Reentrant cavitiy

Resonant cavities used for particle acceleration have many features in common with parallel RLC circuits.

Figura: Reentrant cavity with on axis power input.

With figure we illustrate one of the simpliest resonant cavity, the reentrant cavity.This cavity is used in systems with space constraints, such as klystrons. It oscillates at relatively low frequency for its size.

The reentrant cavity can be divided into predominantly capacitive and predominantly inductive regions. In the central region, there is a narrow gap. The capacitance is large and the inductance is small. A harmonic voltage generator connected at the center of the cavity induces displacement current. The enlarged outer region acts as a single-turn inductor. Real current flows around the wall to complete the circuit. If the walls are not superconducting, the inductor has a series resistance. Assume that there is a load, such as a beam, on the axis of the cavity. If the generator frequency is low, most of the input current flows around the metal wall (leakage current). The cavity is almost a short circuit. At high frequency, most of the current flows across the capacitor as displacement current. At the resonance frequency of the cavity, the cavity impedance is infinite and all the generator energy is directed into the load. In this case, the cavity can be useful for particle acceleration. When the cavity walls have resistivity, the cavity acts as a high impedance in parallel to the beam load. The generator must supply energy for cavity losses as well as energy to accelerate the beam.

The resonant frequency of the reentrant cavity can be estimated easily. The capacitance of the central region is $C=\epsilon_0 \pi R_0^2/d$ and the induction is $L \approx \mu_0\pi a^2/ (2\pi (R_0+a))$ . The resonant angular frequency is

$$\omega_0 \approx C \sqrt{\frac{2(R_0+a)d}{R_0^2a^2\pi}}$$

In figure we have an example of commercial multistage reentrant cavities used in klystron for experiment in nuclear physics.

Figura: Commercial reentrant cavities used for charged particle preacceleration.