multipact.tex 14.2 KB
 snuverink_j committed Sep 06, 2017 1 2 3 4 5 6 7 \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. snuverink_j committed Sep 07, 2017 8 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. snuverink_j committed Sep 06, 2017 9 snuverink_j committed Sep 08, 2017 10 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}. snuverink_j committed Sep 06, 2017 11 snuverink_j committed Sep 06, 2017 12 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. snuverink_j committed Sep 06, 2017 13 14 15 16 17 18 19 \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} snuverink_j committed Sep 08, 2017 20 Here, the horizontal axis is the energy of impacting electron, the vertical axis is the SEY value $\delta$, defined as \ref{Furman-Pivi}: snuverink_j committed Sep 06, 2017 21 22 23 24 25 \delta = \frac{I_s}{I_0} \label{eq:SEY} 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. snuverink_j committed Sep 08, 2017 26 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}. snuverink_j committed Sep 06, 2017 27 snuverink_j committed Sep 06, 2017 28 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}. snuverink_j committed Sep 06, 2017 29 30 31 32 33 34 35 36 37 38 39 \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. snuverink_j committed Sep 08, 2017 40 The Vaughan's secondary emission model is based on a secondary emission yield formula \ref{Vaughan, VaughanRv}: snuverink_j committed Sep 06, 2017 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 \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