Cockcroft Institute 2-6 July 2012 Accelerator Physics
Lecture n°2
Magnet Basics for Accelerators
Tuesday 3 July 2012
Olivier Napoly CEA-Saclay, France
3 July 2012 Lectures at Cockcroft Institute 1 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/ 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 July 2012 Lectures at Cockcroft Institute 2 Transfer Matrix of a Magnetic Quadrupole
courtesy N. Pichoff
3 July 2012 Lectures at Cockcroft Institute 3 Mid-plane symmetry: normal magnets
y y Mid x-plane symmetry x x Sy : x y y x B B : x x Sy By By
y B-field lines should be reversed
Magnets with mid-plane x symmetry (usually the horizontal plane) are called x x normal magnets Sx : y y 3 July 2012 Lectures at Cockcroft Institute 4 Mid-plane symmetry: skew magnets
y y Mid x-plane symmetry x x Sy : x y y x B B : x x Sy By By
B-field lines should be reversed Magnets with mid-plane anti-symmetry (usually the horizontal plane) are called skew magnets
3 July 2012 Lectures at Cockcroft Institute 5 “Horizontal” Mid-Plane Symmetry
Most accelerators are designed with simplifying assumptions : 1. a planar “horizontal” reference trajectory 2. an electromagnetic circuit symmetric with respect to the “horizontal” plane: x x Sy : y y
Under these assumptions, the 6D Transfer Map M ( s 1 , s 2 ) satisfies: ( X ) (X ) Mt1t2 Sy Sy Mt1t2
The Linearized Transfer Map (a matrix R) commutes with the matrix Sy 1 0 0 2 with R S y S y R S y 0 12 0 0 0 12
3 July 2012 Lectures at Cockcroft Institute 6 Transfer Matrix with Horizontal Mid-Plane Symmetry
The planar symmetric 6D transfer matrix simplifies to the following 2x2 blocks: R 0 D x x R 0 Ry 0 Ez 0 Rz in such a way that the vertical motion is decoupled from the two other planes.
Proof: (RS y ) xy R12 (S y ) yy Rxy
(S y R) xy (S y ) xx Rxy Rxy
Rxy 0
3 July 2012 Lectures at Cockcroft Institute 7 Polar symmetry: quadripole magnets
1st step: z-plane symmetry y x x y Sz : y y z z x x
Sz : J z J z
y B-field lines should be reversed B B : x x x Sz By By Magnets symmetric w.r.t. the combination of z-plane symmetry and /2 rotation, are called quadrupoles
3 July 2012 Lectures at Cockcroft Institute 8 Mid-Plane and Polar symmetries • These two symmetry operations are independent: they can be used separately to classify 2D magnet configurations. • They lead to two independent eigen-space decompositions of the most general 2D magnet configuration: • Mid-plane symmetry {normal} {skew} • zpolar symmetry {multipoles} • These two symmetries are ‘engineering-able’: they can be used separately to design and wind real 2D magnets.
y • They can also be combined: most magnet designers =45° are combining them automatically, hence the 2 so- called ‘quadrupole symmetry’ axis’: • Symmetry w.r.t = 0° axis x • Antisymmetry w.r.t. = 45° axis. =0°
I believe that there is a loss of information!
3 July 2012 Lectures at Cockcroft Institute 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…
3 July 2012 Lectures at Cockcroft Institute 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 longitudinal 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 a harmonic function obeying Laplace’s equation: A(x, y) 0
3 July 2012 Lectures at Cockcroft Institute 11 Identities in Vector Calculus Geometric: Differentials: A(BC) (AC)B (A B)C ( V ) V V () 0 (V ) 0 Stokes’s theorem: V dl V nˆd C Green-Ostrogradsky’s theorem: V nˆd dx3 V V V
2 July 2012 Lectures at Cockcroft Institute 12 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 anti-symmetric (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) (A(x, y) A(x,y)) / 2 AN (x,y)
AS (x, y) (A(x, y) A(x,y)) / 2 AS (x,y) • The application of this mid-plane symmetry simplifies the design of accelerators since it eliminates (x,y) transverse coupling by construction.
