1. RF Cavity Design Part 2/3 Combined slides of
Erk Jensen (CERN BE/RF) CAS 2013
Mauritzio Vretenar (CERN BE/RF) CAS 2013

2. Accelerating gap
Erk Jensen

3. Gap of PS cavity (prototype)
We need a voltage over gap
Gap of PS cavity (prototype)

4. Drift Tube Linac (DTL) – how it works
For slow particles! e.g. few MeV
The drift tube lengths can easily be adapted.
electric field
Erk Jensen

5. Drift tube linac – practical implementations
Erk Jensen

6. Accelerating gap We want a voltage across the gap
It cannot be DC, since we want the beam tube on ground potential.
Use 𝐸 ∙𝑑 𝑠 =− 𝑑 𝐵 𝑑𝑡 ∙𝑑 𝐴
For an empty cavity frequency is determined its diameter
Frequency if the lowest accelerating mode TM010 is 𝜔 010 = 𝑐 a
To get small cavity size short wavelengths are used
Electron linacs typically λ=10 cm (f=3 GHz) and cavity radius a=3.38cm
Storage rings λ=60cm (f=500MHz), a = 23 cm
But for small low energy rings lower frequencies are needed
Below 1 MHz
Photo: PS 19 MHz cavity (prototype, 1966)
Erk Jensen
Erk Jensen
Erk Jensen
Erk Jensen

7. Ferrite Cavities
For a pillbox cavity filled with a material frequency for the lowest accelerating mode TM010 is 𝜔 010 = 𝑐 𝜖𝜇 a
Where ε and μ are dielectric and magnetic permeabilities
So the frequency can be lowered by using high μ material
Material that increases inductance, acts as “open circuit”
Materials typically used:
ferrites (depending on f-range)
magnetic alloys (MA) like Metglas®, Finemet®, Vitrovac®…
Resonantly driven with RF (ferrite loaded cavities)
or with pulses (induction cell)
Price to pay: large power losses due hysteresis and eddy currents, the material must be cooled
gap voltage
Erk Jensen
Erk Jensen
Erk Jensen
Erk Jensen

8. Large Frequency Range
Ferrite cavities have another advantage for low energy rings: large frequency tuning range
Frequency must increase until beam becomes relativistic
Applying bias field changes its effective permeability, and hence the frequency
Graph:
Ferrite B-H loop with added ac field.
Incremental permeability μr=Bac/ μoHac
varies with operating point.

9. Simplified model of ferrite cavity

10. Los Alamos / TRIMF single gap 46 - 61 MHz

11. CERN PS Booster Cavity
3 – 8.4 MHz
RF voltages in anti-phase

12. CERN Lear Cavity Single gap 0.38 – 3.5 MHz
RF voltage on the bias winding is made zero at the bias supply feed point by splitting the winding at the mid voltage point
The two windings are crossed and the parallel bias field is in opposite directions in each half of the cavity

13. CERN PS Booster 1998 0.6 – 1.8 MHz < 10 kV gap NiZn ferrites
Erk Jensen

14. Finemet® Cavity
Finemet® is a magnetic alloy exhibiting wideband frequency response and large magnetic field saturation level
It allows building wide-band cavities
Compared to ferrite cavities, no bias is needed!
Example: the CERN PSB 5-gap system:
Single Finemet® ring on its cooling plate
frequency response: (0.6 … 4) MHz
5 gap prototype
… installed in PSB ring 4
Erk Jensen

15. Finemet RF System MedAustron
6-gap finemet cavity
Large instantaneous bandwidth!
0.2 ÷10 MHz, 1 kW solid state amplifier

16. Characterizing a cavity
Erk Jensen

17. Desired Cavity Properties
Delivers high accelerating voltage Vacc
Delivers uniform field across the bunch extent
Requires minimum RF power Pin
Has low power losses Ploss
Has large aperture for the beam
Does not induce higher order modes (HOMs)
Does not develop multipacting
Has low break-down rate
Is easy and cheap to manufacture
Piotr Skowronski

18. Stored Energy W
The electric and magnetic stored energy oscillate in time 90 degrees out of phase.
In practice, we can use either the electric or magnetic energy using the peak value.
Power dissipation
Where
Rs is the surface resistance, s is the dc conductivity d is the skin depth

19. Quality Factor
Quality factor describes how much energy can be stored in cavity for a given input power

20. Acceleration voltage Define 𝑉 𝑎𝑐𝑐 = 𝐸 𝑧 𝑒 j 𝜔 𝛽𝑐 𝑧 d𝑧 .
The exponential factor accounts for the variation of the field while particles with velocity 𝛽𝑐 are traversing the gap (see later).
With this definition, 𝑉 𝑎𝑐𝑐 is generally complex
This becomes important with more than one gap.
For the time being we are only interested in 𝑉 𝑎𝑐𝑐 .
Attention, different definitions are used!
Erk Jensen

