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. Circular Acceleration Synchrotron Radiation from an electron in a magnetic field: Energy loss per turn of a machine with an average bending radius  : Energy loss must be replaced by RF system cost scaling $  E cm 2

9.  V ss VpVp Phase stability is given by off-crest acceleration Stable region Overvoltage To provide a stable bunch you often will accelerate off crest. This means the particles do not experience the maximum beam energy. V b =V c cos(  s ) = V c q Where V c is the cavity voltage and V b is the voltage experienced by the particle,  is the phase shift and q is known as the overvoltage.

10. 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 2  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.

11. 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 B T 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. (  /c) 2 =(m  /a) 2 + (n  /b) 2 + (p  /L) 2 L The resonant frequency of a rectangular cavity can be given by Where a, b and L are the width, height and length of the cavity and m, n and p are integers a

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

13. Cavity Modes TE 1,1 TE 0,1 TM 0,1 TE 2,1 TE r,θ r θ Cylindrical (or pillbox) cavities are more common than rectangular cavities. The indices here are m = number of full wave variations around theta n = number of half wave variations along the diameter P = number of half wave variations along the length The frequencies of these cavities are given by f = c/(2  * (  /r) Where  is the n th root of the m th bessel function for TE modes or the n th root of the derivative of the m th bessel function for TE modes or

14. TM 010 Accelerating mode Electric Fields Magnetic Fields Almost every RF cavity operates using the TM 010 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.

15. TM 010 Dipole Mode E H Beam Z 0 =377 Ohms

16. A standing wave cavity

17. Accelerating Voltage Position, z 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 Position, z Ez, at t=z/v 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

18. 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.

19. 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 E z0 is the peak axial electric field.

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

21. 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. Electric Field Magnitude For a pillbox

22. 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. However if the conductivity is finite the fields will not be completely shielded at the surface and the field will penetrate into the surface. This causes currents to flow and hence power is absorbed in the surface which is converted to heat. Skin depth is the distance in the surface that the current has reduced to 1/e of the value at the surface, denoted by Current Density, J. x. The surface resistance is defined as For copper 1/  = 1.7 x 10 -8  m

23. Power Dissipation The power lost in the cavity walls due to ohmic heating is given by, R surface 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.

24. 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)

25. TM010 Shunt Impedance

26. 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.

27. TM010 R/Q

28. 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 R s is frequency dependant.

29. Q factor Pillbox

30. Frequency Scaling R surf ~ f 0.5 normal conducting R surf ~ f 2 superconducting Q o ~ f 0.5 normal conducting Q o ~ f 0.5 superconducting R s ~ f 0.5 normal conducting R s ~ f -1 superconducting R/Q ~ f 0 normal conducting R/Q ~ f 0 superconducting

31. Multicell It takes x4 power to double the voltage in one cavity but only x2 to use two cavities/cells to achieve the same voltage (R s ~number of cells). To make it more efficient we can add either more cavities or more cells. This unfortunately makes it worse for wakefields (see later lectures) In order to make our accelerator more compact and cheaper we can add more cells. We have lots of cavities coupled together so that we only need one coupler. This however adds complexity in tuning, wakefields and the gradient of all cells is limited by the worst cell.

32. 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. (Limit is ~ 1 MW but 500 kW is safer) 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.

33. 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.

34. Field Emission 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.

35. 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. h 2b E local =  E 0

36. 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.

37. 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.

38. Maximum Gradient Limits All the limiting factors scale differently with frequency. They also mostly vary with pulse length. The limiting factor tends to be different from cavity to cavity. For a CW machine the gradient is limited by average heating instead. Also need to think about the electricity bill as 1 MW is £200 per day.

39. Lecture 2

40. Capacitor – E-Field The electric field of the TM 010 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

41. 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.

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

43. Equivalent circuits 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 V c =V 0 T To increase the frequency the inductance and capacitance has to be increased.

44. 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.

45. 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.

