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Analytical considerations for Theoretical Minimum Emittance Cell Optics 17 April 2008 F. Antoniou, E. Gazis (NTUA, CERN) and Y. Papaphilippou (CERN)

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F. Antoniou/NTUA2 17/4/2008 Outline CLIC pre-damping rings design Design goals and challenges Theoretical background Lattice choice and optics optimisation Analytical solutions Open issues

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F. Antoniou/NTUA 3 17/4/2008 The CLIC Project Compact Linear Collider : multi-TeV electron- positron collider for high energy physics beyond today's particle accelerators Center-of-mass energy from 0.5 to 3 TeV RF gradient and frequencies are very high 100 MV/m in room temperature accelerating structures at 12 GHz Two-beam-acceleration concept High current “drive” beam, decelerated in special power extraction structures (PETS), generates RF power for main beam. Challenges: Efficient generation of drive beam PETS generating the required power 12 GHz RF structures for the required gradient Generation/preservation of small emittance beam Focusing to nanometer beam size Precise alignment of the different components

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CLIC Injector complex 4 17/4/2008 F. Antoniou/NTUA

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CLIC Pre-Damping Rings (PDR) Pre-damping rings needed in order to achieve injected beam size tolerances at the entrance of the damping rings Most critical the positron damping ring Injected emittances ~ 3 orders of magnitude larger than for electrons CLIC PDR parameters very close to those of NLC (I. Raichel and A. Wolski, EPAC04) Similar design may be adapted to CLIC Lower vertical emittance Higher energy spread PDR ParametersCLICNLC Energy [GeV]2.4241.98 Bunch population [10 9 ]4.57.5 Bunch length [mm]105.1 Energy Spread [%]0.50.09 Long. emittance [eV.m]1210009000 Hor. Norm. emittance [nm]6300046000 Ver. Norm. emittance [nm]15004600 Injected Parameterse-e- e+e+ Bunch population [10 9 ]4.76.4 Bunch length [mm]15 Energy Spread [%]0.071.5 Long. emittance [eV.m]1700240000 Hor.,Ver Norm. emittance [nm]100 x 10 3 9.7 x 10 6 L. Rinolfi 17/4/2008 5F. Antoniou/NTUA

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Equations of motion Accelerator main beam elements Dipoles (constant magnetic field) guidance Quadrupoles (linear magnetic fields) beam focusing Consider particles with the design momentum. The Lorentz equations of motion become with Hill’s equations of linear transverse particle motion Linear equations with s-dependent coefficients (harmonic oscillator) In a ring (or in transport line with symmetries), coefficients are periodic Not straightforward to derive analytical solutions for whole accelerator 17/4/2008 6F. Antoniou/NTUA

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Dispersion equation Consider the equations of motion for off-momentum particles The solution is a sum of the homogeneous equation (on-momentum) 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 7 17/4/2008 7 F. Antoniou/NTUA

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Generalized transfer matrix Dipoles : Quadrupoles : Drifts: 8 M =M = The particle trajectory can be then written in the general form: X px y py Δp/p X i+1 = M X i Where X= Using the above generalized transfer matrix, the equations can be solved piecewise 17/4/2008 8 F. Antoniou/NTUA

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Betatron motion The linear betatron motion of a particle is described by: and α, β, γ the twiss functions: Ψ the betatron phase: The beta function defines the envelope (machine aperture): Twiss parameters evolve as 17/4/2008 9F. Antoniou/NTUA

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General transfer matrix From equation for position and angle we have Expand the trigonometric formulas and set ψ (0)=0 to get the transfer matrix from location 0 to s with: For a periodic cell of length C we have: Where μ is the phase advance per cell: 17/4/2008 10F. Antoniou/NTUA

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For isomagnetic ring : 17/4/2008 11 F. Antoniou/NTUA Equilibrium emittance The horizontal emittance of an electron beam is defined as: the dispersion emittance One can prove that H ~ ρθ and the normalized emittance can be written as: 3 ε = γ ε = F C (γθ) nxlatticeq 3 Where the scaling factor F depends on the design of the storage ring lattices lattice

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Low emittance lattices Double Bend Achromat (DBA) Triple Bend Achromat (TBA) Quadruple Bend Achromat (QBA) Theoretical Minimum Emittance cell (TME) dispersion FODO cell: the most common and simple structure that is made of a pair of focusing and defocusing quadrupoles with or without dipoles in between There are also other structures more complex but giving lower emittance: Only dipoles are shown but there are also quadrupoles in between for providing focusing

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Using the values for the F factor and the relation between the bending angle and the number of dipoles, we can calculate the minimum number of dipoles needed to achieve a required normalized minimum emittance of 50 μm for the FODO, the DBA and the TME cells. bend Θ = 2π/Ν F FODO = 1.3 N FODO > 67 N CELL > 33 F DBA = 1/(4√15Jx) N DBA > 24 N CELL > 24 F TME = 1/(12√15Jx) N TME > 17 N CELL > 17 Straightforward solutions for FODO cells but do not achieve very low emittances TME cell chosen for compactness and efficient emittance minimisation over Multiple Bend Structures (or achromats) used in light sources TME more complex to tune over other cell types We want to parameterize the solutions for the three types of cells We start from the TME that is the more difficult one and there is nothing been done for this yet. Cell choice

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Optics functions for minimum emittance

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Constraints for general MEL Consider a general MEL with the theoretical minimum emittance (drifts are parameters) In the straight section, there are two independent constraints, thus at least two quadrupoles are needed Note that there is no control in the vertical plane!! Expressions for the quadrupole gradients can be obtained, parameterized with the drift lengths and the initial optics functions All the optics functions are thus uniquely determined for both planes and can be minimized (the gradients as well) by varying the drifts The vertical phase advance is also fixed!!!! The chromaticities are also uniquely defined There are tools like the MADX program that can provide a numerical solution, but an analytical solution is preferable in order to completely parameterize the problem

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Quad strengths F. Antoniou/NTUA16 17/4/2008 The quad strengths were derived analytically and parameterized with the drift lengths and the emittance Drift lengths parameterization (for the minimum emittance optics) 2 solutions: The first solution is not acceptable as it gives negative values for both quadrupole strengths (focusing quads) instability in the vertical plane The second solution gives all possible values for the quads to achieve the minimum emittance l1=l2=l3 l1>l2,l3 l2>l1,l3 l3>l1,l2

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…Quad strengths F. Antoniou/NTUA17 17/4/2008 Emittance parameterization (for fixed drift lengths) F = (achieved emittance)/(TME emittance) All quad strength values for emittance values from the theoretical minimum emittance to 2 times the TME. The point (F=1) represents the values of the quand strengths for the TME. The horizontal plane is uniquely defined F=1 F=1.2 F=1.4 F=1.6 F=1.8 F=2

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F. Antoniou/NTUA18 17/4/2008 The vertical plane is also uniquely defined by these solutions (opposite signs in the quad strengths) Certain values should be excluded because they do not provide stability to both the planes The drift strengths should be constrained to provide stability The stability criterion is: Trace(M) = 2 cos μ Abs[Trace(M)] < 2 The criterion has to be valid in both the planes

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Open issues F. Antoniou/NTUA19 17/4/2008 Find all the restrictions and all the regions of stability Parameterize the problem with other parameters, like phase advance and chromaticity Lattice design with MADX Follow the same strategy for other lattice options Non-linear dynamics optimization and lattice comparison for CLIC pre-damping rings

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