Magnets and Lattices. - Accelerator building blocks - Transverse beam dynamics - coordinate system

Transcription

1 Magnets and Lattices - Accelerator building blocks - Transverse beam dynamics - coordinate system

2 Both electric field and magnetic field can be used to guide the particles path. r F = q( r E + r V r B ) Magnetic field is more effective for high energy particles, i.e. particles with higher velocity. - For a relativistic particle, what kind of the electric field one needs to match the Lorentz force from a 1 Telsla magnetic field?

3 Dipoles: uniform magnetic field in the gap - Bending dipoles - Orbit steering Quadrupoles - Providing focusing field to keep beam from being diverged Sextupoles: - Provide corrections of chromatic effect of beam dynamics Higher order multipoles

4 Two magnetic poles separated by a gap homogeneous magnetic field between the gap Bending, steering, injection, extraction g B = m 0 NI g

5 F = gm v 2 r = qr v B r For synchrotron, bending field is proportional to the beam energy beam rigidity: Br = p q ρ ; where p is the momentum of the particle and q is the charge of the particle

6 Quadrupole Magnetic field is proportional to the distance from the center of the magnet B x = ky; B y = kx y Produced by 4 poles which are shaped as xy = ±R 2 /2 x Providing focusing/defoucsing to the particle Particle going through the center: F=0 Particle going off center

7 Quadrupole magnet Theorem ò r B dl = m 0 m r I Pick the loop for integral ò 0 R B'rdr = m 0 m r NI For the gap is filled with air, B'= 2m 0 m r NI R 2 B'[T /m] = 2.51 NI R[mm 2 ]

8 Sextupole B x = mxy B y = 1 2 m(x 2 - y 2 ) Focusing strength in horizontal plane: B' y = mx } Place sextupole after a bending dipole where dispersion function is non zero

9 Focusing from quadrupole x Δx x f = l r = l qb y gmv = l qb' gmv x 1 f = qb'l gmv = kl Required by Maxwell equation, a single quadrupole has to provide focusing in one plane and defocusing in the other plane Ñ rb = 0 B x = f B y and B y = B x r F = q(-vb'x ŷ + vb' y ˆx) s

10 Transfer matrix of a qudruploe Thin lens: length of quadrupole is negligible to the displacement relative to the center of the magnet Dx' = - l r = -l qb y gmv = - qb'l gmv x = -klx æ x ö = è x' ø æ ö æ x ö 1 è f ø è x' ø

11 Transfer matrix of a drift space Transfer matrix of a drift space Δx x L s æ x ö æ = 1 L ö è x' ø è 0 1ø æ x ö è x' ø

12 Lattice Arrangement of magnets: structure of beam line Bending dipoles, Quadrupoles, Steering dipoles, Drift space and Other insertion elements Example: FODO cell: alternating arrangement between focusing and defocusing quadrupoles f -f L L One FODO cell

13 FODO lattice æ è x x' æ ö = ø è æ = è f ö æ è ø 1 L L2 2 f 2 2L(1+ L 2 f ) - L 2 f 2 1- L2 2 f 2 æ ö ø è ö æ è ø f x x' ö ø ö æ è ø 1 L 0 1 æ ö ø è f ö æ è ø x x' ö ø Net effect is focusing!

14 FODO lattice Net effect is focusing Provide focusing in both planes! y y' æ è ö ø = f 1 æ è ö ø 1 L 0 1 æ è ö ø f 1 æ è ö ø 1 L 0 1 æ è ö ø f 1 æ è ö ø y y' æ è ö ø = 1- L2 2 f 2 2L(1- L 2 f ) - L 2 f 2 1- L2 2 f 2 æ è ö ø y y' æ è ö ø

15 Curverlinear coordinate system Coordinate system to describe particle motion in an accelerator Moves with the particle r (s) ˆ y ˆ x r 0 (s) s ˆ Set of unit vectors: s ˆ (s) = dr 0 (s) ds x ˆ (s) = -r dˆ s (s) ds y ˆ (s) = x ˆ (s) s ˆ (s)

16 x Equation of motion s Equation of motion in transverse plane x dˆ s (s) ds dˆ x (s) ds r (s) = r 0 (s) + xˆ x (s) + yˆ y (s) dˆ y (s) ds = - 1 r ˆ x (s) = 1 r ˆ s (s) = 0

