USPAS June 2007, Superconducting accelerator magnets Unit 2 Magnet specifications in circular...

USPAS June 2007, Superconducting accelerator magnets

Unit 2Magnet specifications in circular accelerators

Soren Prestemon and Paolo FerracinLawrence Berkeley National Laboratory (LBNL)

Ezio TodescoEuropean Organization for Nuclear Research (CERN)

Unit 2 – Magnet specifications in particle accelerators 2.2


QUESTIONS

Order of magnitudes of the size of our objects: why ?High energy circular accelerators

Length of an accelerator: Km15 m

1.9 Km

Main ring at Fermilab, Chicago, US

41° 49’ 55” N – 88 ° 15’ 07” W

1 Km

40° 53’ 02” N – 72 ° 52’ 32” W

RHIC ring at BNL, Long Island, US

46° 14’ 15” N – 6 ° 02’ 51” E



QUESTIONS

Order of magnitudes of the size of our objects: why ?High energy linear accelerators

Length of a linear accelerator: Km - but we will not deal with them 15 m

Linear accelerator at Stanford, US

46° 14’ 15” N – 6 ° 02’ 51” E

3.5 Km

37° 24’ 52” N – 122° 13’ 07” W



QUESTIONS

Order of magnitudes of the size of our objects: why ?High energy circular accelerators

Length of an accelerator magnet: 10 mDiameter of an accelerator magnet: mBeam pipe size of an accelerator magnet: cm

15 m

A stack of LHC dipoles, CERN, Geneva, CH

46° 14’ 15” N – 6 ° 02’ 51” E

Dipole in the LHC tunnel, Geneva, CH

0.6 m 6 cm



CONTENTS

1. Principles of synchrotron2. The arc: how to keep particles on a circular orbit

Relation between energy, dipolar field, machine length

3. The arc: size of the beam and focusing Aperture requirements for arc quads and dipoles (size of the beam)Gradient requirements (focusing force) for arc quadrupoles

4. The arc: a flow chart for computing magnet parametersExample: the LHC

5. The interaction regions: low-beta magnet specificationsHow to squeeze the beamGradient and aperture requirements for low-beta quadrupoles

6. The interaction regions: detector specifications



1. PRINCIPLES OF A SYNCHROTRON

Electro-magnetic field accelerates particlesMagnetic field steers the particles in a closed (circular) orbit

To drive particles through the same accelerating structure several timesAs the particle is accelerated, its energy increases and the magnetic field is increased (“synchro”) to keep the particles on the same orbit

Limits to the increase in energyThe maximum field of the dipoles (proton machines)The synchrotron radiation due to bending trajectories (electron machines)

Colliders: two beams with opposite momentum collideThis doubles the energy !One pipe if particles collide their antiparticles (LEP, Tevatron)Otherwise, two pipes (ISR, RHIC, HERA, LHC)



1. PRINCIPLES OF A SYNCHROTRON

The arcs: region where the beam is bentDipoles for bendingQuadrupoles for focusingCorrectors

Long straight sections (LSS)Interaction regions (IR) where the experiments are housed

Quadrupoles for strong focusing in interaction pointDipoles for beam crossing in two-ring machines

Regions for other servicesBeam injection (dipole kickers)Accelerating structure (RF cavities)Beam dump (dipole kickers)Beam cleaning (collimators)

ArcArc

ArcArc

LSS

LSS

LSS

LSS

A schematic view of a synchrotron

The lay-out of the LHC



CONTENTS









2. THE ARC:HOW TO KEEP PARTICLES ON A CIRCLE

Kinematics of circular motionRelativistic dynamics

Lorentz (?) force

Putting all togetherHyp. 1 - longitudinal acceleration<<transverse acceleration

vmp

2

2

1

1

c

v

vdt

dv

dt

d

2v

dt

vd

vdt

dmv

dt

dmp

dt

dF

Hendrik Antoon Lorentz, Dutch

(18 July 1853 – 4 February 1928), painted by Menso Kamerlingh

Onnes, brother of Heinke

vdt

dmp

dt

dF

2vm

dt

vdmF

BveF

evBF pvmeB

eBp




Relation momentum-magnetic field-orbit radiusPreservation of 4-momentum

Hyp. 2 Ultra-relativistic regime

Using practical units for particle with charge 1, one has

magnetic field in Tesla …Remember 1 eV=1.60210-19 JRemember 1 e= 1.60210-19 C

2242 cpcmE 42222 cmcpE

2mcpc pcE

][][3.0][ mTBGeVE r [m] B [T] E [TeV]FNAL Tevatron 758 4.40 1.000DESY HERA 569 4.80 0.820IHEP UNK 2000 5.00 3.000SSCL SSC 9818 6.79 20.000BNL RHIC 98 3.40 0.100

