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The TRIUMF nine-cell SRF cavity for ARIEL Kolb, Philipp Ulrich 2016

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The TRIUMF nine-cell SRF cavity for ARIELbyPhilipp Ulrich KolbDipl. Phys., Goethe University Frankfurt, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2016c© Philipp Ulrich Kolb 2016iiAbstractModern physics research relies on particle accelerators and available beam timeis a very limited resource. The ARIEL eLINAC will strengthen the rare isotopeprogram at TRIUMF by providing an alternative way to create rare isotopebeams (RIB). A possible way to add additional use to this machine is to createa return beam line and use the beam to excite a free electron laser (FEL). Theremaining beam can be used to drive fields in the SRF cavities to reduce therequired RF power.One limitation of these energy recovery LINACs (ERL) is beam-break up.Higher order modes (HOM), especially dipole modes, have a negative influ-ence on the beam which can lead to beam loss. The design of the SRF cavityhas to accommodate this to make sure a beam current of up to 10mA can beused for both RIB production and ERL operation.This thesis will go through the design process of the ARIEL 1.3 GHz nine-cellcavity. The design relies on simulations to calculate the fields inside the cavityand with it the shunt impedance of HOMs.The investigations showed that resistive beam line absorbers can be used toreduce the shunt impedance of HOMs sufficiently without interfering with theaccelerating mode. The performance of the absorber material has been verifiedin dedicated low temperature measurements, while the HOM field distributionhas been measured via beadpulling on a copper model of the cavity. Thesemeasurements showed good agreement with the simulations.The power dissipation in the SRF cavities is of vital importance. The cryogenicsystem is a significant part of the capital investment for the accelerator andsets the power budget for each cavity to around 10 W. This corresponds to aQ0 value of 1 · 1010 at an operational temperature of 2 K. The gradient goal is10 MV/m to reach the design energy of 50 MeV with five cavities. Both Q0 andEacc specifications have been met in the first two cavities that are installed incryomodules. Two more cavities have been built and are in their qualificationphase.iiiPrefaceThe presented dissertation is done in an effort to build the eLINAC for theARIEL project at TRIUMF, a large project with many people working on it.The design of the ARIEL SRF cavity, shown in chapter 3, is the original workby the author and was published in [1]. The beam dynamics calculations toestablish the HOM shunt impedance limit were done by Y.C. Chao. The coldtests of the HOM absorber material (chap. 3.9) were done by the author andhave been published in [2].The theoretical description of cavities and the beam-cavity interaction presentedin chapter 2 is textbook knowledge and is repeated to provide context.The vertical performance measurements and the needed preparations for thosetests, presented in chapter 4 and published in [3, 4], were performed by theauthor with assistance from various people for different tasks: etching via BCPas well as 120◦C baking was done by P. Harmer, J. Keir and R. Smith; flatnessand frequency tuning as well as cavity assembly in the clean room were doneby the author with the help of B.S. Waraich; handling of the cryogenic systemis done by D. Kishi and H. Lui; the LLRF system is a development of K. Fongand Q. Zheng.The idea for the multiline HOM beadpulling is from the author and fabricatedby B. Amini, while the measurements were done by the author and publishedin [5].The high temperature degassing of the cavities was done by A. Rowe and histeam at FNAL.One of the single cell cavities in chap. 4.3.1 was tested at FNAL by A. Gras-sellino and A. Romanenko.The cavities themselves were fabricated by PAVAC Industries Inc.The horizontal performance measurements (chap. 4.4) were done in collabora-tion by Z. Yao, Y. Ma and the author, while the cooldown and temperaturecontrol was handled by D. Kishi, H. Lui and R. Nagimov. The beam energymeasurements were done by M. Marchetto and the eLINAC commissioning teamof which the author was a member.All images, unless otherwise noted, were created by the author.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Abbreviations and Units . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Particle Accelerators and Modern Physics . . . . . . . . . . . . . 11.2 ARIEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Similar Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Particle Acceleration in RF Cavities . . . . . . . . . . . . . . . . 82.1 Resonant Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 SRF Cavity Archetypes . . . . . . . . . . . . . . . . . . . 172.2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 The Superconducting State . . . . . . . . . . . . . . . . . 202.2.2 Surface Resistance Rs . . . . . . . . . . . . . . . . . . . 232.3 Higher Order Modes and Beam-Break Up . . . . . . . . . . . . . 252.3.1 Monopole Modes . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Dipole modes . . . . . . . . . . . . . . . . . . . . . . . . 262.3.3 Beam-Break Up . . . . . . . . . . . . . . . . . . . . . . . 273 Design of the ARIEL Cavity . . . . . . . . . . . . . . . . . . . . 293.1 General Cavity Design Considerations . . . . . . . . . . . . . . . 293.2 ARIEL eLINAC Requirements . . . . . . . . . . . . . . . . . . . 303.3 Eigenmode Simulation Codes . . . . . . . . . . . . . . . . . . . . 313.4 Power Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 HOM Damping Techniques . . . . . . . . . . . . . . . . . . . . . 36Table of Contents v3.5.1 HOM couplers . . . . . . . . . . . . . . . . . . . . . . . . 363.5.2 Beam Line Absorbers . . . . . . . . . . . . . . . . . . . . 383.6 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6.1 Beam Line Absorber Design . . . . . . . . . . . . . . . . 403.6.2 HOM Polarization . . . . . . . . . . . . . . . . . . . . . . 433.6.3 Modifying the Cavity . . . . . . . . . . . . . . . . . . . . 463.6.4 The 39-48 Cavity Variant . . . . . . . . . . . . . . . . . . 493.6.5 Modifying the Inner Cells . . . . . . . . . . . . . . . . . . 523.7 Manufacturing Tolerances Analysis . . . . . . . . . . . . . . . . 533.8 HOM Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.9 HOM Absorber Tests . . . . . . . . . . . . . . . . . . . . . . . . 553.10 Final Design Overview . . . . . . . . . . . . . . . . . . . . . . . 584 Cavity RF Performance Measurements . . . . . . . . . . . . . . 604.1 Performance Limitations . . . . . . . . . . . . . . . . . . . . . . 604.2 Cavity Treatments and Preparations . . . . . . . . . . . . . . . . 644.3 Vertical Performance Measurements . . . . . . . . . . . . . . . . 684.3.1 Single Cell Cavities . . . . . . . . . . . . . . . . . . . . . 704.3.2 Nine-Cell Cavities . . . . . . . . . . . . . . . . . . . . . . 714.3.3 Extracting BCS and Residual Resistance . . . . . . . . . 754.4 Horizontal Performance Tests . . . . . . . . . . . . . . . . . . . 764.4.1 EINJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4.2 EACA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5 Higher Order Mode Measurements . . . . . . . . . . . . . . . . 804.5.1 HOM Beadpulling on a Seven-Cell Model . . . . . . . . . 804.5.2 HOM Frequency and Q Measurements . . . . . . . . . . 854.6 Performance Measurement Summary . . . . . . . . . . . . . . . 885 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A Increased Inner Iris Radius . . . . . . . . . . . . . . . . . . . . . 100B Dipole HOM Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 102C Manufacturing Tolerance Study Data . . . . . . . . . . . . . . . 105D Flatness Measurements . . . . . . . . . . . . . . . . . . . . . . . . 107E BCS Resistance Fitting . . . . . . . . . . . . . . . . . . . . . . . . 110F Beam Based HOM Measurements . . . . . . . . . . . . . . . . . 121viList of Tables1.1 Beam parameters for the RIB [28] and ERL [29] beam. . . . . . . 42.1 Critical fields for niobium at 0 K (in mT/µ0) [41]. . . . . . . . . 233.1 TESLA cavity RF parameters [31]. . . . . . . . . . . . . . . . . . 313.2 Geometric parameters of the three different half-cells of the TESLAcavity. All dimensions are in mm. . . . . . . . . . . . . . . . . . 403.3 48/48 cavity variant. All dimensions are in mm. The end cellsare tuned for field flatness. . . . . . . . . . . . . . . . . . . . . . . 413.4 48/48 variant RF parameters. . . . . . . . . . . . . . . . . . . . . 413.5 48/55 cavity variant. All dimensions are in mm. The end cellsare tuned for field flatness. . . . . . . . . . . . . . . . . . . . . . . 473.6 48/55 variant RF parameters for the operational TM010 mode. . 473.7 48/39 cavity variant. All dimensions are in mm. The end cellsare tuned for field flatness. . . . . . . . . . . . . . . . . . . . . . . 513.8 48/39 variant RF parameters for the operational TM010 mode. . 513.9 If considered resonant with the beam, these six dipole HOMswould dissipate each more than 1 W into the cavity. . . . . . . . 543.10 RF parameters of the ARIEL 39-48 cavity compared to the TESLAcavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59B.1 Dipole HOM mode data for the ARIEL eLINAC cavity. . . . . . 102C.1 Tolerance study data for dipole modes between 1.5 and 3.5 GHz. 105E.1 Results of the fits of the Q vs T data to extract the BCS andresidual resistance. All data have been taken at low gradientsbetween 1 and 3 MV/m. . . . . . . . . . . . . . . . . . . . . . . . 110viiList of Figures1.1 The ISAC facility at TRIUMF. Graphic courtesy of TRIUMF. . 21.2 Five 1.3 GHz SRF cavities are used to accelerate 10 mA e− to-wards a target station to produce rare isotopes. . . . . . . . . . . 31.3 Upgraded with a recirculating beam line and an undulator, theaccelerator can be used as light source for intense infrared lightin addition to the RIB production. . . . . . . . . . . . . . . . . . 51.4 The transverse momentum received in a RF cavity by a dipolemode results in a transverse offset from the beam axis when thebeam returns to the cavity. In green is the movement path of thedeflected bunch. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 The Cornell ERL design reuses the existing half-mile circumfer-ence underground synchrotron tunnel (the blue circle). The lin-ear extensions to the right of the circle are additional tunnelscontaining two superconducting linear accelerators. The arrowsshow positions of x-ray stations. Graphic courtesy of Cornell. . . 72.1 Field distribution in a pillbox cavity for the TM010 mode: (a)magnetic field in X-Y cutplane, (b) electric field in X-Z cutplanewith Z as the beam axis. . . . . . . . . . . . . . . . . . . . . . . . 112.2 Cut-plane views of basic examples of low to medium β cavitieswith the electric (blue) and magnetic fields (red): the quarter-wave resonator (a) can be described as a coaxial transmissionline with one end shorted and the other end open, while thehalf-wave resonator (b) has both ends shorted, resulting in theirname giving field distributions. Spoke cavities (c) look similar toa wheel with one spoke, and have a similar field distribution ashalf-wave cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 The geometry of a half-cell of an elliptical cavity can be describedby seven parameters: the length L is depending on the cavityfrequency and particle velocity β, the equator radius Requatorhas a strong influence on the frequency of the cavity, while theiris radius Riris strongly affects the R/Q. The ellipse parametersare used to optimize the peak fields on the surface. . . . . . . . . 19List of Figures viii2.4 Field Distribution in a single cell elliptical cavity for the TM010mode. The electric fields reach from iris to iris with a strongamplitude on the beam axis while the magnetic field curls aroundin the equator region. . . . . . . . . . . . . . . . . . . . . . . . . 202.5 (a) The magnetization for type I superconductors (assumed as ainfinitely long solid cylinder) rises as the applied field increasesuntil the superconducting state breaks down. (b) Until Hc1 isreached, type II superconductors are in the Meisser-state liketype I superconductors. Between Hc1 and Hc2, it is energeticfavourable to allow flux in the superconductor in a vortex state.Above Hc2 the superconducting state breaks down Hc marks thethermodynamic critical field. . . . . . . . . . . . . . . . . . . . . 212.6 Field distribution in a pillbox cavity of the TM110 mode: (a)magnetic field in the X-Y cut-plane, (b) electric field in the X-Zcut-plane, with Z being the beam axis. . . . . . . . . . . . . . . . 263.1 3D model of a nine cell cavity variant generated with CST MWS. 323.2 Needed generator power as function of beam current, detuningbandwidth and external Q. The minimum is reached, when gen-erator power equals the gained beam power. . . . . . . . . . . . . 343.3 Simulations for the coupler position: (a) The parameter Lc isdefined as the distance between the center of the coupler and thebeginning of the closest cell. (b) Coupling strength as a functionof the distance Lc. Optimal coupling is reached with Qext = 106,resulting in a distance Lc = 43.5mm . . . . . . . . . . . . . . . . 363.4 Shunt impedance spectrum of dipole modes of the TESLA cavity[56, 57]. Several modes are close or above the threshold of 1·107 Ω,notably, a mode around 2.56 GHz. The displayed chart does notinclude all dipole modes in this frequency range as there is noliterature data for the QL of some of the modes. . . . . . . . . . 373.5 HOM coupler used in the TESLA cavity. . . . . . . . . . . . . . . 383.6 Beam line absorber (in green) mating with the cavity (in grey). . 393.7 Several dipole modes of the initial 48/48 cavity variant are signifi-cantly above the HOM limit of 10 MΩ. Some form of Q reductionis needed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.8 The Q of the beam line absorbers rises exponentially with thedistance from the cavity. z is the distance between the iris of theend half-cell and the start of the absorber. The integration overthe H-field to calculate the power loss extends far into the distance. 423.9 The effective length of the absorber is independent of the conduc-tivity. A length of 60 mm absorber length is required to saturatethe Q for the TM010-pi mode. The Q for CESiC is higher com-pared to the SS due to the different positions of the absorberswith respect to the beam pipe. . . . . . . . . . . . . . . . . . . . 43List of Figures ix3.10 The used CESiC absorbers on the beam pipes only affect thefew very high shunt impedance modes. The 2.56 GHz mode isuntouched at 2 · 107 Ω. . . . . . . . . . . . . . . . . . . . . . . . . 443.11 The TM111 mode around 2.56 GHz is trapped inside the cavity:no significant fields in the region of the beam line absorbers. . . . 443.12 A different polarization of the same mode results in a slightlydifferent shunt impedance. The vertical polarized mode haveoverall a higher shunt impedance compared to the horizontalmodes, therefore this polarization is considered more dangerousand taken as baseline in the following simulations. . . . . . . . . 453.13 The different polarizations of one mode couple differently to thefundamental power couplers. . . . . . . . . . . . . . . . . . . . . 453.14 Dipole spectrum of the 48/55 cavity variant with and withoutbeam line absorbers. Most data points overlap, showing no effectof the beam line absorbers for those modes. The 2.56 GHz modegets reduced in shunt impedance from 108 Ω to just below 107 Ω. 463.15 The shunt impedance of the accelerating TM010 mode decreasesas the pick-up side beam pipe and end-cell iris increases, makinga smaller iris desirable. . . . . . . . . . . . . . . . . . . . . . . . . 483.16 The geometric shunt impedance of the dangerous 2.56 GHz modeslightly decreases with increasing the beam pipe radius. . . . . . 493.17 The trapping parameter K for the 2.56 GHz mode shows no cleardependence on the pick-up side end group. . . . . . . . . . . . . . 503.18 The frequency also changes only minimally while changing thepick-up side end group. . . . . . . . . . . . . . . . . . . . . . . . 513.19 The combined absorber Q is dominated by the coupler side ab-sorber for a fixed position of the absorbers. . . . . . . . . . . . . 523.20 HOM spectrum for the 39/48 cavity variant. A strong suppres-sion of all HOM up to 4.4 GHz can be found when using CESiCand stainless steel absorbers.(make legend bigger) . . . . . . . . . 533.21 Sensitivity of HOM parameters f , Q and Rd/Q to imperfectionsin the manufacturing process. On the X-axis is plotted the aver-age HOM frequency and on the Y-axis the sensitivity parameterσx/x¯. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.22 Sensitivity of HOM parameters f , Q and Rd/Q vs average HOMfrequency for selected cavities with high shunt impedance of theaccelerating TM010 mode. . . . . . . . . . . . . . . . . . . . . . . 553.23 Simulation results for three different electrical conductivities tomatch the room temperature Q measurements of the TM010mode. The center curve with σ = 4500 S/m is the best fit tothe data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56List of Figures x3.24 (a) A 1.3 GHz Niobium Cavity used for this test. The cavity issuspended from the top and a copper cooling line attached aroundthe beam pipe. Temperature sensors are located at the top andbottom flange. (b) Schematic of the assembly: the sample restson the sample holder and is centered by it. The input antenna isinserted from the top and the pick-up antenna from the bottom.The top flange has an additional hole to evacuate the RF space. 573.25 At a fixed position, the sample conductivity increases exponen-tially with a increasing ratio of QL/Q0. The plot for 3 and 7 mmcorrespond to the uncertainties in the position measurement ofthe sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.26 Cross-section through the final mechanical model of the ARIEL1.3 GHz SRF cavity. . . . . . . . . . . . . . . . . . . . . . . . . . 584.1 The cavity performance on the peak magnetic surface fields islimited by several effects and influenced by the surface treatment[not to scale]: a) at low surface fields a small increase in Q0 canbe measured (LFQS); b) at surface fields between 20 and 100 mTa gentle decrease in Q0 takes place; c) at higher gradients a sharpdrop in Q0 can be observerd (HFQS); d) HFQS can be cured witha 120◦C UHV bake; e) newer treatments like nitrogen dopingshow a higher Q0 with higher gradients but seem to be limitedto medium gradients before a quench occurs; f) multipacting canbe overcome with conditioning; g) Q-disease causes a strong Qslope at low to medium gradients; h) defects cause a very low Q0and an early quench. . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Left: a multi-cell cavity is etched with hydroflouric acid in aprocess called buffered chemical polishing (BCP). Right: A singlecell cavity is being cleaned with high pressure water. . . . . . . . 654.3 The multi-cell tuning station adjusts the frequency and flatnessto the required values. . . . . . . . . . . . . . . . . . . . . . . . . 664.4 The initial flatness and frequency tuning of ARIEL1 before BCPtreatment increased the flatness from 90 % to 98 % and changedthe frequency to be within the goal range. . . . . . . . . . . . . . 674.5 The 120 µm BCP removal caused the flatness to decrease to89.9 %, which could be recovered to 98.9 %. . . . . . . . . . . . . 674.6 During a RF cold test of a cavity, a low level RF system con-trols the frequency of the RF signal sent to an amplifier. A bi-directional coupler extracts the incident and reflected power, anda probe on the other side of the cavity picks up the transmittedsignal. The voltage, power and frequency of this signal is ana-lyzed and sent to the amplifier to keep the RF signal matched tothe resonant frequency of the cavity, creating a self-exited loop. . 70List of Figures xi4.7 The performance of the second single cell cavity at 2 K mea-sured at Fermilab and TRIUMF showed the same high Q0 upto 25 MV/m when a high field Q drop occurs, proving the cleanroom procedures at TRIUMF are up to standards to assemblecavities free of field emissions. . . . . . . . . . . . . . . . . . . . . 714.8 The coupling range of the multi-cell coupler allows for criticalcoupling at 4.2 and 2 K. . . . . . . . . . . . . . . . . . . . . . . . 724.9 Q0 performance measurements of ARIEL1 at 2 K showed a clearimprovement from the initial tests of ARIEL1 to the degassedcavity. The performance measurement is limited to Pcav ∼ 25 Wdue to limitations of the cryo system. . . . . . . . . . . . . . . . . 734.10 An in-situ 120◦ C bake-out step was attempted for ARIEL2 toraise the Q0 after the initial measurement. Insufficient heaterpower prevented this and the performance did not change. Afterdegassing high gradients were reached with a slightly reduced Q0.The Q0 recovered after a 120◦ C bake and HF rinse. The perfor-mance measurement is limited to Pcav ∼ 25 W due to limitationsof the cryo system. . . . . . . . . . . . . . . . . . . . . . . . . . . 734.11 The performance of ARIEL3 recovered after a defect caused a lowQ0 and field emissions during the first vertical test. Subsequentprocessing steps increased the performance further. . . . . . . . . 744.12 Testing at 2.3 K prevented a super-fluid leak on the cavity. The2 K result is calculated by using the difference in BCS resistanceshown in fig. E.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.13 Performance test of the fourth multicell cavity for ARIEL at 4.2and 2 K. This cavity will be the second cavity in the first accel-erating cryomodule EACA. . . . . . . . . . . . . . . . . . . . . . 764.14 Extracting the BCS and residual resistance with Q0 vs T datafor ARIEL1 after degassing. (a): The temperature dependentBCS resistance is linearized and the data is fitted to the naturallogarithm of Rs − Rres. (b): The fitted BCS resistance (greenline) is subtracted from the Rs data (red markers) to reveal theresidual resistance Rres (blue markers). The uncertainties givenin RBCS and Rres follow the uncertainties in the fit parameters. 774.15 The injector cryomodule fully assembled and installed on thebeamline with RF wave-guides, cryogenic supply and return linesand all other diagnostics connected to it. . . . . . . . . . . . . . . 794.16 The calibration of the pick-up voltage is done by measuring thebeam energy after the cryomodule. Shown are four different beamenergy measurements at different pick-up voltages. The measure-ment is done by sweeping the current through a dipole magnetafter the cryomodule and a Faraday cup reads out the beam in-tensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.17 The RF performance of the injector was limited by field emissionsto a maximum gradient of 5 MV/m. The Q0 is at the same levelas the vertical test at low gradients before the field emissions start. 81List of Figures xii4.18 Visible damage on the beam pipes was observed after the cavityfor the ICM was opened. . . . . . . . . . . . . . . . . . . . . . . . 814.19 After an additional BCP treatment on the EINJ cavity to repairdamage on the RF surface the performance reached Q0 = 1010up to 12 MV/m. The gradient was limited by the multipactingin the couplers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.20 The EACA cavity 1 reached the performance specification ofQ0 = 1 · 1010 at 10 MV/m. The increase in Q0 compared tothe vertical test is a result of 20 µm surface removal via BCP. . 824.21 The seven-cell copper model cuts two inner cells from the nine-cellcavity and is used to confirm the HOM simulations. . . . . . . . 834.22 The mode number of the simulation results have been adjusted tocorrespond with the measured frequencies. While the simulationcan be force into dipole modes of a particular polarization, themeasurements account for all modes including monopole modes. . 844.23 Bead pull measurements on four HOMs with different azimuthalpositions of the string. . . . . . . . . . . . . . . . . . . . . . . . . 854.24 The new multi-line beadpulling setup produces a clean azimuthaldependence of the field amplitude of a HOM at 2498 MHz, whichis now identified as a dipole mode. . . . . . . . . . . . . . . . . . 864.25 Measurement (left) and simulation (right) show a very similarlongitudinal field distribution of the electric field of this particularmode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.26 Frequency measurements on the EINJ cavity at 2 K show a goodmatch to simulated frequencies of dipole modes. . . . . . . . . . . 874.27 The transmission signal of the accelerating mode is fitted to aLorentz function. A QL of 2.6 ·106±13% corresponds well to theset value of 3 · 106. . . . . . . . . . . . . . . . . . . . . . . . . . . 884.28 Q fitting results for HOMs in the EINJ and EACA cavity 1 upto 2.5 GHz compared to simulation results. . . . . . . . . . . . . 895.1 The performance of the cavities in the cryomodules meets thespecifications of Q0 ≥ 1 · 1010 at Eacc = 10 MV/m. . . . . . . . . 91A.1 The R/Q of the accelerating mode decreases with increased irisapertures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.2 The shunt impedance of the 2.56 GHz mode generally follows asimilar trend like the accelerating mode. . . . . . . . . . . . . . . 101D.1 ARIEL1 flatness tuning results. . . . . . . . . . . . . . . . . . . . 107D.2 ARIEL2 flatness tuning results. . . . . . . . . . . . . . . . . . . . 108D.3 ARIEL3 flatness tuning results. . . . . . . . . . . . . . . . . . . . 108D.4 ARIEL4 flatness tuning results. . . . . . . . . . . . . . . . . . . . 109D.5 Frequency tuner simulation on the warm tuning stand revealedthat the flatness changes are minimal (within 1%-point). . . . . . 109List of Figures xiiiE.1 BCS fit for ARIEL1 after 120 µm BCP. (a) RBCS fit, (b) Rres fit. 111E.2 BCS fit for ARIEL2 after 120 µm BCP. (a) RBCS fit, (b) Rres fit. 112E.3 BCS fit for ARIEL2 after 85◦ C baking for 48h. (a) RBCS fit,(b) Rres fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113E.4 BCS fit for ARIEL2 after degassing. (a) RBCS fit, (b) Rres fit. . 114E.5 BCS fit for ARIEL2 after 120◦ C baking for 48h and an HF rinse.(a) RBCS fit, (b) Rres fit. . . . . . . . . . . . . . . . . . . . . . . 115E.6 BCS fit for ARIEL3 after 120 µm BCP. (a) RBCS fit, (b) Rres fit. 116E.7 BCS fit for ARIEL3 after additional 30 µm BCP. (a) RBCS fit,(b) Rres fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117E.8 BCS fit for ARIEL3 after additional 60 µm BCP. (a) RBCS fit,(b) Rres fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118E.9 Fit of Q0 vs T data to extract the BCS resistance for ARIEL3to extrapolate a performance test from 2.3 K to 2.0 K. (a) BCSresistance fit, (b) residual resistance fit. . . . . . . . . . . . . . . 119E.10 BCS fit for ARIEL4 after 120 µm BCP. (a) RBCS fit, (b) Rres fit. 120F.1 The envelope of the BPM reading after the beam is kicked bythe 2.57 GHz mode corresponds to the QL of the HOM whilethe maximum kick is proportional to the R/Q. Calculations arebased on a 3 MeV beam with Qb0 = 10 pC, δx0 = 10 mm and adrift length of 3.5 m. The HOM is simulated with a R/Q of 76 Ωand a QL of 105. