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Muon spin rotation characterization of superconducting niobium for applications in high field superconducting… Buck, Terry J. 2016

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 Muon Spin Rotation Characterization of Superconducting Niobium for Applications in High Field Superconducting Radio Frequency Cavities by Terry J. Buck   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate and Postdoctoral Studies  (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)   March 2016  © Terry J. Buck, 2016 AbstractµSR was used to investigate the effect of sample preparation on the magnetic field penetration of high purityniobium to improve the fabrication and preparation of superconducting niobium radio frequency (RF) cavitiesfor use in particle acceleration. The sample preparations tested were (1) electro-polish etching, (2) bufferedchemical polish etching, (3) nitrogen doping, (4) plating with Nb3Sn, (5) baking at 120◦C, (6) baking at800◦C, and (7) baking at 1400◦C. Three different sample geometries and two different applied magneticfield orientations were used in order to observe the effect of sample shape on the µSR measurements and tominimize the effect of the demagnetization factor on the results. The results showed that etching caused fluxto enter the center of the samples at a lower applied magnetic field; however, a 120◦C bake caused the etchedsamples to reach higher field before experiencing flux penetration. These results correlate with RF cavitytest results using the same treatment method. Higher heat treatments caused a reduction in the pinningstrength of the niobium samples and caused flux to enter the center of the sample at lower applied magneticfields.Impurities and vacancies in a sample were suspected of acting as pinning centers and increasing thepinning strength; certain impurities and vacancies are also thought to prevent hydride formation in samplesand prevent high field RF losses in cavities. If the same impurities that prevent RF losses in cavities alsocreate pinning centers in the µSR samples, it could explain why the DC field µSR measurements are showingsimilar results to AC field RF cavity tests.The perpendicular field results for the Nb3Sn plated and nitrogen doped samples showed no differencecompared with regular niobium samples that had undergone similar heat treatments; however, the parallelfield measurements of the Nb3Sn plated sample show an increase in the field of first flux entry. Parallel fieldmeasurements are less affected by pinning strength than the perpendicular field measurements and give abetter indication of when the sample first experiences flux entry. Plating niobium with Nb3Sn could increasethe effective HC1 and thereby accelerating gradient of cavities.iiPrefaceThe work presented in this thesis was completed in conjunction with both the CMMS andRF acceleration groups at TRIUMF. I preformed all the data analysis contained in this thesiswith the help and advice from both groups, with lots of one on one advice from Martin Dehn,Dr. Rob Kiefl, and Dr. Bob Laxdal.Chapter 5 contains a picture taken by Syed Haider AbidiiiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Properties of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Theory of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Radio Frequency Cavity Fundamentals . . . . . . . . . . . . . . . . . . . . . 71.4 SRF Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Superconducting RF Resistances . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Why Niobium is Used for SRF Cavities . . . . . . . . . . . . . . . . . . . . . 111.7 Motivation for the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 122 µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Muon Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Muon Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Muon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19iv3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1 Setting α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Zero Field Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Phase Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1 Sample Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Sample Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Demagnetization and Field Penetration . . . . . . . . . . . . . . . . . . . . 396.1 Demagnetization Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Flux Entry into Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.1 Geometry Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2 Flat Samples Perpendicular Field Results . . . . . . . . . . . . . . . . . . . . 477.3 Formed Samples Perpendicular Field Results . . . . . . . . . . . . . . . . . . 497.4 Flat Samples Parallel Field Results . . . . . . . . . . . . . . . . . . . . . . . 507.5 Ellipsoid Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66vList of Tables5.1 Niobium samples used in the experiment and the corresponding treatments preformed onthem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38viList of Figures1.1 Electrical resistivity vs temperature of conductors and superconductors . . . . . . . . . 21.2 The top image shows a superconductor in the normal state in the presence of an appliedmagnetic field, the bottom shows a superconductor in the Meissner state in the presence ofan applied magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Illustration of a superconductor in the mixed state showing magnetic flux penetrating thesample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Left: phase diagram for a type I superconductor. Right: phase diagram for a type IIsuperconductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Quality factor “Q” vs B for a niobium cavity from RF tests . . . . . . . . . . . . . . 122.1 Examples of spin precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Angular distribution of positrons from a muon decay . . . . . . . . . . . . . . . . . . 182.3 Simplistic example of positron detector orientation for asymmetry measurements usingµSR showing a muon counter, the sample, the muon polarization direction, magnetic fielddirection, the upward positron detector, and the downward positron detector. . . . . . 192.4 Asymmetry measurements taken from muons experiencing a 20mT field with a dampingfunction σ parameter of ∼0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Relaxation function for various correlation times . . . . . . . . . . . . . . . . . . . . 222.6 Transverse field relaxation function for various τ values . . . . . . . . . . . . . . . . 223.1 The resulting depth profiles from simulating muons with 3.6, 3.8, and 4.1MeV energy inTRIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Example of a transverse field µSR setup showing the silver mask, muon detectors, electronicclock, left and right positron detectors, the direction of muon momentum, muon spin, andapplied field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24vii3.3 Detector orientation for asymmetry measurements showing the muon direction to the right,polarization upward, two positron scintillators above the sample which make the upwarddetector, two positron scintillators below the sample which make the downward detector,the collimator furthest up the beamline, then the total muon counter, the silver mask, thesample muon counter and the sample. . . . . . . . . . . . . . . . . . . . . . . . . 254.1 Measurements of α at various fields fit to a linear function for the CF1 sample. Measure-ments were taken at T>TC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Example of how the zero-field asymmetry lowers as the applied field is increased due tofluxoids entering the center of the sample . . . . . . . . . . . . . . . . . . . . . . . 284.3 Example of zero field asymmetry plotted against applied field . . . . . . . . . . . . . . 294.4 Typical zero field µSR measurement of asymmetry vs time on a niobium sample . . . . 304.5 µSR measurement showing complete penetration of applied transverse field in a non mag-netic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 A depiction of how the component of relative asymmetry of the Kubo-Toyabe function inthe direction of the detectors changes as the cosine of the phase shift in the muons spin; forexample at 90 degrees the asymmetry signal would be zero even with no field penetrationinto the sample. The conditions of the experiment correspond to phase shifts of less than45 degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.7 Plot of zero field asymmetry vs applied field showing the affect of phase correction on µSRresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1 Geometry of a flat sample, a) showing the side view of the disc with the beam directionperpendicular to the circular surface and b) showing the front view of the disc . . . . . 345.2 Geometry of a formed sample, a) side view showing the curvature of the sample with respectto the beam direction b) front view showing the face of the sample perpendicular to thebeam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Geometry of an ellipsoid sample showing a) side view of the ellipsoid with the major axisparallel to the beam direction and the 8mm diameter beamspot indicated in blue. b) afront view of the ellipsoid with the 8mm diameter beamspot indicated in blue . . . . . . 36viii5.4 Picture showing a) front view of a flat sample, b) top view of a flat sample, c) front viewof a formed sample d) top view of a formed sample, e) front view of a ellipsoid sample, f)top view of a ellipsoid sample. Photo was provided courtesy of Seyd Haider Abidi . . . . 366.1 Magnetic field distribution in the midplane of a disc for increasing applied field (solidlines), and decreasing feild (dashed lines) starting from the center of the disc, x axis = 0,and moving radially outwards to the outer edge of the disc at r=a . . . . . . . . . . . 427.1 Zero field asymmetry measurements taken for different sample geometries and field orienta-tions plotted against the applied field. The red and black runs were taken in perpendicularfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.2 Zero field asymmetry measurements taken for different sample geometries and field orien-tations plotted against the applied field over the theoretical field of first flux entry for eachgeometry as calculated in section 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . 447.3 a) shows the zero field cooled µSR signal for the H1 sample with no mask at 0mT. b) showsthe zero field cooled µSR signal for the H1 sample with no mask at 40mT. (c) shows theearly time behaviour of (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.4 The image on the left shows the zero field cooled µSR signal for the H1 sample at 0mTusing the 8mm diameter mask. The image on the right shows the zero field cooled µSRsignal for the H1 sample at 40mT using the 8mm diameter mask. . . . . . . . . . . . 467.5 Relative asymmetry vs applied magnetic field for perpendicular field measurements of flatsamples with various treatments for comparison. The treatments being compared are basetreatment, BCP (100µm), BCP then baked at 120◦C, and BCP then baked at 120◦C thenHF rinsed for 5 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 Relative asymmetry vs applied magnetic field for perpendicular field measurements of theTR5 sample before and after being annealed at 1400◦C for 4 hours . . . . . . . . . . . 487.7 Relative asymmetry vs applied magnetic field for perpendicular field measurements of ni-trogen doped, Nb3Sn coated and untreated samples . . . . . . . . . . . . . . . . . . 487.8 relative asymmetry vs applied magnetic field for perpendicular field measurements of theannealed at 1400◦C, Nb3Sn plated, N doped, and 800◦C degassed samples . . . . . . . 49ix7.9 relative asymmetry vs applied magnetic field for perpendicular field measurements of formedsamples with various treatments for comparison. The treatments being compared were basetreatment, BCP (100µm), baked at 800◦C (degas), and annealed at 1400◦C . . . . . . . 507.10 relative asymmetry vs applied magnetic field for parallel field measurements of flat sam-ples with various treatments for comparison. The treatments being compared were basetreatment, Nb3Sn plated, nitrogen doped, and TR5 annealed . . . . . . . . . . . . . 517.11 Zero field asymmetry µSR results vs Applied field for two ellipsoid samples with the basetreatment described in chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.12 Zero field asymmetry µSR results vs Applied field for an ellipsoid sample with the basetreatment and with a roughened surface as described in chapter 5 . . . . . . . . . . . . 