\input{header}
\chapter{Multipacting}
\label{chp:multpact}
\index{Multipacting}
Multiple electron impacting (multipacting) is a phenomenon in radio frequency (RF) structure that under certain conditions (material and geometry of the RF structure, frequency and level of the electromagnetic field, with or without the appearance of the magnetic field \ldots), electrons secondary emission yield (SEY) coefficient will be larger than one and lead to exponential multiplication of electrons.
Besides the particle tracker in \textit{OPAL}, the computational model for solving multipacting problem contains an accurate representation of 3D geometry of RF structure by using triangulated surface mesh see~Chapter~\ref{geometry,femiss}, an efficient particle-boundary collision test scheme, two different secondary emission models, and necessary post-processing scripts.
As we use a triangulated surface mesh to represent the RF structure, our particle-boundary collision test scheme is based on line segment-triangle intersection test. An axis aligned boundary box combined with surface triangle inward normal method is adopted to speedup the particle-boundary collision test \ref{WangHB2010}.
The SEY curve is a very important property of the surface material for the development of a multipacting in a RF structure. Figure~\ref{typicalSEY} shows a typical SEY curve.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\linewidth,angle=0]{figures/Multipacting/SEY_curve.pdf}
\caption{Typical SEY curve}
\label{fig:typicalSEY}
\end{center}
\end{figure}
Here, the horizontal axis is the energy of impacting electron, the vertical axis is the SEY value $\delta$, defined as \ref{Furman-Pivi}:
\begin{equation}
\delta = \frac{I_s}{I_0} \label{eq:SEY}
\end{equation}
where $I_0$ is the incident electron beam current and $I_s$ is the secondary current, i.e., the electron current emitted from the surface. Usually the SEY value $\delta$ appeared in an SEY curve is the measured SEY with normal incident, i.e., the impacting electron is perpendicular to the surface. The energy $E_1$ and $E_2$ are the first crossover energy and the second crossover energy respectively, where the SEY value $\delta$ exceed and fall down to $\delta = 1$ at the first time. Obviously, only the energy range of $\delta>1$, i.e., $E \in (E_1,E_2)$ can contribute to multipacting.
Both Furman-Pivi's probabilistic secondary emission model \ref{Furman-Pivi} and Vaughan's formula based secondary emission model \ref{Vaughan} have been implemented in \textit{OPAL} and have been benchmarked see~Section~\ref{RunPP}.
The Furman and Pivi's secondary emission model calculates the number of secondary electrons that result from an incident electron of a given energy on a material at a given angle (see Figure~\ref{incident electrons}). For each of the generated secondary electrons the associated process: \emph{true secondary}, \emph{rediffused} or \emph{backscattered} is recorded, as is sketched in Figure~\ref{incident electrons}.
\begin{figure}
\centering
\input{figures/Multipacting/HB_Fig4.tikz}
% \includegraphics[width=3 in]{incident_diagram.pdf}
\caption{Sketch map of the secondary emission process.}
\label{fig:incident electrons}
\end{figure}
This model is mathematically self-consistent, which means that (1) when averaging over an infinite number of secondary-emission events, the reconstructed $\delta$ and $\uglyder{\delta}{E}$ are guaranteed to agree with the corresponding input quantities; (2) the energy integral of $\uglyder{\delta}{E}$ is guaranteed to equal $\delta$; (3) the energy of any given emitted electron is guaranteed not to exceed the primary energy; and (4) the aggregate energy of the electrons emitted in any multi-electron event is also guaranteed not to exceed the primary energy. This model contains built-in SEY curves for copper and stainless steel and the only thing user need to set is to choose the material type, i.e., copper or stainless steel, as long as the surface material of user's RF structure has the same SEY curve as built-in SEY curves.
Although a set of parameters in the model can be adjusted to model different SEY curves without breaking the above mentioned mathematical self-consistency, it is easier to use Vaughan's formula based secondary emission model if user has to model a different SEY curve.
The Vaughan's secondary emission model is based on a secondary emission yield formula \ref{Vaughan, VaughanRv}:
\begin{subequations}
\label{allequations}
\begin{eqnarray}
\delta(E,\theta)&=&\delta_{max}(\theta)\cdot (v e^{1-v})^k,\ \text{for}\ v \le 3.6 \label{eq:VaughanA}
\\
\delta(E,\theta)&=&\delta_{max}(\theta)\cdot 1.125/v^{0.35},\ \text{for}\ v > 3.6 \label{eq:VaughanB}
\\
\delta(E,\theta)&=&\delta_0,\ \text{for}\ v \le 0 \label{eq:VaughanC}
\end{eqnarray}
\end{subequations}
where
\begin{eqnarray*}
v=\frac{\displaystyle E-E_0}{\displaystyle E_{max}(\theta)-E_0},
\end{eqnarray*}
\begin{eqnarray*}
k=0.56,\ \ \text{for}\ v<1,
\end{eqnarray*}
\begin{eqnarray*}
k=0.25,\ \ \text{for}\ 1