Magnetic Flux Trapping Investigation on 6 GHz Bulk …...Electrical superficial resistance is...

UNIVERSITÀ DEGLI STUDI DI PADOVA

Dipartimento di Fisica e Astronomia Dipartimento di Ingegneria Industriale

ISTITUTO NAZIONALE DI FISICA NUCLEARE

Laboratori Nazionali di Legnaro

MASTER THESIS

in

Surface Treatments for Industrial Applications

Magnetic Flux Trapping Investigation on 6 GHz Bulk Niobium Cavity

Supervisor: Dr. Cristian Pira

Student: Luca Zanotto Matr.: 1214211

Academic Year 2018-2019

I

Abstract

Magnetic flux trapping is known as one of the major issue related to energy

dissipation in superconducting radiofrequency cavities (SRF) for particle

accelerators. During last decades, studies around this problematic have been more

consistent due to the large benefits that can derive from the reduction of the

ambient magnetic field during Maiβner effect.

Electrical superficial resistance is directly correlated to the power

dissipated by a SRF during its working and it is made up by three contributions.

The first term is described in Bardeen-Schrieffer-Cooper theory and resemble

effect of finite inertia possessed by Copper Pairs, quasi-particles derived by the

interaction of two electrons acting as supercurrents carriers. Cooper pairs, under an

oscillating RF field, generate an electrical field that perturbs electron movement

resulting in energy dissipation by Joule effect. BCS resistance is the only

temperature dependent terms: it exponentially decreases with temperature and at

the usual cryogenic operating temperatures of the particle accelerators (2 K) it has

a lower value respect to the two remaining contributions of the superficial

resistance. Residual resistance is an intrinsic property of the material and it is

temperature independent. Typical contributions to this resistance derive from grain

boundaries and normalconductive precipitates. The last term is the surface

resistance generated by the oscillations of fluxons in a RF field. Fluxons are quanta

of background magnetic field trapped in the superconductive material. Trapping

occur in pinning centers, punctual defects of the material, which hinder a complete

Maiβner effect during the transition to superconductive state SRF cavity.

Recent studies have proved that the cryogenic cooling procedure of

superconducting cavities impact upon the superficial resistance deriving from the

trapped flux. Even if is reported in the literature a direct correlation between this

resistance and the intensity of the environmental magnetic field that is trapped

during the transition from the normal conductive to the superconducting state, this

phenomenon has not yet been fully understood. Nowadays, studies revolve around

bulk niobium as preferential material for building SRF cavities thanks to the

highest temperature of transition to superconductivity state among elemental

superconductors, high-purity grade availability, reasonably high thermal

conductivity and good mechanical workability. After 50 years of advancement,

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several limits on SRF cavities performance have been exceeded and niobium bulk

cavities reach accelerating fields close to 50 MV/m, not far from the theoretical

limit.

Impurity-doping is an exciting new technology in the SRF field which

enable the production of accelerating cavities with record-high quality factor.

Unfortunately, sensitivity to trapped magnetic flux, which increases RF losses, is

greatly enhanced for doped cavities. Flux trapping phenomenon is a brand new

topic that is still not well understood.

The purpose of this master thesis will be to investigate the effect of flux

trapping on RF performance on accelerating cavities produced by different

techniques, starting from bulk niobium SRF cavities. To study this phenomenon, a

new stand for the RF test of SRF cavities capable of changing the magnitude of the

trapped magnetic field has been built. The research will be conducted on 6 GHz

SRF cavities in order to work on a system that is analogous to real high-gradient

accelerators. Furthermore, 6 GHz cavities are cheap and easy to produce and to

test, resulting in fast data collecting and low operational costs.

The new stand for RF test is characterized by an active magnetic field that

can modulate the magnetic field coaxial with the cavity tested. Magnetic field is

trapped during transition to superconducting state due to partial Maiβner effect.

Evaluating the cavity behavior with different intensities of the trapped magnetic

field will indicate the sensibility of the substrate to flux trapping.

Contents

Abstract .......................................................................................................... I

Chapter 0: Introduction ............................................................................... 1

Chapter 1: SRF Overview ............................................................................ 3

1.1 Superconductivity ........................................................................... 3

BCS Theory of Superconductivity ........................................................... 6

DC and AC Electrical Resistance, Residual Resistance ......................... 7

1.2 SRF Cavities ................................................................................... 8

Figures of Merit .................................................................................... 10

SRF Cavities Performance Limitations ................................................. 11

1.3 Trapped Magnetic Field ................................................................ 14

Fluxoid Structure .................................................................................. 15

Losses Due to Trapped Flux ................................................................. 16

Reduction of Flux Trapping .................................................................. 17

Chapter 2: Magnetic Flux Trapping Studies ............................................ 21

2.1 Active Shield Apparatus ............................................................... 21

2.2 6 GHz Cavities Characterization Procedure ................................. 25

Fundamental Equations for RF Test ..................................................... 26

Cryogenic Apparatus and Procedure for Magnetic Flux Trapping ...... 29

Chapter 3: Results & Discussion ............................................................... 34

3.1 RF Measurements ......................................................................... 34

3.2 Cooling Procedure ........................................................................ 37

3.3 Surface Resistance Calculation ..................................................... 39

3.4 Discussion ..................................................................................... 43

Chapter 4: Summary & Outlook ............................................................... 45

Bibliography ................................................................................................ 47

List of Figures

Figure 1.1: Adaptation of the original plot of the electrical resistance of Hg versus temperature by H. K. Onnes. Within 0.01 K, the resistance decrease from 0.10 Ω to immeasurably less (lower than 10-6 Ω) at the critical temperature of 4.2 K. ..... 3

Figure 1.2: Perfect diamagnetism of type-I (in red) and type-II (in blue) superconductors in a magnetic field. In this graph, critical magnetic fields are displayed [7]. ........ 5

Figure 1.3: Critical fields of a type-II superconductor. Above Hc2, the material is in normalconductive state. Below Hc2, the magnetic field is quantized and below Hc1, the magnetic field is expelled. Pinning sites (dislocations or normal conducting precipates) may hinder the expulsion. They lead to flux pinning and hence an incomplete Meiβner phase. From the diagram is clear that Meißner effect is not only a consequence of null electrical resistance, but it is a basic property of superconductors since a sufficient strong magnetic field can inhibit it [6]. ......................................................................................................................... 5

Figure 1.4: Example of how surface resistance (Rs, shown in gray) splits into BCS resistance (RBCS, shown in light blue) and residual resistance (Rres, shown in dark blue). Data measured at low fields on a 1.3 GHz niobium cavity prepared with electropolishing and thermal treatment at 120°C for 48 hours [10]. ..................... 8

Figure 1.5: Classification of accelerating structures. ........................................................ 9 Figure 1.6: Time evolution of the electric and magnetic fields for the TM010 mode of a

pillbox cavity with beam tubes. Image property of Nicholas Valles [12]. ............ 9 Figure 1.7: TESLA type elliptical cavity structure. ........................................................ 10 Figure 1.8: Principal limitation in a superconducting resonant cavity that restrict the

achievable Q0 and accelerating field Eacc [13]. .................................................... 12 Figure 1.9: Trajectory of secondary electrons generated by multipacting....................... 13 Figure 1.10: Example of field emitters. .......................................................................... 14 Figure 1.11: Schematic illustration of a magnetic vortex structure. ................................ 15 Figure 1.12: RF measurements for AES024 bulk Niobium cavity for different

magnitudes of the background magnetic field. The measurements were conducted at 2 K. .................................................................................................................. 19

Figure 1.13: Various HIE-ISOLDE cavies RF performences at 4.5 K. The cooling of the cavities was performed in Earth magnetic field. The star symbol represents the nominal achievable accelerating gradient peak; its value is around 6 MV/m. ..... 19

Figure 1.14: QSS2 RF performances in a background field of 50 mG. The star symbol represents the nominal achievable accelerating field peak for cavities without compensating background field during cooling procedure. ................................. 20

Figure 2.15: Axisymmetric drawing of the active shield. ............................................... 22 Figure 2.16: FEMM simulation output with density plot of the flux density of the

solenoid produced with Copper 21 AWG magnetic wire (@ 10 mA). ................ 22 Figure 2.17: (left) Helium tank and (right) Helium tank cross section. In the figures, the

three cavity stands and the low temperature screens are visible. ......................... 30 Figure 2.18: Sketch of the new RF vertical stand configuration. .................................... 31 Figure 2.19: FEMM simulation for an applied magnetic field of 50 mGauss.

