NPAC 2010-2011 Accelerator Physics

Lecture n°3

Magnet Basics for Accelerators

Wednesday 24 November 2010

Olivier Napoly CEA-Saclay, France 3rd Version 6 December 2010

1 References

Books:

An Introduction to the Physics of High Energy Accelerators D. A. Edwards, M. J. Syphers (Wiley Series in Beam Physics and Accelerator Technology)

Principles of Charged Particle Acceleration Stanley Humphries (John Wiley and Sons, New York, 1986)

Web based lectures: US Particle Accelerator Schools : http://www.lns.cornell.edu/~dugan/USPAS/

CERN Particle Accelerator Schools: http://cas.web.cern.ch/cas/

Nicolas Pichoff SFP : http://nicolas.pichoff .perso .sfr .fr/index _fichiers/slide0001 .htm

22 Collider Parameters, Physical Constants and Notations

E , energy and p , momentum c = 299 792 458 m/s B , magnetic rigidity (B = p/e) e = 1.602 177 33 10-19 C 2 L , luminosity mec = 510 999 eV µ = 4 10-7 N A-2 permeability Q , bunch charge 0 =1/= 1/ 2 permittivity 0 µ0c N , number of particles in the bunch (N = Q/e) -15 re = 2.818 10 m e 2 nb , number of bunches in the train 2 with m e c  4 0 re frep , pulse repetition rate

x and y , horizontal and vertical emittances * * x and y , rms horizontal and vertical beam sizes at the IP z , bunch length ,, , Twiss parameters  , tune  , phase advance,

3 Transfer Matrix of a Drift Space

courtesy N. Pichoff 4 Transfer Matrix of a Thin Lens

courtesy N. Pichoff 5 (1/1) Transfer Matrix of a Magnetic Quadrupole

courtesy N. Pichoff 6 Transfert Matrix Through a Quadrupole

We consider a straight line reference trajectory  R ref (s)  s sˆ with s  v0t  y     passithhdlflthing through a quadrupole of length L with a fie ld : B  G  x  which is null along the reference trajectory (no curvature, h  0 ).  0  d    The equations of motion (  m v )  q v  B can be simplified since: dt 1. the energy is constant (magnetic fields do accelerate, but do not ‘work’); 2. the reference frame ( xˆ , y ˆ , sˆ ) is constant;  x' v0  d d ds d  d    3.   v0 and v  x   y ' v 0  ; ddt ds ddt ds ddt  v  p  p c 2 v 2  1  z  4. 0 0 ;    2 2  1  v  v0 (1  O ( )) p 0 c v  1 2 2 2 2 2 2 5. v z  v  v0 ( x'  y' )  v0  O ( , x' , y' )

7 Example : Transport Matrix Through a Quadrupole

At the first order in the variables ( x , x ' , y , y ' , z ,  ) the equations of motion lead to:

• Transverse motion:  x"  G   x    x        K 1    y" p 0 q  y   y 

• Longitudinal motion : The energy is constant :  '  0 The path length difference is given by: d dt (s) d x(s) z'  v (t (s)  t(s))  1  v 0 ds ref 0 d x(s) ds v  1  0 1  x'2  y'2  0  O ( , x'2 , y'2 ) v

8 Example : Transport Matrix Through a Quadrupole

This equation can be easily solved to obtain the following transfer matrix:  R 0 0   x  R   0 R y 0     0 0 12 

 cos( K L) sin( K L) K  R   1 1 1  x   with  K1 sin( K1 L) cos( K1 L)   cosh( K l ) sinh( K L) K  R   1 Q 1 1  y    K1 sinh( K1 L) cosh( K1 L)  assuming G > 0.

For G < 0 , the matrices R x and R y are interchanged.