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123xivList of Abbreviations andUnitsAbbreviationsRF Radio-FrequencySRF Superconducting Radio-FrequencyBCS Bardeen, Cooper, Schrieffersc superconductingnc normal conductingHOM Higher Order ModeARIEL Advanced Rare IsotopE LaboratoryRIB Rare Isotope BeamLINAC LINear ACceleratoreLINAC electron LINACICM Injector CryoModuleACM Accelerating CryoModuleEINJ eLINAC codename for the ICMEACA eLINAC codename for the first ACMISAC Isotope Separator and ACceleratorFEL Free Electron LaserBBU Beam-break upERL Energy Recovery LINACTTF Transit Time FactorTE Transverse ElectricTM Transverse MagneticMWS MicroWave StudioACE3P Advanced Computational Electromagnetic 3D ParallelTESLA Tera-electron volt Energy Superconducting Linear AcceleratorLN2 Liquid NitrogenLHe Liquid HeliumCESiC Carbon-fiber rEinforced Silicon CarbideSS Stainless SteelLFQS Low Field Q SlopeList of Abbreviations and Units xvMFQS Medium Field Q SlopeHFQS High Field Q SlopeBCP Buffered Chemical PolishingEP ElectroiPolishingHF Hydro-Flurid acidHPR High Pressure water RinsingNb NiobiumNbTi Niobium-TitaniumFNAL Fermi National Accelerator LaboratoryBPM Beam Position MonitorUnitsHz HertzW WattV VoltA AmpereJ Joules secondsΩ OhmS Siemensm MeterK KelvinC CelsiusT TeslaxviAcknowledgmentsI acknowledge that UBC and TRIUMF are seated on the traditional, ancestraland unceded territory of the Musqueam nation.I would like to take the opportunity to express my gratitude towards the peoplewho made my time in Vancouver and at TRIUMF enjoyable and who supportedme in many ways during my time here. I’ve met many great people here, whoinfluenced me for the rest of my life.First and foremost, I want to thank my supervisor, Bob Laxdal, who took meinto his team and guided me in my research. I started without any experiencein the SRF field and have learned so much from you, theoretical and practical,that it would be futile to list everything. Thank you for all the time discussingall the different topics and having an open door for me.Thanks go to Vladimir Zvyagintsev, David Longuevergne, Zhongyuang Yao andYanyun Ma for their expertise and discussions on SRF topics and Tobias Jungin-ger for his expertise in superconductivity and his help in revising the pages onthis topic in this thesis.Bhalwinder Waraich helped a lot with all the clean room work and assemblingcavities for cold tests and has my admiration for his knowledge and attitude.Thanks go the the mechanical team consisting of Peter Harmer, Devon Lang,James Keir, Ryan Smith, Clint Laforge and Christoph Schaub. Thank you all,for fixing all the equipment I broke and all the mechanical help over the years.Without the cryogenics team, David Kishi and Howard Liu, non of the coldmeasurements were possible. Thank you for delivering the countless liters ofliquid helium and the long cavity test days.I want to thank Ramona Leewe, for your friendship and everything. I’ll neverforget our time after the final exam in my first year at TRIUMF.Thanks go my dear friend Anna Grassellino. Thank you for all you have donefor me, especially including my into your circle of friends who over the yearshave become good friends to me as well.All the people I’ve met outside and inside TRIUMF, but not worked directlywith, I want to thank you for your friendship: Kelvin Lee, Sun Nee Tan, AndreaTeigelhfer, Doug Storey, Ania Kwiatkowski, Vanessa Chen and everyone else Idid to mention. You made living here a pleasure and I’m glad to have met youthrough proximity, common friends or just pure chance.To my sisters Johanna and Sarah, thank you for your unending support andcare packages.Last, but not least, I would like to thank my parents, Hedwig and Burkhard,who gave me all the support I needed to reach this point in my life.Acknowledgments xviiThanks to all of you and many more, I am the person I am today and I’m veryhappy to have met and spend time with all.xviiiTo my parents and sisters, for their never-ending support.1Chapter 1Introduction1.1 Particle Accelerators and Modern PhysicsPhysics research and applications have and will continue to use particle accel-erators as an important tool. There are many examples of this in both purescience and applied areas. In particle physics, many of the most importantdiscoveries have been directly the result of improvements in accelerator tech-nology. One recent example is the discovery [6, 7, 8] of the Higgs particle atthe Large Hadron Collider (LHC) at CERN. In contrast to the pure sciencedone at CERN, cancer therapy with proton or heavier ion beams has becomean established treatment, that successfully treats patients in a less harmful waythan conventional radiation based therapy or chemotherapy. One example ofsuch a dedicated treatment center is the Heidelberg Ion-Beam Therapy CenterHIT [9] in Germany. Both these accomplishments have become become possibleby the development of new and better particle accelerators.Particle accelerators are used extensively to generate intense beams of x-rays[10, 11], neutrons [12, 13, 14] and muons [15] to study a wide range of sub-jects. In the particle physics, the International Linear Collider (ILC) [16] willbe instrumental in high precision measurements on the recently discovered Higgsboson. This will hopefully lead to the discovery of new physics beyond the cur-rent standard model of physics.The applications of particle accelerators are not limited for scientific or medicalpurposes. A potential world changing application for particle accelerators couldbe in the energy sector. Sub-critical nuclear reactors can become critical bythe use of high energy particle beams and then generate useful energy out ofnuclear fission in so called accelerator driven systems [17]. And particle accel-erators could be used to transmute the highly radioactive waste products fromnuclear power plants [18], creating shorter lived isotopes, that are less dangerousto the environment.Another aspect, where particle accelerators are of significant importance, is rareisotope science. The rare isotope program ISAC (Isotope Separator and ACcel-erator [19], facility shown in fig. 1.1) at TRIUMF relies on accelerators to createexotic isotopes, which cannot be found otherwise. These isotopes are then usedin a number of different experiments. At TRIUMF, by shooting an energeticproton beam on a target, many unstable isotopes are created. In experimentslike TITAN [20] and β-NMR [21] as examples, these rare isotopes are eitherstudied or used as a probe. The TITAN experiment performs high precisionmass and half-life measurements to make conclusions about the nuclear struc-1.2. ARIEL 2Figure 1.1: The ISAC facility at TRIUMF. Graphic courtesy of TRIUMF.ture. β-NMR uses spin-polarized 8Li ions from ISAC as a probe to measurelocal magnetism at interfaces and surfaces for condensed matter studies. Thoseare just two examples of the experiments that can be done with rare isotopes.Around the world, there are a number of facilities besides ISAC, which alreadyproduce rare isotopes or are under construction. Two other examples besidesISAC at TRIUMF are the RIKEN Rare Isotope Beam Factory (RIBF) in Japan[22] and the Facility for Rare Isotope Beams (FRIB) at MSU, Michigan [23].All those experiments and many more are enabled by the use of particle ac-celerators. Therefore, the advancement and development of new accelerators isessential to increase the available beam time for experiments, which is a verylimiting factor.1.2 ARIELARIEL (Advanced Rare IsotopE Laboratory) [24] complements the existingrare isotope program at TRIUMF. Limitations in available beam time for ex-periments from the existing ISAC facility demonstrate a need for an expandedrare isotope facility. Presently, the 500 MeV proton beam from the TRIUMFcyclotron bombards a target to produce rare isotope beams (RIB) that are de-livered to one of three experimental areas, characterized by the energy of there-accelerated RIB. With ARIEL all three experimental areas may receive beamsimultaneously with the potential to triple the scientific output of the TRIUMFRIB program.The core of ARIEL is a 500 kW electron accelerator, the eLINAC. The elec-tron beam hits a converter target to create photons which in turn bombard an1.2. ARIEL 3E-GunEINJEACA EACB to RIB target station500kW beam powerproduces 10mA 300keV e- beam, 650MHz5 TTF style cavities at 1.3GHzup to 10MeV/cavity gainFigure 1.2: Five 1.3 GHz SRF cavities are used to accelerate 10 mA e− towardsa target station to produce rare isotopes.actinide target to produce radioactive species through photo-fission [25]. Theproduction cross-section of this process starts to saturate at an electron energyof 50 MeV. While the energy required is relatively low, the photo-fission processis relatively inefficient so a high intensity of electrons is required to yield com-petitive RIB rates. A current of 10 mA is required to produce approximately2 · 1013 fissions per second. This process also allows for creation of previouslyinaccessible neutron rich isotopes.To accelerate the electrons, superconducting radio-frequency (SRF) cavities areused. Superconductivity allows for efficient acceleration with low RF powerlosses due to the lower RF surface resistance as compared to room temperaturecavities (10’s of nΩ vs a few mΩ). The disadvantage of SCRF technology is theneed for cryogenic systems and an overall more complicated technical design,but the overall reduction in electric power is substantial, especially for continu-ous wave (cw) operation like in the eLINAC.A schematic of the layout of the eLINAC is shown in fig. 1.2. The beam startsat a 300 kV thermionic electron gun. An RF modulation at 650 MHz is ap-plied to a DC biased grid at the cathode. This RF modulation provides thebunch structure of the beam, that is needed for RF acceleration. After a lowenergy beam transport section, the beam is accelerated in the injector cryomod-ule (ICM) with one nine-cell SRF cavity operating at 1.3 GHz and providing10 MV of accelerating voltage to the beam. The voltage is limited to 10 MVto keep the beam loaded power requirement at a beam current of 10 mA to100 kW. After the ICM follows the medium energy beam transport. The finalacceleration to 50 MeV is done in two accelerating cryomodules (ACM). EachACM houses two identical nine-cell cavities. After the beam reaches its finalenergy it is then guided towards the RIB target station, where the photo-fissionprocess takes place. The created rare isotopes are then delivered to an experi-ment in the ISAC facility, for example to β-NMR.Future upgrade plans for the eLINAC include a recirculating arc (shown infig. 1.3), which takes the accelerated beam after the second ACM and guidesit back to the first ACM. The recirculating ring can be used for either furtherincreasing the beam energy as a variable for RIB production development, orfor an energy recovery LINAC based free electron LASER (FEL) with a highbunch charge beam. During the creation of the FEL light the energy of the1.2. ARIEL 4Table 1.1: Beam parameters for the RIB [28] and ERL [29] beam.Parameter RIB Beam ERL BeamBunch charge q [pC] 16 100Bunch length σz [mm] 3.3 0.3Bunch repetition frequency [MHz] 650 100beam is reduced by only a few %. The remaining energy of the beam can begiven back to the SRF cavities by decelerating the ERL beam. This can be doneby injecting the beam with a phase shift of pi from the accelerating RF phase.Since the bunch frequency and the resonant frequency of the accelerating modematch, the beam interacts strongly with this mode. Due to this phase shift theelectron bunch of the ERL beam passes through a decelerating voltage and theenergy lost by the beam is converted into the electromagnetic fields of the ac-celerating/decelerating mode. With this energy recovery operation the neededRF power to accelerate the intense electron beam is reduced significantly. A de-tailed design and optimization of the recriculating beam line and the ERL withrespect to the beam transport can be found in [26]. It is intended that both RIBand ERL operation run in parallel with two different beams in the accelerator.In the combined mode, with the RIB and ERL beams running simultaneously,the RIB beam gets sent to the photo-fission production target while the ERLbeam gets sent to a low energy beam dump after the deceleration. A RF dipolecavity [27] operating at 650 MHz is used for the separation of the various beams- the RIB beam to the target station, the accelerated ERL beam to the recicu-lating arc and the decelerated ERL beam to the low energy beam dump.Since the RIB beam is operating at 650 MHz, only every second RF periodaccelerates beam. The remaining periods (also called RF buckets) can be usedfor a second beam, the ERL beam, which is estimated to operate at a bunchrepetition frequency of 100 MHz. Other differences in the baseline design ofthe two beams are listed in tab. 1.1. Noteworthy is the significantly increasedbunch charge in the ERL beam. This increased bunch charge is used to increasethe intensity of the resulting Laser beam. Using a particle accelerator with es-sentially two different beams for two different purposes is unique and creates itsown challenges and requirements.In a similar way that a passing beam transfers energy to the acceleratingmode in ERL operation, it is possible to transfer energy to any of the infinitenumber of resonant eigenmodes in a RF cavity. Other modes than the accelerat-ing/decelerating mode are called higher order modes (HOM) and have negativeinfluence on the beam. For example dipole modes have a deflecting field, thatgives a particle bunch an unwanted transverse momentum. In a recirculatingLINAC, this additional transverse momentum will result in an offset from thebeam axis when the bunch returns to the RF cavity (see fig. 1.4). Since theinteraction between a dipole mode and beam is stronger for bigger offsets, moreenergy is transferred between the beam and the electromagnetic fields in thecavity. This can lead to an instability when the deflection becomes so strong1.3. Similar Projects 5RF separatorBunch compression chicaneERL beam dumpFEL UndulatorN S N S N S N S N S N S N SNS N S N S N S N S N S N SE-GunEINJEACA EACBto RIB target station500kW beam powerproduces 10mA 300keV e- beam, 650MHz5 TTF style cavities at 1.3GHzup to 10MeV/cavity gainFigure 1.3: Upgraded with a recirculating beam line and an undulator, theaccelerator can be used as light source for intense infrared light in addition tothe RIB production.that the beam cannot be guided through the accelerator anymore and hits thebeam pipe. This is called beam-break up (BBU) [30]. A more detailed descrip-tion of the phenomenon is given in chapter 2.3.BBU depends on a number of parameters related to the beam, the cavity HOMand the beam transport back to the cavity. A higher beam current or a highershunt impedance of the HOM leads to a stronger excitement of the dipole mode.The beam-transport matrix and the time it takes to recirculate the beam alsoinfluence the BBU threshold. If the frequency of a HOM matches a harmonic ofthe bunch repetition rate, large amounts of power can be transferred betweenthe beam and the fields, resulting in large cryogenic loads, overloading the cryo-genic system. The effects of HOMs on the beam and the cryogenic load haveto be evaluated and the design of the RF cavity for ARIEL has to reflect theseconcerns.For the eLINAC with a design current of 10 mA, a BBU design threshold cur-rent of 20 mA is chosen. This dissertation will first introduce the fundamentalsof RF cavities in chapter 2 and then discuss the design of the SRF cavity forthe ARIEL eLINAC under consideration of HOMs and their mitigation in chap-ter 3. Vertical performance measurements of the fundamental mode in a testcryostat and horizontal measurements in the final cryomodule as well as HOMmeasurements are presented in chapter 4.1.3 Similar ProjectsThere are many new accelerator projects being developed globally. Some involveERL accelerators with similar challenges as the ARIEL eLINAC.Cornell University is building an ERL with stricter HOM requirements thanARIEL: instead of 10 mA the design current is 100 mA with an energy up to5 GeV. This leads to stricter requirements on the cavity with respect to HOMs.Beam line absorbers are used to reduce the quality factor Q of HOMs. In1.3. Similar Projects 6Beam trajectoryElectric field Magnetic fieldSRF cavityFigure 1.4: The transverse momentum received in a RF cavity by a dipole moderesults in a transverse offset from the beam axis when the beam returns to thecavity. In green is the movement path of the deflected bunch.addition, the cavity shape has been heavily modified away from the standardTESLA design [31]: a reduction from nine to seven cells and modifications to thecell shape (bigger iris, bigger beam pipes) reduce dipole mode shunt impedances,but at the cost of reduced performance for the accelerating mode (lower shuntimpedance, lower accelerating length) [32].KEK in Japan also wants to build a 5 GeV, 10 to 100 mA ERL to drive anx-ray free-electron laser [33]. A compact ERL (cERL) will function as a scaleddemonstration (60 to 200 MeV) for this electron accelerator. The main cavityfor the cERL is a heavily customized nine cell cavity with unique features likeeccentric fluted beam pipes to target quadrupole modes. Increasing the iris andbeam pipe diameters help decrease the dipole shunt impedances and are requiredfor this very high beam intensity [34]. In addition to changing the cavity shape,HOM beam line absorbers made out of a ferrite are used to reduce the Q ofHOMs [35]. No HOM couplers are used in this cavity while the TESLA cavityuses two dedicated HOM couplers.BERLinPRO [36] in Germany is comparable in energy (50 MeV) to the ARIELeLINAC, but accelerates a higher beam current of 100 mA. This acceleratorwill be used for FEL excitation in contrast to ARIEL where the main purposeis rare isotope production and FEL operation is a future upgrade plan. Again,due to the high beam current a different cavity will be used than in ARIEL.During the design, two cavity geometries were considered for BERLinPRO:either the JLAB 1.5 GHz 5-cell design scaled to 1.3 GHz, or the Cornell 7-cell 1.3 GHz cavity. Because of higher gradient requirements the choice wentto the 7-cell Cornell cavity, due to a better peak field ratios than the JLABcavity. To fit the specifications, the Cornell cavity will be further modified to usewave-guide couplers, which act in a similar way to the more traditional coaxialHOM couplers. These wave-guides extract HOM power out of the cavity and1.3. Similar Projects 7Figure 1.5: The Cornell ERL design reuses the existing half-mile circumferenceunderground synchrotron tunnel (the blue circle). The linear extensions to theright of the circle are additional tunnels containing two superconducting linearaccelerators. The arrows show positions of x-ray stations. Graphic courtesy ofCornell.guide it to a room temperature load. A benefit of wave-guides over traditionalHOM couplers is that they can naturally reject the accelerating mode, whiletraditional HOM couplers need a frequency filter to avoid a negative influenceon the accelerating mode. A challenge lies in preventing additional heat load onthe cryosystem due to heat transfer from room temperature to the 2 K cooledcavity through the wave-guides.While these projects have similar HOM requirements because of their ERLapplication, the ARIEL eLINAC is unique in its way of using a complicatedbeam structure through one accelerator for RIB production and simultaneouslydriving a FEL based light source with energy recovery. This creates a uniqueset of requirements that the SRF cavity has to fulfill which will be discussed inthis dissertation.8Chapter 2Particle Acceleration in RFCavitiesParticle accelerators use RF cavities to transfer energy from a source to thebeam. These cavities resonate at a certain frequency with a certain field dis-tribution, that is determined by the shape of the cavity. In this chapter, thefoundations of RF cavities are described in section 2.1, as well as the basics ofsuperconductivity needed for SRF cavities can be found in section 2.2 and theinteractions between the beam and the electromagnetic fields are then describedin section 2.3.2.1 Resonant CavitiesIn a similar way that a guitar string responds to excitement with a specific pitch,a RF cavity responds to a driving force of a specific frequency with a specificelectric-magnetic field distribution, which can be used to accelerate chargedparticles. Just like the tension and length of the string in a guitar, the boundaryconditions determine the resonance frequency and field distribution inside thecavity. In the case of RF cavities, the boundary conditions are set by the shapeof the cavity itself. In the following, the basics of RF cavity eigenmodes areshown in section 2.1.1, the characteristic figures of merit to describe RF cavitiesare presented in section 2.1.2 and a short description of various types of cavitiescan be found in section 2.1.3.2.1.1 EigenmodesTo accelerate charged particles, a electric field is created inside a RF cavity.The passing particle experiences an accelerating force according to the LorentzforceF = q · (E+ v×B), (2.1)where q is the charge of the particle, E and B the electric and magnetic fieldsinside the cavity and v the velocity of the particle in question. The fields arecreated by using resonant cavities, that are designed in a way to create a strongelectric field along the beam axis.2.1. Resonant Cavities 9Solving Maxwell’s equations in empty space∇ ·E = 0 (2.2)∇×E = −∂B∂t(2.3)∇ ·B = 0 (2.4)∇×B = 1c2∂E∂t(2.5)leads to the wave equations for the electric and magnetic fields∇2E− 1c2∂2E∂t2= 0 (2.6)∇2B− 1c2∂2B∂t2= 0 (2.7)with c being the speed of light in vacuum. The solution to the wave equationsare plane waves [37], here written in cylindrical coordinatesE(r, t) = E(r, φ)ei(kz−ωt) (2.8)B(r, t) = B(r, φ)ei(kz−ωt). (2.9)The electric conducting walls in a cavity set the boundary conditions for thefields. The electric field has to be perpendicular to the surface and the magneticfield has to be parallel, which can be expressed with the normal vector n asn×E = 0, n ·B = 0. (2.10)These boundary conditions create the environment for standing waves to formin cavities at certain resonant frequencies ω0, where the stored energy inside thecavity is oscillating between the electric and the magnetic field back and forth,equivalently to a LC circuit. The simplest of RF cavities, the so called pillboxcavity a hollow cylinder with radius R and height L, can be derived from sucha LC circuit. The end plates represent the capacitive element with the electricfield between them. The side wall connecting the end plates can be seen as theinductive element with the magnetic field curling parallel to the side wall.The resulting eigenmodes are categorized into three sets: transverse magnetic(TM) or E-modes, transverse electric (TE) or H-modes and transverse electric-magnetic (TEM) modes. The field components for a pillbox cavity are given inequations 2.11 through 2.16 for TM modes and 2.17 through 2.22 for TE modes.2.1. Resonant Cavities 10Transverse magnetic (TM) modes:Ez = E0Jm(xmnrR)cos(mφ) cos(ppizL)eiω0t (2.11)Er = − ppiRLxmnE0J′m(xmnrR)cos(mφ) sin(ppizL)eiω0t (2.12)Eφ = −ppimR2Lx2mnrE0J′m(xmnrR)sin(mφ) sin(ppizL)eiω0t (2.13)Bz = 0 (2.14)Br = iωmR2√µ0/0x2mnrc2E0J′m(xmnrR)sin(mφ) cos(ppizL)eiω0t (2.15)Bφ = iωR√µ0/0xmnc2E0J′m(xmnrR)cos(mφ) cos(ppizL)eiω0t (2.16)Transverse electric (TE) modes:Bz = B0Jm(x′mnrR)cos(mφ) sin(ppizL)eiω0t (2.17)Br =ppiRLx′mnB0J′m(x′mnrR)cos(mφ) cos(ppizL)eiω0t (2.18)Bφ =ppimR2Lx′2mnrB0J′m(x′mnrR)sin(mφ) cos(ppizL)eiω0t (2.19)Ez = 0 (2.20)Er = −iω√µ0/0mR2x′2mnrB0J′m(x′mnrR)sin(mφ) sin(ppizL)eiω0t (2.21)Eφ = −iω√µ0/0RxmnB0J′m(x′mnrR)cos(mφ) sin(ppizL)eiω0t (2.22)with ω0 as the resonance frequency, Jm the m-th Bessel function with its deriva-tive J ′m and xmn/x′mn the n-th root of Jm/J′m. E0 andB0 describe the respectiveamplitude of the electric and magnetic field. The integers m,n and p along withthe categories TM or TE classify the modes and stand for the number of nodesof the corresponding field in the intervals 0 ≤ φ < pi and 0 ≤ r ≤ R and thenumber of half-waves in the interval from z = 0 to z = L. Modes with m = 0have no variation along φ and are called monopole modes (an example is shownin fig. 2.1). m = 1 are dipole modes, m = 2 quardupole modes and so on. Forperfectly axial-symmetric cavities, no preference for the polarization of the non-monopole modes is set. In reality, cavity imperfections and ancillaries to thecavity like power couplers break this polarization and two orthogonal polarizeddipole modes exist with a small separation in frequency.Resonant cavities can be described as harmonic oscillators with an externaldriving force. The stored energy in the cavity is transferred between the E andB fields. This transfer is realized by a phase shift of 90◦ between E and B.While one field is at its maximum amplitude, the other field’s amplitude is zero.2.1. Resonant Cavities 11B-Field(a) E-Field(b)Figure 2.1: Field distribution in a pillbox cavity for the TM010 mode: (a)magnetic field in X-Y cutplane, (b) electric field in X-Z cutplane with Z as thebeam axis.Once the driving force, in this case a signal generator tuned to the resonancefrequency of the cavity, is turned off, the stored energy decays exponentially dueto losses in the cavity walls. In the time domain, the fields inside the cavity showthis exponential decay superimposed on the sinusoidal oscillations. When thissignal is transformed into the frequency domain via a Fourier transformation,this signal is represented by a Lorentz function where the width of the curvecorresponds directly to the decay constant and to the losses on the cavity walls.A more detailed description follows in chap. 2.1.2.From eq. 2.20 it is clear that TE modes are not useful for particle accelerationsince there is no longitudinal electric field in those modes. TM modes are neededto create the needed longitudinal field. Figure 2.1 shows the fundamental TMmode of a pillbox cavity with the strongest electric field along the center of thecavity from one end to another while the magnetic field curls around the sidewall. The directions of the fields change signs back and forth with the resonantfrequency of this mode.For the wave numbers k the following relationships are setk2x + k2y + k2z = k2 (2.23)k2 = k2c + k2z (2.24)For standing wave TM modes in a pillbox cavity, where L is aligned with thez-axis, the Eφ and Er field components have to vanish at z = 0 and z = L. Thissets the condition for p:kz =ppiL, p = 0, 1, 2 . . . (2.25)2.1. Resonant Cavities 12The same condition works for TE modes.The resonant frequency of the modes is given by formulas 2.26 for TM modesand 2.27 for TE modes.fTM = c ·√(xmn2piR)2+14( pL)2(2.26)fTE = c ·√(x′mn2piR)2+14( pL)2(2.27)Since there are no limits on m,n and p besides their integer nature, it is easy tosee that there is an infinite number of modes in a cavity. More complex cavitiesthan the pillbox cavity follow the same boundary conditions but cannot be cal-culated analytically anymore and numerical codes have to be used to calculatethe field distributions.2.1.2 Figures of MeritA number of important figures of merit describe each cavity in terms of suitabil-ity for particle accelerators. In this section, the most important and commonlyused are briefly introduced. References [38] and [39] describe them in moredetail.Frequency fThe resonant frequency f0 (or ω0 = 2pif0) has to match the bunch frequency ofthe accelerator. In the case of a pillbox cavity operating in the TM010 modethe frequency depends only on the radius R of the cavity:f0 =c2pi2.405R. (2.28)Elliptical cavities follow this trend; the frequency largely depends on the radialdimension of the cavity.The frequency is an important parameter for the cavity design as it sets thelength g of the RF cell. The particle bunch has to travel through the cell withinhalf a RF period, just in time to arrive at a node of the longitudinal electricfield when the direction of the field switches. This is the so called Wideroe [40]condition for particle acceleratorsg =βλ2(2.29)with the normalized velocity of the particle β = v/c and the wavelength of theRF wave λ = c/f . For electron accelerators, it is common to assume β = 1 sinceonly a moderate voltage is required to make the beam fully relativistic due tothe significantly lower mass of electrons compared to protons/heavy ions. Inthe case of protons or heavy ions, β changes throughout the acceleration and sothe accelerating gap and/or the frequency have to be modified to match.2.1. Resonant Cavities 13Quality Factor Q0The quality factor of a cavity is defined as the ratio of stored energy U to powerloss per RF cycle P/ω0Q =Stored EnergyPower loss per RF cycle=ω0UPcav. (2.30)The losses come from a finite electrical conductivity of the resonator walls, evenin a superconducting state (more on this in section 2.2) withPcav =∫SRs2|H2|dS. (2.31)The quality factor Q0 is a measure of dissipated power in the cavity walls.Equation 2.30 can be rewritten asdUdt= − ω0Q0U (2.32)⇒ U(t) = U0e−t/τ (2.33)with τ = Q0/ω0 as the decay constant. As soon as the driving signal is turnedoff, the stored energy in the RF fields decays exponentially. The Fourier trans-formation of this decaying cos signal with the decay described in eq. 2.33 fromthe time domain to the frequency domain results in a Lorentz curve with thewidth ∆ω, which is proportional to the inverse of the decay constant τ . ∆ωis the bandwidth of the resonance at the half power point and can be used tocalculate the Q0 withQ0 =ω0∆ω. (2.34)Loaded Quality Factor QLSimilar to the unloaded quality factor Q0, the loaded quality factor is definedby the power loss in a cavity through loads such as power couplers, pick-upcouplers, HOM couplers or beam loads. The exponential decay of the storedenergy in the cavity, when the RF drive is turned off, as described in eq. 2.33,is still true and a new loaded decay constant τL takes into account the totalpower that is flowing out of the cavity via the couplers and dissipated in anyconducting wall. All losses are added together to calculate the total dissipatedpower PtotPtot = Pcav + Pcoupler + Ppick−up + . . . (2.35)With this the loaded quality factor is simplyQL =ω0UPtot. (2.36)2.1. Resonant Cavities 14It is easy to see that to each loss mechanism a Q factor can be defined and thattheir relationship to QL can be written as1QL=1Qcav+1Qcoupler+1Qpick−up+ . . . (2.37)The ratio of unloaded to loaded quality factor is called the coupling factor β1β =Q0QL(2.38)Geometric Factor GThe energy density in an electromagnetic field is given byu =12(E2 + µH2). (2.39)As can be seen in the field equations 2.11 to 2.22 the electric and magneticfields have a 90 degree phase shift and the stored