527.13 Zero field asymmetry µSR results vs Applied field for an ellipsoid sample with the basetreatment, degassed at 800◦C, high temperature baked at 1200◦C and annealed at 1400◦C 537.14 Zero field asymmetry µSR results vs Applied field for various surface treatments on ellipsoidsamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.15 Zero field asymmetry µSR results vs Applied field for the CB1 sample at various temperatures 55xAcknowledgementsI’d like to thank my two supervisors, Professor Rob Kiefl and Lia Merminga, without whomthis thesis would not be possible. Both have done more for me than can be expressed in thisparagraph. They are both not only very knowledgeable people, but they were patient withme and extremely understanding. I consider myself very fortunate to have had such goodsupervisors.Many people have contributed to the project and helped me in various ways throughoutthe course of this thesis; whether it be when something goes wrong while on experiment in themiddle of the night, preparing samples, or discussing measurements and results after shiftsare over. For helping out with all aspects of the project I’d like to thank Rob Kiefl and BobLaxdal. For help on shifts and discussing results I would like to thank Gerald Morris, BassamHitti, Andrew MacFarlane, Donald Arseneau, Rick Baartman, Philipp Kolb, Iain McKenzie,Martin Dehn, Suresh Saminathan, Haider Syed Abidi, Yanyun Ma, Zhongyuan Yao, YangLiu, and Shayan Gheidi. For fabricating and preparing samples, and for discussing interestingsample treatments I’d like to thank Bhalwinder Waraich, John Wong, James Keir, DevonLang, Anna Grassellino (N-dope), Matthias Liepe and Sam Posen (Nb3Sn). For generaldiscussion about results and related studies I’d like to thank Alex Romanenko.xi1 IntroductionSuperconducting radio frequency (SRF) cavities are frequently used in particle acceleration.The main advantage to using SRF cavities is that they have extremely low power dissipation.The low power dissipation of SRF cavities means that, even though cryogenics are neededto reach the superconducting state, less AC power is consumed in continuous wave or largeduty cycle acceleration [1]. SRF performance; however, typically falls short of it’s theoreticalcapabilities and various effects that alter the cavity performance are not well understood.Niobium (Nb) is a commonly used material in cavity construction. In this thesis variousniobium samples were examined using µSR as a local magnetic probe with the goal of findingout what cavity preparations best prevent early flux entry and associated heat generationresulting in a decrease in the quality factor for the cavity.1.1 Properties of a SuperconductorSuperconductivity was first discovered by H. Kamerlingh-Onnes in 1911 while taking lowtemperature resistance measurements of mercury, lead, and tin [2]. He observed the materialslose all of their DC electrical resistance below some critical temperature Tc. Since then,many superconducting materials have been discovered and all share the same remarkableproperties.When cooled below the critical temperature Tc, superconductors undergo a phase tran-sition most evident by the fact that their electrical resistance drops to zero [3] as illustratedin Figure 1.1.The heat capacity and entropy also change notably upon cooling below Tc. The entropyof all superconductors drops significantly indicating that the superconducting state is a1Figure 1.1: Electrical resistivity vs temperature of conductors and superconductors taken from [4]more ordered state [2]. Another property of a material in the superconducting state isthat it acts as a perfect diamagnet in the presence of a magnetic field below some criticalvalue. If an external field is applied to a superconductor it generates surface currents whichexactly cancel the applied field deep inside the material [2]. Also the defining feature of asuperconductor that distinguishes it from a perfect conductor is the Meissner effect, wherebythe magnetic field is expelled from the bulk of the sample when the material is cooled belowits superconducting transition temperature [3] as illustrated in Figure 1.2.There are two types of superconductor which are referred to as type I and type II. Intype I superconductors, when an applied field reaches some critical value Hc, the Meissnerstate is completely destroyed and the material is no longer in the superconducting state [2].Type II superconductors have two critical fields, a lower critical field Hc1, below which thematerial is in the Meissner state, and an upper critical field Hc2. Between Hc1 and Hc2the superconductor enters what is called the mixed or vortex state. In the mixed state themagnetic flux is present in the material in the form of a lattice of magnetic vortices. Eachfluxoid contains a quantized amount of flux, the higher the field, the more fluxoids enterthe material [5]. Above Hc2 the superconductivity is completely destroyed and the materialbecomes a normal conductor. A phase diagram for type I and II superconductors can beseen in Figure 1.4.2Figure 1.2: The top image shows a superconductor in the normal state in the presence of an appliedmagnetic field, the bottom shows a superconductor in the Meissner state in the presence of an appliedmagnetic fieldFigure 1.3: Illustration of a superconductor in the mixed state showing magnetic flux penetrating thesample3Figure 1.4: Left: phase diagram for a type I superconductor. Right: phase diagram for a type II super-conductor.41.2 Theory of SuperconductivityIn order to theoretically model the Meissner effect, Fritz and Heinz London proposed that, inthe superconducting state, the vector potential of a local magnetic field A (where B=∇×A)was proportional to the current density j [2, 6].∇× j = − 1µ0λ2LB (1.1)Where λL is a parameter called the London penetration depth. Using the Maxwell equation:∇×B = µ0j (1.2)then taking the curl of each side∇×∇×B = µ0∇× j (1.3)using the vector identity∇×∇×B = ∇ (∇ ·B)−∇2B (1.4)∵ ∇ ·B = 0 (1.5)∇×∇×B = −∇2Band substituting − 1µ0λ2LB for ∇× j from (1.1)yields:∇2B = Bλ2L(1.6)The solution to which shows that in the Meissner state the magnetic field decays exponen-tially into the material away from the surface at z=0 within a thin surface layer.B (z) = B (0) e−z/λL (1.7)Where B (z) is the field at depth z and the London penetration depth can be calculatedusing the following formula.λL =√0mc2nq2(1.8)Where n is the concentration of electrons with effective mass m and charge q.5Later, Bardeen, Cooper and Schrieffer (BCS) proposed a quantum theory of supercon-ductivity in their seminal 1957 paper [7]. In BCS theory the occurrence of superconductivityis attributed to the condensation of conduction electrons causing the formation of correlatedpairs (Cooper pairs) below Tc. Roughly speaking, the attractive interactions between theelectrons is caused when the first electron interacts with the lattice of a material and de-forms it. The deformation in the lattice behind the first electron leaves a slightly positiveregion which makes it energetically favourable for the second electron to follow the first. TheCooper pairs can be thought of as a single entity with double the charge and mass of a singleelectron and with a pairing energy of 2∆. The resulting density of states has an energy gap,of width ∆, at the fermi energy. The width of the energy gap is temperature dependent andscales with the critical temperature. Within the weak coupling limit, the zero temperatureenergy gap is given by the following formula [3].∆ (0) = 1.76kBTC (1.9)Note, this energy gap is of a different nature than the energy gap of an insulator or semicon-ductor. In insulators the periodic potential exerted on the electrons from the ions is whatgives rise to energy bands and the energy gap between the valance band and conduction band.In superconductors the energy gap is the result of electron-electron interactions (which inlow Tc superconductors originates from the electron phonon interaction) that orders theelectrons in k space [2].The approximate range over which the electron-electron interactions of superconductingelectrons occurs is called the coherence length, ξ0 [8].ξ0 =2~νFpi∆(1.10)Where νF is the Fermi velocity. A ratio of the London penetration depth to the coherencelength, called the Ginzburg-Landau parameter κ, indicates whether a superconductor is typeI or type II [9].κ =λLξ(1.11)If κ< 1√2, the superconductor is type I. If κ> 1√2the superconductor is type II.61.3 Radio Frequency Cavity FundamentalsRadio frequency (RF) cavities, or resonators, are used to transfer energy to a beam ofcharged particles by providing an oscillating electric field [10]. The cavity is constructedso the magnitude of the field varies along the length of the cavity in such a way that theparticles are always gaining energy as they traverse the cavity and are thus accelerated. Thesingle most important measure of the effectiveness of a cavity is how large of an acceleratinggradient it can generate. Accelerating gradient is defined as the amount of energy gained perunit length (1.12) [9]; the larger the accelerating gradient, the smaller the cavity needs tobe to achieve the same acceleration and the lower the operating cost of the cavity, all otherthings being equal.Eacc =Vaccd(1.12)Using the z axis as the axis of symmetry for a given cavity, Vacc is found using (1.13)Vacc =∣∣∣∣∫ d0Ez (z) eiωzc dz∣∣∣∣ (1.13)where ω is the mode of angular frequency at which the cavity resonates. Energy; however,can be dissipated in the cavity wall. This dissipation is characterized by the surface resistanceRS [10].dPds=12RS |H|2 (1.14)where P is power dissipation, s is surface area, and H is the amplitude of the local magneticfield. A common way of characterizing the RF cavity losses is via the quality factor, Q0which is defined as 2pi times the energy stored in the cavity, U , over the energy dissipatedin the walls per oscillation [9].Q0 =ω0UP(1.15)Where U is given by (1.16) below:U =12µ0∫V|H|2 dV (1.16)rearranging (1.14) to get:P =12∫sRS |H|2 dS (1.17)7Q0 becomes:Q0 =ω0µ0∫V|H|2 dV∫sRS |H|2 dS(1.18)Where RS has been kept inside the integral since it may have some field dependence whichmeans the value of Q0 needs to be solved numerically. In the case of low fields, RS can beapproximated to have no field dependence and Q0 can be simplified to an analytic equation:Q0 =ω0µ0∫V|H|2 dVRS∫s|H|2 dS (1.19)=GRS(1.20)Where the top integral is taken over the volume V , the bottom integral is taken over thesurface area S, and G is a geometric factor and is dependant solely on the shape of thecavity.1.4 SRF FieldsRF/SRF cavities use time dependant electromagnetic fields to transfer energy to the chargedparticles and accelerate them. The beam is pulsed so that the particles only experience theaccelerating field in one direction. When the field switches, the particles are in a region ofnegligible field [10,11].The distribution of electric and magnetic fields in a cavity are governed by Maxwell’sequations listed in (1.21).∇ · ~E = ρ0(1.21)∇ · ~B = 0∇× ~E = −∂~B∂t∇× ~B = µ0 ~J + µ00∂~E∂t8Combining Maxwell’s equations and using vector identity (1.4)∇×∇× ~E = ∇×−∂~B∂t∇(∇ · ~E)−∇2 ~E = ∂∂t(µ0 ~J + µ00∂ ~E∂t)Were µ00 = 1/c2 and ~J = ρ = 0, which gives:∇2 ~E − 1c2∂ ~E∂t= 0 (1.22)similarly it can be shown that∇2 ~B − 1c2∂ ~B∂t= 0 (1.23)which gives the wave equation (1.24).