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Introduction to RF Cavities for Accelerators Dr G Burt Lancaster University Engineering TM010 Accelerating mode Electric Fields Almost every RF cavity operates using the TM010 accelerating mode. This mode has a longitudinal electric field in the centre of the cavity which accelerates the electrons. The magnetic field loops around this and caused ohmic heating. Magnetic Fields Accelerating voltage • An electron travelling close to the speed of light traverses through a cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude, L/ 2 E eV e Ez z, t ei z / c dz L/ 2 • To receive the maximum kick the particle should traverse the cavity in a half RF period. c L 2f Transit Time Factor • An electron travelling close to the speed of light traverses through a cavity. During its transit it sees a time varying electric field. V • 1 maximum possible energy gain during transit e To receive the maximum kick the particle should traverse the cavity in a half RF period. L c 2f • We can define an accelerating voltage for the cavity by L / 2 V Ez z, t ei z / c dz Ez 0 LT cos t L / 2 • This is given by the line integral of Ez as seen by the electron. Where T is known as the transit time factor and Ez0 is the peak axial electric field. Peak Surface Fields • The accelerating gradient is the average gradient seen by an electron bunch, Eacc V d • The limit to the energy in the cavity is often given by the peak surface electric and magnetic fields. Thus, it is useful to introduce the ratio between the peak surface electric field and the accelerating gradient, and the ratio between the peak surface magnetic field and the accelerating gradient. Eacc E max Eacc Bmax Electric Field Magnitude Power Dissipation • The power lost in the cavity walls due to ohmic heating is given by, Pc 1 2 Rsurface H dS 2 Rsurface is the surface resistance • This is important as all power lost in the cavity must be replaced by an rf source. • A significant amount of power is dissipated in cavity walls and hence the cavities are heated, this must be water cooled in warm cavities and cooled by liquid helium in superconducting cavities. Cavity Quality Factor • An important definition is the cavity Q factor, given by U Q0 Pc Where U is the stored energy given by, 1 2 U 0 H dV 2 The Q factor is 2p times the number of rf cycles it takes to dissipate the energy stored in the cavity. t U U 0 exp Q0 • The Q factor determines the maximum energy the cavity can fill to with a given input power. Geometry Constant • It is also useful to use the geometry constant G RsurfaceQ0 • This allows different cavities to be compared independent of size (frequency) or material, as it depends only on the cavity shape. • The Q factor is frequency dependant as Rs is frequency dependant. Shunt Impedance • Another useful definition is the shunt impedance, 2 1V R 2 Pc • This quantity is useful for equivalent circuits as it relates the voltage in the circuit (cavity) to the power dissipated in the resistor (cavity walls). • Shunt Impedance is also important as it is related to the power induced in the mode by the beam (important for unwanted cavity modes) Geometric shunt impedance, R/Q • If we divide the shunt impedance by the Q factor we obtain, 2 V R Q 2U • This is very useful as it relates the accelerating voltage to the stored energy. • Also like the geometry constant this parameter is independent of frequency and cavity material. Higher Order Modes • There are a number of modes other than the TM010 mode. They have the same notation as waveguide modes with the addition indice, p, notating the longitudinatl variation. • TE/Mmnp • Modes are often classified by their m, indice. In circular cavities m is the azimuthal variation. • Monopole modes, m=0, accelerate and decelerate the beam • Dipole modes, m=1, kick the beam transverely • Quadropole modes, m=2 can also kick the beam but are weak near the axis. Higher Order Modes • Monopole modes, include the accelerating mode and have m=0 (no azimuthal variation) • There are TM and TE monopole modes, TM monopoles decelerate the beam and are a problem. • TE monopole mode are low loss and are useful for energy storage, they have little interaction with beams. TM011 Beam Dipole Modes E E Beam TM110 Dipole Mode TE111 Dipole Mode H H Beam Multi-cell structures In a multi-cell structure the coupling between the cells causes each mode to split into a number of modes equal to the number of cells. The Pendulum f0 1 2p LC The high resistance of the normal conducting cavity walls is the largest source of power loss P.E or E P.E or E K.E or B Resistance of the medium (air << Oil) Capacitor The electric field of the TM010 mode is contained between two metal plates E-Field – This is identical to a capacitor. This means the end plates accumulate charge and a current will flow around the edges Surface Current Inductor B-Field Surface Current – The surface current travels round the outside of the cavity giving rise to a magnetic field and the cavity has some inductance. Resistor Surface Current This can be accounted for by placing a resistor in the circuit. In this model we assume the voltage across the resistor is the cavity voltage. Hence R takes the value of the cavity shunt impedance (not Rsurface). Finally, if the cavity has a finite conductivity, the surface current will flow in the skin depth causing ohmic heating and hence power loss. Equivalent circuits To increase the frequency the inductance and capacitance has to be increased. 1 LC 2 Vc Pc 2R CVc U 2 2 The stored energy is just the stored energy in the capacitor. The voltage given by the equivalent circuit does not contain the transit time factor, T. So remember Vc=V0 T Equivalent circuits These simple circuit equations can now be used to calculate the cavity parameters such as Q and R/Q. U C Q0 R Pc L R V2 1 L Q0 2U C C In fact equivalent circuits have been proven to accurately model couplers, cavity coupling, microphonics, beam loading and field amplitudes in multicell cavities. Cavity Coupling Probe coupling to E-field Capacitive coupling Higher penetration higher coupling Loop couples to the B-field Inductive coupling Higher penetration lower coupling Couplers The couplers can also be represented in equivalent circuits. The RF source is represented by a ideal current source in parallel to an impedance and the coupler is represented as an n:1 turn transformer. External Q factor Ohmic losses are not the only loss mechanism in cavities. We also have to consider the loss from the couplers. We define this external Q as, P Q U Qe Pe e Pc 0 Qe Where Pe is the power lost through the coupler when the RF sources are turned off. We can then define a loaded Q factor, QL, which is the ‘real’ Q of the cavity 1 1 1 QL Qe Q0 U QL Ptot Beam Loading • In addition to ohmic and external losses we must also consider the power extracted from the cavity by the beam. • The beam draws a power Pb=Vc Ibeam from the cavity. • Ibeam=q f, where q is the bunch charge and f is the repetition rate • This additional loss can be lumped in with the ohmic heating as an external circuit cannot differentiate between different passive losses. • This means that the cavity requires different powers without beam or with lower/higher beam currents.