energy in the mode oscillatesbetween the two fields. For the total stored energy in the volume V, this meansthatU =12µ0∫V|H|2dV = 120∫V|E|2dV. (2.40)With this and equation 2.31, the quality factor 2.30 can be rewritten asQ0 =ω0µ0∫V|H|2dV∫SRs|H2|dS ≈ω0µ0∫V|H|2dVRs∫S|H2|dS (2.41)where it is assumed that the RF surface resistance is approximately uniformover the cavity surfaces. The two integrals are only determined by the cavitygeometry, any loss parameters are taken out. From this follows the definitionof the geometric factor G:G =ω0µ0∫V|H|2dV∫S|H2|dS (2.42)so thatQ0 = G/Rs. (2.43)The geometric factor G is useful since it does not depend on the size of the cavitybut only on the shape. This makes it useful to compare different cavity shapesto each other independent from surface losses that are process dependent.Another use for G is to scale the quality factor for superconducting cavities.Normal conducting cavities have surface resistances in the order of mΩ, whilesuperconducting cavities have much lower Rs of a few nΩ. Numerical calcula-tions provide a value for Q0 based on a non-superconducting conductivity σ,and with the knowledge of the superconducting surface resistance, it is possibleto calculate the Q at low temperatures when the cavity is superconducting.1Not to be confused with the normalized velocity β = v/c.2.1. Resonant Cavities 15Accelerating Voltage Veff , Transit Time Factor TTF and GradientEaccWhen phased correctly, the charged particle should reach the center of the ac-celerating gap when the field amplitude is at its maximum to achieve maximumacceleration. Depending on the length of the gap the field will rise and fall sothat the effective voltage acquired by the particle in crossing an acceleratinggap with length g is given by the integralVeff =∫ g/2−g/2Ez(z) cos(ω0t)dz =∫ g/2−g/2Ez(z) cos(ω0zβc)dz. (2.44)The crossing of an accelerating voltage is velocity dependent. The ratio of thevoltage gained by a particle that travels with speed v = β · c to a particle withinfinite speed (= instantaneous crossing of the accelerating E-field) is called thetransit time factor TTF:TTF (β) =∫ g/2−g/2Ez(z) cos(ω0zβc)dz∫ g/2−g/2Ez(z)dz. (2.45)The transit time factor is always smaller than 1. It is also noteworthy to pointto the dependence of TTF on the gap length g. As has been stated earlier, thereis an optimal g for each β (see 2.29). Since electrons can be considered fullyrelativistic with β = 1, the gap length g is only depending on the wavelength ofthe RF frequency g = λ/2.This factor is used to calculate the effective gradient. The instantaneous gradi-entE0 =1L∫ L0|Ez(z)|dz, (2.46)multiplied by the transit time factor TTF results in the effective acceleratinggradient EaccEacc = E0 · TTF. (2.47)This is a useful figure to compare the performance of different cavities. Itnormalizes the effective voltage of a cavity over a distance of 1 m. The effectivelength of a cavity L can be defined in different methods, which vary in numbers.For some cavity types this difference can be quite significant. The most commonand useful for TM010 mode cavities like the elliptical cavities used in electronaccelerators isL = n · g (2.48)with n as the number of gaps/cells in the cavity. This definition will be usedthroughout this dissertation.The effective voltage multiplied by the charge-state of the accelerated particle2.1. Resonant Cavities 16leads directly to the energy gain. The energy gain is also depending on the ar-rival time of a bunch with respect to the maximum voltage. This time differencecan be expressed as a phase difference. While maximum acceleration is reachedwith a phase difference of 0, deliberately setting the phase difference to a valueof around 30◦ can reduce the energy spread within the particle bunch. With aphase set like this, each bunch sees a rising voltage and the head and tail of thebunch pass through slightly different voltages, causing the reduction in energyspread.Shunt Impedance Rsh and Geometric Shunt Impedance R/QThe shunt impedance is an important factor for cavities. It is a measure ofhow efficient a resonator uses the available RF power to create the acceleratingvoltage. It is defined asRsh =V 2effPcav. (2.49)Note that the effective voltage is used to include the velocity dependence.To normalize the shunt impedance, the surface resistance independent geometricshunt impedance R/Q can be defined asRQ0=V 2effPcav· Pcavω0U=V 2effω0U. (2.50)This can be used to calculate the shunt impedance of a mode with different kindof couplings.Peak Surface Fields Ep/Eacc and Bp/EaccHigh electric surface fields are not a fundamental limit for cavities, but theycan lead to field emission of electrons that reduce the performance of the cavity.Therefore the unit-less ratio of peak surface electric field to the acceleratinggradient Ep/Eacc is a useful figure of merit to compare different shapes. Ingeneral, a low ratio is preferred. For elliptical cavities, the highest electric fieldson the surface can be found around the iris region with a ratio of between 2 and3. TEM cavities like quarter-wave cavities tend to have higher ratios up to avalue of 10.In the following section on superconductivity, it will be shown that supercon-ductivity breaks down in high magnetic fields. For niobium, the critical fieldis around 200 mT at 2 K. A high ratio of the magnetic field on the surfaceto accelerating gradient (usually in units of mT/(MV/m)) limits the reachablegradient. For the TESLA cavity, an elliptical cavity designed for low surfacefields, the ratio is 4.2 mT/(MV/m), setting the fundamental gradient limit at2 K to about 50 MV/m. A lower magnetic surface field is also beneficial toavoid thermal breakdown caused by defects.2.1. Resonant Cavities 17Magnetic fieldElectric field Beam Axisa) b) c)Figure 2.2: Cut-plane views of basic examples of low to medium β cavities withthe electric (blue) and magnetic fields (red): the quarter-wave resonator (a) canbe described as a coaxial transmission line with one end shorted and the otherend open, while the half-wave resonator (b) has both ends shorted, resulting intheir name giving field distributions. Spoke cavities (c) look similar to a wheelwith one spoke, and have a similar field distribution as half-wave cavities.2.1.3 SRF Cavity ArchetypesMany different types of SRF cavities are used in particle accelerators, especiallyin proton and heavy ion accelerators. The high particle mass causes the rel-ativistic velocity β = v/c of the particle to change relativity slowly comparedto electrons, which are accelerated to highly relativistic velocities (β ≈ 1) afterpassing a very moderate voltage. Since RF acceleration depends on synchronismbetween the accelerating field and the time of arrival, different cavity structuresare needed to efficiently accelerate as the velocity increases. Due to the changingβ, the optimal accelerating gap g = βλ/2 changes as the particle gains energy.To compensate for lost efficiency, the geometry can be changed, which can in-volve a frequency change to optimize the efficiency for higher energy beams.For low β (≈ 0.04− 0.15) regimes, quarter-wave resonators (QWR) are usuallyused at frequencies between 50 and 150 MHz. Half-wave resonators (HWR)with frequencies between 150 to 250 MHz are suitable for β ≈ 0.12− 0.2. BothQWR and HWR are very similar structures operating in a transverse electric-magnetic (TEM) mode. Both cavities can be modeled as a coaxial transmissionline. The QWR is shorted at one end and open on the other end as can be seenin fig. 2.2 a). The length of the inner conductor roughly equals to a quarterof the wavelength, giving this type of resonator its name. The magnetic field2.1. Resonant Cavities 18curls around the inner conductor with the maximum magnitude at the shortedend. The electric field reaches its maximum amplitude at the open end of theinner conductor where the beam is accelerated. The inner conductor is outfittedwith a drift tube where the particle does not see any fields while the sign of thevoltage in both gaps changes.The HWR works in a similar way to the QWRs. For this cavity type, the innerconductor is shorted on both sides (fig. 2.2 b) and its length corresponds toλ/2. The strongest electric fields in this cavity are at the center of the innerconductor and the beam ports and drift tube are placed in this area.Spoke cavities are a typical choice for medium β particles (≈ 0.2 − 0.6) withfrequencies ranging from 200 to 400 MHz. The geometry of a spoke cavity canbe described as a wheel with a single spoke (fig. 2.2 c), giving this type of cavityits name.The details of each cavity design vary depending on the optimization param-eter. No cavity design is optimal for all purposes and compromises betweenparameters like peak surface field ratios and geometric shunt impedance haveto be made. Detailed descriptions of the various cavity types can be found in[41] and many other sources.Elliptical CavitiesFor high β (> 0.6) particles, elliptical cavities are the typical cavity of choice dueto the good acceleration efficiency and low peak surface fields. Elliptical cavitiescan be described as an evolved pill-box cavity that has its corners roundedsignificantly. Each cell consists of two axial symmetric half-cells, which can bedescribed with seven geometry parameters (see fig. 2.3): The length L, theequator radius Req, the iris equator Riris and the four half-axis values of theellipses describing the curvature A, B, a and b with A and B for the ellipse atthe equator and a and b for the ellipse at the iris. Two half cells are connectedat the equator to form a single cell cavity. To increase the active accelerationlength2, identical cells are coupled together at the iris. A n-cell cavity canbe described as n coupled resonators, which can oscillate in n modes for eachresonance. The modes in those pass-bands differ in frequency and in cell-to-cellphase advance. For the TM010 pass-band, the highest frequency mode has aphase advance of pi between two cells (the so called ”pi-mode”) and thereforefulfills the Wideroe condition (eq. 2.29) if the distance between the center oftwo neighboring cells is equal to βλ/2. For relativistic electrons, this sets thecell spacing at λ/2.Due to the change in periodicity at both ends of the cavity, the end half-cellsgenerally have slightly different geometric parameters than the inner half cells.This is done to reach an even field distribution in each cell, or ’field flatness’.Perfect flatness is reached when the peak amplitude in all cells reaches the same2Active acceleration length differs from the physical length of the cavity: parts like beamtubes, warm cold transition pieces are not considered.2.1. Resonant Cavities 19LRirisRequatorbaBAFigure 2.3: The geometry of a half-cell of an elliptical cavity can be describedby seven parameters: the length L is depending on the cavity frequency andparticle velocity β, the equator radius Requator has a strong influence on thefrequency of the cavity, while the iris radius Riris strongly affects the R/Q.The ellipse parameters are used to optimize the peak fields on the surface.magnitude. The flatness is defined [42] asfield flatness =(1− Ec,max − Ec,min1N∑Ni Ec,i)(2.51)with Ec being the field amplitude in the center of each cell. The field amplitudecan be measured with a bead-pulling technique [39]. A small metallic beadis pulled through the cavity on the beam axis while the resonance frequency isrecorded. The bead causes a perturbation of the electric field, resulting in a shiftin frequency. This frequency shift is proportional to the local field amplitude.In an elliptical cavity the electric fields are focused onto the beam axis due to theslope of the walls, while the magnetic field curls around the symmetry axis withits highest fields in the equator region. An example of the field distribution forthe accelerating mode can be seen in fig. 2.4 for a single cell. Multi-cell cavitieshave the same field distribution in each cell. In the accelerating pi-mode thedirection of the field vectors switches from cell to cell. In addition to the cellsan elliptical cavity also has beam tubes at the entrance and exit which are sizedto strongly attenuate the passage of the fundamental mode to keep the fieldin the cells and increase R/Q. The beam tubes also contain the fundamentalpower couplers, a pick-up probe and HOM couplers if needed.The TESLA nine-cell cavity [31, 43, 44] is such a cavity and the baseline cavityfor many electron accelerators, including the ARIEL eLINAC.2.2. Superconductivity 20Magnetic fieldElectric fieldbeam tube cell beam tubeEquatorIrisFigure 2.4: Field Distribution in a single cell elliptical cavity for the TM010mode. The electric fields reach from iris to iris with a strong amplitude on thebeam axis while the magnetic field curls around in the equator region.2.2 SuperconductivitySuperconductivity was discovered in 1911 by Kamerlingh Onnes after the lique-faction of helium. He observed, that the electrical resistance of mercury vanishedwhen cooled down to below 4.2 K. This behaviour was also discovered in manyother elements such as lead and niobium. Superconductors are useful for RFcavities to reduce the overall power requirements of an accelerator, even whenthe surface resistance does not vanish completely for AC cases. The supercon-ducting state and parts of the theory behind it are introduced in section 2.2.1,while the loss mechanism and the consequences for the surface resistance aredescribed in section 2.2.2.Detailed explanations can be found in many textbooks, for example [45] and[46]. This section will summarize the most important concepts and phenomena.2.2.1 The Superconducting StateSuperconductivity was first explained on a microscopic level in the very success-ful BCS theory, named after Bardeen, Cooper and Schrieffer [47], while othertheories like the London theory [48] and Ginzburg-Landau (GL) theory [49]work on a phenomenological basis.Each superconductor is characterized by its transition temperature Tc, which isthe upper temperature bound for the superconducting state and is characteris-tic to each material. One particularly interesting phenomenon is the expulsionof magnetic fields from the superconductor when it is cooled below its criticaltemperature in the presence of an ambient magnetic field. This is called theMeissner-effect [50] and cannot be explained by perfect conductivity alone.Figure 2.5 shows the demagnetization due to the Meissner-effect for two types of2.2. Superconductivity 21HcHc-MagnetizationApplied Magnetic Field(a) Type IHcHc1 Hc Hc2-MagnetizationApplied Magnetic Field(b)Meissner VortexType IIFigure 2.5: (a) The magnetization for type I superconductors (assumed as ainfinitely long solid cylinder) rises as the applied field increases until the super-conducting state breaks down. (b) Until Hc1 is reached, type II superconductorsare in the Meisser-state like type I superconductors. Between Hc1 and Hc2, it isenergetic favourable to allow flux in the superconductor in a vortex state. AboveHc2 the superconducting state breaks down Hc marks the thermodynamic crit-ical field.superconductors, both assumed as a infinitely long cylinders. The induced mag-netization exactly cancels the applied field for type I superconductors, shownin fig. 2.5(a), until Hc is reached. This is the thermodynamic critical field. Forfields above Hc, the normal-conducting state is energetic favourable.Type II superconductors have a different magnetization behaviour as can beseen in fig. 2.5(b). Up to the lower critical field Hc1, the superconductor is inthe Meissner-state and fully expels the ambient magnetic field. Between Hc1and the upper critical field Hc2, it is energetically favourable to have magneticflux in the superconductor in the form of vortices. Each vortex carries a mag-netic flux of Φ0 and has a normal conducting core, which causes losses. Thisphase is called the Vortex-phase. At H > Hc2 the superconducting state breaksdown completely and flux fully enters the now normal conductor. Hc2 can besignificantly higher than the thermodynamic critical field Hc.If the applied field exceeds Hc (Hc2 in type II superconductors), the super-conducting state breaks down and the normal conducting state is restored. Hcis temperature dependent and at T = Tc the critical field Hc is zero. As Tapproaches zero, the critical field follows approximatelyHc(T ) = Hc(T = 0)[1−(TTc)2]. (2.52)In the absence of a magnetic field there is no latent heat and the order pa-rameter develops continuously through the superconducting transition. This istypical of a second order (or continuous) phase transition. In the presence of amagnetic field, there is a small latent heat, so technically the phase transition is2.2. Superconductivity 22of first order. However, the order parameter evolves more or less continuouslythrough the transition. From calculating the free energy difference between thenormal conducting and superconducting states, the condensation energy can becalculated to be equal to µ0H2c /2 [45].Bean and Livingston [51] show how the surface of a superconductor providesan energy barrier for flux to enter the superconductor, even if the external fieldis above Hc1. A more energetically favourable state could be reached, but theenergy needed to move flux into the superconductor prevents this. This barrieris formed between Hc1 and Hc2. This meta-stable state above Hc1 is often calledsuperheated state and persists up to Hsh. Once Hsh is crossed, the supercon-ductor stays in the Vortex-state until the flux is fully expelled again.The fundamental concept of the BCS theory is the formation of so called Cooper-pairs. Two electrons of opposite spin and momentum pair up to form a spin0 quasi-particle, the Cooper-pair. It can be imagined, that one electron causesa polarization in the atomic lattice by moving through the lattice. A secondelectron is attracted by this polarized lattice and can reduce its energy by mov-ing in the polarized track. A reduction in total energy of the electron pair ispossible when the particles have opposite spin and momentum [52].The lattice polarization and the reduction of the electron energy can be de-scribed as an exchange of a virtual phonon between the electrons. This phononinteraction is strong enough in the superconducting state to overcome repulsiveforces between the electrons. The electrons interact over a distance ξ0, which iscalled the BCS-coherence length. This length can be interpreted as the size ofthe Cooper-pair. A second coherence length ξGL comes out of the GL-theoryand describes how the order parameter Ψ of the GL-theory changes. |Ψ|2 canbe understood as the density of superconducting electrons. One further impor-tant characteristic length is the London-penetration length λL, which describeshow the magnetic field decays when it enters a superconductor. The ratio ofλL/ξGL is the Ginzburg-Landau parameter κ, which can be used to differentiatebetween type I and II superconductors.Since the Cooper-pair quasi-particle has spin 0, it does not follow the usualFermi-Dirac statistics for electrons, but the Bose-Einstein model: cooper-pairscan be in the same quantum state as other cooper pairs and with that can oc-cupy the same energy levels provided the density is not too high. A temperaturedependent energy gap 2∆(T ) forms between the new ground state of the electronpairs and the Fermi-energy for unpaired electrons. At temperatures T > 0 K,some electrons are thermally activated. This breaks the Cooper-pairs and theseelectrons lose the superconducting characteristics. The unpaired electrons areresponsible for a remaining resistivity in case of AC currents as will be shownin the following section.SRF cavities are typically made out of the type II superconductor niobium. Nio-bium has a relatively high3 transition temperature of 9.2 K and therefore canbe effectively cooled with liquid helium (LHe), which has a boiling temperature3High for pure elements. There are superconductors with a higher Tc, but those have otherunfavourable characteristics for the use in particle accelerators.2.2. Superconductivity 23Table 2.1: Critical fields for niobium at 0 K (in mT/µ0) [41].Hc1 Hc2 Hsh174-190 390-450 240of 4.2 K at atmospheric pressure. The critical fields of niobium, extrapolatedto 0 K, are listed in table 2.1. Niobium has other very favourable qualities suchas its availability in its pure form, a high thermal conductivity and is workableto form cavities.2.2.2 Surface Resistance RsNormal ConductivityIn a normal conducting state (and T = 0 K) electrons fill the energy levels up tothe Fermi energy and follow the Fermi-Dirac statistics with the Pauli-exclusionprinciple: no two electrons can be in the same quantum state. At T > 0 Ksome electrons are above the Fermi energy and are mobile on the surface of theFermi sphere with speed vf . The electrons interact with the atomic lattice andscatter due to impurities and lattice vibrations. The electrical conductance σdepends on the mean free path λmfp of those conductance electrons withσ =ne2λmfpmevf. (2.53)For high frequency fields the induced surface currents are limited to a small layeron the surface, which is known as the skin-effect [37] and is frequency dependent.The skin-depth δ is defined as the attenuation constant of the current densityj(x) = j0e−x/δ (2.54)withδ =1√pifµoσ. (2.55)The surface resistance Rs follows from this withRs =1σδ=√ωµ02σ. (2.56)For copper and for frequencies common for particle accelerators (10’s of MHzto ∼1 GHz) the surface resistance takes values of around 1 to 10 mΩ and thelosses originate from the surface currents within the skin depth.2.2. Superconductivity 24Superconductivity and the Two-Fluid ModelIn the DC case, superconductivity reduces the resistance completely. But inthe AC case, a finite surface resistance remains. This can be explained by atwo-fluid model. The two components are the friction-less moving cooper pairsand the thermally activated, normal conducting electrons.If a time varying magnetic field Bx = B0 exp(iωt) is applied parallel to thesurface (assumed to be in the x-y plane) of a superconductor, currents on thesurface are induced that shield the interior from the magnetic field. Thesecurrents penetrate into the superconductor and cause the magnetic field to decayexponentially withB = B0e−z/λL (2.57)with λL corresponding to the London-penetration depth.Starting from Faraday’s law of induction (eq. 2.3), the electric field, which isinduced by this magnetic field, can be calculated toE = −iωλLB. (2.58)By averaging the field over time, the dissipated power can be calculated byevaluatingP =∫Vσ|E|2dV = −λLσ∫S|E|2dS (2.59)with σ as conductivity of the normal fluid. Combining equations 2.59 and 2.58results inP = ω2λ3Lσ∫s|B|2dS (2.60)This can now be compared to equation 2.31 to result in an expression for thesurface resistance in the superconducting stateRs =12ω2λ3Lσµ20. (2.61)The dependence of eq. 2.61 on the normal state conductivity shows that thelosses in the superconducting state are caused by the unpaired, normal conduct-ing electrons.The number of normal conducting electrons nn is proportional to exp(−∆/kbT )as with decreasing temperature the number of paired, superconducting elec-trons increases. The normal conducting currents jn, are proportional to nn andtherefore proportional to exp(−∆/kbT ) as well. This leads to a temperaturedependence of Rs withRs ∝ jn ∝ nn ∝ e−∆/kbT . (2.62)2.3. Higher Order Modes and Beam-Break Up 25An approximate formula for the temperature dependent BCS resistance for fre-quencies less than 2∆/h(≈ 1012 Hz for Nb) is given byRBCS(f, T ) =ATf2e−∆/kbT , (2.63)where A is dependent on the London penetration depth, coherence length andnormal conductivity. The frequency dependence in eq. 2.63 shows that lowerfrequency cavities (≈ 50-200 MHz) have a significantly lower RBCS than highfrequency cavities (≈ 1-1.5 GHz) at the same temperature. Lowering the tem-perature from 4.2 K (atmospheric LHe) to 2 K reduces the RBCS of high fre-quency cavities significantly (from ≈800 nΩ to ≈10 nΩ), greatly improvingperformance while low frequency cavities do not benefit as much.There are additional contributions to the surface resistance due to a num-ber of different mechanisms including, but not limited to inclusions, surfacepollution, trapped flux and Q-disease due to hydrides. An overview of thosemechanisms is given in chapter 4.1. These factors are combined into the tem-perature independent residual resistance Rres, so that the final surface resistancecalculates asRs = RBCS(T ) +Rres. (2.64)An understanding of both RBCS(T ) and Rres is important to reach a highquality factor. At T = 2 K and 1.3 GHz a BCS resistance RBCS of around10 nΩ is expected for elliptical cavities made out of high quality niobium.This significantly reduced surface resistance compared to a normal conductingcavity allows to operate high gradients in a continuous wave mode. For a TESLAlike cavity, operating at a cavity voltage of 10 MV, the dissipated power reducesfrom 3 MW to 10 W. Of course, the power consumption of the cryosystem,that provides the 2 K liquid helium, has to be included in the total powerconsumption of the accelerator but in many cases the savings in power aresignificant, not to mention that cooling away 3 MW on relatively small surfacearea is unrealistic. A normal conducting variant of the eLINAC would need to bebuild out of more cavities with lower gradients to distribute the thermal load intomanageable pieces. This would increase the length of the machine significantlyand therefore superconducting cavities are the choice for the eLINAC.2.3 Higher Order Modes and Beam-Break UpAs has been shown in section 2.1.1, RF cavities have an infinite number of modesand in general only one of those is useful. All other modes are considered para-sitic and some can have an unwanted influence on the beam. In this section themost important consequences from the fundamental theorem of beam loadingare described. A more detailed derivation can be found in many textbooks, forexample in [38] and [39].2.3. Higher Order Modes and Beam-Break Up 26B-Field(a) E-Field(b)Figure 2.6: Field distribution in a pillbox cavity of the TM110 mode: (a) mag-netic field in the X-Y cut-plane, (b) electric field in the X-Z cut-plane, with Zbeing the beam axis.2.3.1 Monopole ModesMonopole modes can contribute significantly to the power losses in the cavitywalls. The loss factor kn is defined askn =ωn4(RQ)n=V 2c4U(2.65)and can be used to calculate the amount of energy that is deposited in the cavityby a passing bunch with charge q viaU = knq2. (2.66)An important consequence of the finite bunch length σt(= v · σz) is that onlymodes with ωn > 1/σz can be excited. The Fourier spectrum of the bunch doesnot have any components that would excite these high frequency modes. An-other important parameter is the difference between bunch repetition frequencyωb and the HOM frequency ωn. In case the frequencies match to a integer mul-tiple of 2pi, the induced voltage increases compared to a unmatched situation.This situation causes additional power losses and should therefore be avoided.2.3.2 Dipole modesDipole modes do not have any Ez component on the beam axis. Therefore abeam traveling perfectly straight on the beam axis would not excite those modes.But in reality beams have finite transverse emittance and can be misaligned2.3. Higher Order Modes and Beam-Break Up 27with respect to the cavity axis and thus are able to excite dipole modes like theTM110 mode shown in fig. 2.6.This absence of the on axis field has to be compensated and commonly an off-axis by distance a voltage Va along the cavity is defined. With this the lossfactor can be defined equivalently to the monopole mode case withkn =V 2a4U= a2(ωnc)2 ωn4RdQ. (2.67)This uses the dipole mode definition of Rd withRd =V 2a(ωn/c)2a2Pc. (2.68)The Panofsky-Wenzel theorem [53] relates the longitudinal fields with a trans-verse force on the passing particleiω∫ d0F⊥eiωz/vdzv= q[E⊥eiωv/z]d0− q∫ d0∇⊥Ezeiωz/vdz (2.69)with the index ⊥ indicating a transverse component of the vector and ∇⊥ =∇− ∂/∂z.2.3.3 Beam-Break UpBeam-break up happens when dipole modes are excited by a passing beam.This can happen when the beam passes slightly off-axis through the RF cavity.The first bunch initiates the excitement of the mode with following bunches thenbuilding up the amplitude. Following bunches also pass through the fields of thedipole mode and get deflected, resulting in the so called cumulative beam-breakup. 