CERN LHC 2801 8.33 7.000CERN LEP 2801 0.12 0.100

eBp

ceBE




Nikolai Tesla (10 July 1856 - 7 January 1943)Born at midnight during an electrical storm in Smiljan near Gospić (now Croatia)Son of an orthodox priestA national hero in Serbia

CareerPolytechnic in Gratz (Austria) and PragueEmigrated in the States in 1884Electrical engineerInventor of the alternating current induction motor (1887)Author of 250 patents

MiscellaneousStrongly against marriage [brochure of Nikolai Tesla Museum in Belgrade (2000)]

Considered sex as a waste of vital energy [guardian of Nikolai Tesla Museum in Belgrade, private communication (2002)]

Tesla, man of the year




Relation momentum-magnetic field-orbit radius

][][3.0][ mTBGeVE

0.01

0.10

1.00

10.00

100.00

0 5 10 15

Dipole field (T)

Ene

rgy

(TeV

)

Tevatron HERASSC RHICUNK LEPLHC

=10 km

=3 km

=1 km

=0.3 km




The magnet that we need should provide a constant (over the space) magnetic field, to be varied with time to follow the particle acceleration

This is done by dipoles

As the particle can deviate from the orbit, one needs a linear force to bring it back

We will show in the next section that this is given by quadrupoles

01

x

y

B

BB

GyB

GxB

x

y



CONTENTS









3. THE ARC:SIZE OF THE BEAM AND FOCUSING

The force necessary to stabilize linear motion is provided by the quadrupoles

Quadrupoles provide a field which is proportional to the transverse deviation from the orbit, acting like a spring

One can prove that the motion equation in transverse space (with some approximations) is

where

yysxx BBsByc

eBv

c

eF )(

B

G

x

B

BK

y

11

0)(12

2

xsKds

xd

GyB

GxB

x

y




A sequence of focusing and defocusing quadrupoles with the same (opposite) strength and spaced by L is a providing linear stability to the beam – this is called a FODO cell

Let L be the distance between two consecutive quadrupoles

The equations of transverse motion areWhere the term K is zero in dipoles, and

in focusing quadrupoles, in defocusing quadrupoles

0)(12

2

xsKds

xd

0)(12

2

ysKds

yd

BG

K 1 BG

K 1




The motion equation in the transverse space is similar to a harmonic oscillator

where the force depends on time …

Solution: a oscillator whose amplitude and frequency are modulated

and give the beam size

x y are the invariants (emittances) [m rad]

x and y are the beta functions [m]

is the phase advance, related to the beta function

The beta functions oscillate

along the ring, reaching

maxima and minima

in the quadrupoles

s

t

dts

0 )()(

0)(12

2

ysKds

yd

))(cos()()( sssy yyy

))(cos()()( sssx xxx

0

50

100

150

200

0 50 100 150 200 250 300Length (m)

fun

ctio

n (m

)

Betax

Betay

QF QF QFQD QD QD

L

0)(12

2

xsKds

xd




Relations for a FODO cell: beam size vs cell lengthLet 2L be the cell length – we consider it for the moment as an independent variableWe define (2L) as the phase advance per cell

A typical cell has (2L)=/2 (90° phase advance) – for this cell one has

LLf 4.3)22(

LLd 6.0)22(

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300Length (m)

f

unct

ion

(m)

Betax

Betay

QF QD QD

L

Beta functions in a FODO cell with L=50 m

Beta functions in a FODO cell with L=100 m

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300Length (m)

fu

ncti

on (

m)

Betax

Betay

QF QF QFQD QD QD

L




Example of the LHC: L=50 m, f =170 m, d =30 m

The beta functions are in metersthey are related, but not equal to the beam size

Pay attention ! f =170 m does not mean that the beam size is 170 m !!It is not easy to “feel” the dimension of a beta function

