Transcript Study of incoherent beam size blow

Chaos and Emittance growth due to
nonlinear interactions in circular
accelerators
K. Ohmi (KEK)
SAD2006
Sep. 5-7. 2006 at KEK
Emittance growth
•
•
•
•
External incoherent diffusion, radiation,
Intrabeam etc.
Coherent motion, instability
Nonlinear diffusion
Nonlinearlity coupled to external diffusion
(noise)
1. Incoherent electron cloud
2. Beam-beam limit
3. Space charge limit
Diffusive or not
•
Which system does not have emittance
growth?
1.
2.
Integrable system
System with two degrees of freedom
•
Which system can have emittance
growth?
1.
Nonintegrable system with three or more degrees of
freedom
External diffusion: noise, radiation excitation… The
external diffusion is amplified due to nonlinear
interaction
2.
Nondiffusive system
• Integrable system – needless to say
• System with two degrees of freedom. Particles
do not across torus layers.
System fall into global stochastic regime may be diffusive
even for two degrees of freedom, but the diffusion is limited
in the regime.
Diffusion in three or more degrees
of freedom
• Motion in a degree of freedom gives
modulation
• Particles can get over KAM boundary
through additional freedom.
Round beam:
Example of non-diffusive system
• Equal tune, no synchrotron oscillation -- equivalent to a
system with 2 degrees of freedom (r-s).
 px2  p y2
1
H   

x2  y 2
2
2




   P ( s )U (r , z )

 
2
2

p

r
2


p


  P (s)U (r , z )
 r
2 
2 
r  2
(x,y)
pr  r'  px cos  py sin 
p  r 2 '  r  px sin   py cos 

z  s  ct
• H does not include , therefore p is a constant of motion.
• Trajectories on a torus r-pr-s
• Poincare cross-section is mapped on two dimensional space.
• Motion in r-pr space.
1
Jr 
2
1
 p dr   
 r  cos
r
1
rmax
rmin
r2
2E
2
E
 2  2 dr 

  r
2 2
 p d  p
  
r 2   (2 J r  )
 (2 J r  ) 2 
1
J 
2
2
Model- round beam interacting with electron
cloud
• Transverse beam size depends on z; sx(z), sy(z).
• Assume transverse Gaussian charge distribution.
Applicable not only beam-beam but also electron cloud
and space charge issues.
• Strong localized force x, y ~sx, sy

x2
y2 
exp   2
 2


2s x  u 2s y  u 
Nre 

U ( x, y;s x ( z ), s y ( z ))  
du

2
2
0

2s x  u 2s y  u
x(s  0)  exp   : U : x(s  0)
x  ( x, px , y, py , z,  )t
The force for a round charge distribution
• The force depends on z, because of sr(z) of strong
beam or electron cloud.

r2 

exp  2
2s r  u 
re 

U (r;s r ( z ))   
du
2
0

2s r  u
 r 2 
2re 1 
U (r ; s r ( z ))

Fr  

1  exp 
2 
r
 r
 2s r 
 r 2  ds r2
2re 1
U (r;s r ( z ))
U (r;s r ( z )) ds r2
Fz  


exp  2 
2
2
z
s r
dz
 2s r
 2s r  dz
• Example, LHC
• L=26700 m, nx=0.28, ny=0.31, ns=0.006
• ex=ey=8x10-9 m, x= x=100 m
• sx=sy=0.89 mm, sz=0.13 m,
Model of cloud
• sr (cloud) and tune shift
• Ne L=1, 2, 4, 6 x1011. Interact at a point in the ring.
Two degree of freedom
• 4 variable (x, px, s, H’). 1 integral for H’,
Poincare cross section (certain s) 2
variable.
• Poincare plot, x-px plot at a certain s.
• When one more integral, J(x,px)=constant,
the system is solvable. This relation gives
a curve in x-px phase space.
Poincare plot
• r-pr
Dn=0.06, 0.13, 0.27, 0.38
Note sr=0.89 mm
Diffusion rate
• Diffusion rates are very small compare than
those for 3 degrees of freedom, see later.
• T0=89 s.
Resonance overlap
• Fourier expansion for r of U.

U  U k cos k r
k 0
• Resonance position, Jr,R .
U 0
 ( J r )  r 
J r
 ( J r , R )  2
n
k
• Motion near the resonance position,
pendulum motion, separatrix.
Ir 
J r  J r ,R
k
1 2  2U 0
H k
2
J r 2
  m
I r2   P ( s )U k cos 
J r ,R
Overlap condition, Chirikov criterion
• Resonance width
  U0 
DJ r  4 U k 
2 

J
 r 
2
1
• Resonance separation
J r , R (k , n  1)  J r , R (k , n) 
2   U 0 


k  J r2 
2
1
• Overlap condition, width>separation
 2U 0 
k Uk

2
J r
2
Synchrotron oscillation and symplectic diffusion
2
 2 p2   r 2   2    2 
H    pr  2  
  pz    z    P ( s)U (r , z )

2 
r  2 2 
 


• Add synchrotron motion, 3 degrees of freedom
• The structures of tori are different for each z.
• Are particles return the same torus after one synchrotron period.
s
z
???
Are there different
worlds for every z?
pr
r
pr
r
Poincare plot for several z
Note sr=0.89 mm
Poincare plot for several z
• Z=2, 4, 6, 10 cm
Adiabatic invariant?
Solvable or not
•
•
•
•
6 variables (x,p,y,p,s,H’)
One integra H’, Poincare cross section at s.
4 variables
When 2 integrals exist, the system is solvable. The
solution is represented by a surface in 3 dimensional
space, r-pr-z.
• When a surface is not seen, the system is nonsolvable:
i.e., emittance growth occurs.
Nonsolvable
Solvable
Blue: no synchrotron
motion at z=0
Separatrix crossing
• U depends on z. Fourier expansion for
synchrotron phase.


