Introduction to Design Issues and Optimization of Low Emittance Rings Yunhai Cai FEL & Beam Physics Department September 17, 2014 Low Emittance Rings 2014 Workshop INFN-LNF, Frascati (RM), Italy

3rd Generation Light Sources Existing Under construction ESRF NLSL-II Range of energy and emittance Mention undulator SSRF Max-IV 11/22/2018 Yunhai Cai, SLAC

Storage Ring Light Sources Range of energy and size Trend Courtesy of R. Bartolini, Low Emittance Rings Workshop, 2010, CERN Frontier is at 100 pm emttance. Challenge is to lower it even more down to 10 pm without increase of ring size. 11/22/2018 Yunhai Cai, SLAC 3

Design Issues Lower emittance Collective effects Stronger sextupoles DBA, TBA, …, MBA cells, more dipoles and smaller dispersion Stronger focusing, more quadrupoles Reasonable cost, size, or complexity Stronger sextupoles Mitigation of structure resonances Compensation of chromatic aberration Mechanical and electrical design, alignment tolerance, correction scheme Alternative injection scheme Collective effects Intra-beam scattering and Touschek lifetime Resistive-wall impedance and multi-bunch feedback system Coherent synchrotron radiation 11/22/2018 Yunhai Cai, SLAC 4

Synchrotron Radiation Electron beam in undulator Photon spectral flux in 0.1% BW N S nth harmonic wavelength: K is about 3, lambda_u is at cm, lambda_n is 1 angstrom, E0 about a few GeV. Q1 is at an order of 1, there is not much room to increase Fn. 11/22/2018 Yunhai Cai, SLAC

Spectral Brightness Brightness of electron beam radiating at nth (odd) harmonics in a undulator is given by Spectral brightness of PEP-X If the electron beam phase space is matched to those of photon’s, the brightness becomes optimized An ultimate limit is applicable to ERL as well. Increase in brightness is linear with respect to the emittance. Linear in beam current: I. Finally, even for zero emittances, there is an ultimate limit for the brightness A diffraction limited ring at 1 angstrom or 8 pm-rad emittance 11/22/2018 Yunhai Cai, SLAC

Energy Spread and Emittance Balance between the quantum excitation and radiation damping results in an equilibrium Gaussian distribution with relative energy spread sd and horizontal emittance ex: where and Stronger focusing and reduce the dispersion lead to even stronger sextupoles. The quantum constant Cq = 3.8319x10-13 m for electron g is the Lorentz factor (energy) 11/22/2018 Yunhai Cai, SLAC

Minimization of Emittance For an electron ring without damping wigglers, the horizontal emittnace is given by where Fc is a form factor determined by choice of cell and q is bending angle of dipole magnet. In general, stronger focusing makes Fc smaller. Often there is a minimum achievable value of Fc for any a given type of cell. For example, we have Dispersion is zero at an end of dipole Do not forget about two strategies of reduce the emittance. Dynamic aperture is proportional to the bending angle. Dispersion is at a minimum at the center of dipole There is a factor of three between the minimum values of DBA and TME cells. That’s the price paid for an achromat, namely fixing the dispersion and its slop at one end of dipole. 11/22/2018 Yunhai Cai, SLAC

MAX-IV Light Source Multi-bend achromat 7 Bend Achromat, at 3 GeV MAX-IV dipole + 2 sextupoles in one block Innovations: Multi-bend achromat Compact and combined function magnets Octupoles 11/22/2018 Yunhai Cai, SLAC

PEP-X Layout & Parameters Energy, GeV 4.5 Circumference, m 2199.32 Natural emittance, pm 11 Beam current, mA 200 Emittance at 200 mA, x/y, pm 12 / 12 Tunes, x/y/s 113.23 /65.14/0.007 Bunch length, mm 3.1 Energy spread 1.25x10-3 Energy loss per turn, MeV 2.95 RF voltage, MV 8.3 RF harmonic number 3492 Length of ID straight, m 5.0 Wiggler length, m 90.0 Beta at ID center, x/y, m 4.92 / 0.80 Touschek lifetime, hour 10 Dynamic aperture , mm 10 An ultimate storage ring To be Built with 4th-order geometrical achromats in the PEP tunnel. 11/22/2018 Yunhai Cai, SLAC

PEP-X 7 Bend Achromat 29 pm at 4.5 GeV It follows from MAX-VI. Talk about common and different features. Sextupole positions. Ten families. But no octopoles. Cell phase advances: mx=(2+1/8) x 3600, my=(1+1/8) x 3600. 11/22/2018 Yunhai Cai, SLAC

Presentations for Magnetic Elements Lie factors engine in MARYLIE ( A. Dragt) violates symplecticity when evaluates Dragt-Finn engine in TRANSPORT, MAD, COSY (K. Brown and M. Berz), simple R-matrix but high-order one violates Taylor map TPSA Symplectic Integrator engine in TEAPOT, SAD, TRACY, LEGO, PTC (E. Forest, R. Ruth, and K. Hirata) preserves symplecticity simple and based on several known solutions emphasis on numerical process 11/22/2018 Yunhai Cai, SLAC

Cancellation of All Geometric 3rd and 4th Resonances Driven by Strong Sextupoles except 2nx-2ny Third Order Fourth Order How about skew sextupoles? Any cancellation? Talk about speed compare to the tune scan. General, does not depends where are the sextupole, how many families of sextupole. Work for thick sextupoles as well. K.L. Brown & R.V. Servranckx Yunhai Cai Nucl. Inst. Meth., A258:480–502, 1987 Nucl. Inst. Meth., A645:168–174, 2011. There are still three tune shift terms. 11/22/2018 Yunhai Cai, SLAC

