# Presentation: 591606

## Superconductivity for Accelerators

**Superconductivity for Accelerators**

**Soren Prestemon**
**Lawrence Berkeley National Laboratory**

**Lecture No. 12**

## 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	

- 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

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## 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

- 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

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## 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

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## History – High Temperature Superconductors

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

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## 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

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## 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

- 

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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 **** 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*

*k**B*=Boltzmann constant =1.38x10-23
- D=Debye frequency
- ep =electron-phonon coupling
- =euler constant=0.577

*(Uncertainty principle)*

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## Diamagnetic Behavior of Superconductors

- What differentiates a “perfect” conductor from a diamagnetic material?

- A perfect conductor apposes any change to the existing magnetic state

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## 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 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*

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## ClassifyingSuperconductors

- The density of states ns of the Cooper pairs decreases to zero near a superconducting /normal interface, with a characteristic length   (coherence length, first introduced by Pippard in 1953)The two length scales  andL 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 ************ and ************L ******are functions of temperature***

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## 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 L.

**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 ~ “surrounded” by supercurrents shielding the superconducting material.

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## 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 B2/20. 
 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):

- 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 *B**2**/2****0*.
-  The thermodynamic critical field at *T=0* corresponds to the balancing of the superconducting and normal Gibbs energies:
- The BCS theory states that *H**c**(0)* can be calculated from the electronic specific heat (Sommerfeld coefficient):

**Thermodynamic****Critical Field**

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## 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*

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## Hc1: critical field defining the transition from the Meissner state

Hc: Thermodynamic critical field

Hc2: Critical field defining the transition to the normal state

*H**c**: *Thermodynamic critical field
*H**c2**: *Critical field defining the transition to the normal state

**Critical Field****Definitions, T=0**

*M*

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## 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 >>1

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## 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*

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## SuperconductingMaterials Critical Surfaces

- The critical surface *J**c**(B,T,****)* 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**

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## 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

- We have seen the field profiles in RF cavities
- For normal conductors, the equations with *j=****E* yield:

*Assume 1D*

*Skin depth*

*H**z** *and *J**z* follow the same distribution

- Note influence of skin depth

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## 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 ***n**<<****s*
- Valid for *T<<Tc*
- Nb: *T~1.9*K better than *4.2*K

*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*

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## SuperconductingCavity Examples

*From Proch*

| 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*

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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*

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## 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 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,*
*Nb**3**Sn*

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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]**

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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 *J**C* up to 3000 A/mm2 at 4.2 K and 12 T.

**A. Godeke**

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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 *J**C* up to 2300 A/mm2 at 4.2 K and 12 T.

**A. Godeke**

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## Multifilament Wires Fabricationof Nb3Sn Multifilament Wires

- Reaction of a PIT wire:

**A. Godeke**

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## 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*

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## 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 *N**wire*
- Wire diameter *d**wire*
- Cable mid-thickness *t**cable*
- Cable width *w**cable*
- Pitch length *p**cable*
- Pitch angle ***cable  *(tan***cable *= 2 *w**cable* / *p**cable*)
- Cable compaction (or packing factor) *k**cable*
- 	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).

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## 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

- 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 (*B*1 for dipoles, *B*2 for quadrupoles)
- We introduce a reference radius *R*ref to have dimensionless coefficients
- We factorize 10-4 since the deviations from ideal field are 0.01%
- The coefficients *b*n, *a*n are called _normalized multipoles_
*b*n are the _normal_, *a*n are the _skew_ (adimensional)

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## 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)

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## Now we can compute the multipoles of a current line

**Field Harmonics****of a Current Line**

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## 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

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## 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
*B**1 *** current density (obvious)
*B**1 *** coil width *w* (less obvious)
*B**1* is independent of the aperture *r* (much less obvious)
- For a cos**,

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## 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

- 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**

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

- First allowed multipole *B*3 (sextupole)
- for =/3 (i.e. a 60° sector coil) one has *B*3=0
- Second allowed multipole *B*5 (decapole)
- for =/5 (i.e. a 36° sector coil) or for =2/5 (i.e. a 72° sector coil)
- one has *B*5=0
- With one sector one cannot set to zero both multipoles ... but it can be done with more sectors!

- wedge

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## 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

- 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

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## 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 (*F**y, **F*** < 0)
- Outwards in the radial-horizontal direction (*F**x, **F**r* > 0)

- Tevatron dipole

- HD2

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## 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 (*F**y, **F*** < 0)
- Outwards in the radial-horizontal direction (*F**x, **F**r* > 0)

- TQ

- HQ

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## Lorentz Force:Solenoids

- The Lorentz forces in a solenoid tend to push the coil
- Outwards in the radial-direction  (*F**r* > 0)
- Towards the mid plane in the vertical direction (*F**y, *< 0)

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## 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

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## Overview of Nb3SnCoil Fabrication Stages

- Cured with matrix                    Reacted                  Epoxy impregnated

**After winding**

**After reaction**

**After impregnation**

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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 Pointfor Superconductor Instabilities 

- Ex. RECOVERY of a potential Quench

*Thanks to Matteo Allesandrini, Texas Center for Superconductivity,  for these calculations and slides*

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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]

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## Analysis of SQ02:Quench Propagation

**Analysis of SQ02:****Quench Propagation**

- Hot Spot temp. profile

- Tcritical

- Tsharing

- QUENCH with 1 [mJ]

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## Overview of AcceleratorDipole Magnets

**Tevatron**

**HERA**

**SSC**

**RHIC**

**LHC**

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