(left)Magnetic field density plot of a cavity in normalconductive state. ............. 32 Figure 3.20: Flux expulsion ratio BSC/BNC of cavities prepared in various ways as a

function of iris-to-iris temperature difference during cooling procedure. A ratio of 1.8 represents perfect flux expulsion, while a ratio of 1.0 represents full flux trapping [20]. ....................................................................................................... 38

Figure 3.21: Sensitivity as a function of the accelerating gradient measured for the cavities studied. Graphs (a), (b), and (c) show the data acquired for bulk Niobium cavities treated with 120°C, EP/BCP, and 2/6 N-doping, respectively [23]. ....... 43

List of Graphs

Graph 1.1: Axial magnetic field uniformity along nine different positions inside one of the cryostat pits. Data have been collected on a fictitious circumference resembling the cavity equator. Data reported in Table 2.1. ................................ 17

Graph 1.2: Radar chart displaying Axial magnetic field uniformity along nine different positions inside one of the cryostat pits. .............................................................. 18

Graph 2.3: FEMM plot of the magnitude of the axial flux density produced by the active shield solenoid (@ 10 mA) at cell equator and at cut-off level. “Cell” and “Shield” lines are relative to the respective distance from the cell vertical axis of symmetry. ............................................................................................................ 23

Graph 2.4: Comparison between background cryostat magnetic field before and after the activation of the active shield. The red line displays the magnitude of the magnetic field in the case of an ideal solenoid. Data reported in Table 2.1. ...... 24

Graph 2.5: Radar chart displaying axial field uniformity for active shield and cryostat Mumetal passive shield. Ideal solenoid (uniform field) is also displayed. Data reported in Table 2.1. .......................................................................................... 25

Graph 3.6: Bulk Niobium RF test at 4.2 K and 1.8 K. No compensating field has been applied. ................................................................................................................ 34

Graph 3.7: Bulk Niobium RF test at 4.2 K and 1.8 K @ 50 mG background field. ........ 35 Graph 3.8: Bulk Niobium RF test at 4.2 K and 1.9 K @ 100 mG background field. RF

Test has been performed at 1.9 K instead of 1.8 K due to technical difficulties during the cavity cooling procedure.................................................................... 35

Graph 3.9: Bulk Niobium RF test at 4.2 K and 1.8 K @ 180 mG background field. ...... 36 Graph 3.10: RF measurements performed at 4.2 K for different magnitudes of applied

magnetic field. ..................................................................................................... 36 Graph 3.11: RF measurements performed at 1.8 K for different magnitudes of applied

magnetic field. ..................................................................................................... 37 Graph 3.12: Spatial thermal gradient for bulk Niobium RF test @ 50 mG background

field. .................................................................................................................... 38 Graph 3.13: Spatial thermal gradient for bulk Niobium RF test @ 100 mG background

field. .................................................................................................................... 39 Graph 3.14: Spatial thermal gradient for bulk Niobium RF test @ 180 mG background

field.. ................................................................................................................... 39 Graph 3.15: Flux trapped resistance at 1.8 K for different magnitudes of applied

magnetic field. ..................................................................................................... 41 Graph 3.16: Flux trapped resistance at 1.8 K for different magnitudes of applied

magnetic field at low accelerating gradient. ....................................................... 41 Graph 3.17: Sensitivity as a function of the accelerating field for different background

magnetic fields. ................................................................................................... 42 Graph 3.18: Sensitivity for different background magnetic fields at low accelerating

gradient. .............................................................................................................. 42

List of Tables

Table 2.1: Axial magnetic field data collected along nine different positions inside one of the cryostat pits lying on the cell equator. ........................................................... 24

Table 2.2: Estimated magnitude of magnetic field expelled due to complete Meiβner effect. The integral of the magnetic field has been calculated over the volume occupied by the fluxgate probe. .......................................................................... 32

Table 3.3: Superconductive Niobium material parameters inputs for SRIMP code. ....... 40 Table 3.4: Determination of R0 from the RF measurement data done at residual cryostat

magnetic field. ..................................................................................................... 40 Table 3.5: Estimated values of Rfl(B) for different background magnetic fields. ........... 41 Table 3.6: Estimated values of flux trapping sensitivity S for different background

magnetic fields. ................................................................................................... 42 Table 3.7: Spatial temperature gradient for different background magnetic fields. ........ 43

1

Chapter 0:

Introduction

The International Committee for Future Accelerators (ICFA) suggests superconducting radio frequency (SRF) cavities as the future path for Linear Collider design since normal conductive (NC) cavities, typically made of Copper, are limited in their application due to the large surface resistance.

In the last 50 years Niobium superconductive (SC) cavities have represented a standard choice for SRF applications. Despite being able to provide many benefits, the main limits of bulk Niobium cavities are connected to significant raw material and cryogenic operational costs. New proposed Linear Accelerators (LINACS) require the use of thousands of cavities, making the cost reduction a factor that must be no longer neglected.

An answer to this problem has already been developed in the 1980s by Benventi at CERN. The solution consists in the physical deposition (PVD) of several microns of niobium on the internal walls of a copper resonant cavity. The Nb/Cu cavities take advantage of both the low-cost availability of Copper substrate and the superconductive properties of Niobium film. Nb films show clear advantage in particular for low frequency cavities or for operation at 4.2 K. However, a lower quality factor and large Q-slope affect these cavities, limiting their use in high field accelerators. Since the early 1990s researchers of many laboratories tried to understand and solve the Q-slope problem in thin films cavities but until now it remains unsolved.

At LNL laboratories, 6 GHz resonating cavities are being used to obtain statistical significant pools of data in a reasonable amount of time. 6 GHz cavities are characterized by reduced size, especially if compared with 400 MHz prototype cavities for CERN’s future circular collider. Minor dimensions lead to inferior costs but above all, the time needed for cavities performances estimation is considerably decreased.

Two new approaches are being studied at LNL laboratories to improve Nb/Cu cavities performances. The first approach revolves around the Q-slope problem. Thick Nb film deposition is under investigation. Assuming that the purity index of Nb increases with the thickness of the deposited film, the main topic is to inspect the thermal boundary resistance at Nb/Cu interface. According to a model recently proposed by Vaglio and Palmieri, the Q-slope is related to local enhancement of the thermal boundary resistance at the Nb/Cu interface due to poor thermal contact between film and substrate. In thick films, the heat flux derived from RF power dissipation can easily bypass these thermal defects at the interface. In such way, thermal breakdown and the consequent localized loss of the superconductive state might be avoided. The second approach arises from the power loss on the cavity surface due to magnetic field trapping. The

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sensitivity of Nb/Cu cavities to flux trapping is known to be lower than Niobium sheet cavities [1]. This is the main reason why for HIE-ISOLDE installation in CERN no magnetic shielding was foreseen [2]. Still, the surface resistance of Nb/Cu ISOLDE Quarter-Wave Resonators (QWRs) was found to be strongly dependent on the thermal gradient during the cooling to superconductive state. A possible interpretation is trapped flux produced by thermoelectric effect [3]. The effect of trapped magnetic field over cavities performances can be no more neglected.

The influence of the flux trapping over the superconductivity properties of accelerating cavities must be studied. Active shielding offers the opportunity to adjust the magnitude of the trapped magnetic flux simply varying the external magnetic field. 6 GHz cavities are the ideal testing ground for this innovative study because they offer the unique possibility to compare the effect of flux trapping over different SRF cavity typologies. Bulk Niobium, thin films, thick films, multilayer approach can be all easily tested over cavities owning a well fixed geometry and frequency. Furthermore, ambient magnetic field, if trapped during the transition to SC state, leads to an increasing surface resistance and therefore limits the accelerating field. To contain this phenomenon, several approaches are possible. In order to reduce static magnetic fields, shielding of the cavity with sophisticated Mu-metal® structures is a common practice but it can result expensive and laborious. Nevertheless, a new procedure is presented in this thesis. During the transition to Meiβner state, cavities are exposed to a magnetic field that can be manipulated using electromagnetic coils. By generating a magnetic field that is equal but contrary to the preexisting one, it is possible to nullify the external field thus leading to zero flux trapping and superior SRF performances.

3

Chapter 1:

SRF Overview

The main subject of this thesis concerns the improvement of SRF performances of superconducting cavities for particle acceleration. Accounting the cryogenics operative costs, a superconducting (SC) cavity dissipates half of the power than a normal conducting (NC) one. Nevertheless a SC cavity allows to reach high accelerating gradients, ~50 MV/m in continuous wave mode (CW), due to the low power dissipation of the resonant cavity walls. A NC Copper cavity working in CW can reach only few MV/m before failure caused by the high heat generated.

1.1 Superconductivity

The discovery of “a new state which owing to its particular electrical properties can be called the state of superconductivity” is attributable at H. K. Onnes. In 1908 the Dutch physicist succeeded in liquefying helium, a fundamental premise for his discovery. During his studies on the dependence of electrical resistance of Mercury with the temperature, he observed a suddenly drop to zero resistance. This phenomenon was observed at a temperature, later defined as critical temperature (Tc), of 4.2 K, the boiling point of liquid helium at ambient pressure.

