text:
\section{\label{sec:intro}Introduction}
Predicting the high-gradient performance limit imposed by vacuum breakdown is an important part of the design process of many high-voltage vacuum components such as radio frequency particle accelerator structures. The breakdown limit is defined as the highest RF power, with corresponding field levels, at which a structure can be operated stably. Stability in high-gradient applications is often defined by a maximum breakdown rate, where the rate is given by number of breakdowns per pulse. To minimise the need for multiple iterations of designing, building, and testing, a reliable physical model for RF breakdown levels is an essential design tool for high-gradient linear accelerator applications.

Criteria involving the surface electric field $E$, such as Kilpatrick's criterion \cite{KilpatrickOriginal}, magnetic field $H$ \cite{PerformanceLimitingEffects}, or both, such as the ratio of power to iris circumference $P/C$ \cite{Grudiev2009}, or the modified Poynting vector $S_c$ \cite{Grudiev2009}, quantifying breakdown performance already exist, but their range of applicability is limited and are consistent with only certain measurements. Evidence is presented in Section \ref{sec:crab_results} for inconsistency between the predictions of the modified Poynting vector and experimental results. The new model proposed in this paper aims to be more general with a more complete fundamental physical basis than previous work and apply to a broader range of experimental results, whilst still being relatively straightforward in concept and simple to compute. In this paper, existing models will be reviewed and inconsistencies discussed in Sec. \ref{sec:review}, and the new model detailed in \ref{sec:proposal}. The results of the model will be explored for generic RF structure geometries in Sec.\ref{sec:c3}, and for some experimentally-tested exampled in Sec. \ref{sec:comparison_with_experiments}. Additional aspects of the model and its properties will be discussed in Sec.\ref{sec:extensions}, followed by its implications on optimal structure design in Sec. \ref{sec:implications}.

\section{\label{sec:review}Review of Existing Models}
It has been shown that a breakdown criterion that is purely a function of the surface electric field magnitude (such as Boyd's formulation of Kilpatrick's criterion often used in the field of accelerator physics \cite{KilpatrickBoyd}) does not fully describe the breakdown behaviour of accelerating structures, as experiments have shown that structures made of the same material and operating at the same RF frequency, but with different geometries, can reach very different peak surface electric field values. In addition to electric-field-based limits, alternatives based on global and local power flow as well as peak magnetic field and associated pulsed surface heating have been proposed \cite{PerformanceLimitingEffects}.

$S_c$ and $P/C$ depend on the RF power flowing through the structure rather than the electric field. They have been shown to produce predictions of maximum gradient that are more consistent with experimental results than the peak surface electric field alone. $P/C$, however, does not take into account the exact shape of the structure and thus cannot provide detailed insight into how to optimise the geometry for breakdown performance. It is also not applicable to standing-wave structures, in which the real power flow is low.

Tests in the XBox test stands at CERN \cite{Woolley2015HighCavity} have shown cases in which the strongest clustering of breakdown craters was found to be in distinctly different locations from those predicted by the spatial distribution of $S_c$, and also magnetic field. One such case was the Compact Linear Collider (CLIC) Crab Cavity prototype \cite{CrabCavity_Report}, a backward travelling-wave constant-impedance structure with a group velocity of $0.029c$. Being a deflecting cavity, it was operated in the $TM_{110}$ mode as opposed to the $TM_{010}$ mode as would typically be the case with accelerating structures. This mode produces distinctly different electric field, magnetic field, and $S_c$ patterns. As can be seen in Figs.~\ref{fig:crab_post_mortem_e} and \ref{fig:crab_post_mortem_sc}, the spatial distribution of the breakdown craters observed in a \emph{post-mortem} study matched the spatial distribution of the surface electric field magnitude much more closely than that of $S_c$ and magnetic field \cite{CrabCavity_Report}. A large number of breakdowns even occurred at a location at which the $S_c$ value was close to zero. This signifies that neither $S_c$ nor magnetic field fully encapsulates the relationship between structure geometry and breakdown performance. Fig.~\ref{fig:crab_post_mortem_e} also shows that the breakdowns did not occur in locations of high magnetic field, which is inconsistent with models based on magnetic field and pulsed heating. The maximum input power and corresponding surface fields were, however, roughly in line with those predicted by $S_c$. The inconsistency in predicted breakdown locations motivates revisiting the question of high-gradient limits. In addition, there has been an improved understanding of the mechanisms of vacuum arcs which will be used to get a better limit.
Another shortcoming of Kilpatrick's criterion, $S_c$ and $H$ as predictors of breakdown performance is that they cannot be applied to direct-current (DC) setups, in which there is no steady-state power flow or surface currents. This implies that breakdowns should never occur in such systems under steady-state conditions, which is contrary to observations.

\begin{figure}[htbp]
\includegraphics[width=0.35\textwidth]{crab_post_mortem_cell2iris1_e_noscale.png}
\includegraphics[width=0.1\textwidth]{crab_post_mortem_cell2iris1_e_scale.png}
\caption{\label{fig:crab_post_mortem_e}Breakdown locations, represented by black dots, vs. position transverse to the beam in the second cell of the CLIC Crab Cavity, obtained from \emph{post-mortem} analysis \cite{CrabPostMortem}. Colours represent the surface electric field vs. position transverse to the beam. The magnetic field has a similar distribution, but rotated by 90\degree compared to the electric field.}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=0.35\textwidth]{crab_post_mortem_cell2iris1_sc_noscale.png}
\hspace{0.02\textwidth}
\includegraphics[width=0.1\textwidth]{crab_post_mortem_cell2iris1_sc_scale.png}
\caption{\label{fig:crab_post_mortem_sc}Breakdown locations, represented by black dots, vs. position transverse to the beam in the second cell of the CLIC Crab Cavity, obtained from \emph{post-mortem} analysis \cite{CrabPostMortem}. Colours represent $S_c$ value vs. position transverse to the beam.}
\end{figure}

Nevertheless, $S_c$ has seen success in predicting the field levels in accelerating structures operating in the $TM_{010}$ mode and has become a widely used quantity in RF design as evidenced by the numerous citations of \cite{Grudiev2009}, suggesting that available power does play an important role in determining the breakdown limit.

\section{\label{sec:proposal}Loaded Electric Field Model}
\subsection{\label{sec:outline}Outline and Reasoning}
The proposed new model is intended to encompass the idea of power flow determining breakdown rate whilst also resolving the discrepancy between the location of the maximum $S_c$ value and that of breakdown craters discussed in Sec. \ref{sec:review}.

It is believed that an RF breakdown begins with a field-emission site \cite{KovermannThesis}, which is likely a nanometer-scale feature on the surface of an accelerating cavity subject to high electric field. These sites are believed to have a field-enhancing geometric protrusion, locally lower workfunction, or other characteristic that causes them to emit a greater field-emission current density than the surrounding area. At any given time, there are many field-emission sites inside a structure that may suddenly begin a runaway process in which the emitted current grows by many orders of magnitude due to the increase in temperature of the emitter and formation of a plasma spot and develop into a breakdown.

Under the new model, it is proposed that, depending on the amount of RF power available, a field-emission site that has begun this thermal runaway process may either fully develop into an observable breakdown or extinguish itself partway. It is assumed that since the event in the latter case is very short in duration and results in a relatively small perturbation in the RF field compared to a normal breakdown, it is not normally noticed in normal operation, since the field do not collapse, reflected power is not produced, and stored energy is not discharged. The method is an attempt to quantify this threshold, by determining the surface electric field at the breakdown site when loaded by the rapidly increasing electron current associated with the developing breakdown. The quasi-equilibrium loaded and unloaded electric field amplitudes at the location of the breakdown site are denoted $E^*$ and $E_0$ respectively. Due to the loading effect of the electrons, $E^* \leq E_0$. The reduction in electric field caused by the loading depends on the electromagnetic power available from the power source, with less power resulting in a greater reduction in field. This means that breakdowns are most likely to occur in locations with a large surface electric field and high power flow, but are unlikely to occur in locations with a low surface electric field even if the power flow is very large.

$E^*$ therefore determines the breakdown performance of a given geometry, and should not exceed a given threshold value determined by the maximum allowed breakdown rate anywhere in the geometry, analogously to $S_c$. The calculation of $E^*$ for a particular breakdown site involves an equivalent circuit model in which the breakdown site is represented by a load impedance and the RF structure and power source are represented by a voltage source with a series impedance $\hat{Z}_{bd}$. For convenience, the calculation is performed in terms of electric field rather than voltage. The coupling of RF power to the breakdown is modelled by a short monopole antenna located on the surface of the structure at the location of the breakdown. The relationship between the surface electric field $\hat{E}$ at the breakdown site and the emitted current $\hat{I}$, is thus given by:

\begin{equation} \label{eqn:e_loading_function_complex}
    \hat{E} = \hat{E_0} - \hat{I}\hat{Z}_{bd}/l_{ant},
\end{equation}

where $l_{ant}$ is the antenna length. Here, $\hat{E}$, $\hat{E_0}$, $\hat{I}$, and $\hat{Z}_{bd}$ are complex phasor quantities. The emitted current is likely a nonlinear function of the surface electric field. Therefore, if the applied electric field varies sinusoidally in time, as is typical for resonant structures, the emitted current will have a periodic waveform in phase with the electric field. In this paper, only the fundamental frequency component of this waveform will be considered, since this component is likely to have the greatest effect on the peak loaded electric field. Calculating the loaded electric field is important as the basic premise of the model is that progression to full breakdown can only occur if there is a sufficiently high electric field remaining with the loading present. Since we are concerned with the reduction in peak electric field, only the real part of the electric field perturbation is relevant. Given that $\hat{E}$ and $\hat{I}$ have already been assumed to be in phase, only the real part of $\hat{Z}_{bd}$ is needed. Eq.\eqref{eqn:e_loading_function_complex} can thus be reduced to the scalar equation:

\begin{equation} \label{eqn:e_loading_function}
    E = E_0 - IR_{bd}/l_{ant},
\end{equation}

which will be referred to as the load line, with $R_{bd}$ being the real part of $\hat{Z}_{bd}$, and $E$, $E_0$ and $I$ being scalar values indicating the magnitude of $\hat{E}$, $\hat{E_0}$, and $\hat{I}$ respectively. $R_{bd}$ can be formally defined as:

\begin{equation} \label{eqn:bd_impedance_approx}
    R_{bd} = \Re(\hat{Z}_{bd}) \approx \Re\left(\frac{\Delta E l_{ant}}{\Delta I}\right),
\end{equation}

where $\Delta E$ is the perturbation in electric field resulting from a current $\Delta I$ in the antenna. Because of the linearity of Maxwell's equations in vacuum \cite{MicrowaveEngineering}, and if the geometry of the antenna is not affected by the Lorentz force, $\hat{Z}_{bd}$ is independent of the field levels. Given an emission function $I = f(E)$ that describes the load impedance presented by the breakdown, $E^*$ can be determined by solving:

\begin{equation} \label{eqn:e_loading_solution}
    E^* = E_0 - f(E^*)R_{bd}/l_{ant}
\end{equation}

\begin{figure}[htbp]
\includegraphics[width=0.5\textwidth]{example_impedance.png}
\hspace{0.02\textwidth}
\caption{\label{fig:example_impedance}Calculation of $E^*$ on a plot of surface electric field in MV/m vs. current emitted by a breakdown. Black: emission function as given by Eq. \eqref{eqn:child_langmuir_emission}. Blue: a breakdown site with an unloaded field of 150 MV/m and a low breakdown resistance. Red: a breakdown site with an unloaded field of 200 MV/m and a larger breakdown resistance. }
\end{figure}

