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' gy , 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 (gy)'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 Tw o 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 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)) 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 n1 n 1 bn n n AN (z, z) (z z ) 2 n1 n the vec tor po tenti al can be expan de d as fo llows 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- 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 ) n1 n of the Normal Field: an n (n) (n) AS (r, ) r sin(n ) AS (r, ) AN (r, ) n1 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 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 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 : 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 in the 2nd semester, 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 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 “ver tica l” field w hic 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 (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 18 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 .