Yichao Jing 11/11/2010. Outline Introduction Linear lattice design and basic parameters Combined function magnets study and feasibility Nonlinear dynamics

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

Beam Dynamics Workshop Yichao Jing

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.


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.


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


)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:


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.


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


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.


Phase stability diagramFor a fixed Kc, by varying drift space length, we

obtain necktie diagram.


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.


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.


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


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.


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.


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.


Documents

Yichao Jing 11/11/2010. Outline Introduction Linear lattice design and basic parameters Combined function magnets study and feasibility Nonlinear dynamics