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Lecture A3: Damping Rings Linear beam dynamics overview Damping rings, Linear Collider School 2015 Yannis PAPAPHILIPPOU Accelerator and Beam Physics group Beams Department CERN Ninth International Accelerator School for Linear Colliders 26 October – 6 November 2015, Whistler BC, Canada 1 Outline – Transverse transfer matrices Hill’s equations Derivation Harmonic oscillator Transport Matrices Matrix formalism Drift Damping rings, Linear Collider School 2015 Thin lens Quadrupoles Dipoles Sector magnets Rectangular magnets Doublet FODO 2 Equations of motion – Linear fields Consider s-dependent fields from dipoles and normal quadrupoles The total momentum can be written With magnetic rigidity gradient and normalized Damping rings, Linear Collider School 2015 the equations of motion are Inhomogeneous equations with s-dependent coefficients Note that the term 1/ρ2 corresponds to the dipole week focusing The term ΔP/(Pρ) represents off-momentum particles 3 Hill’s equations Solutions are combination of the ones from the homogeneous and inhomogeneous equations Consider particles with the design momentum. The equations of motion become George Hill Damping rings, Linear Collider School 2015 with Hill’s equations of linear transverse particle motion Linear equations with s-dependent coefficients (harmonic oscillator with time dependent frequency) In a ring (or in transport line with symmetries), coefficients are periodic Not straightforward to derive analytical solutions for whole accelerator 4 Harmonic oscillator – spring Consider K(s) = k0 = constant Equations of harmonic oscillator with solution u u Damping rings, Linear Collider School 2015 with for k0 > 0 for k0 < 0 Note that the solution can be written in matrix form 5 Matrix formalism General transfer matrix from s0 to s Note that which is always true for conservative systems Damping rings, Linear Collider School 2015 Note also that The accelerator can be build by a series of matrix multiplications S1 S0 S2 S3 … Sn-1 from s0 to s1 Sn from s0 to s2 from s0 to s3 from s0 to sn 6 Symmetric lines System with normal symmetry Damping rings, Linear Collider School 2015 S System with mirror symmetry S 7 4x4 Matrices Damping rings, Linear Collider School 2015 Combine the matrices for each plane to get a total 4x4 matrix Uncoupled motion 8 Transfer matrix of a drift Consider a drift (no magnetic elements) of length L=s-s0 L Damping rings, Linear Collider School 2015 Position changes if particle has a slope which remains unchanged. After u’ Before u’L 0 L Real Space u s Phase Space 9 (De)focusing thin lens Consider a lens with focal length ±f Slope diminishes (focusing) or increases (defocusing) for positive position, which remains unchanged. u’ u Damping rings, Linear Collider School 2015 Before After 0 f After u Before 0 u’ f 10 Quadrupole Consider a quadrupole magnet of length L = s-s0. The field is Damping rings, Linear Collider School 2015 with normalized quadrupole gradient (in m-2) The transport through a quadrupole is u’ u 0 L s 11 (De)focusing Quadrupoles For a focusing quadrupole (k>0) Damping rings, Linear Collider School 2015 For a defocusing quadrupole (k<0) By setting Note that the sign of k or f is now absorbed inside the symbol In the other plane, focusing becomes defocusing and vice versa 12 Sector Dipole Consider a dipole of (arc) length L. Damping rings, Linear Collider School 2015 By setting in the focusing quadrupole matrix transfer matrix for a sector dipole becomes the L with a bending radius In the non-deflecting plane and θ This is a hard-edge model. In fact, there is some edge focusing in the vertical plane Matrix generalized by adding gradient (synchrotron magnet)13 Rectangular Dipole ΔL θ Damping rings, Linear Collider School 2015 Consider a rectangular dipole with bending angle θ. At each edge of length ΔL, the deflecting angle is changed by i.e., it acts as a thin defocusing lens with focal length The transfer matrix is with For θ<<1, δ=θ/2 In deflecting plane (like drift), in non-deflecting plane (like sector) 14 Quadrupole doublet Consider a quadrupole doublet, i.e. two quadrupoles with focal lengths f1 and f2 separated by a distance L. x Damping rings, Linear Collider School 2015 L In thin lens approximation the transport matrix is with the total focal length Setting f1 = - f2 = f Alternating gradient focusing seems overall focusing This is only valid in thin lens approximation 15 FODO Cell Consider defocusing quad “sandwiched” by two focusing quads with focal lengths ± f. Symmetric transfer matrix from center to center of focusing quads L L Damping rings, Linear Collider School 2015 with the transfer matrices The total transfer matrix is 16 Outline – Betatron functions General solutions of Hill’s equations Floquet theory Betatron functions Transfer matrices revisited General and periodic cell Damping rings, Linear Collider School 2015 General