9 Quadrupole Magnets: the Optician Standpoint

• For the beam optician a Quadrupole is a device which creates a Dipole Gradient G, that is: 1) does not bend the reference trajectory, 2) gently kicks particles with small offsets from the reference trajectory 3) strongly kicks particles with large offsets:  y 1 GL y'  gy   , with g   Q Q f f B Hence  Quadrupole = Dipole Gradient (along a straight line)   • In the same spirit, from the beam optics standpoint a Sextupole is a device which  creates a Quadrupole Gradient G’ in the transverse plane: G' L y'  ( y' )' y  (g y)' y  g'y 2 with g' S  S Q B Hence Sextupole = Quadrupole Gradient Hexapôle = Gradient Quadripolaire (in French) etc…

10 Quadrupole Magnets: the Mathematician Standpoint y Two Dimensional Static Magnetism in an Electrical Current Free Region (representing the accelerator beam pipe) zˆ B(x,y)  x

• We specialize on Time Independent Transverse Magnetic Fields,  B(x, y)  Bx (x, y) xˆ  By (x, y) yˆ which is realized by infinitely long magnets with a uniform field distribution along the longitudinal axis, and no end-fields.   • Such a field B ( x , y ) derives from a longgpitudinal vector potential A  A(x, y) zˆ    B    A   y A xˆ   x A yˆ   • Maxwell-Ampère’s equation in the vacuum   B(x, y)  0 imposes that the scalar field A ( x , y ) is an harmonic function obeying Laplace’s equation:  A(x, y)  0

11 Some Symmetry : Normal and Skew Magnetic Fields

• Symmetry with respect to the ‘horizontal’ plane ( x , y )  ( x ,  y ) : y y B Bx (x, y)  Bx (x,y) zˆ zˆ B (x, y)  B (x,y)   y y x B x A(x, y)  A(x,y)

• Transverse magnetic fields can be decomposed into the sum of symmetric (normal) and antisymetric (skew) components:    B(x, y)  BN (x, y)  BS (x, y)

A(x, y)  AN (x, y)  AS (x, y) such that AN (x, y)  AN (x,y)

AS (x, y)  AS (x,y) • The implementation of this symmetry simplifies the design of accelerator since it eliminates (x,y) transverse coupling by construction (cf. Beam Optics section).

12 Solving the Vector Potential A

• We introduce the complex variable z  x  iiy such that:  1   (  i )  z 2 x y     4 z z 1   (  i )  z 2 x y

• The solution of Laplace’s equation  A ( x , y )  4  z  z A ( z , z )  0 is A(z, z)  a(z)  a'(z) and, for a real potential A  R : A(z, z)  a(z)  a(z) with a an analytic complex function (essentially, an infinite polynomial). •Byyy the symmetr y ( x , y )  ( x ,  y )  z  z

AN (z, z)  (A(z, z)  A(z, z)) 2  Re a(z)  Re a(z)   AS (z, z)  (A(z, z)  A(z, z)) 2  i(Ima(z)  Ima(z))

13 Harmonic Expansion of the Vector Potential A

• Introducing the notation 1  1 n (b ,a ) a(z)    (bn  ian ) z with n n real coefficients 2 n1 n   1 bn n n AN (z, z)    (z  z )  2 n1 n the vec tor pot enti al can b e expan de d as fo llows    1 an n n AS (z, z)    (z  z )  2i n1 n • In polar coordinates ( r,  ), the vector potential can be expanded in the multi-  pol ar h arm oni c ex pan si on:  b Nota Bene  n n A Skew Field derives from a Rotation AN (r, )   r cos(n )  n1 n of the Normal  Field:    an n (n) (n)  AS (r, )   r sin(n ) AS (r, )  AN (r,  )  n1 n 2n

14 (1/2) Harmonic Expansion of the Magnetic Field B

• Introduce the complex magnetic field Bc  By  iBx such that

B  (  i )A  2 A , c x y z   n1 Bc,N (z)  bn z  n1 the complex field can be expanded as follows    n1  Bc,S (z)  i an z  n1

  • In polar coordinates ( ,  ), B  e  i ( B  iB ) and the magnetic field can be r c  r expanded in the multi-polar harmonic expansion:     N n1 N n1 Br (r, )  bn r sin(n ) , B (r, )  bn r cos(n)  n1 n1     S n1 S n1 Br (r, )   anr cos(n ) , B (r, )   an r sin(n)  n1 n1

15 (2/2) Harmonic Expansion of the Magnetic Field B

• This harmonic expansion is nothing else that the Fourier expansion of the magnetic field B cin the azimuth angle  :  n1 Bc (z)  Bc,N (z)  Bc,S (z)  (bn  ian )z n1  n1  (bn  ian )r exp(i(n 1) ) n1

• For reasons which will be explained in the 2nd semester, this harmonic expansion   B (z)  B(n) (z)  (b  ia )z n1 c  c  n n n1 n1 (n) is also called “multipole” expansion, with the following categories of multipolesB : c Dipole (n=1), Quadrupole (n=2), Sextupole (n=3), Octupole (n=4), Decapole (n=5), Dodecapole (n=6), …, 2n-pole (n), etc...

