ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2018

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1638

From Macroscopic to MicroscopicDynamics of SuperconductingCavities

ANIRBAN BHATTACHARYYA

ISSN 1651-6214ISBN 978-91-513-0253-9urn:nbn:se:uu:diva-343704

Dissertation presented at Uppsala University to be publicly examined in Polhemsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 20 April 2018 at 09:30 for thedegree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Jean Delayen (Old Dominion University).

AbstractBhattacharyya, A. 2018. From Macroscopic to Microscopic Dynamics of SuperconductingCavities. Digital Comprehensive Summaries of Uppsala Dissertations from the Facultyof Science and Technology 1638. 75 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-513-0253-9.

Superconducting (SC) radio frequency (RF) cavities are at the heart of many large-scaleparticle accelerators such as the European Spallation Source (ESS), the X-ray Free ElectronLaser (XFEL), the Linac Coherent Light Source (LCLS)-II and the proposed InternationalLinear Collider (ILC). The SC cavities are essentially resonant structures with very highintrinsic quality factors (Q0) of the order of 1010. The high Q0 of the cavities leads to increasedreflection during charging of the cavities to nominal voltage because the bandwidth of the signalexceeding that of the cavity. This results in high energy losses in case of pulsed machines.In this thesis I explore and present a novel technique to optimally charge the superconductingcavities with the particular example of the spoke cavities to be used for the ESS project in Lund,Sweden. The analysis reveals that slow charging with hyperbolic sine cavity voltage profilematches the signal bandwidth to that of the cavity which leads to energy efficient filling.

However, a filling rate lower than some particular value is counter-productive. The energyexpended in cryogenic cooling to evacuate the heat due to ohmic losses in the cavity startsto dominate the lost energy. Such cryogenic losses are dependent on cavity Q0 throughthe residual resistance. The residual resistance changes with the applied electromagneticfield due to the pair-breaking mechanism of Cooper-pairs. Hence, methods for accuratemeasurement of the cavity Q0 are essential for accurate characterization and operation ofthe superconducting cavities. In this thesis I propose a novel method to accurately measureQ0 as a function of the applied electromagnetic field and present experimental results fromthe prototype spoke cavity in the Facility for Research Instrumentation and AcceleratorDevelopment (FREIA), at Uppsala University.

The cavity quality factor (Q0) is also dependent on the material’s purity and the trappedmagnetic flux in the superconducting material. Recent studies have revealed that the rate ofcooling of materials through the critical temperature has an effect on the residual flux trappedin the material. In this thesis I use the time-dependent Ginzburg-Landau equations to modelthe process of state transition from a normal to a superconducting state. This theoretical studymay allow an explanation of the experimentally observed results from the basic principles ofthe general theory of state transitions as proposed by Ginzburg and Landau.

Keywords: superconducting cavity, superconductivity, self-excited loop, Ginzburg-landau,vortex, optimization, quality factor, microwave

Anirban Bhattacharyya, Department of Physics and Astronomy, High Energy Physics, Box516, Uppsala University, SE-751 20 Uppsala, Sweden.

© Anirban Bhattacharyya 2018

ISSN 1651-6214ISBN 978-91-513-0253-9urn:nbn:se:uu:diva-343704 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-343704)

Dedicated to my parents

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Minimization of power consumption during charging ofsuperconducting accelerating cavitiesA. K. Bhattacharyya, V. Ziemann, R. Ruber, and V. GoryashkoNuclear Instruments and Methods in Physics Research Section A:Accelerators, Spectrometers, Detectors and Associated Equipment,801:78-85, 2015.

II Time Domain Characterization of High Power Solid State Amplifiersfor the Next Generation Linear AcceleratorsL. H. Duc, A. K. Bhattacharyya, V. Goryasko, R. Ruber, A. Rydberg, J.Olsson and D. DancilaMicrowave and optical technology letters (Print), ISSN 0895-2477,E-ISSN 1098-2760, Vol. 60, no 1, p. 163-171.

III A method for high-precision characterization of the Q -slope ofsuperconducting RF cavitiesV. A. Goryashko, A. K. Bhattacharyya, H. Li, D. Dancila, and R. RuberIEEE Transactions on Microwave Theory and Techniques,64(11):3764-3771, Nov 2016.

IV Precise measurements of hot S-parameters of superconducting cavities:Experimental setup and error analysisA. K. Bhattacharyya, L. H. Duc, V. Ziemann, T. Lofnes, H. Li, R.Ruber, and V. GoryashkoTechnical Note under preparation, Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, Detectorsand Associated Equipment

Reprints were made with permission from the publishers.

List of other publications

The following publications are not included in this thesis

I A Method for High-Precision Measurements of SuperconductingCavitiesV. A. Goryashko, A. K. Bhattacharyya, H. Li, D. Dancila, and R. RuberProceedings of the 8th International Particle Accelerator Conference(IPAC 2017), Copenhagen, Denmark, May, 2017.

II Wave propagation in a fractal wave guideA. K. Bhattacharyya et alProceedings of the 8th International Particle Accelerator Conference(IPAC 2017), Copenhagen, Denmark, May, 2017.

III Beam-based alignment studies at CTF3 using the octupole componentof CLIC accelerating structuresJ. Ögren et alProceedings of the 8th International Particle Accelerator Conference(IPAC 2017), Copenhagen, Denmark, May, 2017.

IV First High Power Test of the ESS Double Spoke CavityH. Li et alFREIA report, 2017.

V Cryogenic Synopsis from the Testing of the Fully Equipped ESS DoubleSpoke Cavity RomeaR. S. Kern et alFREIA report, 2017.

VI RF Test of the ESS Double Spoke CavityH. Li et alFREIA report, 2016.

VII ESS RF Source and Spoke Cavity Test PlanA. K. Bhattacharyya et alFREIA report, 2015.

VIII Progress at the FREIA LaboratoryM. Olvegard et alProceedings of IPAC’15, JACoW: The Joint Accelerator ConferencesWebsite, 2015.

IX Test Characterization Of Superconducting Spoke Cavities At UppsalaUniversityH. Li et alProceedings of the 17th International Conference on RFSuperconductivity(SRF 2015), Vancouver, Canada, Sept, 2015.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Applications of neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Neutron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Particle accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Power sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 RF Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Accelerating cavity parameters and cavity treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Resonant cavity RF parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Surface resistance (RS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Dissipated power (Pd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.3 Shunt Impedance (Ra) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.4 Quality factor (Q0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.5 Geometrical factor (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.6 Cavity-shape constant (R/Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.7 SC cavity issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.8 Lorentz force detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Cavity treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Centrifugal Barrel Polishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Etching and Buffered Chemical Polishing . . . . . . . . . . . . . . . . . . . . . 222.2.3 Electropolishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.4 High pressure water rinsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 ESS and FREIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 The ESS Linac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Ion source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Low Energy Beam Transport (LEBT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.3 Radio Frequency Quadrupole (RFQ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.4 Medium Energy Beam Transport (MEBT) . . . . . . . . . . . . . . . . . . . . 263.1.5 Drift Tube Linac (DTL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.6 Spoke section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.7 Medium and High β section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.8 High Energy Beam Transport (HEBT) . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.9 Target station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.10 RF Power sources and beam instrumentation . . . . . . . . . . . . . . . . 29

3.2 The FREIA Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.1 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 The cryostat HNOSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 RF power generation and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Charging of superconducting Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 The Cavity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 Transit time factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Cavity RLC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 The step charging profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 The optimal charging profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Characterization of superconducting cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1 Self-excited Loop (SEL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Cavity Q0 and reflection coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Phenomenon of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.1 Phenomenon of superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.1.1 The London equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.1.2 The Ginzburg-Landau (G-L) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2 Flux trapping in superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.3 Flux trapping in presence of impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Sammanfattning på svenska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

1. Introduction

Neutrons are the uncharged sub-atomic building blocks of matter. The factthat they do not carry a charge allows neutron scattering microscopy to be ahighly sensitive research tool which provides high penetration into materialsto allow non-destructive testing and meet future technological needs. Sincethe de Broglie wavelength of thermal neutrons is about an Angström, which isthe typical distance between atoms in matter, it allows imaging at the molecu-lar scale without interfering with the electro-magnetic properties of the mate-rial/process under test.

1.1 Applications of neutron scatteringNeutron scattering allows imaging of materials and dynamic processes suchas structural transformations in materials. Thus, neutron scattering finds wideapplication in the life sciences for drug development, in physics and materialscience to probe superconductivity and magnetism, and provides a guide todevelop new materials. Neutron scattering techniques are applied to determinethe dynamic behavior of materials because of their large penetration lengthsin many substances, specifically in metals (see Figure 1.1).

Figure 1.1. Comparison of neutron and x-ray scattering cross-sections.

Due to the large scattering cross-section of neutrons from hydrogen [46, 33](see Figure 1.1), neutron scattering experiments are very sensitive to water andcells. Hence, the neutron scattering technique is an ideal tool to probe andidentify hydrogen-rich materials and to reveal information about their crys-talline structures. An important application in non-destructive testing of sam-ples lies in the study of structural integrity of energy conversion materials

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like Solid Oxide Fuel Cells (SOFCs). The biggest hindrance for SOFCs frombecoming easily marketable is their poor durability. Neutron probing allowsmore insight into the actual processes to be gained. It can be used to determinethe micro-structure and performance of SOFCs. This allows the study of thephenomena by which such materials develop strains and how defects developin cells thus degrading their lifetime.

Life sciences research requires an understanding of the processes that oc-cur, from the large cellular scale down to the atomic scale. In some casesresearchers need to probe large and complex macromolecules, while in othercases they probe the function of water in various mechanisms dealing withenzymes or different drugs. Thanks to the very high sensitivity of neutronscattering to water-rich materials [46, 33], they are extremely effective probesfor the study of biological samples.

1.2 Neutron sourcesNuclear reactors have been used as the source of slow neutrons for the imagingactivities described above. However, since the late 1970s and early 1980saccelerator driven sources have started to be used for neutron scattering. Theseaccelerator driven sources are excellent for better control of the generation offast neutrons which can then be slowed down. This allows the generationof high flux neutron beams. The accelerator community developed proton-driven neutron sources using various kinds of accelerator technologies such ascyclotrons, synchrotrons or linear accelerators.

Proton-driven sources employ the spallation process which generates sig-nificantly less heat per effective neutron flux than the nuclear fission reactionand thus have significant technological advantage over the most intense re-search reactors. In addition, such sources can be operated in pulsed modewhich leads to generation of neutrons in pulses. This provides peak bright-nesses during a very short pulse time which far exceeds that available fromreactors. The pulsed sources also enable time-of-flight measurements whichsimplifies spectrometry. The very heart of these proton-driven neutron sourcesis the particle accelerator.

1.3 Particle acceleratorsSince the development of the first 9-inch cyclotron at the University of Califor-nia, Berkeley in 1931, particle accelerator technology has developed substan-tially even to this day. Compared to the first research accelerators deliveringa beam of around 1 to 5 MeV energy, the present ones are giants with beamenergies in the TeV range, like the LHC at CERN. While the most promi-nent accelerators delivering very high energy beams (> 1 GeV) are meant for

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fundamental research in physics, majority of them are used in scientific ef-forts like radiotherapy, industrial processing like non-destructive imaging, ionimplantation, and for biomedical and other low-energy research.

An important parameter of accelerators is the beam power. The beam powerdepends on the beam energy and the number of particles being acceleratedper beam pulse [48]. Hence, a higher beam power means a larger numberof particles per pulse and enhanced interaction with the sample. Thus, theimaging quality is improved.

1.3.1 Power sourcesThe low energy machines used for X-ray imaging or particle therapy usu-ally employ DC sources using Cockcroft-Walton or Van de Graaff generators.However, in the high energy accelerators used for basic research the particlesare accelerated by time-varying (AC), longitudinal electric fields in dedicatedresonating structures called cavities. The AC fields are produced by an appa-ratus which generates fast-changing fields in the microwave/radio-frequency(RF) range of the electromagnetic spectrum. The generated fields are thentransferred to the accelerating cavities by means of distribution systems em-ploying waveguides and coaxial lines. The accelerating RF cavities are de-signed such that they efficiently transfer the energy from the electromagneticfield to the accelerated particles.

1.3.2 RF CavitiesOwing to its comparatively low cost, easy malleability and high conductivity,copper has been used to construct RF cavities since the first ones used by RolfWideroe in 1928 [49]. However, with the increasing demand for higher beamenergies, the requirement on the energy transfer rate from the cavity to thehigh-current particle beam becomes more stringent. This calls for very strongEM fields in the cavities, which cannot be supported by copper cavities be-cause of high ohmic losses and corresponding thermal loading [64, 60]. Now,for high energy machines, employing long-pulse linear accelerators (linacs),using only copper cavities is no longer an efficient option. The present stateof the art approach to reduce losses in cavities is the use of superconducting(SC) materials. The cavities can thus be of two types depending upon theiroperating conditions, normal conducting or superconducting. While most ac-celerators use cavities operating at normal temperatures, in 1965 lead-platedcavities were first used at SLAC at extremely low temperatures which madethem superconducting [44].

As the demand for high energy, long pulse or continuous wave machinesarose, it was observed that for optimum cost effectiveness an accelerator mustinclude a superconducting section. Since SC cavities can withstand a larger

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Figure 1.2. Relative cost breakdown for components to determine optimal operatinggradient of an accelerator facility [59].

accelerating field gradient than normal conducting ones [60], the overall con-struction cost can be reduced since higher energy beams can be produced inshorter machines. However, SC cavities require cryogenic cooling, which in-creases in cost with larger accelerating field gradient [64, 59] (Figure 1.2).This has lead to a lot of development of SC RF technology to develop SCcavities with reduced losses in the cavity walls.

