L12 Superconductivity

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Fundamental Accelerator Theory, Simulations and Measurement Lab – Michigan State University, Lansing June 4-15, 2007 Superconductivity for Accelerators Soren Prestemon Lawrence Berkeley National Laboratory Lecture No. 12

References: 

References Many thanks to my colleagues Paolo Ferracin and Ezio Todescu, who provided their course notes from USPAS 2007 Also: Padamsee, Knobloch, and Hays, “RF Superconductivity for Accelerators” Padamsee, Topical Review, “The science and technology of superconducting cavities for accelerators”, Super. Sci. and Technol., 14 (2001) Ernst Helmut Brandt, “Electrodynamics of Superconductors exposed to high frequency fields” Martin Wilson, “Superconducting Magnets” Alex Gurevich, Lectures on Superconductivity Marc Dhallé, “IoP Handbook on Superconducting Materials” (preprint) Arno Godeke, thesis: “Performance Boundaries in Nb3Sn Superconductors”, and for many beautiful photographs and fruitful conversations Feynman “Lectures on Physics” A. Jain, “Basic theory of magnets”, CERN 98-05 (1998) 1-26 Classes given by A. Jain at USPAS MJB Plus, Inc. “Superconducting Accelerator Magnets”, an interactive tutorial. K.-H. Mess, P. Schmuser, S. Wolff, “Superconducting accelerator magnets”, Singapore: World Scientific, 1996. “LHC design report v.1: the main LHC ring”, CERN-2004-003-v-1, 2004. R. Gupta, et al., “React and wind common coil dipole”, talk at Applied Superconductivity Conference 2006, Seattle, WA, Aug. 27 - Sept. 1, 2006. L. Rossi, “Superconducting Magnets”, CERN Academic Training, 15-18 May 2000. S. Wolff, “Superconducting magnet design”, AIP Conference Proceedings 249, edited by M. Month and M. Dienes, 1992, p. 1160-1197. L. Rossi, “The LHC from construction to commisioning”, FNAL seminar, 19 April 2007. P. Schmuser, “Superconducting magnets for particle accelerators”, AIP Conference Proceedings 249, edited by M. Month and M. Dienes, 1992, p. 1100-1158. A. Devred, “The mechanics of SSC dipole magnet prototypes”, AIP Conference Proceedings 249, edited by M. Month and M. Dienes, 1992, p. 1309-1372. T. Ogitsu, et al., “Mechanical performance of 5-cm-aperture, 15-m-long SSC dipole magnet prototypes”, IEEE Trans. Appl. Supercond., Vol. 3, No. l, March 1993, p. 686-691. K. Artoos, et al., “Status of the short dipole model program for the LHC”, IEEE Trans. Appl. Supercond., Vol. 10, No. l, March 2000. K. Koepke, et al., “Fermilab doubler magnet design and fabrication techniques”, IEEE Trans. Magn., Vol. MAG-15, No. l, January 1979. C.L. Goodzeit, “Superconducting Accelerator Magnets”, USPAS, January 2001. Superconductivity for Accelerators S. Prestemon

Outline : 

Outline Superconductivity for accelerators: Basics of superconductivity Some historical perspectives The energy gap and electron-phonon coupling Distinguishing perfect conductors and superconductors: the Meissner state Type I and II superconductors, the flux quantum “Pinning” the flux quantum for useful conductors Using superconductivity for accelerators Using the Meissner state for RF applications Using type II superconductors for magnets Review of magnetic multipoles, and the inverse problem: how to create “perfect” multipole fields Design and fabrication issues with real accelerator magnets Examples of accelerator magnets Superconductivity for Accelerators S. Prestemon

History: 

History 1911: Kamerlingh Onnes discovery of mercury superconductivity: “Perfect conductors” A few years earlier he had succeeded in liquifying Helium, a critical technological feat needed for the discovery 1933: Meissner and Ochsenfeld discover perfect diamagnetic characteristic of superconductivity Kamerlingh Onnes, Nobel Prize 1913 Superconductivity for Accelerators S. Prestemon

History - Theory: 