Radius of the beam in the arc (1 sigma)LHC: n =3.75 10-6 m rad

High field E=7 TeV, =7460 - =0.29 mmInjection E=450 GeV, =480 - =1.2 mm

Beam size depends on cell length, energy and normalized emittance

nn L

0

50

100

150

200

0 50 100 150 200 250 300Length (m)

fun

ctio

n (m

)

Betax

Betay

QF QF QFQD QD QD

L




Focusing in a FODO cellThin lens approximation: focusing strength in a 90° FODO cell is

The focusing strength is related to K1

and to the quadrupole length ℓq

and the quadrupole gradient is

LHC: at high field B=8.33 T, =2801 m, L=50 m, G ℓq=660 T

For a 60° phase advance the same linear dependence on L, with different constants

It looks worse: same beam size, 50% more focusing required

L

Lf 7.0

2

LLf 2.133

2 LLd 4.332 Lf

qq G

B

Kf

1

1

L

B

f

BG q

2



CONTENTS









4. THE ARC:FLOWCHART FOR MAGNET PARAMETERS

Arc magnets aperture

Collision energy

Total length of the dipoles Ld

Injection energy Emittance(injectors)

Cell length (free)

Maximum field (technology)

Integrated gradient in

quadrupoles

Maximum gradient in

quadrupolesQuadrupole length

Number of quadrupoles, and total length of the arc

Field errors,beam stability




Collision energy



Cell length (free)



quadrupoles

Maximum gradient in








Input 1. Collision energy Ec

Gives a relation between the dipole magnetic field B and the total length of the dipoles Ld

Technology constraint 1. Dipole magnetic field BDoes not depend on magnet apertureBt <2 T for iron magnets

Bt <13 T for Nb-Ti superconducting magnets (10 T in practice)

Bt <25 T for Nb3Sn superconducting magnets (16-17 T in practice)

Output 1. Length of the dipole part

Length in m, B in T, energy in GeV

][][3.0][ mTBGeVE

tBB

B

ELd 3.0

22




Collision energy



Cell length (free)



quadrupoles

Maximum gradient in








Input 2. Injection energy Ei

Determines the relativistic factor, that affect the beam size

Constraint 2. Normalized beam emittance n

Determined by the beam properties of the injectors

Semi-cell length LThis is a free parameter that can be used to optimizeDetermines the beta functions

Output 2. Aperture of the arc magnets (also determined by field errors and beam stability)

Size of the beam at injection

Magnet aperture

LLf 4.3)22(

fn

i

n

i

fna

Lbababa

22




Collision energy



Cell length (free)



quadrupoles

Maximum gradient in








Technology constraint 1. Quadrupole magnetic field vs aperture

Output 3. Gradient of the quadrupoles

Semi-cell length LAlso determines the focusing, i.e. the integrated gradient

Output 4. Length of the quadrupoles

L

E

L

BL

L

BG d

q 3.0

2

2

22

ta B

G

2

a

tat

BBGG

2

),(

GL

E

GL

BL

GL

B dq 3.0

2

2

22




Collision energy



Cell length (free)



quadrupoles

Maximum gradient in





Output 5. Number of semi-cells and arc lengthEqual to the number of quadrupoles

q

dq L

Ln

LnL qa




Example: Large Hadron ColliderE=7000 GeVNb-Ti magnets, dipole field B=8.3 T

Ld=17600 m

Cell length L=50 mf =170 m

n=3.7510-6m rad

Injection energy 450 GeV, =480Beam size =0.0012 m (at injection)

2*10=0.024 m, i.e., much less than the available aperture of 0.056 m Aperture is larger then needed to have the beam at injection in the zone of “good field”

B

ELd 3.0

22

LLf 4.3)22(

fn




Example: Large Hadron ColliderArc magnets aperture and technology constraint determine quadrupole gradient:8.3 T at 28 mm radius gives 300 T/m for Nb-Ti at 1.9 K – large safety margin taken, operational gradient chosen at 220 T/m