The European Spallation Source (ESS) in Lund will be a source of slowneutrons of extreme brightness and unparalleled power which will make it aprobe of immense scientific performance. The first step in the process of neu-tron production is accelerating protons to an energy of 2.0-2.5 GeV using alinear particle accelerator. To achieve this, the accelerator will have differentsections starting with an ion source to generate the protons followed by struc-tures to chop, condition and accelerate the beam. The accelerated beam willbe directed onto a tungsten target to produce the neutrons which will be usedby the user community to achieve their scientific requirements. Along withthe accelerator, it will house a host of imaging instruments, each dedicatedto a scientific requirement using different types of spectrometers and diffrac-tometers. In addition to achieving all these scientific targets, the facility isplanned to be carbon neutral and “green” and to be a pioneer in the domain offuture sustainable accelerators. In this respect, the facility is looking for waysto minimize its carbon footprint by being as energy efficient as possible.

1.4 Thesis contributionsThis thesis investigates some of the challenges related to operation and char-acterization of SC RF cavities which are being increasingly used in present

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day pulsed high energy machines. In particular, it investigates the character-istics of double spoke cavities to be used for the proton linac at the ESS. Thework includes theoretical analysis and modeling of physical phenomena re-lated to superconductivity and resonant structures as well as experiments witha prototype double spoke cavity developed at IPN-Orsay. The experimentswith the cavity were performed at the Facility for Research Instrumentationand Accelerator Development (FREIA) at Uppsala University.

The spoke cavities are made of niobium and are operated in superconduct-ing state at cryogenic temperatures. They operate in pulsed mode in the ESSlinac. Superconducting cavities are resonators with an extremely small band-width in the order of 1 Hz. This makes it difficult to excite fields within thecavity. At the start of the RF pulse any energy sent to the cavity is entirelyreflected back, because the signal bandwidth exceeds that of the cavity. Thisresults in a considerable amount of energy being wasted. In chapter 4, I ad-dress the problem of minimizing the reflected energy while exciting RF fieldsin the SC cavities of pulsed machines and analyze the results with respect todifferent RF power sources that are widely used in particle accelerators.

Operating cavities in the superconducting state with higher acceleratinggradients reduces cost in RF power (Figure 1.2). However, since the cavitiesrequire cryogenic cooling additional costs are incurred. For the accelerator tobe cost effective, it is necessary for the power dissipated in the cavity to meetdesign requirements. The power dissipation depends on the quality factor (Q0)of the cavities. Thus, the Q0 of the cavity needs to be accurately measured asthis validates whether the manufactured cavities meet the design requirements.A novel automated method and experimental setup to perform such measure-ments has been developed at FREIA by me and is discussed in chapter 5 of thethesis.

These measurements of Q0 give some indication that the quality factor ofSC cavities depends on the cooldown rate and spatial temperature gradientacross the cavity. Such dependencies have also been observed before at theFermi National Accelerator Laboratory, USA [12, 11], Helmholtz ZentrumBerlin [24, 63] and Conseil Européen pour la Recherche Nucléaire (CERN)[26] but the physics is not well understood. This observation motivates anexercise into understanding the effect of spatial temperature gradients on theonset of superconductivity in the presence of trapped magnetic fluxes. Thetheoretical framework used for such an investigation is derived from the modelof superconductivity proposed by Ginzburg and Landau. The model and firstresults are presented in chapter 6.

However, before going into such details, a discussion regarding the param-eters of accelerating structures is necessary to understand the physics motiva-tions of using superconducting technology.

15

2. Accelerating cavity parameters and cavitytreatment

In this chapter we recapitulate some of the RF parameters used to characterizethe performance of accelerating RF cavities. Some of these parameters aredependent on whether normal conducting or superconducting technology isused and define the energy requirements for the linac. Analyzing the cavityperformance and choosing the right technology allows minimizing the energyconsumption.

An RF cavity is a metallic chamber that contains an electromagnetic (EM)field in the form of cavity modes. The shape of the cavity is selected so thata particular mode can efficiently transfer its energy to a charged particle. Anaccelerating cavity needs to provide an electric field in a longitudinal directionwith the velocity of the particles being accelerated. The associated magneticfields, however, only provide deflection but no acceleration.

In normal conducting cavities the alternating EM fields penetrate into thecavity walls. This is termed as the “skin effect”, since the field reduces intothe material and remains confined to the surface. In contrast, in SC cavitiesweak EM fields are completely expelled from the bulk of the superconductordue to the Meissner-Ochsenfeld effect [55].

2.1 Resonant cavity RF parametersThe RF parameters of accelerating cavities can be cast into two categories: theparameters dependent on material properties of cavities, and, the parameterssolely determined by the geometry. The latter can be used to compare cavitiesregardless of whether it is in a SC or normal conducting state.

2.1.1 Surface resistance (RS)In normal conducting cavities the surface resistance to RF fields is a result ofthe “skin effect” while in their superconducting counterparts the “skin effect”is additionally modified by the Meissner effect.

In a normal conducting state, the change in the magnetic field creates anelectric field which opposes the change in the induced magnetic field. Thisforces the conducting electrons to the outside of the conductor and keeps them

16

confined to the skin depth, δ . The EM wave amplitude is damped exponen-tially into the material at the distance δ . The skip depth is inversely propor-tional to the conductivity, σ . These two quantities are connected together by aconstant of proportionality termed as the surface resistance, RS [61]. Thus,

RS(Ω) =1

σδ=

√πμ0μr f

σ, (2.1)

where, f is the frequency in Hz. The surface resistance of typical materialsused for normal conducting cavities is several milli-ohms.

For SC cavities, due to the persistent supercurrents, the B-field is confinedto the surface. The B-field decays exponentially into the material and reaches1/e at the London penetration depth. This results in surface resistance whichdepends on temperature. The surface resistance decreases exponentially withtemperature, but it increases quadratically with frequency f . A good model ofthe experimental data on the variation of RS for niobium is given by [55, 64]

RS(Ω) = 2×104 1T

(f

1.5

)2

exp(−17.67

T

), (2.2)

where, T is the temperature of the material and f is the RF frequency in GHz.For niobium the surface resistance is of the order of several nano-ohms toseveral tens of nano-ohms.

Thus, RS for superconducting cavities is typically five orders of magnitudelower than that of normal-conducting cavities [58]. However, for supercon-ducting cavities, RS has stronger dependence on frequency and is also affectedby trapped magnetic flux.

2.1.2 Dissipated power (Pd)When an RF signal is applied to cavities, the induced surface currents on thecavity walls along with the surface resistance leads to dissipation of energy[61]. The power density of the energy loss is given by

ρ =12

RS|H|2, (2.3)

where H is the applied magnetic field on the cavity wall. The total dissipatedpower can be obtained by integrating ρ over the whole cavity surface area A,

Pd =12

RS

∫A|H|2dA. (2.4)

Since, for superconducting cavities, RS is several orders of magnitude lowerthan that for normal-conducting ones, so is Pd .

Thus, if the machine is operated with long RF pulses, i.e. it has high dutyfactor, and has large amounts of RF power supplied, then the energy dissipatedcan be very high. In such a case SC technology becomes more favorable forstable and energy efficient machine operation.

17

2.1.3 Shunt Impedance (Ra)The dissipated power in the cavity walls Pd can be related to the acceleratingvoltage V in the cavity along the direction of propagation of the beam bymeans of the shunt impedance Ra [61]. If the beam is moving along the zdirection in an electric field EZ , then the voltage is given by

V =∫

|EZ cos(ωz/βc)|dz, (2.5)

Ra is defined as

Ra =V 2

2Pd, (2.6)

and ω is the angular RF frequency.

2.1.4 Quality factor (Q0)We compare the dissipated power in a cavity, Pd , among cavities using differ-ent technologies, by means of the quantity Q0, which is the intrinsic qualityfactor of the cavity. It is defined as the ratio of the energy stored in the cavityto the energy lost in the cavity per cycle of the RF oscillation [61, 7]. Thus,

Q0 =ωWPd

, (2.7)

where W is the stored energy. The bandwidth of the cavity Δω is defined interms of Q0 as

Δω =ωQ0

. (2.8)

The stored energy W can be calculated from the cavity geometry and the mag-netic field, H, in the cavity of volume V :

W =12

μ0

∫V|H|2dV. (2.9)

Since Q0 is inversely proportional to Pd , it turns out that Q0 of SC cavities isseveral orders of magnitude larger than that of normal conducting ones. WhileQ0 of normal conducting cavities is typically between 103 to 105, that of SCcavities is typically between 107 and 1011.

High-power RF signals are driven into a cavity by means of a power cou-pler from an RF station [18]. The coupler is usually a reciprocal element andcharacterized by the external quality factor Qext . Qext is a figure of merit ofenergy losses, PL, by a cavity to a matched load, in one cycle of the excitedfrequency [7]:

Qext =ωWPL

. (2.10)

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The total energy loss due to these two processes is quantified by the loadedquality factor, QL, and is given by [62]

1QL

=

(1

Q0+

1Qext

). (2.11)

2.1.5 Geometrical factor (G)The geometrical factor of a cavity is defined as [55]

G = RSQ0 =RSωW

Pd=

ωμ0∫

V |H|2dV∫A |H|2dA

. (2.12)

It turns out to be independent of surface resistance RS. Specifically, for a givenfrequency ω and field distribution H, the geometrical factor G ends up beingthe ratio of the cavity volume to the cavity surface area. Thus, it can be usedto compare among cavities using different technologies.

2.1.6 Cavity-shape constant (R/Q)The beam traverses the cavity on-axis where it interacts with the electric fieldand gains in energy. Hence, it is important to characterize the cavity design interms of its effectiveness to focus the electric field on axis. This is measuredby the so-called cavity-shape constant, R/Q, for a probe particle that has thesame velocity as the cavity phase velocity [55]:

R/Q =12

V 2

ωW. (2.13)

Then, using equations (2.5) and (2.9)

R/Q =2[∫ |EZ cos(ωz/βc)|dz]2

ε0ω∫

V |E|2dV, (2.14)

where |E| is the magnitude of the electrical field and β = v/c is the velocityof the accelerated particles with respect to the velocity of light in vacuum.Thus, R/Q is also independent of surface resistance and effectively comparesdifferent cavity designs irrespective of the technology used.

2.1.7 SC cavity issuesFrom the above discussion it might seem obvious that the SC cavities are al-ways a better choice over normal conducting ones. But one must keep in mindthat for the cavities to stay superconducting they need to be cooled with liquid

19

helium, the process of production and maintenance of which itself is an energyintensive process.

It is therefore important to realize when to switch from the use of normalconducting to superconducting cavities. Let us compare the energy require-ments for superconducting and normal conducting cavities.

The superconducting cavities are placed in a Helium bath and dynamic RFlosses, generate heat in the cavity walls which enters the bath. In addition,ambient heat also enters the bath through non-ideal heat-shield around thebath which gives rise to static losses of the order of 1 W. Typical acceleratingvoltage for cavities in around 1 MV. This gives rise to dynamic losses of theorder of 10 W. All this dissipated energy has to be removed by a cryogenicsystem which has very poor Carnot efficiency of the order of 10−3 [21, 34].This results in the total power consumption exceeding 10 kW for continuouswave (cw) operation.

For normal conducting accelerating cavities the accelerating voltage of 1 MVcorresponds to a dynamic ohmic loss of the order of 1 MW. With an RF dutycycle of 1% the loss ends up being of the order of 10 kW. Thus for RF dutycycle less 1% the normal conducting technology is advantageous over the su-perconducting one.

For high duty factor (> 4%) or cw machines SC operation is thus necessary.The use of such technology comes at a price. The operation of SC cavitiesfaces challenges like sensitivity to cavity deformation during cavity filling withRF power due to Lorentz force detuning described below.

2.1.8 Lorentz force detuningThe electromagnetic field in the cavity exerts pressure on the cavity walls. Thepressure is given by [28]

P =14(μ0H2 − ε0E2) , (2.15)

and grows as the accelerating gradient in the cavity increases.In order to provide efficient heat transfer from the inner cavity walls, where

the heat is generated by RF fields, to outer cavity walls, the latter are madethin, typically 4 mm [25]. This makes the radiation pressure substantial forthe cavities and they deform. This variation of shape results in the changeof resonance frequency of the cavities, the so called Lorentz force detuning.Since SC cavities have very small bandwidth, the detuning can be larger thanthe bandwidth, which puts the cavity out of tune of the supplied RF signal.Then the supplied RF power to be partly reflected and the accelerating gradi-ent in the cavity reduces. The change of frequency due to the Lorentz forcedetuning, Δ fLF is given by [28]

Δ fLF = KLE2, (2.16)

20

where, KL is known as the Lorentz force detuning coefficient.While typical bandwidth of loaded SC cavities (see equations (2.11) and

(2.8)) is from 100 to 1000Hz depending on the cavity type, the typical valuesfor KL can be around 5 to 9Hz/(MV/m)2. For accelerating gradients of 10to 25 MV/m this can results in a resonance frequency change of 500 to 6000Hz. Owing to their characteristic shape, the spoke cavities thus have smallerLorentz force detuning which means that the correction to be provided is alsosmall.

2.2 Cavity treatmentNiobium SRF cavities are primarily made from sheets obtained from a highpurity niobium ingot, which are then formed into a shape and electron beamwelded together to obtain the required geometries [17]. As we have alreadydiscussed, the superconducting RF current flows on the cavity surface till theLondon penetration depth. The surface conditions, thus, influence the currentflow which in turn has an effect on the RF characteristics of the cavities, likethe quality factor Q0. In case of niobium cavities the superconducting currentflows till a depth of 40 nm [2]. This requires a high quality surface finish suchthat any surface distortion is smaller than the London penetration depth.

During the mechanical formation of the cavity a damaged layer is formedwhich is about 200 μm. The process of high temperature heat treatment alsointroduces impurities on the surface which can penetrate from 20 to 50 μm.The removal of the contaminated and damaged layers is achieved by mechan-ical polishing processes and also by chemical etching or electropolishing.

