Basics of SRF Cavities 1.1: RF Cavities

Superconducting RF Cavities: A Primer

1.1 RF Cavities

An RF cavity is the device through which power is coupled into the particle beam of an accelerator. In electron-positron colliders, RF accelerating cavities are microwave resonators which generally derive from a "pillbox" shape (right circular cylinder), with connecting tubes to allow particle beams to pass through for acceleration. Figure 1 shows a typical cylindrically symmetric cavity. The fundamental, or lowest RF frequency, mode (TM 010) of the cavity has fields as shown. The electric field is roughly parallel to the beam axis, and decays to zero radially upon approach to the cavity walls. Boundary conditions demand that the electric surface be normal to the metal surface. The peak surface electric field is located near the iris, or region where the beam tube joins the cavity. The magnetic field is azimuthal, with the highest magnetic field located near the cavity equator. The magnetic field is zero on the cavity axis.

Typical_Cavity_With_Field_Patterns

Figure 1. A typical accelerating cavity geometry, showing particle beam and fundamental fields of the RF cavity.

The particle beam traverses the cavity as shown, experiencing an accelerating force along the axis of the cavity due to the electic field. Since the RF fields alternate in time, the particle beam must, of course, be in the proper phase with respect to the fields in order that the force be accelerating rather than decelerating. In addition, since the particles take a finite time to cross the cavity, the accelerating field is the time average of the electric field along the particles flight. The average gradient is defined in equation 1:

integral_of_E_over_one_period___(1)

TRF is the RF period, and E(z,t) is the electric field at the time and position of the particle.

The Q0 of an accelerating cavity is defined as the RF angular frequency (w) times the ratio of the stored energy in the electromagnetic fields (U) to the dissipated power (Pdiss), as shown in equation 2.

Q0=wU/Pdiss___(2)

The relationships between the stored energy and the magnitudes of the electric fields are obtained by numerical solution of Maxwell's Equations for the cavity geometry. Several computer program packages are available to solve the equations for typical cavity geometries, e.g. SUPERFISH[4], URMEL[5] and URMEL-T[6], or MAFIA[7]. Of particular interest in these solutions are the ratios of peak surface electric and magnetic fields to the square root of the stored energy, and the ratio of peak surface electric electric field to the average accelerating field in the cavity, given by:

ratios_of_E_and_H_to_sqrt(U)_and_Ep/Eacc___(3)

In the situation where all cavity losses are due to surface currents, the Q0 can alternately be defined as the ratio of the geometry factor G to the microwave surface resistance Rs, as shown in equation 4.

Q0=G/Rs___(4)

The geometry factor is defined in equation 5. Geometry factors have units of resistance, and generally have values between 200 and 300 Ohms.

G=volume_integral_of_E^2/surface_integral_of_H^2___(5)

The microwave surface resistance in a normal conductor is given by equation 6, and is approximately 15 milliOhms for copper at 3 GHz.

Rs=1/(conductivity*skin_depth)___(6)

where delta is the RF skin depth in a normal conductor.

The surface resistance causes power dissipation by the surface currents which arise in order to support the magnetic fields at the RF surface. Wall losses are the primary reason for investigating superconducting cavities, as the RF surface resistance is five to six orders of magnitude lower than that of a normal conduncting surface. Superconducting surface resistance will be discussed in the next section.

SRF@w4.lns.cornell.edu, Updated 18 September 1996