A theory of superconductivity took time to evolve: 1935: London brothers propose two equations for E and H; results in concept of penetration depth  1950:Ginzburg and Landau propose a macroscopic theory (GL) for superconductivity, based on Landau’s theory of second-order phase transitions History - Theory Ginzburg and Landau (circa 1947) Nobel Prize 1962: Landau Nobel Prize 2003: Ginzburg, Abrikosov, Leggett (the GLAG members) Heinz and Fritz London Abrikosov, With Princess Madeleine Superconductivity for Accelerators S. Prestemon

History - Theory: 

History - Theory 1957: Bardeen, Cooper, and Schrieffer publish microscopic theory (BCS) of Cooper-pair formation that continues to be held as the standard microscopic theory for low-temperature superconductors 1957: Abrikosov considered GL theory for case = Introduced concept of Type II superconductor Predicted flux penetrates in fixed quanta, in the form of a vortex array Superconductivity for Accelerators S. Prestemon

History – High Temperature Superconductors: 

History – High Temperature Superconductors 1986: Bednorz and Muller discover superconductivity at high temperatures in layered materials comprising copper oxide planes Superconductivity for Accelerators S. Prestemon

General Principals: 

General Principals Superconductivity refers to a material state in which current can flow with no resistance Not just “little” resistance - truly ZERO resistance Resistance in a conductor stems from scattering of electrons off of thermally activated ions Resistance therefore goes down as temperature decreases The decrease in resistance in normal metals reaches a minimum based on irregularities and impurities in the lattice, hence concept of RRR (Residual resistivity ratio) RRR is a rough measure of cold-work and impurities in a metal RRR=(273K)/ (4K)) M. Wilson Copper Aluminum Superconductivity for Accelerators S. Prestemon

Basics of Superconductivity: 

Basics of Superconductivity In a superconductor, when the temperature descends below the critical temperature, electrons find it energetically preferable to form Cooper pairs The Cooper pairs interact with the positive ions of the lattice Lattice vibrations are often termed “phonons”; hence the coupling between the electron-pair and the lattice is referred to as electron-phonon interaction The balance between electron-phonon interaction forces and Coulomb (electrostatic) forces determines if a given material is superconducting Electron-phonon interaction can occur over long distances; Cooper pairs can be separated by many lattice spacings x Superconductivity for Accelerators S. Prestemon

Cooper Pairs: 

Cooper Pairs The strength of the electron-phonon coupling determines the energy gap generated at the Fermi surface; we can determine the spatial dimension x0 of the Cooper pairs: The Cooper pairs behave like Bosons, i.e. they condense into a collective wave Current is carried by the ensemble of Cooper pair charges, leading to a slow drift velocity and no scattering from impurities Alex Guerivich, lecture on superconductivity kB=Boltzmann constant =1.38x10-23 D=Debye frequency lep =electron-phonon coupling g=euler constant=0.577 (Uncertainty principle) Superconductivity for Accelerators S. Prestemon

Diamagnetic Behavior of Superconductors: 

Diamagnetic Behavior of Superconductors What differentiates a “perfect” conductor from a diamagnetic material? A perfect conductor apposes any change to the existing magnetic state Superconductivity for Accelerators S. Prestemon

The London Equations: 

Derive starting from the classical Drude model, but adapt to account for the Meissner effect: The Drude model applies classical kinetics to electron motion Assumes static positively charged nucleus, electron gas of density n. Electron motion damped by collisions The penetration depth L is the characteristic depth of the supercurrents on the surface of the material. The London Equations Source of resistance in Drude model; =0 for superconductor First London equation Second London equation Superconductivity for Accelerators S. Prestemon

Classifying Superconductors: 

Classifying Superconductors The density of states ns of the Cooper pairs decreases to zero near a superconducting /normal interface, with a characteristic length x0 (coherence length, first introduced by Pippard in 1953). The two length scales x and lL define much of the superconductors behavior. The coherence length is proportional to the mean free path of conduction electrons; e.g. for pure metals it is quite large, but for alloys (and ceramics…) it is often very small. Their ratio determines flux penetration: From “GLAG” theory, if: x Note: in reality x and lL are functions of temperature Superconductivity for Accelerators S. Prestemon

Type I and II Superconductors: 

