Superconducting Magnets for Accelerators: Design and Applications

1. Lecture 28: Accelerator Magnets Superconducting magnets Peter N. OstroumovProfessor of Physics Michigan State University

2. Superconducting magnets for accelerators • Basics of superconductivity • Type I and Type II • SC wires and cables • Multifilament wires • Magnetization • Quench and protection • Training • Field calculations and coil design • Electromagnetic forces and mechanical stresses • Example of LHC magnets P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

3. Design of superconducting magnets involves many aspects • Material Science • Field generation • Fabrication techniques • Thermal considerations • Mechanical Analysis • Quench Protection • Cryogenics P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

4. 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 BCS breakthrough: Fermi surface is unstable to bound states of electron-pairs kb=Boltzmann constant =1.38x10-23 D=Debye frequency lep =electron-phonon coupling g=Euler constant=0.577 P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

5. Remove B Apply B Cool Apply B Cool Remove B Diamagnetic behavior of superconductors • What differentiates a “perfect” conductor from a diamagnetic material? • All Type I superconductors experience Meissner effect • Type II superconductors experience Meissner effect at fields < B c1 A perfect conductor apposes any change to the existing magnetic state Apply B Cool Remove B Superconductors exhibit diamagnetic behavior: flux is always expulsed - Meissner effect P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

6. Type II 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 • Current can flow only in the surface layer, ~20 nm, no currents allowed in the interior as they generate magnetic field inside the bulk • Superconductors for transport applications are characterized by alloy/composite materials P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

7. Fluxoids in type II superconductors • Fluxoids, or flux lines, are continuous thin tubes characterized by a normal core and shielding supercurrents. • The flux contained in a fluxoid is quantized: • The fluxoids in an idealized material 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: P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

8. Pinning of fluxoids • If fluxoids move, they generate heat • Fluxoidscan be pinned by a wide variety of material defects • Inclusions • Under certain conditions, small inclusions of appropriate materials can serve as pinning site locations; this suggests tailoring the material artificially through manufacturing • Lattice dislocations / grain boundaries • These are known to be primary pinning sites. Superconductor materials for wires are severely work hardened so as to maximize the number and distribution of grain boundaries. • Precipitation of other material phases • In NbTi, mild heat treatment can lead to the precipitation of an a-phase Ti-rich alloy that provides excellent pinning strength. P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

9. Superconductivity makes possible large accelerators with fields well above 2 T • Critical surface: current density depends from the magnetic field and temperature • The surface, determined experimentally, can be fitted with parameterization curves • Critical current density depends on processing P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

10. Introduction and history • Nb3Sn and NbTi, discovered in 1954 and 1961, are the most commonly used type II superconductors (80-90% of all superconducting devices). • Since their critical temperature Tc is 9 K (for NbTi) and 18 K (for Nb3Sn) at 0 T, they are defined as low temperature superconductors. • High temperature superconductors (HTS) have a Tc up to 80-120 K. • For practical applications, superconducting materials are usually produced in small filaments and surrounded by a stabilizer (typically copper) to form a multifilament wire or strand. • A superconducting cable is usually composed by several wires: multistrand cable. Multifilament wire Flat multistrand cable 1mm ~15 mm

11. Example • Superconducting cables winding in a dipole magnet Cross-section Beam P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

12. Superconducting wires LHC • The superconducting materials used in accelerator magnets are • subdivided in filaments of small diameters • 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 • NbTifilament diameters are usually less than 50 µm. • wire diameter = 0.3 - 1.0mm SSC P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

13. Multifilament wires • Interfilament coupling • When a multifilamentary wire is subjected to a time varying magnetic field, current loops are generated between filaments. • If filaments are straight, large loops are generated, with large currents • Big losses • If the strands are magnetically coupled the effective filament size is larger • Flux jumps • To reduce these effects, filaments are twisted with a twist pitch of the order of 20-30 times of the wire diameter. P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

