An Introduction to Magnets for Accelerators

An introduction to Magnets for Accelerators

Attilio Milanese

John Adams Institute Accelerator Course

25 – 26 Jan. 2017 This is an introduction to magnets as building blocks of synchrotrons / transfer lines

// // MADX Example 2: FODO cell with dipoles // Author: V. Ziemann, Uppsala University // Date: 060911

TITLE,'Example 2: FODO2.MADX';

BEAM, PARTICLE=ELECTRON,PC=3.0;

DEGREE:=PI/180.0;

QF: QUADRUPOLE,L=0.5,K1=0.2; QD: QUADRUPOLE,L=1.0,K1=-0.2; B: SBEND,L=1.0,ANGLE=15.0*DEGREE;

FODO: SEQUENCE,REFER=ENTRY,L=12.0; QF1: QF, AT=0.0; B1: B, AT=2.5; QD1: QD, AT=5.5; B2: B, AT=8.5; QF2: QF, AT=11.5; ENDSEQUENCE; 2 These are a few choices for further reading

1. N. Marks, Magnets for Accelerators, J.A.I. Jan. 2015 2. D. Tommasini, Practical Definitions & Formulae for Normal Conducting Magnets 3. Lectures about magnets in CERN Accelerator Schools 4. Special CAS edition on magnets, Bruges, Jun. 2009 5. Superconducting magnets for particle accelerators in U.S. Particle Accelerator Schools 6. J. Tanabe, Iron Dominated Electromagnets 7. P. Campbell, Permanent Magnet Materials and their Application 8. K.-H. Mess, P. Schmüser, S. Wolff, Superconducting Accelerator Magnets 9. M. N. Wilson, Superconducting Magnets 3 According to history, the first electromagnet (not for accelerators!) was built in England in 1824 by William Sturgeon

William Sturgeon

4 The working principle is the same as this large magnet, of the 184’’ (4.7 m) cyclotron at Berkeley (picture taken in 1942)

5 This short course is organized in several blocks

1. Introduction

2. Jargon and mathematical concepts

3. Basics for the design of resistive magnets

4. A glimpse on superconducting magnets

Tutorial – first magnetic simulations with OPERA-2D

6 - 1 -

Introduction

7 There are several types of magnets found in synchrotrons and transfer lines – based on what they do to the beam

dipole solenoid

combined function quadrupole bending

sextupole corrector

octupole skew magnet

kicker undulator / wiggler

8 This is a main dipole of the LHC at CERN: 8.3 T × 14.3 m

9 These are main dipoles of the SPS at CERN: 2.0 T × 6.3 m

10 This is a cross section of a main quadrupole of the LHC at CERN: 223 T/m × 3.2 m

11 These are main quadrupoles of the SPS at CERN: 22 T/m × 3.2 m

12 This is a combined function bending magnet of the ELETTRA light source

13 These are sextupoles (with embedded correctors) of the main ring of the SESAME light source

14 There are several types of magnets found in synchrotrons and transfer lines – based on technology

electromagnet permanent magnet

iron dominated coil dominated

normal conducting superconducting (resistive)

cycled / ramped static fast pulsed slow pulsed

15 - 2 -

Jargon and mathematical concepts

16 Nomenclature

B magnetic field T (Tesla) B field magnetic flux density magnetic induction

H H field A/m (Ampere/m) magnetic field strength magnetic field

-7 m0 permeability of vacuum 4p·10 H/m (Henry/m)

mr relative permeability dimensionless

m permeability, m = m0mr H/m

17 The polarity comes from the direction of the flux lines, from a North to a South pole

in Oxford, on 25/01/2017 |B| = 48728 nT = 0.048728 mT = 0.000048728 T

18 Magnetostatic fields are described by (this version of) Maxwell’s equations, coupled with a law describing the material

