Impedances and Wake Fields: They are forty, but they don’t look it

1. Impedances and Wake Fields:They are forty, but they don’t look it Vittorio Giorgio VaccaroUniversità degli Studi di Napoli “Federico II” andINFN Sezione di Napoli

5. Impedances and Wake Fields: Why? On 1966 the design of a large Intersecting proton Storage Ring (ISR) started. It was supposed to accumulate at 30GeV the largest current ever reached. Bad news were coming from USA. At MURA in the electron storage ring, the accumulator current could not go beyond a limit. Above this threshold there was evidence of transverse instabilities, which were interpreted as the presence of electrodes. This evidence produced in the scientific community the conviction that the in the beam dynamics the interaction with the surrounding equipment should not be neglected (e.m. field produced by image currents).

6. Impedances and Wake Fields: Why? In the ISR a lot of additional equipment was present in the ring (for instance, almost 600 clearing electrode plates). It was not clear what type of interaction the whole complex might have with the beam. In addiction to this, a large number of cavities were foreseen and many other equipment, such as bellows, had to be studied in this respect.

7. An example of beam-equipment interaction RF e.m. field Particle bunches Beam-discontinuity interaction

8. The Devices Considering a Fourier harmonic in the charge distribution of a beam. It will generate a current, interacting with the surrounding medium, will generate electromagnetic fields acting back on beam itself. This action may consist into an enhancement of the perturbation itself. This may trigger an avalanche effect that may limit the beam. This occurrence will be more strong as more high is the charge of beam bunches.

9. The Devices The variation of the cross section in accelerators cannot be avoided. Two examples: the accelerating cavity gap and the bellows. The first one is by definition an essential device for carrying out the function itself of the accelerator. The second one is strictly necessary to absorb the thermal stresses and mechanical fabrication tolerances. Hence, the wake fields are unavoidable.

10. The Devices A DAΦNE accelerating cavity example: Waveguides literally “absorb” HOMs from the cavity

11. The Devices MITSUBISHI example: C-Band choke mode type accelerating structure (Shintake cavity). The “choke” influence cavity modes, damping the High Order Modes (HOMs). It appears to the fundamental mode as a closed cavity (short circuit)

12. The lowest modes which may occurs Instabilities for pedestrians The starting point was the study of instabilities: the beam in a circular accelerator behaves (because it is circular!) as a feedback device. Circularity is the core of the feedback device indeed (main difference with respect to linacs).

13. Instabilities for pedestrians 1) This device, likewise all feedback devices, may be unstable when some conditions are fulfilled. 2) The stability conditions are dictated not only by the dynamical properties of the beam (charge/mass, energy, magnetic focusing, etc.), but also by the type of electromagnetic interaction with the environment (response of the machine, since 1966 called Coupling Impedance).

14. Instabilities for pedestrians The gedanken experiment. An ideal device, fed by a voltage ΔV(Ω), produces a small longitudinal density perturbations in the current I at a frequency Ω Suppose that there is no interaction with surrounding medium and measure the perturbation by means of a reciprocal device which may define the perturbed current ΔI(Ω) where the BEAM ADMITTANCEYB(Ω) depends on the beam properties: charge/mass, focusing properties etc. Of course the perturbed current is proportional to the d.c. current I0

15. Instabilities for pedestrians Taking into account the e.m. interaction on one turn of the beam with the environment, this interaction will produce e.m. forces having the same space and time distribution as the perturbation. These forces can be represented by an equivalent voltage ΔV1’(Ω) which can be meant as produced by the current ΔI(Ω), loading an IMPEDANCE. This impedance, ZM(Ω), represents the overall interaction with surrounding equipment. This voltage now acts back again on the beam, producing an additional perturbation, and so on. After m turns we have:

16. Instabilities for pedestrians The process is circular: • The forcing source ΔV(Ω), acting on the particles, produces the effect ΔI(Ω). (the effect is governed by the particle dynamics) • The effect ΔI(Ω) becomes the forcing source of the electromagnetic fields, namely ΔV1’(Ω), which is the integrated forces over one turn. (the effect is governed by Maxwell’s equations with the appropriate boundary conditions) 3. The effect ΔV1’(Ω) becomes the forcing source…….and so on.

17. Instabilities for pedestrians Intuition suggests that there will be no amplification, if or In previous equations we see that the stability is possible if, at certain frequency, the e.m. interaction with the machine (l.h.s.) is lower than a quantity, which depends on the properties of the beam (r.h.s.) Remember

18. Stability chart below transition energy Stability chart above transition energy Instabilities for pedestriansstability charts for mono-energetic beams

19. Instabilities for pedestriansstability charts for real beams (spread in energy) Stability chart for realistic distribution function A realistic distribution function Landau damping

20. Stability chart for realistic distribution function Instabilities for pedestriansstability charts for real beams (spread in energy) A stability criterion for longitudinal stability: n = harmonic number e = elementary charge I0 = stored current Δp = momentum spread η= slippage

21. Instabilities for pedestrians • The concept of impedance, originally conceived for longitudinal dynamics was extended according to the phenomenon to be studied. Therefore we have: • Transverse impedance • Longitudinal wake field • Transverse wake field • Loss factor • etc.