14. Magnetization • When a superconductor is subjected changing magnetic field, screening currents are induced to flow • Screening currents are in addition to the transport current, which comes from the power supply • They are like eddy currents but, because there is no resistance, they don't decay • dB/dtinduces an electric field E which causes screening currents to flow at critical current density Jc • Uniform Jc means a constant field gradient inside the superconductor • Magnetized cable distorts magnetic field in the aperture • Correction is with properly placed very small permanent magnets P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

15. Multifilament wires: motivation • Quench protection • Superconductors have a very high normal state resistivity. It can be shown that a filament of NbTi, if quenched in free space, could reach very high temperatures in few ms. • If the filament is embedded in a copper matrix, when a quench occurs, the current redistributes in the low-resistivity matrix and the peak temperature can typically be maintained below 300 K. • The copper matrix facilitates quench protection: it allows the quench to propagate and it provides time to act on the power circuit. • In the case of a small volume of superconductor heated beyond the critical temperature (for instance because of a flux jump), the current can flow in the copper for a short moment, allowing the filament to cool-down and recover superconductivity. • The matrix also helps stabilizing the conductor against flux jumps (dynamic stability). • Flux jump: heat generation due to screening currents and uncontrollable temperature increase P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

16. Cable insulation • The cable insulation must feature • Good electrical properties to withstand high turn-to-turn voltage after a quench. • Good mechanical properties to withstand high pressure conditions • Porosity to allow penetration of helium (or epoxy) • Radiation hardness • In NbTi magnets the most common insulation is a series of overlapped layers of polyimide (kapton). • In the LHC case: • two polyimide layers 50.8 µm thick wrapped around the cable with a 50%overlap, with another adhesive polyimide tape 68.6 µm thick wrapped with a spacing of 2 mm. P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

17. Conductor limited quench • The maximum field seen by the conductor in the coil is usually called peak field (Bpeak). • At a given temperature T0, the maximum current that the conductor can reach will be Imax= Ic (Bpeak, T0), where Icis the critical current at Bpeakand T0. • When the magnet quenches, we can have either Iquench= Imaxor Iquench< Imax. • In the first case we have a conductor-limited quench • If the current is lower, we have degradation P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

18. l T A Tcs Tc T0 T0 Conductor length Energy deposited quench • We start considering a wire made purely of superconductor. • Let’s assume that a certain amount of energy E increased the temperature of the superconductor beyond Tcover a length l. The segment l of superconductor is dissipating power given by [W]. • Part (or all) of the heat is conducted out of the segment because of the thermal gradient, which can be approximated as (Tc-To)/(l/2). Therefore, when the power dissipated equals the power conducted away • Which results in • k [Wm-1K-1] is the thermal conductivity P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

19. Energy deposited quenches Point disturbances • The length ldefines the minimum propagation zone (and minimum quench energy ). • A normal zone longer that lwill keep growing (quench). A normal zone shorter than l will collapse. • An example • A typical NbTi 6 T magnet has the following properties • Jc = 2  109 A m-2 •  = 6.5  10-7m (normal conducting state) • k = 0.1 W m-1 K-1 • Tc = 6.5 K • To = 4.2 K • In this case, l = 0.5 m and, assuming a 0.3 mm diameter, the required energy to bring to Tc is 10-9 J. • A wire made purely of superconductor, without any stabilizer (like copper) around, would quench • In order to increase l,since we do not want to reduce Jc, we have to increase k/: we need acomposite conductor! P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

20. Energy deposited quenchesComposite conductor • We now consider the situation where the superconductor is surrounded by material with low resistivity and high conductivity. • Copper can have at 4.2 K • Resistivity  = 3  10-10m (instead of 6.5  10-7 m for NbTi) • k = 350 W m-1K-1 (instead of 0.1 W m-1 K-1for NbTi). • We can therefore increase k/ by almost a factor 107. • A significant improvement was achieved in the early years of superconducting magnet development after the introduction of composite conductor • Both for flux jump and stability viewpoint Copper P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

