Fundamental%20Accelerator%20Theory,%20Simulations%20and%20Measurement%20Lab%20

1
Lecture No. 12
Superconductivity for Accelerators
Soren Prestemon Lawrence Berkeley National
Laboratory
Fundamental Accelerator Theory, Simulations and
Measurement Lab Michigan State University,
Lansing June 4-15, 2007
2
References
Superconductivity for Accelerators S. Prestemon

• 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
  fieldsMartin 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.

3
Outline
Superconductivity for Accelerators S. Prestemon

• 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

4
History
Superconductivity for Accelerators S. Prestemon

• 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
5
History - Theory
Superconductivity for Accelerators S. Prestemon

• A theory of superconductivity took time to
  evolve
• 1935 London brothers propose two equations for E
  and H results in concept of penetration depth ?
• 1950Ginzburg and Landau propose a macroscopic
  theory (GL) for superconductivity, based on
  Landaus theory of second-order phase transitions

Heinz and Fritz London
Abrikosov, With Princess Madeleine
Ginzburg and Landau (circa 1947) Nobel Prize
1962 Landau Nobel Prize 2003 Ginzburg,
Abrikosov, Leggett (the GLAG members)
6
History - Theory
Superconductivity for Accelerators S. Prestemon

• 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

7
History High Temperature Superconductors
Superconductivity for Accelerators S. Prestemon

• 1986 Bednorz and Muller discover
  superconductivity at high temperatures in layered
  materials comprising copper oxide planes

8
General Principals
Superconductivity for Accelerators S. Prestemon

• 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

Copper
Aluminum
M. Wilson
RRR?(273K)/ ?(4K))
9
Basics ofSuperconductivity
Superconductivity for Accelerators S. Prestemon

• 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
10
Cooper Pairs
Superconductivity for Accelerators S. Prestemon

• 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
(Uncertainty principle)
kBBoltzmann constant 1.38x10-23 ?DDebye
frequency lep electron-phonon coupling geuler
constant0.577
11
Diamagnetic Behavior of Superconductors
Superconductivity for Accelerators S. Prestemon

• What differentiates a perfect conductor from a
  diamagnetic material?

A perfect conductor apposes any change to the
existing magnetic state
12
The LondonEquations
Superconductivity for Accelerators S. Prestemon

• 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.

Source of resistance in Drude model 0 for
superconductor
First London equation
Second London equation
13
ClassifyingSuperconductors
Superconductivity for Accelerators S. Prestemon

• 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

Note in reality x and lL are functions of
temperature
x
14
Type I and IISuperconductors
Superconductivity for Accelerators S. Prestemon

• 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.

15
ThermodynamicCritical Field
Superconductivity for Accelerators S. Prestemon

• 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 T0
  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)

16
Fluxoids
Superconductivity for Accelerators S. Prestemon

• 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
17
Critical FieldDefinitions, T0
Superconductivity for Accelerators S. Prestemon

• Hc1 critical field defining the transition from
  the Meissner state
• Hc Thermodynamic critical field
• Hc2 Critical field defining the transition to
  the normal state

M
18
Examples ofSuperconductors
Superconductivity for Accelerators S. Prestemon

• 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
  kgtgt1

19
Aside Uses for Type ISuperconductors
Superconductivity for Accelerators S. Prestemon

• 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
20
SuperconductingMaterials Critical Surfaces
Superconductivity for Accelerators S. Prestemon

• 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.

M.N. Wilson
A. Godeke
21
Outline
Superconductivity for Accelerators S. Prestemon

• 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

22
Basics of RF FieldsNormal Metals
Superconductivity for Accelerators S. Prestemon

• We have seen the field profiles in RF cavities
• For normal conductors, the equations with jsE
  yield

Assume 1D
Skin depth
Hz and Jz follow the same distribution
Note influence of skin depth
23
Superconducting RF
Superconductivity for Accelerators S. Prestemon

• 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 snltltss
• Valid for TltltTc
• Nb T1.9K better than 4.2K

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
We can relate accelerating E-field to surface
magnetic field from equations for TM010 mode Nb
is limited to 57MV/m
24
SuperconductingCavity Examples
Superconductivity for Accelerators S. Prestemon
Type SC Normal
Q0 2x109 2x104
P/L W/m, 1MV/m 1.5 56000
Room temp power kW/m, 1MV/m 0.54 112
Room temp power kW/m, 1MV/m 13.5 2800
Data from Padamsee, Knobloch, Hayes
From Proch
25
Fabrication Issues
Superconductivity for Accelerators S. Prestemon

• 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
26
On to the NextApplication
Superconductivity for Accelerators S. Prestemon

• 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

27
Multifilament WiresMotivations
Superconductivity for Accelerators S. Prestemon

• 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)
NbTi SSC wire (A. Devred)
Godeke, Nb3Sn
Nb3Sn bronze-process wire (A. Devred)
Nb3Sn PIT process wire (A. Devred)
28
Multifilament Wires Fabricationof NbTi
Multifilament Wires
Superconductivity for Accelerators S. Prestemon

