EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN - AB Division
CERN – AB-2005-???
CLIC Note ???
Dynamic Aperture of the CLIC Damping Ring
E.Levichev, P.Piminov*
Abstract
To attain the high luminosity goal in the Compact Linear Collider (CLIC),
damping rings with extremely low equilibrium transverse emittance and
fast damping rate are required. Special unit cells with low dispersion function and damping wigglers in the dispersion free straight section are used
to construct damping rings. Unfortunately, strong chromatic sextupole
magnets and nonlinearities of damping wigglers may lead to reduction of
the dynamic aperture. The paper below presents the results of nonlinear
particle motion simulation and dynamic aperture calculation.
Geneva, Switzerland
September ??, 2005
*
Permanent address: Budker Institute of Nuclear Physics, Novosibirsk 630060, Russia
1
1
Introduction
Nonlinear features of the CLIC damping ring [1] are studied with the help of a general purpose accelerator design code Acceleraticum developed at Budker INP
(Novosibirsk) [2]. This code, beside linear stability analysis and calculation of
many accelerator parameters, provide 6D particle tracking for different sources of
non-linearity including n-pole magnets, magnet edge fields, wigglers, undulators
etc. 6D tracking is performed via the formalism described in [3]. Propagation of the
particle trajectory through the wiggler magnetic field is made with a symplectic integration routine, which can use simulated or measured magnetic field array in the
median plane as input data [4].
2
The CLIC Damping Ring
The damping ring parameters are optimized via taking into account the combined
action of radiation damping, quantum excitation and intrabeam scattering [1]. The
ring has a racetrack shape with two arcs and two long straight sections for the
damping wigglers accommodation. A TME arc cell in the CLIC damping ring
comprises four quadrupoles and a bending magnet. There is a dispersion suppressor
at both ends of the arcs. It contains RF cavities and connects the arcs to the dispersion-free straight sections intended for wiggler FODO cells and injection/extraction
equipment. Main parameters of the CLIC damping ring are listed in Table 1.
Table.1: Main parameters of the CLIC damping ring.
Energy, E
2.424 GeV
Circumference, L
360.63 m
Horizontal tune, Qx
70.128
Vertical tune, Qz
36.416
Compaction factor, α
8.15×10-5
Natural horizontal chromaticity, ξx0
-110.2
Natural vertical chromaticity, ξz0
-134.5
Corrected horizontal chromaticity, ξx
0.59
Corrected vertical chromaticity, ξz
0.46
Injected horizontal emittance
1.8×10-8 m rad
Injected vertical emittance
3.1×10-10 m rad
Equilibrium horizontal emittance
8.2×10-11 m rad
Equilibrium energy spread
7×10-4
Horizontal sigma at injection
122 µm
Vertical sigma at injection
50 µm
Four sextupole magnets per TME cell and a number of sextupoles in the dispersion
suppressors are used to correct natural chromaticity and to optimize the dynamic
aperture. Some interlaced sextupole pairs are used in the arcs. Nine families of sextupoles in three adjacent TME cells are chosen and the –I transformation is imposed
2
over 6 cells providing second-order achromatic conditions and canceling all second-order geometric aberrations (via appropriate choice of the TME cell phase advances).
3
On-energy dynamic aperture without wigglers
One can use azimuthal harmonics of relevant distortion potential in order to perform quantitative estimation of the nonlinear lattice perturbation. In case of sextupole perturbation and lattice mirror symmetry, one should take into account the following 5 types of harmonics ( j  1,3 )
1
3/ 2
  xm
(k 2 l ) m cos( j x   n ) m ,
m
48
1
1/ 2
B1n 
  xm
 zm (k 2 l ) m cos( x    n ) m ,
m
48
1
1/ 2
B n 
  xm
 zm (k 2 l ) m cos(      n ) m ,
48 m
A jn 
(1)
where point-like sextupoles with integral strength (k 2 l ) m are assumed, and notation
 means     x  2 z , etc.
The harmonics (1) represent the following set of resonances, which sextupoles excite in the first order of perturbation
3 x  n ,
x  n,
 x  2 z  n .
Two-fold lattice symmetry cancels odd resonance numbers n, so the resonances
closest to the DR working point ( x  70.128 ,  z  36.416 ) and their strength look
as follows:
Resonance
Table 2: Parameters of the main resonances.
Dimension Detuning Harmonic Strength(m-1)
1
0.128
A1,70
–0.01
 x  70
3 x  210
 x  70
 x  2 z  142
 x  2 z  144
 x  2 z  2
 x  2 z  4
1
0.384
A3,210
0.02
2
0.128
B1,70
0.8
2
0.96
B+,142
–56
2
–1.04
B+,144
72
2
–0.704
B-,-2
46
2
1.296
B-,-4
75
It is clear that due to the –I transform the first harmonic coefficients (A1,70 , B1,70
and A3,210) are suppressed rather efficiently, but all coupling harmonics amplitudes
are large and they really define the transverse dynamic aperture. In Fig.1 the CLIC
3
DR dynamic aperture (in the center of straight section with  xp  1.16 m and
 zp  8.07 m) found by the 1000-turn on-energy particle tracking is depicted in the
units of  x, z at injection. One can see for the pure horizontal motion (initial condi24
x,
tions Az (0)  0 , Ax (0)  0 ) that 1D dynamic aperture is rather large Ax max  15
but as we switch on the second degree of freedom even for very small initial vertical amplitude, the horizontal dynamic aperture shrinks drastically down to
Ax max  54  x due to the sextupole coupling resonances influence.
Fig.1: The on-energy dynamic aperture of the CLIC DR.
A detailed structure (survival plot) of the on-energy dynamic aperture is shown in
Fig.2.
Fig.2: The dynamic aperture survival plot.
4
Black color indicates particles that survived for 1000 revolutions (stable areas)
while other colors show particles lost after less than 1000 turns. A complicated net
of high order resonances (stable islands and stochastic layers) is clearly seen at the
border of the stable region.
A nonlinear amplitude-dependent tune shift is an important characteristic of the dynamical system. For the second-order approximation of the sextupole potential this
tune has the following form:
 z  C zx a x2  C zz a z2 ,
 x  C xx a x2  C xz a z2 ,
(2)
where ax,z is the betatron oscillation amplitude at the azimuth with  xp ,  zp  and
the coefficients are expressed in the following way:
 3 A12n
A32n 