3 July 2012 Lectures at Cockcroft Institute 13 Solving the Vector Potential A
• We introduce the complex variable z x i y 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). • By the symmetry ( 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))
3 July 2012 Lectures at Cockcroft Institute 14 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 n1 n 1 bn n n AN (z, z) (z z ) 2 n1 n the vector potential can be expanded as follows 1 an n n AS (z, z) (z z ) 2i n1 n
• In polar coordinates ( r, ), the vector potential can be expanded in the multi- polar harmonic expansion: b Nota Bene n n A Skew Field derives from a Rotation AN (r, ) r cos(n ) n1 n of the Normal Field: an n (n) (n) AS (r, ) r sin(n ) AS (r, ) AN (r, ) n1 n 2n
3 July 2012 Lectures at Cockcroft Institute 15 (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 n1 Bc,N (z) bn z n1 the complex field can be expanded as follows n1 Bc,S (z) i an z n1
i • In polar coordinates ( r, ), B c e ( B iB r ) and the magnetic field can be expanded in the multi-polar harmonic expansion: N n1 N n1 Br (r, ) bn r sin(n ) , B (r, ) bn r cos(n ) n1 n1 S n1 S n1 Br (r, ) anr cos(n ) , B (r, ) an r sin(n) n1 n1
3 July 2012 Lectures at Cockcroft Institute 16 (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 : n1 Bc (z) Bc,N (z) Bc,S (z) (bn ian )z n1 n1 (bn ian )r exp(i(n 1) ) n1
• For reasons which will be explained later, this harmonic expansion B (z) B(n) (z) (b ia )z n1 c c n n n1 n1 (n) is also called “multipole” expansion, with the following categories of multipoles B : c Dipole (n=1), Quadrupole (n=2), Sextupole (n=3), Octupole (n=4), Decapole (n=5), Dodecapole (n=6), …, 2n-pole (n), etc...
3 July 2012 Lectures at Cockcroft Institute 17 Magnetic Measurements
Magnet fields are measured to checks: 1. the field integral of the main component (Bdl for a dipole, Gdl for a quadrupole) 2. the field harmonics content (~ field homogeneity, or quality)
These measurement are usually performed by a magnetic probe located at a Reference Radius Rref along the magnet bore.
Hence, magnet experts use the following expansion:
n-1 z By (z) iBx (z) Bc (z) bn i an for z Rref n1 Rref
(sometimes with a +)
(n-1) Bc (0) n1 so that b i a R for n 1 n n (n 1)! ref
3 July 2012 Lectures at Cockcroft Institute 18 The Ideal Dipole Magnetic Field : n = 1 y • Normal Component 0 BN(x,y) BN b1 zˆ x
B is a uniform “vertical” field which bends trajectory in the “horizontal” plane 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 dipole 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 !! 3 July 2012 Lectures at Cockcroft Institute 19 The Ideal Quadrupole Magnetic Field : n = 2
• Normal Component
(n2) BN (z, z) b2 z
y (2) BN b2 and b2 G x • Skew Component
(n2) 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 3 July 2012 Lectures at Cockcroft Institute 20 The Ideal Sextupole Magnetic Field : n = 3
• Normal Component
(n3) 2 BN (z, z) b3 z
2xy B(3) b N 3 2 2 x y
• Skew Component
(n3) 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 3 July 2012 Lectures at Cockcroft Institute 21 Misaligned Sextupole Magnet
We consider only the Normal Sextupole Magnet component.
As expected, the sextupole field can be expressed as a superposition of gradients of 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 vertically misaligned sextupole acts as a skew quadrupole.
3 July 2012 Lectures at Cockcroft Institute 22 Quadrupole Magnets: the Engineer Standpoint
Given a Central Axis Oz which coincides with the Reference Trajectory, we introduce an “Engineering Symmetry” RS( ) which allows categorize the Magnets into “Engineering Multipoles”. y ( ) Notations: R R( ) is the rotation with angle around Oz . zˆ S is the symmetry with respect to the Oxy plane. O x
