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Magnet Basics for Accelerators 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' 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… 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.
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