Transverse dynamics. Transverse dynamics: degrees of freedom orthogonal to the reference trajectory

Transcription

1 Transverse dynamics Transverse dynamics: degrees of freedom orthogonal to the reference trajectory x : the horizontal plane y : the vertical plane Erik Adli, University of Oslo, August 2016, Erik.Adli@fys.uio.no, v2.20

2 It s about a beam, in 6D ψ(x, y, z) Any charged particle beam, taken at a given point in time, can be characterized with a distribution in 6D phase space.

3 Transverse distributions The two transverse planes are often to a large degree uncoupled <x y> = 0. However, evidently the position and the angle of particles in a given plane are dynamically coupled, and the correlation <x x >, <y y > will change as the beam evolves in time. Below: the effect of letting the beam propagate in free space, from a time t 1 to a time t 2 : y phase space at t=t 1. <y y > 0 y phase space at t=t 2. <y y > >

4 The reference trajectory An accelerator is designed around a reference trajectory (often called design orbit in circular accelerators) This is the trajectory an ideal particle will follow and consist of a straight line where there is no bending field arc of circle inside the bending field ρ Reference trajectory We will in the following consider transverse deviations from the reference trajectory, and how to control the magnitude of these deviations

5 Bending field Circular accelerators: piecewise circular orbits with a bending radius straight sections are needed for e.g. particle detectors in circular arc sections the magnetic field must provide the desired bending radius The accelerator design specifies a design bending radius, ρ, for the dipole field bending magnets We define 1/ρ as the normalized dipole strength :! F!!! = q( E + v B) = = ρ eb p FE + FB 1 B: bending field [T] 1 eb 1 = ρ 1 [ m ρ In a synchrotron, the bending radius is kept constant during acceleration by synchronization of B and p. p!! p: particle momentum [GeV/c] ρ: design bending radius [m] B[ T] ] = 0.3 p[ GeV / c]

6 Bending field: dipole magnets Dipole magnets provide uniform field in the desired region Several design choices : cos(φ) design LHC dipole magnet design

7 Focusing field: quadrupole magnets Reference trajectory: typically through centers of magnets and structures Desired: a restoring force of the type F x =-kx in order to keep the particles close to the ideal orbit, in both planes A linear field in both planes can be derived from the potential V(x,y) = gxy. Four poles with magnet surfaces shaped as hyperbolas. B x = -gy B y = -gx F x = -qvgx F y = qvgy (focusing) (defocusing) Forces are focusing in one plane and defocusing in the orthogonal plane Opposite focusing/defocusing by rotating the quadrupole by ninety degrees

8 Normalized magnet strengths Analogy to dipole strength: normalized quadrupole strength Quadrupole : k = eg p k[ m 2 ] 0.3 g[ T / m] p[ GeV / c] g: field gradient [T/m] p: particle momentum [GeV/c] k: design normalized quadrupole strength [m -2 ] Dipole : 1 eb 1 = ρ p 1 [ m ρ ] = 0.3 B[ T] p[ GeV / c] B: bending field [T] p: particle momentum [GeV/c] 1/ρ: design normalized dipole strength [m -1 ]

9 Magnetic lenses linear optics Analogy: magnetic lenses (quadrupoles) linear optics Focal length f of a quadrupole: f = 1 / kl where l is the length of the quadrupole Alternating focusing and defocusing lenses will together give total focusing effect in both planes; alternating gradient focusing

10 The Lattice An accelerator is composed of bending magnets, focusing magnets and usually also sextupole magnets The ensemble of magnets in the accelerator constitutes the accelerator lattice

11 Example: lattice magnets CLIC Test Facility 3

12 LHC superconducting magnets

13 Transverse dynamics Equations of motion and transfer matrices

14 We will outline the steps needed to develop the equations of motion in terms of deviations from the reference trajectory. Full derivation, Wille Ch 3.2. Coordinates Co-moving coordinate system : Moves with particle (often v~c), quantities expressed in laboratory frame x, y are small deviations from the reference trajectory x: deviations in the horizontal plane y: deviations in the vertical plane r = ρ + x

15 Linear equations of motion I Curvilinear equations of motion, in a uniform magnetic field :!p = F B: dipole field g: quadrupole field gradient mγ(!!x r! θ 2 ) = er! θ(by gx) In accelerators we use x(s) instead of x(t), and we denote dx/ds = x (s). Furthermore, using v x << v, we write r θ! = v s v!!x(t) = d 2 x ds (ds 2 dt )2 v 2 x ## (s) yielding the equation of motion as function of the accelerator coordinate s : x!! = 1 r(s) e p (B y(s) g(s)x) ( Corresponding equations for y ) p: relativistic momentum