RS ( ) R ( ) S S R ( ) is the combination of the two. y y Note that : B 1) For a transverse B field (pseudo-vector) S B B zˆ zˆ B 2) For a longitudinal electrical current J O O J x J x S J J 3) S RS ( ) RS ( ) R (2 ) 3 July 2012 Lectures at Cockcroft Institute 23 Symmetric Current Distributions y Consider a distribution of longitudinal electrical current J(r,) . J(r,)
If a current distribution is invariant with respect to RS zˆ ( ) J (r, ) J (r, ) x RS 0 O then: y 1) the integrated current through the plane is zero, since J(r,) J (r, ) J (r, ) d J (r, ) 0 r 0 zˆ + O x 2) 2 is a sub-multiple of 2 , since − 0
J (r, ) RS (0 ) RS (0 ) J (r, ) R (20 ) J (r, ) J (r, 20 ) 2 either 2 or J (r, ) J (r) constant zero function of . 0 n
3 July 2012 Lectures at Cockcroft Institute 24 Multipolar Current Distributions We introduce the following categories of current distributions J(r,) : Monopole (n=0): J ( 0 ) ( r ) , azimuthally independent distributions Dipole ( =1), such that: (1) (1) n RS ( ) J (r, ) J (r, ) Quadrupole (n=2): R ( 2) J (2) (r, ) J (2) (r, ) S Sextupole (n=3): R ( 3) J (3) (r, ) J (3) (r, ) S Octupole (n=4): R ( 4) J (4) (r, ) J (4) (r, ) S Decapole (n=5): ( 5) J (5) (r, ) J (5) (r, ) RS …. pole ( ): (n) (n) 2n- n RS ( n) J (r, ) J (r, )
(0) (0) Note that , for the Monopole : RS ( ) J (r) J (r)
3 July 2012 Lectures at Cockcroft Institute 25 Decomposition of Current Distributions The space J of longitudinal current distributions can be decomposed over the sum of symmetric subspaces as follows. 1. Since , one can decompose any current RS ( ) RS ( ) R (2 ) 1 distribution into symmetric (n=1) and antisymmetric components w.r.t. RS():
J ( J RS ( ) J ) 2 ( J RS ( ) J ) 2 J (n1) ( J R ( ) J ) 2 (n1) J J where J is invariant under the rotation R ( ) 2. Since , one can decompose the current distrib- RS( 2) RS ( 2) R ( ) utionJ into symmetric (n=2 ) and antisymmetric components w.r.t. RS(/2): J ( J RS ( 2) J ) 2 ( J RS ( 2) J ) 2 (n2) J ( J R ( 2) J ) 2 J (n2) J etc, etc,… 2 leading to : J J (n1) ( J (n2) (J (n4) J (n8) ...))) J (n0) dipole + quadrupole + octupole + 16-pole + …. + monopole
3 July 2012 Lectures at Cockcroft Institute 26 Natural Harmonics Question: Where are the Sextupoles, and Decapoles, … ? Answer: Sextupoles (n=3) and Decapoles (n=5) are Dipole (n=20) Natural Harmonics: (n3) (n3) 3 (n3) (n3) (n3) RS ( 3) J J RS ( 3) J RS ( ) J J (n5) (n5) 5 (n5) (n5) (n5) RS ( 5) J J RS ( 5) J RS ( ) J J Dodecapoles (n=6) and 20-poles (n=10) are Quadrupole (n=21) Natural Harmonics: ( 6) J (n6) J (n6) 3 ( 6) J (n6) ( 2) J (n6) J (n6) RS R S RS R ( 10) J (n10) J (n10) R 5 ( 10) J (n10) R ( 2) J (n10) J (n10) S S S In general, 2n-poles with n =(2m+1)2p are 2.2p-pole (n=2p) Natural Harmonics: p R ( n) J (n) J (n) R 2m1( n) J (n) R ( 2 ) J (n) J (n) S S S . In other words, using the example of Dipoles, the ensemble of Dipole (RS()- invariant) current distributions includes that of Sextupole (RS(/3)-invariant) ones. As a consequence, Dipole magnets built from a Dipole invariant current distribution most likely contains Sextupole, Decapole, etc… components . Other harmonics are excluded, unless induced by non-invariant fabrication errors.
3 July 2012 Lectures at Cockcroft Institute 27 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
0 I The complex magnetic field BC By iBxis then given by B (z) c 2 z Nota Bene 1: This ‘magnet’ is creating an infinite electric capacitor at z ! Nota Bene 2: Point-like Magnetic Monopoles do not exist on earth !
