# Moden mehrzelliger Resonatoren

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$$f_q = f_{\pi \over 2} \sqrt{ \frac{1}{1-2k \cos{\left( \pi \frac{q}{N-1} \right)}}}$$

$$q=0, ... , N-1$$

## lecture_maurizio_II_mod3.pdf

STANDING WAVE MODES, defined by a phase πq/N corresponding to the phase shift between an oscillator and the next one → adjacent cells (gap) have a fixed phase difference πq/N=Φ .

## Proton_5.pdf

Here, for a 9-cell cavity, the phase shift per cell ranges from 0 to pi in steps of pi/8.

Some Modes for the ANL 7-Cell Cavity

This cavity has 7 cells, so it will have 7 modes, 0, pi/6, 2pi/6, 3pi/6, 4pi/6, 5pi/6, . Each mode has a characteristic frequency.

Each cavity mode will resonate at a different frequency. A plot of the frequency for the N-cavity group is plotted as a function of the phase shift per cell.

For this cavity, the coupling between the cells is primarily through the electric field vector on axis. This results in a higher frequency at a higher phase shift.

For a string of identical cells (no end beam pipes) the dispersion curve takes the form

f(phi)= f(pi/2) SQRT{1−k cos(phi/N)}

Where phi takes on N values from 0 to pi per cell.