Cell length determines focusing strength, i.e. quadrupole length

Quadrupole length → length in the cell available for dipolestogether with total length of dipoles → number of quadrupoles400

is the space for correctors, instrumentation, interconnections

2

Lf

fB

G q 1

mLG

B

fG

Bq 3

2

LnL qa icicq

dq L

Ln

,,

icic ,,



CONTENTS







ArcArc

ArcArc

LSS

LSS

LSS

LSS



5. THE INTERACTION REGIONS:LOW-BETA MAGNET SPECIFICATIONS

We are now in the straight sections of the machineThere are no dipolesOnly quadrupoles to keep the beam focused

In the middle of the straight section one has a free space for the experiment, with the interaction point (IP) where beams collide

Around the experiment the optics must keep two distinct aimsKeep the beam focusedReduce the size of the beam in the interaction point (IP) to increase the rate of collisions (luminosity)

Beam size proportional to () – but is invariant, so act on

n




A system of quadrupoles is used to reach a very low beta function, called *, in the IP (LHC: 0.55 m instead of the 30-200 m in the arcs)Physical constraint: empty space around the IP – distance of the first magnet to the IP, called l*, (LHC: 23 m) – needed for the detectors !

0

1000

2000

3000

4000

5000

6000

0 50 100 150 200Distance from IP (m)

(

m)

-1

0

0

0

0

0

0

Betax

Betay

Q1

Q2

Q3l *

The lay-out of quadrupoles close to the interaction point in the LHC, and the beta functions




Drawback: beta function gets huge in the quadrupoles !But this happens only in collision, where the beam is smaller

In free space around IP (s=0), one has

At the entrance of the triplet one has

In reality, the situation is even worse: the maximum beta function in the LHC triplet is much larger than at the entrance

at the entrance we have

whereas in the triplet we have m =4400 m

*

2*)(

s

s

*

2*

*

2***)(

ll

l

m96055.0

23)(

2

*

2**

ll

*

* ),(

tm

llF




Aperture requirement: a+c/* and dependent on l*, lt

Given the aperture, the technology limits the maximal gradientAt first order, G1/

We will show the limits of the approximation, and a more precise estimate, in Unit 8

The triplet has to focus the beam in the interaction pointThe focusing strength is a function of l*, lt, and is not related to * This gives a requirement on the integrated gradient …… that together with the maximum gradient gives the triplet length

The 4 equations are coupled

*

* ),(

t

c

n

c

mn llFbaba

2)( tBGG




The 4 equations are coupled

For the LHC, one has *=0.55 mm=4400 m

With respect to the arc, m is ~22 times larger, but the is ~16 times larger in collision the aperture is not so different from the cell magnets

= 0.070 m instead of = 0.056 m in the arcsWith a triplet length of 24 m the required integrated gradient of 4800 TThis requires a quadrupole gradient of 200 T/mWith Nb-Ti one can get up to 300 T/m quadrupoles of = 0.070 m – one has a good safety margin

)(GG ),( *tII llGG

),,( ** tll

GlG tI

c

mnba




Example: the LHC interaction regionsBaseline: Nb-Ti quadrupoles, 200 T/m, 70 mm aperture, *

=0.55 mLARP : Nb3Sn quadrupoles, 200 T/m, 90 mm aperture, *

=0.25 m

0

100

200

300

400

0 25 50 75 100 125 150 175 200 225Aperture (mm)

Gra

dien

t (T

/m)

Baseline

LARP program

Nb-Ti 1.9 K

Nb3Sn 1.9 K

l*=23 m

*=55 cm *=14 cm *=7 cm*=28 cm

lq=20 m

lq=25 mlq=30 m

lq=40 mlq=50 m



CONTENTS









6. THE INTERACTION REGIONS:DETECTOR SPECIFICATIONS

Detector magnets provide a field to bend the particlesThe measurement of the bending radius gives an estimate of the charge and energy of the particle

Different lay-outsA solenoid providing a field parallel to the beam direction (example: LHC CMS, LEP ALEPH, Tevatron CDF)

Field lines perpendicular to (x,y)

A series of toroidal coils to provide a circular field around the beam (example: LHC ATLAS)

Field lines of circular shape in the (x,y) plane

Sketch of a detector based on a solenoidSketch of the CMS detector in the LHC




Detector transverse sizeThe particle is bent with a curvature radius

B is the field in the detector magnetRt is the transverse radius of the detector

magnetThe precision in the measurements is related to the parameter bA bit of trigonometry gives

The magnetic field is limited by the technologyIf we double the energy of the machine, keeping the same magnetic field, we must make a 1.4 times larger detector …

R t

b

eBE

E

BReRb tt

22

22




Detector transverse sizeB is the field in the detector magnetRt is the transverse radius of the detector magnetThe precision in the measurements is 1/b

ExamplesLHC CMS: E=2300 GeV, B=4 T, Rl=12.9 m, Rt=5.9 m, b=9 mm

LEP ALEPH: E=100 GeV, B=1.5 T, Rl=6.5 m, Rt=2.65 m, b=16 mmthat’s why we need sizes of meters and not centimeters !