• 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
29
Multifilament Wires Fabricationof Nb3Sn
Multifilament Wires
Superconductivity for Accelerators S. Prestemon

• 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
30
Multifilament Wires Fabricationof Nb3Sn
Multifilament Wires
Superconductivity for Accelerators S. Prestemon

• 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
31
Multifilament Wires Fabricationof Nb3Sn
Multifilament Wires
Superconductivity for Accelerators S. Prestemon

• Reaction of a PIT wire

A. Godeke
32
Superconducting CablesFabrication of Rutherford
Cable
Superconductivity for Accelerators S. Prestemon

• 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 (Turks head). The rolls
  compact the cable and provide the final shape.

Dan Dietderich, Hugh Higley, Nate Liggins
33
Superconducting CablesFabrication of Rutherford
Cable
Superconductivity for Accelerators S. Prestemon

• 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).

34
On to the Next Application
Superconductivity for Accelerators S. Prestemon

• 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

35
Field Harmonics
Superconductivity for Accelerators S. Prestemon

• 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)

36
Field Harmonicsof a Current Line
Superconductivity for Accelerators S. Prestemon

• Field given by a current line (Biot-Savart law)
• using
• !!!
• we get

Félix Savart, French (June 30, 1791-March 16,
1841)
Jean-Baptiste Biot, French (April 21, 1774
February 3, 1862)
37
Field Harmonicsof a Current Line
Superconductivity for Accelerators S. Prestemon

• Now we can compute the multipoles of a current
  line

38
How to Generate a Perfect Field
Superconductivity for Accelerators S. Prestemon

• Perfect dipoles
• Cos theta proof we have a distribution
• The vector potential reads
• and substituting one has
• using the orthogonality of Fourier series

39
How to Build a Good Field Sector Coils for
Dipoles
Superconductivity for Accelerators S. Prestemon

• 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?,

40
How to Build a Good Field Sector Coils for
Dipoles
Superconductivity for Accelerators S. Prestemon

• Multipoles of a sector coil
• for n2 one has
• and for ngt2
• 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

41
How to Build a Good Field Sector Coils for
Dipoles
Superconductivity for Accelerators S. Prestemon

• First allowed multipole B3 (sextupole)
• for ??/3 (i.e. a 60 sector coil) one has B30
• 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 B50
• With one sector one cannot set to zero both
  multipoles but it can be done with more sectors!

wedge
42
On to the Next Issue
Superconductivity for Accelerators S. Prestemon

• 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

43
Design Issues
Superconductivity for Accelerators S. Prestemon

• 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

44
Lorentz ForceDipole Magnets
Superconductivity for Accelerators S. Prestemon

• The Lorentz forces in a dipole magnet tend to
  push the coil
• Towards the mid plane in the vertical-azimuthal
  direction (Fy, F? lt 0)
• Outwards in the radial-horizontal direction (Fx,
  Fr gt 0)

Tevatron dipole
HD2
45
Lorentz ForceQuadrupole Magnets
Superconductivity for Accelerators S. Prestemon

• The Lorentz forces in a quadrupole magnet tend to
  push the coil
• Towards the mid plane in the vertical-azimuthal
  direction (Fy, F? lt 0)
• Outwards in the radial-horizontal direction (Fx,
  Fr gt 0)

TQ
HQ
46
Lorentz ForceSolenoids
Superconductivity for Accelerators S. Prestemon

• The Lorentz forces in a solenoid tend to push the
  coil
• Outwards in the radial-direction (Fr gt 0)
• Towards the mid plane in the vertical direction
  (Fy, lt 0)

47
Stress and Strain Mechanical Design Principles
Superconductivity for Accelerators S. Prestemon

• 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
48
Overview of Nb3SnCoil Fabrication Stages
Superconductivity for Accelerators S. Prestemon
After impregnation
After winding
After reaction
Cured with matrix Reacted
Epoxy impregnated
49
Concept of Stability
Superconductivity for Accelerators S. Prestemon

• 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

50
Calculation of the Bifurcation Pointfor
Superconductor Instabilities
Superconductivity for Accelerators S. Prestemon
Thanks to Matteo Allesandrini, Texas Center for
Superconductivity, for these calculations and
slides
Ex. RECOVERY of a potential Quench
51
Analysis of SQ02
Superconductivity for Accelerators S. Prestemon
QUENCH with 1 mJ
Linear Scale
Quench
Temperature K
Heat deposition
Length m
Time s
52
Analysis of SQ02Quench Propagation
Superconductivity for Accelerators S. Prestemon
QUENCH with 1 mJ
Hot Spot temp. profile
Tcritical
Tsharing
53
Overview of AcceleratorDipole Magnets
Superconductivity for Accelerators S. Prestemon
Tevatron
HERA
SSC
RHIC
LHC