2.2.1 Centrifugal Barrel PolishingOne way of mechanical polishing of the insides of cavities is by means ofcentrifugal barrel polishing (CBP). In this process the cavity is rotated at highspeed while filled with abrasive media. The media can vary during steps of thepolishing process to obtain mirror finish of the internal surface [13, 14, 10, 9].It is an acid-free polishing technique and thus considerably reduces the use ofchemicals for cavity preparation. Approximately 50% of the cavity volumeis filled with a mixture of the abrasive media in combination with deionizedwater and a coolant. This allows cooling of the cavity and removal of ma-terial from the surface to allow further polishing. The polishing media havedifferent shapes, sizes and compositions. The commercially available mediaare designed to be used in a slurry media with water and soap. A differencein choice of the media results in a difference in finally achieved surface fin-ish. CBP results in a surface roughness of 10 nanometers [14]. The longerthe time the CBP process is done, the smoother is the surface. The time re-quired to reduce roughness from 0.1 micron to 0.015 micron is around 4 days.

21

The removal rate depends on the media used. The common materials usedare ceramics, RG-22 cones, a mixture of 800 mesh alumina powder or 40nmcolloidal silica carried by hard wood blocks or corn cobs [9]. An advantage ofthe polishing process is that no harmful by-products like gases are produced.

2.2.2 Etching and Buffered Chemical PolishingA layer of oxide Nb2O5 naturally forms on the surface of Niobium metal.Other oxides and sub-oxides are formed below this layer [22]. The inert layerof Nb2O5 can be dissolved away with hydrofluoric acid (HF) to form watersoluble Niobium Fluoride (NbF5) [1]. The process is termed as etching andit is also useful to remove defects and foreign impurities from the surface.Along with the HF for dissolution of Nb2O5, a strong oxidizing agent likenitric acid (HNO3) is used to re-oxidize the niobium. This re-oxidized layeris again dissolved by the HF, thus removing imperfections from the internalcavity wall. The mixture of the acids is formed of 40% concentrated HF and65% concentrated HNO3 and has a large material removal rate of 30μm/min.During the etching process nitrous gases, oxygen and hydrogen are producedand the reaction is a strong exothermic one.

For a better control of the etching process a buffer substance like phospho-ric acid (H3PO4) is added to the mixture. This process is then called BufferedChemical Polishing (BCP) [22, 1, 2]. The BCP mixture contains 49% concen-trated HF, 69.5% concentrated HNO3, and 85% H3PO4 in 1 : 1 : 2 ratio [1].The mixture is also cooled to 15◦ C, which then reduces migration of hydro-gen into the niobium crystal lattice [22]. Concentrated sulphuric acid (H2SO4)can also be used as a buffer [22].

2.2.3 ElectropolishingThe mechanical polishing is almost always followed by the operation of elec-tropolishing (EP) [56]. The history of electropolishing starts with the patentissued in 1912 by the Imperial German Government to be used for Silver. In1935 it was used for Copper and in 1936-37 for Stainless Steel [56]. Theprocess of electrolytic/electro-polishing is similar to that of etching in an acidbath but with the parallel application of an electric field to the surface [56, 19]operating as the anode. Although there are several hypothesis explaining themechanism of electropolishing of Copper in phosphoric acid solutions, all ofthem deal with the formation of a thin bluish viscous layer of electrolyte onthe anode [56]. The current passes across the electrolyte and through the filmon to the anode. The field enhancement on the sharp tips (surface damages)combined with the higher viscosity and higher electrical resistivity of the film,whose thickness differs from site to site, leads to the faster dissolution of thehigher tips [56, 19]. The surface roughness achieved after electropolishing is

22

on a sub-micron scale [67]. The voltage and current need to be controlled dur-ing the electropolishing process so that the values stay confined to a certainregion. Otherwise the process suffers from either the build up of a strong ox-ide layer or the release of gaseous oxygen both of which lead to the formationof discontinuities on the surface.

2.2.4 High pressure water rinsingHigh pressure water rinsing (HPR) has to be done between various stages ofthe cavity cleaning process. It is done after EP process and also after the BCPsteps. There are different types of systems for water rinsing but all of themshare the general common layout which consists of: (i) an ultrapure (UP) waterplant supplying the HPR water, (ii) a pump to pressurize the ultrapure water,(iii) a filter to hold back particulates, (iv) a cane holding the spraying nozzle,and (v) a motion system to guide the water jets from the nozzles towards allparts of the cavity surface. Most common pumps used are piston or diaphragmbased pumps. However, the problem with these pumps is that they are oftenlubricated with oil and thus run the risk that there is a direct pass of oil to theUP water in case of diaphragm or seal failures. Also, the nonlinear movementof the pistons introduces pulsation to the water jet and can lead to vibrations ofthe spraying cane. For vibration reduction, damping elements are introducedinto the high pressure feed line to the cane [16].

2.3 ConclusionThe heat treatment of the cavity is done after every step of high pressure waterrinsing. Usually, for cavities several of the mentioned treatments are carriedout in sequence with some of them repeating in occurrence. This eventuallyeliminates most of the imperfections on the cavity walls and finally preparesthe cavity for use in accelerators. The treatments allow the cavities to sustainhigher accelerating gradients.

23

3. ESS and FREIA

ESS is going to use a high-power proton accelerator. High power refers to thefact that the product of beam energy and beam current is of the order of MW.Typical applications of such high power machines are nuclear waste transmu-tation, production of rare isotopes and neutron production using high-energyprotons. In ESS the beam will be used to initiate the process of spallation in atungsten target to produce high energy neutrons. The neutrons thus generatedneed to be slowed down from velocities comparable to that of light to thosecomparable to the velocity of sound by means of liquid hydrogen and waterfilled structures.

The facility is planned to be a major research organization using the neutronbeams, which provide insights into the building blocks of matter not achiev-able by other means. The linear accelerator, or linac, of such an accelerator-driven spallation neutron source is of crucial importance.

Figure 3.1. Layout of ESS linac (from [8])

The high level parameters of the ESS linac are given in Table 3.1. Thelayout of the same is shown in Figure. 3.1. From the ion source of protons tillthe Drift Tube Linac (DTL) the linac is normal conducting, whereas the rest ofthe structure from the spoke section to the High β cavities is super conducting[30].

Table 3.1. High level parameters of ESS linac

Parameter Unit ValueAverage beam power MW 5Proton kinetic energy GeV 2Average macro-pulse current mA 62.5Macro-pulse length ms 2.86Pulse repetition rate Hz 14Duty Cycle % 4

24

3.1 The ESS LinacThe linac starts with a proton source from which the beam is transferred tothe radio frequency quadrupole (RFQ) section through the low energy beamtransport (LEBT). The RFQ is used for bunching and is the first stage of accel-eration. The medium energy beam transport (MEBT) section is used to matchthe beam from the RFQ to the normal conducting drift tube linac (DTL). TheDTL is the last in the normal conducting section and leads to the SC spokesection. The RFQ, DTL and spoke sections all operate at 352.21 MHz of RFfrequency, whereas the rest of the SC sections, namely the Medium β andHigh β sections all operate at 704.42 MHz [30]. The linac ends at a tungstentarget which has a novel design and generates neutrons. After introducing theoverall concept we can now take a closer look at each of the individual com-ponents of the linac starting with the ion source, the place where the protonsare extracted.

3.1.1 Ion sourceThe ESS linac requires a proton beam of 62.5 mA. This is achieved using apulsed microwave-discharge source [27, 30] producing protons at 75 keV at2.45 GHz. The ion source has to produce higher current to allow for losses inthe LEBT. It will operate at 14Hz repetition rate with 2.86 msec pulse length,and 99.9% reliability. It also employs a magnetic system with three solenoidswhich are energized independently of one another. This makes the sourcehighly flexible and has a crucial influence on the density of the producedplasma inside the chamber of the source. It also uses two innovative plasmaproducing methods one of which uses RF fields whereas the other employsa short plasma confinement. With a versatile magnetic system it will allowincreased currents and proton counts, with simultaneously reduced emittanceand lower beam halo formation.

3.1.2 Low Energy Beam Transport (LEBT)The LEBT serves two purposes at the same time [30]. First, it focuses thebeam from the source to the sharpness required at the entrance of the next sec-tion, the RFQ. Secondly, it is used to chop the beam. The chopping is requiredas the beam from the ion source lacks sharp edges, and, this is achieved in theLEBT. The chopping allows the beam to be sent into the linac only after theoutput from the ion source has stabilized. The discarded section of the beampulse is dumped at the entrance cone of the RFQ. The beam energy in theLEBT is lower than the Coulomb barrier for nuclear reactions which preventsradio-activation of the choppers [30]. The focusing is done by two solenoidmagnets while the chopping operation is achieved by a pair of electrical de-flection plates. The chopping action thus occurs between the two focusing

25

solenoids. The chopper also has the capability to completely switch off thebeam.

3.1.3 Radio Frequency Quadrupole (RFQ)The RFQ performs the job of accelerating, bunching and focusing the beam.The 4 m RFQ consists of 4 sections each a meter in length and has four vanes[30]. The vanes are electrodes which force the cavity to resonate in the TE210mode. The ESS has stringent requirements on the operational beam currentwhich is as high as 62.5 mA. To meet these requirements the RFQ design isoptimized to reduce beam losses below the limit of 1 W/m when the protonshave an energy of 2 MeV or higher [30]. This is achieved by producing beamswithout any transverse tails in the particle distribution. This prevents the de-velopment of a transverse halo in the later sections of the linac. The RFQincorporates a long section of 120 gaps which is half its length [30]. This sec-tion does not accelerate the beam, but is used purely for bunching the protons.The rate of acceleration in the first half is low. These factors combined withthe large longitudinal acceptance allows for beam transmission, small longi-tudinal emittance and absence of transverse emittance growth. Also the phaseof the particles is matched to the synchronous phase of the adjacent MEBTsection to facilitate the bunching operation.

3.1.4 Medium Energy Beam Transport (MEBT)The MEBT performs four different functions [30]. First, the MEBT is respon-sible for matching and steering the beam from the RFQ to the DTL. Secondly,the section includes an extensive set of beam instrumentation for comprehen-sive diagnostics. Thirdly, it allows collimation of the transverse distribution.Lastly, it has space for an additional beam chopper which acts faster than thechopper in the LEBT. The MEBT has three buncher cavities and 10 quadrupolemagnets to achieve this task. The buncher cavities in the section are designedsuch that the bunches formed are short enough so they see the acceleratingforce with respect to the synchronous phase as a linear force but are longenough to be immune to space charge force that degrades beam quality.

3.1.5 Drift Tube Linac (DTL)The DTL is the first full accelerating section. This section accelerates the beamfrom 3.6 to 90 MeV [20, 30]. It operates at a RF frequency of 352.21 MHzin pulsed mode. The RF pulse length is 2.86 msec with a duty cycle of 4% ata repetition frequency of 14 Hz. It is made up of 5 tanks with a total of 173drift tubes. A constant accelerating field of 3.1 MV/m is planned in each tankand this will require a maximum RF power of around 2.2 MW in each tank.

26

The section uses a focusing-defocussing (FODO) structure with permanentquadrupole magnets used for transverse focusing of the beam. Every seconddrift tube contains steering magnets and beam diagnostic elements. Its overalldesign takes into account practical technological limitations to avoid beamloss along the DTL section. The beam bore radius is higher along the DTL toavoid beam losses. The DTL marks the end of the normal conducting sectionof the linac.

3.1.6 Spoke sectionESS will be among the first accelerator facilities to incorporate spoke cavitiesin its design. The spoke cavities form the first part of the superconductingsection of the linac. While the normal conducting section has a total length ofonly 42.4 m, the length of the spoke section itself is 58.5 m and there will be26 cavities [30]. The spoke section will accelerate the beam from 78 MeV to200 MeV. It is of extreme importance as it prepares the beam for the rest of thesuperconducting section which is 341.8 m in length. The cavities are double-spoke cavities and have 3 accelerating gaps. There will be two such cavitiesin each cryomodule. Quadrupole doublets and beam instrumentation at roomtemperature will be placed in between each cryomodule to monitor and correctany errors in the beam profile. The cavities will be immersed in liquid heliumat 2 K to keep them superconducting and the cryomodule will provide thermalisolation to reduce losses in the 2 K tank. The spoke cavities are designedto be matched to a beam velocity of 0.5 times the velocity of light in freespace (β = 0.5). The cavities will be operated at an accelerating gradient of8 MV/m. This will require on average 250 kW of RF power to be incident on toeach cavity for a beam current of 62.5 mA [30]. All superconducting cavities,including, the spoke cavities, are built with thin walls to easily conduct the heatfrom the inner cavity walls to the outer cavity walls. This allows the cavitiesto easily cool down and remain below critical temperature, which is 9 K forpure niobium. At the same time, because of the thin walls the electromagneticpressure (Lorentz force detuning) and Helium pressure result in the cavitydeformation. This leads to changes in the resonance frequency. However,thanks to their innovative mechanical design, the cavity operating frequencychanges very little due to filling of RF power or Helium pressure fluctuations.

3.1.7 Medium and High β sectionThe medium and high beta sections of the linac will employ elliptical cavities.The medium-β section is composed of 60 cavities which are designed for abeam of β = 0.67. They will accelerate the beam from 216 MeV to 571 MeVoperating at a gradient of 15 MV/m [30].

27

The high-β section will have 120 cavities designed for a beam of β = 0.92.The section will accelerate the beam from 571 MeV to 2 GeV operating at agradient of 18 MV/m. State of the art preparation and cleaning techniques willbe used on the cavities which have been shown to allow elliptical cavities tooperate without break down at gradients of 23.6 MV/m [30].

For both these sections the power couplers will run 30% below rating [30],which will ensure that it is possible to add more power into the cavity forcavity voltage stabilization.