46. 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 P e is the power lost through the coupler when the RF sources are turned off. We can then define a loaded Q factor, Q L, which is the ‘real’ Q of the cavity

47. Scattering Parameters Black Box S 1,1 S 2,1 Input signal input reflection coefficient forward transmission coefficient When making RF measurements, the most common measurement is the S- parameters. The S matrix is a m-by-m matrix (where m is the number of available measurement ports). The elements are labelled S parameters of form S ab where a is the measurement port and b is the input port. The meaning of an S parameter is the ratio of the voltage measured at the measurement port to the voltage at the input port (assuming a CW input). S ab =V a / V b S 11 S 12 S 21 S 22 S =

48. Resonant Bandwidth ω-ω0ω-ω0 P ω0ω0 QLQL 1 tLtL = SC cavities have much smaller resonant bandwidth and longer time constants. Over the resonant bandwidth the phase of S 21 also changes by 180 degrees. 

49. Cavity responses A resonant cavity will reflect all power at frequencies outwith its bandwidth hence S11=1 and S21=0. The reflections are minimised (and transmission maximised) at the resonant frequency. If the coupler is matched to the cavity (they have the same impedance) the reflections will go to zero and 100% of the power will get into the cavity when in steady state (ie the cavity is filled). S11 The reflected power in steady state is given by where

50. Cavity Coupling Cavity Behaviour examples Steady state The most important behaviour we must understand is when the cavity is in steady state (ie when the cavity stored energy is constant and U=U 0 ). We can use the definitions of beta and Q to derive, We can also get voltage by using R/Q (remember the overvoltage). From this equation we can see that the cavity energy is maximum when β=1.

51. Cavity Filling note: No beam! As we vary the external Q of a cavity the filling behaves differently. Initially all power is reflected from the cavity, as the cavities fill the reflections reduce. The cavity is only matched (reflections=0) if the external Q of the cavity is equal to the ohmic Q (you may include beam losses in this). A conceptual explanation for this as the reflected power from the coupler and the emitted power from the cavity destructively interfere. When filling, the impedance of a resonant cavity varies with time and hence so does the match this means the reflections vary as the cavity fills.

52. 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 P b =V c I beam from the cavity. I beam =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.

53. Coupling with Beam Loading The rf source will not see any difference between the power dissipated in the cavity walls and the power extracted by the beam hence we can calculate a new Q factor, Q cb. this Q cb will replace Q 0 when calculating cavity filling. This means the match will change as well as needing more power. Normally we aim for  =1 with beam and have reflections when filling.

54. Typical RF System Low Level RF RF Amplifier Transmission System 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. feedback

55. 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. Electron gun RF Input RF Output Collector RF Cavity RF Power

56. 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 = P rf / P in RF Efficiency= RF Output Power / DC Input Power = P rf / P dc 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.

57. Electron Guns (Diodes) When a cathode is heated, electrons are given sufficient energy to leave the surface. When a high enough voltage is applied, electrons will travel across the voltage gap. A current is then measured on the anode.

58. Triode Guns A grid can be inserted into a diode to control the voltage on the cathode surface. An RF voltage can be applied to the grid to produce bunches of electrons. Electron bunches Grid voltage Time

59. Triodes and Tetrodes The most basic types of RF amplifiers are triodes and tetrodes. These operate by using the grid to bunch the beam and then the beam is collected at the anode. The anodes potential fluctuates with the electron beam hence providing an ac voltage. A tetrode also has a 2 nd grid to screen the control grid from the anode to avoid feedback. These are usually low frequency tubes.

60. Triode Theory The Beam induced from the cathode has a transient current. The current is given by I=I dc +I ac The dc input power is then given by P dc =V anode I dc The ac input power is given by P in =V grid I ac The ac output power is given by P rf =V anode I ac In Class A I dc =I ac Efficiency= P rf / P dc =50% Gain =P rf /P in Using different ratio of AC to DC current we can improve the efficiency at the expense of Gain Class A Class B

61. CERN Tetrode Example Frequency=200 MHz Power= 62 kW Gain=14 dB Efficiency = 64% Cathode Voltage= 10 kV Gain is low so needs a SSPA or IOT driver. This lowers the overall efficiency and increases the cost. A diacrode is a sort of two sided tetrode that doubles the power.

62. Generation of RF Power A B C A bunch of electrons approaches a resonant cavity and forces the electrons within the metal to flow away from the bunch. 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 The lower energy electrons then pass through the cavity and force the electrons within the metal to flow back to the opposite side

63. IOT Schematics Electron bunches Grid voltage Time Density Modulation

64. 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

65. Electron density Electron energy Interaction energy Klystron Schematics

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

67. Combining Tubes IoT’s, tetrodes or SSPA’s are often combined to give a higher power output. This reduces efficiency as the combiners are lossy (perhaps 5-10% less). It is more reliable as if one amplifier breaks you only loose some of the power. Power output limited by heating, normally under 500 kW-1 MW.