,


 xp n    x  n 3 x  n 


B2n
B2n
36   2 A1n B1n

  C zx xp ,
C xz 



 zp n   x  n  x  2 z  n  x  2 z  n 
 zp
C xx  
C zz  
18

(3)
 4 B12n

B2n
B2n

.




 zp n  x  n  x  2 z  n  x  2 z  n 
18

The next order of approximation gives the tune-amplitude dependence  a x4,z and if
a perturbation is large, the contribution of two approximation orders can be comparable. The tune-amplitude dependence for the CLIC DR is shown in Fig.3 and the
relevant detuning coefficients found from tracking are listed in Table 3.
Fig.3: Nonlinear tune shift vs. betatron amplitude.
5
Order
Table 3: Detuning coefficients.
Cxx
Czx
Cxz
Czz
2nd , (mm-2)
–0.05
0.24
0.036
1.40
4th, (mm-4)
–0.14
0.86
0.41
–0.21
Note again suppression of the Cxx coefficient (the main harmonics are A1,70 and
A3,210) in comparison with Czz (the main contributed harmonics are the coupling
ones).
It is well known that properties of the single resonance m x  n are defined by the
Hamiltonian
H  I   ( I )  f ( I ) cos m ,
(4)
where     n / m ,  ( I )  C xx is the nonlinear detuning term and f ( I )  Amn
represents the resonance strength. For small resonance strength (due to the suppressed harmonics in our case), nonlinear detuning stabilizes the resonance and in
the phase space portrait one can see a separatrix curve of finite width. This is a case
of the CLIC DR and Fig.4 demonstrates resonance curves for  x  70 and
3 x  210 surrounded by the stable regular quasi-periodic phase trajectories.
Fig.4: Separatrix of the integer and one-third resonances.
In general, a resonance island inside the stable area is unwanted, since different lattice errors (linear and nonlinear) may make a resonance stochastic layer increase
and the dynamic aperture shrinking.
6
To explore the tune region in the vicinity of the chosen working point, a tune scan
(100×100 points with 300-turn-tracking per point) of the dynamic aperture has been
performed. The maximum horizontal and vertical DA is shown in Fig.5.
Fig.5: A betatron tune scan.
Strong sextupole coupling resonances  x  2 z  142 , 144 and  x  2 z  2 , –4
are clearly seen in Fig.5, meanwhile there is practically no DA reduction at the resonance  x  70 because of good compensation of the harmonics.
4
Off-energy dynamic aperture without wigglers
Two different cases are usually considered for the off-energy dynamic aperture
simulation. In the case of constant momentum deviation (RF cavities are switched
off), the energy aperture is determined by the high order chromatic aberrations (we
assume that the energy acceptance is large due to the appropriate choosing of accelerating voltage). Meanwhile, for synchrotron oscillations excited synchro-betatron
satellites can play important role in some cases.
Fig.6: CLIC DR high order chromaticity.
7
Fig.6 depicts the nonlinear part of the CLIC DR chromaticity while Fig.7 shows the
dynamic aperture for the constant energy deviation with the maximally-achieved
value E / E  10..08 % .
Fig.7: Constant energy deviation dynamic aperture.
Acceleraticum performs synchrobetatron tracking with the help of formalism described in [3] for the following set of variables:

