Introduction to particle accelerators

Walter Scandale

CERN - AT department

Roma, marzo 2006

Lecture III - superconducting devices

topics Limitations of normal conducting dipoles Superconducting material properties

Critical temperature Type I and type II superconductors Theoretical approaches

Meissner effect Cooper pairs and BCS theory

MgB2 and HTS SC dipoles

Current density Magnetization Flux jumping Quenches Wires and cables

SC-RF

Dipoles Iron yoke reduces magnetic reluctance (reduced power and ampere-turns) -> small gap height

Field quality -> determined by the pole shape

Field saturation -> 2 Tesla (BEarth = 3 10-5 Tesla)

B > 2 Tesla -> use SC magnets BLHC = 8.4 Tesla

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B = μμ0H = μ0

N ⋅ Ih

H∫ ds = I ⋅N = h ⋅H0 + l ⋅H iron

Powering a resistive magnet I ≈ 5 kA for 1.8 T I ≈ 3·105 A for 10 T R ≈ 1 m P = R·I2

PLEP = 20 kW/magnet PLHC = 100 MW/magnet (if resistive)

Abolish Ohm’s Law! no power consumption (although do need refrigeration power)

high current density ampere turns are cheap, so we don’t need iron (although often use it for shielding)

Consequences lower power bills higher magnetic fields -> reduced bending radius

smaller rings reduced capital cost new technical possibilities (eg muon

collider) higher quadrupole gradients

higher luminosity

K. Onnes 1911

What is a superconductor

Resistance of Mercury falls suddenly below measurement accuracy at very low temperature

1908 Heinke Kemerlingh Onnes achieves very low temperature producing liquid He (< 4.2 K) 1911 Onnes and Holst observe sudden drop in resistivity to essentially zero SC era starts 1914 Persistent current experiments (Onnes) 1933 Meissner-Ochsenfeld effect observed 1935 Fritz and London theory 1950 Ginsburg - Landau theory 1957 BCS Theory (Bardeen, Coper, Schrieffer) 1962 Josephson effect is observed 1967 Observation of Flux Tubes in Type II superconductors (Abrikosov, Ginzburg, Leggett) 1980 Tevatron: The first accelerator using superconducting magnets 1986 First observation of Ceramic Superconductor at 35 K (Bednorz, Muller) 1987 first ceramic superconductor at 92 K (above liquid Nitrogen at 77 K !) HTS era starts 2003 discovery of a metallic compound the B2Mg superconducting at 39 K (x2 Tc of Nb3Sn)

It took ~70 years to get first accelerator from conventional superconductors. How long will it take for HTS or B2Mg to get to accelerator magnets? Have patience!

Short history of superconductivity

What is a superconductorBelow Tc the B-field lines are expelled out of a superconductor (perfect diamagnetic behaviour)

Type I superconductorsthe superconductivity disappears as T > Tc | B > Bc | J > Jc

Type II superconductorsFor Bc1 < B < Bc2 there is a partial flux penetration through fluxoid vortexes and a mixed phase

Below the critical temperature Tc the resistivity drops

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ρ T( ) = ρ 0 + cT 5

phonon-e- interaction

Cooper pair appearance Meissner 1933

T < Tc

B < Bc

B = 0

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Meissner effect and magnetization

Inside the SC material one has B = 0 E = 0 (otherwise there is an infinite current flowing !) There is a superficial screening current inducing a diamagnetic polarization M = -H/4 = H The B field penetrate with an exponentially decaying intensity B(s) = B(0)exp(-s/L)

BCS theorySuperconducting state

Tc ~ 1/ √Misotopic -> phonons should play a role in superconductivity

Creation of Cooper pairs (over-screening effect) An e- attracts the surrounding ion creating a region of increased positive charge

The lattice oscillations enhance the attraction of another passing by e- (Cooper pair)

The interaction is strengthened by the surrounding sphere of conduction e- (Pauli principle)

In a superconductor the net effect of e-e- attraction through phonon interaction and the e-e- coulombian repulsion is attractive and the Cooper pair becomes a singlet state with zero momentum and zero spin

To break a pair the excitation energy is ∆E = 2∆

Normal conducting state

Predictions of the BCS theory

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Δ 0K( ) ≈1.76 ⋅kBTc

Δ T( ) ≈1.74 ⋅Δ 0K( ) 1−T

Tc

⎛

⎝ ⎜

⎞

⎠ ⎟

12

Hc T( ) = Hc 0K( ) 1− aT

Tc

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ with a constant of the SC

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Δ ≈ hπ ωDe

−1

F ⋅NF

€

h2π ωD ≈ kBTD Debye phonon energy

F effective potential

NF density of Fermi states

BCS theory NOT valid

in the ceramic SC.

