Presentation on theme: "Transverse dynamics Transverse dynamics: degrees of freedom orthogonal to the reference trajectory x : the horizontal plane y : the vertical plane Erik."— Presentation transcript:
1 Transverse dynamicsTransverse dynamics: degrees of freedom orthogonal to the reference trajectoryx : the horizontal planey : the vertical planeErik Adli, University of Oslo, August 2014, v2.07
2 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 ofa straight line where there is no bending fieldarc of circle inside the bending fieldWe will in the following consider transverse deviations from the reference trajectory, and how to control the magnitude of these deviations
3 Bending fieldCircular accelerators: piecewise circular orbits with a bending radiusstraight sections are needed for e.g. particle detectorsin circular arc sections the magnetic field must provide the desired bending radiusThe accelerator design specifies a design bending radius, r, for the dipole field bending magnetsIn a synchrotron, the bending radius is kept constant during acceleration by synchronization of B and p.B: bending field [T]p: particle momentum [GeV/c]r: design bending radius [m]We define 1/r as the normalized dipole strength :
4 Focusing field: quadrupole magnets Reference trajectory: typically through centers of magnets and structuresDesired: a restoring force of the type Fx=-kx in order to keep the particles close to the ideal orbit, in both planesA linear field in both planes can be derived from the potential V(x,y) = gxy. Four poles with magnet surfaces shaped as hyperbolas.Bx = -gyBy = -gxFx = -qvgx (focusing)Fy = qvgy (defocusing)Forces are focusing in one plane and defocusing in the orthogonal planeOpposite focusing/defocusing by rotating the quadrupole by ninety degrees
6 Magnetic lenses linear optics Analogy: magnetic lenses (quadrupoles) linear opticsFocal length f of a quadrupole:where l is the length of the quadrupoleAlternating focusing and defocusing lenses will together give total focusing effect in both planes; “alternating gradient focusing”f = 1 / kl
7 The LatticeAn accelerator is composed of bending magnets, focusing magnets and usually also sextupole magnetsThe ensemble of magnets in the accelerator constitutes the “accelerator lattice”
8 Bending field: dipole magnets Dipole magnets provide uniform field in the desired regionSeveral design choices, including cos(q)" design that allows opposite and uniform field in both vacuum chambersLHC dipole magnet design
11 Transverse dynamics Equations of motion and transfer matrices
12 Coordinates Coordinate system: x, y are small deviations from the reference trajectoryx: deviations in the horizontal planey: deviations in the vertical planer = r + xWe will outline the steps needed to develop the equations of motion in terms of deviations from the reference trajectory. For full derivation, see Wille (2000).
13 Linear equations of motion I Equations of motion in a uniform magnetic field :In accelerators we use x(s) instead of x(t), and we denote dx/ds = x’(s).Furthermore, using vx << v, we writeyielding the equation of motion as function of the accelerator coordinate s :B: dipole fieldg: quadrupole field gradient( assuming g >> 1 )( Corresponding equations for y )
14 Linear equations of motion II Further approximations, and dropping higher order terms :We introduce the normalized magnet strengths:By substituting the above we obtain the linearized trajectory equations:k: normalized quadrupole strength1/r: normalized dipole strengthp0 is the reference momentum, Dp is deviation from ideal momentum
15 Hill’s equationMagnet strength terms are dependent on the position along the reference trajectory, s. We can then write eq. of motion as a Hill's equation: 1/r2(s) – k(s) K(s)We assume first Dp=0, yielding the homogenous Hill’s equation:We write equations for x, analogous for yFor 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 functionsIt is helpful to put the coefficients into a matrix, the transfer matrix :
16 M: drift spaceThe 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
17 Quadrupole transfer matrix k [m-2]Full solution :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 quadrupolevalid if focal length f=1/kl >> lWritten out multiplication:-1/f is a focusing termA defocusing quadrupole in x (rotated 90): -f fl [m]
18 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 :The more common type of dipole found in accelerators is a rectangular dipole, which does not provide the focusing term.
19 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 elementWe solve Hill’s equation by applying the transfer matrix M, from the start to the end of each elementInside dipoles: K(s) = 1/r2Inside quadrupoles: K(s) = +/-kWhere there is no field: K(s) = 0One 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:
20 Quadrupole FODO doublet A FODO quadrupole doublet consist of a focusing quadrupole followed by a drift, a defocusing quadrupole and a driftUsing 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 )
21 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:We may calculate rigorously the stability criterion of a periodic lattice using Courant-Snyder analysis introduced in the next sectionLiming case,L = 4 f
23 Transverse dynamics Courant-Snyder framework Previous section: a straight-forward mathematical framework for computing single particle motionThe next slides : analyze the transverse motion in an accelerators using standard accelerator terminology, allowing for analysis of a beam of particles
24 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 :Reminder: solution of Hill’s equation with K(s) =K harmonic oscillator
25 Reformulation of Hill's equation: beta function We define function varying along the lattice in the solution to the Hill’s equation; the beta function, b(s).The solution is thus a quasi-harmonic oscillator, with amplitude and phase-advance dependent on s.The complete solution is given by solving the betatron equation (left).