An example of a solution of Eq.\eqref{eqn:e_loading_solution} is shown in Fig.~\ref{fig:example_impedance}. Two potential breakdown sites are shown, whose load lines cross the emission function at the same point, resulting in the $E^*$ being the same for both, implying equal breakdown probability. However, due to the larger $R_{bd}$ associated with Case 2, the unloaded surface field $E_0$ is larger than that of Case 1. This example illustrates how, under the loaded-field model, different geometries can achieve different peak field levels for a given breakdown rate.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.155\textwidth]{accelerating_field_arrows_crop.png}
\includegraphics[width=0.155\textwidth]{antenna_field_real_arrows_crop.png}
\includegraphics[width=0.155\textwidth]{antenna_field_imag_arrows_crop.png}
\caption{Electric-field patterns in a $TM_{010}$ cell. Left: Excited by RF power in the input port as in normal operation. Middle: Real part of the field pattern excited by a sinusoidal current in an antenna. Right: Imaginary part of the field pattern excited by a sinusoidal current in an antenna.}
\label{fig:field_patterns}
\end{figure}

An important aspect to note is that the beam loading effect causes a reduction of the field amplitude in the entire cell, and not only in the immediate vicinity of the antenna. Fig. \ref{fig:field_patterns} shows an example of a simulation showing the electric field pattern resulting from a current in an antenna compared with the electric field pattern in normal operation. One can see that the real part of the electric field (i.e. the component in phase with the antenna current) is very similar to the accelerating mode, which implies that a local emitted current affects the fields in the entire cell.

\subsection{\label{sec:emission_function}Emission Function}
Fowler-Nordheim field emission \cite{Wang1997FieldA} is believed to produce insufficient current to bring about significant beam loading in the entire cell. The electrons from an ionised plasma are a more likely candidate to produce sufficient current. Therefore, it is proposed that the emission function $I = f(E)$ should represent the behaviour of a breakdown in the onset phase, in which the breakdown consumes the most power and therefore RF power coupling is the most critical, rather than the pre-breakdown phase in which the current is produced by field emission. The intermediate breakdown stage is modelled here by the Child-Langmuir law \cite{ChildLangmuir}, which applies if the emission is space-charge-limited, which is the case for very large current densities typical of breakdowns \cite{SLAC_Plasma_Model}. The current density $J$ given by the Child-Langmuir law for a 1D diode with an applied voltage $V$ and length $d$ is:

\begin{equation} \label{eqn:child_langmuir}
    J = \frac{\epsilon_0}{9\pi} \left(\frac{2q_e}{m_e}\right)^{1/2} \frac{V^{3/2}}{d^2}
\end{equation}

A 1D model can be justified if the transverse dimensions of the plasma spot are much larger than the thickness of the emission layer, which would mean that most of the emitted electrons follow 1D trajectories that are initially perpendicular to the metal surface.

In the context of this study, the total current, as opposed to the current density, is important since it determines the total power absorbed by the breakdown. The Child-Langmuir law describes a linear vacuum diode in which electrons traverse a drift region of length $d$, after which they are absorbed by the anode and cease to affect the system. This is consistent with the use of a short antenna to model the breakdown, whereby the current flow is constrained to the length of the antenna only (see Sec. \ref{sec:antenna_lengths}). With $d = l_{ant}$ and $V = E l_{ant}$, Eq.\eqref{eqn:child_langmuir} can thus be reformulated as:

\begin{equation} \label{eqn:child_langmuir_emission}
    I = k E^{3/2}l_{ant}^{-1/2},
\end{equation}

where $k$ is a proportionality constant comprised of physical constants and effective emitter area. It should be noted that the use of the Child-Langmuir law implies an assumption that the electron velocities are non-relativistic, which might need reviewing. For simplicity, it is assumed that all potential breakdown sites behave identically, though  future iteration of the model could take into account random variations between breakdown sites.

\subsection{\label{sec:analytical_rbd}Analytical Derivation of Breakdown Resistance}
The use of a short antenna to model the breakdown enables an easy way of quantitatively calculating of the impedance of the RF power source as experienced by the breakdown. This can be performed either analytically or with a finite-element electromagnetic simulation.

\begin{figure}[b]
\centering
\includegraphics[width=0.4\textwidth]{single_cell_rbd.png}
\caption{Equivalent circuit of a single cell in a travelling-wave configuration.}
\label{fig:single_cell_circuit}
\end{figure}

The analytical calculation of $R_{bd}$ involves a circuit model of the structure including the coupling of power sources and loads. The breakdown is represented by a port coupled to one of the cells via an ideal transformer of ratio $N_{bd}:1$. With such a circuit, $R_{bd}$ is the equivalent resistance of the network as observed through the breakdown port. An example for a single cell is shown in Fig. \ref{fig:single_cell_circuit}.

At the resonant frequency of the cell $f = f_r = 1/2\pi\sqrt{LC}$ \cite{MicrowaveEngineering}, the impedances of the inductor and capacitor cancel each other out, leaving only the real resistance $R$. In most practical travelling-wave structures, the impedance of the structure is also dominated by external coupling rather than resistive losses. The power $P$ flowing through the cell, assuming zero breakdown current, can thus be expressed in terms of circuit parameters as:

\begin{equation} \label{eqn:circuit_model_power}
P = \frac{V_{in}^2}{4Z_0}
\end{equation}

It can be easily seen that in this circuit, $V_{acc} = N_{bd}V_{bd}$. The value of $N_{bd}$ depends on the location of the breakdown within the cell, which determines the electric field along the antenna for a given accelerating voltage, as given by:

\begin{equation} \label{eqn:bd_transformer}
    N_{bd} = \frac{V_{acc}}{V_{bd}} = \frac{a}{l_{ant}} \frac{E_{acc}}{E_0},
\end{equation}

where $a$ is the length of the cell, $l_{ant}$ is the length of the antenna, $E_0$ is the surface electric field at the location of the antenna and $E_{acc}$ is the accelerating gradient. With some circuit manipulations, the output impedance of the circuit as seen by the breakdown port can also be expressed in terms of circuit parameters:

\begin{equation} \label{eqn:rbd_circuit}
R_{bd} = \frac{Z_0 N_{cell}^2}{2 N_{bd}^2}
\end{equation}

In the case of no breakdown current, the accelerating voltage can be expressed in terms of the input voltage as $V_{acc} = N_{cell}V_{in}/2$. With this,  Eq. \eqref{eqn:circuit_model_power}, and the relationship between power and group velocity \cite{Wangler2008}, given by:

\begin{equation}
    P = \frac{v_g a}{2 \pi f} \frac{Q}{R} E_{acc}^2,
\end{equation}

where $P$ is the incident power from the source, $f$ is the operating frequency, $v_g$ is the group velocity, $a$ is the length of the cell, $R$ is the shunt impedance, and $Q$ is the quality factor of the structure, the relationship between input power and accelerating voltage can be derived as:

\begin{equation} \label{eqn:rbd_z0}
    \frac{V_{acc}^2}{P} = \frac{2 \pi a f}{v_g} \frac{R}{Q} = N_{cell}^2 Z_0.
\end{equation}

$R_{bd}$ can thus be expressed entirely in terms of quantities obtainable from an eigenmode solution of the electromagnetic wave equation in the cell geometry as:

\begin{equation} \label{eqn:analytical_rbd_single_freq}
    R_{bd} = \frac{\pi f_r l_{ant}^2}{v_g a} \frac{R}{Q} \left(\frac{E_0}{E_{acc}}\right)^2.
\end{equation}

\begin{figure}[b]
\centering
\includegraphics[width=0.45\textwidth]{coupled_cells_rbd.png}
\caption{Equivalent circuit of an infinite array of identical coupled resonant cells, with a port in one cell representing the coupling to a breakdown.}
\label{fig:coupled_cells_rbd_circuit}
\end{figure}

If transient behaviour needs to be taken into account (see sec. \ref{sec:broadband_correction}), the broadband behaviour of the structure needs to be modelled. Fig.~\ref{fig:coupled_cells_rbd_circuit} shows a circuit model of a structure with an infinite number of identical cells coupled with mutual inductances $M$ instead of ideal transformers. One of the cells is coupled to a breakdown port via an ideal transformer as before. The array is assumed to have a travelling wave flowing through it, with a phase advance $ka$ from cell to cell. Thus, the $n$\textsuperscript{th} cell is assumed to have voltage and current values $v_n$ and $i_n$ given by:

\begin{eqnarray} \label{eqn:circuit_model_vn_in}
    v_n &&= v_0 e^{jkan}\\
    i_n &&= i_0 e^{jkan}
\end{eqnarray}

Using circuit theory, one can obtain an expression for the dependence of currents and voltages in adjacent cells:

\begin{eqnarray} \label{eqn:circuit_model_kirchoff}
    &&v_{n+1} = j 2 \pi f M i'_n \nonumber\\&&= j 2 \pi f M \left(i_n - \frac{v_n}{j 2 \pi f  L} - j 2 \pi f  C v_n - \frac{v_n}{R} \right),
\end{eqnarray}

as well as the characteristic impedance $Z_0$ of this transmission line:

\begin{equation} \label{eqn:z0_definition}
    Z_0 = \frac{v_n}{i_n} = -j 2 \pi f  M e^{jka}.
\end{equation}

Eqs. \eqref{eqn:circuit_model_vn_in} and \eqref{eqn:circuit_model_kirchoff} allow the dispersion relation of this transmission line to be expressed as:

\begin{eqnarray} \label{eqn:circuit_model_kirchoff_3}
    &&e^{jka} + e^{-jka} = 2 \cos(ka) \nonumber\\ &&= - j 2 \pi f M \left(\frac{1}{j 2 \pi f L} + j 2 \pi f C + \frac{1}{R} \right).
\end{eqnarray}