transport of betatron functions Drift Beam waist 17 Solution of Betatron equations Betatron equations are linear with periodic coefficients Floquet theorem states that the solutions are Damping rings, Linear Collider School 2015 where w(s), ψ(s) are periodic with the same period Note that solutions resemble the one of harmonic oscillator Substitute solution in Betatron equations 0 0 18 Betatron functions By multiplying with w the coefficient of sin Integrate to get Replace ψ’ in the coefficient of cos and obtain Damping rings, Linear Collider School 2015 Define the Betatron or twiss or lattice functions (CourantSnyder parameters) 19 Betatron motion The on-momentum linear betatron motion of a particle is described by Damping rings, Linear Collider School 2015 with the twiss functions the betatron phase and the beta function is defined by the envelope equation By differentiation, we have that the angle is 20 Courant-Snyder invariant Eliminating the angles by the position and slope we define the Courant-Snyder invariant This is an ellipse in phase space with area πε The twiss functions have a geometric meaning Damping rings, Linear Collider School 2015 The beam envelope is The beam divergence 21 General transfer matrix From equation for position and angle we have Damping rings, Linear Collider School 2015 Expand the trigonometric formulas and set ψ(0)=0 to get the transfer matrix from location 0 to s with and the phase advance 22 Periodic transfer matrix Consider a periodic cell of length C The optics functions are and the phase advance Damping rings, Linear Collider School 2015 The transfer matrix is The cell matrix can be also written as with and the Twiss matrix 23 Stability conditions From the periodic transport matrix and the following stability criterion Damping rings, Linear Collider School 2015 From transfer matrix for a cell we get 24 Tune and working point In a ring, the tune is defined from the 1-turn phase advance Damping rings, Linear Collider School 2015 i.e. number betatron oscillations per turn Taking the average of the betatron tune around the ring we have in smooth approximation Extremely useful formula for deriving scaling laws The position of the tunes in a diagram of horizontal versus vertical tune is called a working point The tunes are imposed by the choice of the quadrupole strengths One should try to avoid resonance conditions 25 Transport of Betatron functions For a general matrix between position 1 and 2 and the inverse Damping rings, Linear Collider School 2015 Equating the invariant at the two locations and eliminating the transverse positions and angles 26 Example I: Drift Consider a drift with length s The transfer matrix is Damping rings, Linear Collider School 2015 The betatron transport matrix is from which γ β α s 27 Simplified method for betatron transport Consider the beta matrix the matrix and its transpose Damping rings, Linear Collider School 2015 It can be shown that Application in the case of the drift and 28 Example II: Beam waist For beam waist α=0 and occurs at s = α0/γ0 Beta function grows quadratically and is minimum in waist γ β α s Damping rings, Linear Collider School 2015 waist The beta at the waste for having beta minimum in the middle of a drift with length L is The phase advance of a drift is which is π/2 when . Thus, for a drift 29 Outline – Off-momentum dynamics Off-momentum particles Effect from dipoles and quadrupoles Dispersion equation 3x3 transfer matrices Periodic lattices in circular accelerators Damping rings, Linear Collider School 2015 Periodic solutions for beta function and dispersion Symmetric solution 3x3 FODO cell matrix 30 Effect of dipole on off-momentum particles Damping rings, Linear Collider School 2015 Up to now all particles had the same momentum P0 What happens for off-momentum particles, i.e. particles with momentum P0+ΔP? Consider a dipole with field B and ρ+δρ bending radius ρ ρ Recall that the magnetic rigidity is θ and for off-momentum particles P0+ΔP P0 Considering the effective length of the dipole unchanged Off-momentum particles get different deflection (different orbit) 31 Off-momentum particles and quadrupoles Consider a quadrupole with gradient G Recall that the normalized gradient is P0+ΔP P0 Damping rings, Linear Collider School 2015 and for off-momentum particles Off-momentum particle gets different focusing This is equivalent to the effect of optical lenses on light of different wavelengths 32 Dispersion equation Consider the equations of motion for off-momentum particles Damping rings, Linear Collider School 2015 The solution is a sum of the homogeneous equation (onmomentum) and the inhomogeneous (off-momentum) In that way, the equations of motion are split in two parts The dispersion function can be defined as The dispersion equation is 33 Dispersion solution for a bend Simple solution by considering motion through a sector dipole with constant bending radius ρ Damping rings, Linear