The magnetic field is limited by technologyBut fields are not so high as for accelerator dipoles (4T instead of 8 T)Note that the precision with BRt

2 – better large than high field …

Detector longitudinal sizeseveral issues are involved – not easy to give simple scaling laws

E

BReRb tt

22

22

GeV15.0

2

E

BRb t



SUMMARY

We gave the principles of a synchrotronThe problem is not only accelerating …but also keeping on a circle !Magnets are needed for keeping particle on the orbit

Arcs: dipoles for bending and quadrupoles for focusingHow to determine apertures, fields and gradientsInput: machine energy and beam emittance (injectors)Free parameter: cell lengthOutput: dipole field, quadrupole gradient, magnet lengths and numbers (i.e. machine length, excluding IR regions)

Interaction regionsHow to squeeze the beam sizeDetermination of the aperture, gradient and length of the IR quads



COMING SOON

During the next days: How these technological limits are determined ? What is the physics and the engineering behind?

0

100

200

300

400

0 25 50 75 100 125 150 175 200 225Aperture (mm)

Gra

dien

t (T

/m)

Baseline

LARP program

Nb-Ti 1.9 K

Nb3Sn 1.9 K

l*=23 m

*=55 cm *=14 cm *=7 cm*=28 cm

lq=20 m

lq=25 mlq=30 m

lq=40 mlq=50 m



REFERENCES

Beam dynamics - arcsP. Schmuser, et al, Ch. 9.F. Asner, Ch. 8.K. Steffen, “Basic course of accelerator optics”, CERN 85-19, pg 25-63.J. Rossbach, P. Schmuser, “Basic course of accelerator optics”, CERN 94-01, pg 17-79.

Beam dynamics - insertionsP. Bryant, “Insertions”, CERN 94-01, pg 159-187.

Beam dynamics - detectorsT. Taylor, “Detector magnet design”, CERN 2004-08, pg 152-165.



ACKNOWLEDGEMENTS

J. P. Kouthcouk, M. Giovannozzi, W. Scandale for discussions on beam dynamics and opticswww.wikipedia.org for most of the picturesThe Nikolai Tesla museum of Belgrade, for brochures, images, and information, and the anonymous guard I met in August 2002F. Borgnolutti for listening all my dry talksB. Bellesia for providing the slides template



APPENDIX A: DEPENDENCE ON THE CELL LENGTH

Example: Large Hadron ColliderLarger L → larger beta function → larger beam size → larger magnet aperture, butLarger L → small number of cells → smaller focusing strength

→ smaller number of quadrupoles

0.045

0.050

0.055

0.060

0.065

0 20 40 60 80Cell length (m)

Ape

rtur

e (m

m)

0100200300400500600700800

Num

ber

of q

uads

Magnet aperture (mm)Number of quads




Example: Large Hadron ColliderDipoles contribute for around 17.5 KmWith a cell length of 30 m quads are 3.5 Km long (20%), with 70 m quads are 1 Km long (6%) – baseline is 50 m, giving 1.3 Km

17000

18000

19000

20000

21000

22000

0 20 40 60 80Cell length (m)

Arc

leng

th (

m)

dipoles

dipoles+quadrupoles




Example: Large Hadron ColliderThe amount of the cable needed for dipoles and quadrupoles can also be estimated – equations will be derived in Unit 8The quantity of cable is roughly independent of the cell length, with a minimum around 50 m (but this was not the criteria used to select L!)

0

5

10

15

20

0 20 40 60 80Cell length (m)

Supe

rcon

duct

or v

olum

e (m

3 )

total

dipoles

quadrupoles

USPAS June 2007, Superconducting accelerator magnets Unit 2 Magnet specifications in circular...

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