U ( J , r , J z , z )  U kl cos  k r  l z 
k 0 l 0
• Resonance separation (narrower than n-n+1)
J r , R (l  1)  J r , R (l ) 
4k
z
z   U0 
2


k  J r2 
 U0
U kl
1
2
J r
2
1
Diffusion due to synchrotron motion
•
nｚ=0.006
=0.012
Tune difference
• 3 degrees of freedom
H  x
 px2  x 2 
2
 y
 p y2  y 2 
2
  P ( s)U (r , z )
 2 p2   r 2
r2
   pr  2  
  cos 2   P ( s)U (r , z )
2 
r  2


• p is not constant. Variation of p for small tune
p
difference, nx=0.285 & ny=0.295.
pr
r
• KAM for various p for
equal tune,.
• p =1.2x10-9, 4.8x10-9, 2x10-8.
Note sr=0.89 mm
Motion in the phase space and
tune space – example I
• Near integrable trajectory---nondiffusive
Motion in the phase space and
tune space - example II
• Chaotic trajectory --- diffusive
Tune scan for 3 degrees of
freedom
• Tune scan without synchrotron motion
0.45, 0.45
0.05, 0.05
• Pumping machanism
• Resonance
• Separatrix crossing (n~0)
More
• 4 degrees of freedom – actual weak-strong
model.
2
2
2
2



p

r
r
 2  2
2

H    pr  2  
  cos 2   pz    z    P ( s)U (r , z )

2 
r  2

2 
 

• Colliding beam – 4x2xN+- degrees of freedom
• 3 degree of freedom+synchrotron motion.
Modulation diffusion, stochastic pumping with
separatrix crossing.
•
•
•
•
4 degrees of freedom
Tune scan with synchrotron motion, nｚ=0.006.
Vertical emittance growth.
Resonance, m ny=n is seen.
Emittance growth is large at cross points of resonances
0.45, 0.45
0.05, 0.05
Importance of Lattice
• Nonlinearity of beam-cloud interaction
• Integrated the nonlinear terms with multiplying 
function and cos (sin) of phase difference
M  e:U1: e:F12 : e:U2 : e:F23: e:U3: e:F34 : e:U4 : e:F45: e:U5: ...e:Fn1:
n


 e:F11: exp   : Ui (e:F1i :x) :   M exp( : UT :)
 i 1

kxm  k im / 2 J m / 2 cos(mD1i )
F: lattice transformation
Nonlinear term should be evaluated with considering the beta function
and phase of position where electron cloud exists.
Unphysical cancel of nonlinear term may be caused by simple increase
of interaction point.
Beam-beam limit
• 4 degree of freedom
• Interaction during collision. If sz~y, Df =1 rad,
4th-order term 4Df.

x2
y2 
exp   2
 2


2s x  u 2s y  u 
re 

U ( x, y;s x ( z ), s y ( z ))   
du
2
2
0

2s x  u 2s y  u
e
:F11:
n


:F1i :
exp   : Ui (e x) :   M exp( : UT :)
 i 1

H  x J x   y J y  z J z   P (s)UT ( x, px , y, py , z)
Integrability near half integer tune
Reduction of the degree of freedom.
• For nx~0.5, x-motion is integrable.
(work with E. Perevedentsev)
lim DC , y  0
n x 0.5
if zero-crossing angle and no error.
1
L
Dn x  (crossing angle)+(coupling)+(fast noise)
• Dynamic beta, and emittance
lim
n x 0.5
x
2
s
2
x ,0
• Choice of optimum nx
lim
n x 0.5
p
2
x

X-px plot near half integer in x
• nx=0.503, 0.510, 0.520, 0.540
Crossing angle
• Calculate UT using Taylor map (Diff. algebra)
• Taylor coefficient ~ Fourier coefficient, Uklm.
UT 

U
klm 0
klm
cos(k x  l y  m z )
4-th order Coefficients due to crossing angle
• Short bunch sz=3mm, sx/fsz=1 at 2x15 mrad, original super KEKB.
4-th order Coefficients as a function of crab
sextupole strength, short bunch sz=3mm, sx/fsz=1
• H=K x py2/2, theoretical
optimum, K=1/xangle.
• Clear structure- 220,121
• Flat for sextupole
strength- 400, 301, 040
Crab crossing and crab waist
• Crossing angle induces synchro-beta and
odd coupling resonance terms.
• Merit of Crab crossing is the absence of
the terms.
• Crab waist reduces the odd coupling
resonance term, but keeps the synchrobeta term.
• In the both method, Luminosity
performance is improved.
Space charge limit
• Similar as electron cloud, integrate the
nonlinear interaction along the ring.
• UT: Gaussian?
e
:F11:
n


:F1i :
exp   : Ui (e x) :   M exp( : UT :)
 i 1

H  x J x   y J y  z J z   P (s)UT ( x, px , y, py , z)
Summary
Keyword of SAD
Dynamic aperture
Emittance