3 July 2012 Lectures at Cockcroft Institute 28 Dipoles Have Two Poles A Dipole magnet can be designed as follows: 1. Define a current distribution J(r,) over a 180° wide sector : 2, 2 nd 2. Rotate it by the symmetry RS() over the 2 sector 2,3 / 2 y y J (1)(r,) J (2)(r,) J (1)(r,) − − RS() − + x + + x
This magnet is the linear superposition of elementary dipole fields of various orientations: y y J (2)(r,) J (1)(r,) – B – − − + + x x + +
3 July 2012 Lectures at Cockcroft Institute 29 Imposing the Mid-Plane Symmetry
Mid-Plane horizontal symmetry can be enforced on the current lines: y y J (1)(r,) J (2)(r,) J (1)(r,) − + − + RS() ─ + −+ x ─+ −+ x
For uniform current distribution, the beam current densities cancel in their intersections → simplification of the Polar Pieces fabrication: y y (2)
J (r,) J (1)(r,) B
+ – + – upper polar piece
– – + x x + lower polar piece
3 July 2012 Lectures at Cockcroft Institute 30 The LHC Cos( ) Dipole Magnet An LHC Superconducting Dipole is made of 2 blocks “poles” of closed conducting circuits:
− −− + B B − + + − − + − − B + + − − pole n°1 + + − + − Two dipoles are assembled in a “2 in 1” − − + cold mass structure, to optimize the field pattern
3 July 2012 Lectures at Cockcroft Institute 31 A ‘Dipole’ Magnet The simplest Dipole Field configuration is given by a pair of linear conductors: y 0 I 0 I Bc (z) 2 (z R0 ) 2 (z R0 )
0 I R0 I +I 2 2 x (z R0 ) zˆ B R0 z R This expression can be expanded for 0 : m I R I I z 2 B (z) 0 0 0 0 c 2 2 2 2 2 (z R0 ) R0 (1 z R0 ) R0 m0 R0 Comparing to the mathematician expressions, one gets : I I m 0 Dipole : b 0 ; m 2 b 0 1 Decapole : 5 5 R0 R0 I I Sextupole: 0 ; (2m+1)-pole : 0 m 1 b3 3 m b2m1 2m1 R0 R0 Nota Bene: This ‘Dipole’ Magnet still has zero curvature radius ! 3 July 2012 Lectures at Cockcroft Institute 32 A Better and Symmetric Dipole Magnet The improved Dipole Field configuration, given by an quadruplet of conductors, as shown on the picture, has 2 features:
1. it is symmetric w.r.t. Sy, the symmetry with respect to the Oxz plane (yy). 2. it has an extra parameter to tune natural harmonics out. y 0 I 0 I Bc (z) i i -I +I 2 (z R0e ) 2 (z R0e ) R 0 I 0 I 0 i( ) i( ) x 2 (z R0e ) 2 (z R0e ) zˆ m -I +I I z 0 (m1) (2cos((m 1) )(1 (1) )) z R0 2 R0 m0 R0 m I z 4cos((m1) ) if m 2k 0 2 R0 m0 R0 0 if m 2k 1 2k 2 0 I z cos((2k 1)) R0 k 0 R0 20 I 20 I k 0 Dipole : b 1 cos( ), k 1 Sextupole : b3 3 cos(3) R0 R0 3 July 2012 Lectures at Cockcroft Institute 33 Quadrupoles Have Four Poles A Quadrupole magnet can be designed as follows:
1. Define a current distribution J(r,) over a 90° wide sector : 0 ,0 2 nd 2. Rotate it by the symmetry RS(/2) over the 2 sector 0 2,0 y y J (1)(r,) J (2)(r,) J (1)(r,) − − − + + + RS(/2) x x
rd 3. Repeat the RS(/2) operation to the 3 sector 0 ,0 3 2 4. Repeat the R (/2) operation to the 4th sector 3 2, 2 y S 0 y 0 J (2)(r,) J (1)(r,) − − – − − + + + + linear superposition + x of elementary quadrupoles + x + + (4) with various orientations + + J (3)(r,) − − J (r,) −– −
3 July 2012 Lectures at Cockcroft Institute 34 Imposing the Horizontal Mid-Plane Symmetry
y
(2) (1)
J (r, )J− (r, ) –
−
+ + − – − +
++ +
+ + +
− + x
− + – + J (3)(r,) −– − J (4)(r,)
y ─ ─
upper left pole upper right pole
+ +
+ + x lower left pole lower right pole ─ ─
3 July 2012 Lectures at Cockcroft Institute 35 LHC Low- Quadrupole Magnets LHC Superconducting Low- Quadrupoles is made of made of 4 blocks “poles” of closed conducting circuits:
Fermilab Low- Quadrupole KEK Low- Quadrupole
pole n°2 pole n°1 −−− − −− ++ ++ + + ++ ++ + + − − pole n°3 − − pole n°4 − −
3 July 2012 Lectures at Cockcroft Institute 36 A Quadrupole Magnet The simplest Quadrupole Field configuration is given by a quadruplet of conductors: 0 I 0 I 0 I 0 I Bc (z) i 2 i i3 2 2 (z R ) 2 (z R e ) 2 (z R e ) 2 (z R e ) 0 0 0 0 m I z 0 i(m1) 2 i(m1)2 2 i(m1)3 2 z R (1 e e e ) 0 2 R0 m0 R0 m y I z 0 m1 m1 m1 I (1 (i) (1) i ) 2 R0 m0 R0 m +I +I 4 if m 4k 1 0 I z x zˆ 2 R0 m0 R0 0 if m 4k 1 R0 I Comparing to the mathematician expressions, one gets :