10 mA is a modest beam current and cumilative beam break up is usuallynot an issue. In addtion beam position monitors downstream of the cavitiesmonitor the beam quality and corrective measures can be taken. For the RIBproduction, beam quality is not as important compared to the ERL beam andtherefore single-pass BBU is an important consideration for the eLINAC.For the ARIEL eLINAC a different aspect of BBU is important. The return ofthe electron beam for a second pass through the LINAC can excite the multi-pass-BBU instability. When an electron bunch gets a transverse kick by dipoleHOMs in a cavity it will return to the same cavity during its second pass throughthe LINAC with a transverse displacement which may pump more energy intothe HOMs. Once HOMs get amplified by the bunch on its second pass, followingbunches get kicked even stronger. When the electron current becomes so large,that more energy is transferred into a HOM by bunches than is taken out bythe HOM couplers, the HOM power will start to grow exponentially and beamloss can ensue.In particle accelerators many cavities are used. But it takes only a single mode2.3. Higher Order Modes and Beam-Break Up 28with a high quality factor QL to drive this instability. The bandwidth and sepa-ration of modes is such that modes don’t overlap in frequency, and so that modescan be considered individually. Even from cavity to cavity the frequency of thesame mode (excluding the accelerating mode which is frequency controlled) canvary significantly due to small variations in the cavity shape.In a single cavity - single mode situation the threshold current Ith can be derived[54] for each HOM n asITh,n = − 2pcq(ωn/c)(Rd/Q)n ·QL,n1M12 sin(ωnTr)(2.70)where q is the bunch charge, p the momentum of the beam at the first entryinto the accelerating cavity, (Rd/Q)n · QL,n is the shunt impedance of moden, Tr the time it takes for the bunch to re-enter the cavity and M12 the beamtransport matrix element connecting the transverse momentum of the beamwhen leaving the cavity to the transverse position when re-entering the cavity4.At this beam current the change in transverse energy of the beam is equal tozero. At a lower current energy is transferred from the beam to the HOM andthen partially dissipated as ohmic losses in the cavity walls before the next buncharrives. The decay constant for the stored energy in the cavity τ is modified bythe beam current I and the threshold current ITh to create the effective decayconstantτeff = τIThIth − I . (2.71)For currents I < ITh, τeff is positive and the stored energy decays. WhenI > ITh, τeff becomes negative and the exponent in 2.33 becomes positive andthe instability grows exponentially and at some point the additional transversemomentum will be so strong that the beam cannot be guided through the latticeof steering and focusing magnets and is lost on the walls of the beam pipe.Avoiding BBU means increasing the threshold current above the goal beamintensity. The average current in an ERL takes the accelerated and deceleratedbeam current into account, so that for the ARIEL eLINAC with a nominal beamcurrent of 10 mA, the threshold current has to reach at least 20 mA. From eq.2.70 follows, that the HOM shunt impedance is a very important parameter toincrease the threshold current as other parameters like the bunch charge q or therecirculating time Tr are set and not variable. Lowering the shunt impedanceof HOMs will be discussed in the next chapter.Equation 2.70 corresponds to a simplified system. Final threshold calculationshave to be done using numerical codes, that model the accelerator and beambehaviour more detailed.4In this case the angle between the polarization of the dipole mode and the beam off-set isassumed to be 0. For any other angle, M12 has to be modified to account for the azimuthalvariation of the dipole mode.29Chapter 3Design of the ARIELCavityThe ARIEL cavity is a 1.3 GHz nine-cell cavity based on the TESLA cavitydesign [31, 43, 44]. The TESLA cavity has many characteristics that are usefulfor the ARIEL eLINAC and is an established design in the SRF community.Many projects will use or are already using this cavity, including LCLS-II atSLAC (280 cavities), EU-XFEL at DESY (800 cavities), and the ILC (16000cavities). Some modifications have to be made in the design, since the TESLAproject and the other mentioned projects have different requirements comparedto the ARIEL eLINAC.3.1 General Cavity Design ConsiderationsThe primary goal of the ARIEL eLINAC is to provide a 50 MeV, 10 mA electronbeam for rare isotope production. This can be reached with a number of differentcavity designs, that are not based on the TESLA cavity that might have certainadvantageous properties compared to the TESLA cavity. For example, thechoice of 1.3 GHz as operating frequency is not a requirement for the photo-fission processes. A lower frequency would reduce the BCS surface resistanceRBCS at a given operating temperature as described in section 2.2.2. For beambreak up a lower fundamental frequency would be beneficial, as lower frequenciesresult in higher threshold currents and less sensitivity to trapped modes. Onthe other hand, a lower frequency increases the radial dimension of an ellipticalcavity. This increased size increases the amount of niobium required to build acavity, thus increasing fabrication costs. A bigger cavity is also more difficult tohandle due to the increased size and weight than a smaller cavity and requiresa bigger cryomodule.Several measures can be taken to reduce the impact of higher order modes. Asdescribed earlier, a multi-cell cavity can be described as n coupled resonatorswith n as the number of cells. This means that for each resonance, n statesexists. For example the accelerating TM010 passband of a TESLA nine cellcavity consists of nine different modes, each with its own frequency and shuntimpedance. Therefore, reducing the number of cells decreases the number ofmodes in a given frequency range. Reducing the number of modes is beneficialto avoid resonances between HOMs and the beam and reduce the risk of trapped3.2. ARIEL eLINAC Requirements 30modes, which do not have any significant fields in areas where HOM couplerscould be effective. The disadvantage of fewer cells is a reduced shunt impedancefor the accelerating mode and a higher cavity gradient is needed to reach thesame voltage. In the case of the ARIEL eLINAC, an effective voltage of 10 MVis required. The gradient of a nine-cell cavity would be around 10 MV/m,which is a very moderate gradient. A seven-cell cavity would need around13 MV/m. At the same time the dissipated power in the cavity wall wouldincrease from 10 W to ∼13 W due to the higher surface fields, assuming thesame Q0 value of 1 · 1010. For five cavities, this results in an added ∼15 Wthat would need to be compensated by the cryoplant. With that in mind, ahigher cell count is desirable. For an even lower cell number, the gradient wouldincrease accordingly. At some point the peak surface fields become a significantchallenge and could potentially reach more fundamental limits.Another way to deal with HOMs is to increase the iris radius of the center cellsin a multi-cell cavity. The iris radius usually has a strong impact on the R/Qof all modes. A smaller iris focuses the electric field onto the beam axis andtherefore increases the R/Q. A bigger iris on the other hand decreases the R/Q.In section 3.6.5, the effect of the iris radius is shown for the accelerating modeand a high shunt impedance mode. The negative consequence of a reducedR/Q is again an increase in dissipated power. A larger iris helps to untrapproblematic modes.The TESLA cavity design is a compromise between all these aspects with theadded benefit of strong developments at the design frequecny of 1.3 GHz, forexamples couplers and klystrons are available. Therefor it chosen as a baselinedesign for the ARIEL eLINAC. The specific requirements for the ARIEL cavityare listed in the following section and analyzed as to how they affect the cavitydesign in details.3.2 ARIEL eLINAC RequirementsThe design of the eLINAC calls for a beam energy of 50 MeV at a beam intensityof 10 mA. This corresponds to a final beam power of 500 kW. Since commerciallyavailable fundamental power couplers are limited to 50 kW, a cavity design withtwo couplers per cavity and a maximum beam loaded RF power of 100 kW percavity needs to be adopted. This corresponds to an energy gain per cavity of10 MV and a modest gradient of ∼10 MV/m given the cavity length of a bitover 1 m (L = n · βλ2 = 9 · 0.115 m = 1.038 m). This gradient is modest, buthas to be reached in an efficient way. A high R/Q for the accelerating mode isrequired.The original TESLA cavity was designed to operate in a pulsed mode with a lowduty cycle and therefore low average power. The TESLA fundamental powercoupler is designed for an average power of less than 10 kW and is therefore notsuitable for the eLINAC cw operation.To allow for a future ERL upgrade, the impact of HOMs has to be suppressedas discussed in the previous chapter. In order to reach a threshold current of3.3. Eigenmode Simulation Codes 3120 mA, beam dynamics calculations [55] indicate a shunt impedance of Rd/Q ·QL < 107Ω for dipole modes is required. Simulations and measurements onthe TESLA cavity show that a few modes are too high in shunt impedance,in particular one of the dipole modes near 2.56 GHz [56, 57]. Therefore theHOM design of the cavity has to be adapted to fulfill this requirement. A moreappropriate BBU limit for the cavity includes the frequency of the HOM aseq. 2.70 is frequency dependent. But within the range of the considered HOMfrequencies, 1 to 4 GHz, and the amplitudes of the shunt impedance, a flat limitis used as a guideline for the cavity design. The final threshold current has tobe calculated using numerical particle tracking codes.These beam related requirements create a unique set of constraints, such thata unique RF design is required for this dual beam operation .In addition to those RF requirements, cryogenic considerations set a limit onthe dissipated power in the cavity walls to 10 W. Assuming the RF parametersof the TESLA cavity, listed in tab. 3.1, this means that a quality factor of1 · 1010 or higher at the design gradient of 10 MV/m is required. Since theTESLA cavity has a geometry factor G of 270 Ω, the surface resistance must beless than 27 nΩ. The BCS resistance at 1.3 GHz and 2 K can be estimated tobe around 10 nΩ, leaving a maximum of 17 nΩ for the residual resistance. Theactual Q0 has to be measured and if the cavity does not fulfill this requirement,additional processing steps have to be taken to raise the Q0 value.Fabrication tools for a TESLA cavity were available before the design started.Table 3.1: TESLA cavity RF parameters [31].f [GHz] Geo. Factor G [Ω] R/Q [Ω] Ep/Eacc Bp/Eacc [mT/(MV/m)]1.3 270 1034 2.0 4.26As a result, changes away from the design of the TESLA cavity would requirenew forming tools and an increased fabrication expense. To avoid this expense,one goal of the design phase of the ARIEL cavity is to rely on the TESLAdesign as much as possible. Specifically, a design restriction is set to not changethe inner cells of the cavity unless the HOM requirement cannot be fulfilledotherwise.3.3 Eigenmode Simulation CodesAs described earlier, the boundary conditions define an eigenmode problemthat is not analytically solvable for any cavity that is more complicated thana simple pill box cavity or a basic coaxial resonator. Computer codes are ableto solve this problem numerically using either a finite element or finite integralmethod. In the following, a selection of the codes used for designing the cavityand calculating the eigenmodes and their characteristic values are described.3.3. Eigenmode Simulation Codes 32Figure 3.1: 3D model of a nine cell cavity variant generated with CST MWS.CST Microwave StudioMicrowave Studio (MWS) [58] is the decedent of a code called MAFIA, whichis one of the pioneers in 3D modeling for RF structures. MWS uses a full 3Dmodel of the cavity and imposes boundary conditions on different surfaces. Afinite integration technique in the eigenmode solver is used to calculate the elec-tic fields inside the cavity. A mesh is layered over the geometry and the solvercalculates the solutions to the wave equations.MWS includes a modeling tool to create the cavity shapes for RF modeling.Each geometric parameter can be set as a variable to facilitate the creation ofnew geometries. This feature is especially useful when optimizing the couplingparameters of the fundamental power coupler and the HOM absorbers (see sec-tion 3.5).MWS is widely used, but still initial simulations of the TESLA cavity wereperformed and compared to values for HOMs found in [56] and a overall goodagreement was found.Despite the powerful solvers, calculating HOMs of frequencies above 3 GHztakes a long time. A high mesh count is required to provide accurate results,increasing calculation time enormously, making a fast turn around impossible.SLANSSLANS [59] is a 2D code with a module that allows for azimuthal variationsin axis-symmetric cavities. Developed in the early 90’s, it is difficult to use asthe modeling functions are not as advanced as in MWS. The advantage lies inthe computation speed, as 2D calculations are much faster than 3D, even witha very high mesh cell count. The input files for this code are simple text filesand a script was coded to generate the needed input files from the geometricparameters. This was tested again against the TESLA cavity to reproduce thesame results before further calculations were performed.This code was used for a manufacturing tolerance study, described in section3.7.The nature of the 2D axis-symmetric code prohibits the calculation of coupling3.4. Power Couplers 33factors as the couplers break the symmetry.ACE3PThe ACE3P suite [60] is probably the most powerful code to calculate eigen-modes in RF cavities. It uses massive parallel computing and the power of aCray supercomputer to either solve large structures like a full cryomodule cavitystring or to solve small scale problems like peak surface fields or multipacting(explained in section 4.1) with great accuracy. It consists of a number of mod-ules, each for different purposes: Omega3P calculates eigenmodes, while S3Pis used for transmission, T3P for wakefields in time domain and Track3P formultipacting calculations.As a benchmark, the results from this code were compared to the MWS simu-lations, which where benchmarked to HOMS of the TESLA cavity and overallgood agreement in all mode metrics was found.Like MWS, this is a 3D code. The massive parallel computing used in Omega3Pdecreases the calculation time of HOMs significantly, allowing for a precise cal-culation of modes particularly for f > 3 GHz, which is not possible with MWS.3.4 Power CouplersThe external quality factor of the coupler has to be set in a specific way tominimize the required generator power. As the natural resonance width of thecavity at a Q0 of 1 · 1010 and a frequency of 1.3 GHz is of the order of 0.1 Hz,very small fluctuations in frequency lead to a significant increase in the requiredgenerator power to keep the cavity at a stable voltage. Fluctuations in RF fre-quency have many causes, for example pressure variations of the helium bath orvibrations originating from the mechanical environment inside the cryomodule.In addition to the environmental detuning, the beam loads the cavity as dis-cussed in section 2.3.1. The beam induced voltage is 180◦ out of phase withthe beam. Assuming an accelerating phase between beam and RF in the cavity,this results in a reduction of the accelerating voltage in the cavity. To keep thethis voltage at a constant level during beam acceleration, more power needs tobe transferred to the cavity. Using a LRC equivalent circuit for the generator-cavity-beam system, it can be shown [38] that the required generator power fora given cavity voltage is given byPgen =Pcav4β((1 + β + b(ϕ))2 + ((1 + β) tan Ψ− b(ϕ)tanϕ))2 (3.1)where b = PbeamPcav =RshI0Vcavcosϕ, β = Q0/Qext and Pcav =V 2cavR/Q·Q0 . The detuningangle Ψ is given bytanΨ = −2 Q01 + β∆ωω0(3.2)3.4. Power Couplers 34 0 50 100 150 200 250 300 350105 106 107 108P Gen [kW]QextI = 1mA0Hz50Hz100Hz150Hz200Hz 0 50 100 150 200 250 300 350105 106 107 108P Gen [kW]QextI = 3mA 0 50 100 150 200 250 300 350105 106 107P Gen [kW]QextI = 10mAFigure 3.2: Needed generator power as function of beam current, detuningbandwidth and external Q. The minimum is reached, when generator powerequals the gained beam power.with ∆ω as the difference between the frequency of the RF signal and the reso-nance frequency of the cavity ω0. As can be seen, the required generator poweris dependent on the coupling factor β, the detuning bandwidth ∆ω and thebeam current I0. Typically either the beam loading or the frequency detuningis dominant. For a high beam loading accelerator like the eLINAC, equation3.1 simplifies to:Pgen ≈ Pcav4β((β + b)2 + 4Q20(δωωc)2)(3.3)with a dynamic frequency detuning range δω. Frequency detuning is caused byenvironmental effects such as micro-phonics and helium pressure fluctuations.Mirco-phonics are caused by vibrations from outside sources, usually vacuum3.4. Power Couplers 35pumps, that cause the cavity to mechanically vibrate, resulting in a change ofthe RF frequency. The pressure from the helium bath on the outside of thecavity deforms the cavity slightly and the cavity reacts with frequency shift.The sensitivity of a nine-cell cavity is around -140 Hz/Torr. Typical pressurefluctuations on a pressure controlled system are of the order of 0.2 Torr, result-ing in frequency fluctuations around ∼20 Hz. The combined frequency detuningfrom micro-phonics and helium pressure fluctuations can be estimated to lessthan 100 Hz.In the case of the eLINAC with Ibeam = 10 mA, Vcav = 10 MV, R/Q ≈ 1000 Ω(for a nine-cell TESLA cavity) and Q0 = 1 · 1010, the optimal Qext is equal to1 · 106 for frequency detuning up to 200 Hz as can be seen in fig. 3.2, where therequired generator power is shown for different beam currents and frequencydetuning magnitudes as a function of the external quality factor. This Qextcorresponds to a bandwidth of 1.3 kHz, a bandwidth significantly higher thanthe expected frequency fluctuations. At this beam current, the beam loadingdominates the frequency detuning from environmental effects.In the heavy beam loading case, the minimal needed generator power is reachedwhen all RF power is delivered to the beam without any reflected wave in thetransmission line. Reflections are caused by an impedance mismatch betweenthe wave-guide and the cavity-beam system. At optimal coupling, there is noreflection and Pgen equals to the product of cavity voltage and beam current.This simple relationship is valid since the losses in the cavity are significantlysmaller than the power transferred to the beam (Pcav ≈ 10 W compared toPbeam = 100 kW).Cornell University has designed and fabricated a two-cell cavity as an injector totheir ERL project [61]. The couplers in this cavity have to deliver 100 kW of RFpower to the beam, similar to the requirements of the ARIEL eLINAC cavities.The coupler setup for this two-cell cavity consists of twin coaxial coupler ports,placed on the beam tube on one end of the cavity, symmetrically positionedopposite each other. Using this setup, each coupler antenna has to deliver onlyhalf of the needed RF power, 50 kW, to the cavity. A second advantage comesthrough the symmetric setup. With a single power coupler, the local fields onthe beam axis are disturbed and cause a small dipole kick to the beam. Thesymmetric power coupler arrangement avoids this by providing a symmetry inthe perturbation of the fields.The standard TESLA cavity uses only one 40 mm diameter coupler port. Thecoupler ports for the two-cell cavity have a diameter of 62 mm. The Cornellinjector cavity uses a 48 mm radius for the beam pipes which is adopted for theinitial design for the ARIEL cavity along with the size of the coupler ports.The external coupling depends on the position of the couplers with respect tothe cells. The distance between the center of the coupler and the end of theclosest cell Lc (see fig. 3.3(a)) is varied in simulations to find the correct positionwhile maintaining the couplers flush to the beam pipe to further reduce the per-turbations to the fields. The exponential increase in external Q with increasingdistance from the cavity (fig. 3.3 (b)) is a result of the wave-guide nature of thebeam pipe. Below a certain frequency, the cutoff frequency, an electromagnetic3.5. HOM Damping Techniques 365*1051*1062*106 40  41  42  43  44  45Q extDistance from iris [mm]Figure 3.3: Simulations for the coupler position: (a) The parameter Lc is definedas the distance between the center of the coupler and the beginning of the closestcell. (b) Coupling strength as a function of the distance Lc. Optimal couplingis reached with Qext = 106, resulting in a distance Lc = 43.5mmwave does not propagate through the wave-guide, and its amplitude decreasesexponentially further into the wave-guide. The cutoff frequency for a circularwave-guide with radius R is calculated byωcutoff,TM =xmnR(3.4)ωcutoff,TE =x′mnR(3.5)for TM and TE modes respectively while xmn and x′mn are again the n-th rootsof the m-th Bessel function Jm(x) and its derivative J′m, (x). For a beampipewith radius r = 48 mm, the cutoff frequency for the TM01 mode is around2.4 GHz. A cutoff frequency higher than the frequency of the operational modeensures that the RF fields are contained within the cavity and do not traveldown the beam line.3.5 HOM Damping TechniquesAs can be seen in fig. 3.4, for the TESLA cavity a few dipole modes are abovethe set limit of (R/Q)d ·QL < 1 · 107Ω and thus could cause beam break-up athigh currents in ERL operation of the eLINAC. Several methods can be usedto increase the coupling and thus reduce the QL of these higher order modes.3.5.1 HOM couplersIn the TESLA cavity, two HOM couplers (shown in fig. 3.5) are used to cou-ple strongly to HOMs. These couplers introduce an additional way to transferpower out of the cavity, which can be quantified with a QHOM. Ideally, the fun-damental TM010 is not affected by this. For this purpose, the HOM couplershave a reject filter tuned to 1.3 GHz.The two HOM couplers are located on the beampipes of the cavity, one on eachside. A rotation in the angular position helps mitigate different polarizations3.5. HOM Damping Techniques 37102103104105106107108 1.5  2  2.5  3  3.5  4  4.5Shunt-Impdance [Ω]Frequency [GHz]Limit = 10 MOhmGOAL = 1 MOhmFigure 3.4: Shunt impedance spectrum of dipole modes of the TESLA cavity[56, 57]. Several modes are close or above the threshold of 1 · 107 Ω, notably, amode around 2.56 GHz. The displayed chart does not include all dipole modesin this frequency range as there is no literature data for the QL of some of themodes.of HOMs. A HOM with a polarization node in the plane of one of the HOMcouplers is guaranteed to have some significant amplitude in the plane of theother HOM coupler.This concept would be difficult to realize with the proposed twin coupler setupas described in section 3.4 for the ARIEL cavity. A third port would be neededat a longitudinal position close to the power coupler ports, which could makesthe structure difficult to fabricate.Another potential downside of HOM couplers is overheating [62]. The fields inthe region of the antenna cause ohmic losses in the antenna and without directcooling, the dissipated power causes an increase in temperature. The antenna isonly cooled through heat conduction and therefore even just moderate gradientscan heat up the antenna above the critical temperature Tc. When this happens,the antenna becomes normal conducing, causing more losses and eventually thecavity will quench. A quench can be very dangerous as the Q0 of the cavitycan drop multiple orders of magnitude. It is possible, that in this situation thecoupling is matched to the cavity and all the power from the generator will betransferred into the cavity. At 100 kW as expected from the eLINAC powersources, this poses danger to the cryogenic system. Due to the very moder-ate gradient, overheating due to the accelerating field is not a concern for theARIEL cavity.Due to the dual coupler setup used in the ARIEL cavity to accommodate thehigh beam loading, space on the beam pipe with the fundamental power coupleris very limited. The distance between the cells and the HOM coupler is impor-tant for the effectiveness of the HOM coupler. On the TESLA cavity, the HOM3.5. HOM Damping Techniques 38Figure 3.5: HOM coupler used in the TESLA cavity.coupler is in roughly the same distance as the fundamental coupler. Assumingthis would be the case in the ARIEL cavity, three coupler port would be at thesame distance from the cavity on the beam pipe. While not impossible, thiswould raise the difficulty of fabrication and it was decided to work with beamline absorbers to reduce the shunt impedance of the HOMs.3.5.2 Beam Line AbsorbersA different concept of reducing the QL for the HOMs are beam line absorbers.A section of the beam pipe on both sides of the cavity is made out of a lowelectric conductivity material and the HOM magnetic field induces ohmic lossesin this section. The resistive losses provide an additional Qext, which reducesthe overall QL.Since the beam line absorbers are cylindrical symmetric, there is no dependenceof the effectiveness of the absorber on the polarization of the HOM. This repre-sents a clear advantage over the HOM couplers. In addition, the absorber canbe part of a warm-cold transition between the cavity at 2 K and the outsideof the cryomodule at 300 K. In this warm-cold transition, a temperature stageof 77 K (liquid nitrogen boiling temperature) can be used to directly cool theabsorbers and transfer power in the form of heat out of the cryomodule.The reduction of the HOM Q depends on the surface currents of the HOMrunning through the absorber. The beam pipes represent a coaxial wave guidewith a cutoff frequency depending on the radius of the pipe and the mode con-figuration. The surface currents for modes below the cutoff frequency decreaseexponentially when moving further away from the cells, similar to the funda-mental mode as describe in 3.4. The accelerating mode is the TM010-pi modein the lowest frequency passband. All other modes have higher frequencies andare more likely to either penetrate further into the beam pipe or are above thecutoff frequency and propagate through it. Just as in the case of the require-ment that HOM couplers reject the operating mode, HOM absorbers must havelittle or no impact on the Q of the pi-mode.3.6. Design Process 39Figure 3.6: Beam line absorber (in green) mating with the cavity (in grey).The Q of the absorbers is calculated in the same way as for the cavity Q0 (seesection 2.1.2). The power loss depends on the integrated fields on the surfaceof each absorber.The optimal position for the absorber is chosen to be as close as possible to thecavity while having minimal impact on the Q of the fundamental mode. As acriteria, the external quality factor of the absorber Qabsorber was chosen to beequal or greater than 1 · 1011 combined or 2 · 1011 for either side individually.Since the electrical conductivity of the material has a fundamental influence onthe power loss in the absorber, the choice of material is crucial and has to beconsidered when determining the position of the absorber. V. Shemelin and S.Belomestnykh from Cornell university measured a material called CESiC withvery promising characteristics i.e. a thermal conductivity of 30 W/m·K at 50 Kand an electrical conductivity of 15 kS/m over a wide range of frequencies [63].These values are used for the initial calculations. Tests of the damping qualitiesat cryogenic temperatures will be shown in section 3.9.The calculations for the position of the absorber have to be repeated every timethe radius of the beam pipe is changed during the cavity design as the radiusaffects the penetration of the accelerating mode into the beam pipe.3.6 Design ProcessThe design of the ARIEL cavity started with the TESLA nine-cell cavity vari-ant. The end groups are labeled by the fundamental power couplers or thepick-up coupler that are on opposite sides of the cavity. As can be seen in table3.2 the cavity is slightly asymmetric in the end-half-cells,but symmetric in beam3.6. Design Process 40Table 3.2: Geometric parameters of the three different half-cells of the TESLAcavity. All dimensions are in mm.Inner half-cell Coupler side half-cell Pick-up half-cellReqautor 103.3 103.3 103.3Riris 35 39 39A 42 40.34 42B 42 40.34 42a 12 10 9b 19 13.5 12.8L 57.7 55.7 56.8pipe sizes with R = 39 mm. The RF parameters of this cavity are listed in table3.1. Those values are used as a guideline for the design of the ARIEL cavity.Since TESLA was designed to be operating at 25 MV/m, the surface field ratiosare much more of a concern compared to the ARIEL cavity, as the operatinggradient is more than a factor of two smaller. A sacrifice in surface fields tomeet the HOM requirements is acceptable.For two reasons the TESLA cavity cannot be used as is: the fundamental powercoupler requires a bigger beam pipe and the HOM spectrum exceeds the re-quirement of Rd/Q < 107 Ω (see fig. 3.4). To comply with the power couplerrequirement the beam pipes on both sides of the cavity have been initially in-creased in radius from 39 mm to 48 mm. While technically the larger beam pipeis only needed on the power coupler