3.1.8 High Energy Beam Transport (HEBT)The high energy beam transport section serves the purpose of transporting thebeam to the target which is 4 m above the linac tunnel level underground. Thedifference in the height is to place the target and instrumentation outside theradiation protected area to save on construction cost. The HEBT will alsoserve as a space for additional cryomodules if required for future upgrade ofthe facility [30]. The horizontal section, which will transport the beam to thetarget, will also contain a beam expander system to match the beam so thatit is incident on a predefined portion of the target to handle dissipated powerefficiently. The HEBT will also have an additional beamline leading to thebeam dump and the dumping targets. It will also contain instruments usefulfor tuning, commissioning and measuring the energy spread of the beam. Aneutron catcher to capture the neutrons, which gets back scattered from thetarget, will also installed in this section. Radiation hard normal conductingmagnets and their associated power supplies will be used for beam steeringthroughout the HEBT.

3.1.9 Target stationThe target station converts the high intensity proton beam to several high in-tensity neutron beams. The process of converting the proton beam to neu-tron beams by means of the spallation process using a heavy metal target alsoproduces a lot of heat, radiation and radio-active isotopes as byproduct. Thetungsten target will have a diameter of 2.5 m divided up into 33 sectors. Thecooling of the target will be done by means of helium-gas. The target willrotate such that each sector receives 1 in every 33 beam pulses [30]. This willresult in a large enough target surface area such that passive cooling by radi-ation and heat conduction by stagnant helium gas will keep the target belowdangerous levels even in case of active cooling failure. The produced neutronsaccount for about 10% of the 5 MW power being delivered by the proton beamin the form of kinetic energy. The neutrons, thus, themselves have a very highkinetic energy and need to be cooled down from around 10% of the velocityof light to velocities comparable to the speed of sound. This is achieved by

28

means of a moderator-reflector assembly surrounding the target. The modera-tors are of two types one using liquid hydrogen, termed as the cold moderator,while the other will use water. The reflectors are made of beryllium housed inan aluminum vessel for inner moderators, while, the outer ones will be madeof steel. About 7000 tons of steel will form a radiation shielding system toprevent the gamma radiation and fast neutrons created in the target from es-caping into the working environment. All these components will be housedinside a 10 m high and 12 m diameter cylinder.

3.1.10 RF Power sources and beam instrumentationApart from the accelerating sections, the facility also uses an RF system whichincludes various sources, like klystrons, tetrodes and solid-state amplifiers,modulators, the RF distribution system, and, an RF control system to effi-ciently convert power from the grid to RF energy for acceleration of the pro-tons [30]. There is also a dense array of beam instrumentation for effectivemonitoring of beam position and beam quality to meet the design require-ments.

To ensure that the components for ESS procured from the industry meetthe design requirements they need to be characterized at some independentlaboratories. The Facility for Research Instrumentation and Accelerator De-velopment (FREIA) at Uppsala University is one such laboratory where thesuperconducting spoke and elliptical cavities as well as Low Level RF controlsystems and tetrode based high power RF stations are characterized.

3.2 The FREIA LaboratoryAccelerator physics in Uppsala started in the late 1940s and since then it out-grew into a valuable part of the academic environment. At present, the accel-erator and instrumentation physics sub-department FREIA conducts researchon beam physics and light generation with charged particles, accelerator tech-nology and instrumentation with a focus on teaching-research nexus.

FREIA has a highly qualified team of over 30 researchers, engineers, PhDstudents and post doctoral fellows. FREIA is equipped with high-power RFsources and dedicated diagnostics equipment, a helium liquefier, 2 K cryostats,vacuum systems, control electronics, and radiation protected areas (more than50 m2 in total) housed in a 1000 m2 experimental hall.

3.2.1 CryogenicsThe laboratory has a Linde L140 liquefier which can deliver over 140 l/h ofliquid helium into a 2000 l storage dewar. The system provides 4.5 K helium

29

Figure 3.2. Layout of the FREIA hall with bunker and associated instrumentation

to FREIA and to external users within and outside the university. The plantalso has a cold box with a 20 K absorber for helium purification, a 100 m3

gas bag and three 27 m3/h compressors for helium recovery. In addition it hasa 20000 l liquid nitrogen storage dewar and a filling station providing liquidnitrogen for the helium liquefier pre-cooling and also for filling of mobiledewars [29, 23].

3.2.2 The cryostat HNOSSOne of the dedicated instruments at FREIA for accelerator development is theversatile horizontal cryostat system for testing superconducting cavities calledHorizontal Nugget for Operation of Superconducting Systems (HNOSS) [29,23]. It allows simultaneous characterization of two superconducting cavities,each equipped with its own helium tank, at desired RF power levels.

The HNOSS has an internal warm magnetic shielding of mu-metal of 1×10−3 m thickness. It can operate from 1.8 to 4.5 K at a pressure of 16 to1250 mbar. It can extract 90 W of power of at 1.8 K. The cavity vacuum ismaintained by a pumping system consisting of an ion pump, a turbo-molecularpump and an oil free roughing pump [29, 23].

3.2.3 RF power generation and distributionThe laboratory is also equipped with two high-power RF amplifiers able todeliver 400 kW pulsed RF power with 5% duty factor and one high-power RF

30

amplifier able to deliver 50 kW in continuous wave (CW) mode. Both operateat a frequency at 352.21 MHz. Apart from the high-power infrastructure, italso has 1 kW CW amplifiers which can be used for bare cavity testing withantenna.

FREIA has also developed in-house high efficiency amplifier modules us-ing solid-state transistors. Eight of these can be combined using an in-housedeveloped high-efficiency planar combiner to deliver 10 kW power. A com-pact 100 kW cavity combiner was also developed and tested. In addition, theresearch staff at FREIA also collaborates with various industry and researchpartners on similar developments at several other frequencies, (100 MHz,750 MHz and 1300 MHz) and power levels (1 kW, 10 kW and 100 kW),relevant for accelerator applications [29, 23].

The availability of HNOSS with the diverse power sources makes FREIAideal for testing and characterization of the superconducting cavities to be usedat ESS. The first step towards making ESS a sustainable future particle accel-erator is to efficiently characterize the cavities so that they are operated effi-ciently and the energy utilization of the facility can be minimized.

31

4. Charging of superconducting Cavities

ESS is expected to run for 40 years, 8000 hours per year at 14 Hz pulse ratewhich may be extended to 28 Hz or 56 Hz with upgrades and additional tar-gets. The power on target is about 5 MW and to produce this beam around11.5 MW of power would be required from the power grid [8]. In this regard,it is important to investigate techniques that can make the linac more energyefficient.

The cavities need to be charged to the designed accelerating voltage beforethe beam is injected. When the RF sources start to drive the field into thecavities, at first all the power fed to the cavities is reflected because the signalbandwidth exceeds that of the cavity. The reflected energy is dissipated ina load, and is thus lost. While the cavities gradually start to fill, the signalbandwidth is gradually reduced and the reflection reduces. To this end, thepossibility of minimizing the reflection is important for ESS in comparison tostep filling of cavities, a strategy which is extensively used. The effect of suchminimization of reflected energy on power source requirements, cavity fillingtime and transit time factor of cavities is also important.

The successful design of such a filling scheme requires information of cav-ity models and parameters, which are discussed in section 4.1. The derivationand comparison of optimal filling scheme with the step filling scheme is in-vestigated in paper I. Practical conditions with regard to gain and efficiencycharacteristics of power sources are also modeled and investigated.

4.1 The Cavity ModelThe simplest accelerating structure that can be considered is one formed fromtwo infinite parallel plates, separated by a distance L (say along reference di-rection z) with a sinusoidal voltage distribution (Figure 4.1a). We assume theE-field in the gap is perfectly sinusoidal and neglect any field perturbationsin the cavity apertures, through which a test particle of charge q is injectedalong the reference direction z. Then, the standing E-field can be given as asuperposition of two waves traveling in opposite directions

Ez = E0 [cos(kz+ωt +φ)+ cos(kz−ωt +φ)] (4.1)

where, E0 is the field amplitude, k = nπL is the wave number, ω is the angular

frequency of the E-field. φ is the relative phase between the accelerating fieldand the accelerated particle (Figure 4.1). The number of accelerating gaps isn and the particle is injected at t = 0.

32

0 0.2 0.4 0.6z(m)

-10

-5

0

5

10E

(MV

/m)

gap

(a) Single gap accelerating structure

0 0.2 0.4 0.6z (m)

-10

-5

0

5

10

E (

MV

/m)

gap

(b) Spoke cavity with 3 gapsFigure 4.1. Acceleration of charged particle in gaps

4.1.1 Transit time factorIf the gain in velocity v of the particle is small, i.e. particle velocity v is almostconstant, then the transit time of a test particle is t = z

v =z

βc .The energy gained by the particle as it passes through the said gap is

ΔW =q∫ L

2

− L2

E0

[cos

(kz+ω

zβc

+φ)+ cos

(kz−ω

zβc

+φ)]

dz

=qE0 cosφ

⎡⎣sin

{L2

(k+ nπβg

Lβ

)}k+ nπβg

Lβ

+sin

{L2

(k− nπβg

Lβ

)}k− nπβg

Lβ

⎤⎦ (4.2)

where βg = ωLnπc is the phase velocity of the wave which corresponds to the

maximum energy gain in the cavity (also called geometrical beta).The transit time factor gives a measure of interaction between the field and

the accelerated particle and reads

T =ΔW∫ L

2− L

22E0 cos(kz)dz

=1

4sin(nπ

2

)⎡⎣sin

{nπ2

(1+ βg

β

)}1+ βg

β

+sin

{nπ2

(1− βg

β

)}1− βg

β

⎤⎦

(4.3)

Now replacing k = nπL we get the variation of T with respect to β

βgas shown

in Figure 4.2, which shows the velocity acceptance of accelerating structureswith different number of gaps. From the graph in Figure 4.2 we can concludethat for a spoke cavity with three accelerating gaps, like the ones at ESS, allparticles with velocities in the range from around 0.6 to 3 of βg can be accel-erated. For a medium-β elliptical cavity with 6 accelerating gaps the velocity

33

0 0.5 1 1.5 2

/opt

0

0.2

0.4

0.6

0.8

1

No

rma

lize

d T

ran

sit T

ime

Fa

cto

r

n = 1

n = 2

n = 3

n = 4

n = 5

Figure 4.2. Transit time factor variation with ratio of particle velocity to phase velocityof structure

acceptance is much smaller and this makes it advantageous to use the spokecavities at the beginning of the long SC section of ESS.

The parameters of accelerating spoke cavities for ESS are listed below inTable 4.1 and the gap structure of the spoke cavities is shown in Figure 4.1b.The spoke section of the linac has 28 cavities, all designed for βg = 0.5. Asthe beam is accelerated along the spoke linac the particle velocity increasesleading to an increase in beam energy which leads to the distribution of β asshown in Figure 4.3a. The beam enters the spoke section with β around 0.4.Hence, initially, the interaction between the field and the accelerated particleis sub-optimal as the cavity is designed for βg = 0.5. So, the value of T is low.As the beam gains in energy the β increases to 0.5 around cavity number 11and 12 which account for the high values of T for these cavities. As the beamgains further in energy the value of T drops again. This leads to a variation ofT as shown in Figure 4.3b.

Table 4.1. List of cavity parameters

Parameter Symbol ValueGeometric beta βg 0.5

Accelerating gaps n 3Bare cavity Quality factor Q0 1.2×1010

External Quality factor Qext 1.76×105

Cavity shape constant R/Q 213ΩDC beam current Ib,DC 62.5 mA

4.1.2 Cavity RLC modelAn electromagnetic (EM) field of a closed metallic cavity is represented byan infinite countable set of TE and TM eigenmodes corresponding to differ-ent spatial field distributions and being solutions to the source-free Maxwell

34

0 10 20 30

Number of Cavity

0.4

0.5

0.6

0 10 20 300

100

200

300

Be

am

En

erg

y (

Me

V)Beam energy

(a) Variation of β and beam energy alongspoke linac

0 10 20 30Number of Cavity

0.8

0.9

1

Tra

nsit t

ime

fa

cto

r

0 10 20 3020

25

30

Syn

ch

ron

ou

s p

ha

se

(d

eg

)

Transit time factor

Synchronous phase

(b) Variation of T and φ along spoke linac

Figure 4.3. Variation of cavity parameters along spoke linac

equations. Each eigenmode is characterized by a quality-factor, Q0, and a fre-quency of temporal oscillations, ω0. In the presence of a source, a generalsolution for a field excited in the cavity can be represented as an expansionover the eigenmodes with unknown scalar field amplitudes [38]. If the modesare not degenerated, then the amplitudes in question evolve in time accordingto the second-order differential equation of the oscillator-type. The excitationsource (driving force) in this equation is given by the overlap integral of thespatial distributions of a physical excitation source (beam or antenna) and theeigenmode. The overlap integral is proportional to the strength of the excita-tion source and determines the cavity coupling to the outside world.

It is advantageous to describe the amplitude of the TM eigenmode – usedfor particle acceleration – in terms of a voltage, V , seen by a test chargedparticle traversing the cavity. Then, the equation for the voltage reads [62]

V +ω0

QLV +ω2

0V = 2ω(R/Q)Ig. (4.4)

Recall that QL = (Q−10 +Q−1

ext )−1 is the total quality factor, (R/Q) is the cavity-

shape constant and Ig(t) is the excitation (incident) current. One must distin-guish between the total QL and intrinsic Q0 quality factors, and it is the latterthat is of interest for characterization of the RF surface impedance of SC RFcavities. In general, the surface impedance varies with the magnitude of thecavity voltage [55], hence Q0 does the same.

The reflected current is then given by [62]

Ir(t) =V (t)

2(R/Q)T

(1

Qext− 1

Q0+2i

Δωω

)− dV (t)

dt1

ω(R/Q)T, (4.5)

where, is the relative detuning, Δω = ω0 −ω .