(∇2 − 1c2∂2∂t2){EH}(1.24)A good approximation to make when solving this equation for a niobium cavity is to assumethat it behaves as a perfect conductor, meaning it needs to satisfy the following boundaryconditions [5].nˆ× E = 0 ; nˆ ·H = 0 (1.25)Where nˆ is the normal vector to the cavity surface. The wave equation (1.24) has threefamilies of solutions, one is called transverse electric (TE), the second is transverse magnetic(TM) [5, 10]. In TE modes, the solution yields no longitudinal component to the electricfield. In a transmission line the “longitudinal” direction is the direction of propagation. In acavity a standing wave is generated which can be considered a wave moving back and forthalong the cavity direction (longitudinal direction). In TM modes there is no longitudinalcomponent to the magnetic field. A third family of solutions exists called TEM modes inwhich there is no longitudinal magnetic or electric field components; however, it exists onlyfor specific cavity geometries.Numerical methods are typically required to solve the wave equation subject to complexboundary conditions imposed by the walls. Cavities provide the electric fields used to accel-erate particles by sustaining standing electromagnetic waves, thus a standing wave solution issought. Since an electric field is also usually required to be on axis for accelerating purposes,the lowest TM eigenmode is normally chosen.91.5 Superconducting RF ResistancesAs stated in section 1.1, a material in a superconducting state exhibits a zero DC electricalresistance. This can be understood by considering a two fluid model, one fluid being a super-fluid of electrons in Cooper pairs, the other a normal fluid with “free” electrons. The fractionof normal conducting electrons is given by the Boltzmann factor e−(∆/kbT ) [3]. Provided thatthe thermal energy of the phonons remains under the 2∆ required to break the Cooper pair(T < TC), the supercurrent flows with zero DC resistance parallel to the normal fluid, thusall the current flows through the superfluid, shielding the normal electrons from the electricfield. With RF currents; however, the superconductor has a finite resistance. Time varyingfields prevent the Cooper pairs, which have inertial mass, from fully shielding the normalelectrons causing dissipation. The induced current in the normal fluid of electrons is givenby (1.26) [5].jn ∝ nnE ∝ nndHdt∝ nnωH (1.26)Where E is the time varying electric field acting on the normal fluid due to incompletescreening by the superfluid induced by the time varying magnetic field and nn is the densityof normal conducting electrons. The dissipated power is then given by (1.27).P ∝ jnE ∝ nnω2H2 (1.27)Power dissipation caused by normal electrons can be characterized by a surface resistance.Substituting P from (1.14) gives an expression for Rs.Rs ∝ nnω2 ∝ ω2e−(∆/kbT ) (1.28)Normal electrons are those that are excited above the energy gap and, due to the Boltzmanfactor involving the energy gap, the density falls exponentially. BCS theory expands on thisrelation and can be used to calculate the surface resistance based on various superconductingparameters [10].RBCS = A(1T)σnλ3f 2 exp(∆(T )kbT)(1.29)Where f is frequency, λ = λL√0/ is the effective penetration depth, σn is the normalconductivity and ∆(T ) is the temperature dependant energy gap. An approximation of the10BCS resistance of niobium for T < TC/2 and f < 1012Hz can be seen below.RBCS (Ω) ∼ 2× 10−4 1T(f (GHz)1.5)2exp(−17.67KT)(1.30)The temperature dependant BCS resistance is not the only thing that contributes to thesurface resistance, Rs. Rs is a combination of BCS resistance and a residual resistance dueto impurities, trapped magnetic flux, and defects.Rs = RBCS +Rres (1.31)Though BCS resistance is a theoretical resistance based on frequency, temperature, and theresulting density of normal electrons; the residual resistance is a temperature independentresistance based on impurities and defects and can be affected by cavity preparation.1.6 Why Niobium is Used for SRF CavitiesDue to their extremely low surface resistance, superconducting RF cavities are advantageousin continuous wave or long-pulse acceleration with large electric fields (over a million voltsper meter). This is because the ohmic power loss in the cavity walls increases as the squareof the accelerating voltage making regular conductors uneconomical [1]. Several factorsare taken into account when choosing the type of superconductor best suited for particleacceleration. It needs to have minimal surface resistance while providing as much tolerancefor inherent material flaws as possible. From equation (1.30), materials with high ∆ and lowλ have a lower BCS surface resistance. Furthermore, materials with large coherence lengthsare less susceptible to defects, since defects smaller than the coherence length only weaklyeffect the surface resistance. Niobium is a good choice for building SRF cavities since it has arelatively short magnetic penetration depth (32-44nm), a long coherence length (30-60nm),and has the highest Tc of the elemental superconductors (9.2K) [5]. It is also easier to makea complex structure out of a simple element or metal than out of a compound. The surfacechemistry with air and water must also be stable since it is impractical to keep the cavity inUHV making niobium an ideal choice.The purity of niobium used in the production of SRF cavities is quantified by it’s “residualresistivity ratio” (RRR) [5]. The RRR value is a ratio of the resistivity of a sample at room11temperature over the resistivity at 4.2K, where the resistivity at 4.2K is measured above HC2so the sample is not in the superconducting state. This measures the change in resistancedue to impurity scattering since phonon and electron-electron interactions drop as 1/T 3 and1/T respectively, RRR is a purity measurement.RRR =ρ300Kρ4.2K(1.32)An acceptable purity of niobium used in cavity construction typically has a RRR value of200-300.1.7 Motivation for the ExperimentThe performance of high purity (RRR300) niobium SRF cavities degrades due to RF fielddependant losses in the cavity walls. The field dependence of the losses is characterized bythe quality factor Q of the cavity. The degradation in quality due to surface resistance isreferred to as Q-slope as seen in Figure 1.5 [5].Figure 1.5: Quality factor “Q” vs B for a niobium cavity from RF tests taken from [12]The overall behaviour of Q versus magnetic field is not well understood. In low fieldthe quality factor raises when the field is increased, at medium field it slowly drops with12increased field and then at high field the quality factor drops abruptly. A 120◦C bake isknow to reduce the high field drop off but it is uncertain why.In a recent study [12], a correlation was revealed between the behaviour of Q versusBpeak and the DC field penetration measured with µSR when various surface treatmentswere applied to the niobium. µSR uses spin polarized muons to probe the local magneticfields within the bulk of a sample. Since then, many studies have been done using µSR toattempt to find what mechanism is responsible for high field Q-slope, to find the optimummethod of preparing SRF Cavities, and to find new materials to coat the cavity with toimprove performance.µSR is used for several reasons. µSR is the best way to probe the local internal magneticfields of a sample, making it a good method of investigating flux entry. Samples used in µSRcan also be small (2cm diameter) so it doesn’t require a full cavity to be constructed andtreated for each test. µSR can also be used to probe specific regions of the sample (e.g thecenter) and it provides more information than a magnetometer.In this study, µSR was used to examine various fine grain RRR 300 niobium sampleswhich had undergone different surface and heat treatments to determine what effect eachhad on the µSR results. These results were used to try to find the cause of loss in the cavitiesin order to determine which future treatments could have the greatest positive affect on theaccelerating gradient of niobium SRF cavities.To understand the effect that surface treatments have on niobium, and in turn deter-mine which method of sample preparation might best benefit SRF cavities, different samplegeometries were also used to try to minimize the demagnetization factor of the samples onµSR results and better approximate magnetic flux as seen by a cavity.132 µSRMuons are fundamental particles with spin 1/2 which are closely related to electrons (fornegative muons) or positrons (for positive muons). Muons, like positrons or electrons, areleptons and as such don’t interact strongly; however, muons are unstable and have decayproperties that make them useful as a sensitive probe of magnetic fields in matter [13, 14].Muons are typically created from pion decays which, due to maximal parity violation in weakinteractions, are created in a helicity eigenstate. For a positively charged muon the spin is100% polarized anti-aligned to the momentum direction in the rest frame of the pion nearthe surface of the production target [15]. The spin polarized positive muons are transporteddown a beam line and stop at interstitial sites within a sample. The muon then precessesabout the local magnetic field and decays into a positron and two neutrinos such that thepositron is emitted preferentially in the direction of its spin [15]. The asymmetry betweentwo opposing detectors is a direct measure of how the muon polarization evolves in thesample, which is determined by the local internal magnetic field.2.1 Muon ProductionIn order to create muons, a high energy proton beam is fired into a production target madeof a light element such as carbon or beryllium, etc. [14, 15]. The collision with a proton inthe target nucleus produces a pion via the following reaction:p+ + p+ → pi+ + p+ + n (2.1)The pion is a short lived meson (τpi ∼ 26ps) and undergoes a parity violating decay fromrest in the surface layer of the target, emitting a muon and a neutrino [13, 15]. The kinetic14energy of the emitted muon is about 4.2MeV in the rest frame of the pion.pi+ → µ+ + νµ (2.2)Since the pion decay is a parity violating decay, the muons produced are nearly 100% spinpolarized if one selects muons emitted in a particular direction.2.2 Muon SpinAs mentioned previously, muons are spin 1/2 particles. The magnetic moment of a spin 1/2muon is given by (2.3).µˆB = ge2mµSˆ (2.3)where g is a dimensionless number called the g factor (for muons g∼2) [16], e is the magnitudeof the charge of an electron, and mµ is the mass of a muon. The Zeeman energy for a muonin an applied field:Hˆ = µˆB ·B = g e2mµSˆ ·B (2.4)For a static, uniform magnetic field applied in the z direction the energy becomes (2.5).Hˆ = ge2mµSˆzBz = ωSˆz (2.5)Where ω is defined in such a way that the difference in the two energy eigenvalues is ~ω.This is the phase factor which has to be multiplied by the initial state to get the time evolvedstate.exp[−iEˆt~]= exp[−iωSˆzt~](2.6)The time dependant state for a muon which started in the |Sx+ > state (Sx = ~/2), andwas placed in a perpendicular magnetic field in the z direction can be written as a linearcombination of the two possible energy eigenstates:|Sx±〉 = C1(t) |Sz+〉 ± C2(t) |Sz−〉 (2.7)Where C1(t) and C2(t) are the time dependent coefficients. At time t = 0, for the muon inthe |Sx+ > state, C1(t) and C2(t) are 1/√2 giving an initial state:|Sx+; t = 0〉 = 1√2|Sz+〉+ 1√2|Sz−〉 (2.8)15usingSz |Sz±〉 =(±~ω2)|Sz±〉 (2.9)and the time evolution given by (2.6), C1(t) and C2(t) can be written as:C1(t) =1√2exp[−iωt2]C2(t) = exp[iωt2](2.10)The time evolution of the |Sx+ > state becomes|Sx; t〉 = 1√2exp[−iωt2]|Sz+〉+ 1√2exp[+iωt2]|Sz−〉 (2.11)For a muon initially in the |Sx+ > state, the probability of it being in the |Sx+ > state atsome later time t becomes:|〈Sx ± |Sx+; t〉|2 =∣∣∣∣[( 1√2)〈Sz+| ±(1√2)〈Sz−|]·[(1√2)exp(−iωt2)|Sz+〉+(1√2)exp(+iωt2)|Sz−〉]∣∣∣∣2 (2.12)Due to orthogonality 〈Sx + |Sx−〉 = 〈Sx − |Sx+〉 = 0 and equation (2.12) becomes:|〈Sx ± |Sx+; t〉|2 =∣∣∣∣12 exp(−iωt2)± 12exp(+iωt2)∣∣∣∣2 (2.13)Using the trig identities:12exp (ix) +12exp (−ix) = cos(x)12exp (ix)− 12exp (−ix) = sin(x)The probability of the muon spin being in the Sx+ state is therefore:Px+ = cos2 (ωt/2) (2.14)and the probability of the muon being in the Sx− state is:Px− = sin2 (ωt/2) (2.15)the expectation value of Sx at some time t is then:〈Sx〉 = (Sx+)Px+ + (Sx−)Px−=(~2)cos2(ωt2)+(−~2)sin2(ωt2)(2.16)=(~2)cos (ωt)16This shows that, in the case where the muon spin is perpendicular to the applied field (Sxvs Bz), the expectation value of Sx will oscillate at a frequency proportional to the appliedmagnetic field. Using the muon values from [16] to solve for ω/B, a ratio between the fieldexperienced by the muon and the frequency with which it oscillates can be found.ωB=2pifB= ge2mµ(2.17)= 8.5× 108 Ckgrad= 8.5× 108T−1s−1radfB=8.5× 108T−1s−12pi= 1.35× 108T−1s−1= 13.5kHzG= γµ (2.18)Where γµ is referred to as the gyromagnetic ratio and can be used to determine the field themuon experienced based on the rate at which the spin oscillates.Figure 2.1: Examples of spin precession172.3 Muon DecayA muon has a lifetime of τµ = 2.2µs and decays at time t with a probability proportional toe− tτµ . A muon decays in a parity violating three body decay emitting a fast decay positron(or electron in the case of a negative muon) and two neutrinos, as seen below [13].