side of the cavity, the symmetry is advanta-geous since less transitions between different beam pipe sizes would be required.To reach an even field distribution the end half-cells had to be modified resultingin the geometry listed in table 3.3 with RF parameters in table 3.4. Increasingthe radius of the iris at both ends decreases the shunt impedance slightly andincreased both peak field ratios compared to the TESLA cavity. As can be seenin fig. 3.7 many modes between 1.4 GHz and 3 GHz are significantly abovethe BBU limit of 10 MΩ and some form of reduction of the shunt impedance isneeded. Due to the earlier discussed limitations on HOM couplers, the choicewent to beam line absorbers to avoid the need to design a third port on thepower coupler end of the cavity.3.6.1 Beam Line Absorber DesignThe effectiveness of the beam line absorber depends on the beam pipe radius, theposition with respect to the cavity as described in section 3.5 and the strengthof the mode in that section of beam pipe. The absorber is modeled as a smoothcontinuation of the beam pipe. In reality there will be a transition and theabsorber forms a coaxial structure with the beam pipe as can be seen in fig.3.6.The calculated external Q of one absorber for the TM010 pi-mode is shown in3.6. Design Process 41Table 3.3: 48/48 cavity variant. All dimensions are in mm. The end cells aretuned for field flatness.Inner half-cell Coupler side half-cell Pick-up half-cellReqautor 103.3 103.3 103.3Riris 35 48 48A 42 45 45B 42 40.5 40.5a 12 10 10b 19 13.5 13.5L 57.7 56 56Table 3.4: 48/48 variant RF parameters.f [GHz] Geo. Factor G [Ω] R/Q [Ω] Ep/Eacc Bp/Eacc [mT/(MV/m)]1.3 289 984 2.39 4.5910110210310410510610710810910101011 1.4  1.6  1.8  2  2.2  2.4  2.6  2.8  3  3.2Shunt Impedance [Ω]Frequency [GHz]Limit = 10 MOhmGOAL = 1 MOhmFigure 3.7: Several dipole modes of the initial 48/48 cavity variant are signifi-cantly above the HOM limit of 10 MΩ. Some form of Q reduction is needed.3.6. Design Process 421051061071081091010101110121013101410151016 0  0.05  0.1  0.15  0.2  0.25Q damperz [m]CESIC, 15 kS/mStainless Steel at 4 K, 1.4 MS/mFigure 3.8: The Q of the beam line absorbers rises exponentially with thedistance from the cavity. z is the distance between the iris of the end half-celland the start of the absorber. The integration over the H-field to calculate thepower loss extends far into the distance.fig. 3.8 for two different choices of material with significantly different electricalconductivity. From this, the position of the absorber with respect to the cavityis determined. The length of the absorber is important as too short an absorbermight not be effective in reducing the QL of dangerous HOMs. To estimate theeffective length the change in Q with the length of the absorber is calculatedand displayed in fig. 3.9 for the TM010 mode again. The plot shows that alength of 40mm is sufficient to saturate the damping. HOMs closer to their cut-off frequency or above their cutoff frequency, penetrate further into the beampipe or even propagate fully into it. The effectiveness of the absorber for thosemodes needs to be considered as well. While the effect of the absorber saturatesat an absorber length of 40 mm for the TM010 mode, a length of 60 mm ischosen to suppress modes penetrating further into the pipe. The estimation ofthe length of the absorber is common for different pipe radii and is not neededto be repeated for other beam pipe variants. With the beam line absorber setup in this way the shunt impedance of several high Rd modes are significantlyreduced as can be seen in fig. 3.10. Most of the previously dangerous modeshave their shunt impedance reduced below the limit set by beam dynamics andfrom considerations of manufacturing tolerances (see section 3.7). One dipolemode around 2.56 GHz still is significantly above the shunt impedance goal of1 MΩ. This mode is the same mode mentioned in [57] with a high Rd/Q of70 Ω, while most dipole modes have an Rd/Q of ≤ 1 Ω. It turns out that themode is trapped inside the cavity, since the field amplitude of this mode is closeto zero in the beam pipe and is unaffected by the HOM absorbers. The fieldplot is shown in fig. 3.11. Note that the absorbers cannot be moved closer tothe cavity without interfering with the accelerating TM010 mode, modifications3.6. Design Process 43101110121013 0  0.02  0.04  0.06  0.08  0.1Q AbsorberAbsorber Length [m]CESIC, 15 kS/m, z0 = 0.14 mStainless Steel at 4 K, 1.4 MS/m, z0 = 0.11 mFigure 3.9: The effective length of the absorber is independent of the conduc-tivity. A length of 60 mm absorber length is required to saturate the Q for theTM010-pi mode. The Q for CESiC is higher compared to the SS due to thedifferent positions of the absorbers with respect to the beam pipe.to the cavity shape have to be made to increase the coupling of this mode tothe HOM absorber.3.6.2 HOM PolarizationWhile the beam line absorbers are polarization independent, the axial symmetryof the cavity is broken by the power couplers. The two couplers are set in thehorizontal plane with respect to the outside world, and force the polarizationof the modes. Additionally, the couplers create a degeneration of dipole modes:two possible polarizations with equal solutions to the wave equations.In 3D simulations the polarization can be forced by selecting appropriate sym-metry conditions along the two longitudinal (horizontal and vertical) symmetryplanes. The boundary conditions define that the transverse field component foreither the electric or magnetic field vanishes in the symmetry plane. Depend-ing on the combination of symmetry plane and symmetric field only certainmodes are calculated. For magnetic horizontal and magnetic vertical symmetry,monopole modes and quadrupole modes of a certain polarization are calculated.Magnetic horizontal and electric vertical symmetry result in dipole modes likethe example shown in fig. 2.6. Flipping the symmetry conditions to electric-magnetic rotates the dipole mode by 90◦.Calculating the shunt impedance spectrum for dipole modes (fig. 3.12) indi-cates a clear difference between the two polarizations. For low frequency modesin the first and second dipole pass-bands between 1.5 GHz and 2 GHz the ver-tical polarization has a lower shunt impedance than the horizontal polarization.3.6. Design Process 4410110210310410510610710810910101011 1.4  1.6  1.8  2  2.2  2.4  2.6  2.8  3  3.2Shunt Impedance [Ω]Frequency [GHz]w/o AbsorberUsing CESiC HOM AbsorberLimit = 10 MOhmGOAL = 1 MOhmFigure 3.10: The used CESiC absorbers on the beam pipes only affect the fewvery high shunt impedance modes. The 2.56 GHz mode is untouched at 2·107 Ω.Figure 3.11: The TM111 mode around 2.56 GHz is trapped inside the cavity:no significant fields in the region of the beam line absorbers.3.6. Design Process 45100101102103104105106107108 1.6  1.8  2  2.2  2.4  2.6  2.8  3  3.2Shunt Impedance [Ω]Mode Frequency [GHz]Vertical (E-H)Horizontal (H-E)Figure 3.12: A different polarization of the same mode results in a slightlydifferent shunt impedance. The vertical polarized mode have overall a highershunt impedance compared to the horizontal modes, therefore this polarizationis considered more dangerous and taken as baseline in the following simulations.Higher frequency modes between 2.8 and 3.2 GHz reverse this trend with slightlylower Rd for vertically polarized modes. The change in shunt impedance for thetwo frequency regimes is a result of stronger coupling to the power couplers,resulting in a lower QL. The lower frequency modes are generally dominated bythe coupling to the power couplers, as the cutoff frequencies for a beam pipe ofradius R = 48 mm are at 1.83 GHz for TE11p modes and 3.8 GHz for TM11pmodes. The first dipole mode pass-band is below these frequencies and is atten-uated in the beam pipe, similar to the TM010 mode discussed in section 3.4.Further evaluation with 3D codes during the design process takes into accountthat the vertical polarization in general has a higher shunt impedance for manymodes and this polarization is chosen as the standard in further calculations.Figure 3.13: The different polarizations of one mode couple differently to thefundamental power couplers.3.6. Design Process 4610010210410610810101012 1.4  1.6  1.8  2  2.2  2.4  2.6  2.8  3  3.2Shunt Impedance [Ω]Frequency [GHz]Without AbsorbersUsing CESIC HOM AbsorberLimit = 10 MOhmGOAL = 1 MOhmFigure 3.14: Dipole spectrum of the 48/55 cavity variant with and withoutbeam line absorbers. Most data points overlap, showing no effect of the beamline absorbers for those modes. The 2.56 GHz mode gets reduced in shuntimpedance from 108 Ω to just below 107 Ω.3.6.3 Modifying the CavityDesign modifications to the cavity focused on reducing the 2.56 GHz mode. Thismode has a high geometric shunt impedance and is trapped in a symmetricalcavity like the TESLA cavity or the earlier discussed 48-48 variant. Furthermodifications to the cavity shape have to be made to reduce shunt impedanceof this mode. Some constrains on possible modifications are already in effect.The shape of the inner cells is locked to the TESLA cavity to reduce fabrica-tion expenses. The coupler end group is optimized to match the couplers. Theremaining part of the cavity is the pick-up side end group, consisting of the lasthalf-cell and the beam pipe.Altering the beam pipe radius changes the cutoff frequency accordingly andpotentially allow the mode to propagate into the beam pipe or at least to pen-etrate far enough into the beam pipe to allow the absorbers to work. Since thisis a TM11 mode, the cutoff frequency with a beam pipe of 48 mm is around3.8 GHz. Getting the cutoff frequency to be below 2.56 GHz would require theradius to go up to over 70 mm, twice the size of the TESLA cavity. This wouldlead to high electric surface fields in the end half-cell and potentially an untune-able cavity. A fully propagating 2.56 GHz mode is impractical but increasingthe mode penetration into the beam pipe seems achievable. As a half-measurea cavity with a beam pipe radius of 55 mm (f cuttoff,TM11 = 3.32 GHz) was sim-ulated. The geometry parameters for a tuned cavity can be found in table 3.5with the RF parameters in table 3.6. The peak surface field ratios increase sig-nificantly compared to the 48-48 cavity geometry. 61 mT surface B-field at theoperational gradient of 10 MV/m is still below the on-set for high field losses3.6. Design Process 47Table 3.5: 48/55 cavity variant. All dimensions are in mm. The end cells aretuned for field flatness.Inner half-cell Coupler side half-cell Pick-up half-cellReqautor 103.3 103.3 103.3Riris 35 48 55A 42 45 48B 42 40.5.5 36.25a 12 10 9b 19 13.5 12.8L 57.7 56 57Table 3.6: 48/55 variant RF parameters for the operational TM010 mode.f [GHz] Geo. Factor G [Ω] R/Q [Ω] Ep/Eacc Bp/Eacc [mT/(MV/m)]1.3 291 964 2.9 6.1which usually start around 80 to 100 mT. In addition to the increased peaksurface field ratios, the R/Q decreases a small amount from 986 Ω to 964 Ω.The dipole shunt impedance spectrum (fig. 3.14) shows a reduced shunt im-pedance for select high Rd modes including the trapped 2.56 GHz mode. Thismode sees a reduction from 108 Ω to just below 107 Ω. Strictly speaking, thiswould satisfy the BBU limit set by the beam dynamics, but not the strongerrequirement modified by the manufacturing tolerances (see sec. 3.7).A systematic approach to changing the pick-up side iris radius was used todetermine an effective cavity design. While changing the pick-up side radius,the flatness (and with it the optimal R/Q value) of the accelerating mode ismaintained by varying the ellipse parameters of the pick-up side half cell. Theiris radius was changed between 39 mm and 48 mm. This range was chosen toincorporate the TESLA design values. The 2D code SLANS was used for thisstudy to increase calculation speed, as a full 3D calculation have taken severaldays. During these calculations the absorber position and length was kept fixedto see the changes on the shunt impedance of the 2.56 GHz mode.Reducing the end cell iris radius and the beam pipe radius with it increases thegeometric shunt impedance of the fundamental mode linearly as can be seen infig. 3.15. The smaller iris focuses the electric field onto the axis better than abig iris, resulting in the observed trend. A high shunt impedance is desired soa lower iris radius is preferred.A similar trend can be found for the Rd/Q value of the 2.56 GHz mode as canbe seen in fig. 3.16. With bigger beam pipes the geometric shunt impedancedecreases slightly1. Going from 39 mm to 48 mm the Rd/Q changes from 71 Ω1The dipole mode simulation with Riris = 45 mm was found later to have a corruptedoutput file and did not produce accurate results. Figure A.2 shows a steady reduction of theRd/Q for this mode for cavities with bigger inner irises. A similar trend has to be assumedin this case.3.6. Design Process 48 972 974 976 978 980 982 984 986 988 990 38  40  42  44  46  48R/Q [Ω]Pick-up Side Beam Pipe Radius [mm]Figure 3.15: The shunt impedance of the accelerating TM010 mode decreasesas the pick-up side beam pipe and end-cell iris increases, making a smaller irisdesirable.to to about 62 Ω, a decrease by a factor of 0.8. In the 48/48 cavity design dis-cussed earlier, the final shunt impedance needs to be reduced by over one orderof magnitude. To reach a Rd/Q of about 7 Ω, the beam pipe radius would haveto be increased to about 96 mm if this trend would continue linearly. This wouldbe about the size of the equator. Clearly this is not a viable way to reduce thedipole shunt impedance spectrum and other ways have to be found as it alsowould significantly reduce the shunt impedance of the accelerating mode.The strength of a trapped mode can be estimated by measuring the frequencyshift, when changing the boundary conditions at the ends of the beam pipesfrom a perfect electric conducting wall to a magnetic boundary [64]. A smallchange in frequency is the result of a small field amplitude at the end of thebeam pipes, in other words a trapped mode. As a figure of merit the ratio offrequency shift to average frequencyK = 2|fM − fE |fM + fE, (3.6)with fM/E as the frequency with either magnetic or electric boundary at theend of both beam pipes, can be used to evaluate if a geometry change has aneffect on the trapped status of a mode.The dependence of the trapping parameter K for the 2.56 GHz mode on thepick-up side end half-cell iris is shown in fig. 3.17. No clear trend is found whilechanging the end cell. It is not surprising as the mode is mostly confined intothe inner cells, which have not been changed in the simulations.Similar to the trapping parameter, the actual frequency (fig. 3.18) of this modedoes not change significantly in the given range for the pick-up side end groupvariations. Since this mode is relatively close to a higher harmonic of the beam3.6. Design Process 49 62 63 64 65 66 67 68 69 70 71 72 38  40  42  44  46  48R d/Q [Ω]Iris Radius [mm]Figure 3.16: The geometric shunt impedance of the dangerous 2.56 GHz modeslightly decreases with increasing the beam pipe radius.frequency, only about 40 MHz difference, it is important to carefully track thismode’s frequency.Despite the trapped status of this mode, it shows a slight asymmetry in thefield amplitudes in the beam pipes. In fig. 3.19 the Q values of the couplerside and pick-up side are shown in addition to the combined Qabsorber. At lowerradii, the absorber Q on pick-up side of the cavity decreases significantly whichcorresponds to a higher field amplitude in the beam tube on that side of thecavity.3.6.4 The 39-48 Cavity VariantThe 39-48 cavity has good properties of the accelerating TM010-pi mode, listedin table 3.8. The design has very similar parameters to the TESLA cavity, withonly slightly higher peak surface field ratios and a slightly lower R/Q. In addi-tion to those attractive parameters the pick-up side end half-cell has identicalgeometric parameters to the TESLA cavity (tables 3.2 for the TESLA cavityand 3.7 for the 39-48 variant).The HOM absorber positions for a Qabsorber ≥ 2 · 1011 have been determinedto be 125 mm for stainless steel and 151 mm for CESiC on the coupler side ofthe cavity and 88 mm (stainless steel) / 105 mm (CESiC) on the tuner side.Those are the distances between the end of each cell on either end of the cavityand the beginning of a 60 mm long beam line absorber section. The differencein distances on different sides of the cavity are due to the different exponentialdecay of the field amplitude, which is caused by different beam pipe sizes andthe corresponding cutoff-frequencies.During the investigation it has been determined that stainless steel absorbers3.6. Design Process 500.0*1005.0*10-51.0*10-41.5*10-42.0*10-42.5*10-43.0*10-4 38  40  42  44  46  48Trapping Parameter KIris Radius [mm]Figure 3.17: The trapping parameter K for the 2.56 GHz mode shows no cleardependence on the pick-up side end group.on the coupler side provide sufficient loading to the HOM that are propagatingout of the coupler end of the cavity. Stainless steel is a preferred material asit is robust compared to materials like CESiC. These characteristics are usefulas they can prevent particulate contamination in the SRF cavity in case theabsorber gets damaged. In addition stainless steel is easy to prepare with stan-dard clean room techniques. CESiC can be prepared to meet class 10 cleanroom specifications, but extra steps are needed.Assuming stainless steel absorbers on the coupler side and CESiC absorbers onthe pick-up side, the dipole spectrum is calculated as shown in fig. 3.20. Allmodes up to 4.4 GHz are sufficiently suppressed well below the limit of 107 Ω setby beam dynamics calculations and even further to around 106 Ω. The 2.56 GHzmode, which was considered the most dangerous, has been reduced to a shuntimpedance of just over 106 Ω, not quite fulfilling the stricter goal set with man-ufacturing tolerances in mind. But since this mode is only minimally above thegoal, this is acceptable. Other designs might have better HOM characteristicsas a bigger beam pipe reduces the Rd/Q for the 2.56 GHz mode, but the higherperformance in the accelerating mode is desirable and the HOM suppression issufficient in this design.During ERL operation a different bunch frequency might be chosen to facili-tate both ERL and RIB operation. This means for the cavity design that thereare potentially more resonances between the cavity and the beam as a lowerbunch frequency creates a different harmonic spectrum with more harmonicresonances. The design beam for RIB production has a frequency of 650 MHzwith harmonics of 1300 MHz, 1950 MHz, 2600 MHz and so on. No mode, dipoleor monopole, in the investigated frequency range is close to those harmonic fre-quencies, even considering a frequency spread of 1%.For a beam with fb = 100 MHz, the situation is slightly different. One monopole3.6. Design Process 51 2500 2550 2600 2650 2700 38  40  42  44  46  48Frequency [MHz]Iris Radius [mm]Figure 3.18: The frequency also changes only minimally while changing thepick-up side end group.Table 3.7: 48/39 cavity variant. All dimensions are in mm. The end cells aretuned for field flatness.Inner half-cell Coupler side half-cell Pick-up half-cellReqautor 103.3 103.3 103.3Riris 35 48 39A 42 45 42B 42 40.5.5 42.275a 12 10 9b 19 13.5 12.8L 57.7 56 57mode around 2.477 GHz has the potential of dissipating a significant amountof power in the cavity, of the order of 23 kW and could be resonant with thebeam given the frequency uncertainty from fabrication of 1%. Calculating theprobability that the mode will be close (within its bandwith) to 2.5 GHz showsa probability of 5.6·10−9, smaller than winning lotto 6/49. This is an acceptablerisk, especially as the HOM frequency can be changed by detuning the cavityand then tuning it to its operational frequency again [65].Table 3.8: 48/39 variant RF parameters for the operational TM010 mode.f [GHz] Geo. Factor G [Ω] R/Q [Ω] Ep/Eacc Bp/Eacc [mT/(MV/m)]1.3 272 989 2.1 4.43.7. Manufacturing Tolerances Analysis 52104105106107 38  40  42  44  46  48QIris Radius [mm]Couper side Absorber QPick-up side Absorber QCombinedFigure 3.19: The combined absorber Q is dominated by the coupler side absorberfor a fixed position of the absorbers.3.6.5 Modifying the Inner CellsFor completeness of the design, modifications of the inner cells of the cavity havebeen simulated to see the effect on the RF performance, with the impact on the2.56 GHz mode of particular interest. The same procedure as described in theprevious section has been applied. The parameters changed are the inner irisradius and the pick-up side iris radius with the attached beam pipe of the sameradius. To keep the flatness as close as possible to its optimal value, the ellipseparameters were adjusted. The inner iris radius was varied from its nominalvalue of 35 mm to 39 mm in 0.5 mm steps.The R/Q of the accelerating mode decreases linearly with increased inner irisat a rate of around 17 Ω/mm. This is related to the fact that a smaller irisprovides stronger electric fields on the beam axis, where the R/Q is calculated.The inner iris radius has a strong impact on the shunt impedance of the 2.56 GHzmode. A bigger iris reduces the Rd/Q to almost half the value when changingthe iris from 35 mm to 39 mm when the end group iris is at 39 mm. For biggerend group iris sizes the same trend is found, with slightly different absolutevalues.This analysis lead to a number of cavities that fulfill the HOM requirements, butwould lead to additional fabrication costs due to the modification of the innercells, which would need new fabrication tools and delay the manufacturing ofthe cavity. The plots demonstrating the correlation with iris size this can befound in appendix A.3.7. Manufacturing Tolerances Analysis 531001011021031041051061071081091010 1.5  2  2.5  3  3.5  4  4.5Shunt-Impdance [Ω]Frequency [GHz]w/o Absorberswith AbsorbersLimit = 10 MOhmGOAL = 1 MOhmFigure 3.20: HOM spectrum for the 39/48 cavity variant. A strong suppressionof all HOM up to 4.4 GHz can be found when using CESiC and stainless steelabsorbers.(make legend bigger)3.7 Manufacturing Tolerances AnalysisManufacturing tolerances create uncertainties in the final shape of the cavity.Variations in the cavity shape impact RF parameters like frequency and R/Qvalues for all modes. To understand and quantify this influence the code SLANSwas used. Each geometric parameter defining the shape was randomly changedwithin a Gaussian distribution centered around the design value. The range ofthe change was chosen to be ±5σ with σ = 0.1 mm. A script was coded tocreate the input files for unique cavity shapes into SLANS.In total 60 different cavities were simulated with this method. The values offrequency f , R/Q and Q0 were recorded and analyzed. The average x¯ and thestandard deviation σx for each of those figures of merit are calculated and theratio of σx/x¯ is taken as a measure of how much the modes vary.As can be seen in fig. 3.7 (data in appendix C), the frequency for each dipoleHOM varies less than 1% around its average value. At frequencies of the orderof 2 GHz, 1% corresponds to 20 MHz. Therefore when evaluating if a HOM is inresonance with a harmonic of the beam a 20 MHz window should be considered.The geometric shunt impedance seems to vary quite strongly within the toler-ances. This motivates a change in the HOM requirement fromRd/Q·QL < 107 Ωto < 106 Ω. This tighter requirement is still fulfilled with the 39/48 design. Thehigh Rd/Q mode at 2.56 GHz varies only by about 8%, significantly less thanother dipole modes.Looking at the accelerating TM010-pi mode, the calculations result in an averageR/Q of 700 Ω with a standard deviation of 180 Ω. Since tuning a cavity wouldincrease the R/Q close to the optimal value of around 1 kΩ (a flatness of > 95%is required for efficient acceleration), a more realistic choice of samples are those3.8. HOM Power 54 0.0001 0.001 0.01 0.1 1 10 1.5  2  2.5  3  3.5norm. stand. DeviationFrequency [GHz]Mode FrequencyRd/QQFigure 3.21: Sensitivity of HOM parameters f , Q and Rd/Q to imperfections inthe manufacturing process. On the X-axis is plotted the average HOM frequencyand on the Y-axis the sensitivity parameter σx/x¯.Table 3.9: If considered resonant with the beam, these six dipole HOMs woulddissipate each more than 1 W into the cavity.Frequency [GHz] 2.431 2.439 2.477 2.745 2.755 2.76Pdiss [W] 2.7 34 23000 1.7 384 276with a high R/Q of the accelerating mode. Six of the simulated cavities havean R/Q of 900 Ω or higher. Reducing the sample size to those six cavities doesnot change the spread in frequency or geometric shunt impedance of the dipoleHOMs significantly as can be seen in fig. 3.22.3.8 HOM PowerAs described in section 2.3, higher order modes can cause significant powerlosses in the cavity, especially if a HOM frequency matches an integer harmonicof the bunch repetition frequency. As the frequency difference between theHOM resonant frequency and a harmonic of the bunch repetition frequency ∆ωapproaches zero, the induced power increases significantly. Therefore it is veryimportant that these resonances are avoided.Worst case calculations with all modes assumed to be in a resonant conditionwith the beam predict HOM power of over 23 kW of dissipated power in thecavity. This is dominated by six modes with power dissipation over 1 W, whichare listed in table 3.9. All other calculated modes would contribute less than1 W combined, which is tolerable. Fortunately, all of those modes are far awayfrom the main RIB production beam harmonics. The closest harmonics are at3.9. HOM Absorber Tests 55 0.0001 0.001 0.01 0.1 1 10 1.5  2  2.5  3  3.5norm. VarianceFrequency [GHz]Mode FrequencyRd/QQFigure 3.22: Sensitivity of HOM parameters f , Q and Rd/Q vs average HOMfrequency for selected cavities with high shunt impedance of the acceleratingTM010 mode.1.95 GHz and 2.6 GHz, so that no HOM is in resonance with this beam.For the ERL beam, a different repetition frequency is considered. A strong can-didate is 100 MHz, which would be close to the HOM at 2.477 GHz. Accordingto the manufacturing tolerance study, the frequency varies with up to 1% ofthe calculated frequency. This means, there is a non-zero probability, that thismode is in resonance with a 100 MHz beam. The probability can be calculatedassuming a Gaussian distribution of the resonance frequency and taking intoaccount the bandwidth of this mode. The bandwidth is a result of the Q of themode and taken from the average of the simulated cavity Ql. Using these as-sumptions, the probability calculates to ∼ 10−9, a very low value. Other modesare further away from resonances.3.9 HOM Absorber TestsThe HOM absorbers were designed assuming an electrical conductivity of 15 kS/m.To test this conductivity at the operational temperature of 77 K, cold measure-ments were carried out using a Q pertubation method. Inserting a sample insidea RF cavity changes the measured Q depending on the position and conductiv-ity of the sample. The measurement results are then compared to simulationswhere the conductivity is a free parameter.The sample is a cylindrical tube, 60 mm in length, with an outer diameter of74 mm and an inner diameter of 72 mm made out of HB-CESiC [66] material. Aniobium 1.3 GHz single cell cavity is used to provide the RF field. The sample isplaced into one of the beam tubes (inner diameter 78 mm) and held concentricwith the beam tube and insulated from the beam tube with wraps of Teflontape.3.9. HOM Absorber Tests 56 0 0.2 0.4 0.6 0.8 1 1.2-60 -40 -20  0  20  40  60  80Rel. change in QDistance from Iris [mm]Simulation (4500 S/m)Simulation (10000 S/m)Simulation (1000 S/m)MeasurmentFigure 3.23: Simulation results for three different electrical conductivities tomatch the room temperature Q measurements of the TM010 mode. The centercurve with σ = 4500 S/m is the best fit to the data.The measurements focused on the first two monopole modes TM010 at 1.3 GHzand TM011 at 2.4 GHz. COMSOL 2D was chosen to simulate the RF fields. Inthe simulations the position of the sample is changed with respect to the transi-tion point from cell to beam pipe. The conductivity is varied in post processingfor the best fit to the measurement results. The ratio of the QL loaded with thesample to the unloaded Q0 is compared to the simulations of the TM010 modeand shown in fig. 3.23. The penetration of the TM010 mode into the beampipe is clearly visible. Two simulation curves with conductivities of 1 kS/mand 10 kS/m indicate the sensitivity of the measurement technique. The bestfit to the data shows a conductivity of 4500 S/m at room temperature. Thesemeasurements indicate that the front edge of the sample should be placed nearthe iris to get roughly 50% reduction in Q. The sample holder for the coldmeasurement of the TM010 mode is designed to hold the sample at a positionwhere the sample edge is located 5 mm from the iris. The cavity is cooled bothwith and without the damping material to get both Q0 and QL.The experimental setup is shown in fig. 3.24. The cavity is cooled with liquidnitrogen via copper plates attached to the beam pipes on both sides of the cell.The whole assembly is lowered into a cryostat, which then is evacuated to pro-vide thermal insulation from the outside. Inside the cavity is the sample holdersitting on the bottom flange with the sample on top of it. RF connections leadto the top of the cryostat and to a network analyzer measuring the transmissionsignal through the cavity to determine the Q via the 3 dB method.In fig. 3.25 the dependence of the conductivity as a function of Q ratio isshown for the designed sample position of 5 mm away from the iris. Includedin the plot is the conductivity for 3 and 7 mm distance to show the measure-ment uncertainty. A ratio of 11350/19800 = 57% results in a conductivity of3.9. HOM Absorber Tests 57Figure 3.24: (a) A 1.3 GHz Niobium Cavity used for this test. The cavity is sus-pended from the top and a copper cooling line attached around the beam pipe.Temperature sensors are located at the top and bottom flange. (b) Schematicof the assembly: the sample rests on the sample holder and is centered by it.The input antenna is inserted from the top and the pick-up antenna from thebottom. The top flange has an additional hole to evacuate the RF space.3.10. Final Design Overview 58 0 2000 4000 6000 8000 10000 30  35  40  45  50  55  60  65  70Conductivity [S/m]QL/Q0 [%]7mmFIT (7mm): 3.003*exp(0.1028*QL/Q0)5mmFIT (5mm): 4.711 * exp(0.1040*QL/Q0)3mmFIT (3mm): 17.18*exp(0.09344*QL/Q0)Figure 3.25: At a fixed position, the sample conductivity increases exponen-tially with a increasing ratio of QL/Q0. The plot for 3 and 7 mm correspondto the uncertainties in the position measurement of the sample.Figure 3.26: Cross-section through the final mechanical model of the ARIEL1.3 GHz SRF cavity.