div 퐵 = 0

퐵 ∙ 푑푆 = 0 푆

rot 퐻 = 퐽

퐻 ∙ 푑푙 = 푗 ∙ 푑푆 = 푁퐼 퐶 푆

James Clerk Maxwell 퐵 = 휇0휇푟퐻

19 The Lorentz force is the main link between electromagnetism and mechanics

퐹 = 푞 퐸 + 푣 × 퐵 퐹 = 퐼ℓ × 퐵 for charged beams for conductors

Oliver Heaviside Hendrik Lorentz Pierre-Simon, marquis de Laplace 20 In synchrotrons / transfer lines magnets the B field as seen from the beam is usually expressed as a series of multipoles

∞ 푦 푟 푛−1 퐵 퐵 = 퐵 sin 푛휃 + 퐴 cos 푛휃 휃 푟 푅 푛 푛 푛=1 퐵푟

∞ 푟 푟 푛−1 퐵 = 퐵 cos 푛휃 − 퐴 sin 푛휃 휃 휃 푅 푛 푛 푛=1 푥 direction of the beam (orthogonal to plane)

∞ 푧 푛−1 퐵 푧 + 푖퐵 푧 = 퐵 + 푖퐴 𝑖휃 푦 푥 푛 푛 푅 푧 = 푥 + 푖푦 = 푟푒 푛=1

21 Each multipole term corresponds to a magnetic flux distribution, which can be added up

B1: normal dipole B2: normal quadrupole B3: normal sextupole

A1: skew dipole A2: skew quadrupole A3: skew sextupole

22 The field profile in the horizontal plane follows a polynomial expansion

∞ 푥 푛−1 푥 푥2 퐵 푥 = 퐵 = 퐵 + 퐵 + 퐵 + ⋯ 푦 푛 푅 1 2 푅 3 푅2 푛=1

퐵푦 퐵푦 퐵푦

푥 푥 푥

B1: dipole B2: quadrupole B3: sextupole

퐵 휕퐵푦 퐺 = 2 = 푅 휕푥

23 Usually, for optics calculation, the field or multipole component is given, together with the (magnetic) length; these are a few definitions from MAD-X Dipole bend angle a [rad] & length L [m]

k0 [1/m] & length L [m] obsolete k0 = B / (Br) B = B1

Quadrupole 2 quadrupole coefficient k1 [1/m ] × length L [m] k1 = (dBy/dx) / (Br) G = dBy/dx = B2/R

Sextupole 3 sextupole coefficient k2 [1/m ] × length L [m] 2 2 k2 = (d By/dx ) / (Br) 2 2 2 (d By/dx )/2! = B3/R 24 We can now compute magnetic quantities from MAD-X entries, and vice versa

BEAM, PARTICLE=ELECTRON,PC=3.0; DEGREE:=PI/180.0; QF: QUADRUPOLE,L=0.5,K1=0.2; QD: QUADRUPOLE,L=1.0,K1=-0.2; B: SBEND,L=1.0,ANGLE=15.0*DEGREE;

(Br) = 109/c*PC = 10^9/299792485*3.0 = 10.01 Tm

dipole (SBEND) B = |ANGLE|/L*(Br) = (15*pi/180)/1.0*10.01 = 2.62 T

quadrupole G = |K1|*(Br) = 0.2*10.01 = 2.00 T/m

25 The harmonic decomposition is used also to describe the field quality (or field homogeneity), that is, the deviations of the actual B vs. the ideal one (normal) dipole

퐵𝑖푑 푥, 푦 = 퐵1푗

퐵푦 푧 + 푖퐵푥 푧 = 퐵 푧 푧 2 푧 3 = 퐵 + 1 푖푎 + 푏 + 푖푎 + 푏 + 푖푎 + 푏 + 푖푎 + ⋯ 1 10000 1 2 2 푅 3 3 푅 4 4 푅