Figure 1.1: Adaptation of the original plot of the electrical resistance of Hg versus temperature by H. K. Onnes. Within

0.01 K, the resistance decrease from 0.10 Ω to immeasurably less (lower than 10-6 Ω) at the critical temperature of 4.2 K.

Proceeding with his research work, Onnes discovered that other

materials exhibit superconductive properties but at different critical temperatures than Mercury. The highest critical temperature between pure metals is shown by Niobium (9.25 K) and the lowest has been found by

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Tungsten (0.0154 K) [4]. Critical temperature was concluded to be an intrinsic characteristic of the material.

In 1914 subsequent investigations about superconductive properties were managed by Onnes. He showed that superconductivity can be destroyed not only by increasing the temperature, but also by applying a sufficiently strong magnetic field. The critical magnetic field (Hc) in which the superconductivity is destroyed decreases with the increment of temperature. The following formula describes empirically the dependence of the critical magnetic field on temperature [4]

Hc(T) = Hc(0) 1 − TTc2

[1]

Superconductivity can also be limited by a strong electric current.

According to Silsbee Rule, the critical current (ic) is reached when the field due to that current reaches the critical field of the superconductor. Additionally, the critical current is governed by the critical current density jc and the geometry of the superconductor.

The next significant step in understanding superconductivity occurred in 1933. W. Meiβner and R. Ochsenfeld learned that, a weak magnetic field, present in the initially normalconductive sample, is expelled when the material turns superconducting. The transition to the superconducting state is accompanied by the appearance of supercurrents on the surface which cancels the magnetic field in the interior [5].

A related aspect of the magnetic behavior of a superconductor is flux exclusion, as distinct from expulsion [5]. An external field (lower than the critical one), switched on after the phase transition to Meiβner State, is hindered to penetrate the material. The field is excluded from the volume of the superconductor, so that magnetic flux density is null. This feature is different from an ideal perfect conductor, which can be called a flux conserving material [6].

Superconductors are divided in two categories, depending on their response to an external magnetic field. The previous observations are accurate only for type-I semiconductors, mainly comprised of pure metals and metalloids. Type-II superconductors, elemental Niobium and all metal alloys, respond in a different way to an applied external magnetic field. They can be distinguished by the presence of an intermediate state between the superconductive behavior and the normalconductive one. Meißner State, corresponding to the flux expulsion, collapses upon reaching a certain value of the applied magnetic field (Hc1). For higher fields, the material enters in the Shubnikov phase and the external magnetic field starts to enter the material in the quantized form of flux lines. Each line contains one elementary flux quantum (φ0) and magnetic flux lines partially enter the material. Still, superconductivity behavior is not hindered. The area in between the flux lines remains superconducting while the core of each flux line is normal conducting. Above a second, upper critical field (Hc2),

5

superconductivity behavior breaks down and the field penetrates all the material. Both critical fields are temperature dependent and can be approximated as [6]

Hc(T) ≈ Hc(0)(1− T2) [2]

Figure 1.2: Perfect diamagnetism of type-I (in red) and type-II (in blue) superconductors in a magnetic field. In this graph,

critical magnetic fields are displayed [7].

Figure 1.3: Critical fields of a type-II superconductor. Above Hc2, the material is in normalconductive state. Below Hc2, the magnetic field is quantized and below Hc1, the magnetic field is expelled. Pinning sites (dislocations or normal conducting

precipates) may hinder the expulsion. They lead to flux pinning and hence an incomplete Meiβner phase. From the

diagram is clear that Meißner effect is not only a consequence of null electrical resistance, but it is a basic property of

superconductors since a sufficient strong magnetic field can inhibit it [6].

6

BCS Theory of Superconductivity

For decades, a comprehensive knowledge of superconductivity eluded the many scientists working on this topic. Then, in the 1950s and 1960s, a remarkably complete and satisfactory theoretical picture of the classic superconductors emerged.

The physics of superconductivity was first motivated by the London equations, developed by the London brothers in 1935. Their phenomenological theory, based no Maxwell’s equations, can describe null resistance and Meiβner Effect. A theory aimed at explaining superconductivity in depth was proposed by Ginzburg and Landau in 1950 as a theory based on phase transitions. Ginzburg-Landau theory looks at the macroscopic effects of superconductors without examining the microscopic behavior. In 1957, a theory published by Bardeen, Cooper, and Schrieffer (BCS theory) explained the microscopic details of superconductivity. Finally, in 1959, Gor’kov showed that the Ginzburg-Landau equations could be derived from the microscopic BCS theory.

The central feature of the BCS theory is that two electrons in the superconductor are able to form a weak bound, known as Cooper pair. Electrons carrying an electrical current through a metal conductor encounter resistance. Electrical resistivity is caused by collisions and scattering as electrons move through the vibrating lattice of metal atoms. As metal is cooled down to low temperatures, lattice vibrations decrease. A moving electron attract nearby metal atoms, distorting the ions lattice, which create a positively charged wake behind the electron. This wake will eventually attract another nearby electron. This case will lead to the pairing of the electrons into a so called Cooper pair. The weak bond between the two electrons, formed due to electron-phonon interaction, dominates over the repulsive Coulomb force. Even if the size of a Cooper pair is related to the coherence length ξ0, larger in value than the lattice constant, the pair encounters less resistance than two electrons moving separately. While electrons are fermions that underlie the Fermi-Dirac statistics, Cooper pairs are composite bosons that underlie the Bose-Einstein statistics. Hence, all Cooper pairs in a superconductor have the same quantum mechanical ground state, which alters the density of states below its transition temperature Tc. Just as other bosons particles, the overall spin of these two paired electrons is zero so many of these pairs can co-exist coherently. If a pair is scattered by an impurity, it will quickly get back in step with another pairs. This allows the electrons to flow undisturbed through the lattice of metal atoms without resistance, so that no power is dissipated. The number of Cooper pairs is temperature dependent and only at 0 K all conduction electrons are condensed into Cooper pairs [9]. The superconducting electrons co-exist with their normalconducting counterparts. The number of normalconducting electrons nn is given by the Boltzmann factor

7

nn(T → 0) ≈ ns(0) exp µ − ∆(T)

KBT [3]

Since the number of Cooper pairs is temperature dependent and only

at 0 K all conduction electrons are condensed into Cooper pairs. Only under this condition, BCS electrical resistance contribution is null.

DC and AC Electrical Resistance, Residual Resistance As explained above, not all conduction electrons form Cooper pairs

at temperatures above 0 K. Hence, both contributions to RF dissipation by normalconducting electrons and Cooper pairs must be evaluated according to the two fluid model. The conduction electrons which are paired into Cooper pairs, carry a current that flow with null resistance. Even if the normal current flows in parallel with the superconductive one, the latter can carry the entire current. To exhibit resistance in a DC field, all existing Cooper pairs have to be stopped. Therefore the DC resistance of a superconductor is exactly zero.

Quite the opposite happens for RF dissipation in superconducting material. The surface resistance for RF currents is non-zero, even though it is still very small compared with normalconductive materials. Due to inertial mass of Cooper pairs, they cannot follow the AC electromagnetic fields switch instantly.

Because of their inertia, the Cooper pairs do not screen the applied

field perfectly, so the penetrating field can interact with the normal electrons causing electrical resistance. Remaining residual field, present in a skin layer, accelerates the normal electrons, leading to power dissipation. A frequently used approximate expression for RBCS, valid at temperature T lower than Tc/2 is given by

RBCS(T, f) = Af2

Texp −

∆gapkBT

[4]

f is the cavity resonant frequency, Δgap is the energy gap at the temperature T, kB is the Boltzmann constant and A is a fore factor which depends on material parameters such as the coherence length (ξ), the London penetration depth (λL) and the electrons mean free path [5].

As observed experimentally, even at low temperatures, the expected electrical resistance is higher than the one predicted by BCS theory. The excess of resistance is accounted by the residual resistance Rres term

Rs(T, B) = RBCS(T) + Rres(B) = RBCS(T) + R0 + Rfl(B) [5]

Figure 1.4 shows the measured Rs, RBCS, and Rres for a 1.3 GHz

TESLA shape cavity as a function of temperature at low accelerating fields.

8

At high temperature (above 4.2 K) Rs is dominated by RBCS. As temperature decreases, RBCS becomes less significant and eventually drop to almost zero at 1.5~1.6 K. In this condition, Rres take over the RBCS term and becomes the main source of superficial resistance.