35

Then, explicitly writing the voltage and current in terms of their complexparts, i.e. V (t) =VI(t)+ iVQ(t) and Ig(t) = II

g(t)+ iIQg (t) in equation (4.4) and

taking Laplace transform we get[VIVQ

]= 2ωQL(R/Q)T

[ ω+2QLs(ω+2QLs)2+(2ΔωQL)2

−2ΔωQL(ω+2QLs)2+(2ΔωQL)2

2ΔωQL(ω+2QLs)2+(2ΔωQL)2

ω+2QLs(ω+2QLs)2+(2ΔωQL)2

][IIg

IQg

](4.6)

When the detuning is negligible, e.g. Δω ≈ 0, Eq. (4.6) reduces to[VIVQ

]= 2ωQL(R/Q)T

[1

(ω+2QLs) 00 1

(ω+2QLs)

][IIg

IQg

](4.7)

from which we can see that the cavity filling time is

tF =2QL

ω. (4.8)

It gives a measure of the time the cavity takes to charge to (1−1/e) times thecavity voltage set point. For the spoke cavities tF ≈ 160 μ sec while for theLow-β and High-β cavities tF ≈ 307 μ sec and tF ≈ 320 μ sec respectively.Correspondingly for Δω ≈ 0, the reflected current is given by

Ir(t) =V (t)

2(R/Q)T

(1

Qext− 1

Q0

)− dV (t)

dt1

ω(R/Q)T, (4.9)

while the total reflected energy before beam injection at t = ti is

Er(ti) =12(R/Q)T 2QL

∫ ti

0|Ir(t)|2 dt (4.10)

4.2 The step charging profileTo demonstrate the problem of reflected energy, let us first consider the caseof the standard step charging of cavity. Let the beam injection time be ti andV (t) must reach Vc at time t = ti. The generator current starts at t = 0, i.e.Ig(t) = I0

gU(t), where U(t) is the unit step function. The Lorentz force de-tuning occurs at a milli-second scale and is much slower than the process ofcavity charging. Then, equation (4.4) can be solved with Δω ≈ 0 to obtainV (t) as

V (t) = 2ZLT I0g {1− exp(−τ)} , (4.11)

From [6] we obtain that the optimal beam injection time, such that the beaminduced voltage gradient is perfectly compensated by the gradient of the RFsource induced voltage, is generally given by,

ti = tF ln

(I0g

I0b

), (4.12)

36

while, the required generator current is given by

I0g = I0

b +Vc

2T ZL. (4.13)

Here, I0b is the DC beam current which is one half of a bunched RF current for

short bunches of charged particles. The effect of the beam, I0b , is accounted

for in the determination of both ti and I0g .

Now, for the calculated value of generator current for t � ti, the reflectedcurrent for such step filling is obtained by solving equation (4.9) to get

Isr (t) = I0

g {1−2exp(−τ)} . (4.14)

Then by inserting equation (4.14) into equation (4.10),

Esr (τi) =

(Vc +2T ZLI0b )

2

8ZLtF{

τi +4e−τi −2e−2τi −2}, (4.15)

where τi = ti/tF . The instantaneous power required from the generator reads

Psi (τ) =

12

T 2ZL

(I0b +

Vc

2T ZL

)2

(4.16)

while the total incident energy from the generator is

Esg(τi) =

tFτi

2T 2ZL

(I0b +

Vc

2T ZL

)2

. (4.17)

Figure 4.4 shows the results for the step charging of cavities, a strategywhich is usually followed for particle accelerators. The total energy lost inreflection is the area under the reflected power curve. If tF is large, then aconsiderable amount of energy is lost during the reflection process.

For ESS with 14 Hz pulse rate of the RF filling process this results in around30 MWhrs of energy being wasted for the spoke cavities during each yearof operation. For the medium-β and high-β cavities the energy wasted isaround 150 MWhrs and 550 MWhrs respectively for each year of operation.All this wasted energy results in an added electricity cost amounting to aroundSEK 50 million over the entire lifetime of ESS.

Since ESS wants to be sustainable and energy efficient we look into waysto reduce the amount of reflected energy lost and this is studied in publicationI. We apply the concept of the “Principle of Least Action” from variationalcalculus to the reflected energy and obtain the optimal voltage profile corre-sponding to the minimum reflected energy.

4.3 The optimal charging profileTo minimize the total reflected energy given by equation (4.10), for t ≤ ti, wesubstitute equation (4.9) into it and get an expression, (4.18), analogous to the

37

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

Ca

vity V

olta

ge

(M

V)

50

100

150

200

250

300

350

Po

we

r (k

W)

Figure 4.4. Typical cavity voltage and reflected power profile during charging of aSpoke superconducting cavity

action integral in classical mechanics

S[V (t)] =

12

Qext

(R/Q)T 2

∫ ti

0

∣∣∣∣V (t)2

(1

Qext− 1

Q0

)− dV (t)

dt1ω

∣∣∣∣2

dt.(4.18)

The integrand can be treated as the Lagrangian, L(t,V,V

), in the principle

of least action [32] with V and V as the dynamical variables. Then using L weconstruct the Euler Lagrange equation (4.19)

∂L∂V

− ddt

∂L∂V

= 0 (4.19)

which results ind2V (t)

dt2 =1t2F

V (t) (4.20)

with boundary conditions V (t) = 0 at t = 0, and, V (t) =Vc at t = ti. This givesthe optimal voltage profile during cavity filling, t � ti, as in equation (4.21) inwhich the beam injection time, ti, can be treated as a parameter, which we canchoose as required,

V o(τ) =Vcsinh(τ)sinh(τi)

, (4.21)

where τ = t/tF and τi = ti/tF . The optimal current profile is then obtained byapplying equation (4.21) to equation (4.4) to obtain

Iog (τ) =

Vc

2T ZL

expτsinh(τi)

. (4.22)

The reflected current for the optimal filling profile is given by

Ior (τ) =− Vc

2T ZL

exp(−τ)sinh(τi)

. (4.23)

38

The total reflected energy, while t � ti, may be represented by

Eor (τi) = tF

V 2c

4ZL

exp(−2τi)

1− exp(−2τi). (4.24)

The instantaneous power required from the generator reads

Poi (τ) =

V 2c

2ZL

exp(2τ)4sinh2 (τi)

(4.25)

while the total incident energy from the generator is

Eog (τi) =

tF4ZL

V 2c

1− exp(−2τi). (4.26)

0 0.5 1 1.5 20

100

0 0.5 1 1.5 20

2

Re

fle

cte

d P

ow

er

(kW

)

(a) Input and reflected power profiles

0 0.5 1 1.5 20

1

2

3

4

5

6

(b) Cavity voltage profileFigure 4.5. Optimal charging profiles for the spoke superconducting cavity

The instantaneous power being sent to the cavity and the reflected powercan be seen in Figure 4.5a with the corresponding cavity voltage profile inFigure 4.5b. Comparing Figure 4.5a to Figure 4.4 we can see that the initialpower loss for the optimal charging scheme is two orders of magnitude lowerthan for the step charging scheme and continues to reduce as the cavity charg-ing is completed. The total energy lost during charging is thus minimized.

4.4 Discussion of the resultsThe optimal charging scheme for the cavities thus obtained does not includethe characteristics of the power sources typically used in accelerator facilities.These sources suffer from low efficiency at low output power and saturationbeyond a certain level of output power, as seen in Figure 4.6. The optimalscheme demands low output power from the source, which causes the source

39

0 1 2

Input Power (Normalized)

0

0.5

1

1.5O

utp

ut

Po

we

r (N

orm

aliz

ed

)

Tetrode

Solid state Doherty power amplifier

Klystrode

(a) Gain of typical sources

0 0.5 1

Output Power (Normalized)

0

20

40

60

80

Eff

icie

ncy (

%)

Tetrode

Solid state Doherty amplifier

Klystrode

(b) Efficiency of typical sourcesFigure 4.6. Gain and efficiency normalized to rated output and corresponding inputpowers. The sources have low efficiency at low output power and the gain saturatesbeyond nominal output power.

to be working at sub-optimal efficiency for a long duration. Thus, to make thefilling scheme energy optimal, with respect to the energy required from theelectricity grid, the source characteristics need to be considered in the designas well. The charging time ti is a free parameter in the obtained optimal charg-ing scheme. In my publication the effect of amplifier saturation and efficiencyis accounted for to determine the choice of optimal ti for energy optimal filling.

1 2 3 4 5 6

0.8

0.85

0.9

0.95

1

1.05

1.1

Rw

all

Tetrode

Solid state Doherty power amplifier

Klystrode

Figure 4.7. Variation of ratio between energy drawn from wall plug by step and op-timal charging with τi for various power sources. It can be seen that the klystrode(IOT) and the Doherty architecture use less energy during optimal filling (Rwall > 1)for τi ≡ ti/tF ≈ 1.5.

The analysis shows that there can be an overall efficiency gain from the pro-posed optimal filling scheme despite the low efficiency of the power sourcesat low output power. In paper I we define an efficiency parameter Rwall . ForRwall > 1 the optimal filling scheme is more beneficial over the conventional

40

step filling approach. We conclude that for the Klystrode and Solid state Do-herty power amplifier the optimal filling scheme provides a savings in energyconsumption for a cavity charging time of 1.5 times the cavity filling time(ti ≈ 1.5tF ) (see Figure 4.7). However, for the tetrodes considered in the anal-ysis the conventional filling with step signal is advantageous.

Inspired by the need for high power sources which can perform with highefficiencies even at low output powers, the FREIA Laboratory launched re-search and development on solid state amplifier modules [15]. I collaboratedwith my colleagues to characterize the modules and determine their long termstabilities. The aim of the characterization study was to investigate the pulseprofile, drain voltage and current waveforms, efficiency, and the pulse-to-pulse (P2P) amplitude and phase stability. The amplifiers were operating ata kilowatt-level output power. The details of the experimental setup and themeasured characteristics are contained in paper II.

The cavity quality factor gives an indication of the amount of energy lost inoperation and it is thus important to have accurate estimation of the same todetermine the usability of a cavity. Thus, I turn my attention to develop a testbench for accurate measurement of a superconducting cavity quality factor.

41

5. Characterization of superconductingcavities

Knowledge of the bare cavity (cavity without power coupler) Q0 is necessaryto ascertain the ohmic losses in the cryostat. This is required since the designof the cryostat and the overall running cost of the accelerator is dependent onthe cavity ohmic losses (Figure 1.2). Hence, accurate methods for Q0 mea-surement are needed.

There are many methods to measure the Q0 for normal conducting cavities[50, 41, 39, 52, 40] with an uncertainty in Q0 down to 1% [40]. However, thesemethods do not work for SC cavities with high accelerating gradient because:

• SC cavities have very narrow bandwidth, of the order of a few Hz (seeFigure 5.2).

• As the cavities are filled with RF power, they change their shape, due toLorentz force detuning. The change in the shape of the cavity results inthe change in the cavity resonance frequency.

• The Q0 of SC cavities changes with accelerating gradient, i.e., with theRF power in the cavities.

The reference [57] presents seven methods to determine the resonant fre-quency and quality factor of superconducting cavities. However, none of thesemethods is applicable to a cavity operated at a high accelerating voltage be-cause strong electromagnetic pressure deforms the cavity. This results in ahysteretic type dependence of the excited cavity voltage as a function of fre-quency described by the nonlinear Duffing equation. In addition, the depen-dence of the surface impedance of the SC cavity on an applied electromag-netic field [55, 66] is ignored. In paper III a novel method of SC cavity qualityfactor measurement is proposed while in paper IV the corresponding experi-mental setup and an estimate of the errors associated with the measurementmethod is obtained.

The changing frequency characteristics of the SC cavity can be accountedfor by building an oscillator around the cavity. This oscillator is referred to asthe self-excited loop (SEL) [4]. It allows the field amplitude in the cavity to benaturally stabilized despite variations of cavity frequency caused by externalfactors such as Helium pressure fluctuations.

5.1 Self-excited Loop (SEL)The basic principle of a self-excited loop is shown in Figure 5.1. It includesthe SC cavity (essentially a bandpass filter), an amplifier, a limiter and a phase

42

shifter. The output of the cavity is phase shifted and amplified before beingused as the cavity input.

SC Cavity

|A|ω

φ

Amplifier Phase shifter

Limiter

Figure 5.1. Schematic representation of the self-excited loop.

If the amplifier gain is greater than the total losses in the loop, then theEM signal in the loop will get amplified. The loop is essentially without anyexplicit driving signal. However, the electrical noise in the amplifier starts togrow, but due to the bandpass filter characteristics of the SC cavity in the loopthe signal becomes quasi-monochromatic. The signal frequency is determinedby the phase equation

φc(ω)+φl(ω) = 2πn, (5.1)

where, φc(ω) is the phase shift across the SC cavity, φl(ω) is the phase shiftacross the rest of the loop and n is an integer.

The loop phase, φl(ω), incorporates within itself the phase shifts acrossthe active devices like the amplifier and the limiter and can thus be nonlinear.However, most of the contribution to φl(ω) comes from the coaxial cablesused to complete the loop. The phase contribution by the cables is linear andthis linear component dominates over the small nonlinear contributions fromthe active components. The loop phase can thus be modeled as straight linesas shown in Figure 5.2. Moreover, the slope of the straight line representingφl(ω) is of the order of 10−6 which makes the straight line virtually parallelto the abscissa axis.

If the total electrical length of the loop is L and the phase shift across the SCcavity φc(ω0) = 0 at the cavity resonance frequency ω0, then the frequencieswhich satisfy the SEL resonance condition equation (5.1) is given by

ω0 = 2πcL

n, (5.2)

where, c is the velocity of light in free space. This resonance frequency condi-tion of the loop is satisfied only by discrete frequency values which are muchdistantly placed compared to the typical bandwidth characteristics of a SC cav-ity as seen in Figure 5.2. Thus, due to the extremely narrow bandwidth of SC

43

cavities, the SEL signal is quasi-monochromatic, since only one frequency,ω0, satisfies the conditions (5.1) and (5.2), simultaneously.