σ = 2200 S/m with an uncertainty ranging from 800 to 5300 S/m, which derivesfrom both the position uncertainty and the Q ratio uncertainty of each 10%.A similar analysis is done for the TM011 mode. The conductivity measured at2.4 GHz and 77 K falls into the range between 660 and 3400 S/m, agreeing withthe 1.3 GHz measurement. More details of the measurement can be found in[2]. Due to this lower than assumed conductivity the absorber is moved slightlyfurther out of the cavity to prevent higher losses in the accelerating pi mode.3.10 Final Design OverviewFigure 3.26 show the final design of the bare ARIEL cavity without any powercouplers or HOM absorbers. The inner cells are identical to the original TESLAdesign with stiffening rings welded in between each of the cells. The rings helpto reduce the frequency sensitivity to pressure changes in the helium bath. On3.10. Final Design Overview 59Table 3.10: RF parameters of the ARIEL 39-48 cavity compared to the TESLAcavity.ARIEL TESLAFrequency [GHz] 1.3 1.3Design Gradient [MV/m] 10 25Duty Factor [%] 100 0.7Operational T [K] 2 2Geo. Factor G [Ω] 272 270R/Q [Ω] 989 1034Ep/Eacc 2.1 2.0Bp/Eacc [mT/(MV/m)] 4.4 4.26the right side are two ports to accommodate two 50 kW power couplers, whichwill be set to an external coupling Q of 1 · 106 (βcoupling = 10000). For verticaltests a dedicated variable coupler is used (more on that in section 4.3) on oneof these ports to reach critical coupling. After the cavity passes vertical per-formance tests, a helium jacket is welded onto the transition pieces, which arelocated just where the cells transition to the beam tubes.The asymmetry in the beam pipes helps reduce the shunt impedance of severalHOMs, especially a high Rd/Q mode at around 2.56 GHz. Beam line absorbers(not shown) made out of stainless steel on the coupler side and CESiC on thepick-up side of the cavity transform energy stored in the electromagnetic fieldsof the HOM into thermal energy, which then will be transferred out of the cry-omodule using liquid nitrogen.The RF parameters are summarized in table 3.10. The cavity is very similarto the TESLA cavity, with slight performance decreases in R/Q and peak fieldratios.A manufacturing tolerance study shows HOM variations in frequency below1% and in Rd/Q of a factor of 2. This motivates a stronger HOM requirementof Rd/Q ·QL ≤ 106 Ω to account for these variations. No HOM is in close prox-imity of a harmonic of the RIB bunch frequency of 650 MHz or a number ofsub-harmonics that are considered for the FEL beam (100, 110, 120, 130 MHz).The HOM data for this cavity can be found in Appendix B.Fabrication of this cavity was done by PAVAC Industries, Inc., located in Rich-mond, B.C., close to TRIUMF. PAVAC worked with TRIUMF on the secondphase of the ISAC-II superconducting LINAC.60Chapter 4Cavity RF PerformanceMeasurementsWith the design and fabrication of the cavity done, the performance of the cav-ity has to be measured. One constant research topic in the SRF community is”how can the SRF cavities perform better”, which usually means higher gradi-ents and higher Q0 values. Higher gradients result in less cavities required for aset specific goal energy. For example, the ILC has the highest gradient require-ments with ≥ 31.5 MV/m averaged over 16000 cavities. If the average gradientis increased, fewer cavities can be used, resulting in significant cost reductionsas cavities, cryomodules and high power RF amplifiers are expensive pieces ofequipment.As stated earlier, the Q0 value is a measure of the RF losses in the cavitywalls, with a higher Q0 corresponding to lower losses. Those losses have to becompensated by the cryogenics system, which is another significant expense forparticle accelerators. Therefore high Q0 values are very desirable, especiallyfor cw applications like the eLINAC. As an example, LCLS-II at SLAC willoperate its cavities in cw and requires a Q0 of 2.7 · 1010 at 16 MV/m and 2 K,to significantly reduce the size of the cryogenic plant and therefore the capitalcosts of the accelerator.Over the past 40 years of operation of SRF based accelerators, the Q0 andgradient have constantly improved. This performance increase is probably bestdemonstrated by the CEBAF 12 GeV upgrade, where in 2013, ten new cry-omodules produced the same energy gain as 40 of the old cryomodules from1994 [67, 68].4.1 Performance LimitationsCavity performance in either Q0 or Eacc is limited by fundamental boundaries.The super-heating field of niobium Hsh at ≈240 mT/µ0 sets the limit of thegradient to about 50 MV/m for the ARIEL cavity or any other elliptical cavitywith similar surface field ratios. But several other phenomena usually limit thegradient from reaching this fundamental limit as shown in fig. 4.1. Since theoperational gradient is only 10 MV/m, reaching the fundamental limit is not arequirement for the ARIEL cavities.The Q0 is fundamentally limited by the BCS resistance RBCS . There are anumber of surface treatments that change the parameters that govern RBCS . In4.1. Performance Limitations 61eacbfghdBpeak [mT]Q0Quench∼20 ∼100 ∼200Figure 4.1: The cavity performance on the peak magnetic surface fields islimited by several effects and influenced by the surface treatment [not to scale]:a) at low surface fields a small increase in Q0 can be measured (LFQS); b) atsurface fields between 20 and 100 mT a gentle decrease in Q0 takes place; c) athigher gradients a sharp drop in Q0 can be observerd (HFQS); d) HFQS can becured with a 120◦C UHV bake; e) newer treatments like nitrogen doping showa higher Q0 with higher gradients but seem to be limited to medium gradientsbefore a quench occurs; f) multipacting can be overcome with conditioning; g)Q-disease causes a strong Q slope at low to medium gradients; h) defects causea very low Q0 and an early quench.4.1. Performance Limitations 62addition to the BCS resistance, the temperature independent residual resistanceRres reduces the Q0. This residual resistance is usually in the order of 2 to 30 nΩdepending on the surface treatment and the quality of fabrication.A few of the most common limitations are introduced in this section.Q SlopeOne area of many studies is the field dependence of the cavity Q. The so calledlow, medium and high field Q slope (LFQS, MFQS and HFQS; fig. 4.1 a, band c) describe different phenomena that can be observed in cavities. LFQSdescribes a small increase in Q between 0 and 20 mT peak surface magneticfield (0 to ∼5 MV/m gradient). Up to around 100 mT (∼25 MV/m), a gentledecrease in Q can be observed (MFQS) and finally above these peak surfacefields a sharp decrease in Q is typical without any measurable x-rays (HFQS).The nature of the HFQS is not fully understood [69] although a mild baking ataround 120◦ C for 48 h has been empirically determined to cure the HFQS (fig.4.1 d). A potential explanation of this effect can be found in [70]. Since HFQSappears at significantly higher gradients than the gradients used in the eLINAC,this will not be a major concern in the ARIEL cavity. There has been evidence[71], that new treatments cause the Q0 to increase as the gradient increases (fig.4.1 e), contrary to the typical high field Q-curve (fig. 4.1 a to c). This is a newdevelopment and attempts to understand this theoretically have been proposed[72, 73].Multi-pactingMulti-pacting is a phenomenon where electrons emitted from one surface areaccelerated by the RF fields and hit the same surface or a nearby surface. Im-pacting electrons can release multiple electrons and if the conditions (surfacegeometry, field amplitude, frequency, secondary electron yield) are right, this isa resonant effect limiting the gradient and Q of the cavity (fig. 4.1 f). Multi-pacting can usually be overcome in-situ by conditioning the cavity with rf field.Surface processing including low temperature baking can be used to reduce thesecondary emission yield of the surface.Q-DiseaseQ disease is caused by hydride formation on the RF surface. Between 50 and150 K, hydrogen in the Niobium diffuses to the surface and forms hydrideswhich have poor superconducting properties. This reduces the performance ofthe cavity and can be identified by a strong Q slope at low to medium gradients(fig. 4.1 g). The formation of the hydrides is time and temperature dependent,so a fast cooldown from 150 to 50 K helps mitigate this effect. To reduce thehydrogen content in the cavity walls, a high temperature (600-800◦ C) vacuumdegassing step can be done.4.1. Performance Limitations 63DefectsInclusions of different materials in the bulk niobium are considered defects in thecavity. These usually involve normal conducting electrons, which cause higherlosses than the surrounding pure niobium resulting in a significantly reducedQ0 at low fields. As a consequence to the higher RF losses, the inclusion willheat up and transfer heat to the niobium resulting an expanding normal zoneand the cavity ‘quenches’ at low gradients (fig. 4.1 h).Inclusions can be prevented during fabrication with the use of clean, dedicatedtools. If a defect makes it into the cavity, a chemical etch of the surface canremove it, as long as it is close to the surface.Field EmissionsField emissions tend to occur when the cavity is not cleaned properly beforeinstallation. Particulates with sharp edges and tips provide local field enhance-ment that manifest as nucleation sites for electron field emission. The emittedelectrons have two effects when they hit the cavity surface: local heating andemission of X-rays. While the local heating is most of the time tolerable, theRF field accelerates the emitted electrons, reducing the energy in the RF field,which translates into a reduced measured Q0 of the cavity. X-rays from stoppedelectrons produce a significant amount of radiation. To prevent field emission,particulate contamination has to be prevented. Assembly of a cavity is usu-ally done in a class 10 clean room environment after a prolonged high pressure(1000-1500 psi) rinse with pure water.Background Magnetic FieldWhile the superconductor expels all magnetic flux from its bulk, lattice defectsor impurities trap magnetic flux in the superconductor, causing an increase inlosses [74]. Those losses are proportional to the square-root of the frequency, sohigh frequency cavities are more susceptible to those losses than low frequencycavities. Flux trapping can be prevented by using one or multiple layers ofmagnetic shielding made out of high µr materials. In addition, all parts thatare within this shield cannot contribute to the remnant field near the cavity. Insome cases active magnetic components are used inside a cryomodule for beamfocusing. In this case the cavities must be cooled with the device off and theremnant field caused by the device needs to be within tolerable limits set by theresidual resistance from trapped flux.The added surface resistance increases linearly with the magnetic field at around2.2nΩ/µT, causing about 125 nΩ of added resistance to a 1.3 GHz cavity if itwas exposed to the earth magnetic field of about 50 µT.Thermal Currents and Flux ExpulsionThe cooldown rate and the temperature gradient across the cavity during thetransition into the superconducting state has been shown to influence the Q04.2. Cavity Treatments and Preparations 64of the cavity [75]. Firstly thermal currents cause an induced magnetic field ina cavity with a helium jacket that can cause trapped flux losses.. The bi-metaljoint between the niobium cavity and the titanium jacket on both ends of thecavity cause a voltage across the cavity if the joints are at different temperaturesand thus create a thermal current.Secondly, a fast transition into the superconducting state seems to lead to abetter expulsion of magnetic flux as a clear phase boundary will be moving acrossthe cavity during the transition into the sc state as opposed to a nucleation ofthe sc state without a clear boundary [76]. There is ongoing research into thistopic to shine more light on the effects of the cooldown on the performance ofSRF cavities.4.2 Cavity Treatments and PreparationsBefore the cavity can be tested, several processing steps have to be done withthe goal to improve the performance of the cavity. Big projects like XFELhave a well established recipe for surface preparation to have a consistent cavityperformance. Current research is focused on improving these treatment stepsor coming up with new ideas (like nitrogen doping [71] or a coating with Nb3Sn[77]) to further increase the Q0 or reach higher Eacc. A few, more standardtreatments are introduced in this section.Buffered Chemical PolishingTo produce a smooth surface finish, a process called buffered chemical polishing(BCP) etches the surface away with a mix of HF, HNO3 and H3PO4. Typicallya surface layer of around 120 µm thickness is removed. If defects are found,smaller etching steps of 10 to 30 µm usually help to improve the performanceof the SRF cavity.During the ISAC-II development, a BCP facility had been established and isused for the ARIEL cavities as can be seen in fig. 4.2. As a baseline, 120 µmare removed and if needed, additional etches can be preformed. During theetch, it is important to control the temperature of the niobium cavity. A hightemperature (above 16◦ C) causes a rise in etching rate and the niobium takesin hydrogen which causes Q-disease.Electro-polishingElectro-polishing (EP) fulfills the same role as BCP and usually either BCP orEP is used. EP uses HF and H2SO4 and with a voltage between the cavity anda cathode inside the cavity so that the surface is etched away. EP producesslightly better Q0 values then BCP, but is technically more complex. The dif-ferences are most apparent at higher frequencies and surface fields.4.2. Cavity Treatments and Preparations 65Figure 4.2: Left: a multi-cell cavity is etched with hydroflouric acid in a processcalled buffered chemical polishing (BCP). Right: A single cell cavity is beingcleaned with high pressure water.Frequency and Flatness TuningAfter fabrication, it is unlikely that the cavity has the correct resonant frequencyand an even field distribution. Therefore a tuning step is needed to modify thecavity frequency and increase the flatness. This is done at the warm tuningstation (fig. 4.3) by squeezing or stretching cells that show a high or low fieldamplitude. The field amplitude is measured via beadpulling (see section 2.1.3):a network analyzer measures the frequency shift while a dielectric bead movesthrough the cavity. The frequency shift is proportional to the square-root of thefield amplitude at the position of the bead.The frequency goal at this stage has to account for a changed dielectric constantfrom air to vacuum inside the cavity, thermal shrinkage from room temperatureto 4 K and an outside pressure change from atmospheric pressure to 24 Torrhelium pressure at 2 K. In addition, the frequency tuner in the cavity onlyallows for stretching of the cavity. The tuner should be engaged all the timeto allow active frequency tuning in both directions. As a result, the goal fre-quency in room temperature, under relaxed conditions without vacuum is setto (1298.90 ± 0.05) MHz. Etching increases the volume of the RF space andaccording to Slaters theorem, the frequency decreases for this geometry. Datafrom the TESLA cavity shows a sensitivity of around 10 kHz/µm. As a result,before the 120 µm BCP etch the goal frequency is (1297.70± 0.05) MHz. Sincethe etching might not be completely uniform, the flatness has to be adjustedbefore and after the etching process.In fig. 4.4 and 4.5, the results of the tuning process for the first multi-cell cavity4.2. Cavity Treatments and Preparations 66Figure 4.3: The multi-cell tuning station adjusts the frequency and flatness tothe required values.’ARIEL1’ is shown along with the initial, untuned beadpull measurement. Theinitial results and the pre-BCP tuning (fig. 4.4) show a flatness increase from90 % to 98 % and the frequency changed by 800 kHz to be within the goal rangeat this stage. The following surface removal of 120 µm via BCP reduced theflatness to 89.9 % (fig. 4.5), which could be tuned to 98.9 %.These tuning steps have to be done for all multi-cell cavities before the firstetch, after big etching steps (120 µm) and before the helium jacket is welded on.The jacket prevents access to the cells and flatness manipulation is impossible.Flatness data for the other cavities is shown in appendix D.DegassingA high temperature (usually 600 to 800◦ C for 4 hours) processing step helps toremove hydrogen from the bulk of the niobium and therefore prevents Q disease.This treatment can also be used to increase the Q0 when applied with nitrogendoping [71].120◦C bakingIt has been shown that a mild bake at 120◦ C removes the HFQS at around100 mT peak magnetic surface field. Baking can be done with either resistiveheaters on the cavity or hot gas flowing on the outside of the cavity. It is impor-tant that the RF surface is in an oxygen free environment to prevent oxidation.This bake generally decreases the temperature dependent RBCS , but the tem-perature independent residual resistance Rres increases [78]. The increase inRres is generally assumed to be a result of the bake on the oxidation layer onthe cavity while the reduction in RBCS is due to a reduction in the mean freepath of the cooper pairs.4.2. Cavity Treatments and Preparations 67 0 2 4 6 8 10 12 14 16 0  200  400  600  800  1000  1200  1400E-field [∝ V/m]Z-Position [mm]Initial measurement: f=1.29792GHz, flatness 90%After tuning: f = 1.29873GHz, flatness 98%Figure 4.4: The initial flatness and frequency tuning of ARIEL1 before BCPtreatment increased the flatness from 90 % to 98 % and changed the frequencyto be within the goal range. 0 2 4 6 8 10 12 14 16 18 0  200  400  600  800  1000  1200  1400E-field [∝ V/m]Z-Position [mm]After etching: f = 1.29776GHz, flatness 89.9%After retuning: f = 1.297541GHz, flatness 98.9%Figure 4.5: The 120 µm BCP removal caused the flatness to decrease to 89.9 %,which could be recovered to 98.9 %.4.3. Vertical Performance Measurements 68HF rinseHF rinsing after a 120◦ C bake has shown to improve the cavity Q0 after re-moving a 4-5 nm of of the oxide layer. The HF rinse keeps the decreased RBCSfrom the 120◦ C bake and resets the oxide layer to minimize Rres resulting inthe overall higher Q0 [79].High Pressure Water RinsingHigh pressure rinsing (HPR) is essential to cavity performance tests. It is thefinal processing step before the RF space is sealed and evacuated. The purposeof HPR is to eliminate any particulate contamination which could cause fieldemissions.During the HPR, a rotating wand sprays water on the inside of the cavity witha pressure of 600 to 1500 psi while the cavity moves up and down. Multiplehours of HPR are needed to clean a nine-cell cavity. An automated washingmachine has been developed at TRIUMF for this purpose and can be seen infig. 4.2 on the right.All ancillaries like blank flanges, pick-up probe and main coupler need to gothrough similar cleaning steps to keep the cavity particulate free. The finalassembly then is done in a class 10 clean room environment to prevent contam-ination of the cavity with particulates.4.3 Vertical Performance MeasurementsBefore the helium jacket gets welded onto the cavity, performance tests aredone to show that this cavity can reach its working point of 10 MV/m witha Q0 ≥ 1010. These tests are done in a cryostat with the cavity suspendedvertically, contrary to the installation in the final cryomodule where the cavityis orientated horizontally. Often there is a measurable difference between verti-cal and horizontal tests, which is commonly linked to a difference in magneticshielding and/or cooldown related effects [80].During the vertical tests a variable main power coupler is attached to the cav-ity. The variable coupler provides the ability to match the Qcoupler to the Q0,resulting in the loaded quality factor QL to be twice the unloaded quality factorQ0, a situation called critical coupling. The loaded quality factor is measuredvia the decay time of the RF voltage extracted from the cavity by the pick-upprobe. This signal is also used to measure the resonant frequency of the cavityand the extracted power by the pick-up probe. Together with the incident andreflected power at the main power coupler the dissipated power in the cavitycan be calculated byPcav = Pincident − Preflected − Ppick−up. (4.1)4.3. Vertical Performance Measurements 69Careful measurements of the attenuation of the cables are needed to accountfor the power dissipated in the cables.Once the Q0 and the dissipated power in the cavity are known, the stored energyU in the RF field is directly calculated with the base formula Q = ωU/Pcav.The stored energy in the cavity is proportional to the square of the gradientEacc with a proportionality constant U/E2acc, which is determined in simula-tions. This way, a first calibration point of the Q0-Eacc curve is measured.After the initial Q0 and Eacc are measured via decay time, a calibration factork can be defined, which relates the pick-up voltage or power1 to the gradientEacc. This calibration factor can then be used to avoid repeating the decay timemeasurement. As the quality factor can change with the gradient, the decaytime would change accordingly, making a simple fit to the expected exponentialdecay impossible. The cavity quality factor as a function of field gradient ismeasured by recording the frequency, incident, reverse and pick-up power.During the measurement, the coupler position is changed as needed to keep thecavity as close to critical coupling as possible to achieve a minimum of reflectedpower. Critical coupling makes direct RF measurement possible, since both re-flected and pick-up power are significantly lower than the incident power, usuallyby a factor 100. As will be shown later for the horizontal cryomodule perfor-mance measurements, accurate Q0 cannot be done with a heavily over-coupledcavity (Qcoupler  Q0) as incident and reflected power become a) very big andb) close to each other, making accurate measurements difficult or impossible.A schematic of the RF setup for the vertical performance measurement is shownin fig. 4.6. The low level RF system (software and hardware) is a TRIUMFdevelopment [81] and controls the frequency and amplitude of the RF signal.The position of the variable coupler is controlled via a stepper motor on topof the cryostat. A 500 W 1.3 GHz amplifier provides the needed amplificationof the generated signal. The power and frequency measurements are done withAgilent E4419B power meters and a Agilent 53181A frequency counter. Thepick-up voltage is monitored on a LeCroy 204Xi oscilloscope. A Labview [82]program collects this data via GPIB (or manual transfer in case for the pick-upvoltage decay time measurement) and processes it to extract the Q0 vs Eaccplot.The cryostat for cavity tests is equipped with multiple temperature sensorsto measure the temperature across the cavity and at different points of inter-est. Heaters attached to the cavity are used for cavity warm-up, once the testis completed. Liquid helium level probes measure how much helium is in thecryostat to ensure that the cavity is fully submerged. A heat shield around thecryostat consists of an insulation vacuum with a copper inlay which is cooledto 77 K with liquid nitrogen. Just below the lid of the cryostat, baffle platesprevent heat transfer via radiation from the 300 K environment to the liquidhelium. The RF space of the cavity is evacuated and constantly pumped with aturbopump and an ion-gauge measures the pressure in the cavity. The helium1Pick-up voltage and power are related by a 50 Ω termination4.3. Vertical Performance Measurements 70Figure 4.6: During a RF cold test of a cavity, a low level RF system controlsthe frequency of the RF signal sent to an amplifier. A bi-directional couplerextracts the incident and reflected power, and a probe on the other side of thecavity picks up the transmitted signal. The voltage, power and frequency ofthis signal is analyzed and sent to the amplifier to keep the RF signal matchedto the resonant frequency of the cavity, creating a self-exited loop.vapor pressure of the LHe bath is measured by pressure transducers. Liquid he-lium boils at 4.2K at atmospheric pressure. To reduce the temperature furtherrequires pumps that lower the bath pressure. This pressure is fed into a throt-tling butterfly valve to control the helium vapor pressure, which determines thetemperature of the helium bath. At a pressure of 23.4 Torr, the liquid is at atemperature of 2 K. The pressure control reduces fluctuations of the He vaporpressure to around 0.1 Torr up to an active RF load of around 20 W.4.3.1 Single Cell CavitiesSingle cell cavities are mostly used for R & D purposes. The smaller size,compared to nine-cell cavities, eases handling and due to the smaller effectivelength, less RF power is needed to reach the same gradients. A single cell cavityhas a U/E2acc of 0.0163 J/(MV/m)2 while the geometry constant for the ARIELnine-cell cavity is 0.1261 J/(MV/m)2.During the testing process, two single cell cavities were tested to show that thecavity processing and testing procedures are adequate to reach the gradient andQ0 specifications for ARIEL.The first single cell cavity was used to develop the etching process and cleanroom assembly and testing procedures. The BCP process was quite advanceddue to the experience gained during the ISAC-II development. Modificationshad to be made to holding fixtures and cooling lines due to the different cavitygeometry and bigger surface size. The etching reaction creates heat. An elevatedtemperature of the cavity and acid mixture causes hydrogen to soak into theniobium, which causes Q disease.The second cavity was used to qualify the clean room procedures. Previous4.3. Vertical Performance Measurements 71 1x109 1x1010 5x1010 0  5  10  15  20  25  30Q 0Eacc [MV/m]10W1W0.1W 20WFNAL 11/7/2012)TRIUMF (29/11/2012)Figure 4.7: The performance of the second single cell cavity at 2 K measuredat Fermilab and TRIUMF showed the same high Q0 up to 25 MV/m when ahigh field Q drop occurs, proving the clean room procedures at TRIUMF areup to standards to assemble cavities free of field emissions.tests with only a 100 µm BCP treatment showed a weak performance at 2 K ofQ0 = 1.4 · 109 up to 7 MV/m. Prepared with a light EP of 35 µm and a 5 µmremoval via BCP to remove stains at FNAL 2 the cavity performed well with ahigh Q0 of around 1.5 · 1010 and up to 25 MV/m without a significant slope ingradient. At 25 MV/m, the expected HFQD was observed. At TRIUMF, thecavity was tested without any additional surface processing except HPR. As canbe seen in fig. 4.7, the performance test at TRIUMF compared well to the FNALresult within the determined uncertainties. No conditioning of field emitters wasneeded to reach this performance. This shows that the TRIUMF procedures inthe clean room achieved particulate free cavities. The difference in uncertaintiesbetween the FNAL and TRIUMF results are due to the variable coupler used atTRIUMF compared to the fixed coupling at FNAL. The same RF performancealso shows that the equipment and procedures used to determine Q0 