µ+ → e+ + νe + ν¯µ (2.19)Due to the fact that the decay is a parity violating decay, the muon emits the fastdecay positron preferentially along the polarization direction. Each individual muon emits apositron at an angle θ from the direction of the muon spin with the following probability [15]:P (θ) = 1 + acos (θ) (2.20)Where a is the asymmetry parameter. The asymmetry parameter is highly dependanton positron energy, being unity for the highest positron energy and averaging to ∼0.3 overall energies. In experiments, the asymmetry parameter is limited by the configuration andefficiency of the detector set-up and the average a typically ranges from 0.2-0.3. The followingimage from [15] shows how the anisotropic distribution of positrons is affected by a.Figure 2.2: Angular distribution of positrons from a muon decay taken from [15]The statistical average direction of the muon spin for a group of muons can be obtainedfrom the anisotropic distribution of the decay positrons.182.4 AsymmetrySince the muon decays preferentially in the direction of its spin, the spin direction and it’stime evolution can be determined through measuring the time and direction of the positronemissions over a large number of muons. The signal produced and analysed during µSRmeasurements is called an asymmetry signal. In a simple case for µSR, a sample lies betweentwo oppositely oriented positron detectors.Figure 2.3: Simplistic example of positron detector orientation for asymmetry measurements using µSRshowing a muon counter, the sample, the muon polarization direction, magnetic field direction, the upwardpositron detector, and the downward positron detector.The positron detectors, called up and down detectors in Figure 2.3, detect and count thenumber of muon decays that pass through them. The asymmetry is given by:Asy =U −DU +D(2.21)Where U is the number of counts in the up detector and D is the number of counts in thedown detector. The asymmetry can be affected by solid angles subtended by each detectoras well as the differences and inefficiencies in the detectors and so, in practice, the measuredasymmetry is given by:Asy =U − αDU + αD(2.22)19Where α is a coefficient used to account for differences in detectors design and/or efficiencyand is found through fitting data [17]. For simplicity we assume the asymmetry parameter“a” for each detector is the same.The muon spin relaxes due to interactions with magnetic and/or nuclear fields. In thecase of the muons experiencing an applied transverse field, where a “transverse field” refers tothe fact that the applied field is perpendicular to the muon spin, the positron time spectrumtakes the general form given in (2.23) from [18].Nx = N0 exp (−t/τµ)× [1 + AGx (t) cos (ωµt+ θ)] (2.23)Where θ is the “phase shift”, or angle by which the muon spin is offset from pointing directlyat the up detector at time t = 0. Gx (t) is the envelope of the damping of the Larmorprecession and is referred to as the transverse field relaxation function. The relaxationfunctions are derived in [18] and for the high field static limit, the transverse relaxationfunction is:Gx (t) = exp(−σ2t2) (2.24)Where σ is the line width of a Gaussian damping function and is related to the secondmoment of the field by the following:σ2/γ2µ =< H2x >=< H2y >=< H2z > (2.25)so for transverse fieldsGx (t) = exp(< H2x > σ2µt2)(2.26)An example of how the asymmetry between the U and D detectors evolves as a functionof time, in a magnetic field of 20 mT with σ ∼0.8, is given in Figure 2.4.In the case of an applied longitudinal field, where a “longitudinal field” means the ap-plied field is parallel to the muon spin, the time histogram for a detector along the initialpolarization direction is:Nz (θ, t) = N0 exp (−t/τµ)× [1 + AGz (t) cos θ] (2.27)where Gz (t) is the longitudinal relaxation function. The case where there is zero appliedfield can be treated as a special case of the longitudinal applied field. Assuming the field is20Figure 2.4: Asymmetry measurements taken from muons experiencing a 20mT field with a dampingfunction σ parameter of ∼0.8static at each muon site and the distribution of each component of the field is gaussian witha width sigma, the zero field relaxation function is given by (2.28) [18].Gz (t) =13+23(1− σ2t2) exp(−12σ2t2)(2.28)This function is called the static Kubo-Toyabe function and it gives a 1/3 recovery in theasymmetry, which makes sense for static random fields since the projection of the muon spinon the x axis averages to 1/3. In some cases; however, the muons can hop from site to site ina material or the fields within a sample itself may fluctuate and the muons do not experiencea static field. This gives rise to a dynamic Kubo-Toyabe function shown in figure 2.5, whereτ in figure 2.5 is the correlation time for internal dipolar fields experienced by the muons tochange. This is discussed further in section 4.3.The hop rate also affects the transverse field relaxation function, Figure 2.6 shows thetransverse field relaxation function for various correlation times τ .21Figure 2.5: Relaxation function for various correlation times taken from [18], where once again ∆ = σ isline widthFigure 2.6: Transverse field relaxation function for various τ values taken from [18], ∆ in this image is linewidth and is equivalent to σ223 Experimental ProcedureµSR measurements were taken using TRIUMF’s M15 and M20 muon beamlines, whichdeliver a beam of positively charged muons with a mean energy of about 3.8 MeV intoa sample. The stopping profile (see Figure 3.1) corresponds to a mean depth that is muchlonger than the London penetration depth; meaning the muons stop well beyond the Londonpenetration depth and into the bulk of the sample [15,17]. Results from a simple TRIM [19]simulation of the expected penetration depth can be seen in Figure 3.1.muon depth (um)0 50 100 150 200(muons/cm^3)/(muons/cm^2)01020304050607080903.8 MeV/c4.1 MeV/c3.6 MeV/cMuon Depth ProfileFigure 3.1: The resulting depth profiles from simulating muons with 3.6, 3.8, and 4.1MeV energy in TRIMThe muon beam spot has a Gaussian distribution with a σ of approximately 5mm [12].The muons travel down the beamline and pass through an incoming muon counter whichstarts a precision electronic clock. The muons then pass through a silver mask with an 8mmdiameter hole in the centre, used to restrict the measured muons to the centre of the sample.Muons which do not pass through the hole stop in the silver mask. Muons which passthrough the hole in the silver mask are detected with a second muon counter before entering23and stopping in the sample. These are defined as “sample muons” ie muons that stop in thesample. Figure 3.2 shows the set-up with one small difference; the positron detectors used inthis particular experiment were in the x direction for the given coordinate system, not the ydirection.Figure 3.2: Example of a transverse field µSR setup showing the silver mask, muon detectors, electronicclock, left and right positron detectors, the direction of muon momentum, muon spin, and applied field takenfrom [15]The muons that enter the sample stop at interstitial sites and act as local probes forthe internal magnetic field. Two different set-ups were used in the experiment. The twoset-ups were a “perpendicular field” set-up and a “parallel field” set-up. Both set-ups wereused to perform transverse field µSR measurements, meaning the applied magnetic fieldin each set-up was perpendicular to the muon spin. In the perpendicular field set-up theLAMPF spectrometer was used to apply a DC magnetic field parallel to the direction of themuon beam and perpendicular to the circular surface of the flat and formed samples. In the‘parallel field” set-up, an iron dipole magnet was used to apply a field perpendicular to the24muon velocity and spin, but parallel to the flat surface of the disc samples. A second irondipole magnet (farthest downstream), was used to compensate for the deflection caused bythe stray field of the first dipole magnet so that the muons would stop in the sample.In both set-ups, muons enter the sample region with their spins initially aligned upwardspointing towards the Up telescope, which consisted of two plastic scintillators measuring20cm long, 7.5cm wide and located ∼4cm above the sample center. Good positron eventscause a coincidence between the two detectors. In this way the two scintillators acted as asingle upward decay detector and are henceforth referred to as the up detector (U). A secondset of positron scintillators , the down detector (D), with the same dimensions were located∼4cm below the sample center. Figure 3.3 shows the general set-up and gives a depiction ofthe detector locations relative to the beam direction.Figure 3.3: Detector orientation for asymmetry measurements showing the muon direction to the right,polarization upward, two positron scintillators above the sample which make the upward detector, twopositron scintillators below the sample which make the downward detector, the collimator furthest up thebeamline, then the total muon counter, the silver mask, the sample muon counter and the sample.When a decay positron passes through the up or down detector the electronic clockstarted by the initial muon counter stops, giving the time of the decay with a resolution onthe order of 1ns [15, 17]. The counts from the up and down detectors were binned in timebins with a width of 390ps and used to find the asymmetry signal to interpret the data.254 Data AnalysisThe µSR signal produced was analysed to determine the fraction of muons that did notexperience any applied magnetic field. This signal is referred to as the zero field asymmetrysignal and, for the purpose of this experiment, was analysed using a program called musrFit[20]. In order to obtain meaningful results, the α parameter of the detector set-up, mentionedin section 2.4 was needed for each sample.4.1 Setting αAs shown in section 2.2, if a muon experiences a magnetic field which is transverse to it’s spin,the spin precesses. This precession takes the form of a Gaussian damped cosine. Knowingthis, the alpha of the detector arrangement was found by using the µSR signal from a runtaken in the normal state (the non-superconducting state, where there is no expulsion of fieldfrom the bulk of the sample) with an applied transverse field. α is determined by requiringthat the time dependant asymmetry function (Asy(t)) for the two opposing detectors (givenby Eqn 2.22) oscillates about zero. Early α calibrations were taken at various applied fieldsfrom 20mT to 180mT.For perpendicular field runs, the fitted α values had little variance (average standarddeviation of less than 0.3%) at each field. α was therefore found to be independent of field.The variance was small enough on the initial sample runs that only a few α calibrations wereneeded to determine an accurate α for the remaining samples. Once the α was found fora given sample set-up, it was fixed and used to fit the zero field asymmetry signal in fieldcooled µSR runs.For parallel field runs, the path of the positrons (emitted by muon decays) in the direction26of the detectors were affected by the field, due to the orientation of the detectors with respectto the magnetic field. This means that α is affected by the field when taking parallel fieldmeasurements. To account for this, α calibrations were taken at various applied fields in thenormal state of niobium. The fitted α was consistent with a linear function of field as seenin Figure 4.1. The linear function was then used to determine α when taking data belowTC .Figure 4.1: Measurements of α at various fields fit to a linear function for the CF1 sample. Measurementswere taken at T>TC4.2 Zero Field AsymmetrySince, at zero field, the internal random field distribution in niobium results in a µSR signalthat fits a Dynamic Kubo-Toyabe function (DKTF); the amplitude of the zero-field KuboToyabe signal represents the fraction of muons that see essentially no applied field. Whenthe applied transverse field begins entering the sample, the amplitude of the Kubo-Toyabesignal will drop as the muons begin precessing with the applied field.In order to find out when the field starts penetrating the sample, the sample is cooled to27Figure 4.2: Example of how the zero-field asymmetry lowers as the applied field is increased due to fluxoidsentering the center of the sample∼2K in zero applied field. Measurements that are taken after cooling in zero applied field arecalled zero field cooled (ZFC). Once the sample is cooled in zero field, µSR measurements aretaken at zero field. This establishes a baseline for no field penetrating the sample. Once azero field measurement is made, a small field (20-40mT) is applied and another measurementis taken (this is still a ZFC measurement since the sample has still been cooled with no appliedfield, the field is only applied after the sample is at ∼2K). The field is incremented until thezero-field signal drops off to zero. Figure 4.2 shows for asymmetry measurements taken fromone ZFC sample at four different applied fields. The resulting amplitude of the zero fieldasymmetry is then plotted against the applied field as seen in Figure 4.3.The field at which the zero-field asymmetry starts to fall as well as the location it dropsto zero tells us when the magnetic field starts to penetrate and when no muons see zerofield. We can use this information to make deductions about what treatments impact fieldpenetration and pinning in samples (pinning is described in section 6.2).28Applied Magnetic Field (mT)0 50 100 150 200Relative Asymmetry00.20.40.60.81Zero field asymmetry plotted against applied fieldRelative AsymmetryFigure 4.3: Example of zero field asymmetry plotted against applied field4.3 Curve FittingThe musrfit program developed at PSI [20] was used to fit the µSR data. The programfits a 4 parameter dynamic Kubo Toyabe function (DKTF) of the following form using χ2minimization:12pii∫ γ+i∞γ−i∞fG (s+ Γ)1− ΓfG (s+ Γ) exp (st) ds (4.1)Where fG (s) is given by:fG (s) =∫ ∞0G (t) exp (−st) dt (4.2)And G (t) is the static longitudinal applied field KTF:G (t) = 1− 2σ2(2piν)2[1− exp(−12σ2t2)cos (2piνt)]+2σ4(2piν)3∫ t0exp(12σ2τ 2)sin (2piντ) dτ (4.3)With the 4 parameters being Γ which represents hop rate of muons in the sample. Hoprate is the inverse of the correlation time for muons to hop from site to site, or for the29internal dipolar fields to fluctuate meaning Γ is 1/τ described in section 2.4. σ is the widthof a Gaussian damping of the