21. Quench protection • A superconducting accelerator magnet has a large magnetic stored energy • A quench produces a resistive zone • Current is flowing through the magnet • The challenge of the protection is to provide a safe conversion of the magnetic energy to heat in order to minimize • Peak temperature (“hot spot”) and temperature gradients in the magnet • Peak voltages • The final goal being to avoid any magnet degradation • High temperature => damage to the insulation or stabilizer • Large temperature gradient => damage to the conductor due to differential thermal expansion of materials P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

22. General quench protection diagram P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

23. Quench protection P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

24. Training of magnets • Training is characterized by two phenomena • The occurrence of premature quenches • What are the causes? • The progressive increase of quench current • Something not reversible happens, or, in other words, the magnet is somehow “improving” or “getting better” quench after quench. • Some irreversible change in the coil’s mechanical status is occurring. • In R&D magnets, training may not be an issues. • For accelerator magnets it can be expensive, both in term of time and cost. P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

25. Training causes • Mechanically induced quenches are considered the main causes of training in a superconducting magnets. • Frictional motion of a superconductor • During excitation electromagnetic forces determine conductor motion; any motion of a conductor in a frictional environment produces heat. • After each quench, the coil is partially locked by friction in a new and more secure state which allows the conductors to withstand higher levels of electro-magnetic forces. • Epoxy failure • Under the stress status induced by the mechanical structure and the e.m. forces, the coil stores strain energy. When a crack is initiated and propagates (for example in the resin), part of the original strain energy is dissipated as heat. • Premature quenches produced by epoxy cracking take place when the stresses in the winding exceed the epoxy’s fracture stress. Once the epoxy is locally fractured, further cracking appears only when the e.m. stress is increased. P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

26. Field calculations • How to generate a perfect field • Dipoles: cos, intersecting ellipses • Quadrupoles: cos2, intersecting ellipses • How to build a good field with a sector coil • Dipoles • Quadrupoles P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

27. Perfect dipole field -1 • Pros: • mechanical structure and winding look easy • Cons: • the coil is infinite • truncation gives reasonable field quality only for rather large height the aperture radius (very large coil, not effective) A practical winding with flat cables P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

28. Perfect dipole field - 2 • Intersecting ellipses: a uniform current density in the area of two intersecting ellipses produces a pure dipole field • the aperture is not circular • the shape of the coil is not easy to wind with a flat cable (ends?) • need of internal mechanical support that reduces available aperture • (Homework) Prove that intersecting circles give perfect field within a cylinder carrying uniform current j0, the field is perpendicular to the radial direction and proportional to the distance to the center r: Homework A practical winding with flat cables Intersecting ellipses P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

29. Perfect dipole field - 3 • cos : a current density proportional to cos in an annulus - it can be approximated by sectors with uniform current density • self supporting structure • the aperture is circular, the coil is compact • winding is manageable Cable block wedge A practical winding with three layers and no wedges Artist view of a cosmagnet A practical winding with one layer and wedges An ideal cos P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

30. Perfect quadrupole field • cos2 : a current density proportional to cos2, in an annulus - approximated by sectors with uniform current density and wedges • Two intersecting ellipses • Perfect sextupoles: cos3, or three intersecting ellipses • Perfect 2n-poles: cosn, or n intersecting ellipses Quadrupole as an ideal cos2 Quadrupole as two intersecting ellipses P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

31. Biot-Savart low • Infinite wire carrying current I P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

32. Magnetic field in the complex plane (z-plane) • To avoid coefficients with physical dimensions depending on the order of the multipoles, a reference radius Rref is usually defined • Multipole expansion is • Main (dipole) component is P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

33. Dipole component • 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) P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

34. Sector coils for dipoles • A dipolar symmetry is characterized by • Up-down symmetry (with same current sign) • Left-right symmetry (with opposite sign) • Why this configuration? • Opposite sign in left-right is necessary to avoid that the field created by the left part is canceled by the right one • In this way all multipoles except B2n+1are canceled these multipoles are called “allowed multipoles” • Remember the power law decay of multipoles with order • And that field quality specifications concern only first 10-15 multipoles • The field quality optimization of a coil lay-out concerns only a few quantities ! Usually b3 , b5 , b7 , and possibly b9 , b11 P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

