LOW EMITTANCE STORAGE RING DESIGN Zhenghao Gu Email: guzh@indiana.edu Department of Physics, Indiana University March 10, 2013 Zhenghao Gu Department of Physics, Indiana University 1 / 32
Outline Introduction a 100 pm, 5 GeV ring design based on 4th order geometric achromat(45 degree) Non-interleaved 4th order geometric achromat (180 degree) Conclusion Zhenghao Gu Department of Physics, Indiana University 2 / 32
Introduction Why low emittance A modern storage ring requires a high brightness, where F is photon flux and Σ x,y = F B = 4π 2 Σ x Σ x Σ yσ, (1.1) y σx,y 2 +σλ 2,Σ x,y = σ x,y 2 +σ λ 2 (1.2) Here σ x,y, σ x,y are the rms sizes and divergences of the electron beam;σ λ,σ λ are the sizes and divergences of the photon beam. Therefore to generate high brightness radiation, we should try to decrease the emittance of a ring. Zhenghao Gu Department of Physics, Indiana University 3 / 32
Introduction ultimate storage ring Emittance of a photon: ǫ λ = λ 4π, (1.3) Ultimate Storage Ring: diffraction-limited emittance for hard X- ray photons(10 pm) Compared to FEL, Ultimate Storage Ring has advantages: high average brightness(low peak),high reliability, low cost per user. Zhenghao Gu Department of Physics, Indiana University 4 / 32
Introduction Challenge The theoretical minimum emittance of a dipole is given as: ǫ TME = C qγ 2 θ 3 12 15J x, (1.4) where C q = 3.83 10 13 m,j x 1 is the horizontal dampingpartition number, and θ is the bending angle of the dipole. A good strategy of reducing the emittance is to increase the number of dipole magnets and decrease the bending angle of each dipole. However, large number of dipoles and quadrupoles would bring an issue: large negative chromaticities. Zhenghao Gu Department of Physics, Indiana University 5 / 32
Introduction chanllenge To correct these negative chromaticities, strong sextupole magnets are needed, which would lead to other serious issues: strong resonances and large tune shifts. Then dynamic aperture shrinks dramatically. The main challenge of designing a low emittance ring is the optimization of dynamic aperture(da). A large DA is the key for injection efficiency and beam lifetime. Zhenghao Gu Department of Physics, Indiana University 6 / 32
4th order geometric achromat(45 degree) PEP-X PEP-X in Slac(Y. Cai et al., PRSTAB 15, 054002(2012)). 4th order geometric achromat: all 3rd & 4th order driving terms cancelled except h22000, h00220, h11110, h20020, h02200. Energy Ring circumference Natural emittance 4.5 GeV 2.2km 29 pm-rad Zhenghao Gu Department of Physics, Indiana University 7 / 32
4th order geometric achromat(45 degree) Lie algebra Lie algebra is a useful language to describe nonlinear beam dynamics.(j. Bengtsson, SLS note 9/97.) For a storage ring, the nonlinear components come from quadrupoles and sextupoles. Quadrupole: Sextupole: V i = b 2i 2(1 + δ) (x2 y 2 ) = b 2 2 (1 δ)(x2 y 2 ) + O(δ 2 ) (2.1) In resonance basis or action-angle basis: V i = b 3i 3 (x3 3xy 2 ) (2.2) h x± 2J x e ±iφx = 2J x cosφ x ± i 2J x sinφ x = x ip x. (2.3) Zhenghao Gu Department of Physics, Indiana University 8 / 32
4th order geometric achromat(45 degree) Lie algebra Collecting the terms we find that the Lie generator: h :, the nonlinear driving terms, has to first order the following generic form in the resonance basis h (1) Ī =n hīh i 1 x+ h i 2 x h i 3 y+ h i 4 y δ i 5 (2.4) where Ī = [i 1, i 2, i 3, i 4, i 5 ], Ī = i 1 + i 2 + i 3 + i 4 + i 5 h (2) 1 [ˆV i, ˆV j ] = 1 2 i<j Jx αjβ y Ī = J =n hīh J hi 1 +j 1 x+ hi 2 +j 2 x hi 3 +j 3 y+ hi 4 +j 4 y δi 5 +j 5 (2.5) The problem becomes computing all the coeffiencts h Ī Zhenghao Gu Department of Physics, Indiana University 9 / 32
4th order geometric achromat(45 degree) Lie algebra 4th order driving terms: f 4 = 1 n n [V i, V j ] (2.6) 2 i=1 i<j In normalized coordinate, [V i, V j ] = S i S j β x,i β x,j [sin(µ y,i µ y,j )β y,i β y,j x i x j y i y j + sin(µ x,i µ x,j )(β x,i x 2 i β y,i y 2 i )(β x,ix 2 i β y,i y 2 i )/4] (2.7) In PEP-X, they designed a periodic cell with betatron phase advances,ν x = 4π + π/4 and ν y = 2π + π/4. Every eight of such cells make an identity transformation and an achromat. Zhenghao Gu Department of Physics, Indiana University 10 / 32