signal. ν is a frequency which is proportional the appliedlongitudinal field, since there is no longitudinal applied field in this experiment ν is fixed at0 when fitting data and the static KTF simplifies to equation 2.28 in section 2.4. The otherparameter which this formula is fit to is the initial asymmetry at time t = 0 [20]. A typicalzero field asymmetry fit can be seen in figure 4.4.Figure 4.4: Typical zero field µSR measurement of asymmetry vs time on a niobium sampleAs more magnetic flux enters the sample fewer muons experience just the weak randomfield distribution from nuclear moments resulting in a drop in the relative amplitude of theDKTF. At high transverse fields, when the field has penetrated the sample the zero fieldKTF asymmetry drops to zero and the muons precess at a high frequency close to the Larmorfrequency of the applied field. The asymmetry is then described by a sinusoidal oscillationdampened by a Gaussian distribution of the form:exp(−12(σt)2)cos (ωt+ φ) (4.4)30Figure 4.5: µSR measurement showing complete penetration of applied transverse field in a non magneticmaterial4.4 Phase CorrectionThe initial results showed an immediate drop in zero field asymmetry when even small(∼40mT, well under Hc1 of 190mT) fields were applied. This is because the muons spinprecesses in the stray field of the magnet before reaching the sample. Since the muons startto precess, their spins no longer point directly at the up detector when they enter the sample,resulting in a phase shift. The frequency with which the muon spin precesses increases withfield and corresponding stray field, meaning the phase shift varies in an approximate linearfashion with applied field.This means that the component of the polarization directed along the Up detector whenthe muons enter the sample decreases with increasing magnetic field, independent of any fluxpenetration. This appears as a phase shift which scales with the applied magnetic field. Tocorrect for phase shift, µSR measurements were taken at different fields when the samples31Figure 4.6: A depiction of how the component of relative asymmetry of the Kubo-Toyabe function inthe direction of the detectors changes as the cosine of the phase shift in the muons spin; for example at 90degrees the asymmetry signal would be zero even with no field penetration into the sample. The conditionsof the experiment correspond to phase shifts of less than 45 degrees.were in the normal state (sample temperature was above Tc). Above Tc the field fullypenetrates the samples and the µSR asymmetry oscillates at the Larmour frequency. Thechange in the phase of the oscillating signal is due to spin precession in the stray magneticfield before the muons reach the sample. By taking normal state data at several fields, thephase shift per unit field was measured. The observed asymmetry for a run was then dividedby the cosine of the measured phase shift in order to find the true maximum amplitude ofthe zero field signal at any given applied field. See Figure 4.7 for an example of how thephase correction effected the results.32Applied Magnetic Field (mT)0 50 100 150 200Relative Asymmetry00.20.40.60.81Effect of phase correctionRelative AsymmetryCB1T2K - Nb3Sn coated run @2.5KCF1T2K - Nb3Sn coated flat sample run @2K no phase correctionFigure 4.7: Plot of zero field asymmetry vs applied field showing the affect of phase correction on µSRresults335 SamplesAll of the samples used in this experiment were composed of fine grain RRR300 niobium.Various surface treatments commonly used in SRF cavity preperation where used on thesamples to see how they affected the µSR results. There were also three different geometriestested in order to see the effect of sample shape on the µSR results.5.1 Sample GeometryWe first discuss the three different sample geometries used in the µSR experiment. Onegeometry was a flat disc 20mm in diameter and 3mm thick. The flat disc samples were cutfrom a stock sheet of RRR300 niobium using a water jet and are henceforth referred to as“flat” samples. See Figure 5.1 below.Figure 5.1: Geometry of a flat sample, a) showing the side view of the disc with the beam directionperpendicular to the circular surface and b) showing the front view of the discOther samples were cut from a formed SRF cavity using wire EDM resulting in a disc34that is slightly curved toward the beam direction. These samples are referred to as “formed”samples.Figure 5.2: Geometry of a formed sample, a) side view showing the curvature of the sample with respectto the beam direction b) front view showing the face of the sample perpendicular to the beamThe final geometry used was an ellipsoid which, in Cartesian co-ordinates, is describedby the equation (5.1)x2a2+y2b2+z2c2= 1 (5.1)The ellipsoidal samples were prolate ellipsoids (a = b < c) and they were chosen to limitthe effect of the geometry on the penetration of applied field by reducing the demagnetizationfactor of the sample (discussed in further detail later in section 6.1). The larger the c:a ratioof the ellipsoid the smaller it’s demagnetization factor; however, the field applied by theLAMPF spectrometer is only homogeneous over a finite length. The ellipsoids were createdso that their two minor axes were 9mm long, and their major axis was 22.9mm in order togive a sufficient demagnetization factor while keeping the exposed 8mm diameter portionwithin the homogeneous region of the field (∼2mm deviation in the z axis).On the back of the ellipse, along the major axis there is a 21mm deep 1/4” threaded holewhich was used to secure the ellipsoid samples to the sample holder. A picture of the threedifferent geometries can be seen in Figure 5.4.35Figure 5.3: Geometry of an ellipsoid sample showing a) side view of the ellipsoid with the major axisparallel to the beam direction and the 8mm diameter beamspot indicated in blue. b) a front view of theellipsoid with the 8mm diameter beamspot indicated in blueFigure 5.4: Picture showing a) front view of a flat sample, b) top view of a flat sample, c) front view of aformed sample d) top view of a formed sample, e) front view of a ellipsoid sample, f) top view of a ellipsoidsample. Photo was provided courtesy of Seyd Haider Abidi5.2 Sample TreatmentsStarting from a standard baseline, the samples were prepared using different methods tosee how each treatment influenced flux entry. For the flat and formed samples the baselinewas “no treatment”, and they were run as is after being cut from the sheet or cavity. Forthe ellipsoid samples, after machining, the surface was sanded with successively higher grainsand papers, up to 2000 grit.36When SRF cavities are constructed it is common for them to undergo a bulk etch, of∼100µm, to remove the surface layer which contains mechanical damage caused when formingthe cavities.One of the etching treatments tested was a Buffered Chemical Polish (BCP). Whenetching the samples with a BCP, a standard acid mixture of H3PO4:HNO3:HF was used in a2:1:1 ratio which gave an etching rate of approximately 1µm/minute at a temperature below10◦C.Electro Polish (EP) is another commonly used method of etching and was also tested inthis experiment. For EP a mixture of H2SO4 and HF in a 9:1 ratio was used with an appliedvoltage of 10V and current of 0.5-1A.Etching has been observed to lower the field of flux entry into cavities [21]; however,the field of flux entry is known to recover after the cavity is baked for 48 hours at 120◦C.The effect of baking niobium was also of interest in this experiment. Samples were bakedat various temperatures for various lengths of time. Some were baked at low temperatures,120◦C for 48 hours whereas others were baked at higher temperatures for less time. Somewere “degassed” at 800◦C for 4 hours or “annealed” at 1400◦C for 4 hours.For the high temperature bakes (degassing and annealing), the samples were wrappedin 99.8% pure niobium foil and baked using TRIUMF’s target oven, which is capable ofreaching 1400◦C at a vacuum of 10−7 TORR. Doping the niobium samples with nitrogenwas done by Fermi lab using the procedure described in [22]. Some samples were also platedwith Nb3Sn, which was done at Cornell University [23].37Table 5.1: Niobium samples used in the experiment and the corresponding treatments preformed on themSample ID Geometry TreatmentCF1 Flat Plated with Nb3Sn from Cornell [23]H1 Flat same sample as H1 in [12]TR1 Flat 800◦C bake for 4 hours using TRIUMF target ovenTR2 Flat 800◦C bake for 4 hours using TRIUMF target oven without foilTR3 Flat 100µm BCPTR4 Flat 100µm BCP + 120◦C bake for 48 hoursTR5 Flat 100µm BCP + 120◦C bake for 48 hours + 5µm HF rinseTR5b Flat TR5 + 1400◦C bake for 4 hours using TRIUMF target ovenTR8 Flat UntreatedTR10 Formed UntreatedTR11 Formed 800◦C bake for 4 hours using TRIUMF target ovenTR13 Formed 100µm BCP etchTR16 Formed 1400◦C bake for 4 hours using TRIUMF target ovenCB1 ellipsoid Plated with Nb3Sn from Cornell [23]BR1 ellipsoid Base TreatmentBR2 ellipsoid Base TreatmentBR4 ellipsoid Roughened surface using 240grit sandpaperBR7 ellipsoid 30µm EP + N doped at Fermilab + 5µm EPBR8 ellipsoid 30µm EP + N doped at FermilabBR10 ellipsoid 30µm EP + 120◦C bake for 48 hoursBR11 ellipsoid 30µm EPTR20 ellipsoid 1400◦C bake for 4 hours using TRIUMF target ovenTR21 ellipsoid 1200◦C bake for 4 hours using TRIUMF target ovenTR22 ellipsoid 800◦C bake for 4 hours using TRIUMF target oven386 Demagnetization and Field Penetra-tionWhen a superconducting sample expels an applied magnetic field, the magnetic field linesare forced to bend around the sample causing them to bunch together at the edges of thesample. This creates a demagnetizing field which is larger than the applied field (withoutthe sample being present). The demagnetizing field is geometry dependant, meaning thefield of first flux entry is affected by the geometry of a sample. The larger the ratio ofperpendicular surface area to the length of a sample (where perpendicular surface area isthe area perpendicular to the applied field lines), the larger the demagnetizing field becomesand the earlier the field will enter the sample. Ernst Helmut Brandt has written a seriesof papers on modelling magnetic flux entry into superconducting samples by relating thesample to the applied field through surface screening currents [24–29]. The Brandt modelwas used to simulate the effect of pinning strength in samples in order to compare to theµSR results on samples treated with various surface treatments.6.1 Demagnetization FactorThere were three sample geometries (as described in section 5.1) and two field orientations(chapter 3) used during the experiment. In order to properly understand the results it wasimportant to quantify the effect of the geometry of the samples. This was accomplishedusing the demagnetization factor N . The demagnetization factor is a unit-less tensor, which39for ellipsoids in Cartesian co-ordinates the trace equals 1 (6.1).Nxx +Nyy +Nzz = 1 (6.1)The higher the demagnetization factor, the larger the demagnetizing field, the lower thefield of first flux entry.From [30], the demagnetization factor of a prolate ellipsoid with a major axis of length cand minor axes of length a in an applied field parallel to it’s major axis is given by:Nc =1− 22[12ln(1 + 1− )− 1](6.2)Where  is the prolate eccentricity of the ellipsoid. =[1−(a2c2)] 12; c > a (6.3)It can be seen from (6.2) and (6.3) that as c increases in relation to a the prolate eccen-tricity approaches one, and the demagnetization factor approaches zero. Equation (6.2) and(6.3) give a demagnetization factor of 0.1327 for the 45.8mm long, 18mm diameter prolateellipsoid samples used in this experiment.For non-ellipsoidal superconductors the induced field B (r) can be non-homogeneous,meaning the demagnetization needs to be estimated. One method of estimating the demag-netization factor for disks, like the ones used in this experiment, is to approximate them asoblate ellipsoids (c < a = b). The demagnetization factor for an oblate ellipsoid is given by(6.4)Npar =12− [1− 2]123sin−1  (6.4)Where  in this case is the oblate eccentricity. =[1−(c2a2)] 12; c < a (6.5)Estimating the 20mm diameter, 3mm thick disks to be oblate ellipsoids with a = b = 10mmand c = 1.5mm the samples have an approximate demagnetization factor of 0.803 in the caxis, henceforth referred to as the perpendicular axis (when the field is along the c axis it isperpendicular to the circular face of the disc).40The demagnetization factor for the other 2 axes of the oblate ellipsoid can be found using(6.1), where Nxx in this case would be the demagnetization factor along the c (perpendicular)axis and Nyy = Nzz = Npar would be the two axes parallel to the face of the disc.Nxx +Nyy +Nzz = Nperp + 2Npar (6.6)1 = 0.803 + 2NparNpar = 0.0985Therefore, the approximate demagnetization factor for a disc in a parallel field would be0.0985.6.2 Flux Entry into SamplesEven though a homogeneous field was applied to the samples, the field they experienced wasvaried over the