By substituting $0$ and $\pi$ for the phase advance $ka$ into the dispersion relation \eqref{eqn:circuit_model_kirchoff_3} and defining the resulting values of $f$ as $f_0$ and $f_{\pi}$ respectively, as well as using the definitions of resonance frequency $f_r = 1/2 \pi \sqrt{LC}$ and quality factor $Q = R\sqrt{L/C}$, an expression for $M$ may also be obtained:

\begin{equation} \label{eqn:M}
    M = \frac{2}{\pi} \frac{R}{Q} \frac{f_0}{f_{\pi}^2 - f_0^2},
\end{equation}

leading to an expression of the dispersion relation without circuit values:

\begin{equation} \label{eqn:dispersion_relation}
    f^2 = f_r^2 - \frac{f_{\pi}^2 - f_0^2}{2} \cos(ka).
\end{equation}

Using \eqref{eqn:M} and \eqref{eqn:dispersion_relation}, one can also eliminate the parameters $M$ and $ka$ from the expression for characteristic impedance \eqref{eqn:z0_definition}, yielding:

\begin{equation} \label{eqn:z0_eigen}
    \Re(Z_0(f)) = \frac{R}{Q} \frac{4f f_r}{f_{\pi}^2 - f_0^2} \left(1-\left(2\,\frac{f_r^2 - f^2}{f_{\pi}^2 - f_0^2}\right)^2\right)^\frac{1}{2}.
\end{equation}

The $R_{bd}$ value in this case can be determined from the characteristic impedance $Z_0$ of the structure:

\begin{eqnarray} \label{eqn:analytical_rbd}
    &&R_{bd}(f) = \frac{\Re(Z_0(f))}{2N_{bd}^2}= \nonumber\\
 && \frac{2f f_r}{f_{\pi}^2 - f_0^2} \left(1-\left(2\frac{f_r^2 - f^2}{f_{\pi}^2 - f_0^2}\right)^2\right)^\frac{1}{2} \nonumber\\ &&\times \frac{R}{Q} \left(\frac{l_{ant}}{a}\right)^2 \left(\frac{E_0}{E_{acc}}\right)^2,
\end{eqnarray}

where $f_0$, $f_r$, $f_\pi$ are frequencies corresponding to a cell-to-cell phase advance values of zero, $\pi/2$, and $\pi$ respectively. Standing-wave structures will not be considered specifically in this paper, though it is expected that an analogous approach, using a circuit model with a finite number of cells coupled to a transmission line, should give appropriate results.

\subsection{\label{sec:antenna_lengths}Antenna Lengths}
Both of the analytical models presented in Sec. \ref{sec:analytical_rbd} exhibit a quadratic dependence of $R_{bd}$ on the antenna length, i.e.: $R_{bd} \propto l_{ant}^2$. This resembles the behaviour of the radiation resistance of a small Hertzian dipole whose length $\l_{ant}$ is much smaller than the free-space wavelength $\lambda$ \cite{AntennaTheory}:

\begin{equation} \label{eqn:Hertz_dipole}
    R_{rad} = \frac{2\pi}{3} \zeta_0 \left( \frac{l_{ant}}{\lambda} \right)^2,
\end{equation}

where $R_{rad}$ is the radiation resistance, and $\zeta_0$ is the impedance of free space. The antenna length is the only parameter in Eqs. \eqref{eqn:analytical_rbd_single_freq} and \eqref{eqn:analytical_rbd} not already defined by the geometry of the structure. It is not immediately clear what value of antenna length should be chosen to best model a breakdown. It may not be practical in general to use a particular antenna length in all cases, due to limitations imposed by the geometry in question. The issue of choosing a value for antenna length can be circumvented by introducing a quantity $R_0$ that is independent of $l_{ant}$, such that:

\begin{equation} \label{eqn:r0_definition}
    R_{bd} = R_0 l_{ant}^2
\end{equation}

As per the definition of $R_{bd}$ in Eq. \eqref{eqn:bd_impedance_approx},

\begin{equation} \label{eqn:bd_resistance_normalised}
\begin{aligned}
    \Re (\Delta V) &= I R_{bd} = I R_0 l_{ant}^2\\
    \Re (\Delta E) &= \frac{\Re (\Delta V)}{l_{ant}} = Il_{ant} R_0
    \end{aligned}
\end{equation}

Thus, if the breakdown loading plot (such as the one in Fig. \ref{fig:example_impedance}), is modified such that the axes are the electric field $E$ and the product $Il_{ant}$, the plot and equilibrium solution become independent of antenna length. To suit, the emission function can also be expressed as:

\begin{equation} \label{eqn:bd_emission_characteristic_normalised}
    Il_{ant} = k' E^{3/2},
\end{equation}

where $k = k' l_{ant}^{-1/2}$. The value of $k'$ can be chosen independently of the antenna length. Eq. \eqref{eqn:e_loading_solution} can thus be expressed as the solution to the equation:

\begin{equation}
\label{eqn:eloaded_solution}
    k' R_0 E^{*3/2} + E^* - E_0 = 0,
\end{equation}

which is independent of antenna length. As the electrons emitted during a breakdown continue to absorb energy from the local electric field until they collide with the wall of the structure or escape, the amount of energy they absorb depends on the length of their trajectories. The fact that the quantity $Il_{ant}$ depends on antenna length reflects this behaviour, and picking $l_{ant}$ to be a value representative of the mean distance travelled by the emitted particles appears reasonable. It is difficult to determine the exact physical length of a breakdown during the onset phase in every possible breakdown location in a given geometry without a detailed particle tracking simulation study but it is most reasonable to assume it scales approximately linearly with the size of the cell. If this is the case, the same $k'$ value should be used for structures with the same operating frequency, and thus similar dimensions. It may be the case for some geometries that $l_{ant}$ is independent of frequency. For example, the electron trajectory could be constrained by physical obstacles, such as a narrow waveguide or the vanes of a radiofrequency quadrupole (RFQ). Such effects are briefly considered in Sec. \ref{sec:thin_gaps}, but require further work.

\subsection{\label{sec:broadband_correction}Broadband Effects}
As breakdown is by nature a transient phenomenon, the transient behaviour of the system needs to be considered. Introducing transient behaviour requires information about $R_{bd}$ over the full bandwidth of the structure. In contrast, the previous case of calculating $E^*$ using $R_{bd}$ at a single fixed frequency implies a quasi-steady-state scenario in which the local stored energy plays a relatively minor role, like in high-group-velocity structures.

It was decided to use a scalar impedance-like quantity that is representative of the transient behaviour of the structure. Following an impulse excitation current $i(t) = Q\delta(t)$, $Q$ being the emitted charge and $\delta(t)$ begin the Dirac delta function, the transient voltage across the antenna is given by $Qr_{bd}(t)$, where $r_{bd}(t)$ is the inverse Fourier transform of $R_{bd}(f)$. The total energy $W$ dissipated by the breakdown (approximated by a linear load resistance $R_l$) will thus be given by:

\begin{equation} \label{eqn:broadband_derviation}
    W = \frac{Q^2}{R_l} \int_{-\infty}^{\infty} r_{bd}^2(t) dt = \frac{Q^2}{R_l} \int_{-\infty}^{\infty} R_{bd}^2(f) df.
\end{equation}

A distinction should be made between the energy that results in the heating of the emission site and the total energy absorbed by the breakdown event. Much of the energy consumed by a breakdown results in the acceleration of electrons to high energies which then either escape the structure or impact a site relatively far from the breakdown site, neither of which contribute to the heating of the breakdown site. If it is the latter that is important, then $R_l$ should represent the load resistance of the breakdown site rather than the entire breakdown. The quantity $L$ was defined in order to describe the influence of the structure geometry on the heating of the emission site:

\begin{equation} \label{eqn:square_integral_rbd}
    L = \sqrt{\int_{0}^{\infty}R_{bd}(f)^2 df}.
\end{equation}

While the value of $R_l$ is unknown, it can be absorbed by the proportionality constant $k$ of the emission function, as long as all candidate breakdown sites have similar properties. The square root of the integral is taken to obtain a quantity that scaled linearly with $R_{bd}$. Analogously to $R_0$, a quantity $L_0$ that is independent of antenna length can also be introduced such that $L = L_0 l_{ant}^2$, which cancels out the $l_{ant}^2$ term seen in Eq.\eqref{eqn:analytical_rbd}.

\subsection{\label{sec:numerical_rbd}Numerical Calculation of Breakdown Resistance}
The concept of using a monopole antenna to model the electromagnetic coupling of a breakdown can be implemented directly in a numerical simulation, which allows $R_{bd}$ to be determined for arbitrary geometries and frequencies. In this study, simulations were performed using CST Microwave Studio \cite{CST}. To model the breakdown, a Discrete Current Port was added to the geometry, with one end point located on the surface of the structure at the location of the prospective breakdown site, while the other end point was located a distance $l_{ant}$ away, with the antenna normal to the surface. A frequency-domain simulation was run with the antenna used as a constant-amplitude current source. The $R_{bd}$ value for this location was calculated using Eq.\eqref{eqn:bd_impedance_approx}. Since a current port in Microwave Studio models an ideal current source, the impedance of the port itself is infinite, meaning that no power is absorbed by a port not currently set as the excitation source. Thus, several such ports could be incorporated into the model simultaneously, allowing the $R_{bd}$ to be calculated at multiple locations using the same mesh. The relation in Eq.~\eqref{eqn:r0_definition} can be used to scale results obtained with different $l_{ant}$ values, which means that $l_{ant}$ can be chosen for optimal meshing performance in a specific simulation setup.

The $R_{bd}$ value can also be obtained from a finite-element simulation by using the $Z$-matrix \cite{MicrowaveEngineering} of the simulated geometry:

\begin{equation} \label{eqn:z_param}
    R_{bd} = \Re \left(Z_{a,a} - \frac{Z_{a,i} Z_{i,a}}{Z_0 + Z_{i,i}} \right),
\end{equation}

where $Z_0$ is series impedance of the RF power source, and the subscripts $i$ and $a$ indicate terms of the $Z$-matrix corresponding to the RF input port and the antenna respectively. In CST Microwave Studio, this requires the antennas to be set as S-parameter ports with a finite characteristic impedance. A sufficiently large value of $Z_0$ allows several antennas to be incorporated into the same simulation with negligible loading effects, as with the current sources.

