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





MgB2 and HTS
SC dipoles






Meissner effect
Cooper pairs and BCS theory
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

Abolish Ohm’s Law!
NI
h
 Hds  I  N  h  H 0  l  H iron
B  0 H  0
Powering a resistive magnet






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
I ≈ 5 kA for 1.8 T
 smaller rings
I ≈ 3·105 A for 10 T
 reduced capital cost
R ≈ 1 m
 new technical possibilities (eg muon collider)
P = R·I2
 higher quadrupole gradients
PLEP = 20 kW/magnet
 higher luminosity
PLHC = 100 MW/magnet (if resistive)
What is a superconductor
Resistance of Mercury falls suddenly
below measurement accuracy at very
low temperature
K. Onnes 1911
Short history of superconductivity

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!
What is a superconductor
Below the critical temperature
Tc the resistivity drops
Cooper pair appearance
T  0  cT 5
Below Tc the B-field lines are expelled out of a
superconductor (perfect diamagnetic behaviour)
Meissner 1933
phonon-einteraction

Type I superconductors
the superconductivity disappears as T > Tc | B > Bc | J > Jc
Type II superconductors
For Bc1 < B < Bc2 there is a partial flux penetration through
fluxoid vortexes and a mixed phase
B=0
T < Tc
B < Bc
Meissner effect and magnetization
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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/4p = cH
 The B field penetrate with an exponentially decaying intensity B(s) = B(0)exp(-s/lL)

BCS theory
Normal conducting state
Superconducting state
 Tc ~ 1/ √Misotopic -> phonons should play a role in superconductivity
 Creation of Cooper pairs (over-screening effect)



 In
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)
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∆
Predictions of the BCS theory
  ph  D e
h
2p
1
F  NF
Energy bond of a Cooper pair
 D  kB TD Debye phonon energy
F
NF


effective potential
density of Fermi states
 0K   1.76  kB Tc
 T  2
 T   1.74   0K 1 
 Tc 
  2 
T
H c T   H c 0K 1 a  with a constant of the SC

 Tc  

1


Size of a Cooper pair 100 nm
Lattice spacing 0.1 ÷ 0.4 nm
More on type I and II superconductors
Note: of all the metallic
superconductors, only NbTi
is ductile.
All the rest are brittle
intermetallic compounds
  2 
T

H c T   H 0 T  1 a  

 Tc  

Type I: not good for accelerator magnets Type II: allow much higher fields



Also known as the “soft superconductors”.
Completely exclude the flux lines.
Allow only small field (Bc < 0.1 T).

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.




Also known as the “hard superconductors”.
Completely exclude flux lines up to Bc1
then part of the flux enters till Bc2
but
Examples: NbTi, Nb3Sn
Physics of type I and II superconductors
 “London
Penetration Depth” lL
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)

“Coherence Length” 
the average size of Cooper in the
superconductor (in the range of 10 to
100 nm, I.e. much larger than the interatomic distance typically of 0.1 to 0.3
nm.
Ginzburg-Landau Parameter k
k  1  type I

l
2
L
k    
k  1  type II


2
More on fluxoids
Fluxoids consist of resistive cores with
super-currents circulating round them.
a single fluxoid encloses flux
spacing between the fluxoids
o 
h
 2 1015 Webers
2e
1
2
 2  o 
d  
  22nm at 5T
 3 B 
Fluxoid patter in Nb

Fluxoid motion
due to current
flow in Nb (SC
type II)
The Magnesium Diboride MgB2
The magnesium diboride MgBDiscovered in January 2001 (Akimitsu)
LTS with Tc: ~39 K A low temperature superconductor with high Tc



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)


80

70


50
40
B2212
30
NbTi
Normal Resistivity
20
metallic
10
0
0
20
40
60
80
M agnetic Field
Upper critical field Bc2 (T)
60
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
Flux Flow
100
Critical temperature Tc (K)
Flux Pinning
HTS
Temperature
SC dipole
r1 cos 1  r2 cos 2  d
r1 sin 1  r2 sin 2  0
d
-J
sin n
I 

Br  0
  cos   for r  a
2pr

 0 

+J

J0
Bx 0
real xsection
By   0 J

I
sin n
J  r 

Br  0
  cos   for r  a
2

 0 

d
2
I 
I
B
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

simplest winding uses
racetrack coils
Current density


In pure SC filament -> J ~ 3 kA/mm2
In the real world replace J -> Jeng the 'engineering' current density

Wire: enough copper to provide stability (Cu/SC ≈ 1.7)
 against transient heat loads
 to carry the current in the event superconductor turns normal.
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)
NbTi

insulation
Cu
Current density in wires and cables of Nb3Sn
600
Nb3Sn at 4.2K
NbTi at 1.9K
500
Je A/mm
2
NbTi at 4.2K
B2212 at 4.2K
400
B2212 at 35K
300
200
100
0
0
5
10
B Tesla
15
20
25
Flux jumping
a problem solved using fine SC filaments 