21. Shunt impedance
The square of the acceleration voltage is proportional to the power loss 𝑃 𝑙𝑜𝑠𝑠
The proportionality constant defines the quantity “shunt impedance”
Attention, also here different definitions are used!
Traditionally, the shunt impedance is the quantity to maximize in order to minimize the power required for a given gap voltage.
But, too large value may lead to other unwanted effects like easy higher order mode excitation
Erk Jensen

22. R-upon-Q
The square of the acceleration voltage is proportional to the stored energy 𝑊 The proportionality constant defines the quantity called R-upon-Q:
𝑅 𝑄 = 𝑉 𝑎𝑐𝑐 𝜔 0 𝑊
Attention, also here different definitions are used!
Erk Jensen

23. Cavity resonator - equivalent circuit
Simplification: single mode IG
Beam IB
Vacc P
Generator Z C L R
L=R/(Qw0)
: coupling factor
C=Q/(Rw0)
Cavity
R: Shunt impedance
: R-upon-Q
Erk Jensen

24. Field rotates by 360° during particle passage.
Transit time factor
The transit time factor is the ratio of the acceleration voltage to the (non-physical) voltage a particle with infinite velocity would see.
The transit time factor of an ideal pillbox cavity (no axial field dependence) of height (gap length) h is:
Field rotates by 360° during particle passage.
h/l
Erk Jensen

25. Pillbox cavity field (w/o beam tube)h
Ø 2a
The only non-vanishing field components :
Erk Jensen

26. Reentrant cavity
Nose cones increase transit time factor, Round outer shape minimizes losses.
Example: KEK photon factory 500 MHz R probably as good as it gets -
this cavity optimized
pillbox
R/Q: 111 Ω Ω Q:
R: 4.9 MΩ MΩ
nose cone
Erk Jensen

27. Loss factor t f0 Impedance seen by the beam V (induced) BeamIB
V (induced)
Beam
Energy deposited by a single charge q:
R/b C L R
L=R/(Qw0)
C=Q/(Rw0)
Cavity
Voltage induced by a single charge q: 1 - 1 5 1 1 5 2
t f0
Erk Jensen

28. Summary: relations Vgap,W, Ploss
Gap voltage
Vgap
R-upon-Q
Shunt impedance
Energy stored inside the cavity W
Power lost in the cavity walls
Ploss
Q factor
Erk Jensen

29. Beam loading – RF to beam efficiency
The beam current “loads” the generator, in the equivalent circuit this appears as a resistance in parallel to the shunt impedance.
If the generator is matched to the unloaded cavity, beam loading will cause the accelerating voltage to decrease.
The power absorbed by the beam is , the power loss
For high efficiency, beam loading shall be high.
The RF to beam efficiency is
Erk Jensen

30. Beam Loading – allows efficiency
choice for CLIC main beam
full beam loading – optimum efficiency
eta
acceleration
Erk Jensen

31. Characterizing cavities
Resonance frequency
Transit time factor
field varies while particle is traversing the gap
Shunt impedance
Q factor
gap voltage – power relation
R/Q
independent of losses – only geometry!
loss factor
Circuit definition
Linac definition
Erk Jensen

32. Example Pillbox:
Erk Jensen

33. Pillbox: dipole mode electric field magnetic field TM110-mode
(only 1/4 shown)
electric field
magnetic field

34. Panofsky-Wenzel theorem
For particles moving virtually at v=c, the integrated transverse force (kick) can be determined from the transverse variation of the integrated longitudinal force!
Pure TE modes: No net transverse force !
Transverse modes are characterized by
the transverse impedance in ω-domain
the transverse loss factor (kick factor) in t-domain !
W.K.H. Panofsky, W.A. Wenzel: “Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields”, RSI 27, 1957]
Erk Jensen

35. CERN/PS 80 MHz cavity (for LHC)
inductive (loop) coupling, low self-inductance
Erk Jensen

36. Higher order modes Example shown: 80 MHz cavity PS for LHC.
Color-coded:
Erk Jensen

37. Higher order modes ... ... n1 n2 n3 external dampers R1, Q1,w1IB
Erk Jensen

38. Higher order modes (measured spectrum)
without dampers
with dampers
Erk Jensen

39. Many gaps
Erk Jensen

40. What do you gain with many gaps?
The R/Q of a single gap cavity is limited to some 100 Ω. Now consider to distribute the available power to n identical cavities: each will receive P/n, thus produce an accelerating voltage of The total accelerating voltage thus increased, equivalent to a total equivalent shunt impedance of
P/n
P/n
P/n
P/n 1 2 3 n
Erk Jensen

41. Standing wave multicell cavity
Instead of distributing the power from the amplifier, one might as well couple the cavities, such that the power automatically distributes, or have a cavity with many gaps (e.g. drift tube linac).
Coupled cavity accelerating structure (PIMS: PI Mode Structure)
The phase relation between gaps is important!
Erk Jensen

42. Coupling accelerating cells
1. Magnetic coupling:
open “slots” in regions of high magnetic field  B-field can couple from one cell to the next
2. Electric coupling:
enlarge the beam aperture  E-field can couple from one cell to the next
How can we couple together a chain of n accelerating cavities ?
The effect of the coupling is that the cells no longer resonate independently, but will have common resonances with well defined field patterns.