Harmonic Sextupoles For Tune Shifts and 2nx-2nyResonance Without harmonic sextupoles With harmonic sextupoles 1) Explain how the OPA solution was obtained including chromatic sextupoles and using ten families of sextupoles. OPA is used for optimizing the setting of 10 families of sextupoles. Due to the cancellation of many resonances, the optimization becomes much simpler and easier. OPA is an Accelerator Design Program from SLS PSI developed by A. Streun. 11/22/2018 Yunhai Cai, SLAC

4th Order Geometric Achromat 4th order geometric achromat Chromatic effects 0,0 -57,-89 1332,-150 1) Explain how the OPA solution was obtained including chromatic sextupoles and using ten families of sextupoles. There are 4 families of chromatic sextupoles and 6 families of harmonic ones. The 4th order geometric achromat (f3=f4=0) was obtained with the analytical Lie method. It is published on PRSTAB 15, 054002 (2012) 11/22/2018 Yunhai Cai, SLAC

Tune Scan of Dynamic Aperture PEP-X: Baseline (2008) PEP-X: USR (2011) The dynamic aperture is in unit of sigma of the equilibrium beam size. The USR design is built with 4th-order geometric achromats and therefore no 3rd and 4th order resonances driven by the sextupoles seen in the scan. 11/22/2018 Yunhai Cai, SLAC

Frequency Map Analysis PEP-X: Baseline (2008) PEP-X: USR (2011) Y 1) At design tunes. X [mm] X [mm] The dynamic aperture is in unit of mm at the injection. The baseline design has a factor of ten larger emittance than the one in the USR design. 11/22/2018 Yunhai Cai, SLAC

Order-by-Order Computation of Chromatic Optics (PEP-X) Dispersions Its First Derivatives 3rd order achromat by Karl Brown and Roger Servranckx. There is a price to pay for the high beta insertion. Its Second Derivatives Yunhai Cai, “Symplectic Maps and Chromatic Optics in Particle Accelerator,” September, 2014, SLAC-PUB-16075, Submitted to PRSTAB 11/22/2018 Yunhai Cai, SLAC 18

Order-by-Order Computation of Chromatic Optics (PEP-X) Betatron Tunes Its First Derivatives 1) Stronger focusing in the horizontal plane. Its Second Derivatives 11/22/2018 Yunhai Cai, SLAC 19

High-Order Chromatic Compensations (FODO RING) Betatron Tunes compensated by two families of sextupoles Its First Derivatives compensated by two families of octupoles 1) Stronger focusing in the horizontal plane. Why the second-order keep rising? Its Second Derivatives It is an analytic solution. The next order can be cancelled by decapoles. 11/22/2018 Yunhai Cai, SLAC 20

Intra-Beam Scattering The growth rate in the relative energy spread sd is given by where Nb is the bunch population and (log) the Coulomb log factor and the other factors are defined by Combined with synchrotron radiation Emphasis on gamma^3 dependence. Lower energy worse. Minimize Curly H is good as well. And the horizontal growth rate is given by 11/22/2018 Yunhai Cai, SLAC

Optimization of Energy 1) There is a balance between intra-beam scattering and quantum excitations of synchrotron radiation. 11/22/2018 Yunhai Cai, SLAC

Touschek Lifetime momentum aperture lifetime When a pair of electrons go through a hard scattering, their momentum changes are so large that they are outside the RF bucket or the momentum aperture. This process results in a finite lifetime of a bunched beam. The lifetime is given by momentum aperture with 1) Strong dependence on the momentum aperture. lifetime where dm is the momentum acceptance. 11/22/2018 Yunhai Cai, SLAC

Threshold of Instability Driven by CSR Measured bursting threshold at ANKA See M.Klein et al. PAC09, 4761 (2009) Based on the bunched beam theory, the threshold can be written as where xth is given by (courtesy of M. Klein, xth=0.5 used.) My talk, IPAC 2011, San Sebastian, Spain 11/22/2018 Yunhai Cai, SLAC

General Design Issues Better understanding of beam dynamics Structure resonances driven by stronger sextupoles Chromatic aberration from stronger focusing quadrupoles Compact magnets and tight alignment tolerances Control intra-beam scattering and provide adequate Touschek lifetime Resistive-wall impedance and need of faster bunch-by-bunch feedback system Accurate beam position monitors and better correction algorithm 11/22/2018 Yunhai Cai, SLAC

Conclusion Majority of resonance driving terms up to the 4th order generated by strong sextupoles can be eliminated simply by a proper choice of phase advances in a unit module, or so-called achromat. Few residual terms can be cancelled by selecting a particular settings of sextupoles. The cancellation leads to the 4th order geometrical achromat. This approach significantly simplifies the optimization process and improves the dynamic aperture in storage rings. Perfect chromatic optics is achievable. In particular, up to 3rd order of the momentum deviation, it is achieved by adding octupoles and decapoles at the locations of quadrupoles in FODO cells. Chromatic optics can be computed order-by-order in the momentum deviation. As a result, we should be able to match the optics order-by-order in the future. Yunhai Cai, SLAC 11/22/2018

Acknowledgements For design of PEP-X: Karl Bane, Michael Borland, Robert Hettel, Yuri Nosochkov, Min-Huey Wang, For maps, Lie method, differential algebra: Johan Bengtsson, Alexander Chao, Etienne Forest, John Iwrin, Yiton Yan For microwave instability and coherent synchrotron radiation: Gennady Stupakov, Robert Warnock Yunhai Cai, SLAC 11/22/2018