Figure 1.4: Example of how surface resistance (Rs, shown in gray) splits into BCS resistance (RBCS, shown in light blue) and residual

resistance (Rres, shown in dark blue). Data measured at low fields on a 1.3 GHz niobium cavity prepared with electropolishing and thermal

treatment at 120°C for 48 hours [10]. Residual resistance is made up of two different terms that take in

account of residual losses. Rfl term derives from trapped magnetic flux during cool down through critical temperature; imperfections present in the deposited film can suppress the expulsion of magnetic field during the superconducting transition to Meiβner State. Many reasons regarding the source of residual resistance R0 are instead listened in literature such as: Normal conducting defects such as oxides inclusions and porosity

between the penetration depth. Surface imperfections (cracks, scratches) and surface roughness can

cause thermal breakdown of Meiβner State due to localized hot spots.

Hydrogen absorption during chemical processing of the surface. Hydrogen is usually absorbed by Niobium during chemical polishing, which dissolves or damages the passivating layer of dielectric pentoxide, Nb2O5, on the Niobium surface.

1.2 SRF Cavities

Superconducting RF (SRF) cavities excel in applications requiring continuous wave (CW) or long-pulse accelerating fields above a few million volts per meter (MV/m). The reason for using superconductive materials for accelerating cavity is implied in the usage of AC fields. The surface of the cavity is permeated by surface shielding currents that dissipate energy. While under DC currents superconductors show zero electrical resistance,

9

under AC currents the finite inertia of the Cooper pairs leads to a small but not negligible resistance. Since the ohmic power loss in the walls of a cavity increases as the square of the accelerating voltage, Copper cavities become uneconomical due to increasing demand for high accelerating voltage and resulting refrigeration costs.

Superconducting accelerating cavities can be divided into three major classes: high, medium and low-β (here β is equal to the ratio between the speed of the accelerated particle and the speed of light). High-β structures are used for acceleration of electrons, positrons, or high-energy protons with β~1. A typical high-β accelerating structure consists of a sequence of coupled elliptical TESLA cells working in operating in the TM010 mode (no axial dependence). The TM modes have an electric field component that points along the longitudinal axis, allowing energy to be transferred to charged particles that pass through the structure along this axis [11]. The phase of the instantaneous electric field in adjacent cells is shifted by π to preserve charged particle acceleration while traversing each cell in half an RF period.

Figure 1.5: Classification of accelerating structures.

Figure 1.6: Time evolution of the electric and magnetic fields

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for the TM010 mode of a pillbox cavity with beam tubes. Image property of Nicholas Valles [12].

Single-cell cavities are generally used for SRF R&D. The beam

enters and exits the structure from the beam tubes. Input coupler devices attached to ports on the beam tubes bring RF power into the cavity to establish the field and deliver beam power.

Figure 1.7: TESLA type elliptical cavity structure.

Figures of Merit

During the design for an accelerating cavity for a particular purpose, its shape and size will go through many iterations before obtaining a satisfactory cavity. Therefore, several figures of merit are used to compare different design and determine the optimal solution.

An important figure of merit for SRF cavities is the average accelerating electrical field Eacc, defined as the ratio between the accelerating voltage Vc, to which a charged particle is subjected, and the length d of the cavity

Eacc ≡Vcd

[6]

Despite being superconductive, cavity walls show finite conductivity. This implies non-zero electrical surface resistance Rs and eventually power dissipation Pdiss due to Joule effect

dPdissds

=12

RsH 2

[7]

Pdiss = Rs H 2

Sup𝑑𝑑𝑑𝑑 [8]

11

where H is the external magnetic field in which the cavity is submerged. Another important figure of merit is the quality factor Q0, the

cryogenic efficiency of the cavity. It is related to the energy U stored in the cavity and to the power dissipated Pdiss in one RF period

Q0 ≡ωU

Pdiss [9]

U =12µ0 H

2𝑑𝑑𝑑𝑑 =

Vol

12ε0 E

2𝑑𝑑𝑑𝑑

Vol [10]

Q0 ≡ω0µ0 ∫ H

2𝑑𝑑𝑑𝑑Vol

Rs ∫ H 2𝑑𝑑𝑑𝑑Sup

[11]

The factor Q0 is roughly 2π times the number of RF cycles it takes to

dissipate the energy stored in the cavity. Assuming that the superficial resistance is uniform in the cavity

surface, the quality factor Q0 can also be written as a function of the geometrical constant G

Q0 =GRs

[12]

The constant G depends only on the cavity d but not its size; it is obtained from numerical simulations.

SRF Cavities Performance Limitations The most important figures of merit for an SRF cavity are Q0 and

Eacc; in fact, the performances of a cavity are normally represented by a quality factor versus accelerating gradient curve. As show in the underlying picture, there are several important mechanisms that limit the achievable accelerating gradient and move out the curve from its ideality.

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Figure 1.8: Principal limitation in a superconducting resonant cavity that restrict the achievable Q0 and accelerating field

Eacc [13]. Q-slope

The measured Q of a Niobium cavity is not constant with increasing

Eacc, but shows several anomalies. In general, Q0 increases with increasing Eacc between 0 and 5 MV/m. This phenomenon is known as low field Q slope. Differently, at medium fields, till 25 MV/m, the Q gradually falls. In the end, Q-slope shows a strong drop at high fields: superconductivity quenches either due to surface defect or localized material contamination near RF surface of the cavity or due to phase transition at the local RF critical magnetic field.

Residual Losses Residual losses due to electrical resistance represent the divergence

between the ideal behavior of the cavity, derived from purely theoretical values, compared to the real one, measured through a RF test. Approaches on quality factor Q0 improvement are usually associated to surface treatment processes after manufacturing aimed to enhance the purity of the niobium. The residual-resistance ratio (RRR) is given by the ratio of resistance at 300 K (room temperature) to resistance at 10 K

RRR =ρ (300 K)ρ (10 K)

[13]

The residual resistivity ratio (RRR=300 for a high gradient cavity) is a common indicator of the level of purity. The main interstitially dissolved impurities, oxygen, nitrogen, hydrogen and carbon (O, N, H, C), act as scattering centers for unpaired electrons and reduce the RRR.

13

Multipacting Multiple impact electron amplification in cavities is an avalanche

effect in which a large number of secondary electrons are emitted from the cavity surface by an incident primary electron. The secondary electrons are accelerated and, upon impacting the surface, another generation of electrons is emitted. Then the process repeats itself. Electrons in the high magnetic field region of the cavity travel in circular orbit segments returning to the RF surface near to their point of emission. Secondary electrons generated upon impact travel similar orbits. The colliding electrons lead to a temperature rise that eventually cause localized thermal breakdown of the superconductive state.

Number of secondary emission electrons per impact is very sensitive to the surface conditions.

Figure 1.9: Trajectory of secondary electrons generated by multipacting.

Thermal Breakdown Thermal breakdown is the major phenomenon that limits the

achievable accelerating field. Also known as “quench”, thermal breakdown originates at sub-millimeter size regions of the RF surface. Quenches derive from supercurrents that flows through surface defects, generating Joule heating. Surface power dissipation on the defect increases with the Eacc and results in quench after reaching threshold of thermal instability Temperature exceeds the critical one if the heat generated at the hotspot is not rapidly evacuated to the helium bath. Out of the Meiβner state, the surface now rapidly dissipates the energy stored in the cavity walls.

Field Emission Under normal circumstances electrons are confined inside a metal by

a potential well. The energy of an electron is insufficient to enable it to

14

escape from the metal; additional energy is required to escape the metal. Certain defects, like microprotusions or surface contaminants, enhance the local electrical field. Consequently, due to field enhancement factor, those sharp-tip defects act as electron emitters. The electrons may either hit the cavity or the surrounding ambient causing radiations diffusion and eventually thermal breakdown if electrons are sufficiently energetic.

Figure 1.10: Example of field emitters.

1.3 Trapped Magnetic Field

As said before, one known source of residual resistance is trapped DC magnetic flux. The residual resistance equation can be written as sum of several independent terms

Rres(B) = R0 + Rfl(B) [14]

The intrinsic residual resistance R0 embraces the sources of electrical

resistance that are not affected by the presence of magnetic fields. The term Rfl can be written as

Rfl(B) = S η Bext [15]

whereas S represent the sensitivity of the material to an external magnetic field and η the efficiency of flux expulsion.

If an external magnetic field is present during the transition to Meiβner State, complete flux expulsion cannot occur. In ideal conditions, if the external field is lower than the critical Hc1, the DC flux will be fully expelled from the bulk of a superconductor due to Meiβner Effect. Unfortunately, lattice defects or surface imperfections can hindered the expulsion. Flux lines are pinned to these impurities, so that flux remain trapped into the material. The smallest amount of flux that can penetrate a superconductor corresponds to a flux quantum Φ0 = h ⁄ 2e = 2.07E-15 Tm2; it does so as a vortex (or fluxoid).