17 d r (s) dt d 2 r (s) dt 2 Equation of motion = ds dt [ d r 0 ds + x' x ˆ + x dˆ x ds + y' y ˆ ] = ds dt [(1+ x r )ˆ s + x' x ˆ + y' y ˆ ] r v = ds dt [(1+ x r )ˆ s + x' x ˆ + y' y ˆ ] = v sˆ s + v x x ˆ + v y y ˆ v 2 = v r 2 æ = ds ö è dt ø = ds dt 2 [(1+ x r )2 + x' 2 + y' 2 ] dv r ds» v 2 (1+ x [(x''- r + x r ) x ˆ + x' s r ˆ + y'' y ˆ ] r )2

18 Equation of motion d 2 r (s) dt 2» v 2 (1+ x [(x''- r + x r ) x ˆ + x' s r ˆ + y'' y ˆ ] = qr v B r gm r )2 x''- r + x r 2 = - qb y gmv (1+ x r )2 x''+ q B gmv x = 0 y'' = qb x gmv (1+ x r )2 y''- qb' gmv y = 0

19 Solution of equation of motion Comparison with harmonic oscillator: A system with a restoring force which is proportional to the distance from its equilibrium position, i.e. Hooker s Law: F = d 2 x(t) dt 2 Equation of motion: = -kx(t) Where k is the spring constant d 2 x(t) dt 2 + kx(t) = 0 x(t) = Acos( kt + c) Amplitude of the sinusoidal oscillation Frequency of the oscillation

20 transverse motion: betatron oscillation The general case of equation of motion in an accelerator } For k > 0 x''+kx = 0 Where k can also be negative x(s) = Acos( ks + c) x'(s) = -A k sin( ks + c) } For k < 0 x(s) = Acosh( ks + c) x'(s) = -A k sinh( ks + c)

21 Transfer matrix of a quadrupole For a focusing quadrupole æ 1 æ x ö cos kl = k sin kl è x øout è - k sin kl cos kl ö æ è ø x x ö ø in For a de-focusing quadrupole æ è x x ö ø out æ = è cosh kl 1 k sinh - k sinh kl cosh kl kl ö æ è ø x x ö ø in

22 Hill's equation In an accelerator which consists individual magnets, the equation of motion can be expressed as, x''+k(s)x = 0 k(s + L p ) = k(s) Here, k(s) is an periodic function of L p, which is the length of the periodicity of the lattice, i.e. the magnet arrangement. It can be the circumference of machine or part of it. Similar to harmonic oscillator, expect solution as or: x(s) = A(s)cos(y(s) + c) x(s) = A b x ( s) cos(y(s) + c) b x (s + L p ) = b x (s)

23 Hill s equation: cont d x'(s) = -A b x ( s)y'(s)sin(y(s) + c) + b' x s 2 with y'(s) = 1 b x (s) ( ) A 1/b x b x '' 2 b x - b '2 x 4 + kb 2 x =1 } Hill s equation x''+k(s)x = 0 is satisfied ( s) cos(y(s) + c) x(s) = A b x ( s) cos(y(s) + c) ( ) x'(s) = -A 1/b x ( s) sin(y(s) + c) + b' x s 2 A 1/b x ( s) cos(y(s) + c)

24 Betatron oscillation b x (s) Beta function : Describes the envelope of the betatron oscillation in an accelerator Phase advance: 1 y(s) = ò s ds 0 b x (s) Betatron tune: number of betatron oscillations in one orbital turn

25 Phase space In a space of x-x, the betatron oscillation projects an ellipse where b x x' 2 +g x x 2 + 2a x xx'= e g x e x X a x = b ' x b x g x =1+ a x 2 e x b x X } The are of the ellipse is pe

26 Courant-Snyder parameters The set of parameter (β x, α x and γ x ) which describe the phase space ellipse Courant-Snyder invariant: the area of the ellipse e = b x x' 2 +g x x 2 + 2a x xx'

27 Transfer Matrix of beam transport Proof the transport matrix from point 1 to point 2 is æ æ x(s 2 ) ö = è x'(s 2 ) ø è } Hint: b 2 b 1 (cosy s2 s 1 + a 1 siny s2 s 1 ) b 1 b 2 siny s2 s 1-1+ a 1a 2 b 1 b 2 siny s2 s 1 + a 1 -a 2 b 1 b 2 cosy s2 s 1 b 1 ö æ x(s 1 ) ö è x'(s (cosy s2 s b 1 -a 2 siny s2 s 1 ) 1 ) ø 2 ø x(s) = A b x ( s) cos(y(s) + c) ( ) x'(s) = -A 1/b x ( s) sin(y(s) + c) + b' x s 2 A 1/b x ( s) cos(y(s) + c)

