Introduction to RF for Accelerators

1. Introduction to RF for Accelerators Dr G Burt Lancaster University Engineering

2. Electrostatic Acceleration + - - - - - - - + + + + + +

3. Van-de Graaff - 1930s A standard electrostatic accelerator is a Van de Graaf These devices are limited to about 30 MV by the voltage hold off across ceramic insulators used to generate the high voltages (dielectric breakdown).

4. - + - + - + - + RF Acceleration By switching the charge on the plates in phase with the particle motion we can cause the particles to always see an acceleration You only need to hold off the voltage between two plates not the full accelerating voltage of the accelerator.

5. RF acceleration • Alternating gradients allow higher energies as moving the charge in the walls allows continuous acceleration of bunched beams. • We cannot use smooth wall waveguide to contain rf in order to accelerate a beam as the phase velocity is faster than the speed of light, hence we cannot keep a bunch in phase with the wave.

6. Early Linear Accelerators • Proposed by Ising (1925) • First built by Wideröe (1928) Replace static fields by time-varying periodic fields by only exposing the bunch to the wave at certain selected points.

7. Cavity Linacs • These devices store large amounts of energy at a specific frequency allowing low power sources to reach high fields.

8. Cavity Quality Factor • An important definition is the cavity Q factor, given by Where U is the stored energy given by, The Q factor is 2p times the number of rf cycles it takes to dissipate the energy stored in the cavity. • The Q factor determines the maximum energy the cavity can fill to with a given input power.

9. Cavities • If we place metal walls at each end of the waveguide we create a cavity. • The waves are reflected at both walls creating a standing wave. • If we superimpose a number of plane waves by reflection inside a cavities surface we can get cancellation of E|| and BT at the cavity walls. • The boundary conditions must also be met on these walls. These are met at discrete frequencies only when there is an integer number of half wavelengths in all directions. a L The resonant frequency of a rectangular cavity can be given by (w/c)2=(mp/a)2+ (np/b)2+ (pp/L)2 Where a, b and L are the width, height and length of the cavity and m, n and p are integers

10. Pillbox Cavities Wave equation in cylindrical co-ordinates • Transverse Electric (TE) modes • Transverse Magnetic (TM) modes Solution to the wave equation

11. 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

12. TM010 Dipole Mode H E Beam

13. A standing wave cavity

14. Accelerating Voltage Ez, at t=0 Normally voltage is the potential difference between two points but an electron can never “see” this voltage as it has a finite velocity (ie the field varies in the time it takes the electron to cross the cavity Ez, at t=z/v Position, z The voltage now depends on what phase the electron enters the cavity at. If we calculate the voltage at two phases 90 degrees apart we get real and imaginary components Position, z

15. 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, • To receive the maximum kick the particle should traverse the cavity in a half RF period.

16. 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. • To receive the maximum kick the particle should traverse the cavity in a half RF period. • We can define an accelerating voltage for the cavity by • 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.

17. For TM010 mode Ez, at t=z/v Position, z Hence voltage is maximised when L=c/2f This is often approximated as Where L=c/2f, T=2/p

18. Peak Surface Fields • The accelerating gradient is the average gradient seen by an electron bunch, • 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. For a pillbox Electric Field Magnitude

19. Surface Resistance As we have seen when a time varying magnetic field impinges on a conducting surface current flows in the conductor to shield the fields inside the conductor. Current Density, J. However if the conductivity is finite the fields will not be completely shielded at the surface due to ohms law (J=sE where s is the conductivity) and the field will penetrate into the surface. x Skin depth is the distance in the surface that the current has reduced to 1/e of the value at the surface, denoted by . This causes currents to flow and hence power is absorbed in the surface which is converted to heat. The surface resistance is defined as

20. Power Dissipation • The power lost in the cavity walls due to ohmic heating is given by, 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.

21. Shunt Impedance • Another useful definition is the shunt impedance, • 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)

22. TM010 Shunt Impedance

23. Geometric shunt impedance, R/Q • If we divide the shunt impedance by the Q factor we obtain, • 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.

24. TM010 R/Q

25. Geometry Constant • It is also useful to use the geometry constant • 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.

26. Q factor Pillbox

27. P.E or E P.E or E K.E or B The Pendulum The high resistance of the normal conducting cavity walls is the largest source of power loss Resistance of the medium (air << Oil)

28. Capacitor E-Field The electric field of the TM010 mode is contained between two metal plates – This is identical to a capacitor. This means the end plates accumulate charge and a current will flow around the edges Surface Current

29. 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. –

30. Resistor Finally, if the cavity has a finite conductivity, the surface current will flow in the skin depth causing ohmic heating and hence power loss. 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).

31. Equivalent circuits To increase the frequency the inductance and capacitance has to be increased. 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

32. Equivalent circuits These simple circuit equations can now be used to calculate the cavity parameters such as Q and R/Q. In fact equivalent circuits have been proven to accurately model couplers, cavity coupling, microphonics, beam loading and field amplitudes in multicell cavities.