68. Density modulation direct from the cathode Little velocity spread Higher gap voltage Increased output power Higher efficiency Efficiency is approximately constant for reduced output power Low Gain Grid geometry will not permit IOTs to operate at high frequencies like Klystrons. Technical Data Electron Bunches formed by velocity modulation from the cavities. Several bunching cavities High Gain Long Device Expensive Considerable velocity spread Maximum gap voltage determined by the slower electrons Rapid reduction in efficiency for reduced output power High Gain IOT Klystron

69. Solid State Power Amplifier (SSPA) We can also make a high power amplifier by combining hundreds of low power solid state amplifiers

70. SSPA vs Tubes Advantages No warm-up time High reliability Low voltage (<100 V) Air cooling High stability Graceful degradation Disadvantages Complexity Losses in combiners Failed transistors must be isolated Electrically fragile High I2R losses Low efficiency High maintenance

71. Magnetrons For small industrial accelerators the most common source is the magnetron. This works by having an electron cloud rotate around a coaxial cathode. They are cheap and fairly efficient and can reach powers of 5 MW pulsed or 30 kW CW at 3 GHz (100 kW at lower frequencies). Phase stability is not good enough for large accelerators. It may be possible to phase-lock magnetrons to allow them to be used for larger accelerator.

72. Magnetrons for medical linacs

73. Pulse Compression Power time Compressed Pulse Klystron Pulse For pulse linacs it is often cheaper and easier to produce longer RF pulses and compress them to produce higher peak powers. This is performed by storing the RF in a cavity and switching the external Q of the cavity (or otherwise increasing the output power).

74. When to use what types?

75. When to use what types In the range of 400 MHz to 1.3 GHz you have a choice. There is no right answer different accelerators make different choices. IoTs are higher efficiency but limited to <100 kW and normally need combining. SSPA’s are very low down-time but expensive, inefficient and need a parts replaced a lot. Klystrons are high power and difficult to swap so if one breaks you have trouble. Tetrodes are very low gain so need more amplifiers to drive them. Magnetrons are unstable so are not used for large machines with multiple cavities.

76. Device frequency You can only buy many tubes for accelerators at discrete frequencies hence most accelerators have to use common frequencies. The frequencies are: 200 MHz, 267 MHz, 352 MHz, 400 MHz, 508 MHz, 650 MHz, 704 MHz 1.3 GHz, 2.87 GHz, 3 GHz, 3.7 GHz, 3.9 GHz, 5.6 GHz, 9.3 GHz, 11.424 GHz, 11.994 GHz The frequencies tend to correspond to integer wavelengths in mm and inches and try to avoid frequencies used in broadcast and comms.

77. Lecture 3

78. Generation of RF Current A B C A bunch of electrons approaches a resonant cavity and forces the electrons to flow away from the bunch. The negative potential difference causes the electrons to slow down and the energy is absorbed into the cavity The lower energy electrons then pass through the cavity and force the electrons within the metal to flow back to the opposite side

79. Bunch Spectrum A charged bunch can induce wakefields over a wide spectrum given by, f max =1/T. A Gaussian bunch length has a Gaussian spectrum. On the short timescale (within the bunch) all the frequencies induced can act on following electrons within the bunch. On a longer timescale (between bunches) the high frequencies decay and only trapped low frequency (high Q) modes participate in the interaction.

80. Mode Indices

81. Dipole modes Dipole mode have a transverse magnetic and/or transverse electric fields on axis. They have zero longitudinal field on axis. The longitudinal electric field increases approximately linearly with radius near the axis. Electric Magnetic Wakefields are only induced by the longitudinal electric field so dipole wakes are only induced by off-axis bunches. Once induced the dipole wakes can apply a kick via the transverse fields so on-axis bunches can still experience the effect of the wakes from preceding bunches.

82. If we rearrange Farday’s Law ()and integrating along z we can show Panofsky-Wenzel Theorem Inserting this into the Lorentz (transverse( force equation gives us for a closed cavity where the 1st term on the RHS is zero at the limits of the integration due to the boundary conditions this can be shown to give This means the transverse voltage is given by the rate of change of the longitudinal voltage

83. Multibunch Wakefields For multibunch wakes, each bunch induces the same frequencies at different amplitudes and phases. These interfere to increase or decrease the fields in the cavity. As the fields are damped the wakes will tend to a steady state solution.