퐵2 퐵3 퐴1 퐴2 푏2 = 10000 푏3 = 10000 푎1 = 10000 푎2 = 10000 … 퐵1 퐵1 퐵1 퐵1

26 The same expression can be written for a quadrupole

(normal) quadrupole

1 퐵 푥, 푦 = 퐵 푥푗 + 푦푖 𝑖푑 2 푅

퐵푦 푧 + 푖퐵푥 푧 = 푧 퐵 푧 푧 2 푧 3 = 퐵 + 2 푖푎 + 푏 + 푖푎 + 푏 + 푖푎 + ⋯ 2 푅 10000 2 푅 3 3 푅 4 4 푅

퐵3 퐵4 퐴2 푏3 = 10000 푏4 = 10000 푎2 = 10000 … 퐵2 퐵2 퐵2 27 The so-called allowed / not-allowed harmonics refer to some terms that shall / shall not cancel out for design symmetries

fully symmetric dipoles

allowed: b3, b5, b7, b9, etc. not-allowed: all the others

half symmetric dipoles

allowed: b2, b3, b4, b5, etc. not-allowed: all the others

fully symmetric quadrupoles

allowed: b6, b10, b14, b18, etc. not-allowed: all the others

fully symmetric sextupoles

allowed: b9, b15, b21, etc. not-allowed: all the others 28 The field quality is often also expressed by a DB/B plot

Δ퐵 퐵 푥, 푦 − 퐵 푥, 푦 = 𝑖푑 퐵 퐵𝑖푑 푥, 푦

29 DB/B can (usually) be expressed from the harmonics, this is the expansion for a dipole

퐵푦,𝑖푑 푥 = 퐵1

퐵 푥 푥 2 푥 3 퐵 푥 = 퐵 + 1 푏 + 푏 + 푏 + ⋯ 푦 1 10000 2 푅 3 푅 4 푅

Δ퐵 1 푥 푥 2 푥 3 (푥) = 푏 + 푏 + 푏 + ⋯ 퐵 10000 2 푅 3 푅 4 푅

30 - 3a -

Basics for the design of resistive magnets - preliminaries -

31 The design of resistive magnets is in most cases that of “iron- dominated” magnets, which depends on the BH characteristics of the yoke material

2.5 10000 B [T] mr [/]

2.0 8000

1.5 6000

B-H 1.0 mu_r 4000

0.5 2000

0.0 0 0 500 1000 1500 2000 2500 3000 3500 H [A/m]

curves for typical M1200-100 A electrical steel

32 These are typical values of fields viable in this resistive magnets (examples from machines at CERN)

PS @ 26 GeV combined function bending B = 1.5 T

SPS @ 450 GeV bending B = 2.0 T

quadrupole Bpole = 21.7*0.044 = 0.95 T

TI2 / TI8 (transfer lines SPS to LHC, @ 450 GeV) bending B = 1.8 T

quadrupole Bpole = 53.5*0.016 = 0.86 T

33 This is the transfer function field B vs. current I of the SPS main dipoles

2.4

2.0 B [T] 1.6

1.2

0.8

0.4

0.0 0 1000 2000 3000 4000 5000 6000 I [A]

34 If the magnet is not dc, then an rms power / current has to be considered, for the most demanding duty cycle

푇 1 푃 = 푅퐼2 = 푅 퐼 푡 2푑푡 푟푚푠 푟푚푠 푇 0

퐼푝푒푎푘 for a pure sine wave 퐼푟푚푠 = 2

퐼푝푒푎푘 for a linear ramp from 0 퐼푟푚푠 = 3

35 This will be a cycle to 2.0 GeV of the PSB at CERN after the upgrade

6000

5000 I [A] 4000

3000 rms 2000

1000

0 0 200 400 600 800 1000 1200 t [ms]

36 The material of the coils is most often copper, sometimes aluminum

Cu Al raw metal price ≈ 5000 $/ton ≈ 1500 $/ton electrical resistivity 1.72∙10-8 W/m 2.65∙10-8 W/m density 8.9 kg/dm3 2.7 kg/dm3