sample surface due to supercurrents the sample generates in order to expel theapplied field. Figure 6.1 from [29] shows the induced field in the mid-plane of a disc of radiusa and thickness 2b with a homogeneous applied field . This shows how the field experiencedby the sample is much stronger than the applied field at the edge of a sample. In order toproperly interpret the data taken in this experiment it was important to understand boththe magnitude of the applied field that would cause the sample to experience an effectivefield of Hc1 and how the field was expected to enter the sample. In a simplistic model offield penetration, the approximate demagnetization factors calculated in section 6.1 can beused to calculate the expected field of first flux entry via formula (6.7).HEntry = (1−N)Hc1 (6.7)Where HEntry is the applied field at which, the part of the sample which experiences thehighest field (the edges of the sample for our sample shape), experiences a field of Hc1. Usingan Hc1 of 170mT for RRR300 niobium at 2K [21], formula (6.7) gives an expected field offirst flux entry of approximately 33.5mT for the perpendicular field discs, 147.4mT for theellipsoids, and 153.3mT for the parallel field samples.41Figure 6.1: Magnetic field distribution in the midplane of a disc for increasing applied field (solid lines),and decreasing feild (dashed lines) starting from the center of the disc, x axis = 0, and moving radiallyoutwards to the outer edge of the disc at r=a taken directly from [29]A detailed model of flux entry into superconducting samples is found in a series of papersby Brandt [24–29]. According to Brandt’s model, when the applied field reaches HEntrymagnetic vortices will start forming in the sample and, since it is energetically favourable,the flux lines snap to the center of the sample. This means that even though the transversefield µSR used in this experiment only probed the center of the sample, it could give anaccurate indication of when the flux enters a sample. Superconductors; however, generallyexperience “flux pinning”, whereby the field lines get trapped preventing them from jumpinginto the center of the sample.Brandt models the flux entry into a superconductor by coupling an arbitrary shapedsample to the external field via surface screening currents. This method incorporates theeffect of pinning on the flux entry [29]. The results from this model show that pinning holdsthe field in place and prevents it from snapping to the center. This means that samples withhigher pinning strength would require a larger applied magnetic field before experiencing fluxentry in the center of the sample, and µSR measurements of samples with a higher pinningstrength would therefore show flux entry at higher fields.427 ResultsOne of the novel aspects of this study was that measurements were taken with three differentsample geometries and two different field orientations. This was done so that the fieldexperienced by the sample would better approximate the field seen by an accelerator cavity.The first set of results show the difference in field entry into the base treatment samples foreach geometry. After the base treatment results for each geometry was plotted, the resultsof the various surface treatments for each geometry were analysed. The effect of the silvermask is also presented, unlike any bulk measurement of magnetic properties (e.g. Squid)it is shown that the µSR results isolate the magnetic response of the central region of thesample.7.1 Geometry ComparisonFigure 7.1 shows the µSR results for the base treatments of the flat sample in parallel field,the flat sample in perpendicular field, and the ellipsoid sample. The results were affected byboth the different demagnetization factors of each geometry, as described in section 6.1, andthe pinning strength of the samples, as described in section 6.2. In order to remove the effectof the demagnetization factor from the geometry comparison and get a better depiction ofhow strongly pinning strength affected the µSR results in each geometry, the relative zerofield asymmetry was re-plotted against the applied field over the theoretical Hentery for eachrespective geometry, calculated using equation (6.7) in section 6.2.It can be seen from figure 7.2 that the zero field asymmetry in the parallel field sample(TR8) starts dropping around the theoreticalHentry as predicted from section 6.2 (Ha/Hentry=1)it then quickly drops to zero at about 1.2 times the theoretical entry point. The very flat43Figure 7.1: Zero field asymmetry measurements taken for different sample geometries and field orientationsplotted against the applied field. The red and black runs were taken in perpendicular fieldFigure 7.2: Zero field asymmetry measurements taken for different sample geometries and field orientationsplotted against the applied field over the theoretical field of first flux entry for each geometry as calculatedin section 6.2low field section followed by the abrupt drop in signal indicates that once the field beginspenetrating the sample, it very quickly reaches the penetration depth of the muons and gives44a good indication of the field of first flux entry.The perpendicular field measurements for the flat sample (also TR8) shows an initialdrop in asymmetry at much higher values of the normalized field, at about 3.5 times thetheoretical entry point, and then a more gradual drop to zero at over 5 times the theoreticalentry point. This is due to the fact that the theoretical entry point for the field is at theedge of the sample and the muons are probing the center. This means that, if the field ispinned and held to the outside of the sample, the magnetic field will not be experienced bythe muons until the field makes it’s way to the center of the sample. Furthermore, if there isstrong pinning there will still be some regions of zero field in the center of the sample evenafter some flux has entered.The base ellipsoid sample shows a gradual decline in asymmetry at low field, then anabrupt drop in the zero field asymmetry at about 1.5 times the predicted field of first fluxentry reaching zero at less than twice the predicted entry point. This shows that it is morestrongly affected by pinning than the parallel field flat disc measurements. The combinationof the strong pinning effect and low demagnetization factor allow the ellipsoid samples toreach the highest applied field before experiencing flux entry into the sample center.In order to observe the field of first flux entry into the flat samples in perpendicular field, adata run was taken on the H1 sample both with and without the silver mask which restrictedthe muons to the center of the sample. Figure 7.3 is the unmasked µSR results for the 0and 40mT runs. A second oscillating component(in addition to the Kubo-Toyabe associatedwith muon in zero field) indiates some fraction of the muons see a large internal field whichcan be seen in the 40mT run (40mT is near the predicted field of first flux entry) indicatingthat a perpendicular field is penetrating the sample. Figure 7.4 shows the µSR results forthe masked sample, it is clear no oscillation or significant drop in zero field asymmetry canbe observed in the masked sample. This shows that the field is entering the outer edges ofthe sample and requires higher applied field before it breaks into the center.The 8mm mask was present for all measurements in the results section except the runsused to test the effect of the mask indicated above.45Figure 7.3: a) shows the zero field cooled µSR signal for the H1 sample with no mask at 0mT. b) shows thezero field cooled µSR signal for the H1 sample with no mask at 40mT. (c) shows the early time behaviourof (b)Figure 7.4: The image on the left shows the zero field cooled µSR signal for the H1 sample at 0mT usingthe 8mm diameter mask. The image on the right shows the zero field cooled µSR signal for the H1 sampleat 40mT using the 8mm diameter mask.467.2 Flat Samples Perpendicular Field ResultsSeveral different surface treatments were tested on the flat samples in perpendicular field.Some common surface treatments tested were etching and low temperature baking. Fig-ure 7.5 shows the µSR results for fine grain niobium samples that have undergone thesetreatments.Figure 7.5: Relative asymmetry vs applied magnetic field for perpendicular field measurements of flatsamples with various treatments for comparison. The treatments being compared are base treatment, BCP(100µm), BCP then baked at 120◦C, and BCP then baked at 120◦C then HF rinsed for 5 µmFrom figure 7.5, it appears that the 100µm BCP etch causes the magnetic field to breakinto the sample center earlier. The 120◦C bake pushes flux entry back out towards the baseflux entry point, and the HF rinse has very little effect.Figure 7.6 shows the µSR results from the TR5 sample before and after a 4 hour annealat 1400◦C. This sample shows that the annealing causes the field to enter the center of thesample at lower applied magnetic field.Many recent studies have focussed on doping or plating niobium to try to increase theperformance of niobium in accelerating cavities. The µSR results for flat niobium samplesdoped with nitrogen, and for flat niobium samples plated with Nb3Sn can be seen in figure7.7.Figure 7.7 shows that magnetic flux reaches the central region of the Nb3Sn sample earliest47Figure 7.6: Relative asymmetry vs applied magnetic field for perpendicular field measurements of the TR5sample before and after being annealed at 1400◦C for 4 hoursFigure 7.7: Relative asymmetry vs applied magnetic field for perpendicular field measurements of nitrogendoped, Nb3Sn coated and untreated samples(lowest field), then the nitrogen doped sample, with the field breaking into the center of theuntreated sample the latest (highest field). As stated in section 6.2, perpendicular field µSRresults are greatly affected by pinning strength; it is possible that the early flux entry in thecenter of the plated and doped samples is due to a reduction in pinning strength from heat48treatments involved in plating and doping and not a result of early flux entry at the edgesof the sample. Figure 7.8 shows the results from the annealed at 1400◦C, Nb3Sn plated, Ndoped, and 800◦C degassed samples. The plated and annealed samples are similar as are thedegassed and N-doped samples.Figure 7.8: relative asymmetry vs applied magnetic field for perpendicular field measurements of theannealed at 1400◦C, Nb3Sn plated, N doped, and 800◦C degassed samples7.3 Formed Samples Perpendicular Field ResultsThe formed samples tested were untreated, BCP etched, degassed at 800◦C, and annealedat 1400◦C. It can be seen in figure 7.9 that the results for all the formed samples are fairlyflat at low field. At higher field, around 150mT, the zero field asymmetry in the basetreatment starts to fall, plateaus around 160-180mT, drops again, then jumps up, and dropsagain until finally reaching zero at around 260mT. This asymmetry vs field behaviour is notseen in any of the other samples/geometries. It is likely due to the fact that wire EDM(electrical discharge machining), used to cut the samples from the cavity, introduced defectsand thereby more pinning centres. An interesting result here is that the 100µm BCP, whichis commonly used to remove surface defects caused by machining, seems to reduce the jumpin the signal. The 800◦C degas treatment doesn’t change the signal shape much but fluxenters earlier than the base treatment, which is consistent with other samples results. The491400◦C anneal experiences flux entry into the center of the sample the earliest and showsno signs of the signal jumping up and down indicating the anneal has greatly lowered thepinning strength.Figure 7.9: relative asymmetry vs applied magnetic field for perpendicular field measurements of formedsamples with various treatments for comparison. The treatments being compared were base treatment, BCP(100µm), baked at 800◦C (degas), and annealed at 1400◦C7.4 Flat Samples Parallel Field ResultsFigure 7.10 shows the results for all the parallel field runs taken during the experiment.In parallel field the annealed sample experienced flux entry at the lowest field, followed bythe nitrogen doped sample, then the base treatment sample, and finally the plated samplebreaks in at the highest field. Though the annealed and nitrogen doped results match theperpendicular field results, the Nb3Sn plated results are opposite, breaking in much laterthan the rest of the samples in parallel field as opposed to breaking in much earlier intransverse field. The parallel field results for the plated sample support the results from [31]which indicate plated cavities are capable of reaching higher gradients without quenching.50Figure 7.10: relative asymmetry vs applied magnetic field for parallel field measurements of flat sampleswith various treatments for comparison. The treatments being compared were base treatment, Nb3Sn plated,nitrogen doped, and TR5 annealed7.5 