4th order geometric achromat(45 degree) different canceling schemes Zhenghao Gu Department of Physics, Indiana University 11 / 32
4th order geometric achromat(45 degree) 32-7BA ring 32 cells of 7-bending-achromats. Every 8 cells consist of one 4-th order geometric achromat. Figure 2.1: achromat cell with 7 dipoles Zhenghao Gu Department of Physics, Indiana University 12 / 32
4th order geometric achromat(45 degree) 32-7BA ring Figure 2.2: Twiss parameter of 32 7BA ring. Zhenghao Gu Department of Physics, Indiana University 13 / 32
4th order geometric achromat(45 degree) 32-7BA ring Figure 2.3: Injection section. With this superlarge β x(250m), we can easily obtain large DA in x direction. Zhenghao Gu Department of Physics, Indiana University 14 / 32
4th order geometric achromat(45 degree) 32-7BA ring Table 1: main parameters of the ring Energy 5 GeV Ring circumference 1370m Natural emittance 108 pm-rad Natural horizontal chromaticity -136 Natural vertical chromaticity -95 Horizontal betatron tune 76.8 Vertical betatron tune 48.4 Zhenghao Gu Department of Physics, Indiana University 15 / 32
4th order geometric achromat(45 degree) Optimization Here I used genetic algorithm to optimize dynamic aperture, with sextupoles as knobs. Minimizing those detuning and resonance terms could be the objective. Figure 2.4: Driving terms (Computed with elegant (M. Borland, et al.)). Zhenghao Gu Department of Physics, Indiana University 16 / 32
4th order geometric achromat(45 degree) Dynamic aperture and Diffusion rate Figure 2.5: dynamic aperture. Figure 2.6: Tune footprint Color represents tune diffusion. Tune diffusion d = log( (ν x,1 ν x,2 ) 2 + (ν y,1 ν y,2 ) 2 ) where ν x,1 is horizontal tune computed for the first N turns of tracking data,ν x,2 for the following N turns. Lower diffusion rate indicates more stable particle. Zhenghao Gu Department of Physics, Indiana University 17 / 32
4th order geometric achromat(45 degree) error analysis Figure 2.7: 50 ensembles of random errors with Strength error: rms=0.05%, cutoff=2,transverse coupling: rms tilt angle=5e-4, cut-off=2 Zhenghao Gu Department of Physics, Indiana University 18 / 32
4th order geometric achromat(45 degree) local momentum acceptance Figure 2.8: scan. Local momentum acceptance Figure 2.9: 5 ensembles with errors. Need to be improved. 2% is required for reasonable Touschek lifetime. Zhenghao Gu Department of Physics, Indiana University 19 / 32
Non-interleaved 4th order geometric achromat(180 degree) Non-interleaved 4th order geometric achromat With this design, all 3rd and 4th order geometric terms vanish. So harmonic sextupoles are not necessary. But there is a constraint to apply this scheme. The two sextupoles with µ = π between them have to be non-interleaved, which means no other sextupole allowed in between. This forbids the conventional alternating distribution of sextupoles: focusing, defocusing, focusing, defocusing Zhenghao Gu Department of Physics, Indiana University 20 / 32
Non-interleaved 4th order geometric achromat(180 degree) 32-7BA ring To have a direct comparison between the two schemes, I made a very similar design except the sextupole distribution. Table 2: main parameters of the ring Energy 5 GeV Ring circumference 1.7km Natural emittance 248 pm-rad Natural horizontal chromaticity -125 Natural vertical chromaticity -180 Horizontal betatron tune 77.1 Vertical betatron tune 96.9 Zhenghao Gu Department of Physics, Indiana University 21 / 32
Non-interleaved 4th order geometric achromat(180 degree) 2 types of cells Figure 3.1: focusing cell with 4 focusing sex-figurtupoleing 3.2: defocusing cell with 4 defocus- sextupoles The total phase advance of 1 focusing cell plus 1 defocusing cell is :µ x = 9.5π,µ y = 11.5π. So h20001 term has been cancelled every focusing-defocusing structure. The entire ring consists of 16 focusing cells,16 defocusing cells and injection sections. Zhenghao Gu Department of Physics, Indiana University 22 / 32