Collider School 2015 The dispersion equation becomes The solution of the homogeneous is harmonic with frequency 1/ρ A particular solution for the inhomogeneous is and we get by replacing Setting D(0) = D0 and D’(0) = D0’, the solutions for dispersion are 34 General dispersion solution Damping rings, Linear Collider School 2015 General solution possible with perturbation theory and use of Green functions For a general matrix the solution is One can verify that this solution indeed satisfies the differential equation of the dispersion (and the sector bend) The general betatron solutions can be obtained by 3X3 transfer matrices including dispersion Recalling that and 35 General solution for the dispersion Introduce Floquet variables Damping rings, Linear Collider School 2015 The Hill’s equations are written The solutions are the ones of an harmonic oscillator For the dispersion solution , the inhomogeneous equation in Floquet variables is written This is a forced harmonic oscillator with solution Note the resonance conditions for integer tunes!!! 36 3x3 transfer matrices - Drift, quad and sector bend Damping rings, Linear Collider School 2015 For drifts and quadrupoles which do not create dispersion the 3x3 transfer matrices are just For the deflecting plane of a sector bend we have seen that the matrix is and in the non-deflecting plane is just a drift. 37 3x3 transfer matrices - Synchrotron magnet Synchrotron magnets have focusing and bending included in their body. From the solution of the sector bend, by replacing 1/ρ with Damping rings, Linear Collider School 2015 For K>0 For K<0 with 38 3x3 transfer matrices - Rectangular magnet Damping rings, Linear Collider School 2015 The end field of a rectangular magnet is simply the one of a quadrupole. The transfer matrix for the edges is The transfer matrix for the body of the magnet is like for the sector bend The total transfer matrix is 39 Periodic solutions Consider two points s0 and s1 for which the magnetic structure is repeated. The optical function follow periodicity conditions Damping rings, Linear Collider School 2015 The beta matrix at this point is Consider the transfer matrix from s0 to s1 The solution for the optics functions is with the condition 40 Periodic solutions for dispersion Consider the 3x3 matrix for propagating dispersion between s0 and s1 Damping rings, Linear Collider School 2015 Solve for the dispersion and its derivative to get with the conditions 41 Symmetric solutions Consider two points s0 and s1 for which the lattice is mirror symmetric The optical function follow periodicity conditions Damping rings, Linear Collider School 2015 The beta matrices at s0 and s1 are Considering the transfer matrix between s0 and s1 The solution for the optics functions is with the condition 42 Symmetric solutions for dispersion Consider the 3x3 matrix for propagating dispersion between s0 and s1 Damping rings, Linear Collider School 2015 Solve for the dispersion in the two locations Imposing certain values for beta and dispersion, quadrupoles can be adjusted in order to get a solution 43 Periodic lattices’ stability criterion revisited Consider a general periodic structure of length 2L which contains N cells. The transfer matrix can be written as Damping rings, Linear Collider School 2015 The periodic structure can be expressed as with Note that because Note also that By using de Moivre’s formula We have the following general stability criterion 44 3X3 FODO cell matrix Damping rings, Linear Collider School 2015 Insert a sector dipole in between the quads and consider θ=L/ρ<<1 Now the transfer matrix is which gives and after multiplication 45 Longitudinal dynamics Damping rings, Linear Collider School 2015 RF acceleration Energy gain and phase stability Momentum compaction and transition Equations of motion Small amplitudes Longitudinal invariant Separatrix Energy acceptance Stationary bucket Adiabatic damping 46 RF acceleration Damping rings, Linear Collider School 2015 The use of RF fields allows an arbitrary number of accelerating steps in gaps and electrodes fed by RF generator The electric field is not longer continuous but sinusoidal alternating half periods of acceleration and deceleration The synchronism condition for RF period TRF and particle velocity v p L = vTRF / 2 = bc = bl / 2 wRF 47 Energy gain Assuming a sinusoidal electric field Ez = E0 cos(wRF t + fs ) where the synchronous particle passes at the middle of the gap g, at time t = 0, the energy is g/ 2 z W (r,t) = q ò Ez dz = q ò E0 cos(wRF + f s )dz v -g/ 2 Damping rings, Linear Collider School 2015 And the energy gain is sin Q / 2 = qV T with the transit time and finally DW = qV Q/2 sin(wg/ 2v) T= factor defined as g/ 2 wg/ 2v It can be shown that in general T= ò E(0, z)coswt(z)dz -g/ 2 g/ 2 ò E(0, z)dz -g/ 2 48 Damping rings, Linear Collider School 2015 Phase stability Assume that a synchronicity condition is fulfilled at the phase fs