84. Resonances As you are summing the contribution to the wake from all previous bunches, resonances can appear. For monopole modes we sum Hence resonances appear when It is more complex for dipole modes as the sum is This leads to two resonances at +/-some Δfreq from the monopole resonant condition.

85. Damping As the wakes from each bunch add together it is necessary to damp the wakes so that wakes from only a few bunches add together. The smaller the bunch spacing the stronger the damping is required (NC linacs can require Q factors below 50). This is normally achieved by adding external HOM couplers to the cavity. These are normally quite complex as they must work over a wide frequency range while not coupling to the operating mode. However the do not need to handle as much power as an input coupler.

86. Beampipe cutoff TE 1,1 TM 0,1 TE r,θ r θ In a circular waveguide/beampipes the indices here are m = number of full wave variations around theta n = number of half wave variations along the diameter The cutoff frequencies of these are given by f c = c/(2  * (  /r) Where  is the n th root of the m th bessel function for TM modes or the n th root of the derivative of the m th bessel function for TE modes or (=2.4 for TM01 and 1.8 for TE11) In order to provide heavy damping it is necessary to have the beampipes cutoff to the TM01 mode at the operating frequency but not to the other modes at HOM frequencies.

87. Coaxial HOM couplers ICsCs R HOM couplers can be represented by equivalent circuits. If the coupler couples to the electric field the current source is the electric field (induced by the beam in the cavity) integrated across the inner conductor surface area. V R If the coaxial coupler is bent at the tip to produce a loop it can coupler to the magnetic fields of the cavity. Here the voltage source is the induced emf from the time varying magnetic field and the inductor is the loops inductance. L

88. Loop HOM couplers ICsCs R ICsCs R L L CfCf Inductive stubs to probe couplers can be added for impedance matching to the load at a single frequency or capacitive gaps can be added to loop couplers. Also capacitive gaps can be added to the stub or loop inductance to make resonant filters. The drawback of stubs and capacitive gaps is that you get increase fields in the coupler (hence field emission and heating) and the complex fields can give rise to an electron discharge know as multipactor (see lecture 6). As a result these methods are not employed on high current machines.

89. F-probe couplers F-probe couplers are a type of co-axial coupler, commonly used to damp HOM’s in superconducting cavities. Their complex shapes are designed to give the coupler additional capacitances and inductances. These additional capacatances and inductances form resonances which can increase or decrease the coupling at specific frequencies. The LRC circuit can be used to reduce coupling to the operating mode (which we do not wish to damp) or to increase coupling at dangerous HOM’s. Output antenna Capacative gaps Inductive stubs frequency Log[S 21 ]

90. Waveguide Couplers waveguide 2 waveguide 1 w 2 /2 w 1 /2 Waveguide HOM couplers allow higher power flow than co-axial couplers and tend to be used in high current systems. They also have a natural cut-off frequency. They also tend to be larger than co-axial couplers so are not used for lower current systems. To avoid taking the waveguides through the cryomodule, ferrite dampers are often placed in the waveguides to absorb all incident power.

91. Choke Damping load choke cavity For high gradient accelerators, choke mode damping has been proposed. This design uses a ferite damper inside the cavity which is shielded from the operating mode using a ‘choke’. A Choke is a type of resonant filter that excludes certain frequencies from passing. The advantage of this is simpler (axially- symmetric) manufacturing

92. Beampipe HOM Dampers For really strong HOM damping we can place ferrite dampers directly in the beampipes. This needs a complicated engineering design to deal with the heating effects.

93. Decay in beampipe When a mode is resonant in the cavity but below the cut-off frequency of the beampipe or waveguide dampers the fields decay exponentially in the beampipe. A=exp(-k z *z), where k z = 1/c*sqrt(  c 2 -  2 ) The TM010 mode will also decay and some fields will be absorbed in any absorbers It is necessary to tailor the beampipe size and length to make sure the TM010 mode is sufficiently attenuated but all the HOMs are damped. Often the beampipe can have flutes added to reduce the cutoff of HOMs without affecting the TM01 mode.