Non-interleaved 4th order geometric achromat(180 degree) Dynamic aperture and Diffusion rate Figure 3.3: dynamic aperture with diffusion Figure 3.4: Tune footprint indicated. The value of natural tune (zero amplitude particle) also affects DA. Zhenghao Gu Department of Physics, Indiana University 23 / 32
Non-interleaved 4th order geometric achromat(180 degree) Comparison of DA Figure 3.5: dynamic aperture of 45 degreefigure 3.6: dynamic aperture of 180 degree scheme. scheme. The same β x = 260m,β y = 5.5m for both. The second result is at least 4 times larger in both x and y plane. Zhenghao Gu Department of Physics, Indiana University 24 / 32
Non-interleaved 4th order geometric achromat(180 degree) Comparison of driving terms Figure 3.7: Driving terms of 45 degree scheme. Figure 3.8: Driving terms of 180 degree scheme. Some residual terms in 2nd table are due to the finite length of sextupoles. Most terms are smaller in 2nd table. But it is hard to predict which terms make the difference in DA. Probably it is a collective effect or because of even higher order terms. Zhenghao Gu Department of Physics, Indiana University 25 / 32
Non-interleaved 4th order geometric achromat(180 degree) Comparison of DA with errors Figure 3.9: DA with errors for 45 degreefigure 3.10: The same rms error for 180 degree scheme. scheme. Zhenghao Gu Department of Physics, Indiana University 26 / 32
Non-interleaved 4th order geometric achromat(180 degree) Local momentum acceptance Figure 3.11: Local momentum acceptance scan for 180 degree scheme. The result is not good, either. Zhenghao Gu Department of Physics, Indiana University 27 / 32
Non-interleaved 4th order geometric achromat(180 degree) TME cell One issue of the 180 degree design is the relatively large emittance 250 pm(10 times larger than the theoretical minimum 24 pm for this bending angle of dipole). It is because the emittance is limited by the phase advance of TME cell. The minimum emittance for fixed phase advance(p. Emma and T. Raubenheimer, Phys. Rev. ST Accel.Beams 4, 021001 (2001).): tan µx 2 = 15ǫr + 3 2 ǫ 2 r 1 ǫ 2 r 6 (3.1) where µ x is phase advance of the TME cell and ǫ r is ratio of the real emittance to TME value. In my case, the phase advance of the TME cell is about 90 (180 every 2 cells). So the minimum ǫ r I can achieve is 8.8, which explains my large emittance. Figure 3.12: conventional TME cell(y., Jiao et. al, PRSTAB 14, 054002 (2011)). Figure 3.13: Modified TME cell. Zhenghao Gu Department of Physics, Indiana University 28 / 32
Non-interleaved 4th order geometric achromat(180 degree) TME cell Figure 3.14: conventional TME cell. Figure 3.15: Modified TME cell. Another limit is for modified TME cell, the up limit of phase advance is less than 180 degree. So we have to switch to conventional TME cell if we want to have 180 degree phase advance of one TME cell. Zhenghao Gu Department of Physics, Indiana University 29 / 32
Non-interleaved 4th order geometric achromat(180 degree) Design with conventional TME cell Figure 3.16: 7BA with conventional TME cell Fig. 3.16 is a preliminary result of the new design with conventional TME. It is a 7BA including 10 defocusing sextupoles. The emittance is 57pm. This new design could be improvable: 1. Phase advance 540 every 2 TME? 2. Add more quadrupoles? Zhenghao Gu Department of Physics, Indiana University 30 / 32
Conclusion Conclusion This non-interleaved geometric achomat shows great potential in DA improvement. Pros: No harmonic sextupole required. No need to optimize DA. Only 1 objective(local momentum acceptance) left. Cons: The non-interleaved constraint limits the number of sextupoles. Not alternating distribution. 180 degree phase advance. Zhenghao Gu Department of Physics, Indiana University 31 / 32
Conclusion Next Step Reduce emittance. Reduce circumference. Ultimate storage ring. Zhenghao Gu Department of Physics, Indiana University 32 / 32