Type I and II Superconductors Type I superconductors are characterized by the Meissner effect, i.e. flux is fully expulsed through the existence of supercurrents over a distance lL. Type II superconductors find it energetically favorable to allow flux to enter via normal zones of fixed flux quanta: “fluxoids” or vortices. The fluxoids or flux lines are vortices of normal material of size ~px2 “surrounded” by supercurrents shielding the superconducting material. Superconductivity for Accelerators S. Prestemon

Thermodynamic Critical Field: 

The Gibbs free energy of the superconducting state is lower than the normal state. As the applied field B increases, the Gibbs free energy increases by B2/2m0. The thermodynamic critical field at T=0 corresponds to the balancing of the superconducting and normal Gibbs energies: The BCS theory states that Hc(0) can be calculated from the electronic specific heat (Sommerfeld coefficient): Thermodynamic Critical Field Superconductivity for Accelerators S. Prestemon

Fluxoids: 

Fluxoids Fluxoids, or vortices, are continuous thin tubes characterized by a normal core and shielding supercurrents. The fluxoids in an idealized material subjected to an applied field and in the absence of transport current are uniformly distributed in a triangular lattice so as to minimize the energy state Fluxoids in the presence of current flow (e.g. transport current) are subjected to Lorentz force: Concept of flux-flow and associated heating Solution for real conductors: provide mechanism to pin the fluxoids From Dhalle Superconductivity for Accelerators S. Prestemon

Critical Field Definitions, T=0: 

Hc1: critical field defining the transition from the Meissner state Hc: Thermodynamic critical field Hc2: Critical field defining the transition to the normal state Critical Field Definitions, T=0 M Superconductivity for Accelerators S. Prestemon

Examples of Superconductors: 

Examples of Superconductors Many elements are superconducting at sufficiently low temperatures None of the pure elements are useful for applications involving transport current, i.e. they do not allow flux penetration Superconductors for transport applications are characterized by alloy/composite materials with k>>1 Superconductivity for Accelerators S. Prestemon

Aside – Uses for Type I Superconductors: 

Aside – Uses for Type I Superconductors Although type I superconductors cannot serve for large-scale transport current applications, they can be used for a variety of applications Excellent electromagnetic shielding for sensitive sensors (e.g. lead can shield a sensor from external EM noise at liquid He temperatures Niobium can be deposited on a wafer using lithography techniques to develop ultra-sensitive sensors, e.g. transition-edge sensors Using a bias voltage and Joule heating, the superconducting material is held at its transition temperature; absorption of a photon changes the circuit resistance and hence the transport current, which can then be detected with a SQUID (superconducting quantum interference device) See for example research by J. Clarke, UC Berkeley; Mo/Au bilayer TES detector Courtesy Benford and Moseley, NASA Goddard Superconductivity for Accelerators S. Prestemon

Superconducting Materials Critical Surfaces: 

Superconducting Materials Critical Surfaces The critical surface Jc(B,T,e) defines the boundary between superconducting state and normal conducting state in the space defined by magnetic field, temperature, and current densities. A. Godeke M.N. Wilson Superconductivity for Accelerators S. Prestemon

Outline : 

Outline Superconducting magnets for accelerators: Basics of superconductivity Some historical perspectives The energy gap and electron-phonon coupling Distinguishing perfect conductors and superconductors: the Meissner state Type I and II superconductors, the flux quantum “Pinning” the flux quantum for useful conductors Using superconductivity for accelerators Using the Meissner state for RF applications Using type II superconductors for transport current - magnets Review of magnetic multipoles, and the inverse problem: how to create “perfect” multipole fields Design and fabrication issues with real accelerator magnets Examples of accelerator magnets Superconductivity for Accelerators S. Prestemon

Basics of RF Fields: Normal Metals: 

Basics of RF Fields: Normal Metals We have seen the field profiles in RF cavities For normal conductors, the equations with j=sE yield: Assume 1D Skin depth Hz and Jz follow the same distribution Note influence of skin depth Superconductivity for Accelerators S. Prestemon

Superconducting RF: 

