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
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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
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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
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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
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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
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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
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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
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Basics ofSuperconductivity:

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
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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
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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
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The LondonEquations:

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
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ClassifyingSuperconductors:

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
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Type I and IISuperconductors:

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
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ThermodynamicCritical 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
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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
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Critical FieldDefinitions, 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
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Examples ofSuperconductors:

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
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Aside – Uses for Type ISuperconductors:

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
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SuperconductingMaterials 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
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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
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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
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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
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SuperconductingCavity Examples:

Superconducting Cavity Examples From Proch Data from Padamsee, Knobloch, Hayes Superconductivity
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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
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On to the NextApplication…:

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
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Multifilament WiresMotivations:

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
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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
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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
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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
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Multifilament Wires Fabrication of Nb3Sn Multifilament Wires Reaction of a PIT wire: A. Godeke Superconductivity
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Superconducting CablesFabrication 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
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Superconducting CablesFabrication 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 (tancable = 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
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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
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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
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Field Harmonicsof 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
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Field Harmonicsof a Current Line:

Now we can compute the multipoles of a current line
Field Harmonics of a Current Line Superconductivity
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Overview of Nb3SnCoil Fabrication Stages:

Overview of Nb3Sn Coil Fabrication Stages Cured with matrix Reacted Epoxy impregnated After winding After reaction After impregnation Superconductivity
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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
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Calculation of the Bifurcation Pointfor 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
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Analysis of SQ02:

Analysis of SQ02 Time [s] Linear Scale Length [m] Heat deposition Quench Temperature [K] QUENCH with 1 [mJ] Superconductivity
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Analysis of SQ02:Quench Propagation:

Analysis of SQ02: Quench Propagation Hot Spot temp. profile Tcritical Tsharing QUENCH with 1 [mJ] Superconductivity
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Overview of AcceleratorDipole Magnets:

Overview of Accelerator Dipole Magnets Tevatron HERA SSC RHIC LHC Superconductivity
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