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Design of 10 pm storage ring
Yichao Jing11/11/2010
OutlineIntroduction
Linear lattice design and basic parameters
Combined function magnets study and feasibility
Nonlinear dynamics and dynamical aperture
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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.
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unrrn uny uny '
4u
for hard x-ray 1 Å, ɛx≈10-11m.
zyx
eNB
Transversely coherent!
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.
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32 qun FC with mCq131083.3 and dip
HF
22 ''2 DDDDHdip
minimized lattice is matched
TME is achieved.
The lattice we useWe 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
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)cos1(sin2
)cos1(sin)cos1()'(2
sin)'(2''2)(
0
220
200000
0000022
DD
DDHDDDDH
cos'sin)1('
sin'cos)cos1(
00
00
DD
D
DDD
Emittance minimizationWe obtain <H>dip by averaging H over all phase
across the dipole and calculate extremum:
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0'00
D
H
D
Hdipdip
60
6
1
*0
*0
L
LD
Using small angle approximation
The minimum of beta function and dispersion happen at the center of dipoles
Simulation– MAD resultUsing MAD for lattice calculation with constraint
set as the values calculated above, find an optimal solution when <H>dip is minimized.
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Parameters Value
Circumference 2663m
Energy 5GeV
Biggest quad strenght
31(T/m)
Qx 203.39
Qy 34.325
dE/E 3.8e-4
Edge dipole length 1.3m
Natural emittance
9.1pm
Effort in shortening the CUsing 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
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Good matching but a factor of 4 is unclear!
Sort of “analytical” wayNo 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.
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Phase stability diagramFor a fixed Kc, by varying drift space length, we
obtain necktie diagram.
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The boundary reaches stability limit.
Almost cover phase region from 0 to Pi. Bigger beta function has greater effect in changing tune.
Dispersion and beta functionSimilarly, we have dispersion and beta function
curves. The dashed lines indicate the theoretical result.
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No solution in this case, Kc=0.5 is too small.
First solutionAs 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.
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Calculated B1/B = Kc*ρ≈ 78m-1, not possible for magnet fabrication. It is almost impossible to make a very large dipole with high gradient!
Chromaticity correctionTwo families of sextupoles are used
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SF SD
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.
Dynamical apertureDA 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.
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IBS effect on emittanceTo 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.
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IBS effect is small for this lattice.
To-do list and conclusionMore 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.
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