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In particle accelerators, a common mechanism for accelerating a charged particle beam is via copper resonant cavities in which electric and magnetic fields form a standing wave, the mode of which is designed so that the E field points along the axis of the accelerator, producing forward acceleration of the particles when in the correct phase.

The maximum electric field ${\displaystyle E}$ achievable is limited by a process known as RF breakdown. The reliable limits for various RF frequencies ${\displaystyle f}$ were tested experimentally in the 1950s by W. D. Kilpatrick.

An approximate relation by least-square optimization of the data yields

${\displaystyle f=1.64\,\mathrm {MHz} \cdot \left({\frac {E}{E_{0}}}\right)^{2}\cdot \exp \left(-8.5{\frac {E_{0}}{E}}\right),\quad }$ with ${\displaystyle E_{0}=1\mathrm {\frac {MV}{m}} }$ (megavolts per metre).

This relation is known as the Kilpatrick Limit.

References

1. Kilpatrick, W. D. (1957). "Criterion for Vacuum Sparking Designed to Include Both rf and dc". Review of Scientific Instruments. 28 (10): 824–826. Bibcode:1957RScI...28..824K. doi:10.1063/1.1715731.
2. Wangler, Thomas (2008). RF Linear Accelerators (2nd ed.). Wiley-VCH. ISBN 978-3-527-62343-3.. This form apparently comes from a Los Alamos note:

   T. J. Boyd, Jr., Kilpatrick's criterion, Los Alamos Group AT-1 report AT-1:82-28, February 12, 1982.

• Accelerator physics