28 One Turn Map Transfer matrix of one orbital turn æ (cos2pq æ x(s 0 + C) ö x + a x,s0 sin2pq x ) b x,s0 sin2pq x ö = è x'(s 0 + C) ø - 1+ a 2 æ x(s 0 ) ö x,s 0 sin2pq x (cos2pq x -a x,s0 sin2pq x ) è b è x'(s 0 ) ø x,s0 ø Stable condition 1 Tr(M s,s+c ) = 2cos2pQ x æ x(s + C) ö æ } Closed orbit: = x(s) ö è x'(s + C) ø è x'(s) ø æ x(s + C) ö æ = M(s + C,s) x(s) ö è x'(s + C) ø è x'(s) ø 2 Tr(M s,s+c ) 1.0

29 Stability of transverse motion Matrix from point 1 to point 2 M s2 s 1 = M n L M 2 M 1 } Stable motion requires each transfer matrix to be stable, i.e. its eigen values are in form of oscillation æ ö è 0 1ø M - li = 0 I = 1 0 With l 2 - Tr M ( )l + det M ( ) = 0 and det M ( ) =1 l = 1 2 Tr ( M ) ± 1 4 [Tr ( M )] Tr(M) 1.0 2

30 Dispersion function Transverse trajectory is function of particle momentum ρ ρ+δρ Momentum spread Define Dispersion function

31 Dispersion function Transverse trajectory is function of particle momentum. x''- r + x r 2 = - qb y gmv (1+ x r )2 B y = B 0 + B' x D(s + C) = D(s) é D''+ 1 2 p 0 - p ê ë r 2 p + B' p 0 ù ú D = 1 Br 0 p û r

32 Dispersion function: cont d In drift space 1 r = 0 and B'= 0 Þ D''= 0 dispersion function has a constant slope } In dipoles, 1 r ¹ 0 and B'= 0 D''+[ 1 2p 0 - p ]D = 1 r 2 p r

33 Dispersion function: cont d } For a focusing quad, 1 r = 0 and B'> 0 Þ D''+B' p 0 p D = 0 dispersion function oscillates sinusoidally } For a defocusing quad, 1 r = 0 B'< 0 and Þ D''-B' p 0 p D = 0 dispersion function evolves exponentially

34 Compaction factor } The difference of the length of closed orbit between offmomentum particle and on momentum particle, i.e.

35 Path length and velocity } For a particle with velocity v, L = vt } Transition energy g t : when particles with different energies spend the same time for each orbital turn - Below transition energy: higher energy particle travels faster - Above transition energy: higher energy particle travels slower

36 Chromatic effect Comes from the fact the the focusing effect of an quadrupole is momentum dependent 1 f = kl Particles with different momentum have different betatron tune - Higher energy particle has less focusing } Chromaticity: tune spread due to momentum spread Tune spread momentum spread

37 Chromaticity } Transfer matrix of a thin quadrupole Transfer matrix A B

38 Chromaticity

39 Chromaticity Assuming the tune change due to momentum difference is small x = DQ x Dp / p = - 1 åk i l i b x,i 4p i

40 Chromaticity of a FODO cell β f β d β L L One FODO cell x = - 1 æ 4p b f è 1 f - b d 1 f ö ø

41 Chromaticity correction Nature chromaticity is always negative and can be large and can result to large tune spread and get close to resonance condition Solution: - A special magnet which provides stronger focusing for particles with higher energy: sextupole

42 Sextupole B x = mxy B y = 1 2 m(x 2 - y 2 ) Focusing strength in horizontal plane: m = 2 B y x 2 B' y = mx where and, l is the magnet length Tune change due to a sextupole: DQ x = 1 4p b x,s 0 k sx x k sx = ml Br let x = D Dp p DQ x / Dp p = 1 4p b x,s 0 k sx D x

43 Chromaticity Correction DQ x / Dp p = 1 4p b x,s 0 k sx D x Sextupole produces a chromaticity with the opposite sign of the quadrupole! It prefers to be placed after a bending dipole where dispersion function is non zero Chromaticity after correction x = DQ x Dp / p = - 1 4p åk i b x,i + 1 i 4p åk sx,i i b x,i D x

44 Chromaticity correction ξ=20 ξ=1

45 How to measure betatron oscillation? Ø Excite a coherent betatron motion with a pulsed kicker X X Ø Record turn by turn beam position

46 How to measure betatron oscillation? Turn-by-turn beam position monitor data Kicker betatron tune is obtained by Fourier transform TbT beam position data

47 Beam Position Monitor (BPM) A strip line bpm response Right electrode response And left electrode response b Hence, Let x=rcosθ I R (t)- I L (t) I R (t)+ I L (t)» 4sin(f / 2) bf x

48 Coherent betatron oscillation at RHIC

49 How to measure betatron functions and phase advance?