16 Linear equations of motion II Further approximations, and dropping higher order terms : 1 r = 1 ρ + x 1 ρ (1 x ρ ), 1 p = 1 p 0 + Δp 1 (1 Δp ) p 0 p 0 We introduce the normalized magnet strengths: 1/ρ = eb / p, k = eg/ p By substituting the above we obtain the linearized trajectory equations: x!! (k(s) 1 ρ 2 (s) )x = 1 ρ(s) Δp p 0 k: normalized quadrupole strength 1/ρ : normalized dipole strength p 0 is the reference momentum, Δp is deviation from ideal momentum

17 Hill s equation Magnet strength terms are dependent on the position along the reference trajectory, s. We can then write eq. of motion as a Hill's equation: We lump focusing term into big K, 1/ρ 2 (s) k(s) K(s) We assume first Δp=0, yielding the homogenous Hill s equation: We write equations for x, analogous for y For a given magnet lattice: we take a piece-wise approach to solution: For K(s) = const>0, solutions is: K(s)=const <0: replace with hyperbolic functions xʹʹ + K( s) x = x ʹʹ + K( s) x = 0 It is helpful to put the coefficients into a matrix, the transfer matrix : 1 Δp ρ p x 1 (s) = x 0 cos( Ks) + x " 0 (1/ K )sin( Ks) 0 # x 1 & % ( = $ "' x 1 # cos( Ks) % $ K sin( Ks) (1/ K )sin( Ks) & cos( Ks) ' ( # x & # % 0 ( = M x & 0 % ( $ x " 0 ' $ x " 0 '

18 M: drift space The element with the simplest transfer matrix M: drift space between magnets (no field), with length l : Written out this gives: This simply says that in a drift space x is unchanged, and x drifts = l M x x lx x x x x l x x x x ʹ = ʹ ʹ + = ʹ = ʹ = ʹ M

19 Quadrupole transfer matrix cos( l k ) Full solution : M = k sin( l k ) Thin lens approximation : Real quadrupole may be modeled as a infinitely thin lens that focuses or defocuses, plus the drift space to represent the length of the quadrupole valid if focal length f=1/kl >> l Written out multiplication: x = x 0 1 sin( l k ) k cos( l k ) Mthin = " $ $ $ # f l [m] 1 k [m -2 ] % ' ' ' & xʹ = xʹ 0 1 x f 0-1/f is a focusing term A defocusing quadrupole in x (rotated 90 ): -f f

20 Dipole transfer matrix Bending magnets introduce focusing terms as well. The solution of Hill s equation provides the focusing terms for a idealized sector dipole : Msec = " $ $ $ $ # $ cos l ρ 1 ρ sin l ρ ρ sin l ρ cos l ρ % ' ' ' ' &' The more common type of dipole found in accelerators is a rectangular dipole, which does not provide the focusing term.! # Mrect = 1 ρ sin l # ρ # " 0 1 $ & & & %

21 Hill s equation: solutions for an accelerator The transverse optics of an accelerator can be modelled as composed of elements where K(s) = constant inside the element We solve Hill s equation by applying the transfer matrix M, from the start to the end of each element Inside dipoles: K(s) = 1/ρ 2 Inside quadrupoles: K(s) = +/-k Where there is no field: K(s) = 0 One can find the effect on a particle travelling through the whole, or part, of the lattice by multiplying the M matrices for the various components: M tot = M dipole M F M dipole M D...

22 Quadrupole FODO doublet A FODO quadrupole doublet consist of a focusing quadrupole followed by a drift, a defocusing quadrupole and a drift Using the thin lens approximation we can calculate the total matrix : FODO is focusing in both the horizontal and the vertical planes (since changing plane equals f = -f ) + + = = / / f l f l f l f l l f l l f l f M doublet

23 Stability of a FODO structure A FODO lattice yields focusing in both planes, however too short focal length will give overfocusing, and an unstable trajectory: Liming case, L = 4 f stability f > L / 4 We may calculate rigorously the stability criterion of a periodic lattice using Courant-Snyder analysis introduced in the next section

24 ?