E  E0 
 x, p x , z, p z ,  ( s)  s  c  t (s),  
.
E0 

Transformation of the longitudinal coordinate  (s ) through the lattice element is
given by the following equation

   d / ds  1  1  h(s) x 2  x 2  y  2

1/ 2
,
which was solved for many types of elements. Obtained explicit expressions are
used in the computer code providing rather fast and effective 6D tracking.
For the CLIC DR case the synchrobetatron dynamic aperture is demonstrated in
Fig.8.
8
Fig.8: CLIC DR dynamic aperture with synchrotron oscillations.
5
Influence of errors upon the dynamic aperture
Different kind of magnetic field imperfections can affect the dynamic aperture and
their accurate consideration requires an extensive and time-consuming simulation
and statistical treatment of the results. With some risk of oversimplifying the situation we have performed the following rough estimation of the error influence to the
CLIC DR dynamic aperture:



A single corrector was inserted at a random position and its strength was adjusted to produce a 0.1 mm COD in the DR arcs (separately for the horizontal and vertical planes).
A single quadrupole was shifted transversely to produce a 0.1 mm COD in
the arcs.
A single sextupole was shifted transversely to produce a 0.1 mm COD in
the arcs (and simultaneously to move the tune point by ~10-2).
Two main effects reduce the dynamic aperture due to the above errors: (i) the 2-fold
lattice symmetry is destroyed and non-systematic resonances appear and (ii) betatron tune point shifts randomly for the order of magnitude ~0.01. Fig.9 illustrates
typical COD around the ring while Fig.10 depicts the COD errors effect to the
CLIC DR stable region size.
9
Fig.9 CLIC DR closed orbit distortion.
Fig.10 CLIC DR dynamic aperture shrinking due to the COD errors.
Violet triangles: sextupole shift; blue triangles: quadrupole shift; grin dots: dipole
corrector and red dots: initial aperture.
The influence of the quadrupole errors has been studied via a single quadrupole
strength mismatch to produce a 5% beta function beating along the ring and performing the aperture scan as in the ideal case (Fig.5).
10
Fig.11: A horizontal quadrupole mismatch.
Fig.12: A vertical quadrupole mismatch.
The results of the betatron tune scan for the quadrupole errors are demonstrated in
Fig.11 and Fig.12. An unstable resonance region widening and non-systematic resonance lines are clearly seen. The tune area corresponding to reasonably large dynamic aperture has shrunk considerably.
6
Damping wiggler influence upon the dynamic aperture
Introducing a wiggler magnet into the ring generates more damping and may reduce
the equilibrium emittance defined by quantum fluctuation. The 160-m-long wiggler
magnet that is planned to be installed in the FODO section of the CLIC DR will
reduce the horizontal emittance 4 times and increase the damping integral I2 from
0.7 m-1 to 4.5 m-1.
The parameters of the permanent magnet wiggler that was proposed for the CLIC
DR are listed in Table 4.
11
Table 4: Main parameters of the CLIC damping wiggler.
Period
10 cm
Gap
12 mm
Pole width
50 mm
Length
2m
Field amplitude
1.7 T
10-3
Field quality @1 cm
The wiggler magnetic field can influence the nonlinear beam dynamics and in the
following the simulation procedure and results for the CLIC DR with the wigglers
are described.
Three main sources of the wiggler influence upon the beam dynamics should be
distinguished:
(1) Linear optics distortion and lattice symmetry violation (can be cured by the
quadrupole magnets compensation schemes).
(2) Longitudinal field variation (see Fig.13) that inevitably generates effective
pseudo-octupole component, whose contribution to the amplitude-dependent tune
shift can be estimated for a sine-like wiggler model with the period length  w and
the total length Lw as
 1