Problems with SC type II

Energy bond of a Cooper pair

Size of a Cooper pair 100 nmLattice spacing 0.1 ÷ 0.4 nm

More on type I and II superconductors

Type I: not good for accelerator magnets

Also known as the “soft superconductors”.

Completely exclude the flux lines. Allow only small field (Bc < 0.1 T).

Type II: allow much higher fields Also known as the “hard superconductors”. Completely exclude flux lines up to Bc1

but then part of the flux enters till Bc2

Examples: NbTi, Nb3Sn

In accelerator magnets only Type II Low Temperature Superconductors are used.

NbTi, a ductile material, is the conductor of choice so far to build SC accelerator magnets.

Nb3Sn (higher Bc2) is the only very promising conductor for future higher field magnet. However, Nb3Sn is brittle nature and presents many challenge in building accelerator magnets.

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Hc T( ) = H0 T( ) 1− aT

Tc

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Note: of all the metallic superconductors, only NbTi is ductile. All the rest are brittle intermetallic compounds

Physics of type I and II superconductors “London Penetration Depth” L is the e-fold decay length of the magnetic field from the superconductor skin due to the Meissner effect (in the range of 10 to 103 nm)

Ginzburg-Landau Parameter

€

=Lξ ⇒

k < 12

⇔ type I

k > 12

⇔ type II

⎧

⎨ ⎪

⎩ ⎪

“Coherence Length” the average size of Cooper in the superconductor (in the range of 10 to 100 nm, I.e. much larger than the inter-atomic distance typically of 0.1 to 0.3 nm.

More on fluxoids

Fluxoid patter in Nb

Fluxoid motion due to current flow in Nb (SC type II)

a single fluxoid encloses flux

Fluxoids consist of resistive cores with super-currents circulating round them.

spacing between the fluxoids

Weberse

ho

151022

−×==φ

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d =2

3

φo

B

⎧ ⎨ ⎩

⎫ ⎬ ⎭

1

2= 22nm at 5T

The magnesium diboride MgBDiscovered in January 2001 (Akimitsu)LTS with Tc: ~39 K A low temperature superconductor with high Tc

The Magnesium Diboride MgB2

The basic powder is very cheap, and abundantly available. The champion performance is continuously improving in terms of Jc and Bc.

However, it is still not available in sufficient lengths for making little test coils.

The high temperature SC (HTC)

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100

Critical temperature Tc (K)

Upper critical field Bc2 (T)

NbTi

B2212

many superconductors with critical temperature above 90K - BSCCO and YBCO

operate in liquid nitrogen? Unlike the metallic superconductors, HTS do not have a sharply defined critical current.

At higher temperatures and fields, there is an 'flux flow' region, where the material is resistive - although still superconducting

The boundary between flux pinning and flux flow is called the irreversibility line

metallic

HTS

II

I

B

SC dipole

simplest winding uses racetrack coils

saddle shaped long dipole coils to make more uniform fields

some iron - but field shape is set mainly by the winding

for good uniformity need special winding cross sections

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B r( ) = μ0

I

2πr•

−sinnφ

cosφ

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ for r > a

€

B r( ) = μ0

J ⋅r2

•

−sinnφ

cosφ

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ for r < a

€

r1 cosφ1 − r2 cosφ2 = d

r1 sinφ1 − r2 sinφ2 = 0

€

J = 0

Bx = 0

By = μ0Jd

2

d

+J-J

real x-section

Current density

0

100

200

300

400

500

600

0 5 10 15 20 25B Tesla

Je A/mm

2

Nb3Sn at 4.2K

NbTi at 1.9K

NbTi at 4.2K

B2212 at 4.2K

B2212 at 35K

Wire: enough copper to provide stability (Cu/SC ≈ 1.7) against transient heat loads to carry the current in the event superconductor turns normal.

In pure SC filament -> J ~ 3 kA/mm2

In the real world replace J -> Jeng the 'engineering' current density

NbTi

Cu

insulation Cable: the trapezoidal “Rutherford cable” is made of several round wires (filling factor ~ 0.9)

Coil: it consists of many turns. There must be a turn-to turn insulation (filling factor ~ 0.85)

Current density in wires and cables of Nb3Sn

When the B-field raises, large screening current are generated to oppose the changes.

The current densities are initially much larger than Jc which will create Joule heating.

The large current soon dies and attenuates to Jc, which persist. 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.