20 I 20 I m 1 Quadrupole: b 2 2 ; m 5 Dodecapole: b6 6 R0 R0 3 July 2012 Lectures at Cockcroft Institute 37 A Better and Symmetric Quadrupole Magnet The improved Quadrupole Field configuration, given by an octuplet of conductors, as shown on the picture, has 2 features: 1. it is symmetric w.r.t. Sy,the symmetry with respect to the Oxz plane (yy). 2. it has an extra parameter to tune natural harmonics out. The calculation proceeds like the dipole one leading to: m I z 4cos(m 1) if m 4k 1 0 for z R0 Bc (z) R R 0 if m 4k 1 0 m0 0 y 4k 1 I I 4 0 I z cos(4k 2) +I R R +I 0 k 0 0 R 0 Comparing to the mathematician expressions, one gets : ˆ x z 40 I +I +I k 0 Quadrupole: b cos(2 ) ; 2 2 I I R0 4 I Dodecapole: 0 cancels for k 1 b6 6 cos(6) R0 12 3 July 2012 Lectures at Cockcroft Institute 38 A Sector Quadrupole Magnet This quadrupole is realised with 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) c 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 m 1 Quadrupole : b 0 ln 0 sin(2 ) ; 2 R 0 i J 1 1 m 5 Dodecapole: b 0 sin ( 6 ) , cancels for 6 4 4 0 0 6 6 Ri R0 3 July 2012 Lectures at Cockcroft Institute 39 An Ideal Quadrupole Magnet: Cos(2 ) distribution The engineer meets the mathematician when he tries to build current lines following 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 ) c i 2 (z re ) Ri 0 2 1 0 dr J (r) d cos(2 )ei 2 (1 zei r) Ri 0 2 m z 0 dr J (r) d cos(2 )ei emi z R i 2 m0 r Ri 0 m z 2 z J (r) 0 dr J (r) d cos(2 )ei emi 0 dr 2 m0 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
3 July 2012 Lectures at Cockcroft Institute 40 The Cos(2 ) Quadrupole Magnet Assuming a uniform radial distribution of currents :
J for Ri r Ro J (r, ) J (r)cos(2 ) with J (r) 0 elsewhere the complex magnetic field is given by the following expression:
J R 0 o Bc (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
3 July 2012 Lectures at Cockcroft Institute 41 Superconducting Quadrupoles
3 July 2012 Lectures at Cockcroft Institute 42 The LHC Cos( ) Dipole Magnet LHC Superconducting Dipoles have been wound from the “cos( )” model, using 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 Current Lines: NI = ??? A.t
angular wedges
pole n°1
3 July 2012 Lectures at Cockcroft Institute 43 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: Internal Diameter : = 56 mm Current Density: J = 500 A.mm-2 → Quadrupole gradient : G = 223 T/m Current Lines: NI =??? A.t
pole n°1
3 July 2012 Lectures at Cockcroft Institute 44 LHC Low- Quadrupole Magnets LHC Superconducting Low- Quadrupoles are realized from the “cos(2 )” model. 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
3 July 2012 Lectures at Cockcroft Institute 45 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 4.2 K and 5 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: 1.610-8 .m • in normal state : 5-610-7 .m -10 • at 10 K, 0 T (RRR =100): 1.610 .m • in superconducting state : • at 10 K, 7 T (RRR = 100): 4.310-10 .m 3 July 2012 Lectures at Cockcroft Institute 46 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 SuperPower tape RRP Nb3Sn used in record Nb3Sn 18 Nb-Ti breaking NHMFL Complied from insert coil 2007 ASC'02 and 1000 ICMC'03 papers (J. Parrell OI-ST) 427 filament strand with Ag alloy outer sheath 2212 tested at NHMFL (A/mm²) YBCO Insert Tape (B|| Tape Plane) E
J Maximal JE for entire LHC NbTi 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 Bronze-16wt.%Sn- Courtesy M. Tomsic, 2007 0.3wt%Ti (Miyazaki- Nb3Sn High Sn Bronze Cu:Non-Cu 0.3 MT18-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. 3 July 2012 Lectures at Cockcroft Institute 47 Normal Conducting Magnets
Normal Conducting are generally built with water cooled Cu conductors. 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 Current Density: J = 4 A/mm2 → Quadrupole gradient : G = 0.39 T/m Current Lines: NI =13000 A.t Air Conductors
B B
3 July 2012 Lectures at Cockcroft Institute 48 Normal Conducting Magnets: the Role of the Iron Yoke
The Iron Yoke increases the magnetic field in the Iron thanks to its much 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
3 July 2012 Lectures at Cockcroft Institute 49 Magnetic Saturation Introducing the auxiliary magnetic field H such that H J , one gets 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 permalloy (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.
3 July 2012 Lectures at Cockcroft Institute 50