and that energy increase produces a velocity increase Around point P1, that arrives earlier (N1) experiences a smaller accelerating field and slows down Particles arriving later (M1) will be accelerated more A restoring force that keeps particles oscillating around a stable phase called the synchronous phase fs The opposite happens around point P2 at -fs, i.e. M2 and N2 will further separate 49 RF de-focusing In order to have stability, the time derivative of the Voltage and the spatial derivative of the electric field should satisfy ¶V ¶E >0Þ <0 ¶t ¶z charge In the absence of electric Damping rings, Linear Collider School 2015 the divergence of the field is given by Maxwell’s equations where x represents the generic transverse direction. External focusing is required by using quadrupoles or solenoids 50 Momentum compaction Off-momentum particles on the dispersion orbit travel in a different path length than on-momentum particles The change of the path length with respect to the momentum spread is called momentum compaction P+ΔP Damping rings, Linear Collider School 2015 The change of circumference is D(s)ΔP/P P ρ Δθ So the momentum compaction is 51 Transition energy The revolution frequency of a particle is The change in frequency is From the relativistic momentum we have Damping rings, Linear Collider School 2015 for which and the revolution frequency The slippage factor is given by For vanishing slippage factor, the transition energy is defined 52 Damping rings, Linear Collider School 2015 Synchrotron Frequency modulated but also B-field increased synchronously to match energy and keep revolution radius constant. The number of stable synchronous particles is equal to the harmonic number h. They are equally spaced along the circumference. ESRF Booster Each synchronous particle has the nominal energy and follow the nominal trajectory Magnetic field increases with momentum and the per turn change of the momentum is 53 Damping rings, Linear Collider School 2015 Phase stability on electron synchrotrons For electron synchrotrons, the relativistic is very large and as momentum compaction is positive in most cases Above transition, an increase in energy is followed by lower revolution frequency A delayed particle with respect to the synchronous one will get closer to it (gets a smaller energy increase) and phase stability occurs at the point P2 ( - fs) 54 Energy and phase relation The RF frequency and phase are related to the revolution ones as follows and Damping rings, Linear Collider School 2015 From the definition of the momentum compaction and for electrons c Replacing the revolution frequency change, the following relation is obtained between the energy and the RF phase time derivative c c 55 Longitudinal equations of motion The energy gain per turn with respect to the energy gain of the synchronous particle is The rate of energy change can be approximated by Damping rings, Linear Collider School 2015 The second energy phase relation is written as By combining the two energy/phase relations, a 2nd order differential equation is obtained, similar the pendulum ^ d æ R df ö ceV sin f - sin f s ) = 0 ( ç ÷+ dt è cac h dt ø 2pREs 56 Small amplitude oscillations Expanding the harmonic functions in the vicinity of the synchronous phase Damping rings, Linear Collider School 2015 Considering also that the coefficient of the phase derivative does not change with time, the differential equation reduces to one describing an harmonic oscillator with frequency For stability, the square of the frequency should positive and real, which gives the same relation for phase stability when particles are above transition cos fs < 0 Þ p / 2 < fs < p 57 Longitudinal motion invariant For large amplitude oscillations the differential equation of the phase is written as Damping rings, Linear Collider School 2015 Multiplying by the time derivative of the phase and integrating, an invariant of motion is obtained reducing to the following expression, for small amplitude oscillations 58 Damping rings, Linear Collider School 2015 Separatrix In the phase space (energy change versus phase), the motion is described by distorted circles in the vicinity of fs (stable fixed point) For phases beyond - fs (unstable fixed point) the motion is unbounded in the phase variable, as for the rotations of a pendulum The curve passing through - fs is called the separatrix and the enclosed area bucket 59 Energy acceptance The time derivative of the RF phase (or the energy change) reaches a maximum (the second derivative is zero) at the synchronous phase The equation of the separatrix at this point becomes Damping rings, Linear Collider School 2015 Replacing the time derivative of the phase from the first energy phase relation This equation defines the energy acceptance which depends strongly on the choice of the synchronous phase. It plays an important role on injection matching and influences strongly the electron