33. Beam Loading • In addition to ohmic 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.

34. Average Heating • In normal conducting cavities, the RF deposits large amounts of power as heat in the cavity walls. • This heat is removed by flushing cooling water through special copper cooling channels in the cavity. The faster the water flows (and the cooler), the more heat is removed. • For CW cavities, the cavity temperature reaches steady state when the water cooling removes as much power as is deposited in the RF structure. • This usually is required to be calculated in a Finite Element code to determine temperature rises. • Temperature rises can cause surface deformation, surface cracking, outgassing or even melting. • By pulsing the RF we can reach much higher gradients as the average power flow is much less than the peak power flow.

35. Pulsed Heating Pulsed RF however has problems due to heat diffusion effects. Over short timescales (<10ms) the heat doesn’t diffuse far enough into the material to reach the water cooling. This means that all the heat is deposited in a small volume with no cooling. Cyclic heating can lead to surface damage.

36. Field Enhancement • The surface of an accelerating structure will have a number of imperfections at the surface caused by grain boundaries, scratches, bumps etc. • As the surface is an equipotential the electric fields at these small imperfections can be greatly enhanced. • In some cases the field can be increase by a factor of several hundred. Elocal=b E0 2b h

37. Field Emission • As we saw in Lecture 3, high electric fields can lead to electrons quantum tunnelling out of the structure creating a field emitted current. Once emitted this field emitted current can interact with the cavity fields. Although initially low energy, the electrons can potentially be accelerated to close to the speed of light with the main electron beam, if the fields are high enough. This is known as dark current trapping.

38. Breakdown • Breakdown occurs when a plasma discharge is generated in the cavity. • This is almost always associated with some of the cavity walls being heated until it vaporises and the gas is then ionised by field emission. The exact mechanisms are still not well understood. • When this occurs all the incoming RF is reflected back up the coupler. • This is the major limitation to gradient in most pulsed RF cavities and can permanently damage the structure.

39. Kilpatrick Limits • A rough empirical formula for the peak surface electric field is • It is not clear why the field strength decreases with frequency. • It is also noted that breakdown is mitigated slightly by going to lower group velocity structures. • The maximum field strength also varies with pulse length as t-0.25 (only true for a limited number of pulse lengths) • As a SCRF cavity would quench long before breakdown, we only see breakdown in normal conducting structures.

40. Typical RF System feedback Low Level RF Transmission System RF Amplifier Cavity DC Power Supply or Modulator • A typical RF system contains • A LLRF system for amplitude and phase control • An RF amplifier to boost the LLRF signal • Power supply to provide electrical power to the Amplifier • A transmission system to take power from the Amplifier to the cavity • A cavity to transfer the RF power to the beam • Feedback from the cavity to the LLRF system to correct errors.

41. Transformer Principle • An accelerator is really a large vacuum transformer. It converts a high current, low voltage signal into a low current, high voltage signal. • The RF amplifier converts the energy in the high current beam to RF • The RF cavity converts the RF energy to beam energy. • The CLIC concept is really a three-beam accelerator rather than a two-beam. RF Cavity RF Power RF Input RF Output Collector Electron gun

42. Vacuum Tube Principle RF Vacuum Tubes usually have a similar form. They all operate using high current (A - MA) low voltage (50kV-500kV) electron beams. They rely on the RF input to bunch the beam. As the beam has much more power than the RF it can induce a much higher power at an output stage. These devices act very much like a transistor when small ac voltages can control a much higher dc voltage, converting it to ac. RF Input RF Output Collector Electron gun Bunched Beam DC beam

43. Basic Amplifier Equations • Input power has two components, the RF input power which is to be amplified and the DC input power to the beam. • Gain=RF Output Power / RF Input Power = Prf / Pin • RF Efficiency= RF Output Power / DC Input Power = Prf / Pdc • If the efficiency is low we need large DC power supplies and have a high electricity bill. • If the gain is low we need a high input power and may require a pre-amplifier.

44. Generation of RF Power A bunch of electrons approaches a resonant cavity and forces the electrons within the metal to flow away from the bunch. B The lower energy electrons then pass through the cavity and force the electrons within the metal to flow back to the opposite side A At a disturbance in the beampipe such as a cavity or iris the negative potential difference causes the electrons to slow down and the energy is absorbed into the cavity C

45. IOT Schematics Grid voltage Density Modulation Time Electron bunches

46. IOT- Thales • 80kW • 34kV 2.2Amp • 160mm dia, 800mm long, 23Kg weight • 72.6% efficiency • 25dB gain • 160W RF drive • 35,000 Hrs Lifetime 4 IOT’s Combined in a combining cavity • RF Output Power 300kW

47. Klystron Schematics Interaction energy Electron energy Electron density

48. Klystron • RF Output Power 300kW • DC, -51kV, 8.48 Amp • 2 Meters tall • 60% efficiency • 30W RF drive • 40dB Gain • 35,000 Hrs Lifetime

49. 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.

50. 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, 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