\section{\label{sec:c3}Comparison with $S_c$ for $TM_{010}$ structures}
There is ample experimental data that $S_c$ accurately predicts the breakdown performance for RF structures operating in the $TM_{010}$ mode, so it can be used as a benchmark for the new antenna-based model in this limited parameter space. Thus, the predictions of the $E^*$ model should be in line with those of $S_c$ for such structures. To investigate this, a series of hypothetical $TM_{010}$ travelling-wave accelerating structure geometries was simulated to compare $E^*$ and $S_c$ values inside them.

\begin{table}[]
\caption{\label{tab:c3_params}Parameters common to all of the C3 structures simulated.}
\centering
\begin{ruledtabular}
\begin{tabular}{lcc}
Parameter            & Value & Unit \\ \hline
Operating frequency  & 12    & GHz  \\
Phase advance        & 120   &  \degree \\
Phase velocity / c   & 1     &   -   \\
Periodicity          & 8.333 & mm   \\
Iris thickness     & 2     & mm   \\
Iris rounding radius & 1     & mm   \\
Cell rounding radius & 2     & mm
\end{tabular}
\end{ruledtabular}
\end{table}

Due to the availability of a relatively large body of experimental data on CLIC-like structures, the parameters of the simulated structures were made similar to CLIC structures so that a meaningful comparison could be made. Hence, a travelling wave design with a range of aperture radii varying from 2.5 mm to 6 mm was simulated. The outer radius of the cells was adjusted for each aperture size to maintain a 120\degree phase advance from cell to cell at 12 GHz. The dimensions of each regular cell are shown in Fig~\ref{fig:c3_dimensions}, with values given in Tab.~\ref{tab:c3_params}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.22\textwidth]{cell_params_cropped.png}
\includegraphics[width=0.22\textwidth]{antenna_locations.png}
\caption{Left: dimensions of a regular cell. Right: placement of antennas for numerical $R_{bd}$ calculations, with markings showing the distances from the iris apex in millimetres. Antenna lengths not to scale.}
\label{fig:c3_dimensions}
\end{figure}

\begin{table}[]
\caption{\label{tab:c3_matching_params}Geometrical parameters of each of the simulated C3 structures.}
\centering
\begin{ruledtabular}
\begin{tabular}{cccc}
\multicolumn{1}{l}{$r_{iris}$ {[}mm{]}} & \multicolumn{1}{l}{$r_{cell}$ {[}mm{]}} & \multicolumn{1}{l}{$r_{iris, match}$ {[}mm{]}} & \multicolumn{1}{l}{$r_{cell, match}$ {[}mm{]}} \\ \hline
2.5                                       & 10.083                                    & 4.803                                              & 10.438                                            \\
3                                         & 10.211                                    & 5.316                                              & 10.608                                            \\
3.5                                       & 10.373                                    & 5.765                                              & 10.785                                            \\
4                                         & 10.563                                    & 6.194                                              & 10.975                                            \\
5                                         & 11.026                                    & 6.962                                              & 11.381                                            \\
6                                         & 11.584                                    & 7.660                                              & 11.817
\end{tabular}
\end{ruledtabular}
\end{table}

Since calculating $E^*$ requires a consideration of power flow, a calculation of the eigenmode field patterns was insufficient as it would not be useful to simulate the resonant cells alone without specifying how RF power was coupled into them. Thus, the simulation also had to include matched waveguide ports at each end of the structure, coupled to circular waveguides of 12 mm radius. To  ensure an impedance match between the structure and the circular waveguides, matching cells were incorporated between the regular cells and the input and output waveguides. The radius of the matching cells and the outermost irises was chosen independently of the parameters of the regular cells to minimise the reflected power at 12 GHz, with the optimised parameter values shown in Table  \ref{tab:c3_matching_params}.The structure featured three identical regular cells, with the middle cell containing a number of antennas on its surface. The antennas were placed in a line from the apex of one of the irises bordering the centre cell, to the middle of the wall of the centre cell, as illustrated in Fig. \ref{fig:c3_dimensions}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{full_structure.png}
\caption{Diagram of the full structure simulated. The shaded area represents the region in which antennas were placed for calculating $R_{bd}$.}
\label{fig:c3_full_structure_diagram}
\end{figure}

Three regular cells were used to avoid any distortions of the results caused by evanescent fields from the matching cells bleeding into the cell containing the antennas. This helped make the conditions in the centre cell more similar to those within an infinitely long constant-impedance structure.

A diagram of the full geometry simulated is shown in Fig.~\ref{fig:c3_full_structure_diagram}. The simulations were performed using CST Microwave Studio as described in Sec. \ref{sec:numerical_rbd} to calculate $E_0$ and $R_{bd}$ values at locations inside the structure.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{rbd_vs_ant_l_4mm.png}
\caption{Calculated $R_{bd}$ in $\Omega$ vs. antenna length in mm for antennas in different locations in a C3 structures of 4 mm aperture radius. The distance of the antenna from the iris apex, as shown in Fig.~\ref{fig:c3_dimensions}, is indicated in the legend.}
\label{fig:rbd_vs_ant_l_mesh_comparison}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{rbd_vs_lant_statistics.png}
\caption{Distribution of $log(\Delta R_{bd}) / log(\Delta l_{ant})$, i.e. the slope on a logarithmic plot of $R_{bd}$ vs. $l_{ant}$ for different antenna lengths and aperture sizes.}
\label{fig:rbd_vs_lant_statistics}
\end{figure}

Simulations were performed with various antenna lengths varied to find the optimal value. Fig. \ref{fig:rbd_vs_ant_l_mesh_comparison} shows the dependence of $R_{bd}$ on antenna length for different antenna locations for a 4 mm iris radius. For values of $l_{ant}$ between 0.1 and 1 mm, the relation $R_{bd} \propto l_{ant}^2$ held for most of the antennas. For most of the antennas, the quadratic relationship did not hold outside of this range. It is believed that this was due to numerical noise for very short antennas, and the assumption that $E$ is constant over the length of the antenna no longer being valid for very long antennas. Since an antenna length of 0.1 mm resulted in the closest adherence to the expected $R_{bd} \propto l_{ant}^2$, as illustrated in Fig. \ref{fig:rbd_vs_lant_statistics}, this was the length value at which the final $R_{bd}$ values were obtained.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{e0_profile.png}
\caption{Unloaded surface electric field at unit accelerating gradient vs. position along the surface in mm. The position is defined by the arrow shown in Fig.~\ref{fig:c3_dimensions}, where 0 is the iris apex. Different coloured plots represent C3 structures of different aperture sizes. The discontinuities near the 0.5 mm and 1.5 mm marks on some of the plots are likely the result of the rounded geometry of the iris being discretised as a tetrahedral mesh in the finite-element simulation.}
\label{fig:c3_e0_profile}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{sc_profile.png}
\caption{$\sqrt{S_c}$ at unit accelerating gradient vs. position along the surface in mm. The position is defined by the arrow shown in Fig.~\ref{fig:c3_dimensions}, where 0 is the iris apex. Different coloured plots represent C3 structures of different aperture sizes. The discontinuities near the 0.5 mm and 1.5 mm marks on some of the plots are likely the result of the rounded geometry of the iris being discretised as a tetrahedral mesh in the finite-element simulation.}
\label{fig:c3_sc_profile}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{eloaded_transient_profile.png}
\caption{$E^*$ at an accelerating gradient of 100 MV/m vs. position along the surface in mm. The position is defined by the arrow shown in Fig.~\ref{fig:c3_dimensions}, where 0 is the iris apex. Different coloured plots represent C3 structures of different aperture sizes. The discontinuities near the 0.5 mm and 1.5 mm marks on some of the plots are likely the result of the rounded geometry of the iris being discretised as a tetrahedral mesh in the finite-element simulation.}
\label{fig:c3_transient_eloaded_profile}
\end{figure}

A value of $k' = 2.30\times10^{-17}$ Am\textsuperscript{2}V\textsuperscript{-3/2} was chosen along with a threshold $E^*$ of 105 MV/m, which produced results consistent with experimental data (see Sec.\ref{sec:t24_vs_crab_results}).

$E^*$ and $S_c$ were compared in two ways: the expected breakdown location, i.e. the physical location in the geometry where the maximum value of the quantity is found, and the prediction of the maximum achievable accelerating gradient. The breakdown location is compared in Figs. \ref{fig:c3_e0_profile}, \ref{fig:c3_sc_profile}, \ref{fig:c3_transient_eloaded_profile}, which show $E_0$, $S_c$, and $E^*$ vs. position on the surface of the middle cell respectively. $E_0$ and $S_c$ have been normalised to accelerating gradient, whereas $E^*$, not scaling linearly with gradient, has been plotted for an accelerating gradient of 100 MV/m. Both $S_c$ and $E_0$ have peaks between the 1 mm and 2 mm marks, which corresponds roughly to experimentally observed breakdown locations, i.e. a short distance from the iris apex at the location of maximum $E_0$.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{transient_max_gradient_comparison.png}
\caption{Maximum accelerating gradient without beam loading vs. aperture size and group velocity as a fraction of $c$.}
\label{fig:c3_max_gradient_transient}
\end{figure}

Predictions of the maximum gradient according to various breakdown quantities are represented in Fig. \ref{fig:c3_max_gradient_transient}. For each quantity, a certain maximum permissible value was chosen. The values $S_c = $ 4.52 MW/mm\textsuperscript{2}, $P/C = $ 4.8 MW/mm, and $E_0 = $ 90 MV/m (the Kilpatrick limit at 12 GHz) are used as limiting quantities along with $E^* = $ 105 MV/m (see Sec. \ref{sec:t24_vs_crab_results}). Similar behaviour can be seen for all three quantities for iris radii of 3.5 mm and above. The maximum gradient prediction of $P/C$ diverges for smaller iris radii, and therefore smaller group velocities. This is likely due to the contribution of energy stored in the structure, which is not considered by the $P/C$ quantity, yet becomes more significant as the group velocity decreases.

\section{\label{sec:comparison_with_experiments}Comparison with Experimental Results}
$E^*$ was calculated for two experimentally-tested X-band structure geometries to allow a comparison to experimental results, and to check if the method shows consistent behaviour and its usefulness a predictor of breakdown performance. Both of the structures discussed in this section are, like the idealised structures discussed in Sec. \ref{sec:c3}, 12 GHz travelling-wave structures. The same general approach as before to the numerical calculation of $E^*$ was used, with $k' = 2.30\times10^{-17}$ Am\textsuperscript{2}V\textsuperscript{-3/2}, the same value as used in Sec. \ref{sec:c3}. The $R_{bd}$ values were obtained from simulations performed with $l_{ant} =$ 0.5 mm, followed by scaling to $l_{ant} =$ 0.1 mm to make meaningful comparisons with the results from Sec. \ref{sec:c3}.

In should be noted that in the $E^*$ model, conditioning would correspond to the increase of the $E^*$ value at which breakdown occurs. To model this, the threshold $E^*$ value should therefore be increased with conditioning progress. However, experiments generally show the accelerating gradient tending to some limit as conditioning progresses. The experimentally-tested structures discussed in this section have been fully conditioned and the limiting gradient values will be compared, to ensure that the comparison is made between structures in similar states of conditioning.