C, O, Na, Al, Si

In

15

Fluxoid Structure When a type-II superconductor is cooled down at ambient field,

lower than Hc2, it enters in the Shubnikov phase. The magnetic field is quantized in the form of flux lines, called fluxoids. The structure of the flux lines resemble a vortex in which its DC field decays in a distance λL (penetration depth, decay length of magnetic field in the superconductor) from its normalconductive core of size 2ξ; in the center of the vortex the order parameter |ψ|2 is zero. With increasing distance to the center, the number of Cooper pairs ns = |ψ|2 increases as B decreases.

In Shubnikov phase, fluxoids are metastable. Flux lines interact with each other and, for magnetic fields lower than Hc2, a fluxoids lattice will form due to mutual repulsion between fluxoids.

Figure 1.11: Schematic illustration of a magnetic vortex structure. As said before, flux line lattice is metastable. Neverthless fluxoids

can survive even without an external magnetic field due to pinning phenomenon. Porosity, dislocations, grain boundaries and inclusions of the superconductor constitute pinning centers. This defects are characterized by a lower energy potential compared to the surrounding superconductive locations. A pinning force arose and the flux lines are attracted toward pinning centers, so vortex lattice become distorted. If pinned, fluxoids are not able to leave its pinning site. From now on, even lowering the external filed, vortices are trapped in the superconductor and can survive even in Meiβner state.

B

r

JS

λL

2ξ

pinning center

16

Losses Due to Trapped Flux If a RF field is applied, power is dissipated by flux lines oscillations.

To appreciate the effects of flux trapping, theoretical RF losses due to full trapped flux can be estimated. The trapped field Hext over an area A breaks up into N fluxoids, each with a quantum of flux Φ0

A Htrap = N Φ0 [16]

In the normalconductive core of fluxoids the superconducting order

parameter vanishes. Over a distance ξ0, the order parameter reaches its maximum value. Therefore the contribution to the residual resistance from N fluxoids is the normal state resistance Rn times the fraction of the normalconducting area

Rfl = Nπ ξ02

ARn =

Htrap π µ0 ξ02 Rn

Φ0 [17]

According to the theory of type-II superconductors, the upper critical

field Hc2 is given by

Hc2 =Φ0

2 π µ0 ξ02 [18]

Therefore

Rfl =Hext

2 Hc2 Rn [19]

This prediction is heavily dependent on material parameters,

specifically mean free path, pinning site distance from the surface and pinning force [14]: The higher the mean free path, the more the extent of the oscillation

of the flux line around its pinning center. Sensitivity to flux trapping depends on the pinning site distance

from the surface q0. Superficial pinning (q0 < 700 nm) affect sensitivity but flux trapping is governed by bulk pinning (q0 > 700 nm).

The higher the pinning force, the more constrained the oscillation of the fluxoid.

For Niobium Hc2 = 2400 G, and the classical normal state surface resistance at 1 GHz is Rn ≈ 1.5 mΩ. Substituting these terms into the previous equation results in

Rfl[mΩ] = (3.125E− 4) ∗ Hext [G] ∗ f [GHz] [20]

17

The √f dependence is justified because the normalconductive core

size is of order ξ0 and comparable to the mean free path. Experimental results indicate that the coefficient Rfl is about 1 mΩ.

If a 1 GHz superconductive cavity is not perfectly shielded from external magnetic fields, the maximum achievable Q0 value could be limited below 109.

An interesting consequence is that cavities made of a thin film of Niobium on Copper are much less sensitive to the background DC magnetic field because the upper critical field of sputtered, fine-grained Niobium is between 15000 and 35000 G. Therefore the Q0 of a Nb/Cu cavity is up to a factor of ten less sensitive to the external field than a Niobium cavity made from bulk sheet metal.

Reduction of Flux Trapping

As seen in the residual resistance equation [14] in order to reduce the Rres term, several options are available. In this thesis the chosen approach is to restrain the Rflux term through reduction of the external magnetic field.

In LNL cryostat, a Mumetal® magnetic shield is already installed. The shields consist of high permeability material that bundles the earth field in its walls and hence reduce the field in their midst. One shield is mounted close to the cryostats wall (outer shield) and a second one is mounted directly on the liquid Helium tank of the cavity (inner shield). The main drawback of this system is that with a cylindrical geometry of the cryostat, the shielding efficacy is reduced along the cavity axis. It is also problematic to obtain a good uniformity of the transversal magnetic field as proven by the data collected in one of the three cryostat holes.

Graph 1.1: Axial magnetic field uniformity along nine different

positions inside one of the cryostat pits. Data have been collected on a

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9

Mag

netic

Fie

ld [m

G]

Position

Cryostat Axial Field Uniformity

18

fictitious circumference resembling the cavity equator. Data reported in Table 2.1.

Graph 1.2: Radar chart displaying Axial magnetic field uniformity along nine different positions inside one of the cryostat pits.

Quite the opposite happens with compensation coils, which are

tuned on to create a magnetic field to compensate the existing one. The problem with active compensation method is that coils may cancel the external field only within a certain volume. In addition, wiring to power up the coils, flange feedthroughs and a power supply is required.

Experimental evidences on the possibility to reduce flux trapping impact on the quality factor are reported in literature. At Fermilab, a fine grain 1.3 GHz single cell cavity, denominated AES024, was fabricated using high RRR Niobium. Because of the cavity weak flux expulsion behavior, RF measurements showed substantial vulnerability to Q0 degradation due to flux trapping [15]. AES024 exhibits a substantial impact on the Q0 even if the background field is small.

0123456

1

2

3

4

56

7

8

9

Cryostat Axial Field Uniformity [mG]

19

Figure 1.12: RF measurements for AES024 bulk Niobium cavity for

different magnitudes of the background magnetic field. The measurements were conducted at 2 K.

The results indicate that high RRR Niobium material can be

vulnerable to flux trapping; cavities produced will have substantially degraded RF performances.

Extensive R&D work was done on HIE-ISOLDE high β QWRs DC bias magnetron sputtered Niobium on Copper cavities at CERN. In particular, seamless QSS2 cavity reached unprecedented peak fields for Nb\Cu technology after cooling in a partially screened ambient magnetic field [16].

Figure 1.13: Various HIE-ISOLDE cavies RF performences at

4.5 K. The cooling of the cavities was performed in Earth magnetic field. The star symbol represents the nominal

achievable accelerating gradient peak; its value is around 6 MV/m.

20

Figure 1.14: QSS2 RF performances in a background field of 50 mG. The star symbol represents the nominal achievable accelerating field peak for cavities without compensating

background field during cooling procedure.

21

Chapter 2:

Magnetic Flux Trapping Studies

Ambient magnetic field, if trapped in the penetration depth, represents one of the well-known contributors to the residual resistance Rs of SRF Nb/Cu cavities. It limits the achievable intrinsic Q0 at low and medium accelerating fields. In this thesis work, compensating coils are designed and manufactured to shield 6 GHz SRF cavities.

2.1 Active Shield Apparatus

Generally, active shields are used preferably as the backup module of a permanent magnetic shield. The reason of this choice is related to the fact that passive shields don’t necessitate any external operation in order to work. Contrary, active shields need a power source and additional electronics in order to generate the field that hinders the existing magnetic field [17]. Since with the actual cryostat configuration is not possible to improve the current passive shield, active shielding system has been implemented.

The design of the magnetic shield starts with the investigation of the magnetic field existing inside the cryostat. Using Lakeshore Hall probes it was found that only the axial magnetic field inside the pits is sufficiently intense to affect flux trapping. Each of the three pits for the cavities are individually shielded with a Mumetal passive shield that should limits the geomagnetic field. The main problem with the current configuration is that the region above the cavity is lacking of a permanent shield. This fact greatly simplifies the design of the active shield due to that fact that only the vertical component of the external magnetic field shall be compensated.

The design of the shield is limited due to the spatial constrain of the cryostat pits. In fact, the pits have an internal diameter of 75 mm while the two flanges of the cavity stand have a diameter of 70 mm. According to the geometrical constrains and to the background field present in the pits, a cylindrical solenoid coaxial to the cavity will act as active shield.

22

Figure 2.15: Axisymmetric drawing of the active shield.

A preliminary mathematical simulation of the shield behavior has been implemented by using Finite Element Method Magnetics (FEMM) software. FEMM is a finite element package for solving 2D planar and axisymmetric problems in low frequency magnetics and electrostatics. The main requirement is to generate a uniform compensating magnetic field within the cavity cell, where flux trapping will increase the residual surface resistance. The magnitude of the field must be enough large to overcome the background field of the cryostat.

Figure 2.16: FEMM simulation output with density plot of the

flux density of the solenoid produced with Copper 21 AWG magnetic wire (@ 10 mA).

23

Graph 2.3: FEMM plot of the magnitude of the axial flux density

produced by the active shield solenoid (@ 10 mA) at cell equator and at cut-off level. “Cell” and “Shield” lines are relative to the respective distance from the cell vertical axis of symmetry.

