1. Yichao Jing 11/11/2010

2. Outline Introduction Linear lattice design and basic parameters Combined function magnets study and feasibility Nonlinear dynamics and dynamical aperture Beam Dynamics Workshop Yichao Jing

3. What is 10 pm? Natural emittance with or less than 10 picometer in both planes would greatly enhance the brightness simply due to the decrease in transverse beam size. When beam has such a small emittance, it reaches the diffractive limit. Beam Dynamics Workshop Yichao Jing for hard x-ray 1 Å, ɛ x ≈10 -11 m. Transversely coherent!

4. How to achieve 10 pm? Theoretically, the limit of emittance is given by We choose 5GeV beam, thus bending angle of each dipole must be very small. We use total 440 dipoles. Beam Dynamics Workshop Yichao Jing withand minimizedlattice is matched TME is achieved.

5. The lattice we use We use so-called 11–BA type lattice which has 9 center dipoles and 2 edge dipoles. The dispersion is not closed within each cell while the outer 2 dipoles are adjusted that the dispersion within different cells is for insertion devices. Theoretically, we use the non-acromat minimization for TME calculation. Dispersion and H function can be expressed as Beam Dynamics Workshop Yichao Jing

6. Emittance minimization We obtain dip by averaging H over all phase across the dipole and calculate extremum: Beam Dynamics Workshop Yichao Jing Using small angle approximation The minimum of beta function and dispersion happen at the center of dipoles

7. Simulation– MAD result Using MAD for lattice calculation with constraint set as the values calculated above, find an optimal solution when dip is minimized. Beam Dynamics Workshop Yichao Jing ParametersValue Circumference2663m Energy5GeV Biggest quad strenght 31(T/m) Qx203.39 Qy34.325 dE/E3.8e-4 Edge dipole length 1.3m Natural emittance 9.1pm

8. Effort in shortening the C Using combined function magnets, we can minimize the number of magnets for optics matching. Instead of using quadrupole triplet, we use singlet while make the dipole with gradient. After some data analysis to match the lattice, we find out Beam Dynamics Workshop Yichao Jing Good matching but a factor of 4 is unclear!

9. Sort of “analytical” way No matching process is carried out in the process and we manually change all the parameters(drift length– L; dipole gradient– Kc; matching quadrupole Kq) to search for best solution. We choose one Kc and then vary L and Kq to get beta function and dispersion and tune. And then choose another Kc and do this again. Each Kc would have a set of band plot. Beam Dynamics Workshop Yichao Jing

10. Phase stability diagram For a fixed Kc, by varying drift space length, we obtain necktie diagram. Beam Dynamics Workshop Yichao Jing The boundary reaches stability limit. Almost cover phase region from 0 to Pi. Bigger beta function has greater effect in changing tune.

11. Dispersion and beta function Similarly, we have dispersion and beta function curves. The dashed lines indicate the theoretical result. Beam Dynamics Workshop Yichao Jing No solution in this case, Kc=0.5 is too small.

12. First solution As we gradually increase the Kc value to about 1.0. First solution shows up with beta function matched to 0.15m and dispersion 3e-3m. Cell length is also nice compact– only 2.5m. Emittance is 6.8pm with 440 dipoles. Beam Dynamics Workshop Yichao Jing Calculated B 1 /B = Kc*ρ≈ 78m -1, not possible for magnet fabrication. It is almost impossible to make a very large dipole with high gradient!

13. Chromaticity correction Two families of sextupoles are used Beam Dynamics Workshop Yichao Jing SFSD First order Chromaticity corrected with sextupole strengths: SF= -346 and SD= 222 unit is 1/m^3. Pole tip field about 0.57T when a 2cm bore radius magnet is used.

14. Dynamical aperture DA calculation with 500 turns ELEGANT tracking gives a small aperture with 2.5mm in x and 1.5mm in y. No error or off momentum is included. Beam Dynamics Workshop Yichao Jing

15. IBS effect on emittance To study the IBS effect, we do simulation with ELEGANT. We use 2000 particles tracking of 1000 turns for a demo to see how the emittance depends on the peak current. Beam energy is @ 5GeV. Beam Dynamics Workshop Yichao Jing IBS effect is small for this lattice.

16. To-do list and conclusion More dynamical aperture optimization: higher order sextupole effect, tune dependence on actions, particle diffusion mechanisms in phase space, development of DA tracking codes, etc. More careful calculation of IBS effect involving more particles and more tracking turns. 10 picometer storage ring has the potential to become next generation light source. Beam Dynamics Workshop Yichao Jing