# Design Criteria for High-Gradient Radio-Frequency Linacs
## 2 Overview of High-Gradient Studies for Particle Accelerators  
Experimental work on high-gradient acceleration was a major part of early electron-positron linear collider development; see, for instance, the work of [PERSON] et al. [16] and of [PERSON] et al. [17]. The work of [PERSON] et al. [17] considered that the rf breakdowns were directly linked to the peak rf electric field, while neither the peak magnetic field nor rf power were considered relevant. Therefore, they were studying the peak limit value of the electric field that generates breakdowns as a proxy for the accelerating gradient. This work was carried out at different frequencies: S-band, C-band, and X-band. They found that the maximum peak surface electric field achievable without rf breakdowns grows with the square root of the frequency. This analysis leads to the idea that the maximum accelerating gradient, limited by breakdown, increases with frequency. Later, other research carried out at CERN [18; 19; 20] in the frequency range from 21 to 39 GHz concluded that there is no increase in the maximum achievable gradient at higher frequencies. Both studies considered rf breakdown as a phenomenon generated when the peak electric field exceeds a certain threshold. Recent tests of single accelerating cavities at the S-band and C-band [14] do not show obvious frequency dependence of the achievable gradients. Major work to understand and mitigate the effects of rf breakdown were conducted during the development of the normal conducting 11.424 GHz linear collider NLC/GLC [2; 3; 21], which has been followed by the 12 GHz CERN-based linear collider CLIC [22]. NLC/JLC work looked at both traveling wave (TW) and standing wave (SW) accelerating structures, while CLIC considered only TW structures. During the NLC/JLC work, the statistical nature of rf breakdown became apparent [2; 3; 21; 23], which fundamentally changed our approach to studying rf breakdown phenomena and allowed us to quantify it: for most accelerating structures exposed to the same rf power and pulse shape, the number of rf breakdowns per pulse is nearly steady or slowly decreasing over \(10^{5}\)-\(10^{7}\) pulses. The breakdown probability became one of the main quantitative requirements characterizing the high-gradient performance of rf linacs. For example, the CLIC linear collider requires the rf breakdown probability to be less than \(4\times 10^{-7}\) per pulse per meter for a loaded accelerating gradient of 100 MV/m in order to maintain the electron-positron collision's luminosity. As technology progressed, sophisticated manufacturing and surface preparation techniques and systematic rf processing methods were developed [24; 25; 26; 27]. As a result of this R&D, practical 11.4 GHz TW accelerating structures, which are CLIC prototypes, run at breakdown rates of about \(10^{-6}\) per pulse per meter at unloaded gradients up to 120 MV/m and a \(\sim\)200 ns pulse length [28]. TW structures that include wakefield damping work at about 100 MV/m for similar breakdown rates [28; 29]. Continuing studies of new geometries, materials, and manufacturing methods at 11.424 GHz show the potential to reach 150 MV/m accelerating gradients with similarly low breakdown rates [15; 30; 31; 32; 33]. Advances in understanding limitations on accelerating gradients go beyond linear colliders. Accelerators are used in applications such as inverse Compton scattering gamma ray sources [5; 6], compact free-electron lasers (FELs) [34; 35; 36], and compact medical linacs for hadron therapy [37; 38; 39; 40; 41; 42]. Modern high-gradient devices, such as photo-rf guns [43; 44; 45], fourth harmonic linearizers for FELs [46], rf deflectors [9; 10; 47; 48; 49], and rf undulators [7; 8], all use these technologies and methods, developed and sustained by studies of high-gradient accelerators. Due to large investment into the development of X-band (11-12 GHz) linear colliders, the X-band accelerating cavities are the most investigated in terms of their high-gradient performance [23; 31; 50; 51; 52].