Ellipsoid ResultsThe ellipsoid samples were made to utilize TRIUMF’s existing perpendicular field set up witha low demagnetization factor in order to better approximate the conditions in an acceleratorcavity.Due to the fact that there was considerable machining involved in creating the ellipsoidsamples, the first test done with them was a comparison between two different samplesboth with the same base treatment. The results in figure 7.11 show no significant differencebetween them. This means that the base treatment done to all ellipsoid samples should givethe same results and that the base treatment ellipsoids could be used to establish a baselinefor treatments.One hypothesises as to why flux could be prematurely entering niobium cavities was thatmicroscopic bumps in the cavity wall could create a demagnetization effect about the localpeaks and enhance the local fields. To test this, the results from the roughened ellipsoidwere compared with the base treatment as seen in figure 7.12 The roughened ellipsoid was51Figure 7.11: Zero field asymmetry µSR results vs Applied field for two ellipsoid samples with the basetreatment described in chapter 5Figure 7.12: Zero field asymmetry µSR results vs Applied field for an ellipsoid sample with the basetreatment and with a roughened surface as described in chapter 552very similar to the base, but extended slightly further before reaching zero, it is possiblethat roughing the outside of the ellipsoid increased it’s pinning strength slightly by creatingdefects and extra pinning centres; however, the difference in the two signals was small andit did not cause early flux entry into the center of the sample.Another test preformed on the ellipsoid samples was a heat treatment test. The ellipsoidsamples were baked at 800, 1200, and 1400◦C respectively for 4 hours. Figure 7.13 showsFigure 7.13: Zero field asymmetry µSR results vs Applied field for an ellipsoid sample with the basetreatment, degassed at 800◦C, high temperature baked at 1200◦C and annealed at 1400◦Cthat the field breaks into the annealed (1400◦C) sample first, then the 1200◦C baked sample,followed by the 800◦C baked sample and finally the base treated sample. This shows thatthe high temperature baked samples experience flux entry earlier all other things being equal(ie geometry and field orientation).Figure 7.14 shows many different ellipsoid samples, all treated differently. One notewor-thy result is that, even though BR10 and 11 were only etched by 30µm, the 120◦C bake afteretching still caused the field to break in later than in the case where the sample was onlyetched. Unfortunately the BR10 field scan was not completed; however, it was included inthe results since, no matter where the next point on BR10’s asymmetry vs field plot lies, it53still reaches the largest field before experiencing significant flux entry into the sample centre.Another result included on figure 7.14 was TR22. Even though TR22 was included in figure7.13 it was included here to see the difference between baking at 800◦C and doping withnitrogen. This is because nitrogen doped samples BR7 and BR8 are heated to 800◦C duringthe doping process so TR22 is a better base to compare with than BR1. The nitrogen dopedsamples seem to hold the field outside the center a little better than TR22; however, thereis not a large difference between TR22, BR7, and BR8 all of which reach zero around thesame field. The Nb3Sn coated sample CB1 breaks in abruptly and reaches zero before anyother sample, similar to the flat sample perpendicular field results but by less of a margin.Figure 7.14: Zero field asymmetry µSR results vs Applied field for various surface treatments on ellipsoidsamplesOne of the appealing things about plating niobium with Nb3Sn is that Nb3Sn’s criticaltemperature is much higher than niobium’s, meaning it may operate more effectively at ahigher temperature. The result of a temperature scan, seen in figure 7.15, is somewhatexpected. The field enters the sample center earlier at higher temperatures; however, CB1at 5K still keeps the field from entering the sample centre almost as well as the annealedellipsoid sample.54Figure 7.15: Zero field asymmetry µSR results vs Applied field for the CB1 sample at various temperatures558 DiscussionThere were several factors considered when analysing the µSR results. The first was HC1which is the field at which the sample transitions from the Meissner state to the mixed stateand vortices can begin entering the sample as described in section 1.1. The higher the HC1 ofa sample, the later the field would break into the sample, the later the zero field asymmetryfrom the µSR results would start to drop. Another factor was the demagnetization factorassociated with the shape of the sample and the direction of the applied field, the higherthe demagnetization factor the larger the effective field became at the edges of the samplein comparison with the applied field. This causes the magnetic flux density to reach HC1at the edge of the sample before the applied field reached HC1 and therefore cause the zerofield µSR asymmetry to start dropping at lower applied fields. The theoretical applied fieldat which the Demagnetization factor causes any point on the sample to experience HC1 andthereby experience flux entry was referred to as HEntry and is described in section 6.2. Thefinal property to consider was pinning strength, which controls how easily the vortices atthe edge can move to the center of the sample. Since only the center of the sample wasbeing probed by the muons, samples with higher pinning strength would resist the motionof vortices to the center of the sample and thus extend the ZF signal to higher applied fields.8.1 GeometryFrom the geometry section of the results chapter, section 7.1, it can be seen that field brokeinto the center of the flat and formed perpendicular field samples well above the theoreticalentry point. Figures 7.3 and 7.4 indicate that flux entered the sample around the theoreticalentry point HEntry, but was not immediately experienced at the center of the sample where56the muons were probing. Brandt’s model of flux entry [24–29], discussed in section 6.2, showsthat in a pin-free sample the flux lines will snap to the center immediately upon enteringthe sample, as it is energetically favourable to do so. Thus, in the perpendicular field flator formed samples the demagnetizing field caused the outer edges of the sample to reachHC1 well below an applied field of HC1 and enter the sample. The strong pinning in thesample then held the flux lines out of the center of the sample until the applied field wasraised high enough to overcome the pinning strength. Once the applied field was raised highenough the flux lines began to break in slowly which caused a gradual decrease in the zerofield asymmetry. Since the demagnetization factor is the same for all the flat or formedsamples and the field entered well before it was experienced by the muons, the difference inthe results was dominated by the pinning strength.It was hypothesized in [32] that impurities and vacancies may prevent the formation ofhydrides responsible for high field RF losses. It is possible that the correlation between theperpendicular field µSR results and the RF cavity tests is due to the fact that the samevacancies and impurities that prevent hydride formation and cause RF losses in RF cavitytests also introduce pinning centers in samples and delay flux entry in Perpendicular fieldµSR measurements.The ellipsoid samples were made to limit the demagnetization factor and create a betterapproximation of the field experienced by an RF cavity. The ellipsoid samples were runin the perpendicular field set-up; however, unlike the flat and formed samples, they had avery small demagnetization factor, meaning that flux entry started when the applied fieldwas much closer to HC1. The ellipsoid samples are still affected by pinning. In a pin-freeellipsoid, flux that entered the sample would redistribute itself uniformly once it nucleatedat the edges; however, pinning would cause the field to have a gradient with more flux awayfrom the center. From section 7.1 we see that the flux is held outside the center of the sampleby pinning, until about twice the theoretically predicted Hentry.This experiment was the first time using parallel field µSR measurements on niobiumsamples. The results show that at low field the zero field asymmetry is flat, with no signsof flux entry. At high field the zero field asymmetry drops quickly, reaching zero almostimmediately after exhibiting signs of flux entry. This is likely due to the fact that the field57only needs to penetrate ∼100µm before reaching the implanted muons, as opposed to the6mm it needs to go through in the perpendicular field samples. This greatly limits the effectof pinning on the µSR measurements, which means that the zero field asymmetry dropsat a field much closer to the field of first flux entry than in the perpendicular field set-up.The parallel field results give a more accurate depiction of when the sample first experiencesflux entry. Flat samples in parallel field also have a greatly reduced demagnetization factormeaning the µSR results are much more strongly influenced by changes the field of first fluxentry than in perpendicular field results.8.2 Sample PreparationAs mentioned in section 5.2, one of the treatments tested was a 100µm etch using a BCP.The BCP is known to lower cavity performance in RF tests and cause the field to break intothe sample earlier in previous perpendicular field µSR tests on long grain niobium samplesin [12]. As seen in the results sections 7.2 and 7.3, a 100µm etch from the base treatmentcaused magnetic flux to enter the center of the sample in the flat perpendicular field, formedperpendicular field and ellipsoid fine grain niobium samples at lower applied fields. Pinninghas a large effect on the perpendicular field results and the µSR measurements indicate thatthe pinning strength of the samples is reduced by etching. The intention of etching is toremove any surface defects or damages created during the manufacturing process. Defectscan act as pinning centers, so removing them may have caused a decrease in pinning strength.This would explain why etching the formed sample reduced the strange high field behaviourbelieved to be caused by damages introduced when cutting the samples using wire EDM.Etching could also have introduced contaminants into the first several nanometres of thesamples such as oxides, hydrocarbons, water, and chemical residues.In all of the fine grain niobium 120◦C bake tests (both ellipsoid samples and flat perpen-dicular field samples), baking an etched sample for 48 hours caused flux entry into the centerof the etched sample to occur at a higher field. This matches previous results for long grainsamples from [12]. The mechanism by which the low temperature baking increased the fieldof first flux entry into the center of the sample is not well understood; however, in [32], it58is shown that after baking the London penetration depth of a sample cut from a cavity isincreased, which is typical of a lower purity superconductor.Two different high temperature bakes were tested, one was a 800◦C bake in UHV (referredto as a degas due to the temperature being to low to affect the metallurgic properties ofniobium) and the other was a 1400◦C bake in UHV (referred to as an anneal). A degas isused in UHV to remove impurities and in all cases tested using µSR, the 800◦C degas causedthe flux to enter the center of the sample at lower applied field than the untreated sample.This is likely due to the fact that impurities can act as pinning centers and once they’ve beenremoved, the flux is able to snap to the center of the sample easier. It can also be observedfrom section 7.3 that, although the 800◦C degas caused the flux to enter the center of thesample earlier, it did not remove the erratic high field behaviour of the formed samples. Thisis because the high field behaviour is likely a result of mechanical damages introduced bycutting the samples with a wire EDM and 800◦C is not hot enough to change the metallurgicproperties of niobium. Degassing niobium could also increase the RRR value of the samplewhich can alter the effective HC1.In all cases the 1400◦C anneal caused early flux entry into the sample. A 1400◦C annealis hot enough to affect the metallurgic properties of the niobium. It can change the grainboundary size, remove mechanical defects, and since it is done in UHV it can cause impuritiesto outgas and increase the purity of the sample. The most drastic change in flux entry aftera 1400◦C anneal occurred in the perpendicular field formed and flat disc results seen insections 7.2 and 7.3. In the formed sample results the anneal removed the strange highfield behaviour and caused the flux to enter at a much lower applied field. It is likely thatthe high field behaviour was removed because the anneal eliminated the