\subsection{\label{sec:crab_results}CLIC Crab Cavity}
One of the structures considered here is the CLIC Crab Cavity prototype discussed in Sec. \ref{sec:review}, in order to determine whether or not the $E^*$ model resolves the discrepancy in location of breakdowns between the predictions of $S_c$ and experimental results.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{crab_setup_antenna_locations.png}
\caption{A cutaway view of the internal volume of the CLIC Crab Cavity with the locations of antennas in the coupling cell shown as white lines with yellow circles around them.}
\label{fig:crab_antenna_setup}
\end{figure}

Antennas were placed at various azimuths around the iris of the input coupling cell and first regular cell. When this structure was tested in XBox 2, most of the breakdowns were found to occur in these two cells, with a very small proportion of the total breakdown count located in the remaining cells. This is typical of constant-impedance travelling-wave structures, in which the power and field levels gradually decrease along the structure due to Ohmic losses. Nine antennas were used for each of the two cells, forming a 180\degree arc in each case. The antennas were placed slightly away from the iris apex at the location of maximum surface electric field, in order to characterise the limiting field value of the structure. The location of the antennas in the coupling cell is illustrated in Fig.~\ref{fig:crab_antenna_setup}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{crab_post_mortem_cell1_angle.png}
\hspace{0.02\textwidth}
\includegraphics[width=0.5\textwidth]{crab_cavity_cell1_eloaded_profile_transient.png}
\caption{Top: breakdown locations (represented by blue dots) vs. transverse position in  the input coupling cell of the CLIC Crab Cavity. Bottom: unloaded surface electric field (solid blue) and $E^*$ (dashed blue) in MV/m and $S_c$ (red) in MW/mm\textsuperscript{2} vs. angular position on the iris in degrees, at an input power of 43.0 MW. The definition of the angular position relative to the geometry of the cell is shown in the top plot.}
\label{fig:crab_cell1_eloaded_profile}
\end{figure}

Fig. \ref{fig:crab_cell1_eloaded_profile} shows a comparison of $E_0$, $E^*$, and $S_c$ at the locations of each of the antennas in the coupling cell, as well as a plot of actual breakdown locations in that cell. It can be clearly seen that, unlike $S_c$, $E^*$ exhibits maxima at the 0\degree and 180\degree positions, which coincides with the two clusters of breakdowns. The $E^*$ value of the 180\degree position is also higher than that in the 0\degree position, which is consistent with more breakdowns having occurred close to the former position. The magnetic field has a distribution with maxima rotated 90\degree compared to the electric field, which corresponds to the minimum of the breakdown distribution. Although the $E^*$ distribution in this cell is qualitatively similar to that of $E_0$, its absolute level is important in determining the breakdown performance of the structure, and will be discussed in more detail in Sec. \ref{sec:t24_vs_crab_results}.

\subsection{\label{sec:t24_results}CLIC T24 Structures}
A calculation was also performed on a T24 CLIC structure prototype without Higher-Order-Mode (HOM) damping \cite{T24Design}, that can be considered representative of typical undamped designs for CLIC. A large number of experimental results for this and similar geometries, but with different dimensions and RF parameters, are available, and $S_c$ has been shown to work very well for this class of structure. The coupling iris diameter of the structure is tapered, meaning among other things that the group velocity decreases from 0.018c to 0.009c along its length in order to compensate for the reduction in accelerating gradient caused by the loss of RF power to the beam and wall resistance. It has a mean accelerating gradient of 100 MV/m with no beam loading at its nominal power of 37.5 MW, and was optimised for breakdown performance using $S_c$.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{psi_setup_annotated_antenna_locations.png}
\caption{A cutaway view of the geometry of the T24 structure with the locations of antennas shown as white lines with yellow circles around them.}
\label{fig:t24_antenna_setup}
\end{figure}

As the cells are radially symmetric, there was no need to place antennas at different azimuths. Twelve antennas were placed in various cells along the length of the structure to observe the dependence of $E^*$ on the tapering of the structure. As with the Crab Cavity, each antenna was placed slightly away from the apex of each iris at the location of maximum surface electric field.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{psi1_bd_pos_full.png}
\hspace{0.02\textwidth}
\includegraphics[width=0.5\textwidth]{t24_eloaded_profile_transient.png}
\caption{Top: Accumulated breakdowns vs. longitudinal position in the T24PSIN1 over its entire test run in XBox 2, in units of RF signal propagation time in ns. 0 ns represents the structure input and 65 ns represents the structure output \cite{JanThesis}. Bottom: unloaded surface electric field (solid blue) and $E^*$ (dashed blue) in MV/m and $S_c$ (red) in MW/mm\textsuperscript{2} vs. iris number, at an input power of 42.4 MW.}
\label{fig:t24_eloaded_profile}
\end{figure}

The $E_0$, $E^*$, and $S_c$ values at each antenna location are compared in the bottom plot of Fig. \ref{fig:t24_eloaded_profile}. Due to the tapered design, $E_0$ and $S_c$ increase along the length of the structure and are both larger at the output end than the input end when no beam is present. Despite this, tests at the XBox test stands have shown a concentration of breakdowns near the input end of this and other structures \cite{JanThesis} \cite{TD26CC} \cite{SiC_Results}, for example as shown in the top plot of Fig. \ref{fig:t24_eloaded_profile}. On the other hand, $E^*$, whose value in the input coupler cell is significantly higher than in the regular cells, is consistent with these results.

\subsection{\label{sec:t24_vs_crab_results}Absolute $E^*$ Values}

\begin{table}[]
\caption{\label{tab:t24_vs_psi_results}Comparison of $E^*$ and $S_c$ results for the T24 structure and Crab Cavity. The reference power refers to the RF power at which each structure had a breakdown rate of $10^{-6}$ breakdowns per pulse with 200 ns pulses after conditioning.}
\centering
\begin{ruledtabular}
\begin{tabular}{l|c|c|c}
Parameter            & T24 & Crab & Units \\ \hline
Nominal power without beam & 37.5 &  13.35 & MW \\
$S_c$ at nominal power & 3.50 & 1.71 & MW/mm\textsuperscript{2} \\
$E^*$ at nominal power & 98.6 & 60.8 & MV/m \\ \hline
Reference power & 42.4 & 43.0 & MW \\
$S_c$ at reference power & 3.95 & 5.09 & MW/mm\textsuperscript{2} \\
$E^*$ at reference power & 123 & 105 & MV/m \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}[htbp]
\includegraphics[width=0.48\textwidth]{t24_vs_crab_fitting.png}
\hspace{0.02\textwidth}
\caption{\label{fig:t24_vs_crab_fitting}Red: load lines corresponding to various regular cells of the T24 structure at 42.4 MW. Blue: Load lines corresponding to various locations inside the first cell of the Crab Cavity prototype at 43.0 MW. Black: emission function as given by \eqref{eqn:child_langmuir_emission}.}
\end{figure}

As the premise of the loaded-field model is that the $E^*$ value for a given breakdown rate and material is constant irrespective of the geometry, test results of the T24 and Crab Cavity prototype structures were used as reference values. As was the case with $S_c$, the results were compared to a pulse length of 200 ns and a breakdown rate of $10^{-6}$ breakdowns per pulse (bpp). At the end of its test run, the Crab Cavity was found to run stably under these conditions at a power level of 43 MW \cite{Woolley2015HighCavity}. The T24 structure was run with the same pulse length and a breakdown rate of $3.1\times10^{-7}$ bpp at an accelerating gradient of 103 MV/m \cite{t24_paper}. This was scaled, using the relation $E_{acc}^{30} \propto$ BDR \cite{Grudiev2009}, to $10^{-6}$ bpp and 106.3 MV/m and therefore an input power of 42.4 MW.
If all the breakdowns in the T24 structure did indeed occur in the input matching cell after an initial conditioning period, as predicted by $E^*$ (see Sec. \ref{sec:t24_results}, the effective local breakdown rate would be much higher than the breakdown rate in the remainder of the structure, making it difficult to draw a limiting value for $E^*$ from this data. It was thus decided to consider only the regular cells of the T24 when comparing it to the Crab Cavity to pick a value for $k'$.

The load lines corresponding to the maximum $E^*$ in the Crab Cavity and the regular cells of the T24 cross the point ($E^* =$ 105 MV/m, $I^*l_{ant} = 2.47\times 10^{-5}$ Am), as shown in Fig. \ref{fig:t24_vs_crab_fitting}. The value of $k'$ was chosen such that the emission function crossed this point, thus ensuring agreement in their maximum $E^*$ values. It was decided that this should be considered equivalent to the mean of the $S_c$ values of the two structures at their reference power levels, i.e. $S_c = $ 4.52 MW/mm\textsuperscript{2}.

Results related to the peak $E^*$ and $S_c$ values in the two structures can be found in Table \ref{tab:t24_vs_psi_results}. Although there were significant differences in the design of the two structures, they were made of the same material and a breakdown quantity should have the same value in both structures when fully conditioned and operating at the same breakdown rate and pulse length. Comparing reference values, $S_c$ differed by 28.7\% between the two structures, implying a 13.4\% discrepancy in predicted maximum field values.

If the input matching cell of the T24 is disregarded, the $E^*$ values of the two structures at their reference power levels are identical by virtue of the choice of $k'$. If the matching cell is taken into account, the maximum $E^*$ value in the T24 rises to 123 MV/m. This is a 17.5\% discrepancy in $E^*$ and a 21.4\% discrepancy in predicted unloaded field. It should also be noted that the $k'$ and threshold $E^*$, and the corresponding $S_c$ value used here also exhibited good agreement for the C3 structures as shown in Fig. \ref{fig:c3_max_gradient_transient}.

\section{\label{sec:extensions}Additional Considerations}
\subsection{\label{sec:broadband_vs_single_freq}Comparison of Single-Frequency and Broadband Results}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.49\textwidth]{all_structures_rbd_transient_vs_rbd_trendlines_nopsi.png}
\caption{$L$ vs. single-frequency $R_{bd}$ for C3 structures of different aperture sizes, as indicated on the legend, scaled to an antenna length of  0.1 mm. Results for the T24PSI and the first cell of the CLIC Crab Cavity, scaled to an antenna length 0.1 mm, are also shown. Solid lines represent linear fits to the C3 structure data, whereas the blue dashed line represents a linear fit to the data from the Crab Cavity.}
\label{fig:all_structures_rbd_transient_vs_rbd}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.48\textwidth]{max_gradient_comparison.png}
\caption{Maximum accelerating gradient without beam loading vs. aperture size and group velocity as a fraction of $c$. Blue curve: with a maximum permitted $S_c$ of 4.5 MW/mm\textsuperscript{2}. Red curve: with a maximum permitted $P/C$ of 4.5 MW/mm. Yellow curve: with a maximum permitted $E^*$ of 105 MV/m.}
\label{fig:c3_max_gradient}
\end{figure}

To investigate the difference between the broadband and single-frequency methods, the values of $R_{bd}$($f = 12$ GHz) and $L$ for all of the antenna locations in all of the numerical simulations discussed in this paper were compared and are shown in Fig.~\ref{fig:all_structures_rbd_transient_vs_rbd}. For each of the aperture sizes of the C3 structure, as well as the Crab Cavity, an approximate dependence $L \propto R_{bd}$ can be seen, whose slope depends of the bandwidth of the structure, which is related to the group velocity for travelling-wave structures. The C3 structures with larger aperture sizes and thus larger bandwidths correspond to a steeper slope on the plot. Each of the data points for the T24 structure have different ratios of $L$ to $R_{bd}$, reflecting the tapered design of the structure.