16 The Ideal Dipole Magnetic Field : n = 1 y • Normal Component   0  BN(x,y)   BN    b1  zˆ  x

B is a uniform “ verti ca l” field whi c h ben ds tra jec tory in the “hor izon ta l” p lane N y

• Skew Component BS(x,y)  a1  BS     0  zˆ  x BS is a uniform “horizontal” field which bends trajectory in the “vertical” plane

• Unlike this mathematical 2D model, most of real accelerator dippgole magnets are not straight, but are bent around the reference trajectory with a curvature radius  given by B  pref / q Dipole magnet transfer maps are the most complex elementary map to modelize !! 17 The Ideal Quadrupole Magnetic Field : n = 2

• Normal Component

(n2) BN (z, z)  b2 z

  y (2)    BN  b2   and b2  G  x  • Skew Component

(n2) BS (z, z)  ia2 z

 (2)  x  BS  a2   Magnetic field lines of a quadrupole magnet  y   B is derived B from by a rotation of . S N 4 18 The Ideal Sextupole Magnetic Field : n = 3

• Normal Component

(n3) 2 BN (z, z)  b3 z

  2xy   B(3)  b   N 3  2 2   x  y 

• Skew Component

(n3) 2 BS (z, z)  ia3 z

  x2  y 2   B(3)  a   S 3   Magnetic Field lines of a sextupole magnet  2xy   BS is derived BN from by a rotation of . 6 19 Misaligned Sextupole Magnet

We consider only the Normal Sextupole Magnet component.

As expected, the sextupole field can be expressed as a superposition of gradients ofdlf quadrupole :   2xy   y  x   B(3)  b    b x   b y    B(2)  B(2) N 3  2 2  3   3   N s  x  y   x   y  b2 b3x a2 b3 y but the ‘bad’ surprise is that it includes both normal and skew components.

As a consequence: • the effect of a sextupole on a horizontally misaligned particle is that of a normal quadrupole , • the effect of a sextupole on a vertically misaligned particle is that of a skew quadrupole .

Conversely: • an horizontally misaligned sextupole acts as a normal quadrupole, •a verticallyyg misaligned sextupole acts as a skew qqpuadrupole.

20 The Simplest Monopole Magnet

The simplest Monopole magnet is given by longitudinal current line transporting a total current I ^z . By rotation symmetry :  ˆ . B  B(r)θ 2  I From Ampère’s law : 0 . rd B(r)  0 I  B(r)  0 2 r y B

 I zˆ x

 I The complex magnetic field B  B  iB is then given by B(z)  0 y x 2 z Nota Bene 1: The magggnetic field inside of the conductor is an ideal focusing field ! Nota Bene 2: Point-like Magnetic Monopoles do not exist on earth ! 21 A ‘Dipole’ Magnet The simplest Dipole Field configu ration is giv en by a pair of linear condu ctors: y  I  I B(z)  0  0 2 (z  R0 ) 2 (z  R0 )

0 I R0 I +I     2 2 x  (z  R0 ) zˆ B R0 This expression can be expanded for z  R 0 : m  I R  I  I   z 2  B(z)  0 0   0   0   2 2 2 2  2   (z  R0 )  R0 (1 z R0 )  R0 m0  R0  Comparing to the mathematician expressions, one gets :

0 I 0 I m  0  Dipole : b 1   ; m  2  Decapole : b5   5  R0  R0  I  I Sextupole: 0 ; (2m+1)-pole : 0 m  1  b3   3 m  b2m1   2m1  R0  R0 Nota Bene: This ‘Dipole’ Magnet still has zero curvature radius ! 22  A Quadrupole Magnet The simplest Quadrupole Field configuration is given by a quadruplet of conductors:  I  I  I  I B(z)  0  0  0  0 2 (z  R ) 2 (z  R ei 2 ) 2 (z  R ei ) 2 (z  R ei3 2 ) 0 0 0 0 m  I   z  0 i(m1) 2 i(m1)2 2 i(m1)3 2 z  R     (1 e  e  e ) 0 2 R0 m0  R0  m y  I   z   0 m1 m1 m1 I     (1 (i)  (1)  i )  2 R0 m0  R0  m +I +I    4 if m  4k 1    0 I z x       zˆ 2 R0 m0 R0 0 if m  4k 1    R0 I  Comparing to the mathematician expressions, one gets :