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.
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 temperature
Cure flux jumping by making superconductor in the form of fine filaments
–--> weakens Jc  T  Q
Stabilization of flux jumping
criterion for stability against flux jumping
a = half width of filament
2


1 3 Cc  o 
a  

Jc 
o

1
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 K
so 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 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:
M 
V
I A
V
for cylindrical filaments the inner current boundary is roughly elliptical

for a fully penetrated slab
M
1
a
a

0
J c  x  dx 
Jc  a
2
J
J
J
B
B
down-ramp branch
up-ramp branch
when fully penetrated, the magnetization
per unit volume of filament is
4
M
Jc a
3p
where a = filament radius
Synchrotron injection and field errors
M
don't inject here!
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

much
better
here!
B


Magnetization also produces field error.
The effect is worst at injection because
 B/B is greatest
 magnetization, ie B is greatest at
low field
MBSMS3.V1 and MBSMS3.V4
Degraded
performance and 'training'
Training Curve @ 1.8K (including "de-training" test)
10.20
9.80
St. Steel
collars
9.60
Magnetic Field, B[T]
LHC
short
model
dipole
training
histories:
data from
Andrzej
Siemko
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
10.00
2.07K dI/dt=0
9.40
1.90K dI/dt=0
1.98K dI/dt=0
9.20
Aluminum
collars
9.00
8.80
8.60
8.40
8.20
8.00
0
 most
2
4
6
8
10
12 14
16 18
20
22 24 26
Quench Number
28 30
32 34
36
38 40 42
44
46
magnets do not go straight to the expected quench point *, instead they go
MBSMS3.V4_Run1
MBSMS3.V4_Run2
MBSMS3.V1
Bnom = 8.3 T
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
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: JB 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  small
 make
heat transfer term hP/A large





NbTi has high  and low k
copper has low  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
o
o
o
o
o
o
o
o
o
o
Superconducting wires & cables
all superconducting accelerators to date still use NbTi (45 years after its discovery)
performance of superconductors is described by the critical surface in B J T space,
magnet performance is often degraded and shows ‘training’
SC stability requires making superconductor as fine filaments embedded in a matrix
of copper
magnetic fields induce persistent screening currents in superconductor
flux jumping occurs when screening currents go unstable  quenches magnet
- avoid by fine filaments - solved problem
screening currents produce magnetization  field errors
- reduce by fine filaments
in changing fields, filaments become coupled  increased magnetization
- reduce by twisting
twisted filaments
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
coupled
uncoupled
can get coupling between strands in cables
filament
filament
- causes additional magnetization  field error
- control coupling by oxide layers on wires or resistive core foils
fully transposed
cables
Rutherford
cable
What is an RF cavity
A 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
0 
1
LC
Equivalent circuit
What makes a good RF cavity
 quality factor Q : it measures the
ability of the cavity to store energy
L
U
L
Q  0  0  C
Pc
R
R
Stored energy
U  LI  CV 
1
2
2
0
2
0
1
2
Rsh 
Pc
2 0
L
C
V02
L
L

Q

2Pc
C 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
Rsh
V02

Q 2 0U
 1
Rskin 

2 
V02
Power loss in the cavity wall
V02
2
Pc  i t  R 
2 0 L C
Shunt resistance
V 2 t 
 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

Super/normal conducting RF cavities
RF power loss
HOM
RF power into the beam
Pbeam  e V  Ibeam
RF power into the cavity wall
E acc  Lacc 

Pc 
: accelerating field
Q

: accelerating
length
Eacc
Lacc
 : RF-wave phase
Cavity at 700 MHz - ß = 0.65 - 5 cells - Lacc = 5·0.14 m
Eacc = 10MV/m -  = 0 -> eVacc = eEaccLacc = 7 MeV
5 cell cavity
Rskin
Q
Eacc
Pbeam (Ibeam = 10 mA)
Pc
PRF = Pbeam + Pc
PAC from the plug
Pbeam / PAC
LACC to reach 100
MeV

Nb cavity (2 K)
real
ideal
20 n
3.2 n
10
10
6·1010
10 MV/m 44 MV/m
60 kW
16 W @ 2 K
60 kW
125 kW
48 %
30 m
Cu cavity (300 K)
7 m
3·104
2 MV/m
12 kW
218 kW @ 300 K
230 kW
400 kW
3%
80 m
2
Plot à la ‘Livingstone’ for SRF cavities
SRF cavity limitations
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.
"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.
Lecture III - superconducting devices
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.