37 - 3b -

Basics for the design of resistive magnets - 2D -

38 These are the most common types of resistive dipoles

“C” dipole “H” dipole

window frame dipole window frame dipole (“O” dipole) (“O” dipole) with windings on both backlegs 39 The magnetic circuit is dimensioned so that the pole is wide enough for field quality, and there is enough room for the flux in the return legs

푤푙푒𝑔,1

푤푙푒𝑔,2 h

푤푝표푙푒

푤푝표푙푒 ≅ 푤퐺퐹푅 + 2.5ℎ

푤푝표푙푒 + 1.2ℎ 퐵푙푒𝑔 ≅ 퐵𝑔푎푝 푤푙푒𝑔

40 The Ampere-turns are a linear function of the gap and of the B field (at least up to saturation)

푙퐹푒 NI/2 h NI/2

퐵퐹푒 퐵𝑔푎푝 퐵𝑔푎푝ℎ 푁퐼 = 퐻 ∙ 푑푙 = ∙ 푙퐹푒 + ∙ ℎ ≅ 휇0휇푟 휇0 휇0 퐵ℎ 1 푁퐼 = 휂 = 1 푙 휂휇0 1 + 퐹푒 휇푟 ℎ 41 The same can be solved using magnetic reluctances and Hopkinson’s law, which is a parallel of Ohm’s law

NI V ℛ = R = Φ 퐼

푙 푙 ℛ = R = 휇0휇푟퐴 휎푆

1 휂 = ℛ 1 + 퐹푒 ℛ𝑔푎푝

42 Example of computation of Ampere-turns and current

central field B = 1.5 T total gap 80 mm 휂 ≅ 0.97 NI = (1.5*0.080)/(0.97*4*pi*10^-7) = 98446 A total

low inductance option 64 turns, I ≅ 98500/64 = 1540 A L = 62.9 mH, R = 15.0 mW low current option 204 turns, I ≅ 98500/204 = 483 A L = 639 mH, R = 160 mW checks magnetic energy (1/2*L*I^2) the same inductance scales with (number of turns)^2 43 Besides the number of turns, the overall size of the coil depends on the current density j, which drives the resistive power consumption (linearly)

ex. NI = 50000 A (rms)

air cooled 2 (by conduction on j = 1 A/mm external surface) A = 50000/1 = 50000 mm2

1 – 1.5 A/mm2 (for Cu)

water cooled j = 5 A/mm2 (hollow conductor) A = 50000/5 = = 10000 mm2

j (rms) 44 These are common formulae for the main electric parameters of a resistive dipole (1/2)

퐵ℎ Ampere-turns (total) 푁퐼 = 휂휇0 (푁퐼) current 퐼 = 푁

휌푁퐿푡푢푟푛 resistance (total) 푅 = 퐴푐표푛푑

2 inductance 퐿 ≅ 휂휇0푁 퐴/ℎ

퐴 ≅ (푤푝표푙푒+1.2ℎ)(푙퐹푒 + ℎ)