35. Examples • By using several sectors higher order terms can be zeroed • Let us see two coil lay-outs of real magnets • The Tevatron has two blocks on two layers – with two layers one can set to zero b3and b5 • The RHIC dipole has four blocks Tevatron main dipole - 1980 RHIC main dipole - 1995 P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

36. Forces • Superconducting accelerator magnets are characterized by high fields and high current densities. • As a results, the coil is subjected to strong electro-magnetic forces, which tend to move the conductor and deform the winding. • A good knowledge of the magnitude and direction of the electro-magnetic forces, as well as of the stress of the coil, is mandatory for the mechanical design of a superconducting magnet. • In dipole and quadrupole magnets, the forces are directed towards the mid-plane and outwardly. • They tend to separate the coil from the pole and compress the mid-plane region • Axially they tend to stretch the windings P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

37. Electro-magnetic forceDipole magnets Tevatron dipole P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

38. End forces in dipole and quadrupole magnets • In the coil ends the Lorentz forces tend to push the coil • Outwards in the longitudinal direction (Fz > 0) • Similarly as for the solenoid, the axial force produces an axial tension in the coil straight section. The deflections are enlarged P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

39. Collars • How to make an external structure that • fits tightly round the coil • presses it into an accurate shape • has low ac losses • can be mass produced cheaply • Answer: make collars by precision stamping of stainless steel or aluminum alloy plate a few mm thick • inherited from conventional magnet laminations P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

40. LHC dipole • Usually, in a dipole or quadrupole magnet, the highest stresses are reached at the mid-plane, where all the azimuthal e.m. forces accumulate (over a small area). LHC dipole at 0 T LHC dipole at 9 T 194 mm 400 mm P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

41. Iron yoke • Iron laminations are added around the collared coil • The iron has several impacts • Useful for shielding, can considerably increase the field for a given current – the impact on the performance is small but not negligible • Drawbacks: saturation, inducing field harmonics at high field – can be cured by shaping or drilling holes in the right place • Coil ends – the design must aim at reducing the peak field • Pure iron becomes brittle at low temperature • The tensile forces are therefore taken by a stainless steel shell which is welded around the iron, while still in the press LHC magnet P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

42. LHC magnet P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

43. Parameters of LHC magnet • Operational Field 8.36 T • Coil aperture 56 mm • Magnetic length 14.2 m • Operating Current 11500 A • Operating temperature 1.9K • Distance between aperture axes (cold) 194.0 mm • Outer diameter of cold mass 570 mm • Overall cold mass length 15.14 m • Outer diameter of cryostat 914 mm • Self-inductance, both channel together 119 mH • Stored energy, both channels together 7.4 MJ • Weight of cold mass 31 metric ton P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

44. Conclusion - 1 • Superconductivity offers higher magnetic fields and field gradients, with less energy dissipation than in RT magnets • NbTiis the most common superconducting material and has been used in all accelerators to date. • Superconducting magnets do not use iron to shape the field so must use special winding shapes • Magnets don’t reach their expected current/field first time but show training • Control training by reducing movement, a tension to contraction and increasing Maximum Quench Energy • Persistent screening currents produce magnetization of the superconductor which causes field errors and AC loss – need fine ~ 5 μm filaments • Accelerators need high currents so must use many wires in parallel - a cable P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

45. Conclusion - 2 • Coupling between filament in wire and between wires in cable increases magnetization • Magnets store large inductive energy which is released at quench. The heating must protect • Magnet manufacturing techniques have been developed to ensure accurate winding shape and minimize conductor movement • Large electromagnet forces and stresses are developed during the operation • Proper mechanical design must be provided to avoid premature quenches and file distortions Attend Lucio Rossi (CERN) seminar next Friday, December 7 “Development of SC magnets for LHC high luminosity upgrade” P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

46. 2-Tesla iron dominated SC magnet, FRIB • SC coil is used to reduce electrical power consumption and overall cost of the magnet BH-curve Typical range P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets

47. P.N. Ostroumov PHY862 Accelerator Systems, topic: SC magnets