Fig. \ref{fig:c3_max_gradient} shows the predicted maximum accelerating gradient for C3 structures of various aperture sizes, with $E^*$ calculated using the $R_{bd}$ value at 12 GHz instead of $L$, with a constant $k'$ value. The $E^*$ curve diverges from the $S_c$ curve significantly for low-group-velocity structures, demonstrating the issues with using the single-frequency method with narrowband structures. The agreement between the curves is much better for larger group velocities, indicating that the single-frequency method might be sufficient for such structures.

\subsection{\label{sec:freq_dependence}Scale Dependence}
A good limiting quantity should give consistent results for a wide range of frequencies without the need for re-fitting the free parameters separately for each frequency. $S_c$ has been found to be in general agreement with experimental results for 12 GHz and 30 GHz accelerating structures \cite{Grudiev2009}.

To investigate the dependence on frequency, the effect of scaling a given geometry by a factor $s$ and the RF frequency $f$ correspondingly by a factor $1/s$ will be considered. The field patterns in this geometry, normalised to accelerating gradient and scaling, i.e. $\mathbf{E}(\vec{x}/s)/E_{acc}$ and $\mathbf{H}(\vec{x}/s)/E_{acc}$, where $\vec{x}$ is the position inside the geometry, remain invariant as a consequence of the scale invariance of Maxwell's equations \cite{MicrowaveEngineering}. When normalised in a similar fashion, $S_c(\vec{x}/s) / E_{acc}^2$ is also scale-invariant. This means that the maximum gradient for a given maximum $S_c$ value is scale-invariant.

The same logic can be applied to fP/C: the normalised Poynting vector $\mathbf{S}(\vec{x}/s) / E_{acc}^2$ is scale-invariant. Thus, the total power $P$ flowing into the structure, given by integral of the real part of the axial vector component of the Poynting vector over the area of the iris, i.e. $\int_{\text{iris}}\Re(S_z)dA$, scales as $s^2$. Since the iris circumference $C$ scales as $s$, $fP/C$ is scale-invariant.

The dependence of $E^*$ on scaling can be determined from the equation for $E^*$:

\begin{equation}
\label{eqn:eloaded_solution_freqscaling}
    k l_{ant}^{1/2} L_0 E^{*3/2} + E^* - E_0 = 0.
\end{equation}

Since $k$ should only describe properties of the emission site, it should not depend on $s$. As discussed in Sec.~\ref{sec:antenna_lengths}, it is reasonable to assume $l_{ant}$ should be scaled proportionally to the geometry by factor $s$. As can be deduced from Eqs. \eqref{eqn:analytical_rbd} and \eqref{eqn:square_integral_rbd}, $L_0$ is proportional to the square root of bandwidth, which is in turn proportional to the operating frequency, i.e. $L_0 \propto s^{-1/2}$. This dependence cancels out that of $l_{ant} \propto s$, resulting in the coefficients of $E^*$ in Eq.~\eqref{eqn:eloaded_solution_freqscaling}, and therefore the $E^*$ value that satisfies the equation, being scale-invariant.

The fact that all three quantities are independent of scale suggests that $E^*$ should be valid for RF cavities of other frequencies than 12 GHz without modification, given that $E^*$ and $S_c$ have been shown to show similar results for various $TM_{010}$ geometries.

\subsection{\label{sec:exponent_dependence}Dependence on the Exponent of the Emission Function}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.235\textwidth]{c3_broadband_2_5mm_nsweep.png}
\includegraphics[width=0.235\textwidth]{c3_broadband_6mm_nsweep.png}
\caption{$E^*$ at an accelerating gradient of 100 MV/m vs. position along the surface in mm. The position is defined by the arrow shown in Fig.~\ref{fig:c3_dimensions}, where 0 is the iris apex, for different values of exponent $n$ of the electric field term in the emission function. Left: For an aperture radius of 2.5 mm. Right: For an aperture radius of 6 mm.}
\label{fig:n_sweep}
\end{figure}

Although a justification for the use of the Child-Langmuir law as the emission function was given in Sec. \ref{sec:emission_function}, it may not describe exactly the behaviour of real breakdowns. Reasons for this include the fact that the emitted electrons are not constrained to move in one dimension, as well as relativistic effects. The actual emission function may have an electric field term with a different exponent from the 3/2 as assumed in Eq. \eqref{eqn:child_langmuir_emission}, or have a different functional form altogether.

To verify that the $E^*$ method has some level of robustness to uncertainty in the emission function, calculations of $E^*$ were done for the C3 structures with various exponent values $n$ between 1 and 10 in the emission function. In each case a value of $k'$ was chosen such that the function crossed the point ($E^* =$ 105 MV/m, $I^*l_{ant} = 2.47\times 10^{-5}$ Am). No differences in the maximum predicted gradient vs. aperture radius, as shown in Fig. \ref{fig:c3_max_gradient_transient}, were found for the exponent values tried. Differences in the spatial distribution of $E^*$ on the surface of the cell were noted, however, as shown in Fig. \ref{fig:n_sweep}. For larger values of $n$, the spatial distribution of $E^*$ became flatter across the high-electric-field region, with a sharper drop off to zero outside it, since the larger the exponent value, the closer the emission function becomes to a sharp threshold in $E$. For exponent values of 5 and above, the maximum values of $E^*$ for the 6 mm aperture radius case no longer coincided with the maximum $E_0$, due to the effect of lower $L$ in those regions outweighing the smaller $E_0$ value.

The $E^*$ results for the C3 structures thus appear good for exponent values between 1 and 3, showing a degree of robustness to differences in the emission function.

\subsection{\label{sec:thin_gaps}Behaviour in Thin Parallel-Plate Geometries}

Setups featuring a thin gap between two parallel conducting surfaces are common in the study of vacuum breakdown, particularly in the case of DC experiments, as well as choke-mode RF structures \cite{Shintake_Choke}. A dependence of the maximum electric field on the gap size has been noted in experiments involving both RF \cite{CLIC_Choke_Design} and DC electric fields \cite{gap_dependence_histogram}.

To investigate the behaviour of breakdown impedance in such a setup, one can consider the case of an antenna spanning such a gap. As discussed in Sec. \ref{sec:antenna_lengths}, the electron trajectory, and therefore the antenna length, is assumed to be constrained by the thickness of the gap, i.e. $l_{ant} = d$, with $d$ being the gap thickness. Assuming that the thickness of the gap is much smaller than the free-space RF wavelength $\lambda_{fs}$, one can assume that there is no variation in the electric or magnetic fields with distance normal to the surfaces. This means that the field patterns are symmetric about the plane halfway between the two electrodes, and a conductive plane could therefore be inserted in place of this symmetry plane without perturbing the field patterns. This means that the perturbation in electric field $\Delta E$ caused by a current $\Delta I$ in the antenna is the same for a gap size of $d$ or $d/2$ for any $d \ll \lambda_{fs}$. One can therefore deduce that $\Delta E/\Delta I$ is independent of the gap size, and that $R_{bd} \propto d$. The quadratic scaling law of Eq. \eqref{eqn:r0_definition} does not apply here as the geometry changes with $d$ and thus $R_0$ cannot be assumed constant. This implies:

\begin{equation}
    R_0 = \frac{R_{0,n}}{d},
\end{equation}

with $R_{0,n}$ being the $R_0$ value for unit $d$. Substituting this into Eq. \eqref{eqn:eloaded_solution}:

\begin{equation}
\label{eqn:eloaded_solution_gap_scaling}
    E_0 = k R_{0,n} E^{*3/2} d^{-1/2} + E^*.
\end{equation}

It should be noted that with very small gap sizes of 1 mm of less, the emission function is very likely to be the Child-Langmuir law due to 1D electron trajectories being much more likely, and voltage being too low (assuming electric field levels on the order of 100 MV/m) to cause significant relativistic effects.

Eq. \eqref{eqn:eloaded_solution_gap_scaling} implies that for very large gaps, $\lim_{d \to \infty} E_0 = E^*$, whereas for small gaps, $E_0$ can be much higher than $E^*$. This supports the intuition that a very small gap can restrict the flow of power to a breakdown and allow higher operating electric fields. This equation can be related to a study of the gap size dependence performed in the DC Large Electrode System (LES) at CERN \cite{Iaroslava_MeVArc_2019}. In this experiment, it was found that for a constant breakdown rate, the gap voltage varied as $V_0 \propto d^{0.72}$. This can be compared with the loaded-field model by expressing Eq. \eqref{eqn:eloaded_solution_gap_scaling} in terms of gap voltage as:

\begin{equation}
\label{eqn:eloaded_solution_gap_scaling_voltage}
    V_0 = E_0 d = k R_{0.n} d^{1/2} + E^* d.
\end{equation}

The constant breakdown rate in the experiment implies a constant value of $E^*$. It is likely that with appropriate values of the coefficients $k$, $R_{0,n}$, and $E^*$, a linear combination of the $d$ and $d^{1/2}$ terms would be indistinguishable from a power law with exponent 0.72.

Other experiments involving parallel-plate and near-parallel-plate geometries \cite{gap_dependence_histogram} have shown exponents between 0.1 and 1.1, with 0.7 being the most common. Eq. \eqref{eqn:eloaded_solution_gap_scaling_voltage} can explain a spread of exponent values between 0.5 and 1 by differences in the relative values of $k R_{0,n} E^{*3/2}$ and $E^*$ in each setup.

\section{\label{sec:implications}Implications on Accelerating Structure Design}
The $E^*$ model has implications on the behaviour of vacuum breakdowns and the design of structures for optimum breakdown performance. A key feature of the $E^*$ model is that the breakdown rate depends only on the local electric field, and not the magnetic field, though there is generally a correlation between power coupling and magnetic field levels. $E_0$ should thus be minimised where possible to avoid breakdowns.

A $k'$ value of $2.30\times10^{-17}$ Am\textsuperscript{2}V\textsuperscript{-3/2} and an $E^*$ of 105 MV/m have been shown to produce results consistent with an $S_c$ value of 4.52 MW/mm\textsuperscript{2}. Lower vales of $S_c$ correspond to the same $k'$ with a lower $E^*$. The broadband method is recommended in general, though the single-frequency model could suffice for designs with a sufficiently large group velocity and not much variation in group velocity, offering the advantage of simpler calculations.

Exotic designs such as structures driven with multiple frequencies to reduce the peak surface field have been proposed \cite{MultiFrequencyStructure}. Based on the premise of the $E^*$ model, this could be effective in improving performance, though the model may need modification to handle multi-frequency excitation.

The $E^*$ model shows an explicit dependence of accelerating gradient on group velocity - a correlation which has been observed experimentally and noted in previous work \cite{Adlophsen_Normal_Conducting} \cite{SLAC_Gradient_Limit}.