45 These are common formulae for the main electric parameters of a resistive dipole (2/2)

푑퐼 voltage 푉 = 푅퐼 + 퐿 푑푡

2 resistive power (rms) 푃푟푚푠 = 푅퐼푟푚푠 2 = 휌푉푐표푛푑푗푟푚푠

휌퐿푡푢푟푛퐵푟푚푠ℎ = 푗푟푚푠 휂휇0

1 magnetic stored energy 퐸 = 퐿퐼2 푚 2

46 These are useful formulae for the main cooling parameters of a water cooled dipole

푃 cooling flow 푄 ≅ 14.3 푄 ≅ 푁 푄 푡표푡 Δ푇 푡표푡 ℎ푦푑푟

1000 water velocity 푣 = 푄 15휋푑2

Reynolds number 푅푒 ≅ 1400푑푣

푄1.75 pressure drop Δ푝 = 60퐿 ℎ푦푑푟 푑4.75

47 The table describes the field quality – in terms of allowed multipoles – for the different layouts of these examples

C-shaped H-shaped O-shaped

b2 1.4 0 0

b3 -88.2 -87.0 0.2

b4 0.7 0 0

b5 -31.6 -31.4 -0.1

b6 0.1 0 0

b7 -3.8 -3.8 -0.1

b8 0.0 0 0

b9 0.0 0.0 0.0 -4 bn multipoles in units of 10 at R = 17 mm

NI = 20 kA, h = 50 mm, wpole = 80 mm

48 These are the most common types of resistive quadrupoles

standard quadrupole

figure-of-8 quadrupole quadrupole with half the coils (useful because narrow) (maybe not so common) 49 These are useful formulae for standard resistive quadrupoles

Pole tip field 퐵푝표푙푒 = 퐺푟

퐺푟2 Ampere-turns (per pole) 푁퐼 = 2휂휇0

(푁퐼) current 퐼 = 푁

휌푁퐿 resistance (total) 푅 = 4 푡푢푟푛 퐴푐표푛푑

50 The ideal poles for dipole, quadrupole, sextupole, etc. are lines of constant scalar potential

dipole 휌 sin 휃 = ±ℎ/2 푦 = ±ℎ/2 straight line

quadrupole 휌2 sin 2휃 = ±푟2 2푥푦 = ±푟2 hyperbola

sextupole 휌3 sin 3휃 = ±푟3 3푥2푦 − 푦3 = ±푟3

51 As a example, this is the real pole tip used in the SESAME quadrupoles vs. the theoretical hyperbola

52 This is the lamination of the LEP main bending magnets, with the pole shims well visible

53 - 3c -

Basics for the design of resistive magnets - 3D -

54 In 3D, the longitudinal dimension of the magnet is described by a magnetic length

∞

푙푚퐵0 = 퐵 푧 푑푧

−∞ 55 The magnetic length can be estimated at first order with simple formulae

푙푚 > 푙퐹푒

dipole

푙푚 ≅ 푙퐹푒 + ℎ

quadrupole

푙푚 ≅ 푙퐹푒 + 0.80푟

56 There are many different options to terminate the pole ends, depending on the type of magnet, its field level, etc.

DIAMOND dipole

z

inner

central outer

SESAME dipole 57 Usually two dipole elements are found in lattice codes: the sector dipole (SBEND) and the parallel faces dipole (RBEND)

SBEND top views

RBEND

58 The two types of dipoles are slightly different in terms of focusing, for a geometric effect

SBEND horizontal focusing

RBEND vertical edge focusing

- and anything in between (playing with the edge angles) - 59 - 4 -

A glimpse on superconducting magnets

(thanks to Luca Bottura for the material of many slides)

60 This is a history chart of superconductors, starting with Hg all the way to HTS (High Temperature Superconductors)

61 Superconductivity makes possible large accelerators with fields well above 2 T

10000

1000

Nb-Ti

)

2

m 100

m

/

A

(

y conventional iron

t

i

s

n electromagnets

e

d

10

t

n

e

r

r

u

C

1 0 2 4 6 8 10 Field (T)

62 This is a summary of (somehow) practical superconductors

LTS HTS

material Nb-Ti Nb3Sn MgB2 YBCO BSCCO year of discovery 1961 1954 2001 1987 1988

Tc [K] 9.2 18.2 39 ≈93 95 / 108

Bc [T] 14.5 ≈30 36…74 120…250 ≈200

63 The field in the aperture of a superconducting dipole can be derived using Biot-Savart law (in 2D)

휇 퐼 퐵 = 0 Biot-Savart law for an infinite wire 휃 2휋휌

2휇 sin 휑 퐵 = 0 푗푤 휑 휋 for a sector coil

푟 푟+w 3휇 퐵 = 0 푗푤 휋 푗 for a 60 deg sector coil

64 This is how it would look like one aperture of the LHC dipoles at 8.3 T, with two different current densities (without iron)

