RF cavities and longitudinal phase stability

Define the integrated peak gap voltage before transit averaging. A particle takes finite time to cross the gap, so the electric field changes during passage. A transit-time factor accounts for that reduction for the chosen field profile and speed. Choose sine phase so the crest is at ninety degrees. If an effective cavity voltage already includes transit averaging, multiplying by the same factor again would double-count a loss.

A synchronous particle arrives at the reference phase and receives the scheduled energy gain per passage. In a ring the RF frequency is an integer harmonic of revolution frequency. In a linac, cell lengths and field phases instead maintain synchronism as speed changes. The simple gap formula assumes the trajectory and speed do not change enough inside the cavity to invalidate its chosen transit factor; large changes require integration along the actual orbit.

Let delta denote a small fractional momentum offset. Define momentum compaction as the fractional orbit-length change per delta, and define eta here by the fractional revolution-period change. Because increasing momentum also changes speed, eta is momentum compaction minus inverse relativistic gamma squared. Some accelerator references define slip using revolution frequency instead, which reverses eta. A stability formula copied without translating that definition has the wrong sign.

Positive arrival phase means later arrival at the same cavity. With turn number used as a continuous slow variable, a longer revolution period increases that phase. Linearize about a slowly varying reference orbit and neglect collective effects. The momentum kick follows from the difference between the particle’s RF gain and the synchronous gain, using the relativistic relation between energy and momentum changes. The two coefficients must use the same reference energy and phase definition.

For a small phase displacement, expand the sine around the synchronous phase. Combining the first-order equations produces a harmonic equation only if the negative product of the coefficients and voltage slope is positive. For a positive charge and positive peak voltage, below transition the chosen eta is negative and the rising sine slope is stable. Above transition the eta sign changes, so the stable slope changes as well.

At transition the leading slip term vanishes and this simple small-amplitude approximation loses its restoring frequency. A transition crossing needs a separate strategy and cannot be modeled by smoothly carrying an unchanged stable branch through zero. For larger amplitudes the sine is nonlinear and stable motion occupies a finite bucket bounded by a separatrix. RF noise, beam loading and collective forces further modify capture and lifetime; a stable linear frequency alone does not guarantee that every injected particle is captured.