# Design Criteria for High-Gradient Radio-Frequency Linacs
## 3 RF Breakdown
### Geometric Dependence  
Here, I discuss the results of the cavity geometry studies conducted using short SW accelerating structures. I started with experiments with cavities of a disk-loaded-waveguide type, which are typical for linear accelerators (an example of such an X-band structure with a beam iris radius of 3.75 mm is shown in Figure 3a). I have built and tested cavities with several different beam aperture sizes. The results of these tests are summarized in [31] and shown in Figure 4a,b. The breakdown probability for structures with different iris sizes was correlated with peak rf magnetic fields and varied with the rf pulse length as if dependent on peak pulse surface heating (see Figure 4b)) rather than the peak surface electric field or accelerating gradient (see Figure 4a)).The magnitude of peak pulse heating is approximately proportional to the square root of the rf pulse length. These findings prompted a dedicated study of rf pulse heating [54] and experiments with hard materials [74]. Note that originally high rf magnetic fields and moderate electric fields were found to be the sources of rf breakdowns in the couplers of NLC structures [53]. Meanwhile, [PERSON] et al. made a successful attempt to find a local field quantity that can describe the breakdown behavior observed in many experiments [56]. They found that the peak of a modified Poynting vector correlated to the measured (or scaled) breakdown rate in seventeen TW and four SW cavities, including the cavities that are described above. To understand how the Poynting vector is correlated with the breakdown rate in my experiments, I scaled the peak pulse surface heating with the ratio \(Z^{\prime}=Max(|E\times H^{*}|)/Max(|H^{2}|)\). The results of this scaling are shown in Figure 4c. Because of the similar geometry of the cavities, this ratio is similar in all three structures. Again, because of this similarity, the breakdown rate in these particular structures is correlated with both the peak pulsed heating and the peak Poynting vector. To unwrap this correlation, SW cavities were tested that have the peak value of the Poynting vector increased and decreased, with respect to a \"standard\" cavity, as shown in Figure 3a. The structure with a nose-cone cell shape shown in Figure 3b has been designed with minimized peak surface magnetic fields [75]. However the ratio \(Z^{\prime}\) is almost doubled from 390 Ohm to 632 Ohm. Both of these cavities are fed with rf power on-axis. A third  
Figure 2: Breakdown rate reproducibility for three copper SW structures of the same geometry [71]: (**a**) breakdown rate vs. accelerating gradient and (**b**) vs. peak pulse surface heating. To simulate the beam loading on these structures, rf pulses are used, the shape of which creates a flat top with a duration of 150 ns. structure shown in Figure 3c is fed off-axis [76]. Peak pulse surface heating is increased on the coupling irises, thus decreasing \(Z^{\prime}\) to 301 Ohm. The results of high-power tests for these three cavities are shown in Figure 5. The breakdown rate is more dependent on the Poynting vector than on the accelerating gradient or peak pulse surface heating. Plus, the location of the RF breakdown damage observed during the autopsy of the side-coupled cavity correlates with the region of the highest values of the surface Poynting vector. Because the correlation between the magnitude of the peak Poynting vector and the rf breakdown rate has been derived from numerous experiments, it has remarkable predictive power. However, there are many exceptions that confirm that its physics has not been understood. The SW cavities with a narrow-gap choke [77], three-cell side-coupled structure [76], photonic band gap that has an enhanced magnetic field [78], X-band deflector [49], and S-band high-gradient proton TW accelerator [38] all had higher breakdown rates than would be predicted using the Poynting vector, and the breakdown damage was not located in the region of the maximum of the Poynting vector.