As can be seen in Figure 2.16 and Graphic 2.2, within the cell volume the variation of the produced magnetic field is limited to a maximum of 5.36%. This fact implies the feasibility of a magnetic shield constituted by a simple solenoid. The data extrapolated from the FEMM simulation have been eventually compared with empirical data.

Table 2.2 reports axial magnetic field data collected inside one of the cryostat pits. Data have been collected using a Bartington Mag-01H fluxgate equipped with a Mag-F probe by measuring the field in 9 different positions lying on a fictitious circumference resembling the cavity cell equator. After “Cryostat” data collecting, regarding the background field inside the pit, the active shield has been turn on. Finely tuning the current passing through the solenoid allows compensating the magnetic field. “Compensated - Real” data have been collected upon nullifying the background field at Position 1. Maintaining a constant field during the measurement permits to evaluate the homogeneity of the magnetic field generated by the active shield. Ideal behavior of the solenoid is represented by “Compensated - Ideal” data. “Generated” data, concerning the field generated by the shield to nullify Position 1 background field, has been simply extrapolated by experimental data by subtraction of “Cryostat” by “Compensated - Real” data.

-100

-50

0

50

100

150

0 10 20 30 40

B.n

[mG

]

Distance from cell center [mm]

Axial Flux Density - 21 AWG solenoid

B.n EQ B.n CO

Cell Shield

24

Table 2.1: Axial magnetic field data collected along nine different positions inside one of the cryostat pits lying on the cell equator.

Position Axial Magnetic Field

Cryostat Compensated Generated

Real Ideal Real Ideal # [mG] [mG] [mG] [mG] [mG] 1 5,304 0,000 0,000 5,304 5,304 2 5,390 -0,167 0,086 5,557 5,304 3 5,355 -0,270 0,051 5,625 5,304 4 4,388 -1,097 -0,916 5,485 5,304 5 3,564 -1,783 -1,740 5,347 5,304 6 2,096 -3,321 -3,208 5,417 5,304 7 2,285 -3,225 -3,019 5,510 5,304 8 3,280 -2,203 -2,024 5,483 5,304 9 3,831 -1,694 -1,473 5,525 5,304

Graph 2.4: Comparison between background cryostat magnetic field

before and after the activation of the active shield. The red line displays the magnitude of the magnetic field in the case of an ideal

solenoid. Data reported in Table 2.1.

-4

-2

0

2

4

6

1 2 3 4 5 6 7 8 9

Mag

netic

Fie

ld [m

G]

Position

Compensated Cryostat Axial Field Uniformity

B.n (null@1)

B.n

25

Graph 2.5: Radar chart displaying axial field uniformity for active shield and cryostat Mumetal passive shield. Ideal solenoid (uniform

field) is also displayed. Data reported in Table 2.1.

Data reported Table 2.1 show that the axial magnetic field generated

by the active shield has a maximum variation of about 6%. As to evaluate the magnetic field inside the cryostat, a Bartington

Mag-01H single axis fluxgate equipped with a Mag-F probe, placed on the equator of the cavity cell, will be used to evaluate the field to which the SRF cavity is exposed due to trapped flux. Cryogenic Mag-F probe will allow detecting low magnetic fields from 0 to 0.2 mT (2 G). A common use power supply, that powers the solenoid, completes the active shield system.

2.2 6 GHz Cavities Characterization Procedure

With the excitation curve (Q versus Eacc) of a SRF cavity is possible to evaluate the overall performance of a cavity. To evaluate the response of the cavity to RF fields it is necessary to excite a resonant mode of the cavity. For this purpose a RF source can be used. The RF signal frequency that excites the cavity, produced by the RF generator, is divided into three paths. One goes through the programmable phase shifter; since the input signal (forward signal) can be different from the cavity resonant frequency, a feedback loop is established to correct the mismatch between the two frequencies. Once the cavity frequency is determined, the loop phase is adjusted such that the RF power stored by the cavity (pickup signal) is maximized. A second line goes to the frequency counter; in this way the frequency of the input signal is determined. The third is routed to the power amplifier. The output of the amplifier is used to drive the coupler antenna inside the cavity (forward signal). The cavity reacts as mismatch in the circuit and reflects back to the amplifier part of the incident power (reflected signal). During the RF test, the cavity is monitored and held on resonance by the phase-locked-loop. The coupler antenna allows monitoring the forward

0%

30%

60%

90%

120%1

2

3

4

56

7

8

9

Axial Field Uniformity Comparison

Active Shield

Ideal Solenoid

Cryostat Field

26

and reflected powers immediately at the input to the cavity. Some of the power is reflected back at the input coupler while most of the power is transmitted through the cavity to be detected on the other side by the pickup antenna (pickup signal). Each of the three powers (forward, reflected and transmitted) are then split enabling a scalar measurement to be made using a power meter.

Fundamental Equations for RF Test

When a cavity mode oscillates with a resonant frequency ω0, a stored

energy U and RF losses on the cavity walls, Pdiss, the intrinsic quality factor Q0 can be defined as

Q0 =ωUPdiss

[21]

After the RF source is turned off, the power stored by the cavity will

eventually be lost. The total power lost Ptot can be written as the sum of the power dissipated by the cavity walls Pdiss due to ohmic resistance and the power that leaks out each coupler

Ptot = Pdiss + Pcpl + Ppk [22] where Pcpl designates the power leaking back out the input couples (because the RF source is off) and Ppk the power transmitted out via pick up antenna. Analogous to the intrinsic quality factor Q0, a loaded quality factor QL, that take into account the resonant circuit behavior when it is coupled with an external line, can be defined as

QL ≡ωUPL

[23]

During a free decay, the power lost corresponds to the variation with

time of the stored energy

dUdt

= −Ptot = −ωUQL

[24]

In absence of anomalous losses, the solution to the differential

equation gives

U = U0 exp −ωtQL [25]

where U0 is the energy stored at time t = 0. The energy decays exponentially with a time constant

27

τL =QL

ω [26]

The determination of the decay time constant τL allows calculating

the loaded quality factor of the cavity. The figure of merit Q0 of the cavity is more relevant than the QL since it is not affected by the power leaked through the couplers. For each power loss mechanism is possible to write a quality factor

PtotωU

=Pdiss + Pcpl + Ppk

ωU [27]

1

QL=

1Q0

+1

Qcpl+

1Qpk

[28]

For each dissipated power, an external quality factor for the coupler has been written

Qcpl ≡ωUPcpl

[29]

Qpk ≡ωUPpk

[30]

External coupling parameters can be written as

βcpl ≡Q0

Qcpl [31]

βpk ≡Q0

Qpk [32]

Determining the coupling parameters and the loaded Q, allow to calculate

Q0 = 1 + βcpl + βpkQL [33] The coupling terms β quantify the perturbation that the couplers cause to the resonating cavity

βcpl =PcplPdiss

[34]

βcpl =PcplPdiss

[35]

28

If βs values are high, the power leaking out of the couplers is larger

than the power dissipated by the cavity walls. To transfer all the input power to the cavity, the coupler should satisfy the condition βcpl = 1, named critical coupling. That condition assures a perfect match of the system and the cavity electrical impedances (coupling). In fact, when βcpl = 1 the input power Pi equals the power dissipated in the cavity plus the amount of reflected power Pref and the power Ppk that goes out of the pickup port

Pdiss = Pi − Pref − Ppk [36]

Impedance matching is essential, otherwise a mismatch causes

power to be reflected back to the source from the boundary between the high impedance and the low impedance. The transmission antenna should be sized in order to avoid perturbation of the cavity operation. This condition is reached when βpk ≪ 1; in this way the antenna picks up the bare minimum energy requested for the measurement. Once the superconductive state of the Niobium is reached, the cavity is ready for the RF test. Two main parameters are measured: the quality factor Q0 of the equivalent resonant circuit and the respective accelerating field Eacc. These parameters are determined from a measurement of the decay time τ, incident, reflected and transmitted power (respectively Pi, Pref and Ppk) and the resonant frequency of the cavity ω:

Q0 = 1 + βcpl + βpkQL = 2QL = 2ωτ [37]

To raise the stored energy U of the cavity, the power Pcpl must be increased. The previous equation is not valid anymore, but is possible to evaluate Q0 and Eacc staring from Qpk

Qpk =2ωτPi − Pref − Ppk

Ppk [38]

Q0 =QpkPpk

Pi − Pref − Ppk [39]

The accelerating field is then calculated starting from the value of

the geometric shunt impedance r/q and the electrical length L using the electromagnetic simulation tool called Superfish (82.7 Ω/m and 25 mm respectively)

Eacc = QpkPpkr qL2

[40]

29

Cryogenic Apparatus and Procedure for Magnetic Flux Trapping

During the RF test, the cavity has to be cooled at cryogenic

temperatures in order to reach the superconductive state. The measurement is typically operated at very low temperatures in order to simulate the actual accelerator operating conditions. The cavity is tested at 4.2 K and then at 1.8 K, the actual operational temperature of the TESLA-type superconducting accelerating cavities, where the superconducting properties are enhanced.