The behaviour of the T24 structure in Sec.~\ref{sec:t24_results}, i.e.: $E^*$ and the breakdown rate being the largest in the cell closest to the input and thus receiving the most RF power, suggests that coupling power to each cell individually, such as in the distributed-coupling structure designed at SLAC \cite{TantawiDistributedCoupling}, may improve breakdown performance. This way, a breakdown in any particular cell would only have a small fraction of the total input RF power to the whole structure available to it, which could bring about an improvement in breakdown performance. Local stored energy would still needs to be taken into account in such a design.

\section{\label{sec:conclusions}Conclusions}
A new model to quantify breakdown limits in normal-conducting accelerating structures has been proposed. It describes the coupling of power to a breakdown, and takes into account the perturbation in local $E$ and $H$ fields by the breakdown whilst also removing the intricacies of calculating actual particle trajectories for every possible breakdown location.

$E^*$, the quantity that predicts the probability of a breakdown of a particular geometry at a given input power, is relatively simple to calculate and can be obtained either analytically from standard cavity parameters or numerically from finite-element simulations.

The model resolves issues with the breakdown locations predicted by $S_c$ whilst still predicting maximum gradient values that are consistent with it for geometries that $S_c$ has been well-tested for (i.e. accelerating structures operating in the $TM_{010}$ mode). Like $S_c$, the loaded-field model takes stored energy into account and is thus valid for structures with low group velocities. It also exhibits scale invariance similarly to $fP/C$ and $S_c$, meaning that it should be equally applicable to different frequency cavities without changes of parameters. Because the $E^*$ model does not require steady-state power flow, it is in principle also applicable to DC systems. Although standing-wave structures with zero group velocity have not been considered directly in this paper, obtaining an analogous analytical circuit model to those discussed in Sec. \ref{sec:analytical_rbd} for such structures is expected to be straightforward, while the numerical approach discussed in Sec. \ref{sec:numerical_rbd} can be applied without modification.

redpajama_set_name: RedPajamaArXiv


text:
Texans hire longtime Patriots executive Nick Caserio as new general manager: reports
January 6, 2021 By Paulina Dedaj
The Houston Texans reached an agreement with longtime New England Patriots executive Nick Caserio to serve as the team's new general manager on Tuesday, according to multiple reports.
Caserio, 45, will take over the position after Bill O'Brien was fired in October from both GM and head coaching positions following four straight losses.
FOR NON-PLAYOFF TEAMS, IMPORTANT OFFSEASON IS UNDER WAY
According to ESPN, Caserio was interviewed in Houston on Monday and Tuesday night flew back to New England, where he has served as the Patriots' director of player personnel since 2008.
The decision to go with Caserio should come as no surprise. The Texans previously approached him in 2019 following the firing of Brian Gaine but the move was blocked by the Patriots, who went as far as to threaten tampering charges against Houston because of their process, CBS Sports reported.
Caserio's departure will leave a significant void behind for New England. He first joined the team in 2001 as a personnel assistant before working his way up to director of player personnel in 2008.
The Patriots have three Super Bowls with Caserio and, according to ESPN, he is the highest-ranked staffer in the personnel department to work with coach Bill Belichick.
Caserio will likely be replaced by assistant director of player personnel Dave Ziegler, the NFL Network reported.
Filed Under: National Headlines

redpajama_set_name: RedPajamaCommonCrawl


text:
Jean-Michel Rabeux est un dramaturge et un metteur en scène de théâtre français.

Biographie

Auteur
 Les Charmilles et les morts, Éditions du Rouergue, 2002
 Le Ventre, Clownerie philosophique, Éditions Les Solitaires intempestifs, 2002
 Une magnifique désolation, 50 auteurs de théâtre s'expriment, Collectif, Éditions de l'Amandier, 2005
 Les Nudités des filles, Éditions du Rouergue, 2008
 La Barbe bleue, éditions L'avant-scène théâtre, 2010
 Peau d'Âne suivi de La Petite Soldate américaine, éditions L'avant-scène théâtre, 2014

Comédien
 1971 : Le Songe d'une nuit d'été de William Shakespeare, mise en scène Jean Gillibert, théâtre de Chateauvallon

Metteur en scène
 1976 : Iphigénie de Racine, théâtre Essaïon
 1978 : L'Imitation de Mathieu Benezet, Centre Georges Pompidou
 1979 : Ode pour hater la venue du printemps de Jean Ristat, théâtre Édouard VII
 1981 : Le Malade imaginaire de Molière, théâtre de la Cité Universitaire
 1981 : La Fausse Suivante de Marivaux, théâtre de la Cité Universitaire, Les Gémeaux, Théâtre des Arts de Cergy-Pontoise
 1982 : Vaudeville d'Eugène Labiche, Georges Courteline et Alfred Jarry, Théâtre des Arts de Cergy-Pontoise, Les Gémeaux
 1983 : Déshabillages (Comédie mortelle) de Jean-Michel Rabeux, Centre culturel de Cergy-Pontoise
 1984 : Onanisme avec troubles nerveux chez deux petites filles de Jean-Michel Rabeux d'après le Dr Démétrius Zambaco, Festival d'Avignon, Théâtre des Arts de Cergy-Pontoise
 1985 : La Double Mort d'un criminel ordinaire de Breyten Breytenbach, Théâtre des Arts de Cergy-Pontoise
 1986 : Phèdre de Racine, Maison de la Culture de Bourges, Maison des Arts de Créteil, théâtre national de Strasbourg, Théâtre des Arts de Cergy-Pontoise
 1987 : Ce qui est resté d'un Rembrandt déchiré en petits carrés bien réguliers, et foutu aux chiottes de Jean Genet, Théâtre de l'Atalante, Teatro Due/Festival de Parme, Théâtre des Arts de Cergy-Pontoise
 1987 : Éloge de la pornographie de Jean-Michel Rabeux, théâtre Gérard Philipe
 1988 : Onanisme avec troubles nerveux chez deux petites filles de Jean-Michel Rabeux d'après le Dr Démétrius Zambaco, théâtre national de Chaillot, théâtre national de Strasbourg, TNP Villeurbanne
 1988 : La Républicaine d'Hélène Delavault, Théâtre des Arts de Cergy-Pontoise, théâtre des Bouffes du Nord, Maison de la Culture de Bourges, Atelier Lyrique du Rhin, théâtre de la Bastille, tournée
 1989 : Le vide était presque parfait spectacle sans texte, mise en scène avec Marc Mérigot
 1990 : L'Amie de leur femmes de Luigi Pirandello, Maison de la Culture de Bourges, à l'E.D.A.C. de Poitiers, théâtre de l'Athénée, Théâtre des Arts de Cergy-Pontoise
 1992 : Légèrement sanglant de Jean-Michel Rabeux, Nouveau théâtre d'Angers
 1992 : Le Travail du plâtre spectacle sans texte de Jean-Michel Rabeux
 1994 : Les Charmilles de Jean-Michel Rabeux, La Rose des vents
 1995 : Scènes de naissances - actes IV de Jean-Michel Rabeux et Eugène Durif, La rose des vents
 1996 : Sade, français encore un effort de Sade, Beaubourg
 1997 : Tentatives de Pieta d'après L'Ennemi déclaré de Jean Genet, Festival Trafic au CRDC de Nantes
 1999 : Meurtres hors champ d'Eugène Durif, La Rose des vents, théâtre Ouvert, tournée
 1999 : Les Enfers Carnaval spectacle sans texte de Jean-Michel Rabeux

 2000 : Le Labyrinthe spectacle sans texte de Jean-Michel Rabeux et Sylvie Reteuna, La rose des vents
 2001 : Arlequin poli par l'amour de Marivaux, mise en scène Jean-Michel Rabeux et Sylvie Reteuna, Théâtre Garonne, La rose des vents, théâtre de Gennevilliers
 2001 : L'Homosexuel ou la Difficulté de s'exprimer de Copi, Studio de l'Ermitage, théâtre de la Bastille
 2003 : Déshabillages (Comédie mortelle) de Jean-Michel Rabeux, théâtre de la Bastille
 2004 : Feu l'amour avec trois pièces de Georges Feydeau : On purge bébé, Léonie est en avance, Hortense a dit « j'm'en fous », MC93 Bobigny
 2004 : Mais n'te promène donc pas toute nue ! de Georges Feydeau
 2004 : Le Balcon ou à peu près d'après Jean Genet, Atelier de  des élèves du Conservatoire national supérieur d'art dramatique
 2005 : Le Sang des Atrides d'après Eschyle, Théâtre Daniel Sorano, Théâtre Garonne, théâtre de la Bastille
 2006 : Emmène-moi au bout du monde… ! de Blaise Cendrars, théâtre de la Bastille, théâtre du Chaudron
 2007 : Le songe d'une nuit d'été de William Shakespeare, MC93 Bobigny
 2008 : Onanisme avec troubles nerveux chez deux petites filles de Jean-Michel Rabeux d'après le Dr Démétrius Zambaco, MC93 Bobigny
 2009 : Le Corps furieux de Jean-Michel Rabeux, MC93 Bobigny, théâtre de la Bastille
 2009 : Le Cauchemar de Jean-Michel Rabeux, théâtre de la Bastille

 2010 : La Barbe Bleue de Jean-Michel Rabeux d'après Charles Perrault
 2011 : Ce soir je n'y suis pour personne sauf pour Antonio Lobo Antunes (Livre des chroniques III) d'António Lobo Antunes, lecture mise en espace, MC93 Bobigny
 2011 : La Nuit des rois d'après William Shakespeare, La Rose des vents, MC93 Bobigny
 2012 : Les Quatre Jumelles de Copi
 2012 : Peau d'Âne de Jean-Michel Rabeux d'après Charles Perrault
 2013 : R.& J. Tragedy d'après Roméo et Juliette de William Shakespeare
 2013 : La Petite Soldate américaine de Jean-Michel Rabeux
 2013 : Les Fureurs d'Ostrowsky de Gilles Ostrowsky et Jean-Michel Rabeux
 2014 : Au bord de Claudine Galéa, MC93 Bobigny
 2015 : La Belle au bois dormant de Jean-Michel Rabeux d'après Charles Perrault
 2015 : Aglaé de Jean-Michel Rabeux d'après les mots d'Aglaé

Notes et références

Liens externes

Metteur en scène français
Dramaturge français du XXe siècle
Dramaturge français du XXIe siècle
Date de naissance non renseignée (XXe siècle)

redpajama_set_name: RedPajamaWikipedia


Damaged tooth? A dental crown can give your damaged tooth a second chance. Dental crowns offer many benefits, which makes choosing a dental practitioner who offers them a fabulous idea. Dr. James Bourne at Eagle Dental in Homer, AK, offers a full range of dental services. Read on to find out how dental crowns restore damaged teeth.
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If you need a dental crown, why wait? Start your journey today! Call Eagle Dental at 907-235-8574 today to schedule a dental consultation in Homer, AK. Dental crowns will give you your smile back and restore your self-confidence.