25 Transverse dynamics Courant-Snyder framework Previous section: a straight-forward mathematical framework for single particle motion The next slides : allowing for analysis of a beam of particles

26 Particle motion: Hill's equation We have calculated particle motion for a single particle by solving Hill s equation piece-wise and multiplying transfer matrices M(s). We now seek a general solution of the the equation : x " + K(s)x = 0 Reminder: solution of Hill s equation with K(s) =K harmonic oscillator x( s) xʹʹ + Kx = Asin( = 0 K s +φ 0 )

27 Reformulation of Hill's equation: beta function We will define a function varying along the lattice in the solution to the Hill s equation; the beta function, β(s). " " x + K(s)x = 0 the following reformulation fulfills the original equation : x(s) = εβ(s) sin(φ(s) + φ 0 ), if the following constraints are fulfulled : φ(s) = s s 0 dt β 1 2 β(s)β'' 1 4 β'2 +K(s)β 2 (s) =1 The variable amplitude given by solving the betatron equation, including initial conditions (β(0), β'(0)) The solution is thus a quasi-harmonic oscillator, with amplitude and phaseadvance dependent on s. Derivation: Wille 3.7.

28 Transverse phase-space The transverse phase-space in the horizontal plane is spanned by x and x : x'(s) = x(s) = εβ(s) sin(φ(s)+φ 0 ) ε β(s) (cos(φ(s)+φ 0)+ β '(s) 2 sin(φ(s)+φ 0)) One can eliminate the phase, φ(s), to get an equation for an ellipse with area πε : γ(s)x(s) 2 + 2α(s)x(s)x'(s)+ β(s)x'(s) 2 = ε where we have defined : α(s) -(1/2)β'(s), γ(s) (1+ α 2 (s))/ β(s), β(s), α(s), γ(s): Twiss parameters (also called Courant-Snyder parameters) ε : single-particle emittance Derivation: Wille 3.8.

29 The phase-space ellipse γ(s)x(s) 2 + 2α(s)x(s)x'(s)+ β(s)x'(s) 2 = ε Particles with different phase and equal single-partice emittance. Envelope of particle motion : x(s) = εβ(s)

30 Evolution of the phase-space ellipse The beta function, β (and α, γ) evolve according to the solution of Hill s equation, i.e. they depend on the position along the lattice, s : As β evolves, the area of the ellipse, πε, remains constant : γ(s)x(s) 2 + 2α(s)x(s)x'(s)+ β(s)x'(s) 2 = ε Phase-space area for a particle : invariant with respect to s. From A. Chao Envelope of particle motion : x(s) = εβ(s)

31 Summary : single particle propagation In the Courant-Snyder framework, which information do we need to describe the propagation of a single particle? 1) Twiss parameters. These are given as solutions of the betatron equation. They propagate along the lattice, specification by K(s) as : 2) The particle initial state must be specified, by the initial single particle emittance, ε, and its initial phase φ 0. 3) For the moment, the beta function, β, must also be defined by initial condition β(0) and β'(0) = -2α(0). A more complicated description of single particle motion with respect to the matrix framework. Advantage comes when we now relate the Twiss parameters to the description of the entire beam.

32 Description of a beam

33 The beam sigma matrix (Bivariate Guassian distribution)

34 Gaussian beams

35 Description of beams We have shown that we can describe the second order moments of a beam of particles using the Twiss parameters, which we used before to describe the motion of a single particle. Gaussian beams are fully described by the moments up to second order, this by centroid coordinates and Σ. For Gaussian beams and linear optics, this implies that the Twiss parameters uniquely defines the beam shape along the lattice. The phase-space ellipses may represent the evolution of the entire beam in phase space. Individual particles have differences phases, and different singe-particle emittance, but the same Twiss parameters

36 Evolution of a beam Rms beam size: σ ( s) = ( s) ε rmsβ Beam quality Evolves with lattice

37 Twiss parameters: initial conditions For a non-circular lattice, the solution of Hill s equation, the Twiss parameters, depends on the initial Twiss parameters (the initial beam), β(0) and α(0)=-1/2β (0). Usually the lattice has design Twiss parameters, and one aims to match the incoming beam to the lattice design. Example from the CLIC Test Facility at CERN Example to the right: evolution of Twiss parameters where the initial magnets are adjusted to match the beam into a periodic focusing lattice of FODO-type. F D

38 Evolution of the beta function

39 Calculation of Twiss using M(s) We calculated earlier the evolution of single particles along an accelerator lattice using the transfer matrices: x 1 = M 1 0 x 0 We can calculate the evolution of the beta function using transfer matrices. We define a matrix of Twiss parameters : # B(s) = Σ(s) /ε rms = % $ % β(s) α(s) α(s) γ(s) The solution of Hill's gives: x T B -1 (s) x = const. Substituting x 1 = M 1 0 x 0 gives : B = M B 0 M 1T 0 & ( '(, x = # x % $ x' & ( ' Focusing of a beam, β(s), can be calculated by M(s) Wille 3.10.