Flux jumping

Unstable behaviour is shown by all type II and HT superconductors. The unstable loop is:

reduction in screening currents allows flux to move into the superconductor

flux motion dissipates energy thermal diffusivity in superconductors is low, so energy dissipation causes local temperature rise

critical current density falls with increasing temperatureCure flux jumping by making superconductor in the form of fine filaments

–--> weakens ΔJc ΔT ΔQ

a problem solved using fine SC filaments

Stabilization of flux jumpingcriterion for stability against flux jumpinga = half width of filament

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a =1

Jc

3γ C θc −θo( )μo

⎧ ⎨ ⎩

⎫ ⎬ ⎭

12

typical figures for NbTi at 4.2 K and 1 T

Jc critical current density = 7.5 x 10 9 Am-2

g density = 6.2 x 10 3 kg·m3

C specific heat = 0.89 J·kg-1K-1

q c critical temperature = 9.0 Kso a = 33 m, ie 66 m diameter filaments

Less stable at low field -> Jc is highest when decreasing T -> Jc up and C down

Magnetization

when fully penetrated, the magnetization per unit volume of filament is

aJM c34

=

where a = filament radius

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M =I ⋅AVV

∑

€

M =1

aJc ⋅ x ⋅dx

0

a

∫ =Jc ⋅a

2

When viewed from outside the sample, the persistent currents produce a magnetic moment.

Problem for accelerators because it spoils the precise field shape

We can define a magnetization (magnetic moment per unit volume) as:

for a fully penetrated slab

B

up-ramp branch

down-ramp branch

for cylindrical filaments the inner current boundary is roughly elliptical

B

J JJ

synchrotron injects at low field, ramps to high field and then back down again

note how quickly the magnetization changes when we start the ramp up

so better to ramp up a little way, then stop to inject

M

Bmuch better here!

don't inject here!

Synchrotron injection and field errors

Magnetization also produces field error.

The effect is worst at injection because

ΔB/B is greatest magnetization, ie ΔB is greatest at low field

Degraded performance and 'training'

most magnets do not go straight to the expected quench point *, instead they go resistive - quench - at lower currents

at quench, the stored energy 1/2LI2 of the magnet is dissipated in the magnet, raising its temperature way above critical - must wait for it to cool down and then try again

second try usually goes to higher current and so - known as training

8.00

8.20

8.40

8.60

8.80

9.00

9.20

9.40

9.60

9.80

10.00

10.20

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

Quench Number

St. Steel collars

Aluminum collars

1.98K dI/dt=0

2.07K dI/dt=0

1.90K dI/dt=0

LHC short model dipole training histories: data from Andrzej Siemko

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

Causes of training and some cures

Low Specific Heat: at 4.2K the specific heat of all substances is ~2,000 times less than at room temperature – so the smallest energy release can produce a catastrophic temperature rise.

Cure: work at higher temperatures – but HTS materials don’t yet work in magnets

Jc decreases with temperature: so a temperature rise drives the conductor resistive.

Cure: there isn’t one. Conductor motion: JB force makes conductor move, which releases heat by friction - even 10µm movement can raise the temperature by 3K:

Cures: i) make the coils fit together very tightly, pre-compress them

ii) vacuum impregnate with epoxy resin – but……………….

Resin cracks: organic materials become brittle at low temperature, because of differential thermal contraction they are often under tension – cracking releases heat.

Cure: fill the epoxy with low contraction (inorganic) material, eg silica powder or glass fibre.

Point quenching: even if only a very small section of conductor is driven resistive, the resistive zone will grow by Ohmic heating until it has quenched the magnet.

Cure: make the conductor such that a resistive zone will not grow until a large section has been driven resistive.

Causes of training and some cures

make thermal conductivity k large

make resistivity ρ small make heat transfer term hP/A large

NbTi has high ρ and low k copper has low ρ and high k mix copper and NbTi in a filamentary composite wire

NbTi in fine filaments for intimate mixing

make the windings porous to liquid helium --> superfluid is best

twisted filaments

coupledfilament

uncoupledfilament

Superconducting wires & cableso all superconducting accelerators to date still use NbTi (45 years after its discovery)

o performance of superconductors is described by the critical surface in B J T space,

o magnet performance is often degraded and shows ‘training’o SC stability requires making superconductor as fine filaments embedded in a matrix of copper

o magnetic fields induce persistent screening currents in superconductoro flux jumping occurs when screening currents go unstable quenches magnet - avoid by fine filaments - solved problem

o screening currents produce magnetization field errors - reduce by fine filaments

o in changing fields, filaments become coupled increased magnetization - reduce by twisting

o accelerator magnets need high currents cables- cables must be fully transposed ie every wire must change places with every other wire along the length of the cable - Rutherford cable used in all accelerators to date

o can get coupling between strands in cables- causes additional magnetization field error- control coupling by oxide layers on wires or resistive core foils

Rutherfordcable

fully transposed cables

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ω0 =1

LC

What is an RF cavityA metallic box in which a resonant RF wave generate EM field modes to accelerate charged particles

Acceleration mechanisms

There is a specific resonant frequency of the cavity that one wishes to drive the cavity

The capacitance C and the inductance L of the cavity affect the transfer efficiency of power between the RF amplifying system and the cavity

The most efficient transfer of power would occur when the impedance appears as a simple resistor to the RF amplifying system

The accelerating voltage is V(t) = d·E (t) where d is the effective cavity length

The resonant frequency is

Equivalent circuit

power loss Pc

The high resistance Rskin of the cavity walls is the largest source of power loss.