The cryostat used for the RF test has been designed for operating with three 6 GHz cavities at once. In order to minimize the Helium dissipation during the RF measurement, the cryostat has four coaxial vessels. The exterior one is in vacuum to insulate the Nitrogen vessel from the external environment. A third vessel, in vacuum, insulates the internal Helium tank from the Nitrogen vessel. The Helium tank has three pits that host the 6 GHz cavities. The thermal insulation of the top part of the cryostat is provided by several Copper screens that are cooled with the recovery Helium gas coming from the internal Helium vessel.

The cooling of the cryostat is performed in three stages. A preliminary cooling is achieved by injecting liquid Nitrogen. When the Nitrogen level, measured by a dedicated probe, reaches its maximum, the liquid Helium vessel is filled. This procedure allows to reach a temperature of 4.2 K. To test the cavities at 1.8 K is necessary to reduce the Helium vapor pressure in equilibrium with the liquid phase. During the measurements the cavity is completely immersed in the liquid Helium bath, and sustained with a special stand for the RF vertical test.

30

Figure 2.17: (left) Helium tank and (right) Helium tank cross section. In the figures, the three cavity stands and the low

temperature screens are visible. In order to minimize the possibility of field emission during the measurement, a good grade of cleanness is mandatory for the cavities. Therefore the cavities are mounted inside a class 1000 clean-room with a controlled environment.

The vacuum sealing between the cavity and the flanges is provided by Indium metal gaskets. Since Indium can be easily deformed, it will fill any imperfections of the flanges surface, providing a hermetic seal. Moreover, Indium retains its malleability down to cryogenic temperatures, thus avoiding cracking.

The cavity and the two flanges are assembled on a bellow flange that is connected to the pumping line of the vertical stand (prior to cooling with liquid Helium, the pressure of should reach approximately 10-7~10-8 mbar). On the insert top plate there is a linear BNC feedthrough that, together with the bellow, permits the coupler antenna movement.

31

Figure 2.18: Sketch of the new RF vertical stand configuration.

To investigate 6 GHz cavities performances as a function of the trapped flux, the following experimental procedure has been established. In advance to the normal cooling practice to 4.2 K, the cavity needs to be set in the desired background magnetic field, by turning on the active shield, before passing through Tc. In this way, most of the field will be trapped in the Meiβner state. The current supplied to the solenoid can be adjusted upon reaching the desired background field. The magnitude of the field can be read using the fluxgate magnetometer. Passing through Tc, an increase in the measured field is expected due to the magnetic field expulsion upon reaching Meiβner state. After the phase transition, the solenoid is turned off since the field can be no more trapped. The amount of the magnetic flux trapped in or expelled from the cavity material can be estimated from the ratio between the magnetic field after (BSC) and before (BNC) the superconductive transition. Indeed, in case of complete expulsion of the external magnetic field, this ratio, which has been estimated by FEMM simulations, is about 1.3 at the volume occupied by the fluxgate probe at the cavity equator.

1

2

3 ACTIVE SHIELD MAG-F PROBE CERNOX

THERMOMETER

KEYSIGHT E36106A DC power supply, 100V, 0.4A

BARTINGTON MAG-01H fluxgate magnetometer

LAKESHORE Model 336 cryogenic temperature controller

CONNECTED TO

32

Table 2.2: Estimated magnitude of magnetic field expelled due to complete Meiβner effect. The integral of the magnetic field has been calculated over the volume occupied by the

fluxgate probe.

Integral of B over Fluxgate Block NC SC BSC

BNC radial component

[G·mm3] [G·mm3] [a.u.] 6,029 7,893 1,31

Figure 2.19: FEMM simulation for an applied magnetic field of 50 mGauss. (left)Magnetic field density plot of a cavity in normalconductive state. (right) Magnetic field density plot of a cavity in superconductive state.

Complete expulsion of the magnetic field has been simulated considering the cavity as made of a perfect diamagnetic material. The green rectangular area

represents the space occupied by the fluxgate probe

Contrarily, when the cavity traps all the magnetic field there are no changes in the amount of the background magnetic field. Making a simple proportionality it is possible to estimate the percentage of trapped flux as follow

Btrap = BNC 1 −BSC

BNC − 1

0,3 [41]

33

To appreciate the increase of the quality factor Q0 due to null flux trapping, it will be necessary to generate a compensating field capable of hinder the background magnetic field in the cryostat. Entering Meiβner state in this condition will result in a superconductive material free of power dissipation due to fluxoids oscillations.

The trapped flux sensitivity determines the amount of losses per unit of magnetic field trapped and is calculated by normalizing the trapped flux surface resistance with the amount of trapped magnetic field

S = RflBtrap [42]

For the same amount of trapped flux, higher sensitivity cavity will

be charcterized by higher overall surface resistance since Rfl (RF losses generated by the oscillation of magnetic flux vortices in the superconductor) increases monotonically as a function of trapped flux.

34

Chapter 3:

Results & Discussion

A variable amount of magnetic field has been trapped over a bulk Niobium cavity walls to investigate its effect upon the cavity quality factor. Trapped magnetic field magnitude and sensitivity to flux trapping has been evaluated for each RF measurement.

3.1 RF Measurements The first characterization of the bulk Niobium cavity has been done

in the residual magnetic field present inside the cryostat, which is about 2 mG, without applying any compensating field. This measurement will provide the baseline for the following RF tests with different magnitude of trapped magnetic field; since the residual field is small enough to be considered null, from this data will be extrapolated R0 because of the negligible Rfl value.

Graph 3.6: Bulk Niobium RF test at 4.2 K and 1.8 K. No compensating field has been applied.

1E+7

1E+8

1E+9

0 2 4 6 8 10 12 14

Qua

lity

Fact

or [a

.u.]

Accelerating Field [MV/m]

Residual Cryostat Magnetic Field

1.8 K

4.2 K

35

Graph 3.7: Bulk Niobium RF test at 4.2 K and 1.8 K @ 50 mG background field.

Graph 3.8: Bulk Niobium RF test at 4.2 K and 1.9 K @ 100 mG background field. RF Test has been performed at 1.9 K instead of 1.8

K due to technical difficulties during the cavity cooling procedure.

1E+7

1E+8

1E+9

0 1 2 3 4 5

Qua

lity

Fact

or [a

.u.]


50 mG Background Magnetic Field

1.8 K

4.2 K

1E+7

1E+8

1E+9

0 1 2 3 4 5

Qua

lity

Fact

or [a

.u.]



1.9 K

4.2 K

36

Graph 3.9: Bulk Niobium RF test at 4.2 K and 1.8 K @ 180 mG background field.

Graph 3.10: RF measurements performed at 4.2 K for different magnitudes of applied magnetic field.

1E+7

1E+8

1E+9

0 1 2 3 4 5

Qua

lity

Fact

or [a

.u.]



1.8 K

4.2 K

1E+7

1E+8

0 0.5 1 1.5 2 2.5 3

Qua

lity

Fact

or [a

.u.]


Bulk Niobium Cavity 4.2 K RF Tests

Cryostat50 mG100 mG180 mG

37

Graph 3.11: RF measurements performed at 1.8 K for different magnitudes of applied magnetic field.

3.2 Cooling Procedure Spatial thermal gradient during the cooling procedure to Tc must be

correlated to the sensitivity value since flux expulsion is strongly proportional to the thermal gradient [21]. The first challenge in the data analysis is the definition of relevant cooling parameters. Because of the current RF vertical test stand configuration, it is not possible to regulate the cooling rate of the cavity by heaters. Moreover, liquid Helium transfer cannot be enough easily controlled to adjust the temperature gradient.

Thermoelectrically generated currents during the cooling of the cavity generate a magnetic field that is trapped during the superconductivity phase transition. Since large spatial gradients lead to a stronger flux expulsion, and so to a lower surface resistance [22], the measured trapped magnetic field will be also correlated with the temperature difference between the Cernox thermometers placed at the cut-offs.

1E+7

1E+8

1E+9

0 2 4 6 8 10 12 14

Qua

lity

Fact

or [a

.u.]



Cryostat

50 mG

100 mG

180 mG

38

Figure 3.20: Flux expulsion ratio BSC/BNC of cavities prepared in various ways as a function of iris-to-iris temperature difference during cooling procedure. A ratio of 1.8 represents perfect flux expulsion, while a ratio of 1.0 represents full flux trapping [20].

