_x^2-\sigma_y^2)}}\right)\right ]$$

$$\Delta y'=\frac{K_{sc}Le^{-z^2/(2\sigma_z^2)}}{2\sigma_z\sqr... ...x}{\sigma_y}} {\sqrt{2(\sigma_x^2-\sigma_y^2)}}\right)\right ]$$ (31)

where $w(z)$ is the complex error function

$$w(z)=e^{-z^2}\left [ 1+\frac{2i}{\sqrt{\pi}}\int\limits_0^z e^{\zeta^2}d\zeta\right ]$$ (32)

Equation 31 appear to diverge when $\sigma_x=\sigma_y$. In fact, this is not true, because the expressions inside the square brackets will provide zero too at $\sigma_x=\sigma_y$ to cancel the poles outside. In our code, we calculate this equation at $1.01 \sigma_x$ and $0.99\sigma_x$, and average the total effects.

To invoke the calculation, one must use set up command ``insert_sceffects'' proceed ``run_setup'' and ``Twiss_output'' command proceed ``track''.