22. The Impedances and their estimation • In parallel to the studies been equipment interaction, various problems were tackled. • Transverse stability charts • Bunched beam stability criteria: (single bunch, multi-bunch, etc.) • etc. • Since their birth an intensive effort was done in order to calculate (BEIC-Beam Equipment Interaction Committee) and to measure the coupling impedances. This effort started at CERN around 1968 and is still lasting. • Huebner, Zotter, Palumbo, Ruggiero and many others were active in the calculation tecniques. • In the measurement techniques Sands and Rees, Sacharer and Nassibian and many others were active.

23. Elliptic Vacuum Chamber In the case of relativistic particles the EM fields can be described resorting to a quasi-static representation Therefore all the EM-fields can be derived by the following equation Flatness parameters

24. Elliptic Vacuum Chamber A new definition of impedances 24

25. Elliptic Vacuum Chamber We report the sentence by Morse and Feschbach about the sum "As all these Green's function expansions, the series is only conditionally convergent and should not differentiated unless the poorly convergent part can be condensed into a closed function". Anyway the series has been condensed in a closed form as a linear combination of Jacobi Θ-functions .

26. Elliptic Vacuum Chamber The result of these computations is given, for the image potential, by the following formula,

27. Elliptic Vacuum Chamber

28. Elliptic Vacuum Chamber L. Palumbo, V.G. Vaccaro, “Coupling Impedance Between Circular Beam and a LossyVacuum Chamber in Particle Accelerators”, Il NuovoCimentoVol. 89 (1985). Vittorio G. Vaccaro Green function’s for a lossy elliptical vacuum chamber. Seminar at CERN (1992) Francesco Ruggiero. “Resistive wall impedance as derivative of the electric capacitance for a beam pipe of arbitrary cross section.” Phys. Rev. E 53, 2802 - 2806 (1996)

29. Measurements: Hera Vertex Chamber Impedance measurement setup of “Mazinga”

30. Measurements: Hera Vertex Chamber

31. Measurements: Hera Vertex Chamber MEASUREMENT OF THE LONGITUDINAL COUPLING IMPEDANCE OF THE HERA-B VERTEX DETECTOR CHAMBER V.G. Vaccaro & alii, EPAC96

32. Analytic EM characterization of some structures The fields induced by the beam passing through discontinuities and changes of the vacuum chamber cross section (irises, accelerating cavities, bellows, etc.) may be particularly critical for the beam stability. Therefore in the design of an accelerating machine it is necessary to have simple and accurate numerical tools in order to evaluate the effects of these fields.

33. The adopted method • Wave Matching Technique (WMT): a well tested technique which describes the em-field making use propagating guide modes with orthogonality properties in the cross section (twofold domain). • Mode Matching (MMT): beside the guide modes, it resorts to resonant modes with orthogonality properties in a threefold domain. It is possible to insert losses.

34. B A B B A A B B B A B B B B B A A B B A B An examle of MMT: Shintake the cavity. A zones: cavities B zones: waveguides 34

35. Measurements: Loss Factor of a DUT

36. Measurements: Loss Factor of a DUT Cavity longitudinal loss factor measurement by means of a beam test facility. V.G. Vaccaro & aliiPhy. Rev. S. T. - volume 3, 2000

37. Impedance calculations (FMT) An instructive particular example: abrupt junction. Longitudinal coupling impedance of an abrupt junction in a vacuum chamber V. G. Vaccaro & Al. Nuovo Cimento A 1999

38. Impedance calculations (FMT)

39. Impedance calculations (FMT)

40. Impedance calculations (FMT)

41. Impedance calculations (FMT)

42. Impedance calculations (FMT)

43. Impedance calculations (FMT)

44. Mode Matching Technique The expansion on a complete set The definition of a complete set

45. Mode Matching Technique

46. Mode Matching Technique All methods subdivide the space affected by EM field in subspaces in which the field is representable through a linear combination of orthogonal configurations On the separation surface between subspaces, the field continuity equations are turned in linear system of infinite equations of infinite unknowns. The coefficients of the linear system are the unknowns of the problem. • The numerical codes based on Mode Matching method are more simple to be implemented, but sometimes they show slowness to reach convergence. • The numerical codes based on Modal Matching are more difficult to be implemented, but the convergence is reached more quickly.

47. How to verify the numerical methods • In order to verify on a real device the goodness of simulations, one needs to measure the fundamental parameters as the Loss Factor or the Coupling Impedance: not easy to realize in small labs. • In Loss factor case, a relativistic bunch generator is required in order to shot particles through the device and measure the kinetic energy loss. • Moreover, any measure of energy loss must be done as a difference between two entities very similar to each other, leading to a variance in the same order of the measure or bigger. • An indirect measure is required in order to achieve reliable measurements.

48. The studied models : The Choke mode cavity (Shintake). bunch

49. The models to be studied : The Shintake cavity arrayIt is used for an High performer LINAC allocated as F.E.L. feeder (according to MITSUBISHI) 1999 2008 The maquillage is suspicious: remark the absence of the fundamental mode…

50. Wire method as a test Goodness of numerical methods is achieved “modifying” the Device Under Test (D.U.T.) : a wire, which substitutes the beam, is inserted on the DUT axes . The response to a known signal is measured by means of the scattering matrix (reflection and transmission response). Then the measured scattering parameters are compared with the results of a numerical algorithm applied on the same D.U.T. It is necessary to underline that this method is not useful to determine the parameters of the original device, but it is useful to validate numerical results only. In fact it perturbs the D.U.T. modifying its parameters.