116 mm 656 mm

j = 400 A/mm2 j = 40 A/mm2 w = 30 mm w = 300 mm NI = 1.2 MA NI = 4.5 MA

P = 14.9 MW/m (if Cu at room temp.) P = 6.2 MW/m (if Cu at room temp.) 65 This is the actual coil of the LHC main dipoles (one aperture), showing the position of the superconducting cables

Coil blocks

Magnet bore

Spacers

Superconducting cable

66 Around the coils, iron is used to close the magnetic circuit

gap between coil and yoke

coil

67 The current density is high – though finite – and it depends on the temperature and the field

2 Jc [A/mm ]

T=1.9 K T=4.2 K

100,000

10,000

1,000

100 B [T]

T [K] 5 5 10 15 10 68 The maximum achievable field (on paper) depends on the amount of conductor and on the superconductor’s critical line

Nb-Ti critical surface IC = JC x ASC Nb-Ti critical current IC (B)

7

6

] 2 5

4 [kA/mm

3 5 T bore quench!

2 field Current density density Current 1

2 2 peak 4 4 6 6 field 8 8 10 10 12 14

69 In practical operation, margins are needed with respect to this ultimate limit

current quench IQ

loadline quench Imax

field quench BQ

temperature

quench TCS

70 This is the best (Apr. 2014) critical current for several superconductors Applied Superconductivity Center at NHMFL

71 The forces can be very large, so the mechanical design is rather important

Nb-Ti LHC MB @ 8.3 T

Fx ≈ 350 t per meter

precision of coil positioning: 20-50 μm

Fz ≈ 40 t

72 The coil cross sections of several superconducting dipoles show a certain evolution; all were (are) based on Nb-Ti

Tevatron HERA RHIC LHC (one aperture)

73 Different choices were made for the iron, the mechanical structure and the operating temperature

Tevatron HERA RHIC LHC

76 mm bore 75 mm bore 80 mm bore 56 mm bore B = 4.3 T B = 5.0 T B = 3.5 T B = 8.3 T T = 4.2 K T = 4.5 K T = 4.3-4.6 K T = 1.9 K first beam 1983 first beam 1991 first beam 2000 first beam 2008

74 These are the same magnets in the respective tunnels

Tevatron @ FNAL RHIC @ BNL (Chicago, IL, USA) (Upton, NY, USA)

HERA @ DESY LHC @ CERN (Hamburg, D) (Geneva, CH/FR) 75 - 5 -

extra for the tutorial and your design project

76 These are common programs used for magnetic simulations, among the many available

1. OPERA-2D and OPERA-3D, by Vector Fields 2. ROXIE, by CERN 3. POISSON, by Los Alamos 4. FEMM 5. RADIA, by ESRF 6. ANSYS 7. Mermaid

77 Here are some (possibly useful) reading for your design study

Normal (conducting) 1. J. Bauche, A. Aloev, Design of the Beam Transfer Line Magnets for HIE-ISOLDE, IPAC2014

Super (conducting) 2. E. Weihreter, Compact Synchrotron Light Sources, World Scientific, 1996, in particular the chapter “Superconducting bending magnets for compact rings” 3. M.N. Wilson et al., HELIOS: A compact synchrotron X-ray source, Microelectronic Engineering 11, 1990 R. Anderson et al., Recent developments in HELIOS compact synchrotrons, EPAC 1998 4. D.S. Robin et al., Superconducting toroidal combined-function magnet for a compact ion beam cancer therapy gantry, NIMPR A, 2011 W. Wan et al., Alternating-gradient canted cosine theta superconducting magnets for future compact proton gantries, PRST-AB, 2015

5. The LHC design report, vol. 1, 2004 78