In a superconducting RF cavity Rskin is 106 times smaller than in a normal conducting cavity

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Rskin =μω

2σ=

1

σδ

What makes a good RF cavity quality factor Q : it measures the ability of the cavity to store energy

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Q =ω0

U

Pc

=ω0

L

R=

LC

R

€

U = 12 LI0

2 = 12 CV0

2 =V0

2

2ω0LC

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Pc = i2 t( ) R =V0

2

2ω0LC

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Rsh =V 2 t( )

Pc

=V0

2

2Pc

= QL

C=

L

CR

At the resonant frequency: The shunt resistance Rsh is the resistive input impedance

The ratio Rsh/Q measures the acceleration efficiency per unit of stored energy

Shunt resistance

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Rsh

Q=

V02

2ω0U

Stored energy

Power loss in the cavity wall

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Pbeam = e V ⋅ Ibeam

€

Pc ∝Eacc ⋅Lacc( )

2

Q

Super/normal conducting RF cavitiesRF power into the beam

RF power into the cavity wall

RF power loss

HOM

Cavity at 700 MHz - ß = 0.65 - 5 cells - Lacc = 5·0.14 m Eacc = 10MV/m - = 0 -> eVacc = eEaccLacc = 7 MeV

Eacc: accelerating fieldLacc: accelerating length : RF-wave phase

5 cell cavity Nb cavity (2 K)real ideal

Cu cavity (300 K)

Rskin 20 n 3.2 n 7 mQ 10 10 6 ·1010 3 ·104

Eacc 10 MV/m 44 M /V m 2 MV/mPbeam (I beam = 10 mA) 60 kW 12 kWPc 16 W @ 2 K 218 kW @ 30 0 KPRF = Pbeam + Pc 60 kW 230 kWPAC fr om the plug 125 kW 400 kWPbeam / PAC 48 % 3 %LACC to reac h 100MeV

30 m 80 m

Plot à la ‘Livingstone’ for SRF cavities

SRF cavity limitations

"Q virus" a recently discovered phenomenon, in which excessive hydrogen in high purity

niobium can condense onto the RF surface of the cavity, forming a niobium hydride with poor superconducting property.

The Q virus is characterized by an anomalously low cavity Q (high surface resistance) at low electric field, followed by a rapid Q decrease with increasing fields.

Cure: a vacuum bake to 900 degrees C is sufficient to remove the hydrogen from the niobium, while not damaging the cavity.

Multipacting or resonant electron emission

Electrons emitted follow a trajectory such that they impact back at the surface of the cavity an integral number of RF cycles after emission, causing an avalanche effect, until all available power goes into this process.

Cure: change the cavity cross section from a rectangular to a spherical or elliptical shape.Thermal breakdown, or quench

Twall > Tc, the cavity becomes normal conducting, rapidly dissipating all stored energy. A small, local "defect" in the RF surface dissipates power more rapidly than the surrounding walls can conduct away.

The quench field depends upon thermal conductivity of the bulk niobium, heat transfer from the niobium to liquid helium bath, and size and resistance of the defect.

Cure: improve the thermal conductivity of the niobium, improving the purity of the metal. Residual Resistivity Ratio (RRR), the ratio of the resistivity at 300 K / 4.2 K is a good indicator.

reminder The main reasons to introduce superconducting devices (magnets and RF cavities) in particle accelerators are power saving and increase of performance.

In a superconductor the resistivity drops below the critical temperature.

Type I SC cannot be penetrated by the B-field,instead, type II SC partially can. The latter are the only useful material for SC devices.

We miss full theoretical explanations for SC. Cooper pairs explains the resistivity drop in type I SC. Dynamics of fluxiods explains properties of type II SC. We have no explanations for HTS.

In SC dipoles we need maximizing the current density, and fighting magnetization, flux jumping and quenches (luckily training helps). This imply optimal design of wires and cables for SC coils. Presently SC improves field and gradient performance by a factor 4 respect to NC.

In SC RF we need high-purity Nb, with thermal treatment to deplete H2 and round or elliptic cavities. Presently SC improves Vacc by a factor 10 and Q by 6 orders of magnitude respect to NC.

Lecture III - superconducting devices