43. Drift Tube Linac Maximum coupling, no slots
Standing wave linac structure for protons and ions, b= , f= MHz
Drift tubes are suspended by stems (no net RF current on stem)
The 0-mode allows a long enough cell (d=bl) to house focusing quadrupoles inside the drift tubes!
E-field
B-field

44. The Side Coupled Linac
To operate efficiently in the π/2 mode, the cells that are not excited can be removed from the beam axis
→ they become coupling cells, as for the Side Coupled Structure.

45. Example of Side Coupled Structure
LIBO (= Linac Booster)
A 3 GHz Side Coupled Structure to accelerate protons out of cyclotrons from 62 MeV to 200 MeV
Medical application:
treatment of tumours (proton therapy)
Prototype of Module 1 built at CERN (2000)
Collaboration CERN/INFN/TERA Foundation
Erk Jensen

46. H-mode structures Interdigital-H Structure Operates in TE110 mode
Transverse E-field “deflected” by adding drift tubes
Used for ions, β<0.3
CH Structure operates in TE210, used for protons at β<0.6
High ZT2 but more difficult beam dynamics (no space for quads in drift tubes)

47. IH Cavity

48. Multi-gap coupled-cell cavities
Between the 2 extreme cases (array of independently phased single-gap cavities / single long chain of coupled cells with lengths matching the particle beta) there can be a large number of variations (number of gaps per cavity, length of the cavity, type of coupling) each optimized for a certain range of energy and type of particle.
DTL
SCL CH 48
PIMS
CCDTL

49. More examples of cavities
Erk Jensen

50. Examples of cavities PEP II cavity 476 MHz, single cell,
1 MV gap with 150 kW,
strong HOM damping,
LEP normal-conducting Cu RF cavities,
350 MHz. 5 cell standing wave + spherical cavity for energy storage, 3 MV
Erk Jensen

51. CERN/PS 40 MHz cavity (for LHC)
example for capacitive coupling
coupling C
cavity
Erk Jensen

52. Transverse Deflecting RF Cavities
If an electron is travelling in the z direction and we want to kick it in the x direction we can do so with either
An electric field directed in x
A magnetic field directed in y
As we can only get transverse fields on axis with fields that vary with Differential Bessel functions of the 1st kind only modes of type TM1np or TE1np can kick electrons on axis.
We call these modes dipole modes
𝐹 =𝑒( 𝐸 + 𝑣 × 𝐵 )

53. Dipole Modes
TE111 TM110 H field B field

54. TE Modes
For TE110 mode transverse kick due to electric field cancels out the kick due to magnetic field
Note: for low beta cavities TE modes can be used for deflecting
However, when beam pipe is added, the cavity TM110 mode couples to beam pipes TE11 mode
The electric and magnetic fields are 90 degrees out of phase in both space and time so that their kicks coherently add.

55. Applications of RF Deflectors
Bunch separation and recombination
Bunch length measurements
Emittance exchange
Crab crossing in colliders

56. Single cell crab cavity

57. RFQ

58. The Radio Frequency Quadrupole (RFQ)
At low proton (or ion) energies, space charge defocusing is high and quadrupole focusing is not very effective, cell length becomes small
conventional accelerating structures (Drift Tube Linac) are very inefficient
use a (relatively) new structure, the Radio Frequency Quadrupole. + −
Opposite vanes (180º) − +
Adjacent vanes (90º)

59. RFQ The RFQ uses only electric field to accelerate and focus the beam
The wave equation can be replaced with the Laplace equation in cylindrical coordinates
The general solution
with l+n =2p+1 p=0,1,2…,V/2 the electrode potential, I2n is the modified Bessel function of order 2n and k=2p/bl
Taking only the low order solution

60. RFQ
The first term is the potential of an electric quadrupole (focusing term);
the second, will generate a longitudinal accelerating electric field.
Constants A01 and A10 are determined by imposing the voltage in the electrodes
Increasing m one gets more acceleration
Decreasing a one gets more focusing

61. RFQ
The operation of the RFQ can best be understood by considering a long electric quadrupole with an alternating voltage on it,
Particles moving along the z-axis and staying inside the RFQ for several periods of the alternating voltage, would be exposed to an alternating gradient focusing
If the tips of the electrodes are not flat but 'modulated' a part of the electric field is 'deviated' into the longitudinal direction and this field can be used to bunch and accelerate particles

62. Feeding and tuning the cavity
The transmission of the power between the generator and the cavity is done
Through a coaxial line (short distances, low power < 100 kW)
Through a waveguide. Low losses. Can be cooled
The connection between the waveguide and the cavity is done with a short coaxial line with virtually no-losses
A ceramic window inside the coaxial cable separates the waveguide from the cavity
To bring the cavity into the resonance condition, tuning is done using tuning plungers

63. END of part 2 of 3