Superconducting RF In the case of a superconductor, in the vicinity of the surface the current can be described by a two-fluid model, with J composed of normal and Cooper-pair electrons: This model assumes sn<<ss Valid for T<<Tc Nb: T~1.9K better than 4.2K We can relate accelerating E-field to surface magnetic field from equations for TM010 mode; Nb is limited to ~57MV/m Note: it is essential that the superconductor remain in the Meissner state; any flux penetration will result in unacceptable thermal loads from flux motion, as well as hysteretic behavior associated with pinning Superconductivity for Accelerators S. Prestemon

Superconducting Cavity Examples: 

Superconducting Cavity Examples From Proch Data from Padamsee, Knobloch, Hayes Superconductivity for Accelerators S. Prestemon

Fabrication Issues: 

Fabrication Issues A key issue with any cavity fabrication is cleanliness Defects, dirt, etc. can contribute to surface heating or field emission Typically require semiconductor-class clean-room From Padamsee, Topical Review Superconductivity for Accelerators S. Prestemon

On to the Next Application…: 

On to the Next Application… Superconducting magnets for accelerators: Basics of superconductivity Some historical perspectives The energy gap and electron-phonon coupling Distinguishing perfect conductors and superconductors: the Meissner state Type I and II superconductors, the flux quantum “Pinning” the flux quantum for useful conductors Using superconductivity for accelerators Using the Meissner state for RF applications Using type II superconductors for transport current - magnets Review of magnetic multipoles, and the inverse problem: how to create “perfect” multipole fields Design and fabrication issues with real accelerator magnets Examples of accelerator magnets Superconductivity for Accelerators S. Prestemon

Multifilament Wires Motivations: 

Multifilament Wires Motivations The superconducting materials used in accelerator magnets are subdivided in filaments of small diameters to reduce magnetic instabilities called flux jumps to minimize field distortions due to superconductor magnetization twisted together to reduce interfilament coupling and AC losses embedded in a copper matrix to protect the superconductor after a quench to reduce magnetic instabilities called flux jumps NbTi LHC wire (A. Devred) Nb3Sn bronze-process wire (A. Devred) NbTi SSC wire (A. Devred) Nb3Sn PIT process wire (A. Devred) Godeke, Nb3Sn Superconductivity for Accelerators S. Prestemon

Multifilament Wires Fabrication of NbTi Multifilament Wires: 

Multifilament Wires Fabrication of NbTi Multifilament Wires Monofilament rods are stacked to form a multifilament billet, which is then extruded and drawn down. Heat treatments are applied to produce pinning centers (-Ti precipitates). When the number of filaments is very large, multifilament rods can be re-stacked (double stacking process). A. Devred, [1] Superconductivity for Accelerators S. Prestemon

Multifilament Wires Fabrication of Nb3Sn Multifilament Wires: 

Multifilament Wires Fabrication of Nb3Sn Multifilament Wires Internal tin process A tin core is surrounded by Nb rods embedded in Cu (Rod Restack Process, RRP) or by layers of Nb and Cu (Modify Jelly Roll, MJR). Each sub-element has a diffusion barrier. Advantage: no annealing steps and not limited amount of Sn Disadvantage: small filament spacing results in large effective filament size (100 m) and large magnetization effect and instability. Non-Cu JC up to 3000 A/mm2 at 4.2 K and 12 T. A. Godeke Superconductivity for Accelerators S. Prestemon

Multifilament Wires Fabrication of Nb3Sn Multifilament Wires: 

Multifilament Wires Fabrication of Nb3Sn Multifilament Wires Powder in tube (PIT) process Nb2Sn powder is inserted in a Nb tube, put into a copper tube. The un-reacted external part of the Nb tube is the barrier. Advantage: small filament size (30 m) and short heat treatment. Disadvantage: fabrication cost. Non-Cu JC up to 2300 A/mm2 at 4.2 K and 12 T. A. Godeke Superconductivity for Accelerators S. Prestemon

Multifilament Wires Fabrication of Nb3Sn Multifilament Wires: 

Multifilament Wires Fabrication of Nb3Sn Multifilament Wires Reaction of a PIT wire: A. Godeke Superconductivity for Accelerators S. Prestemon

Superconducting Cables Fabrication of Rutherford Cable: 