storage ring lifetime 60 Stationary bucket When the synchronous phase is chosen to be equal to 0 (below transition) or (above transition), there is no acceleration. The equation of the separatrix is written Damping rings, Linear Collider School 2015 Using the (canonical) variable and replacing the expression for the synchrotron frequency ^ C W = ±2 c qV Es f sin 2p h a c 2 ^ C Wbk = 2 c . For f= , the bucket height is 2p eV Es and the area A = 2 W df = 8W ò bk bk 2p h a c 61 0 Adiabatic damping The longitudinal oscillations can be damped directly by acceleration itself. Consider the equation of motion when the energy of the synchronous particle is not constant Damping rings, Linear Collider School 2015 From this equation, we obtain a 2nd order differential equation with a damping term From the definition of the synchrotron frequency the damping coefficient is 62 Damping rings, Linear Collider School 2015 Outline – Phase space concepts Transverse phase space and Beam representation Beam emittance Liouville and normalised emittance Beam matrix RMS emittance Betatron functions revisited Gaussian distribution 63 Damping rings, Linear Collider School 2015 Transverse Phase Space Under linear forces, any particle moves on ellipse in phase space (x,x’), (y,y’). Ellipse rotates and moves between magnets, but its area is preserved. The area of the ellipse defines the emittance x´ x´ x x The equation of the ellipse is with α,β,γ, the twiss parameters Due to large number of particles, need of a statistical description of the beam, and its size 64 Beam representation Beam is a set of millions/billions of particles (N) A macro-particle representation models beam as a set of n particles with n<<N Distribution function is a statistical function representing the number of particles in phase space between Damping rings, Linear Collider School 2015 1.0 0.8 0.6 0.4 0.2 -4 4 -3 3 -2 2 -1 1 0 0 1 -1 2 -2 3 -3 4 -4 65 Liouville emittance Emittance represents the phase-space volume occupied by the beam The phase space can have different dimensions Damping rings, Linear Collider School 2015 2D (x, x’) or (y, y’) or (φ, Ε) 4D (x, x’,y, y’) or (x, x’, φ, Ε) or (y, y’, φ, Ε) 6D (x, x’, y, y’, φ, Ε) The resolution of my beam observation is very large compared to the average distance between particles. The beam modeled by phase space distribution function The volume of this function on phase space is the beam Liouville emittance 66 Damping rings, Linear Collider School 2015 Vlasov and Boltzmann equations The evolution of the distribution function is described by Vlasov equation Mathematical representation of Liouville theorem stating the conservation of phase space volume In the presence of fluctuations (radiation, collisions, etc.) distribution function evolution described by Boltzmann equation The distribution evolves towards a Maxwell-Boltzmann statistical equilibrium 67 2D and normalized emittance When motion is uncoupled, Vlasov equation still holds for each plane individually The Liouville emittance in the 2D phase space is still conserved In the case of acceleration, the emittance is conserved in the but not in the (adiabatic damping) Considering that Damping rings, Linear Collider School 2015 the beam is conserved in the phase space Define a normalised emittance which is conserved during acceleration 68 Beam matrix Damping rings, Linear Collider School 2015 We would like to determine the transformation of the beam enclosed by an ellipse through the accelerator Consider a vector u = (x,x’,y,y’,…) in a generalized ndimensional phase space. In that case the ellipse transformation is Application to one dimension gives and comparing with provides the beam matrix which can be expanded to more dimensions Evolution of the n-dimensional phase space from position 1 to position 2, through transport matrix 69 Damping rings, Linear Collider School 2015 Root Mean Square (RMS) beam parameters The average of a function on the beam distribution defined Taking the square root, the following Root Mean Square (RMS) quantities are defined RMS beam size RMS beam divergence RMS coupling 70 RMS emittance Damping rings, Linear Collider School 2015 Beam modelled as macro-particles Involved in processed linked to the statistical size The rms emittance is defined as It is a statistical quantity giving information about the minimum beam size For linear forces the rms emittance is conserved in the case of linear forces The determinant of the rms beam matrix Including acceleration, the determinant of 6D transport matrices is not equal to 1 but 71 Beam betatron functions The best ellipse fitting the beam distribution is Damping rings, Linear Collider School 2015 The beam betatron functions can be defined through the rms emittance 72 Gaussian distribution The Gaussian distribution has a gaussian density profile in phase space for which The beam boundary is Damping rings, Linear Collider School 2015 Uniform (KV) Gaussian 73