94. Multicell cavity damping Each coupler removes a given power when a field is applied to it. The Q factor and hence damping is given by Q e =  U/P Multicell cavities have more stored energy hence have higher Q factors. In addition HOMs can be trapped in the middle cells and will have low fields at the couplers. Damping requirements must be carefully balanced vs the length and cost of the RF section. CEBAF = 5 cells, high current but a linac DLS = 1 cell, high current storage ring SOLIEL = 2 cell, high current storage ring ILC =9 cells, high gradient low current

95. SCRF Cavities The power required to keep a cavity on filled to a set voltage is the power extracted by the beam plus the ohmic heating in the walls. In order to increase the efficiency of coupling power to the beam we need to minimise the ohmic heating in the cavity walls. The ohmic heating can be reduced by 5-6 orders of magnitude with the use of a superconducting cavity.

96. RF superconductivity The surface resistance has the following dependence Rs increases with frequency squared Rs increases exponentially with temperature SCRF cavities have higher losses as they increase in frequency, for this reason there are few SCRF cavities above 4GHz.

97. Residual Resistance An SRF cavity will decrease its resistance with temperature in theory. However in practice there is often a minimum resistance due to the effects of normal conducting impurities in the niobium. One of the main effects is flux pinning where magnetic fields are frozen into normal conducting impurities inside the superconductor. This can be avoided by shielding the cavity from magnetic fields during cooldown.

98. RF Critical B field When a magnetic field is applied to a superconductor, supercurrents flow. This increases the free energy of the superconducting state. When the free energy of the superconducting state equals the normal conducting state the flux enters the material. For RF fields the flux continues to be excluded in a metastable state unit the field reaches the critical superheating field (240 mT for Nb) When the electrons condence into cooper pairs the resulting superconducting state is more ordered than the normal-conducting state.

99. SRF Couplers Also a limited power in SRF couplers. WG limited to 500 kW due to multipactor (electron cloud). Coax is limited to a similar amount by limited cooling of inner coax.

100. Microphonics Microphonics: Changes in frequency caused by connections to the outside world Vibrations Pressure Fluctuations This means the cavity is not always on resonance and will require more RF power to fill. Additionally the constantly varying frequency will cause phase errors. To avoid problems we need to artificially broaden the cavity bandwidth by using a lower Q e of at least 10 6

101. Filling factor The cryomodule in an SRF system also takes up significant space. The filling factor is the ratio of the cavity length to cryomodule length. It varies from a factor of 5 for single cells to 1.5 for a 9 cell cavity (typically its 2 extra cells on each side).

102. Cryogenic systems All refrigerators have a technical efficiency, η T of 20%-30% The Carnot efficiency is given by The dynamic heat load, P c, is the rf power dissipated in the cavity walls. A static heat load, P s, adds an additional heating Liquid helium transfer lines require 1W per metre, so total loss is length L (More efficient lines can be used) It is standard to fill to an overcapacity, O Total Power for N cryostats= O N (P c +P s +L) / (η T η c )

103. Circular AcceleratorsLinac (High energy) High HOM damping CW operation  high power couplers required High gradient cavities Multicell Pulsed operation ( often ) RF Cavities for Linacs and Circular Accelerators CESR cavity TESLA cavity

104. SCRF vs NCRF SCRF More efficient (even when including cryogenic losses) Higher CW gradient Long pulse or CW only Complex system needing cryostats and cryogenics Only frequencies below 4 GHz. NCRF Less efficient. Higher pulsed gradient Simpler systems, water cooled More reliable Lower capital costs Smaller apertures mean higher wakefields

105. RF for High Energy Linacs Linear accelerators RF requirements are very different to those of circular acclerators. Circular Accelerator Acceleration over many passes Emphasis on beam current Need to reduce instabilities  HOM damping required CW operation Big SR contribution to RF losses (lighter particles in particular)  few high energy storage rings as SR losses increase with E^4 Linac Acceleration in one pass  High gradients and high efficiency required Beam current limited by source (no stacking) Emphasis on beam energy Often pulsed

106. Putting it all together First we need to know the beam current, how much it needs to be accelerated by, and the overvoltage. Can use this to calculate required power and Q factors for an SRF and/or NCRF system based on pillbox numbers. Investigate possible power sources. Single or multicell? SCRF or NCRF Choose frequency. Model real cavity and look at HOM damping. Adjust calculations using numbers from RF models.