mechanical defectsintroduced through cutting the samples with wire EDM. The fact that the signal falls offmuch earlier is also likely due to the removal of defects and impurities which act as pinningcenters.The µSR results for the Nb3Sn plated samples were the most complicated to interpretas the Nb3Sn coating was only 2µm thick. This means that the demagnetization factorfor the Nb3Sn coating was different than that of the pure niobium samples and when theplated sample temperature was above the Tc of niobium, the demagnetization factor of the59sample was extremely large. All the µSR results above 9K may not accurately representthe flux penetration into the sample. Furthermore, the Nb3Sn plated samples were the onlysamples which had conflicting results from parallel and perpendicular field measurements(see chapter 7). The demagnetization factor is smallest for the parallel geometry. It ispossible that by plating the niobium the effective HC1 and pinning strength of the samplewas affected in different ways. During the plating process the niobium was heated to nearlyannealing temperatures, which µSR results suggest lowers the pinning strength. At the sametime, adding Nb3Sn to the outer layer of the sample changes the effective HC1. Since theflat perpendicular field measurements are the most strongly affected by pinning strengthof the sample and the Nb3Sn plated sample results match the 1400◦C anneal results inperpendicular field (see figure 7.8 section 7.2) it is likely that the effect of heating the Nb3Snplated sample and removing pinning centers in the niobium is dominating the perpendicularfield results. In parallel field; however, pinning has less of an effect on the results and it islikely that the parallel field Nb3Sn results are showing a change in effective HC1. This couldmean that plating niobium with Nb3Sn raises the effective HC1 and thereby the achievablegradient in accelerator cavities as suggested in [31].The N-doped sample results show flux breaking into the sample earlier than in the un-treated sample in all cases indicating that it reduces either the effective HC1, the pinningstrength, or both. During the doping process the sample is heated to 800◦C and, similar tothe Nb3Sn case, the N-Doped flat perpendicular field results matched the flat perpendicularfield results of a sample with the same heat treatment (see figure 7.8 section 7.2). Onceagain this shows a reduction in pinning caused by the high temperature bake; however, italso indicates that nitrogen may not be suppressing the order parameter of the supercon-ductor as much as a vacancy or other impurities resulting in weaker pinning. Unfortunatelythe 800◦C baked samples have yet to be tested in parallel field so it is not possible to see ifthe N-doped samples exhibit any differences than a simple heat treatment in parallel field.609 Future WorkThe ability to make high field parallel µSR measurements is new to TRIUMF; not manysamples have been run in parallel field. More samples, such as a 100µm etch, a 100µmetch followed by a 120◦C bake, an Nb3Sn plated sample at 4-5K, a 100µm etch followed by a120◦C bake at 4-5K and a 800◦C bake need to be run in order to see what differences parallelfield and perpendicular field results yield and which might better correlate to the effect on anRF cavity. The 100µm etch would be interesting to test, as it is believed to reduce the effectof pinning which should have less of an effect on parallel field measurements. The 120◦Cbake is a standard treatment after etching and is known to improve RF test results andperpendicular field µSR results. If this is due to vacancies introduced in baking which act aspinning centers it may not affect parallel field results the same way. The reason for platingniobium with Nb3Sn is that Nb3Sn has a higher Tc and plating niobium with it might allowcavities to maintain sufficient accelerating gradients while operating at higher temperaturessuch as 4K. This would reduce cost and energy consumption as cooling helium from 4K to2K is expensive and more complex. For this reason it would be interesting to see how theNb3Sn plated sample at 4K compares to a niobium sample with a standard treatment of a100µm etch followed by a 120◦C bake at 4K. The 800◦C bake is needed to compare withthe N-Doped sample already run in parallel field to see if it differs from the straight heattreatment in parallel field field as the Nb3Sn plated sample did.TRIUMF is also currently designing an extension to an existing βNMR beamline capableof reaching 2kG DC fields parallel to the sample surface. This would allow parallel fieldβNMR measurements to be taken on niobium samples with the option of depth profiling toobtain a better understanding of what is happening on the sample surface and not just inthe bulk. Furthermore, the βNMR extension uses Helmholtz coils capable of a much more61homogeneous field.Another use for µSR is to examine the temperature dependant hop rate of muons in amaterial. Since muons can behave as light hydrogen atoms and hydrides are a suspectedcause of RF losses, muon hop rate measurements may give insight into which dopants havethe greatest affect on hydride diffusion. It could also be used to determine what temperaturesthe cavity should be cooled through quickly to avoid hydrides diffusion into the material.Some hop rate measurements have already been taken and analysed by other members ofthe RF group but more are needed.6210 ConclusionµSR was used in two different field orientations (parallel and perpendicular to the samplesurface) on samples of three different geometries (flat, formed, and ellipsoidal) in order toinvestigate the effect of sample preparation on the magnetic field penetration of RRR300niobium and Nb3Sn plated niobium.The perpendicular field µSR measurements for the fine grain niobium samples reachedhigher applied fields before experiencing flux penetration in the center of the sample afterreceiving common surface treatments known to improve cavity performance. In the per-pendicular field set-up, the fraction of muons in the sample center that saw zero field washighly affected by the pinning strength of a sample. The pinning strength is affected bythe number of vacancies, impurities and mechanical defects present in the sample, whichcan act as pinning centers and increase the pinning strength. It was hypothesised that thecorrelation between perpendicular field µSR measurements and cavity performance comesfrom the presence of impurities, which prevent the formation of hydrides and therefore highfield RF losses in RF tests, and act as pinning centers which prevents flux entry into thecenter of the samples in µSR measurements.Some newer cavity treatments such as nitrogen doping and Nb3Sn plating were also testedusing perpendicular field µSR. Contrary to the results from common cavity treatments, theperpendicular field µSR results did not show flux entering the sample at higher fields intreatments which increased cavity performance. The perpendicular field µSR results showedno difference between the plated sample and a sample with no plating that had been bakedat 1400◦C, a similar temperature to what the sample reached during the plating process.Similarly, no difference was observed between the N-doped sample and a non-doped samplebaked at 800◦C, which is the temperature a sample is heated to during the doping process.63In the case of the N-doped sample this means that the added nitrogen did not cause asufficient increase in pinning strength to change the µSR results. This may mean that thenitrogen does not suppress the superconductors order parameter as much as vacancies orother impurities. Since N-doped samples seem to preform well in RF cavity tests, it ispossible that impurities which displace hydrides or prevent hydride formation, yet do notsuppress the order parameter of the superconductor, are the best choices for dopants in SRFcavities. In the case of the Nb3Sn plated sample, the muons are still stopping in niobiumbulk and it is possible that the coating is affecting the initial HEntry more strongly than itaffects the pinning; however, the pinning has more influence on the perpendicular field µSRresults and thus the heat treatment has the largest effect on the perpendicular field results.The ellipsoid samples were run in the perpendicular field set-up and showed similar resultsto the perpendicular field flat and formed samples. The main difference between the ellipsoidresults and the flat and formed perpendicular field results is that the ellipsoid samples hada much smaller demagnetization factor, meaning the drop in zero field asymmetry occurredmuch closer to HEntry, and therefore HEntry had a larger affect on the results.The parallel field samples showed a drop in zero field asymmetry much closer to thetheoretically predicted HEntry, and the flux lines only needed to penetrate 100µm to reachthe muons deposited in the material. Because of this, the strength of pinning had less ofan effect on the results. Furthermore, the flux lines in the parallel field set-up were a betterapproximation of the field experienced by a cavity. Since the parallel field set-up is new,only a few measurements have been taken using this set-up; more are needed in order toadequately compare results to RF cavity tests and perpendicular field µSR measurements.One surprising result from parallel field µSR measurements is that, unlike in perpendicularfield µSR, the Nb3Sn plated sample showed flux entry at a higher field than the base sample.This means that the plating has affected the HEntry of the sample and Nb3Sn plated cavitiesmay be capable of reaching higher accelerating gradients.Although µSR measurements are DC field measurements, and are thereby incapable oftesting the BCS resistance, they can give insight into the flux entry and pinning strength ofsuperconducting samples, which could lead to discoveries about types of treatments will helpcavity performance. Furthermore, µSR is capable of examining muon hop rate at various64temperatures, which may be affected by impurities since they can trap the muon. This couldlead to a better understanding of doping and cooling methods for cavities. In the future moreparallel field measurements, doped sample hop rate temperature scans, and βNMR depthprofiling may play a key role in gaining a better understanding of superconducting materialsand thereby SRF cavity performance.65Bibliography[1] H. Padamsee. The science and technology of superconducting cavities for accelerators.SUPERCONDUCTOR SCIENCE AND TECHNOLOGY, 14:R28–R51, 2001.[2] Charles Kittel. Introduction to Solid State Physics. John Wiley & Sons, Inc., New York,7th edition, 1986.[3] N.W. Ashcroft and N.D. Mermin. Solid State Physics. Thompson Learning Inc.,Philadelphia, 1976.[4] Francisca Wheeler and Peter Freilinger. Superconductivity [website], 2001.http://teachers.web.cern.ch/teachers/archiv/hst2001/accelerators/superconductivity/superconductivity.htm, accessed November 2014.[5] Anna Grassellino. 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Muon spin rotation studies of niobium for superconducting rf applications.Phys. Rev. ST Accel. Beams, 16:062002, Jun 2013.[13] S. Cox. Introduction to µsr: What, how, where?, 2005. url:http://www.isis.stfc.ac.uk/groups/muons/muon-training-school/2005-introduction-cox-review7904.pdf, accessed October 2014.[14] S. J. Blundell. Spin-polarized muons in condensed matter physics. ContemporaryPhysics, 40:175–192, 1999.[15] J. Sonier. Muon spin relaxation/rotation/resonance (µsr), April 2002. url:http://cmms.triumf.ca/intro/musr/muSRBrochure.pdf, accessed October 2013.[16] K.A. Olive et al. Review of Particle Physics. Chin.Phys., C38:090001, 2014.[17] S.R. Kreitzman, D.J. Arseneau, B. Hitti, and J.H. Brewer. µsr facility users guide,February 1999.[18] R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo. Zero-and low-field spin relaxation studied by positive muons. Phys. Rev. B, 20:850–859, Aug1979.[19] James F. Ziegler, M. D. Ziegler, and J. P. Biersack. Srim - the stopping and rangeof ions in matter. Nuclear Instruments and Methods in Physics Research Section B,268:1818–1823, 2010.[20] A. Suter and B.M. Wojek. Musrfit: A free platform-independent framework for sr dataanalysis. Physics Procedia, 30:69, 2012.67[21] K. Saito. Critical field limitation of the niobium superconducting rf cavity. The 10thWorkshop on RF Superconductivity, Tsukuba, Japan, 2001.[22] A. Grassellino et al. Nitrogen and argon doping of niobium for super- conducting radiofrequency cavities: a pathway to highly efficient accelerating structures. SuperconductorScience Technology, 26:102001, Oct 2013.[23] S. Posen and M. Liepe. Stoichiometric nb3sn in first samples coated at cornell. Proceed-ings of SRF2011, Chicago, IL USA, pages 886–889, 2011.[24] E.H. Brandt. Superconductors of finite thickness in a perpendicular magnetic field:Strips and slabs. Phys. Rev. B, 54:4246–4264, 1996.[25] E.H. Brandt. 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