20 I 20 I m  1  Quadrupole: b 2   2 ; m  5  Dodecapole: b6   6  R0  R0 23 A Better Quadrupole Magnet An improved Quadrupole Field configuration is given by an octuplet of conductors: y I I  +I   +I  R  0  zˆ x   +I The calculation proceeds like the preceding one +I leading to: I I  m I   z  4cos(m 1) if m  4k 1 B(z)   0       z  R0 R0 m0  R0   0 if m  4k 1 Comparing to the mathematician expressions, one gets :

20 I m  1 Quadrupole: b 2   2 cos(  ) ;  R0 2 I  Dodecapole: 0 cancels for   m  5  b6   6 cos(6)  R0 12 24 A Sector Quadrupole Magnet  This qu adru pole is realised w ith 4 finite sectors of current density J . One can integrate elementary currents by using the preceding result :  R  4k 1 4 00   z  B(z)  0 J drd   cos(4k  2) k 0  r  Ri 0   R 4k 1  4 J  00 z   0   dr   d cos(4k  2)  k 0 r Ri   0 2 J  R  J   1 1  sin(4k  2)  0 ln 0 sin(2 )z  0    0 z (4k 1)   0  4k 4k    Ri   k 1  Ri R0  k(4k  2) z  R 2 J  R  i 0  0  m 1  Quadrupole : b2  ln  sin(2 0 ) ;   Ri   J  1 1   m  5  Dodeca pp, ole: b   0    sin ( 6  ) , cancels for   6  4 4  0 0 6 6  Ri R0  25 An Ideal Quadrupole Magnet: Cos(2 ) distribution  The engineer meets the mathematician when he tries to b uild c urrent lines follow ing a “cos(2 )” angular distribution : J ( r ,  )  J ( r ) cos( 2  ) with J ( r  R i )  0 . The complex magnetic field  is given by the following integral:   2 1 B(z)  0 rdrd J (r)cos(2 )  i 2  (z  re ) Ri 0   2  1   0 dr J (r) d cos(2 )ei 2  (1 zei r) Ri 0   2   m   z    0 dr J (r) d cos(2 )ei emi z  R      i 2 m0  r  Ri 0  m    z  2   z  J (r)   0  dr J (r)  d cos(2 )ei emi   0  dr 2 m0 r 2 r Ri   0 Ri This is a Pure Quadrupole, in the mathematician sense:   J (r) m  1  Quadrupole b   0 dr 2 2  r Ri

26 The Cos(2 ) Quadrupole Magnet Assuming a u niform radial distribu tion of cu rrents :

 J for Ri  r  Ro J (r, )  J (r)cos(2 ) with J (r)   0 elhlsewhere the complex magnetic field is given by the following expression:

0 J  Ro  B(z)   z ln  2  Ri 

This is a Pure Quadrupole, in the mathematician sense:  J  R  Quadrupole 0  o  m  1  b2  G   ln  2  Ri 

27 Superconducting Quadrupoles

28 The LHC Cos( ) Dipole Magnet LHC Supercon duc ting Dipo les have been woun d from the “cos( )” modldel, us ing two layers of cables and anglular wedge to fit the “cos( )” distribution.

Internal Diameter : = 56 mm Current Density: J = 500 A.mm-2 → Dipole field : B = 8.33 T Curr ent Lin es: NI = ??? A.t

angular wedges

pole n°1

29 LHC Arc Quadrupole Magnet LHC Superconducting Arc Quadrupoles have been wound from the “sector” model. Note that current sectors encompass the [0°, 30°] azimuth interval to eliminate the b6 “dodecapole” harmonics. Their main parameters are: ItInternal lDi Diame ter : = 56 mm Current Density: J = 500 A.mm-2 → Quadrupole gradient : G = 223 T/m Current Lines: NI =??? A.t

pole n°1

30 LHC Low- Quadrupole Magnets LHC Supercon duc ting Low- QdQuadrupol es are realize d from the “cos(2 )” modldel. Internal Diameter : = 70 mm Current Density: J = 545-700 A . mm-2 → Quadrupole gradient : G = 215 T/m Current Lines: NI = ??? A.t Fermilab Low- Quadrupole KEK Low- Quadrupole

pole n°2 pole n°1

pole n°3 pole n°4

31 NbTi Superconducting Cables

NbTi ingot (136 kg)  mono-filament rod in Cu can (10 km)  multi-filaments cable in Cu can multi-filament cable  multi-wire Rutherford cable

mono-filament hex. rods

~1mm  NbTi Wire (Courtesy Alstom/MSA)