and Eaccare adequate to measure the performance of SRF cavities.4.3.2 Nine-Cell CavitiesThe vertical performance measurements are done in a bath cryostat, where thecavity is fully submerged in LHe. To measure the Q0 precisely, a variable coupleris attached to one of the two power coupler ports on the cavity. This variablecoupler provides a coupling range from 107 and 1011 as can be seen in fig. 4.8.This range is useful, as the expected Q0 is around 4 · 108 for 4.2 K and 1 · 1010at 2 K. The low end of the coupling range at 107 is useful for high power pulsedconditioning. A strong coupling (low QL) provides a short time constant τ for2thanks to the Technical Division and testing by A. Grassellino and A. Romanenko4.3. Vertical Performance Measurements 72107108109101010111012 0  5  10  15  20  25  30  35  40Q extCoupler Position [mm]IN OUTFigure 4.8: The coupling range of the multi-cell coupler allows for criticalcoupling at 4.2 and 2 K.the RF to reach its maximum amplitude. A fast rise of the cavity voltage helpsto break through multi-pacting barriers and conditioning of field emitters.The results of various performance measurements for the first four multi-cellcavities for ARIEL are discussed in the following paragraphs. In each perfor-mance measurement plot, curves of constant power are shown as reference. Eachsurface preparation was done to raise the Q0 to meet the specification of 1 ·1010at 2 K. For most tests, Q0 data during the cooldown from 4.2 to 2 K was takenat a fixed gradient to extract the values for the temperature dependent BCSresistance and the temperature independent residual resistance, as described insection 4.3.3 to determine the limitation of the Q0. In most cases the Q0 waslimited by a high residual resistance.ARIEL 1 (fig. 4.9) did not meet the performance specification after the verticaltests. With a Q0 of around 6 · 109, the expected active heat load on the LHesystem will be around 20 W. The cryogenic capacity of the 4K/2K system ofheat exchanger and JT-valve in the cryomodules is designed for the heat loadgenerated by two cavities. This allows for a lower Q0 in the injector cavitythan the ACM cavities. To fully characterize the cryomodule design, the cavitywas equipped with its helium vessel and assembled into the cryomodule, EINJ.Further performance measurements on this cavity are shown in section 4.4.1.The final surface preparation, a 30 µm etch after a 120◦ C bake and HF rinse,for ARIEL2 (fig. 4.10) could not be measured in the cryostat due to repeatedsuperfluid leaks. The sealing surface on the power couplers wore out after re-peated tests. In the cryomodule, the seal at this location separates the beamlinevacuum from the thermal isolation vacuum and the problem of superfluid he-lium leaking into the cavity is non-existent. Before this, a similar performanceto ARIEL1 with a Q0 at 6 · 109 up to 10 MV/m was measured. The cavitywas welded into its helium jacket and then assembled into the first ACM due to4.3. Vertical Performance Measurements 731091010 0  2  4  6  8  10  12  14Q 0Eacc [MV/m]10W5W2W 20W1W120um BCPAdditional cleaningDegassed cavityFigure 4.9: Q0 performance measurements of ARIEL1 at 2 K showed a clearimprovement from the initial tests of ARIEL1 to the degassed cavity. Theperformance measurement is limited to Pcav ∼ 25 W due to limitations of thecryo system.1*1091*1010 0  2  4  6  8  10  12Q 0Eacc [MV/m]10W5W2W 20W1W120um BCP85C bakedegassed120C bake + HF rinseFigure 4.10: An in-situ 120◦ C bake-out step was attempted for ARIEL2 toraise the Q0 after the initial measurement. Insufficient heater power preventedthis and the performance did not change. After degassing high gradients werereached with a slightly reduced Q0. The Q0 recovered after a 120◦ C bake andHF rinse. The performance measurement is limited to Pcav ∼ 25 W due tolimitations of the cryo system.4.3. Vertical Performance Measurements 741091010 0  2  4  6  8  10  12Q 0Eacc [MV/m]10W5W2W 20W1W120 um BCP+30um BCP+60um BCP+degassed +30um BCPFigure 4.11: The performance of ARIEL3 recovered after a defect caused alow Q0 and field emissions during the first vertical test. Subsequent processingsteps increased the performance further.time constrains. The first ACM, EACA, was assembled with only one RF cavityand one dummy cavity (a straight beam pipe with a helium vessel around it, inorder to characterize the cryogenic engineering quality of the cryomodule. Theperformance of the cavity in the cryomodule is shown in section 4.4.2.ARIEL3 showed initially a very low Q0 (fig. 4.11) with a high residual resis-tance. This was caused by a defect, which took multiple etches to fully removeand reach a performance similar to ARIEL1 and ARIEL2. After repeated assem-blies, the sealing surfaces on the power coupler ports wore out as has happenedwith ARIEL2. To prevent this leak, the final vertical test was performed at2.3 K. At this temperature, Helium is not in its superfluid state and the sealholds. Using Q0 vs T data, the BCS resistance at 2.0 K could be estimatedto (11 ± 2) nΩ with an Rres = (37.2 ± 0.3) nΩ (fig. E.9). By subtractingthe difference in BCS resistance between 2.3 and 2.0 K (about 10 nΩ) from theQ0(2.3K) data, a baseline Q0 of 6·109 at 9.5 MV/m was estimated for T = 2.0 K(fig 4.12). This assumes that the difference in BCS resistance between 2.3 Kand 2.0 K does not change with increased gradient. Given that fig. 4.12 showsonly a moderate slope, this is a reasonable assumption. In niobium on coppercavities, it has been shown [83] that the surface resistance changes more withincreased gradient when the helium bath is above the lambda point at 2.17 K.This would lead to a flatter curve at 2.0 K for ARIEL3 at this stage. The max-imum gradient reached was 9.5 MV/m. the limitation was due to the cryogenicheat load on the cryostat. At a lower temperature of 2.0 K and the associatedlower RF losses, a gradient of 10 MV/m or higher should have been possible.ARIEL3 is planned for a second injector cryomodule, that is being assembledfor the collaboration with VECC, Kolkata (India).Figure 4.13 shows the performance of the fourth nine-cell cavity, ARIEL4, at4.3. Vertical Performance Measurements 751091010 1  2  3  4  5  6  7  8  9  10Q 0Eacc [MV/m]10W5W2W 20W1W210um BCP + degassed + 30um BCP, 2.0K+30um BCP, 2.3Krecalulated for 2.0KFigure 4.12: Testing at 2.3 K prevented a super-fluid leak on the cavity. The2 K result is calculated by using the difference in BCS resistance shown in fig.E.94.2 and 2 K. As can be seen, a high performance of this cavity up to 11 MV/mwith a Q0 of 8 · 109 comes close to the specifications for the cryomodules. Aslightly higher residual resistance of 22 nΩ (see fig. E.10) was measured, whichshould be reduced in the degassing treatment, currently being done at FNAL.All final performance measurements have been limited by the cryogenic load onthe cryostat. Neither field emissions, multipacting or quench limited the cavitiesin performing at 10 MV/m or higher. If the Q0 were higher, higher gradientscould have been measured due to the reduced power dissipation in the cavitywalls.4.3.3 Extracting BCS and Residual ResistanceThe BCS and residual components of the surface resistance Rs can be extractedby collecting Q0(T ) data at constant gradient with varying temperature. Thesurface resistance Rs(T ) is calculated using the geometric factor G. The equa-tion for the BCS resistance (eq. 4.2) is linearized by using 1/T as the indepen-dent variable and taking the natural logarithm of the data. This way the fittingfunction becomes a simple linear equation with an additional ln(x) term (eq.4.3).RBCS(T ) = A · 1Texp(−m · 1T))(4.2)⇒ ln(RBCS)(x) = −B · x+ ln(A) + ln(x). (4.3)By adjusting the estimate for Rres, the best fit of ln(Rs − Rres)(T ) to eq. 4.3is found (fig. 4.14 a). Once a set of fit parameters A and B for the temperaturedependence of RBCS(T ) are found, the difference between Rs(T ) and the fitted4.4. Horizontal Performance Tests 761081091010 0  2  4  6  8  10  12Q 0Eacc [MV/m]10W5W2W 20W1WMay 15, 2015: 120um BCP, 2KMay 15, 2015: 120um BCP, 4KFigure 4.13: Performance test of the fourth multicell cavity for ARIEL at 4.2 and2 K. This cavity will be the second cavity in the first accelerating cryomoduleEACA.RBCS(T ) results in Rres. The best fit is reached when the initially estimatedand resulting residual resistance resistance match as shown in fig. 4.14 b.As an example, fig. 4.14 shows the final fit for data taken from the last verticaltest of ARIEL1. A BCS resistance RBCS of (17± 2) nΩ at 2 K and a residualresistance Rres of (28.6 ± 0.7) nΩ were found using this method. The uncer-tainties given in RBCS and Rres follow the uncertainties in the fit parameters.Before the degassing the BCS resistance was found to be (14 ± 5) nΩ (see fig.E.1).Reference [38] estimates a BCS resistance to about 10 nΩ at 1.3 GHz and 2 K,showing a substantial BCS resistance along with the high residual resistance inARIEL1 after the degassing. Before the degassing the residual resistance wasmeasured to be (58 ±1) nΩ, proving that the increase in Q0 is due to the de-crease in Rres. Figures and values for the other fits can be found in appendixE.4.4 Horizontal Performance TestsAfter the helium jacket is welded onto the cavity, the cavity is hermeticallysealed with the high power couplers, pick-up probe, beam line absorbers andwarm-cold transition pieces. Following the assembly, the cavity is installed inthe cryomodule. Details of the cryomodule design can be found in [84]. Afterthe cavity is cooled down to 2 K, the RF measurements can proceed.Due to the heavy overcoupling, precise RF measurements of the power lossesin the cavity are not possible. Due to the absence of beam loading, both theincident and the reflected power are of the order of 1 to 20 kW, while theestimated power loss in the cavity of the order of 1 to 20 W, making a RF based4.4. Horizontal Performance Tests 77 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -18.22 ± 0.07b = 12.65± 0.02(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 200 400 600 800 1000 1200 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [28.49 ± 0.70] nΩRBCS(2K) = [17.21 ± 1.98] nΩ20131128(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =28.49Figure 4.14: Extracting the BCS and residual resistance with Q0 vs T datafor ARIEL1 after degassing. (a): The temperature dependent BCS resistance islinearized and the data is fitted to the natural logarithm of Rs−Rres. (b): Thefitted BCS resistance (green line) is subtracted from the Rs data (red markers)to reveal the residual resistance Rres (blue markers). The uncertainties givenin RBCS and Rres follow the uncertainties in the fit parameters.4.4. Horizontal Performance Tests 78measurement very inaccurate. For this case, calorimetric measurements on theliquid helium volume are done to measure the power loss in the cavity. Byclosing the supply of liquid helium, the liquid levels drop in the reservoir overthe cavity. By calculating the rate of volume change due to boil-off, the powertransferred into the liquid can be calculated. These measurements have to bedone with the RF off as well to determine the amount of static load on the 2 Kreservoir, coming from heat radiation or conduction from warmer sections of thecryomodule. The cryomodule is designed to minimize the static load with theuse of a thermal shield cooled with LN2, multi-layer insulation and struts withlow thermal conductivity, but a reduction of the static load to 0 is not possible.The pick-up probe measuring the field strength in the cavity is calibrated bymeasuring the energy gain of the electron beam. An analyzing dipole magnetdeflects the beam to a diagnostic leg of the beamline. The field strength of themagnet can be used to calculate the energy gain of the beam and the voltage inthe cavity. The beam has to be accelerated on crest to receive the full voltage.If there is a difference in phase between the RF in the cavity and the arrival ofthe bunch in the center of the cavity, a lower energy gain is measured.The fully assembled and installed injector cryomodule is shown in fig. 4.15.Helmholtz coils are surrounding the ICM and the low energy beam transportsection to reduce the significant fringe magnetic fields from the nearby 500 MeVcyclotron.4.4.1 EINJThe initial gradient measurements relied on the previously determined pick-upQ and a calibration factor of 8.87 (MV/m)/V (fig. 4.16) was used to measurethe gradient. The initially measured Q0 is identical with the Q0 measured inthe vertical tests as can be seen in fig. 4.17. The gradient was limited to below5 MV/m. Above this gradient the RF signal would break down and quench thecavity. Measurements with a portable X-ray monitor showed significant levelsof radiation during operation at high gradients, up to 1 mSv/h. Pulsed con-ditioning over multiple days did not decrease these field emissions. By movingthe x-ray monitor around the crymodule, the source of the x-rays was estimatedto be on the coupler side of the cavity. The cavity had to be taken out of thecryomodule and inspected for any damage.Once the cavity was opened, visible damage on the coupler end of the cavitywas observed. A circular arc, about 1 cm into the cavity, was visible. Thecoaxial aligned stainless steel damper on this side of the cavity touched downduring the assembly phase with enough force to damage the surface and gener-ate particulate in the cavity causing the observed field emissions.To repair the damage, a 20 µm etch was performed on this cavity, which re-moved the visible damage to the RF surface. After the hermetic sealing wasre-established, the cavity was once again installed in the injector cryomodule.As can be seen in fig. 4.19, the performance exceeds all previous vertical testswith a Q0 of over 1 · 1010 up to 12 MV/m. The limitation in gradient is aresult of multipacting in the coupler assembly. This can be overcome by further4.4. Horizontal Performance Tests 79Figure 4.15: The injector cryomodule fully assembled and installed on thebeamline with RF wave-guides, cryogenic supply and return lines and all otherdiagnostics connected to it.conditioning. Due to the heavy overcoupling, a gradient of 10 MV/m requires10 kW of generator power in absence of beam loading. The couplers had beenconditioned up to this power in a dedicated room temperature conditioning sta-tion [85]. Further increases of the gradient will require further conditioning ofthe couplers, which has to be done in-situ.4.4.2 EACAThe first accelerating cryomodule EACA was assembled initially with only oneof the two planned cavities. This was done to get an early test of the cryogenicengineering and installed cavity performance while waiting for the completionof ARIEL4.The first cavity in the EACA, ARIEL2, showed a decent performance in verticaltests with a Q0 of 6·109 up to 10 MV/m. The performance with a 20 µm removalvia BCP could not be measured in the vertical test cryostat due to a superfluidleak. In the cryomodule, the cavity performed exceptionally well with a Q0 of1 · 1010 up to 12 MV/m as can be seen in fig. 4.20. No further treatment of thiscavity is needed as it meets the performance specifications. The gradient waslimited by multipacting in the couplers, similar to the EINJ.4.5. Higher Order Mode Measurements 80 0 0.2 0.4 0.6 0.8 1 1.2 2.5  3  3.5  4  4.5  5  5.5  6Intensity [a.u.]Beam Energy [MeV]Pick-up Voltage375mV436mV500mV572mVFigure 4.16: The calibration of the pick-up voltage is done by measuring thebeam energy after the cryomodule. Shown are four different beam energy mea-surements at different pick-up voltages. The measurement is done by sweepingthe current through a dipole magnet after the cryomodule and a Faraday cupreads out the beam intensity.4.5 Higher Order Mode MeasurementsIn order to verify the design of the cavity with respect to the HOM shown insection 3, a number of measurements have been done on a copper model of thecavity as well as the installed cavities inside the cryomodules.4.5.1 HOM Beadpulling on a Seven-Cell ModelFabricated initially to prototype the fabrication of the nine-cell cavity, a coppermodel (shown in fig. 4.21) of the real cavity was used to verify the field distri-bution calculations of the dipole modes. For this copper prototype the numberof the identical inner cells has been reduced from seven to five. This results ina seven-cell cavity with the same fundamental coupler ports like the nine-cellcavity. Instead of a small pick-up port two bigger ports with an inner diameterof 40 mm are added. Those ports are at an angle of 150◦ to each other withone of them aligned with the coupler plane. In this way, different polarizationsof HOMs can be measured if needed.Measurements are carried out using bead pulling. A bead pulling stand for theTRIUMF nine cell cavity was designed and built with the ability to move thestring radially and azimuthally in addition to the necessary longitudinal beammotion. TM110 dipole modes like the 2.56 GHz mode discussed in chapter 3do not have any electric field on the beam axis. Therefore, it is necessary tomeasure the field distribution with an offset to the beam axis. Since the beadprobes the field amplitude, a dipole mode can be identified by a sin2(α) depen-4.5. Higher Order Mode Measurements 811091010 0  2  4  6  8  10  12  14Q 0Eacc [MV/m]10W5W2W 20W1Wlast vertical testCryomodule testFigure 4.17: The RF performance of the injector was limited by field emissionsto a maximum gradient of 5 MV/m. The Q0 is at the same level as the verticaltest at low gradients before the field emissions start.Figure 4.18: Visible damage on the beam pipes was observed after the cavityfor the ICM was opened.4.5. Higher Order Mode Measurements 8210910101011 0  2  4  6  8  10  12  14Q 0Eacc [MV/m]10W5W2W 20W1WInitial cryomodule performacne+30 um BCPFigure 4.19: After an additional BCP treatment on the EINJ cavity to repairdamage on the RF surface the performance reached Q0 = 1010 up to 12 MV/m.The gradient was limited by the multipacting in the couplers.1*1091*10103*1010 0  2  4  6  8  10  12Q 0Eacc [MV/m]10W5W2W 20W1WVertical TestHorizontal test: + 120C bake, HF rinseFigure 4.20: The EACA cavity 1 reached the performance specification ofQ0 = 1 · 1010 at 10 MV/m. The increase in Q0 compared to the vertical test isa result of 20 µm surface removal via BCP.4.5. Higher Order Mode Measurements 83Figure 4.21: The seven-cell copper model cuts two inner cells from the nine-cellcavity and is used to confirm the HOM simulations.dence of the maximum amplitude, quadrupole modes with sin2(2α) and so on,where α is the azimuthal angle with respect to an arbitrary starting point.This bead pulling stand has been used to measure the field distribution of modesin the seven-cell copper cavity. The string is aligned to the beam axis on thecenter of the cavity using targets on both sides of the cavity. After the string isaligned to the optical axis of the cavity, the string is moved parallel by 1 cm toestablish an offset.A stepper motor controls the longitudinal motion of the bead. The motor andthe fixtures on the other side of the cavity are mounted in a way to allow themto rotate without changing the alignment of the string with respect to the beamaxis. The stepper motor is controlled by a computer via Labview, which alsocontrols the HP network analyzer used for this measurement. The softwarerecords Q, frequency and reference phase of the rf signal for the selected mode,and then proceeds to the actual bead pulling. During the bead pull measure-ment the phase of the RF signal is recorded. The phase shift from the referencephase is proportional to the electric field (using dielectric beads) or electric andmagnetic field combined (using metallic beads). For a flatness tuning of theaccelerating TM010 pi-mode metallic beads can be used as this mode does nothave any magnetic field on the beam axis and the metallic bead gives the trueelectric field. However for dipole modes this is not true and dielectric beadsshould be used to isolate the electric field on the chosen axis.Before modes can be identified, their resonant frequencies need to be mea-sured. A network analyzer measures the transmission of an RF signal throughthe cavity as a function of frequency and is used to find the resonances of high4.5. Higher Order Mode Measurements 84 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 0  10  20  30  40  50  60  70  80  90Frequency [GHz]Mode #MeasurementDipole SimulationFigure 4.22: The mode number of the simulation results have been adjusted tocorrespond with the measured frequencies. While the simulation can be forceinto dipole modes of a particular polarization, the measurements account for allmodes including monopole modes.transmission. The frequencies of the measured modes are compared to the sim-ulated frequencies of dipole modes in fig. 4.22. The boundary and symmetryconditions in the the simulation force the resulting modes to be dipole modeof a certain polarization. The uncertainties in the frequency measurements areusually very precise, of the order of 10s of kHz, which result in relative uncer-tainties in the order of 0.1 % and therefore are not shown in fig. 4.22.Overall the measured frequencies match the simulated eigenmode frequencies.In the range from 1.6 GHz to 1.9 GHz the match is very close. At higher fre-quencies slight differences appear. This can be attributed to tuning differences.These frequency measurements act as a guideline to show which modes couldbe dipole modes.To measure the field distribution the string holding the bead is rotated aroundthe optical axis in 30◦ steps with an offset of 1 cm from the beam axis. Fig-ure 4.23 shows the azimuthal dependence of the electric field of four HOMs(1620 MHz, 1734 MHz, 1832 MHz and 2498 MHz. While slight variations arevisible, no clear variation of the amplitude in the expected pattern emerged inthe peaks.The slight variations hint that the polarization changes when the string withthe bead changes position. A new measurement setup to avoid this shift of thepolarization uses multiple strings through the cavity. Arranged in 30◦ stepsaround a circle, the polarization change caused by the rotating of the stringwith the bead on it is avoided. One additional line is added to the center of thecircle to align the strings to the optical axis of the cavity.Using this multi-line setup the earlier discussed azimuthal amplitude depen-dence of a dipole mode at 2498 MHz could be measured as shown in fig. 4.24,4.5. Higher Order Mode Measurements 85                 1620 MHz0deg 30deg 60deg 90deg 120deg 150deg                1734 MHz              E0 [∝ V/m]1832 MHz                    Position2498 MHzFigure 4.23: Bead pull measurements on four HOMs with different azimuthalpositions of the string.where the relative field amplitude at a peak is plotted against the azimuthalposition of the bead. The horizontal coupler plane corresponds to the plane of0◦ and 180◦, so the polarization of this mode is in the vertical direction.The final step in verifying the simulations lies in comparing the measured fielddistribution to the simulations. Figure 4.25 shows the same longitudinal fielddistribution in the measurements and the simulation.4.5.2 HOM Frequency and Q MeasurementsTransmission measurements on the cold cavities in the cryomodules can revealthe frequency and potentially the QL of HOMs. The transmission signal of a res-onance is fitted to a Lorentz-function to extract the frequency and bandwidth.Once both values are obtained, the Q value can be easily calculated using eq.2.34. For the data acquisition a Labview program was written to automaticallycollect the transmission data from a network analyzer for a narrow frequencyband and after the signal is sufficiently stable, move on to higher frequencies.The resonant frequencies were measured using this method. To improve theresolution for high Q modes, high precision measurements of each resonancefollowed with a narrower frequency span. Since the frequency is in the rangeof 1 to 3 GHz, the bandwidth for modes with Qs of 107 and higher is of theorder of 100 Hz and lower. This makes measurements difficult as the resolutionof the network analyzer is limited and frequency fluctuations caused by smallHelium pressure changes or physical vibrations broaden the resonance. Thissets the limitations of the measurements to Qs of 107 to 108. Significant signalattenuation occurs in the wave-guides, directional couplers and cables used totransfer the signal from the cavity to the network analyzer. Signal amplifica-tion to reduce the signal-to-noise ratio and strong signal averaging are used to4.5. Higher Order Mode Measurements 86-2.8-2.6-2.4-2.2-2-1.8-1.6-1.4-1.2-1 0  50  100  150  200  250  300  350PhaseshiftAngle [deg]Multi lineSingle lineFigure 4.24: The new multi-line beadpulling setup produces a clean azimuthaldependence of the field amplitude of a HOM at 2498 MHz, which is now iden-tified as a dipole mode. 0 1 2 3 4 5 6 0  100  200  300  400  500  600  700  800  900  1000E-Field [∝ V/m]Z-Position [mm]1694 MHz - Measurements0deg30deg60deg90deg120deg150degFigure 4.25: Measurement (left) and simulation (right) show a very similarlongitudinal field distribution of the electric field of this particular mode.4.5. Higher Order Mode Measurements 87 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 0  10  20  30  40  50  60  70  80  90  100Frequency [GHz]Mode #MeasurementsSimulationsFigure 4.26: Frequency measurements on the EINJ cavity at 2 K show a goodmatch to simulated frequencies of dipole modes.compensate.Figure 4.26 shows both the measured frequencies and simulated frequencies.The frequency measurements are similar to those measured on the copper modelin the previous section, but in this case an even better match of the measuredfrequencies to the simulated frequencies was recorded. No modes are close tothe beam frequency harmonics at 1.95 GHz and 2.6 GHz.As a benchmark the measured transmission of the accelerating TM010 modewas used. During commissioning of the accelerator, low beam currents with alow duty cycle are used and the adjustable coupler was set to 3 · 106 to reduceforward power requirements. As can be seen in fig. 4.27, a good match wasfound using the Lorentz function fit with a QL of 2.6 · 106 ± 13%. The uncer-tainty is calculated by the uncertainty of the fit parameters f and the width ofthe Lorentz function at the half power point.As can be seen in fig. 4.28, the measured Q values are significantly higherthan the expected Q values. In the range from 1.5 GHz to 2 GHz both theEINJ cavity and the EACA cavity Q measurements generally agree with eachother and the frequencies match to simulations. But the Q values are abouttwo orders of magnitude higher than the Q coming from simulations. Higherfrequency modes could not be measured reliably in Q.A possible cause for the difference in measured and simulated QL could be dueto simplifications made in the simulations. During the simulations the absorberswere simplified as continuations of the beam pipes while in reality they are coax-ial with the beam pipe with a small difference in radius of about 1 mm betweenthe beam pipe and the HOM absorber. As can be seen in the dipole mode fielddistribution shown in fig. 2.6(a), dipole modes in a pillbox cavity (and circularwave-guides like the beam pipes) have their highest magnetic fields not on thesurface of cavity or wave-guide, but roughly at 2/3 of the radius. Having the4.6. Performance Measurement Summary 88-90-85-80-75-70-65-60-55-4000 -2000  0  2000  4000S21 [dB]Δf = f - f0 [Hz]QL = 2.60e+006 ± 13 %f0 = (1.3005 ± 0.0002) GHzMeasurment DataFitFigure 4.27: The transmission signal of the accelerating mode is fitted to aLorentz function. A QL of 2.6 · 106 ± 13% corresponds well to the set value of3 · 106.absorber in slightly higher fields should increase the dissipated power and thusdecrease the Q factor. In addition simulations with the coaxial structure (butsimplified bellows) did not show a reduction of the effectiveness of the absorbers.Another cause for the discrepancy could come from the measurement setup. Pos-sible capacitive or inductive loading of the measurement signal could decreasethe observed bandwidth and therefore increase the fitted QL values. Furthermeasurements are needed to verify this.4.6 Performance Measurement SummaryAt the time of writing four nine-cell cavities have been fabricated. Three ofthose cavities have passed their vertical testing phase and two of those cavi-ties are installed in cryomodules. While the vertical performance in those threecavities could be higher compared to similar cavities, constant improvementson the cavity performance have been made and the horizontal performance isup to specification for the ARIEL project. The fourth cavity shows promise tooutperform the previous cavities as its Q0 is higher with fewer processing stepsthan the other cavities.The field distribution of higher order modes have been measured on a seven-cellcopper prototype and match the simulations, giving confidence in the simu-lations