Figure 5.2. Typical bandwidth characteristics of a super-conducting cavity. The cavitybandwidth is only a few Hertz. The phase changes π radians across the cavity reso-nance frequency. The resonance frequency itself changes as the cavity changes shapedue to Lorentz force detuning. As the loop phase, φl(ω) is changed the straight linemoves in the vertical direction and two particular situations are shown in cases (a) and(b).

As the resonance frequency of the cavity changes dynamically due to Lorentzforce detuning, the loop tracks the change according to the condition (5.1). Itshould be emphasized here that dynamic change of cavity resonance frequencymeans the entire curve in Figure 5.2 shifts such that the condition (b) of Figure5.2, i.e., φc(ω0) = 0 is satisfied.

If the loop phase φl(ω) is changed then as long as the loop signal frequencyis within cavity bandwidth, the loop signal is alive, but reduced in magnitudedue to reduced gain characteristic of the cavity away from resonance, as shownby condition (a) in Figure 5.2.

Thus, changing the loop phase, φl(ω), allows changing of the cavity excita-tion frequency and excursion of the cavity gain characteristic. It also changesthe power incident on the cavity and thus the Lorentz detuning related defor-mation and the amount of power reflected by the cavity. The SEL thus allowsmeasurement of the cavity reflection coefficient, Γ, over the entire operationalbandwidth of the cavity.

44

5.2 Cavity Q0 and reflection coefficientAs was described before, the resonant modes of standing electromagnetic(EM) waves in cavities can be described by an equivalent forced oscillatormodel [62, 31]. An EM wave not only enters the cavity through the input an-tenna, but also leaks out through it forming a reflected wave. The reflectioncoefficient Γ, or S11 parameter, which quantifies the reflected wave is definedas the ratio of the reflected voltage to the incident one and is given by

Γ =κ −1+ iQ0δκ +1− iQ0δ

(5.3)

where, κ = Q0Qext

is the coupling coefficient and

δ =ω0(t)

ω− ω

ω0(t)

≈ Qext +Q0

Q0Qexttan(φc)

(5.4)

is the relative detuning of the cavity with ω as the excitation signal frequency [31].In general the cavity quality factor Q0 depends on the cavity voltage [55].Therefore, the coupling coefficient κ depends on the cavity voltage V as well.

Rewriting equation (5.3) to separate real and imaginary parts we find

iΓ′′+Γ′+1 =− 2iQ0

Q0 (Qextδ − i)− iQext, (5.5)

which can be further simplified using equation (5.4) and eliminating the phaseshift across the cavity θc, to obtain(

Γ′+1

1+κ

)2

+Γ′′2 =(

κ1+κ

)2

(5.6)

which is the equation of a circle with center at (− 11+κ ,0) and radius of r = κ

1+κ .Then the coupling coefficient can be calculated using

κ =1

1r −1

. (5.7)

The circle traced out by the reflection coefficient on the complex plane of theSmith-Chart is termed as the Q circle. The radius of the Q circle can thus beused to calculate the coupling coefficient κ . Once Qext is measured by someindependent method, one can obtain Q0.

This method however cannot be used if κ changes significantly when chang-ing the cavity voltage V , which is the case for superconducting cavities. Thedependence of κ on V causes the variation of the reflection coefficient whichdeviates from a circle (see Figure 5.3) and the method needs to be modified.

45

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Increasing dependence

on V

Figure 5.3. Variation of Q circle with in-creased dependence on cavity voltage V .Since the reflection coefficient becomes afunction of the cavity voltage, conventionalmethods of measurement of reflection co-efficient fail in such cases.

Figure 5.4. The variation of reflection co-efficient with κ . The Q-surface is shownin the figure. Measurement of the reflec-tion coefficient of a super conducting cav-ity gives us the locus of points on this sur-face and as a result it does not remain acircle in general.

For fixed cavity voltage, κ is constant and the reflection coefficient tracesa circle, while for varying cavity voltage V it traces a surface in the 3-D κ- Γ space which we call Q-surface as shown in Figure 5.4. This means thatwhen we do measurements and observe the reflection coefficient, we observethe projection of a path on the Q-surface on the Γ′-Γ′′ plane. In order to as-semble the circles we simultaneously need to deduce the cavity voltage fromexperiments. In the next section we present the experimental set up to obtainthe voltage dependent κ values and eventually Q0.

5.3 Experimental SetupThe self-excited loop [5], with the schematic shown in Fig. 5.5, consists ofthe SC cavity, amplifiers and limiters and the digital loop delay. The vectornetwork analyzer and the directional couplers are added for diagnostics.

Starting immediately after the cavity we have a directional coupler that di-rects part of the transmitted signal to the VNA but most of the power is passedthrough a sequence of amplifier-limiter-amplifier. This configuration allowsthe limiter to operate in its preferred range and thus provides a good signal-to-noise ratio while protecting the delicate electronics of the digital loop delay.Subsequently the signal is passed through a sequence of amplifier, limiter, and

46

Vector Network Analyzer

PF PR

Amplifier 1

PT

Digital Loop Delay

Superconducting Cavity

LimiterAmplifier 2 Amplifier 2

LimiterAmplifier 2

Bandpassfilter

Load

Load

Figure 5.5. Setup for self-excited loop based measurement of Q circle. The vector net-work analyzer working as a superheterodyne receiver, tuned to the cavity bandwidth,measures the forward, reflected and transmitted signals to and from the cavity undertest. The signal picked up from the cavity is sent through amplifier-limiter-amplifiercombination to maintain high signal-to-noise ratio and then delayed by means of thedigital loop delay. Variation of the delay, changes the phase shift across the cavity andscans the reflection coefficient as a function of cavity phase. This changes the loopresonance frequency, which might differ from the cavity resonance frequency and thusresult in variation of the forward, reflected and transmitted signals. Using this data theQ surface can be constructed.

power amplifier before a directional coupler directs a small fraction of the for-ward and reflected power signals to the VNA. The dominant part of the signalis directed to the cavity.

An important ingredient of the experimental setup is the digital loop de-lay which introduces the phase delay that determines the operating frequencyof the SEL. The schematic of digital loop delay or phase shifter is shown inFigure 5.6. Our implementation uses the super-heterodyne principle for radiotransceivers. The dominant mode of the cavity is 352.21 MHz. The first stepis to convert the analog loop signal to a digital one. This is achieved usingan IF transceiver with AC-coupled option with Dual 14-bit, 250 MS/s inputshoused in a PXIe chassis [53]. Since the sampling rate is lower than that re-quired by Nyquist criterion, the technique used here is that of under-sampling.However, since the signal of interest is the only signal present in the loop,this causes no problem downstream if proper filtering and frequency scaling isused. Under-sampling implies that in the digital system, 352.21 MHz appearsat around 102.21 MHz. Once digitized, the signal is mixed with a digital sig-nal of 102.21 MHz and then sequentially low-pass filtered and decimated toreduce the sampling rate from 250 MS/s to 1 MS/s. This is implemented insoftware using an NI FlexRIO FPGA module programmed using LabVIEW.

47

The time delay or phase shift algorithm on the FPGA is based on rotat-ing the in-phase (I) and quadrature (Q) component of the signal in basebandaccording to

Ir = I cos(δφ)−Qsin(δφ)Qr = I sin(δφ)+Qcos(δφ).

(5.8)

where we converted the time delay δ t to a phase delay by δφ = ωδ t, Ir andQr are the phase-shifted signals. The signal is then digitally mixed with thesine and cosine components of the digital local oscillator and converted backto analog in a Dual 16 bit, 500 MS/s digital-to-analog converter that is part ofthe IF transceiver.

The output is thus under-sampled as well which means that there is an out-put signal not only at 352.21 MHz but also at the mirror frequency 147.79 MHz(500 MHz - 352.21 MHz). The 147.79 MHz component of the signal is re-moved by three circulators [36] connected in series, acting as a band-pass filteraround 352.21 MHz. After that the signal is sent to the power amplifier.

The cavity is placed in a bunker inside the cryostat and the amplifiers andmeasurement equipment are located in the control room for radiation safetyreasons. This makes it necessary for long co-axial cables to be used for theloop. The RF signal lost in the cables is compensated during calculation of thereflection coefficient, Γ.

5.4 ResultsThe loop delay is varied and measurements are repeated at different amplifiergain settings. This permits recording of the power transmitted by the cavityto the input of the amplifier, PT , the forward powers to the cavity, PF and thepower reflected form the cavity, PR. Let the φT , φF and φR be the phases ofthe corresponding voltage signals of PT , PF and PR. The reflection coefficientΓ = Γ′+ iΓ′′ can then be computed from PR and PF using equations (5.9).

|Γ|=√

PR

PF

Γ′ = |Γ|cos(φR −φF)

Γ′′ = |Γ|sin(φR −φF)

(5.9)

The Γ′ and Γ′′ values thus computed are shown in Figure 5.7. From PT we canestimate cavity voltage, V , using equation (5.10),

V =√

PT (R/Q)QT , (5.10)

where QT is the quality factor of the cavity transmission antenna obtainedfrom the manufacturer.

48

Analog

Signal

Analog

toDigital

Con

version

×

352.21

MHz

Dow

nSam

ple

1MHz

ILow

passFilter

Delay

Delay

toPhase

at352.21

Mhz

Rotatesign

alSam

ple

rate

250MHz

Sam

ple

rate

Signal

at352.21

MHz

Digital

toAnalog

Con

version

500MHz

Sam

ple

rate

Delayed

signal

×

-sincos

Mixed

Dow

nSam

ple

Low

passFilter

1MHz

Sam

ple

rate

Signal

× ×

352.21

MHz

Signal

+

Q

δφ

Signal

Mixed

Signal

Upconversion

andFilter

Rotatesign

al

Upconversion

andFilter

-sin cos

Q I

Figure 5.6. Digital down-conversion scheme for loop delay introduction.

49

Figure 5.7. Reflection coefficient mea-sured after cable compensation. Thecurves are not circles and this points to thefact that the cavity quality factor is depen-dent on cavity voltage.

Cavity Voltage (MV)

-11

-0.5

0

0.5

1

0 32

1-1 0

Figure 5.8. The raw data of variation ofreflection coefficient with cavity voltage.

This method permits us to assemble data points that cover a very large frac-tion of the Q-surface as shown in Fig. 5.8. This operation of Q-surface scantakes about 24-hours, depending upon the density of points.

To extract Q circles from the 3-D data set involving Γ′ Γ′′ and V we bin thedata along the V axis and fit circles to the points in each bin. For the data ob-tained from the experiments the minimum and maximum variation of voltagebins is from 0.005 MV to 0.1 MV. After the binning the Q circle variation canbe displayed with respect to V as shown in Fig. 5.9.

Figure 5.9. The reflection coefficient measurements are grouped by cavity voltagevalues to obtain variation of Q-circle with accelerating voltage.

50

The obtained curves are then used to calculate the coupling coefficient κ =(Q0/QL)− 1 using equation (5.6) and equation (5.7) and plotted along withthe cavity voltage. The variation of κ with cavity voltage as shown in Figure5.10 can thus be obtained.

The external quality factor, QL, can be obtained from decay measurements[55]. The SEL is thus operated in pulsed mode with a period of 10 sec andthe transmitted signal was recorded as a function of time with a time step of0.1 msec using a digital storage oscilloscope. Two cycles of the pulsed SELoperation are shown in Figure 5.11. Under ideal conditions, the cavity voltagedecays in an exponential fashion and we can fit a straight line to the logarithmof the voltage as it is depicted in Figure 5.11. We perform the fit only on ashort time interval of 10 ms to make sure that the cavity quality factor doesnot change because of the Q-slope. The decay measurement was performed ata cavity voltage of 3 MV/m and yielded QL of 0.84×109. Thus the variationof Q0 with accelerating gradient was obtained.

0 0.2 0.4 0.6 0.80

1

2

3

4

Figure 5.10. Variation of Q0 with acceler-ating gradient. The two curves are for twodifferent cooling rates.

Figure 5.11. Transmitted signal vs time.(adapted from publication III c©2016IEEE)

5.5 ConclusionThe method was used to measure the Q0 of a prototype single spoke cavityas well. During the measurements it was observed that two different coolingrates produced two slightly different behaviors of the Q0 variation as is seen inFigure 5.10. The effect of cool-down rate and temperature gradient on cavityquality factor has been observed before at the Fermi National Accelerator Lab-oratory, USA (Fermilab) [12, 11], Helmholtz Zentrum Berlin (HZB) [24, 63]and Conseil Européen pour la Recherche Nucléaire (CERN) [26]. Hence, Istart an investigation for possible explanation of the effect of cooling using thetime-dependent Ginzburg-Landau model of superconducting state transition.

51

6. Phenomenon of Superconductivity

The advantage of SRF cavities over normal conducting ones is related to thereduced surface resistance of the SRF cavities which leads to increased cavityquality factor (Q0). The increased Q0 leads to decreased RF losses (ideallynone) and enables the operation of accelerators with long RF pulses.

The surface resistance of a superconductor has two contributing compo-nents, the Bardeen - Cooper - Schrieffer (BCS) resistance, RBCS, and, theresidual resistance, Rres. The superconducting cavities, are usually operatedat very low temperatures (≈ 1.8K), around which RBCS almost vanishes.

Rres, however, is weakly temperature dependent and has contributions frommagnetic flux trapped in the material as quantised vortices, and becomes a ma-jor contributing factor to the surface resistance [45]. It is thus quite importantto be able to reduce quantised vortices in superconducting surfaces in an effortto reduce Rres and the total surface resistance.

Even though cavities are shielded from ambient magnetic fields there isalways some residual magnetic field incident on the cavity surface. The pres-ence of surface impurities further aggravates the problem as these impuritysites trap the magnetic flux. It has been observed that cool-down conditionsaffect the number of trapped vortices. Larger spatial temperature gradients arefound to lead to better flux expulsion even in the presence of surface impuri-ties.