“Rutherford” type cable

(Courtesy Oremet Wah Chang) (Courtesy Furukawa Electric Co.) 2 High performance mono-filament rods achieve JC of 3000 A/mm at 42K4.2 K and 5T5 T. Copper is necessary to flow the electric current in case of a magnet quench :  J = 500 A/mm2 at 2 K on average in the conductor. Electric Resistivities: Copper NbTi • at room temperature: 1.610-8 .m • in normal state : 5-610-7 .m -10 • at 10 K,,( 0 T (RRR =100) : 1.610 .m • in superconducting state :  • at 10 K, 7 T (RRR = 100): 4.310-10 .m 32 B vs. JC Limitations of SC Conductors

Material TC (K) 10000 YBCO B Tape Plane Nb 9.2 YBCO B|| Tape Plane NbTi 9.5 SPSuperPower t ape RRP Nb3Sn used in record Nb3Sn 18 NbNb--TiTi breaking NHMFL Complied from insert coil 2007 ASC'02 and 1000 ICMC'03 papers (J. Parrell OIOI--ST)ST) 427 filament strand with ²) Aggy alloy outer sheath mm 2212 tested at NHMFL (A/m YBCO Insert Tape (B|| Tape Plane) E

J Maximal JE for entire LHC Nb- Ti YBCO Insert Tape (B  Tape Plane) strand production (– 100 ) CERN-T. Boutboul MgB2 19Fil 24% Fill (HyperTech) '07, and (- -) <5 T Bronze MgB2 data from Boutboul Nb Sn et al. MT-19, IEEE- 3 2212 OI-ST 28% Ceramic Filaments TASC’06) NbTi LHC Production 38%SC (4.2 K)

4543 filament High Sn Nb3Sn RRP Internal Sn (OI-ST) 18+1 MgB2/Nb/Cu/Monel BronzeBronze--16wt.%Sn16wt.%Sn-- Courtesy M. Tomsic, 2007 0.3wt%Ti (Miyazaki-(Miyazaki- Nb3Sn High Sn Bronze Cu:Non-Cu 0.3 MT18MT18--IEEE’04)IEEE’04) 10 0 5 10 15 20 25 30 35 40 45 Applied Field (T) Nb3Sn conductors allow for increase of field and gradient, but: 1. Cables are more expensive  5-10 and production volume  100, 2. Fabrication of accelerator magnets still at the R&D stage. 33 Normal Conducting Magnets

Normal Condu cting are generally bu ilt w ith w ater cooled Cu condu ctors. In “Air magnets”, the magnetic field is generated by the current lines through the Maxwell-Ampère’s equation:     B  0 J “Air magnets” are very ineffective, as illustrated by this example: Internal Diameter : =300 mm CtDitCurrent Density: J = 4A/4 A/mm2 → QdQuadrupol e gradi ditent : G = 039T/0.39 T/m Current Lines: NI =13000 A.t Air Conductors

 B  B

34 Normal Conducting Magnets: the Role of the Iron Yoke

The Iron Yoke increases the magnetic field in the Iron thanks to its m uch higher Permeability µ     B   J with   6000  0 , It is shaped to optimize the Magnetic Circuit through the Air internal chamber, and to reduce the Stray Fields at large radii. → Quadrupole gradient : G = 5.7 T/m Iron Yoke Conductors

 B  B

35 Magnetic Saturation   Int rod uci ng th e auxili ary magne tic field H suchthth that   H  J , one ge ts     B  H   H where is the relative permeability. r 0 µr 2 T Saturation occurs around 2 T 10% for high quality electrical steel or ppyermalloy (nickel-iron).

First magnetization curve, then hysteresis takes over. µ The relative permeability can reach r max. values from 5000 to 10000. H In the saturation regime, B vs. J ‘linearity’ is lost and increasing the electric current is not effective any more. As a consequence: 1. Normal Conducting Magnets are limited to ~2 T fields 2. In Superconducting Magnets, the Iron Yoke is mostly used to contain the transverse stray fields.

36