of the full nine-cell cavity. Measuring the HOMs on the cryomoduleswas only partially successfull. Frequency measurements fit well, while QL mea-surements indicated higher than expected values. To finalize the HOM charac-terization, beam based measurements (see appendix F) need to be done. Therealization of these measurements still needs to be developed.4.6. Performance Measurement Summary 89101102103104105106107108109 1  1.5  2  2.5  3  3.5  4Q LFrequeny [GHz]SimulationARIEL1ARIEL2Figure 4.28: Q fitting results for HOMs in the EINJ and EACA cavity 1 up to2.5 GHz compared to simulation results.90Chapter 5ConclusionsThe ARIEL eLINAC is a 50 MeV, 500 kW electron accelerator based on SRFtechnology that complements the existing rare isotope beam (RIB) program atTRIUMF. To accelerate the electrons, superconducting RF cavities based onthe 1.3 GHz TESLA cavity design are used. A future upgrade plan to add arecirculating loop for energy recovery operation adds to the requirements tomake a unique set of constraints for the cavity.As has been shown in section 3.4, the design of the cavity is modified to ac-commodate the requirements caused by the 10 mA electron beam in cw. Highbeam loading requires 100 kW or RF power to be transferred into the cavity.The TESLA cavity and its fundamental power couplers were designed for a lowduty cycle beam and corresponding low average power needs, and are thereforenot suitable for the ARIEL eLINAC.Future upgrade plans for the eLINAC include energy recovery operation to drivea free electron LASER (FEL) for development towards a fourth generation lightsource. In this scheme, the beam is guided through an undulator to create thelight and then through the SRF cavities again 180◦ out of phase with the crest ofthe RF. During the creation of the FEL light only a small fraction of the beamenergy is lost. The remaining energy can be returned to the electromagneticfields in the RF cavities, which reduces the power consumption significantly.In multi-pass ERL accelerators, beam-break up (BBU) is a strong concern ashas been shown in section 2.3.3. Dipole modes excited by the beam cause thefollowing bunches to receive a transverse momentum. Under the right circum-stances, this leads to an exponential growth of the dipole higher order mode(HOM) field and a strong deflection of the beam, leading to the loss of thebeam. The threshold of the HOM shunt impedance for this effect at a beamcurrent of 10 mA was determined to be Rd = 1 · 107Ω. A manufacturing toler-ance study (section 3.7) indicated a spread in shunt impedance by a factor of2. As a safety margin, the goal for the HOM dipole shunt impedance was setto 1 · 106 Ω or less.These HOM considerations caused further changes in the cavity geometry. Inthe TESLA cavity, a high shunt impedance dipole mode at a frequency of around2.56 GHz mode is trapped within the inner cells of the cavity. This mode wouldhave reduced the threshold current for BBU significantly below the design cur-rent of 10 mA. To untrap this mode dedicated simulations have shown thatdifferent beam pipe sizes change the field distribution enough so that this modepropagates out of the cavity.To reduce the HOM shunt impedance beam line absorbers are used. ThoseChapter 5. Conclusions 911091010 0  2  4  6  8  10  12  14Q 0Eacc [MV/m]10W5W2W 20W1WICMACMFigure 5.1: The performance of the cavities in the cryomodules meets the spec-ifications of Q0 ≥ 1 · 1010 at Eacc = 10 MV/m.consist of resistive materials, for the eLINAC stainless steel on one side and thecarbon fiber reinforced silicon carbide (tradename CESiC) on the other side ofthe cavity are chosen. Those absorbers are cooled with liquid nitrogen to pre-vent any heat load on the liquid helium system. The effectiveness of CESiC hasbeen measured at 80 K in dedicated measurements to ensure adequate perfor-mance (section 3.9). A single cell cavity was used to provide a reference and theQ of this cavity with and without the absorber was measured. The reduction inQ corresponds to the electrical conductivity. The measurements showed a con-ductivity between 600 and 5300 S/m at frequencies of 1.3 and 2.4 GHz. Thisperformance in conjunction with the adjusted cavity design provides a substan-tial reduction in dipole HOM shunt impedance.Using this cavity design with the beam line absorber for Q reduction, a thresh-old current of around 30 mA has been found. In addition, the beam transportdesign found a sufficient tuning range in the beam transport matrix elementM12 to further mitigate unwanted excitation [26].Performance measurements of the accelerating TM010 mode are fundamentalto ensure the cavity performs up to the specified gradient and within the limitof power dissipation. Limitations to the performance are described in section4.1. All four fabricated cavities exceeded the required accelerating gradient of10 MV/m in their vertical tests (section 4.3), limited only by the active loadlimitation of the cryostat. This means all cavities met the gradient specifica-tions set for the RIB beam. The Q0 values, indicating the power losses in thecavity walls, of all four cavities have been slightly below the specifications withvalues between 5 and 8 · 109 at 10 MV/m.Two of these four cavities have been installed in their respective cryomodulesand, after additional surface treatments, are performing with the desired Q0measured up to 11 and 12 MV/m respectively (fig. 5.1). The gradient is cur-Chapter 5. Conclusions 92rently limited by the conditioning status of the fundamental power couplers.Once the conditioning has been done to higher power levels, it is estimated thatthe cryogenic system will be the bottleneck at 30 to 40 W of active load on the2 K system.On the HOM side, the simulated field distributions have been verified by beadpulling on a copper model of the ARIEL cavity. 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For both mode types, thecutoff frequency increases with increased radius, allowing more communicationbetween the individual cells. In the case of the accelerating TM010 mode, theiris radius is responsible to focus the electric field onto the beam axis. Lessfocus with a bigger iris would reduce the R/Q.This would be a way to reach the HOM shunt impedance goal, if no possiblesolution with the TESLA inner cells could be found. 900 910 920 930 940 950 960 970 980 990 38  40  42  44  46  48R/Q [Ω]Pick-up Side Beam Pipe Radius [mm]Riris = 35mmRiris = 35.5mmRiris = 36mmRiris = 36.5mmRiris = 37mmRiris = 37.5mmRiris = 38mmRiris = 38.5mmRiris = 39mmFigure A.1: The R/Q of the accelerating mode decreases with increased irisapertures.Appendix A. Increased Inner Iris Radius 1010.0*1005.0*1051.0*1061.5*1062.0*1062.5*1063.0*1063.5*106 38  40  42  44  46  48R d/Q * QL [Ω]Pick-up Side Beam Pipe Radius [mm]Riris = 35mmRiris = 35.5mmRiris = 36mmRiris = 36.5mmRiris = 37mmRiris = 37.5mmRiris = 38mmRiris = 38.5mmRiris = 39mmFigure A.2: The shunt impedance of the 2.56 GHz mode generally follows asimilar trend like the accelerating mode.102Appendix BDipole HOM DataThe dipole HOM spectrum of the final cavity design, listed in table B.1, showsno modes on a harmonic of the bunch frequency of 650 MHz. Modes only upto 4 GHz were considered. At higher frequencies, modes will propagate into thebeam pipes. The QL in table B.1 considers only the cavity Q0 and the couplerQext, for the final QL both have to be added properly.Figure 3.20 is based on this data.Table B.1: Dipole HOM mode data for the ARIEL eLINAC cavity.f [GHz] Q0 Rd/Q [Ω] QL QAbsorbers1.577 1.54E+10 5.11E+00 1.27E+03 5.31E+061.620 1.29E+10 2.04E-01 1.40E+05 5.29E+061.630 1.32E+10 1.21E+00 3.94E+04 5.03E+061.647 1.34E+10 2.39E+00 2.13E+04 4.62E+061.669 1.37E+10 2.55E+00 1.46E+04 4.13E+061.695 1.39E+10 3.41E+01 1.10E+04 3.61E+061.724 1.41E+10 1.32E+02 8.59E+03 3.09E+061.753 1.41E+10 4.50E+01 6.71E+03 2.59E+061.780 1.36E+10 3.27E+00 4.97E+03 2.10E+061.797 1.30E+10 6.75E+00 2.19E+04 1.95E+061.837 1.85E+10 2.85E+00 1.44E+04 1.84E+061.852 1.76E+10 1.20E+00 1.08E+04 1.68E+061.865 1.73E+10 3.52E+01 1.20E+04 1.55E+061.874 1.72E+10 5.97E+01 1.67E+04 1.47E+061.881 1.72E+10 1.94E+01 2.84E+04 1.42E+061.886 1.72E+10 7.96E-02 5.89E+04 1.40E+061.889 1.72E+10 1.20E+00 1.58E+05 1.39E+061.890 1.72E+10 3.57E-02 7.38E+05 1.39E+062.015 6.11E+09 1.10E+00 4.00E+01 3.06E+042.253 4.83E+09 2.35E+00 1.56E+04 1.67E+042.338 1.36E+09 7.18E-01 1.36E+09 6.39E+032.380 5.07E+09 3.09E-01 4.42E+03 6.12E+032.472 8.70E+09 2.02E-01 7.05E+04 6.14E+032.482 6.22E+09 1.08E-01 2.08E+04 5.82E+032.497 4.41E+09 1.37E+00 1.38E+04 5.27E+03Appendix B. Dipole HOM Data 103f [GHz] Q0 Rd/Q [Ω] QL QAbsorbers2.497 4.41E+09 1.37E+00 1.38E+04 5.46E+042.515 3.44E+09 6.80E-01 1.32E+04 2.22E+042.534 2.97E+09 1.94E+00 1.63E+04 1.30E+042.551 2.78E+09 2.51E+00 2.42E+04 8.97E+032.565 2.72E+09 2.65E+00 4.52E+04 6.85E+032.574 2.79E+09 7.73E+00 1.18E+05 5.59E+032.579 3.44E+09 6.98E+01 7.52E+05 4.95E+032.253 4.83E+09 2.35E+00 1.56E+04 4.08E+032.338 1.36E+09 7.18E-01 1.36E+09 2.95E+032.380 5.07E+09 3.09E-01 4.42E+03 2.91E+032.684 5.69E+09 2.36E+00 1.14E+02 2.88E+032.732 1.02E+10 8.64E-04 9.67E+06 2.91E+032.786 9.87E+09 3.23E-03 9.23E+09 2.94E+032.787 9.88E+09 2.94E-03 7.70E+09 2.94E+032.787 9.90E+09 4.53E-05 5.83E+09 2.94E+032.788 9.92E+09 2.65E-04 4.07E+09 2.94E+032.788 9.94E+09 5.34E-06 2.94E+09 2.94E+032.788 9.96E+09 3.83E-05 2.58E+09 2.94E+032.789 9.97E+09 3.62E-07 3.75E+09 2.94E+032.821 6.32E+09 1.22E-01 1.59E+04 2.91E+032.827 2.92E+09 3.25E-01 7.78E+03 2.70E+032.839 1.75E+09 1.01E+00 6.19E+03 2.35E+032.856 1.91E+09 4.05E-01 4.79E+03 2.11E+032.882 2.66E+09 6.12E-01 3.81E+03 1.98E+032.916 3.61E+09 1.72E-01 3.17E+03 1.91E+032.916 3.61E+09 1.72E-01 3.17E+03 4.68E+042.955 4.61E+09 2.37E+00 2.77E+03 2.77E+042.991 5.71E+09 3.90E+00 3.25E+03 2.16E+043.016 5.92E+09 4.23E-01 7.81E+03 1.87E+043.022 3.99E+09 7.61E-02 1.72E+04 1.81E+043.032 6.77E+09 5.52E-01 1.24E+06 1.60E+043.055 8.99E+09 4.84E-05 1.79E+06 1.61E+043.062 1.49E+10 7.16E-01 6.85E+03 1.59E+043.067 9.02E+09 2.41E-05 9.02E+09 1.59E+043.069 9.00E+09 1.34E-07 7.38E+09 1.59E+043.070 9.03E+09 1.31E-06 5.60E+09 1.59E+043.070 9.06E+09 3.38E-06 5.04E+09 1.59E+043.071 9.11E+09 1.18E-05 5.47E+09 1.59E+043.072 9.15E+09 2.65E-05 6.58E+09 1.59E+043.072 9.18E+09 2.23E-05 7.91E+09 1.59E+043.073 9.20E+09 4.66E-07 8.89E+09 1.59E+043.079 1.55E+10 2.01E-01 1.18E+09 1.59E+043.079 1.57E+10 5.67E-02 2.73E+06 1.59E+043.080 1.56E+10 3.08E-02 6.21E+05 1.59E+043.082 1.56E+10 8.60E-01 1.92E+05 1.59E+043.086 1.55E+10 3.20E+00 5.62E+04 1.59E+043.094 1.50E+10 1.22E+00 1.49E+04 1.57E+043.110 1.43E+10 1.13E-02 5.32E+03 1.53E+043.131 6.64E+09 1.56E-01 3.96E+02 1.22E+04Appendix B. Dipole HOM Data 104f [GHz] Q0 Rd/Q [Ω] QL QAbsorbers3.268 7.14E+08 2.23E-01 3.17E+08 4.83E+033.268 7.14E+08 2.23E-01 3.17E+08 7.89E+033.131 6.64E+09 1.56E-01 3.96E+02 6.83E+033.342 8.94E+09 1.84E-01 1.16E+06 7.05E+033.347 8.52E+09 8.87E-05 1.68E+07 7.05E+033.348 7.87E+09 2.89E-02 3.28E+08 7.00E+033.349 8.33E+09 5.28E-02 1.47E+08 6.98E+033.352 8.52E+09 9.77E-02 7.78E+09 6.98E+033.355 8.56E+09 6.01E-01 3.29E+09 6.98E+033.359 8.56E+09 3.49E-01 1.12E+09 6.98E+033.362 8.56E+09 1.25E-02 2.12E+09 6.98E+033.364 8.56E+09 3.71E-03 3.61E+09 6.99E+033.437 8.34E+09 3.00E-02 2.35E+05 6.87E+033.470 5.67E+09 8.15E-02 4.69E+04 6.46E+033.522 4.23E+09 3.95E-02 2.07E+04 5.94E+033.580 3.46E+09 1.44E-03 1.38E+04 5.38E+033.640 2.96E+09 4.87E-02 1.45E+04 4.84E+033.670 6.75E+09 2.77E-08 6.19E+09 4.86E+033.698 2.60E+09 6.25E-02 3.15E+04 4.35E+033.712 6.63E+09 8.94E-08 6.36E+09 4.36E+033.737 4.84E+09 1.37E-03 8.83E+03 4.37E+033.755 2.53E+09 2.49E-03 1.27E+05 3.95E+033.809 3.01E+09 1.32E-01 9.94E+04 3.70E+033.831 4.90E+09 3.60E-04 2.72E+09 3.71E+033.835 4.84E+09 7.02E-05 9.05E+06 3.71E+033.837 4.89E+09 1.68E-04 2.91E+06 3.72E+033.837 4.89E+09 1.68E-04 2.91E+06 3.85E+083.841 4.94E+09 5.37E-04 1.36E+06 9.34E+073.845 5.00E+09 1.07E-03 9.06E+05 3.72E+073.849 5.06E+09 4.38E-03 4.28E+05 1.11E+073.853 5.10E+09 1.10E-02 2.39E+05 3.78E+063.854 2.10E+09 4.47E+00 1.09E+03 2.05E+043.855 5.17E+09 1.75E-02 1.30E+07 2.05E+043.863 2.51E+09 3.72E+00 2.30E+03 1.13E+043.809 3.01E+09 1.32E-01 9.94E+04 9.28E+033.831 4.90E+09 3.60E-04 2.72E+09 9.30E+033.835 4.84E+09 7.02E-05 9.05E+06 9.31E+033.898 7.38E+09 1.37E+00 2.12E+05 9.04E+033.953 3.31E+09 1.14E+00 2.57E+02 7.85E+034.007 6.62E+09 4.24E-02 4.48E+03 7.88E+034.028 4.76E+09 5.12E-01 1.28E+03 7.33E+034.072 4.97E+09 3.18E-01 7.78E+03 7.21E+034.084 3.98E+09 7.64E-01 2.79E+03 6.87E+034.101 3.81E+09 6.89E-01 1.70E+03 6.51E+034.121 3.91E+09 1.33E-02 1.00E+03 6.20E+034.133 7.10E+09 2.57E-03 7.80E+07 6.21E+034.133 3.63E+09 9.97E-01 1.08E+02 5.70E+034.143 3.90E+09 4.10E-01 1.18E+03 5.45E+034.156 6.21E+09 9.46E-06 8.29E+07 5.46E+034.161 6.63E+09 4.05E-03 7.51E+06 5.46E+034.162 3.59E+09 2.96E-02 3.54E+03 5.19E+03105Appendix CManufacturing ToleranceStudy DataIn the manufactoring tolerance study described in chapter 3.7, the ideal geom-etry parameters and randomly varied them for each halfcell within the giventolerances. A side condition is, that the connection between two half-cells issteady, i.e. if two half-cells connect at the iris, this radius is the same for both.Similar if the half-cells connect at the equator.Table C.1 shows the average values for frequency, Q0 Rd/Q and the associatedstandard deviation for dipole modes between 1.5 and 3.5 GHz.Table C.1: Tolerance study data for dipole modes between 1.5 and 3.5 GHz.f [MHz] σ(f) [MHz] Q0 σ(Q0) Rd/Q [Ω/cm] σ(Rd/Q) [Ω/cm]1553.9645 12.42 1.1e10 8.9e7 0.925 0.0651611.335 5.01 9.7e9 1.6e8 0.464 0.2671623.88925 3.69 9.4e9 9.6e7 0.297 0.1421639.7655 3.80 9.5e9 5.3e7 0.120 0.0951660.8655 2.75 9.7e9 5.0e7 0.982 0.2611686.5475 2.54 9.8e9 4.4e7 1.658 0.4561715.11125 2.14 9.9e9 4.6e7 21.85 0.7131743.803 1.94 9.8e9 5.1e7 17.18 1.0361770.24625 1.54 9.5e9 4.5e7 0.046 0.0471788.05325 1.05 9.0e9 2.2e7 1.391 0.2611833.281 1.32 1.4e10 6.4e7 0.446 0.1261848.11775 0.90 1.1e10 4.1e7 0.157 0.0831860.733 0.75 1.3e10 2.4e7 5.046 0.4011870.7095 0.65 1.3e10 1.7e 10.322 0.1221877.932 0.56 1.3e10 1.9e7 5.0611 0.286Appendix C. Manufacturing Tolerance Study Data 106f [MHz] σ(f) [MHz] Q0 σ(Q0) Rd/Q [Ω] σ(Rd/Q) [Ω]1882.89175 0.50 1.3e10 2.3e7 0.1950 0.0831886.062 0.61 1.3e10 2.4e7 0.1691 0.0811888.01825 0.66 1.3e10 2.9e7 0.0240 0.0201986.71075 10.16 9.1e9 1.3e8 0.2525 0.0222271.28475 15.64 6.7e9 1.8e9 0.0841 0.0482293.934 6.54 8.7e9 1.8e9 0.1485 0.0442414.372 11.95 6.5e9 7.0e7 0.2576 0.0152476.0265 3.46 7.4e9 5.8e7 0.0441 0.0282488.18625 3.23 7.2e9 5.6e7 0.1145 0.0632504.51725 2.90 7.0e9 5.0e7 0.2378 0.0552524.18425 2.51 6.8e9 4.8e7 0.4399 0.1112542.7755 2.28 6.5e9 3.6e7 0.3519 0.1202558.0865 2.67 6.3e9 3.9e7 2.0284 1.0722569.978 2.88 6.1e9 3.5e7 1.6819 1.4702578.02075 3.47 6.0e9 2.8e7 18.158 1.4842657.796 5.85 9.8e9 1.4e9 0.6796 0.2342670.42725 9.42 8.4e9 1.2e9 0.7661 0.1962814.9625 4.05 6.4e9 7.6e7 0.0043 0.0042825.2525 4.22 6.5e9 6.1e7 0.0118 0.0052843.018 3.31 6.8e9 4.7e7 0.0273 0.0082868.7705 2.79 7.2e9 4.4e7 0.0108 0.0072901.643 2.66 7.6e9 4.9e7 0.1721 0.0282937.9025 2.53 7.9e9 6.8e7 0.3180 0.0392972.03475 2.84 8.3e9 8.0e7 1.8366 0.0573001.51175 2.70 8.8e9 8.2e7 0.1046 0.0343023.27675 3.06 9.2e9 9.2e7 0.0058 0.0043059.008 2.96 1.1e10 1.0e8 0.1969 0.0223073.4885 2.80 1.1e10 4.0e8 0.0760 0.0423076.48025 2.44 1.1e10 7.4e8 0.0900 0.0773078.69075 2.53 1.1e10 5.4e8 0.1134 0.0853080.98075 2.44 1.1e10 6.9e8 0.1838 0.1253083.33275 2.52 1.1e10 6.5e8 0.5504 0.1993087.44625 2.94 1.0e10 6.3e8 0.7254 0.1363095.21225 2.84 1.0e10 9.3e8 0.2120 0.1203107.59125 2.29 1.1e10 6.3e8 0.0376 0.0193118.352 3.70 1.1e10 3.2e8 0.0447 0.0223338.18575 2.12 6.6e9 3.7e7 0.0641 0.0143346.572 2.07 6.4e9 2.9e7 0.0122 0.0113350.67825 2.08 6.4e9 2.3e7 0.0365 0.0253353.209 1.59 6.4e9 2.3e7 0.0514 0.0413355.9395 1.90 6.4e9 3.3e7 0.1092 0.0543358.46 1.68 6.4e9 3.3e7 0.1096 0.0473362.38525 3.73 6.4e9 9.1e8 0.0581 0.0403386.14675 15.01 6.8e9 2.9e8 0.0235 0.0223426.87775 10.00 7.7e9 2.2e8 0.0117 0.015107Appendix DFlatness MeasurementsThe flatness of a cavity is a measure of how even the field distribution is in eachcells. The flatness is calculated with equation 2.51. As the flatness directlyinfluences the R/Q, a high flatness (≥ 95%) is important to keep a high R/Q.Figures D.1, D.2, D.3 and D.4 show the improvements made on the flatness.The flatness can be tuned by plastic deformation of the cell length. To tune aslightly unflat cavity, a overall length change of 1 to 2 mm is necessary. Thecavity frequency can be influenced in the same way. The averaged frequencysensitivity is around 0.3 kHz/mm, but strongly depends on the field distribution.Figure D.5 demonstrates, that the frequency tuner in the cryomodule does not 0 2 4 6 8 10 12 14 16 0  200  400  600  800  1000  1200  1400E-fieldZ-Position [mm]Pick-up side Coupler sidef=1.29792 GHzafter tuning, f = 1.2974 GHzafter degassing, f = 1.29763 GHz, 94.1%after degassing+tuning, f = 1.29757 GHz, 97.3%after jacketing, f = 1.29742 GHz, 90.3%Figure D.1: ARIEL1 flatness tuning results.change the flatness significantly. The frequency tuner grabs the cavity by thebeam tubes and stretches the whole cavity, resulting in a increased frequency.This behaviour has been simulated on the warm tuning station by stretchingthe cavity to get a 200 kHz frequency change. The flatness changes within 1percentage point, which is acceptable.Appendix D. Flatness Measurements 108 0 2 4 6 8 10 12 14 16 18 0  200  400  600  800  1000  1200  1400E-field [∝ V/m]Z-Position [mm]Pick-up side Coupler sideAs recieved, f = 1.29705 GHz, flatness 75%Pre-BCP, f = 1.29880 GHz, flatness 96%Post-BCP, f = 1.29804 GHz, flatness 90%Post-BCP tuned, f = 1.29756 GHz, flatness 96%With He-Jacket: f = 1.29770 GHz, flatness 86.7%Figure D.2: ARIEL2 flatness tuning results. 0 2 4 6 8 10 12 14 16 18 0  200  400  600  800  1000  1200  1400E-field [∝ V/m]Z-Position [mm]Pick-up side Coupler sideAs recieved, f = 1.29705 GHz, flatness 13%Pre-BCP, tuned, f = 1.29835 GHz, flatness 96%Post-BCP, untuned, f = 1.29735 GHz, flatness 72%Post-BCP, tuned, f = 1.29735 GHz, flatness 72%degased, tuned, f = 1.29692 GHz, flatness 97%Figure D.3: ARIEL3 flatness tuning results.Appendix D. Flatness Measurements 109 0 2 4 6 8 10 12 14 16 18 0  200  400  600  800  1000  1200  1400E-field [∝ V/m]Z-Position [mm]Pick-up side Coupler sideAs recieved, f = 1.29761 GHz, flatness 44%Pre-BCP tuned, f = 1.29846 GHz, flatness 97%Post-BCP, untuned, f = 1.297423 GHz, flatness 88%Figure D.4: ARIEL4 flatness tuning results. 0 2 4 6 8 10 12 14 16 18 0  200  400  600  800  1000  1200  1400E-field [∝ V/m]Z-Position [mm]Pick-up side Coupler sidenormal: f = 1.29770 GHz, flatness 86.7%free floating: f = 1.29770 GHz, flatness 86.4%streched 200kHz: f = 1.29790 GHz, flatness 85.1%free floating again: f = 1.29769 GHz, flatness 86.4%Figure D.5: Frequency tuner simulation on the warm tuning stand revealedthat the flatness changes are minimal (within 1%-point).110Appendix EBCS Resistance FittingIn chapter 4.3.3, the extraction of the BCS and residual component of the surfaceresistance is discussed. Table E.1 summarizes the performed cavity tests on theARIEL nine-cell cavities with their respective treatment and lists their BCS andresidual resistances. Following figures show the individual fits withTable E.1: Results of the fits of the Q vs T data to extract the BCS and residualresistance. All data have been taken at low gradients between 1 and 3 MV/m.Cavity Treatment RBCS(T=2K) [nΩ] Rres [nΩ] FigureARIEL1 120 µm BCP 14 ± 5 58 ± 1 E.1cleaning no datadegassed 17 ± 2 28.6 ± 0.7 4.14ARIEL2 120 µm 19 ± 4 30.3 ± 0.7 E.285◦C bake 13 ± 4 33.3 ± 0.7 E.3degassed 6 ± 4 45 ± 2 E.4120◦C bake and HF rinse 12 ± 7 39.5 ± 0.9 E.5ARIEL3 120 µm BCP 10 ± 1 98.5 ± 0.2 E.6+30 µm BCP 14 ± 1 29.4 ± 0.3 E.7+60 µm BCP 23 ± 2 23.5 ± 0.2 E.8degassed+30 µm BCP no data+60 µm BCP 11 ± 2 37.3 ± 0.3 E.9ARIEL4 120 µm BCP 12.6 ± 1.3 22.8 ± 0.2 E.10Appendix E. BCS Resistance Fitting 111 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -18.86 ± 0.22b = 12.76± 0.07(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 200 400 600 800 1000 1200 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [58.14 ± 1.18] nΩRBCS(2K) = [13.98 ± 5.09] nΩ20130711(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =58.14Figure E.1: BCS fit for ARIEL1 after 120 µm BCP. (a) RBCS fit, (b) Rres fit.Appendix E. BCS Resistance Fitting 112 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.53 ± 0.14b = 12.40± 0.04(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 200 400 600 800 1000 1200 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [30.31 ± 0.73] nΩRBCS(2K) = [18.97 ± 4.25] nΩ20131004(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =30.31Figure E.2: BCS fit for ARIEL2 after 120 µm BCP. (a) RBCS fit, (b) Rres fit.Appendix E. BCS Resistance Fitting 113 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.58 ± 0.21b = 12.18± 0.07(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 800 900 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [32.17 ± 0.58] nΩRBCS(2K) = [14.81 ± 5.16] nΩ0131029(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =32.17Figure E.3: BCS fit for ARIEL2 after 85◦ C baking for 48h. (a) RBCS fit, (b)Rres fit.Appendix E. BCS Resistance Fitting 114 0 1 2 3 4 5 6 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -20.08 ± 0.43b = 12.58± 0.14(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 800 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [45.15 ± 1.94] nΩRBCS(2K) = [6.30 ± 4.40] nΩ20131212(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =45.15Figure E.4: BCS fit for ARIEL2 after degassing. (a) RBCS fit, (b) Rres fit.Appendix E. BCS Resistance Fitting 115 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.47 ± 0.36b = 11.89± 0.12(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [39.52 ± 0.89] nΩRBCS(2K) = [11.70 ± 7.10] nΩ20140324(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =39.52Figure E.5: BCS fit for ARIEL2 after 120◦ C baking for 48h and an HF rinse.(a) RBCS fit, (b) Rres fit.Appendix E. BCS Resistance Fitting 116 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.92 ± 0.06b = 12.04± 0.02(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [98.54 ± 0.20] nΩRBCS(2K) = [10.88 ± 1.12] nΩ20140617(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =98.54Figure E.6: BCS fit for ARIEL3 after 120 µm BCP. (a) RBCS fit, (b) Rres fit.Appendix E. BCS Resistance Fitting 117 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.51 ± 0.04b = 12.11± 0.01(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 800 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [29.45 ± 0.30] nΩRBCS(2K) = [14.29 ± 0.98] nΩ20140725(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =29.45Figure E.7: BCS fit for ARIEL3 after additional 30 µm BCP. (a) RBCS fit,(b) Rres fit.Appendix E. BCS Resistance Fitting 118 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -16.59 ± 0.06b = 12.16± 0.02(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 200 400 600 800 1000 1200 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [23.53 ± 0.23] nΩRBCS(2K) = [23.74 ± 2.28] nΩ20141114(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =23.53Figure E.8: BCS fit for ARIEL3 after additional 60 µm BCP. (a) RBCS fit,(b) Rres fit.Appendix E. BCS Resistance Fitting 119 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.95 ± 0.12b = 12.08± 0.04(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [37.25 ± 0.33] nΩRBCS(2K) = [11.14 ± 2.23] nΩ20150722(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =37.25Figure E.9: Fit of Q0 vs T data to extract the BCS resistance for ARIEL3 toextrapolate a performance test from 2.3 K to 2.0 K. (a) BCS resistance fit, (b)residual resistance fit.Appendix E. BCS Resistance Fitting 120 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0.25  0.3  0.35  0.4  0.45  0.5ln(R s - Rres,est) [nΩ]1/T [K-1]m = -17.73 ± 0.07b = 12.09± 0.02(a)fitting function: ln(Rs[nΩ]) = m*x +b + ln(x)Rs = G/QBCS resistance fit (1/T = 0.25...0.4 K-1) 0 100 200 300 400 500 600 700 800 0.25  0.3  0.35  0.4  0.45  0.5R s [nΩ]1/T [K-1]RRes = [22.75 ± 0.24] nΩRBCS(2K) = [12.62 ± 1.34] nΩ20150515(b) Rs = G/QBCS fitRRes = Rs - RBCSResidual Fit (1/T = 0.45...0.5 K-1) =22.75Figure E.10: BCS fit for ARIEL4 after 120 µm BCP. (a) RBCS fit, (b) Rresfit.121Appendix FBeam Based HOMMeasurementsTo finalize the HOM characterization and verify or disprove in chapter 4.5 dis-cussed Q measurement results beam based measurements are needed. By pur-posefully exciting a HOM with a carefully prepared beam allows measurementsof the HOM frequency, its shunt impedance Rd, its loaded quality factor QLand polarization for modes other than monopole modes.In order to excite a HOM the beam has to be prepared to match the HOMfrequency. Reference [86] discusses three methods to excite HOMs.A straight forward way is to use the frequency control system of the cavity tochange the HOM frequency to fulfill the conditionfHOM = n · fb, (F.1)with fb as the bunch repetition frequency and n as any integer. Unfortunatelyfor the ARIEL beam parameters fb is rather large with 650 MHz comparedto the tuning range of 400 kHz for the accelerating mode1. Due to the designof the e-gun changing the bunch frequency drastically is impossible due to thedielectric transmission line that prevents the transmission of lower frequencysignals.Since dipole modes do not interact with a beam that is perfectly on axis, thebeam has to be injected into the cavity with an offset δx0. This can be usedto create the resonant condition by modulating this offset with a modulationfrequency fmod. The transverse injection position of bunch m would oscillatearound the beam axis asδxn,0 = δx0 sin(nωmodTb). (F.2)This offset modulation creates side-bands to the bunch frequency and the newresonant condition becomesfHOM = n · fb ± fmod. (F.3)The position of the bunch has to be measured with a beam position monitor(BPM) after the dipole mode kicks the beam. Once the mode m is fully excited,the BPM reading of the position looks likeδxn,BPM − δxn,0 ∝ δx0 IbE0Rm, (F.4)1HOMs might have larger tuning ranges, but still small compared to fb.Appendix F. Beam Based HOM Measurements 122with the beam current Ib, the injection energy E0 and the shunt impedance Rmof mode m.Similarly, the HOM can by excited by modulating the bunch charge Qb. If thebunch charge is modulated asQbn = Qb0 (1 + λ sin(nωmodTb)) (F.5)with a modulation amplitude λ, side-bands are created and can excite the HOM.This bunch charge modulation method has been used recently on the Cornellmain LINAC cavity [87].The modulation frequency has to be scanned from 0 to half of the bunch repe-tition frequency to catch all HOMs. Once a HOM is found, the frequency thencan be measured by looking at the signal on the pick-up. The measured kickcontains information about the QL and the R/Q of the mode. The QL can beextracted from the time constant of the envelope of the BPM reading, whichcorresponds to the decay constant of the mode (see eq. 2.33). The maximumkick is related to the R/Q of the mode. To fully characterize the mode an az-imuthal scan of the beam position has to be done, similar to the HOM beadpullmeasurements described in section 4.5.1. In this way, it can be determined if aHOM has monopole, dipole or higher order character and its polarization. Themaximum kick is also proportional to the average bunch charge (higher bunchcharges get kicked more), the drift distance (a longer drift with some transversemomentum results in bigger displacements) and inversely proportional to theinjection energy E0 (higher energy beams are stiffer and deflect less than lowenergy beams). Figure F.1 shows calculations for the added displacement afterthe beam has been received transverse momentum from a HOM. The chosen pa-rameters correspond to a 3 MeV beam with an average bunch charge of 10 pCand a drift length of 3.5 m. These correspond to settings reachable in the firstaccelerating cryomodule. It is important that the beam is fully relativistic as allcalculations assume β = 1. With an energy of 3 MeV the electron beam reaches0.99c, enough to consider the difference negligible.Unfortunately, it is currently impossible to create either a position or a bunchcharge modulation with the installed equipment. The bunch charge modula-tion seems straight forward by modulating the RF signal sent to the e-Gun.But a narrow bandwidth (∼2 MHz) matching section in the e-Gun would at-tenuate any of the side-bands created. The position modulation would requiretwo strong RF dipoles installed in the beam line with their respective neededancillaries and controls.Appendix F. Beam Based HOM Measurements 123-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0  5000  10000  15000  20000  25000  30000x BPM - x0 [mm]Bunch numberFigure F.1: The envelope of the BPM reading after the beam is kicked by the2.57 GHz mode corresponds to the QL of the HOM while the maximum kick isproportional to the R/Q. Calculations are based on a 3 MeV beam with Qb0 =10 pC, δx0 = 10 mm and a drift length of 3.5 m. The HOM is simulated witha R/Q of 76 Ω and a QL of 105.

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