Graph 3.12: Spatial thermal gradient for bulk Niobium RF test @ 50 mG background field.

0

20

40

60

80

100

120

7.07.58.08.59.09.510.010.511.0

Ther

mal

Gra

dien

t [m

K]

Temperature [K]

Spatial Thermal Gradient - 50 mG

39

Graph 3.13: Spatial thermal gradient for bulk Niobium RF test @ 100 mG background field.

Graph 3.14: Spatial thermal gradient for bulk Niobium RF test @ 180 mG background field..

3.3 Surface Resistance Calculation From the excitation curve (Q versus Eacc) of a SRF cavity is possible

to extract the surface resistance Rs by using Equation 12, with a geometry factor G equal to 287 Ohm for 6 GHz cavities. For TESLA cavity, the proposed method has a low miscalculation rate since the distribution of the magnetic field is nearly a constant in the equator region and decays rapidly to the iris region [21].

-10

10

30

50

70

90

110

130

150

7.07.58.08.59.09.510.010.511.0

Ther

mal

Gra

dien

t [m

K]

Temperature [K]


0.0E+0

2.0E+2

4.0E+2

6.0E+2

8.0E+2

1.0E+3

1.2E+3

1.4E+3

7.07.58.08.59.09.510.010.511.0

Ther

mal

Gra

dien

t [m

K]

Temperature [K]


40

Within BCS theory, no simple formula can be derived to calculate RBCS for a superconductor. Even if several approximated formulas can be found in the literature, SRIMP numerical code, which incorporates the full BCS theory, has been used in this thesis work. Written by J. Halbritter, SRIMP program, developed by Cornell, can derive the BCS surface resistance starting from several parameters required to describe the superconductor (transition temperature Tc, energy gap, London penetration depth λ @ 0 K, coherence length ξ @ 0 K, electron mean free path l). The program simulated a 6 GHz TESLA cavity in 1.8 K Helium bath.

Table 3.3: Superconductive Niobium material parameters inputs for SRIMP code.

Parameters Bulk Niobium

Frequency [MHz] 6000 Tc [K] 9,2

Energy gap 1,82 λ [Å] 320 ξ [Å] 390 RRR 300

Because of BCS surface resistance dependence from the working

temperature of the cavity, this term prevail on the residual resistance term at 4.2 K. As can be realized by comparing Graph 3.10 and Graph 3.11, the impact of the resistance term linked to the trapped flux on the quality factor is prevalent only at low operational temperature. In fact, by changing the temperature of operation, BCS resistance change from 1,5E-5 Ohm at 4.2 K to 1,1E-7 Ohm at 1.8 K. For this reason, the following data analysis will take into account only to 1.8 K (or 1.9 K) RF measurements results.

RF test reported in Graph 3.6 will be assumed as reference to calculate the intrinsic residual resistance R0 term since Rfl(B) can be considered negligible due to the low background magnetic field. In this case, from Equation 5 will be Rres = R0.

Table 3.4: Determination of R0 from the RF measurement data done at residual cryostat magnetic field.

Q0 Rs RBCS R0 [a.u.] [Ω] [Ω] [Ω]

9,24E+8 3,11E-7 1,01E-7 2,10E-7

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Table 3.5: Estimated values of Rfl(B) for different background magnetic fields.

Bback Q0 Rs RBCS Rres Rfl [mG] [mG] [Ω] [Ω] [Ω] [Ω] 50 6,41E+8 4,48E-7 1,01E-7 3,47E-7 1,37E-7

100 4,68E+8 6,13E-7 1,60E-7 4,54E-7 2,44E-7 180 3,96E+8 7,25E-7 1,01E-7 6,24E-7 4,14E-7

Graph 3.15: Flux trapped resistance at 1.8 K for different magnitudes of applied magnetic field.

Graph 3.16: Flux trapped resistance at 1.8 K for different magnitudes of applied magnetic field at low accelerating gradient.

0

1

2

3

4

5

6

7

0 1 2 3 4 5

Flux

Tra

pped

Res

ista

nce

[μΩ

]



50 mG

100 mG

180 mG

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Flux

Tra

pped

Res

ista

nce

[μΩ

]



50 mG100 mG180 mG

42

From Equation 41 and Equation 42 is possible to derive the trapped flux sensitivity of the Niobium cavity.

Table 3.6: Estimated values of flux trapping sensitivity S for different background magnetic fields.

Bback BNC BSC Btrap S [mG] [mG] [mG] [mG] [nΩ/mG] 50 53,96 54,33 52,77 2,60

100 96,27 96,55 95,37 2,56 180 190,92 198,02 168,02 2,46

Graph 3.17: Sensitivity as a function of the accelerating field for different background magnetic fields.

Graph 3.18: Sensitivity for different background magnetic fields at low accelerating gradient.

01020304050607080

0 1 2 3 4 5

Sens

itivi

ty [n

Ω/m

G]



50 mG

100 mG

180 mG

0

2

4

6

0 1 2 3 4 5

Sens

itivi

ty [

nΩ/m

G]



50 mG100 mG180 mG

43

3.4 Discussion Three major conclusions can be drawn from the collected data. First, the

conducted RF tests demonstrated how trapped flux affects the cavity performances. From Graph 3.11, it is clearly visible how increasing the trapped field causes the reduction of the achievable quality factor and maximum accelerating gradient, represented by “Cryostat” curve.

Second, no cooling conditions were found to appreciably change the percentage of flux that was trapped. It can be concluded that a larger range of cooling conditions must be explored. In facts, spatial thermal gradients reported in literature (Figure 3.20) have an order of magnitude of tens of Kelvin degree. As reported in Table 3.5, differences of hundreds of mK may not be sufficient to highlight different behaviors of flux expulsion.

Table 3.7: Spatial temperature gradient for different background

magnetic fields.

Bback T ∆T [mG] [K] [mK] 50 9,25 96

100 9,24 133 180 9,25 628

Third, there are some inconsistencies with the calculated sensitivity.

Literature suggests that the power dissipated in the cavity increases due to the reduced mean free path and the associated high sensitivity to trapped flux. Since the variation in the mean free path is achieved by different cavity preparations, a cavity should have a single value of sensitivity independently to the trapped field. Even if sensitivities calculated in Table 3.6 are quite consistent there is not a directly correlation between them and literature data.

Figure 3.21: Sensitivity as a function of the accelerating gradient

measured for the cavities studied. Graphs (a), (b), and (c) show the data acquired for bulk Niobium cavities treated with 120°C, EP/BCP,

and 2/6 N-doping, respectively [23].

In all graphs of Figure 3.21, it is clear that the higher is the

frequency of the cavity, the larger is the sensitivity to trapped field,

44

independent of the surface treatment. Since the RF measurements reported in this thesis work were conducted on a 6 GHz EP bulk Niobium cavity, from Figure 3.21 (b) it is reasonably to suppose that sensitivity should be around 0.5 nΩ/mG at 4 MV/m. Instead, from Graph 3.18, it is clear that the sensitivity of the measured bulk Niobium cavity is far higher than the supposed one. The calculated sensitivity, evaluated at the low accelerating gradient, is comparable to 2/6 N-doping 3.9 GHz bulk Niobium cavity from Figure 3.21 (c). Nitrogen doped cavities are characterized by a low mean free path (2 nm for EP + 120°C baked Niobium and 40 nm for nitrogen-doped Niobium) which involve energy loss from Cooper pairs elastic scattering on impurities. It could be assumed that the tested cavity is affected by high impurity content. High temperature baking under ultra-high vacuum (800°C for 3 hours) and in-situ 120°C baking are common procedures to reduce the amount of such impurities, mainly N, O, C, H.

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Chapter 4:

Summary & Outlook

Understanding the dependence of sensitivity of residual resistance to RF losses from trapped flux is crucial now as Q0 performance and specifications keep increasing.

During this thesis work, a 6 GHz vertical RF stand has been equipped with an active shield and a fluxgate magnetometer in order to investigate flux trapping effect on different types of SRF cavity. Active shielding offers the opportunity to adjust the magnitude of the trapped magnetic flux simply varying the external magnetic field. 6 GHz cavities are the ideal testing ground for this innovative study because they offer the unique possibility to compare the effect of flux trapping over different SRF cavity typologies.

A first bulk Niobium cavity has been tested at different magnitude of trapped magnetic fields. Promising results have been obtained but some drawbacks arose after repeated consecutive RF tests. Future works will involve a reprocessing of the tested Niobium cavity to recover its initial performances, degraded during several repeated RF measurements. A new high vacuum RF induction system will be realized to post-process the cavity.

Flux trapping studies on 6 GHz cavities will proceed with RF

measurement at different background fields for Niobium on Copper thin film and thick film technology toward a more complete understanding of trapped flux for various SRF cavities production approaches.

47

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