Superconducting Cables Fabrication of Rutherford Cable Rutherford cables are fabricated by a cabling machine. Strands are wound on spools mounted on a rotating drum. Strands are twisted around a conical mandrel into an assembly of rolls (Turk’s head). The rolls compact the cable and provide the final shape. Dan Dietderich, Hugh Higley, Nate Liggins Superconductivity for Accelerators S. Prestemon

Superconducting Cables Fabrication of Rutherford Cable: 

Superconducting Cables Fabrication of Rutherford Cable The final shape of a Rutherford cable can be rectangular or trapezoidal. The cable design parameters are: Number of wires Nwire Wire diameter dwire Cable mid-thickness tcable Cable width wcable Pitch length pcable Pitch angle cable (tancable = 2 wcable / pcable) Cable compaction (or packing factor) kcable i.e the ratio of the sum of the cross-sectional area of the strands (in the direction parallel to the cable axis) to the cross-sectional area of the cable. Typical cable compaction: from 88% (Tevatron) to 92.3% (HERA). Superconductivity for Accelerators S. Prestemon

On to the Next Application…: 

On to the Next Application… Superconducting magnets for accelerators: Basics of superconductivity Some historical perspectives The energy gap and electron-phonon coupling Distinguishing perfect conductors and superconductors: the Meissner state Type I and II superconductors, the flux quantum “Pinning” the flux quantum for useful conductors Using superconductivity for accelerators Using the Meissner state for RF applications Using type II superconductors for transport current - magnets Review of magnetic multipoles, and the inverse problem: how to create “perfect” multipole fields Design and fabrication issues with real accelerator magnets Examples of accelerator magnets Superconductivity for Accelerators S. Prestemon

Field Harmonics: 

Field Harmonics We have seen that the field can be expanded as a power series: It is common to rewrite this as We factorize the main component (B1 for dipoles, B2 for quadrupoles) We introduce a reference radius Rref to have dimensionless coefficients We factorize 10-4 since the deviations from ideal field are 0.01% The coefficients bn, an are called normalized multipoles bn are the normal, an are the skew (adimensional) Superconductivity for Accelerators S. Prestemon

Field Harmonics of a Current Line: 

Field Harmonics of a Current Line Field given by a current line (Biot-Savart law) using !!! we get Jean-Baptiste Biot, French (April 21, 1774 – February 3, 1862) Félix Savart, French (June 30, 1791-March 16, 1841) Superconductivity for Accelerators S. Prestemon

Field Harmonics of a Current Line: 

Now we can compute the multipoles of a current line Field Harmonics of a Current Line Superconductivity for Accelerators S. Prestemon

How to Generate a Perfect Field: 

How to Generate a Perfect Field Perfect dipoles Cos theta: proof – we have a distribution The vector potential reads and substituting one has using the orthogonality of Fourier series Superconductivity for Accelerators S. Prestemon

How to Build a Good Field: Sector Coils for Dipoles: 

How to Build a Good Field: Sector Coils for Dipoles We compute the central field given by a sector dipole with uniform current density j Taking into account of current signs This simple computation is full of consequences B1  current density (obvious) B1  coil width w (less obvious) B1 is independent of the aperture r (much less obvious) For a cos, Superconductivity for Accelerators S. Prestemon

Slide40: 

Multipoles of a sector coil for n=2 one has and for n>2 Main features of these equations Multipoles n are proportional to sin ( n angle of the sector) They can be made equal to zero ! Proportional to the inverse of sector distance to power n High order multipoles are not affected by coil parts far from the centre How to Build a Good Field: Sector Coils for Dipoles Superconductivity for Accelerators S. Prestemon

How to Build a Good Field: Sector Coils for Dipoles: 

How to Build a Good Field: Sector Coils for Dipoles First allowed multipole B3 (sextupole) for =/3 (i.e. a 60° sector coil) one has B3=0 Second allowed multipole B5 (decapole) for =/5 (i.e. a 36° sector coil) or for =2/5 (i.e. a 72° sector coil) one has B5=0 With one sector one cannot set to zero both multipoles … but it can be done with more sectors! wedge Superconductivity for Accelerators S. Prestemon

On to the Next Issue…: 