These forums haven't gotten as much use as they used to. Primarily because there's new competition for discussion with live chat sites like Discord. And of course there's always chat in the tearoom or elsewhere within MUD2 itself. But the forums DO have some advantages over chatting in MUD2 and on Discord. Here, it's possible to have lengthy, multi-paragraph statements on a variety of MUD2 and sort of related topics (gaming, fantasy, role-playing), inviting further in-depth discussion. We can have lengthy light-hearted humor threads, or more serious in-depth discussion on MUD2's game mechanics. We can discuss in-game events or MUD2 article sites such as Muddled Times in-depth as well. And we can do it without having to repeat ourselves for the benefit of those who weren't in the tearoom (or in MUD2), or on Discord when we had the initial discussions. People can browse and read through threads and get insights from months or even years ago. I've been going back and re-visiting a lot of the older threads from years ago recently. And it's surprising how many of them are still very relevant today. Some are so relevant that one could post in them and get the discussion started again almost seamlessly. The downside of forums is that they have to be constantly moderated and looked after, to keep spammers under control. As I've said elsewhere, forums are like gardens. They have to be tended to on a somewhat regular basis, or else the weeds will take over. In any case, forums generally have a greater longevity than live chat. Because unless someone logs the chats and posts them to forums, people who weren't there don't get to see the points being made. Okay, I'm rambling. If anyone else has any points to make on the value of the forums, feel free to comment.
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Pensacola/Panhandle/South Alabama guys interested in a Smoke Day?
It's been so long since I fired up the mini or 18.5 WSM I don't know if I'll remember how to do it. Lousy winds and rain for the last month and a half here and no end in site.
At least with the rains no fire restrictions yet, so come Hell or high water Barb and I will smoke something on smoke day.
I'll be smoking on the 24th.....having a big party for the holiday and for my birthday. Pretty excited.
I'm in. I'm not sure what I'll cook yet. We are planning to have a couple of friends over for some good food, a few too many drinks, and a fire in our new fire pit out back, weather permitting. Good times!
Trying to get back into the Smoke Day groove. We'll be camping at the Aspen group site in the Rock Creek campgound near Tom's Place, North of Bishop, CA. I'm bringing pork ribs and will go BRITU style. My camping buddie is talking about pork loins, so that may jump in the smoker also. We may experiment with working on some chicken. and will definitely smoke some ABT's. Have a nice holiday everyone.
Yes, of course. You can cook anytime leading up to Smoke Day or shortly afterwards and share your experience with us!
Not being able to take part in next weekend's Smoke day, (back to work today for 28 days), I would like to to enter my Smoked Beef Cheeks. (Just wanna be a part of this year's smoke day).
Thanks. And everyone have a great WSM Smoke Day 2015.
Eat a balanced diet.....a piece of pork belly in each hand.
We're watching the weather here. They're forecasting storms for the holiday weekend.
Brisket, Texas style. Plenty of salt, pepper and mesquite (and beer).
Last edited by Mark J; 05-23-2015 at 07:46 AM.
We have great weather forecast for tomorrow, and I picked up some baby back ribs a little while ago. The grocery store has a coupon vending machine at the entrance which I always ignore, but at my wife's suggestion I swiped my card and got a coupon for $3 off a $20 pork purchase - obviously an omen!


I'm feeling sad today, because my cat just died. Fritzi wasn't really 'my' cat, but she lived in my back yard for sixteen years. She appeared one April day with three kittens, wild as a hare. The neighbors adopted the kittens, but the mom was unapproachable, and produced another litter in short order. Don built a large cage with one-by-ones and chicken wire, and we lured her into it with cat food and slammed the door. With the help of a broom, I managed to get her into a cat carrier and took her to a vet who was willing to spay a feral cat. We kept her in the big cage until she was all better, then I opened the door to let her go, thinking she'd get the hell out of Dodge after what we did to her.
But she recognized a good gig when she saw one, and never left. As the years went by, she became much more approachable, rubbing up against our legs and loving a good back rub, but she would never let us pick her up. She seldom ventured from our back yard, savoring her little condo that we fixed up for her on the back porch. When she grew old, she never left the yard at all. In truth, she had a nice little life, and was one very lucky feral cat.
Last week, I could tell she wasn't feeling well. She was moving slowly, and in a day or two, grew uninterested in eating. When I tried to check her out, she wouldn't let me get near her. Yesterday (Saturday), I went by the local pet hospital and asked if they would treat my feral cat if I could trap her, which was going to be difficult, since she wasn't eating and I wasn't going to be able to lure her with food. As I was explaining the situation to the nice young lady behind the counter, I suddenly began bawling like a baby, great heaving sobs that surprised me even more than they did the poor nurse (is that what you call a vet assistant?) She was very calm and sympathetic and let me stand there and weep for a while, and we decided that if I could somehow get her into a carrier today, I could bring her in on Monday and the vet would look at her.
But this morning, she was already gone. I cried a little bit, but to tell the truth, I was relieved and rather glad that it was over. Don was a brick. It was he who dug a hole in the yard, picked her up and wrapped her in an old towel and put her in a box, then tripped over the garden hose and fell on his face, getting a nice bruise on his chest. I ended up burying the cat, but then he had done the hard work, and I was glad that I was able to do something for her.
So here's to Fritzi, who was an etherial little presence in my back yard for a lot longer than I would ever have imagined. I don't ever want an animal again. It's too hard when they join the choir eternal.
This entry was posted on Sunday, September 7th, 2008 at 7:51 pm	and is filed under Writer Rant. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
What a poignant post about your cat, Fritzi. Loved it. I read it with wet eyes, chuckling the while. Yeah, here's to Fritzi, from our feral, Henry, so like her. Almost unapproachable but still enjoying his soft life.
Thanks, Irene. I'm still feeling a bit blue about it and wondering why. I couldn't "love on her", as my grandma used to say, but I loved her, and I think she loved me in her own catly way.
My condolences to you on the loss of your cat. We enjoyed the decision of a stray tabby to stay with us when we built our house many years ago. She gradually went from feral to living downstairs with access daily to the outdoors.
She presented us with three litters before we caught her and had her spayed. My mother still has one of her cats and we have two left (both 14)that we love dearly.
How do cats know where they belong??
I enjoyed "The Old Buzzard Had It Coming" very much. Your characters have life and depth. Thank you.


Thank you for donating to Sandra Kennedy Ministries. Your support finances the Healing Ministry of SKM and allows us to broadcast our TV program worldwide. With your help we are able to reach more people around world as we share the good news of Jesus Christ.


One man only in this orchestra does not allow himself any such diversion. Wholly intent upon his task, all energy, indefatigable, his eye glued to his notes and his arm in perpetual motion, he would feel dishonored if he were to miss an eighth note or incur censure for his tone quality. By the end of each act he is flushed, perspiring, exhausted; he can hardly breathe, yet he does not dare take advantage of the respite offered by the cessation of musical hostilities to go for a glass of beer at the nearest bar. The fear of missing the first measures of the next act keeps him rooted at his post. Touched by so much zeal, the manager of the opera house once sent him six bottles of wine, "by way of encouragement." But the artist, "conscious of his responsibilities," was so far from grateful for the gift that he returned it with the proud words: "I have no need of encouragement." The reader will have guessed that I am speaking of the man who plays the bass drum.


They've had a meeting about renovating the entire structure of county cricket. They've resolved to repaint one of the bedrooms and maybe replace some of the sealant around the bath.
In 1890, it was suggested that county cricket be divided into first, second and third classes with eight teams in each.
We never did get first, second and third class cricket and over a hundred years later, these sorts of discussions still pan out much the same. So many people have their say that the status quo or some sort of bizarre half-baked compromise are the only likely outcomes.
First-class, T20 Blast-playing counties cannot countenance any kind of erosion of the standard of cricket they are seen to play. They would rather play a poor standard of first-class cricket than a better standard labelled second-class cricket. They would rather be one of 16 mediocre Twenty20 sides in a sprawling, diluted competition than a less visible part of a more concentrated event.
Just as at international level, there's no thought to blurring boundaries and giving the have-nots a route to progress on merit. The first-class counties will remain first-class counties and they will also remain the only first-class counties – forever.
The first-class counties are in the club and they aren't voting anyone else into that club or themselves out of it. Any changes made must therefore be within these parameters.
We long ago grew weary of mooted county cricket reforms. It's not that what eventuates ultimately fails to match what was proposed – that much is inevitable. It's that what eventuates is so far removed from what was proposed as to be all but unrelated.
County cricket is camel stew.
As a Leicestershire fan, I'm pretending to like football instead these days. And occasionally rugby.
I'm not sure what standard I'd call Leics' cricket. Bog, perhaps.
I understand why you might be following football more closely this year- Leicester Nirvana are having a great year in the ChromaSport & Trophies United Counties League.
I think the other Leicester team are doing quite well too.
English cricket has benefited, it's fair to say, from Durham's elevation to FC status. So the boundaries of "First Class" haven't quite been set in stone.
On the other hand, it would probably be logical to strip the remaining university matches of FC status but that doesn't seem likely to happen any time soon (tradition etc).
What galls me the most is, as noted on the last thread, that since 2nd division cricket isn't "proper cricket", they can apparently make do with a "champion" who hasn't played every other county home and away. That just seems wrong to me. It really can't be fair that some counties get two chances to play, oh I don't know, let's say Leics twice, while others do it just the once. What kind of "league" table is that? Pathetic.
I'm with Bailout on the revised Division 2 modus. It simply doesn't feel like a proper, fair competition, yet that all-important promotion will hinge on it. That's a muddle.
I'm not with KC on the matter of camel stew. I realise he is alluding to the idiom that a camel is a race-horse designed by committee. Even that idiom doesn't really work if you look at camel racing in Arabia.
…they should have gone to Specsavers.
(No doubt most self-appointed humorous writers about cricket will try that joke over the next few days – KC seems to have neglected to spot the opportunity).
My only solace is that, by reducing the size of Div 1, they haven't made the grand prize, the championship itself, so dependent on the whims of the schedule.
But it does seem bizarre that promotion – which in some ways is arguably more valuable a reward – will do so.
Perhaps it's a sign they want three uncommercial teams to drop dead.
Gary Keedy made it and we thought that was probably a decent sign we should pass.
Writing the camel stew bit, in the back of our mind we suspected someone would take issue. We're not going to say that we were 100 per cent sure it would be you, but it was certainly high-90s.
I'll put a message out to Lidgates to see if they can supply camel meat as they did with Iberico streaky bacon.
2014 was the year of the Lord's Thordkin. Could 2016 be the year of the Lord's Camel Stew?
It's not entirely out of the question.
A mate of ours scored some camel biltong once (it wasn't called biltong, but that's basically what it was).
Are you sick of 50 over cricket?