Although a lot of experimental results support this finding and a substantialamount of development has been made in experimental studies [12, 11, 24,63, 26, 65, 47, 54], there is still limited progress in the theoretical domain, asthe only studies found are by Kubo [45]. In the present chapter we look atthe Time Dependent Ginzburg-Landau (TDGL) equations to model the phasetransition and magnetic flux dynamics, and compare the results with those ofKubo. This study will enable us to describe the optimal temperature gradientfor flux expulsion and also provide a theoretical understanding of process fromthe first principles of state transition.

6.1 Phenomenon of superconductivitySuperconductivity was discovered by Kamerlingh Onnes in 1911 with the ob-servation that the electrical resistance of metals like Pb, Hg, Sn, Al and Indisappears completely below a certain temperature. The temperature value

52

is termed as critical temperature, Tc, and it is dependent on the material inquestion [3].

The next important discovery in the phenomenon of superconductivity wasthe made by Meissner and Ochensfeld in 1933. They observed that a magneticfield is completely expelled when a material becomes superconducting. Alsoif a magnetic field is incident on a material which is already in the supercon-ducting state, then the field is excluded from the material and the field insidethe superconductor vanishes [3].

These observations pointed to the fact that the superconducting transitioncan be modeled using principles of state transition from thermodynamics whichwas put forward by F. and H. London [3] in 1935. However, in 1950 Ginzburgand Landau identified the shortcomings of the London phenomenological model.The London model did not allow computation of the surface tension at thesuperconducting-normal conducting interface. Also the surface energy con-nected with the magnetic field and the supercurrent obtained from solving theLondon equations is negative, while the observed surface energy is positive.The London theory also does not enable the destruction of superconductivityby a current.

Ginzburg and Landau indentified that the state transition of superconductingmaterials around Tc in the absence of a magnetic field is a second order phasetransition [35, 3]. The general theory of such phase transitions, as proposedby Landau and Lifshitz, involves some parameter which differs from zero inthe ordered phase and equals to zero in the disordered phase. This parameterwas termed as the order parameter, ψ , by Ginzburg and Landau. However,before going into the details of the Ginzburg-Landau model of phase transi-tions a brief discussion of the London model is essential to identify some keyparameters of superconductivity.

6.1.1 The London equationsThe earliest macroscopic theory of superconductivity in a constant magneticfield was based on including to the model of a superconducting current density,Js, along with the Maxwell’s equations [35]. The relations included are

�×ΛJs =−1c

B, (6.1)

�×B =4πc

Js, (6.2)

in cgs units, where, c is the speed of light in free space, Λ is a quantity depend-ing only on the temperature and B is the induced magnetic field. Together with

53

equations (6.1) and (6.2) the conditions �.B = 0 and �.Js = 0 produces

�2B =1

λ 2 B, (6.3)

�2Js =1

λ 2 Js. (6.4)

Here λ =√

Λc2

4π is a phenomenological parameter with the dimensionality oflength and is called the London penetration depth. It is a temperature depen-dent quantity and a characteristic property of a material. The equations (6.3)and (6.4) also display that at a plane interface between the superconductor andvacuum, the magnetic field and the superconducting current decay exponen-tially into the superconducting material with a spatial decay constant of λ . Itwas shown by Ginzburg and Landau [35] that

λ =c2e

√m

πns, (6.5)

where e2/m is the ratio between the square of charge and mass of a free elec-tron and ns is defined as the concentration of superconducting electrons.

The London model is valid for weak magnetic fields when the incident fieldH � Hc, where Hc is the critical magnetic field strength which destroys super-conductivity [35]. The London model successfully predicted the existence ofa temperature dependent penetration length, the presence of which was laterproved experimentally. However, it does not explain the destruction of super-conductivity due to strong magnetic fields (H � Hc) or currents. It also doesnot account for the dependence of the penetration depth of the field on the fieldstrength.

6.1.2 The Ginzburg-Landau (G-L) modelThe basic idea behind the G-L model of superconductivity is the existence ofthe order parameter, ψ [3, 35]. It assumes a value of ψ = 0 during normalconducting state and ψ = 0 in the superconducting state. ψ can be consideredto represent an effective wave function of the superconducting electrons suchthat their density in the London equations is given by the local relation

ns = |ψ|2. (6.6)

The Helmholtz free energy for the superconducting state can be written interms of ψ as

Fs = Fn +α|ψ|2 + β2|ψ|4 + 1

2ms

∣∣∣∣(−ih�− esA

c

)ψ∣∣∣∣2

+1

8π|H|2 , (6.7)

54

where, α and β are temperature dependent material properties, es = 2e is theeffective charge of a “superelectron”, ms is the effective mass, Fn is the freeenergy density in the normal state, the induced magnetic field is B = �×Aand H is the applied magnetic field. Fs is assumed to be analytic near |ψ|= 0and T = Tc

The total free energy is then given by

F =∫

vFsd3r, (6.8)

where, v is the entire volume of the superconductor. Treating the integrandas the Lagrangian and minimizing Fs with respect to ψ and A we obtain theEuler-Lagrange equations. Combining them with the Maxwell’s equations,gives us the two G-L equations.

αψ +β |ψ|2 ψ +1

2ms

(ih�+

esAc

)2

ψ = 0, (6.9)

Js =c

4π�×B =

es

ms

{ψ∗

(ih�+

esAc

)ψ}

(6.10)

If we consider a one-dimensional problem then equation (6.9) in the absenceof an induced magnetic field, B, i.e. A = 0, reduces to

h2

2ms

∂ 2ψ∂x2 −αψ −β |ψ|2 ψ = 0 (6.11)

The equation (6.11) defines a parameter called the coherence length

ξ =h√

2ms|α| (6.12)

which is also a temperature dependent material property of a superconductor.The coherence length defines the distance over which the order parameter canvary [3]. If we consider an interface between a normal conductor and a mate-rial in a superconducting state, then the order parameter ψ at the interface willbe reduced. ξ then gives a measure of the length over which ψ recovers to itsmaximum value inside the superconductor.

The ratio of the London penetration depth and the coherence length

κ =λξ

(6.13)

turns out to be temperature independent and is called the Ginzburg-Landauparameter. It determines the type of the superconducting material.

If ξ > λ , the surface energy at the normal superconducting boundary ispositive and such materials are termed as type-I superconductors. They are

55

characterized by abrupt loss of superconductivity above a certain value of theapplied magnetic field Hc.

The condition ξ < λ defines type-II superconductors. If a magnetic fieldis applied on them with magnitude below a critical field Hc1 (termed as thelower critical field), then the entire field is expelled. Such state of the materialis termed as the Meissner state. These materials enter a vortex state withquantized magnetic fluxes entering into the material as the applied field goesabove Hc1. If the applied field goes above the upper critical field, Hc2(> Hc1),then these materials enter the normal conducting state.

However, this G-L model is limited to a steady-state equilibrium situation.In addition, the G-L model is only valid if the spatial variation is slow near thesecond-order transition [3].

There have been many attempts to derive a time-dependent generalizationof G-L equations. The ones that were able to produce experimentally verifiableresults were based on the assumption of a small order parameter just below thesecond order transition to the normal state [3]. While Gor’kov developed themicroscopic theory superconductivity, the macroscopic one has been devel-oped by Schmid, Caroli and Maki and Gor’kov and Eliashberg. The idea isa phenomenological approach from non-equilibrium thermodynamics whichreveals a dissipative process in the superconductor connected with the timevariation of the order parameter.

The state transition of a superconducting material from normal to super-conducting state and vice-versa, including the phenomenon of vortex forma-tion for type-II materials, can be described by the equations which have beenderived from microscopic theory by Gor’kov and Eliashberg for a gaplesssuper-conductor containing a high concentration of paramagnetic impurities[37, 42, 43]. Setting h = c = kB = 1 the equations are given by

1D

(∂∂ t

+ i2eφ)

ψ =−(

∇i−2eA

)2

ψ − 1ξ (T )2

(|ψ|2 −1)

ψ, (6.14a)

J = σ(−∇φ − ∂A

∂ t

)+

18πeλ (T )2 Real

[ψ∗

(∇i−2eA

)ψ]. (6.14b)

In equation (6.14b) J is the total current.The formation of vortices in the mixed state is described by the interaction

force between a pinning centre and a magnetic flux line which is the gradientof the free energy. The energy of the flux line is given by

F ′ =Hc2(T )2

2κ2

{−|ψ |2 + 1

2|ψ|4 +ξ 2 (∇|ψ|)2

}, (6.15)

where, we have disregarded the energies of the magnetic field and currentwhich are large, but less important [51]. In equation (6.14) D is the normal-state diffusion constant, φ is a scalar potential and ψ is normalised order pa-rameter. The order parameter is normalized by its equilibrium value in the

56

absence of the magnetic field,

ψ∞ = π{

2(T 2

c −T 2)} 12 , (6.16)

where T is the temperature and Tc is the critical temperature. σ is the normalstate conductivity while the temperature dependent magnetic field screeninglength

λ (T ) =(8πστs)

− 12

ψ∞. (6.17)

The temperature dependent coherence length

ξ (T ) =(6D/τs)

12

ψ∞, (6.18)

where τs is the spin-flip scattering time. In equation (6.14) the difference be-tween the scalar potential and the electrochemical potential has been neglected[42, 43]. Then

ξ (0) =(6D/τs)

12

π√

2Tc, (6.19)

while

λ (0) =(8πστs)

− 12

π√

2Tc. (6.20)

(1 + δ)Tc

(1 + δ − ∂T∂xL)Tc

0 L

(1 + δ)Tc

Tem

purature

Length of sample along x-axis

Tc

√1−H

H > Hc2

H > Hc2

Tc

√1− κ

√2H

H > Hc1

Figure 6.1. Temperature distribution along the sample being cooled to a gradient of∂T∂x . The length of the sample is L. The applied magnetic field is H. The figure showsthe regions which are subjected to fields above the lower and upper critical fields forthe respective sample temperatures. Initially the entire sample is above Tc by δ .

57

Figure 6.2. A superconducting material cooled down from right to left. The gray andblue regions represent the regions with T > T c and T < T c, respectively. The originof the x-axis is fixed at the interface between these two regions (taken from [45]).

6.2 Flux trapping in superconductorsIn [45] Kubo has presented some results on the phenomenon of magnetic fluxexpulsion as observed during the cooling down of SC accelerating cavitieswith a spatial temperature gradient. The temperature gradient of the materialas it cools down below the phase transition temperature Tc can seen in Figure6.1. The same is also taken from [45] and shown in Figure 6.2.

In the thesis I will present some first results of the cool-down and flux ex-pulsion process modeled using the time dependent Ginzburg-Landau (TDGL)equations (6.14). The parameters have been synthetically manipulated to en-able faster simulation so that the results are easily comparable to results al-ready published in the literature. The parameters are shown in Table 6.1.

Table 6.1. TDGL equation simulation parameters

Parameter Valueξ (0) 100 ÅTc 1 Kκ 2H 0.01Hc2(0)L 125ξ (0)∂T∂x 5.12×10−4Tc/ξ (0)δ 0.005

The temperature dependent lower and upper critical fields can be expressedas

Hc2(T ) = Hc2(0)(

1− T 2

T 2c

)(6.21)

Hc1(T ) =Hc2(T )√

2κ. (6.22)

The sample of superconducting material is placed under a magnetic field ofmagnitude H > Hc2(0) as shown in Figure 6.1. Initially it is at a tempera-

58

ture above Tc. The sample is then cooled from left to right such that it is atlowest temperature at x = 0 and at the highest temperature at x = 125ξ (0). Ifthe temperature gradient is along the x-axis ( ∂T

∂x ), then the final temperaturedistribution is shown in Figure 6.1.

In this case since κ > 1 the superconducting material is type-II. Due to thefixed magnitude of the applied magnetic field, some parts of the material willbe subjected to fields above Hc1(T (x)) while others parts will be subjected tofields above Hc2(T (x)). The region where Hc1(T (x)) < H < Hc2(T (x)) is inthe vortex or mixed state with magnetic vortices penetrating into the materialwhile where H > Hc2(T (x)) is in the normal conducting state. The width ofthis vortex state is given by

δx =Tc∂T∂x

(√1− H

Hc2(0)−√

1−κ√

2H

Hc2(0)

). (6.23)

A qualitative visualization of the situation is given in [45] by Kubo and thesame is reproduced here in Figure 6.3. It is then quite obvious that as thesample cools down as suggested in [45] the Meissner and the vortex statetravel across the sample with the velocity of the applied temperature front.However, [45] does not elaborate on the motion of each trapped vortex.

The same situation is simulated by solving the time dependent G-L equa-tions (TDGL) (6.14). In the case of the parameters considered in our exampleδx ≈ 18 ξ (0). From the simulation we can see in Figure 6.5 that the processof vortex formation or nucleation sets in at t = 2640t0 as the cold front sweepsacross the material. Here, t0 = 4τsπ2T 2

c ≈ 10−13 sec is the characteristic time.However, the vortices are completely formed only at t = 4947t0. This meansthat there is a delay of 2307t0 between the time when nucleation sets in andwhen the vortices are fully formed. Then as the material stays subjected to thetemperature gradient the produced vortices move from the low temperatureregion (left) to the high temperature region (right) as can be seen in Figure6.4 and Figure 6.6 in concurrence with [45]. As the vortex migration reachessteady state, the region where the conditions T (x)< Tc and H < Hc1(T (x)) aresatisfied simultaneously, is in Meissner state and is superconducting. The re-gion with T (x)< Tc and Hc1(T (x))< H < Hc2(T (x)) has quantized magneticflux penetrating into the material and is in the mixed state while the rest is innormal conducting state.

However, what is interesting is the velocity of the individual vortices inthe sample. From Figure 6.4 we can see the movement of the vortices fromtime t = 5850t0 to t = 14730t0 for a temperature gradient of 51200 K/m. Thisgives us a vortex velocity of around 140−210 m/sec which is smaller than thepropagation of the temperature front at around 1700 m/sec.