40 Twiss parameters: evolution M tot = M dipole M F M dipole M D... Example from the CLIC Test Facility at CERN F D

41 M parameterized using Twiss We may also express the transfer matrix between two locations in an accelerator in terms of Twiss parameters at the locations. This is especially interesting when studying periodic lattices. Using and we can derive :

42 Periodic lattices

43 Periodic lattices Consider a periodic accelerator lattice, with M s s+l (s) from point s to point s+l. For example, one full turn of a circular accelerator may be the period. The Twiss parameters must obey the following condition : B(s + L) = M s s+l B(s)(M s s+l ) T ", B(s) = $ # $ β(s) α(s) α(s) γ(s) % ' &' The parameters are in this case uniquely defined by M s s+l (s) : α = α(s) = α(s + L) β = β(s) = β(s + L) γ = γ(s) = γ(s + L) Circular accelerators: periodic lattice by default. Existence of periodic solutions follows from Floquet Theory. Wille 3.14.

44 Twiss parameters: period lattice Start with the general parameterization : For a periodic lattice, with period L, this reduces to : where Φ = ψ(s 2 ) ψ(s 1 ) is the phase-advance from s 1 to s 2. The lattice Twiss parameters may then be calculated from the periodic transfer matrix :

45 Example: thin-lens FODO f L Example from A. Chao

46 Betatron oscillations and phase advance Solution of Hill s equation : Example solution in a periodic lattice : x(s) = εβ(s) sin(ϕ(s)+ϕ 0 ), ϕ(s) = s s 0 dt β A particle will undergo betatron oscillations around the reference trajectory. The lattice parameter β(s) defines the envelope for the particle motion φ(s) is the particle phase-advance from point s 0 to point s in a lattice In a FODO structure the beta function is at maximum in the middle of the F quadrupole and at minimum in the middle of the D quadrupole In the figure : about 5 FODO cells for a full oscillation, yielding a phaseadvance per FODO cell of φ cell 70º.

47 Periodic lattice, tune Reminder: even if the beta function is periodic, the particle motion is in general not periodic; after one revolution the initial phase φ 0 is altered, since the particle has advanced by a certain phase, φ turn. Phase advance per turn may be given as the number of periodic cells, and the phase-advance per cell, φ turn = N cell φ cell. We define the tune, Q, as the number of betatron oscillations per turn : Q = N cell φ cell / 2π (From M. Sands)

48 Tune diagram In order to avoid driving resonant instabilities in a circular accelerator, the fractional part of the tune, Q, must not be mq x + nq y = N, where m,n,n may be 0,1,2,3.(up to a number depending on the performance requirements). These instabilities originate from imperfections of real magnets; beam undergoes small kicks, which if they add up coherently may blow up beam size. Integer and N-integer tunes must be avoided

49 Stability of a periodic lattice Requirement for the lattice structure to be stable: motion must be bounded Let M(s) be the matrix for one periodic cell, M(s) N is the matrix for the whole accelerator,. M(s) nn corresponds to n turns in the accelerator Stability requires that the elements of M(s) nn remain bounded as n Applying this criterion on the FODO cell gives the condition : stability f > L / 4 Liming case, L = 4 f

50 Evolution of a beam Rms beam size: σ ( s) = ( s) ε rmsβ Beam quality Evolves with lattice

51 Emittance preservation We have shown that the rms emittance is a preserved quantity for Gaussian beams, in drift and with linear magnetic lenses More generally the phase-space area, emittance, is preserved if only conservative forces do work on the particles (Liouville s theorem) Particle acceleration by an RF-field is not conservative, the emittance will shrink if the beam is accelerated : The emittance will shrink proportional to the beam energy increase, given by βγ,. We define a normalized emittance, conserved under acceleration : NB: β γ are here the normalized velocity and the Lorentz factor respectively. They are not related to the Twiss parameters. ε N,rms = γβ ε rms Independent quantities in each plane, x, y.

52 Summary: Transverse parameters Betatron oscillations: particle oscillations in the transverse planes, due to the focusing, for example alternating gradient focusing β(s): beta function square of envelope of the particle motion defined by the lattice ε : emittance, measurement of beam quality; conserved under conservative forces Q = N cell φ cell / 2π : accelerator tune Integer and half-integer tunes etc. will lead to instabilities and must be avoided An alternative expression for luminosity: L = f n n 1 2 * * 4 ε xβ xε yβ y

53 Acknowledgements We have to a large part followed K. Wille (2000) in the derivations. Part of the material presented here is based on Alex Chao s USPAS lecture notes and Volker Ziemann s lecture notes.