On to the Next Issue… Superconducting magnets for accelerators: Basics of superconductivity Some historical perspectives The energy gap and electron-phonon coupling Distinguishing perfect conductors and superconductors: the Meissner state Type I and II superconductors, the flux quantum “Pinning” the flux quantum for useful conductors Using superconductivity for accelerators Using the Meissner state for RF applications Using type II superconductors for transport current - magnets Review of magnetic multipoles, and the inverse problem: how to create “perfect” multipole fields Design and fabrication issues with real accelerator magnets Examples of accelerator magnets Superconductivity for Accelerators S. Prestemon

Design Issues: 

Design Issues Superconducting magnets store energy in the magnetic field Results in significant mechanical stresses via Lorentz forces acting on the conductors; these forces must be controlled by structures Conductor stability concerns the ability of a conductor in a magnet to withstand small thermal disturbances, e.g. conductor motion or epoxy cracking, fluxoid motion, etc. The stored energy can be extracted either in a controlled manner or through sudden loss of superconductivity, e.g. via an irreversible instability – a quench In the case of a quench, the stored energy will be converted to heat; magnet protection concerns the design of the system to appropriately distribute the heat to avoid damage to the magnet Superconductivity for Accelerators S. Prestemon

Lorentz Force: Dipole Magnets: 

Lorentz Force: Dipole Magnets The Lorentz forces in a dipole magnet tend to push the coil Towards the mid plane in the vertical-azimuthal direction (Fy, F < 0) Outwards in the radial-horizontal direction (Fx, Fr > 0) Tevatron dipole HD2 Superconductivity for Accelerators S. Prestemon

Lorentz Force: Quadrupole Magnets: 

Lorentz Force: Quadrupole Magnets The Lorentz forces in a quadrupole magnet tend to push the coil Towards the mid plane in the vertical-azimuthal direction (Fy, F < 0) Outwards in the radial-horizontal direction (Fx, Fr > 0) TQ HQ Superconductivity for Accelerators S. Prestemon

Lorentz Force: Solenoids: 

Lorentz Force: Solenoids The Lorentz forces in a solenoid tend to push the coil Outwards in the radial-direction (Fr > 0) Towards the mid plane in the vertical direction (Fy, < 0) Superconductivity for Accelerators S. Prestemon

Stress and Strain Mechanical Design Principles: 

Stress and Strain Mechanical Design Principles LHC dipole at 0 T LHC dipole at 9 T Usually, in a dipole or quadrupole magnet, the highest stresses are reached at the mid-plane, where all the azimuthal Lorentz forces accumulate (over a small area). Displacement scaling = 50 Superconductivity for Accelerators S. Prestemon

Overview of Nb3Sn Coil Fabrication Stages: 

Overview of Nb3Sn Coil Fabrication Stages Cured with matrix Reacted Epoxy impregnated After winding After reaction After impregnation Superconductivity for Accelerators S. Prestemon

Concept of Stability: 

Concept of Stability The concept of stability concerns the interplay between the following elements: The addition of a (small) thermal fluctuation local in time and space The heat capacities of the neighboring materials, determining the local temperature rise The thermal conductivity of the materials, dictating the effective thermal response of the system The critical current dependence on temperature, impacting the current flow path The current path taken by the current and any additional resistive heating sources stemming from the initial disturbance Superconductivity for Accelerators S. Prestemon

Calculation of the Bifurcation Point for Superconductor Instabilities : 

Calculation of the Bifurcation Point for Superconductor Instabilities Ex. RECOVERY of a potential Quench Thanks to Matteo Allesandrini, Texas Center for Superconductivity, for these calculations and slides Superconductivity for Accelerators S. Prestemon

Analysis of SQ02: 

Analysis of SQ02 Time [s] Linear Scale Length [m] Heat deposition Quench Temperature [K] QUENCH with 1 [mJ] Superconductivity for Accelerators S. Prestemon

Analysis of SQ02: Quench Propagation: 

Analysis of SQ02: Quench Propagation Hot Spot temp. profile Tcritical Tsharing QUENCH with 1 [mJ] Superconductivity for Accelerators S. Prestemon

Overview of Accelerator Dipole Magnets: 

Overview of Accelerator Dipole Magnets Tevatron HERA SSC RHIC LHC Superconductivity for Accelerators S. Prestemon