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SCMULT

Tracks through a zero length multipole to simulate space charge effects
Parallel capable? : yes
Parameter Name Units Type Default Description
GROUP   string NULL Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup





This element simulates transverse space charge (SC) kick using K.Y. Ng's formula ( K.Y. Ng, FNAL, private communication). N.B.: This element is not designed for space charge calculations in guns or linacs. It is only intended for simulating space charge in rings.

The linear SC force is given by:

\begin{displaymath}
\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)}
\end{displaymath}


\begin{displaymath}
\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)}
\end{displaymath} (93)

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:

\begin{displaymath}
\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 ]
\end{displaymath}


\begin{displaymath}
\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 ]
\end{displaymath} (94)

where $w(z)$ is the complex error function
\begin{displaymath}
w(z)=e^{-z^2}\left [ 1+\frac{2i}{\sqrt{\pi}}\int\limits_0^z e^{\zeta^2}d\zeta\right ]
\end{displaymath} (95)

Equation 94 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''.


next up previous
Next: SCRAPER Up: Element Dictionary Previous: SCATTER
Robert Soliday 2014-06-26