59

Figure 6.3. Schematic view of the vicin-ity of the phase transition fronts. There ex-ist three domains: the normal conductingdomain (x ≤ xc2), the vortex state domain(xc2< x≤ xc1), and the Meissner state do-main (x > xc1). As the material is cooleddown, the vortex state domain togetherwith the phase transition fronts sweep thematerial from right to left as shown in (a),(b), and (c) (taken from [45]).

0 100 200

X

0

50

100

150

200

250

Y

t = 5850t0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(86,123)

(66,166)

(43,213)

(a) Vortex distribution at t = 5850t0

0 100 200

X

0

50

100

150

200

250

Y

t = 14730t0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(123,126)

(90,167)

(67,213)

(b) Vortex distribution at t = 14730t0Figure 6.4. Movement of vortices un-der temperature gradient and incident mag-netic field in a superconductor.

6.3 Flux trapping in presence of impuritiesThe same experiment as before is repeated but this time with an impurity zoneembedded at the center of the superconducting substance. The impurity ismodeled simply by maintaining the temperature of the said region above Tceven as the rest of the sample is gradually cooled down. The impurity region is50ξ (0)×50ξ (0) embedded in a sample of size 150ξ (0)×150ξ (0). To matchthe magnetic field strength with that of earth the value of H = 0.0005Hc2(0).

From Figure 6.7 we can see that the magnetic flux induced vortex movestowards the edge of the impurity (Figure 6.7a to 6.7b) where it is sucked intothe impurity zone. But as the superconducting front propagates, which can beseen in Figure 6.8, the field is expelled out of the bulk (Figure 6.7c to 6.7f)and pushed towards the edge of the sample.

60

0 100 200

X

0

50

100

150

200

250Y

t = 2640t0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Order parameter

0 100 200

X

0

50

100

150

200

250

Y

t =2640t0

9.9992

9.9994

9.9996

9.9998

10

10.0002

10-3

(b) Magnetic fieldFigure 6.5. Nucleation setting in as the sample cools down. The time is t = 2640t0,where t0 = 10−13 sec. is the characteristic time. It is interesting to note that thereis no appreciable change in the order parameter but the magnetic filed has started toquantize into vortices.

6.4 ConclusionIn this chapter I have applied the time dependent G-L equations to model theeffect of temperature gradient on phase transition. The developed computa-tional model and simulation code allows for studying the vortex depinningmechanism by means of thermal gradients. The model correctly predicts theflow of the quantized magnetic flux lines and successfully recreates resultspreviously published in the literature. It also allows modeling the absorptionof vortices by surface impurities and expulsion of the trapped flux. The studythus proves promising to be pursued further to explain the effect of temper-ature gradients on the flux expulsion phenomenon as has been observed inexperiments by [12, 11, 24, 63, 26, 65, 47, 54].

61

0 100 200

X

0

50

100

150

200

250Y

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vortex

N

o

r

m

a

l

M

e

i

s

s

n

e

r

0 100 200

X

0

50

100

150

200

250

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7. Conclusions

Particle accelerator technology has developed with leaps and bounds over thepast eight decades. Their sizes have increased from the first 9 inch cyclotron tothe present day 27 Km ring of the Large Hadron Collider at CERN. The beamenergies have increased from 1 MeV to 7 TeV. The SC cavities have playedan important role in this development. They have allowed higher acceleratingfields to be integrated into accelerator design. This development in high energymachines has allowed the use of accelerator technology to contribute towardsthe development in the field of materials and medical sciences, and imagingtechnology.

The European Spallation Source has taken the next step in the developmentof accelerator technology. It is designed to be carbon neutral and pioneerthe movement towards sustainable particle accelerators. In this respect thesuperconducting technology provides a means by reducing energy dissipationin the cavity walls. The ability to sustain higher accelerating gradients in theSC cavities also reduces cost of construction by reducing the length of themachine.

However, the use of SC cavities introduces complications in the process ofmachine operation. The first problem, in this respect is the fact that due totheir high quality factors the SC cavities take longer to reach nominal gradi-ent than their normal conducting counterparts. This is because when the RFsources start to drive the field into the cavities, in the beginning the signalbandwidth exceeds that of the cavity and all the power fed to the cavities isreflected. This reflected energy is dissipated in a load, and is thus lost. Whilethe cavities gradually start to fill, the signal bandwidth is gradually reducedand the reflection reduces as well. In the thesis I have investigated a novelcavity charging scheme which can reduce the total reflected energy during fill-ing. The investigation employed the Principle of Least Action used widely inClassical Mechanics. It revealed that a slow charging rate following a hyper-bolic profile is most efficient in reducing total reflected energy. However, itwas also pointed out that owing to the fact that the power sources commonlyused to drive cavities suffer from low efficiency at low output power such ascheme is yet to be entirely useful. Thus research and development is requiredto obtain sources which can provide higher efficiency at low output powers.This has inspired research into the development of high efficiency high powersolid state amplifiers at the FREIA laboratory.

The quality factor of SC cavities is dependent on accelerating gradient inthe cavity and this poses a challenge to be accurately measured. The measure-ments of the SC cavities also demonstrated that Q0 reduces with increasing

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cavity gradient. Associated with this there is the fact that the resonance fre-quency of SC cavities changes with the gradient and is also sensitive to heliumpressure fluctuations. This makes the proposed method of cavity characteris-tics measurement by means of the self-excited loop important. The proposedmethod allows stabilization of the cavity voltage even in the presence of dis-turbances like Lorentz force detuning or helium pressure fluctuations as longas the loop delay is kept constant. Changing the loop delay allows one tochange the cavity excitation frequency which changes the accelerating voltageinduced in the same. Changing the loop delay thus allows scanning the entireoperational range of the cavity and this allows for accurate quality factor mea-surements. The method and the experimental setup provide the opportunityto measure the mechanical modes and Lorentz force detuning along with thequality factor of SC cavities. The quality factor is measured with increasedaccuracy than the conventional method of the single point measurement. Thesetup thus allows for accurate characterization of SC cavities.

The quality factor of SC cavities also depends on the magnetic flux trappedin the surface impurities of the cavity. The experiments at Fermilab, HZB andCERN have demonstrated the effectiveness of cooling the SC cavities withhigh temperature gradients to expel trapped magnetic flux. However, there islimited understanding of the phenomenon by which such expulsion occurs. Itis demonstrated in the thesis that the Ginzburg-Landau model of superconduc-tivity can be used to explain the flux expulsion in presence of surface impu-rities and temperature gradient. The development of such a theoretical modelwill allow decisions about the optimal cooling rate of the cavities, such thatcomplete expulsion of trapped magnetic flux is achieved. This will lead toincreased quality factor of the SC cavities.

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Sammanfattning på svenska

Partikelacceleratortekniken har utvecklats enormt mycket under de senaste åt-ta årtiondena. Partikelacceleratorer storlek har ökat från den första 9 tumscyklotronen till dagens 27 Km långa Large Hadron Collider (LHC) vid CERN.Strålenergin har ökat från 1 MeV till 7 TeV. Supraledande (SC) kaviteter harspelat en viktig roll i denna utveckling. De har gjort det möjligt att integre-ra högre accelerationsfält i acceleratordesign. Denna utveckling av maskinermed hög energi har även gjort det möjligt att använda acceleratortekniken föratt bidra till utvecklingen inom material- och medicinskvetenskap samt bild-teknik.

European Spallation Source (ESS) har tagit nästa steg i utvecklingen av ac-celeratortekniken. Den är utformad för att vara koldioxidneutral och leder ut-vecklingen mot miljövänliga partikelacceleratorer. I detta avseende ger den su-perledande tekniken ett medel att minska energiförlusterna i accelerationska-viteterna. Förmågan att uppnå högre accelerationsgradienter i supraledandekaviteter minskar också byggkostnaden genom att minska maskinens totalalängd.

Användningen av SC kaviteter introducerar emellertid komplikationer vidanvändandet av maskinen. Det första problemet är i det avseendet att SC-kaviteterna på grund av deras höga kvalitetsfaktor tar längre tid att nå nominellgradient än sina icke supraledande motsvarigheter. Detta beror på att när RF-källorna börjar lägga ut en signal till kaviteternaöverskrider signalens band-bredd kavitetens egna bandbredd initialt, och all kraft som matas till kavite-terna reflekteras. Den reflekterade energin dissiperas i ett belastningsmotståndoch förloras därmed. Medan kaviteterna gradvis börjar fyllas minskar signal-bandbredden och reflektionen minskar. I avhandlingen har jag undersökt en nymetod av fylla accelerationskaviteterna som kan minska den totala reflektera-de energin. Undersökningen använde Principle of Least Action som användsallmänt inom klassisk mekanik. Det visade sig att en långsam laddningshas-tighet efter en hyperbolisk profil är mest effektiv för att minska den totalareflekterade energin. Det påpekades emellertid att på grund av det faktum attde kraftkällor som vanligtvis används för att driva kaviteter lider av låg effek-tivitet vid låg uteffekt, är ett sådant system ännu inte helt användbart. Därförkrävs forskning och utveckling för att erhålla källor som kan ge högre effek-tivitet även vid låga uteffekter. Detta har inspirerat forskning kring utvecklingav högeffektiva halvledarbaserade förstärkare vid FREIA-laboratoriet.

Kvalitetsfaktorn för supraledande kaviteter är beroende av accelerationsgra-dienten i kaviteten vilket gör noggranna mätningar av kvalitetsfaktorn utma-nande. Mätningarna av SC-kaviteterna visade också att Q0 minskar med ökad

67

kavitetsgradient. Orsaken till detta är det faktum att resonansfrekvensen hosSC-kaviteter förändras med gradienten och är också känslig för heliumtrycks-fluktuationer. Detta gör den föreslagna metoden för mätning av kavitetsegen-skaper med hjälp av en självsexciterande slinga viktig. Den föreslagna meto-den möjliggör stabilisering av kavitetsspänningen även i närvaro av störningarsom Lorentz-krafter eller heliumtrycksfluktuationer så länge som slingfördröj-ningen hålls konstant. Genom att ändra slingans fördröjning är det möjligt attändra kavitetens excitationsfrekvens som i sin tur förändrar kavitetens fältstyr-ka. Ändring av slingans fördröjning möjliggör sålunda att skanna hela kavite-tens operativa intervall och utföra korrekta kvalitetsfaktormätningar. Metodenoch den experimentella uppställningen ger möjlighet att mäta både de me-kaniska moderna och Lorentz force detuning samt kavitetens kvalitetsfaktor.Kvalitetsfaktorn mäts därmed med högre noggrannhet än den konventionellametoden baserad på mätningar av enstaka punkter. Uppställningen möjliggörsålunda exakt karakterisering av supraledande kaviteter.

Kvalitetsfaktorn för supraledande kaviteter beror också på magnetiska flö-den som fångas i orenheter i kavitetens yta. Experimenten hos Fermilab, HZBoch CERN har visat hur användandet avhögtemperaturgradienter vid nedkyl-ning av supraledande kaviteter kan användas för att tvinga bort infångat mag-netiskt flöde i ytan. Det finns emellertid en begränsad förståelse av exakt hurtemperaturgradienten under nedklyningen verkar på mängden fångade fält iden supraledande ytan. Avhandlingen visar hur Ginzburg-Landau-modellenför supraledare kan användas för att förklara utstötningen av magnetiska fälti samband med ytföroreningar och temperaturgradienter. Utvecklingen av ensådan teoretisk modell kommer att möjliggöra beslut om kaviteternas optimalakylningshastighet så att en fullständig utstötning av fångade magnetiskta flö-den uppnås. Detta kommer att leda till ökad kvalitetsfaktor för supraledandekaviteter.

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Acknowledgements

This thesis is a culmination of the last four years which have been a greatlearning experience. I will take this opportunity here to thank the people whohave made it possible.

The first person who I would like to thank is my supervisor Vitaliy. Sincethe first day I met him outside of Ångströmlaboratoriet four and a half yearsago, I have been working with him. I have learned a lot from him and yeta lot more still remains to be learned. His knowledge of electromagnetism,microwave physics, optics and mathematics is something I aspire to obtain.

The next person is my co-supervisor Volker. He is someone with inter-est in and knowledge of varied subjects ranging from electronics to finance.The discussions with him and the ideas from your strong physics intuition tosolve problems I have faced over the course of my work will always be aninspiration. It was his idea that helped win the Rektors resebidrag fran Wal-lenbergstiftelsen and made the trips to the USPAS possible.

I would like to thank Roger, my co-supervisor, for all the help, supportand guidance that he has provided me throughout these years. His review ofmy work always pointed out overlooked issues, fixing which invariably madethe final results stronger, the papers more readable and the presentations morestreamlined.

My colleagues at FREIA have always been inspiring and providing guid-ance throughout the duration of the PhD, particularly during my experiments.Particularly I would like to thank Tor for all the support with the digital self ex-cited loop hardware. His knowledge of LLRF systems is vast and discussionswith him on the subject always turn into an enjoyable learning experience. Ihave had the pleasure of working with Han. She is always patient to answerall my questions and I learned a lot from her significant knowledge of super-conducting cavity operation. Magnus was always willing to help with the RFmeasurements and provided me with the materials to build my copper cavity.He was extremely kind during the thesis writing process with the summaryin Swedish. The FREIA team including Rocio, Lars, Maja and Tor deservespecial mention for all the help in the control room and making it possible torun the experiments smoothly.

I would like to thank my office mates Mike and Jim for discussions, the tripsto CERN, USPAS and conferences and the enjoyable afterworks. Long andDragos are a pleasure to work with and I look forward to future collaborationswith them.

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I would like to thank Marzieh for all the inspiration she gave me wheneverI felt down. She has always helped be bounce back and face the challengeswith renewed energy and a positive outlook.

This thesis and everything else that I have achieved in my life would not bepossible without my parents. They supported me every step of the way andhave sacrificed so much, all so that their son can go around following his ownwhims. The entire length of this thesis is too small a space to express what Ifeel towards them.

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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1638

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A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally throughthe series Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)

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