 z  
Lw k 4 m2  z2  J z ,
 8

(5)
where Jz is the vertical action variable, k  2 / w ,  m is the wiggler deviation angle and  z is the average beta.
(3) Transverse nonlinear components of the wiggler field, which can be controlled
with the help of pole shimming.
12
Fig.13: Longitudinal distribution of the wiggler magnetic field.
Before the particle tracking, the linear optics distortion introduced by the wigglers
was effectively corrected with the quadrupole magnets in the dispersion suppressor
sections of the DR (both the beta beating and betatron tunes).
To track electron through a complicated 3D wiggler magnetic field, a special
symplectic integrator was applied [4]. This integrator is based on the well known
Verlet integration scheme [5], when the coordinates ( qi, pi) at the instant t are
changed to the coordinates qi , pi  at the instant t  t with the help of the canonical transformation
F
H ( q, p )
F
H ( q, p )
pi 
 pi  t
qi 
 qi  t
,
qi
qi
pi
pi
with the generating function F ( q, p, t )  q  p  tH ( q, p ) . Simple and fast explicit
expressions for the above canonical transformation in the first-order approximation
were found for the paraxial Hamiltonian of a relativistic particle in the wiggler
magnetic field. The median plane magnetic field map (found either from magnetic
field simulation or from measurement) is used as the input parameters for the tracking routine.
The main conclusion is that the wiggler influence upon the beam dynamics is negligible compared with the chromatic sextupoles. For instance, the wiggler detuning
coefficient (5) found from tracking is C zzw  1.7  10 3 mm-2 (cf. the pure sextupole
case from Table 3). The transverse dynamic apertures for different combinations of
the wiggler and chromatic sextupoles are shown in Fig.14 and Fig.15.
13
Fig.14: The dynamic aperture for the wiggler alone and wiggler+sextupoles.
Fig.15: The dynamic aperture for chromatic sextupoles and wiggler+sextupoles.
7
Conclusions
A nonlinear beam dynamics of the CLIC damping ring has been considered for different sources of perturbation: chromatic sextupoles, some kind of lattice errors and
damping wigglers. The dynamic aperture of the ring is determined by the strong
chromatic sextupoles while the damping wiggler nonlinearities do not affect its
size. The stable motion area is not large, especially when imperfections are taken
into account. So, the sextupole optimization procedure seems to be desirable.
We thank Maxim Korostelev for the CLIC DR lattice calculations, Hans Braun and
Frank Zimmerman for support and helpful discussions.
14
8
References
1. M.Korostelev, F.Zimmerman. Optimization of CLIC damping ring design parameters, Proc. of EPAC 2002, Paris, France, pp.536-538.
2. P.Piminov. Master’s thesis, BINP, Novosibirsk, 2000 (in Russian).
3. G.Ripken – DESY Report 85-084, August 1985.
4. E.Levichev, P.Piminov. Symplectic integrator for particle tracking in complex
magnetic field. Proc. of EPAC 2002, Paris, France, pp.1655-1657.
5. L.Verlet. Phys. Rev. 159, 98 (1967).
15