26 Transverse phase-space The particle phase-space in the horizontal plane is spanned by x and x‘ :By eliminating the phase, f, we get equation for an ellipse with area pe :where we have defined :a(s) -(1/2)b'(s),g(s) (1+ a2(s))/ b(s),b(s), a(s), g(s): Twiss parameters(in the USA: called Courant-Snyder parameters)e : single particle emittanceFor a given single particle emittance, e, the phase-space ellipse area is an invariant (with respect to the s coordinate).
27 The phase-space ellipse Envelope of particle motion :Particles with different phase and equal emittance.
28 Evolution of the phase-space ellipse Particles populate trajectories on ellipses with the shape determined by the Twiss parameters, b, a, g.The Twiss parameters evolve according to the solution of Hill’s equation, i.e. they depend on the position along the lattice, sAs they evolve, the area of the ellipse, pe, remains constantFrom A. Chao
29 Single particle propagation In the Courant-Snyder framework, which information do we need to describe the propagation of a single particle?1) We need the Twiss parameters. These are given as solutions of the betatron equation. How they propagate depends only on the lattice specification, K(s) :2) In addition, the particle initial state must be specified, by the initial single particle emittance, e, and its initial phase f0.This is hardly a simplification of the single particle motion, however, we shall now relate the Twiss parameters and the description of the entire beam.
30 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), b(0) and a(0)=-1/2b’(0).Usually the lattice has design Twiss parameters, and one aims to match the incoming beam to the lattice design.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.Example from the CLIC Test Facility at CERNF D
31 Betatron oscillations and phase advance Solution of Hill’s equation :Example solution in a periodic lattice :A particle will undergo betatron oscillations around the reference trajectory. The lattice parameter b(s) defines the envelope for the particle motionf(s) is the particle phase-advance from point s0 to point s in a latticeIn 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 quadrupoleIn the figure : about 5 FODO cells for a full oscillation, yielding a phase-advance per FODO cell of fcell 70º.
34 Evolution of beamsThe importance of this is that we can describe the transport of the 2nd order moments of the beam in the same way as we describe the transformation of the Twiss parameters along the lattice. For Gaussian beams and linear optics, this means that the Twiss parameters uniquely defines the beam shape along the lattice.
35 Relation b(s) and M(s)We calculated earlier the evolution of single particles along an accelerator lattice using the transfer matrices: x1= M10x0We can calculate the evolution of the beta function using transfer matrices as well. We define a matrix with the Twiss parameters :The solution of Hill's gives: xTB-1 (s) x = const.Substituting x1 = Mx0 gives :
36 Emittance preservation We have shown that the rms emittance is a preserved quantity for Gaussian beams, in drift and with linear magnetic lensesMore 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 bg,. We define a normalized emittance, conserved under acceleration :NB: b g on this slide is the normalized velocity and the Lorentz factor respectively. They are not related to the Twiss parameters.eN,rms = gb ermsIndependent quantities in each plane, x, y.
37 Periodic latticesConsider a periodic accelerator lattice, with Mss+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 :The parameters are in this case uniquely defined by Mss+L(s) :Circular accelerators: periodic lattice by default.Existence of periodic solutions follows by Floquet Theory.
38 Twiss parameters: period lattice Using the solution of Hill’s equation we may re-write the transfer matrix between two locations in an accelerator in terms of Twiss parameters as :For a periodic lattice, with period L, this reduces to :where F = y(s2) – y(s1) is the phase-advance from s1 to s2. The lattice Twiss parameters may then be calculated from the periodic transfer matrix :
40 Stability of a latticeAbsolute requirement: the lattice structure of an accelerator must be stable: motion must be boundedLet 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 acceleratorStability requires that the elements of M(s)nN remain bounded as n Applying this general criterion on the FODO cell gives the condition :Liming case,L = 4 f
41 Particle motion in a periodic lattice, tune Reminder: even if beta function is be periodic, the particle motion itself is in general not periodic; after one revolution the initial phase f0 is altered, since the particle has advanced by a certain phase, fturn. Phase advance per turn may be given as the number of periodic cells, and the phase-advance per cell, fturn = Ncell fcell.We define the tune, Q, as the number of betatron oscillations per turn :Q = Ncell fcell / 2p(From M. Sands)
42 Integer and N-integer tunes must be avoided Tune diagramIn order to avoid driving resonant instabilities in a circular accelerator, the fractional part of the tune, Q, must not be mQx + nQy = 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
43 Evolution of a beamRms beam size:Beam qualityEvolves with lattice
44 Summary: Transverse parameters Betatron oscillations: particle oscillations in the transverse planes, due to the focusing, for example alternating gradient focusingb(s): beta function square of envelope of the particle motionin periodic accelerators: defined uniquely by the latticee : emittance, measurement of beam quality; conserved under conservative forcesQ = Ncell fcell / 2p : accelerator tuneInteger and half-integer tunes etc. will lead to instabilities and must be avoidedAn alternative expression for luminosity:
45 AcknowledgementsPart of the material presented here is based on Alex Chao’s USPAS lecture notes and Volker Ziemann’s lecture notes.