Further Studies on longitudinal dynamics in the CR
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Further Studies on longitudinal dynamics in the CR
T. Agoh*, D. Alesini, C. Biscari, R. Corsini**, A. Ghigo,
LNF, INFN, Frascati
*KEK, Japan
** CERN
11th CTF3 Collaboration Meeting – 16-17 January 2007
Contents
• CSR in Combiner Ring (Tom Agoh, KEK)• Transverse effects of rf deflectors (David Alesini)• Possible Measurements (Roberto Corsini, Andrea Ghigo)
Previous studies on CSR in CR
• Roberto (1999) analytical and simulations for first CR design
• Mikhail analytical approach (2001) with present dipoles and chambers
To be on the safe path Bunch length of the order of 2 mm to limit energy
spread below 1%
R56 tuning chicane
T. Agoh at Frascati for SUPERB studies – Nov 06
has dedicated some time to simulate the CR dynamics and validate the previous analytical approach results
CSR calculation using paraxial approximation
• Parabolic equation
• Comparison with analytic solutions
CSR effect in the CTF3 combiner ring
• Wakefield, Impedance, Loss factor
• Energy spread growth due to CSR
Numerical method *
(1) Begin with Maxwell equations in vacuum (E, B)
(2) Fourier transform EM field w.r.t z
Frequency domain :
Time domain :
(3) Approximate these equations in Paraxial approximation (chamber cross section much smaller than radius of curvature)
(4) Solve them by finite difference
Beam pipe = boundary condition
(5) Inverse Fourier transform Back to the time domain
*T.Agoh, K.Yokoya, PRST-AB, 7, 054403 (2004)
Role of beam pipe
The light, emitted from a bunch, cannot deviate from the s-axis due to the reflection on the pipe wall.
The radiation always propagates near around the axis.The assumption is the condition so
that the radiation field can be a paraxial ray.
Equation of Evolution
(parabolic equation)
Equation to describe CSR:
Algorithm
Discretize the equation by central difference:
Solve initial condition at the entrance of bending magnet (radius=∞)
Proceed field evolution step by step along s-axis
Solve equation of evolution with boundary condition
Free space or very large vacuum chamber
• EM field is no longer a paraxial ray.
Chamber structure so that backward waves are produced
• Bellows, Cavity Chamber wall must be smooth.
Ultra-short bunch, or fine structure in the bunch
• Fine mesh is required to resolve the field. (expensive)
The shortest bunch length I computed is 10 microns in 6cm pipe.
• Bunch profile with sharp edge, e.g. rectangular, triangular, etc
Bunch profile must be smooth.
Examples to which this approach cannot be applied
• Vacuum chamber (aluminium)height =
36mm width = 90mm
Beam energy: E 150 MeV
Bunch charge : NeNumber of particles: N
2.3 nC
1.45×1010
Bunch length : z 1~10 mm
Circumference : C 84 m
Bending radius : 1.075 m
Dipole length : Lbend 0.56 m
Number of bends : Nbend 12
Number of turns : Nturn 0.5 ~ 4.5
Rectangular cross section in CSR calculation
• Combiner ring parameters
• Requirement< 1% after 4.5
turns
Longitudinal wakefield for single turn
z=2mm
z=1mm
z=3mm
Solid lines = simulation (Agoh)
Dotted lines = analytic solution (parallel plates model)
Es ~ ±0.03 MeV/mE ~ ±0.2 MeV/turn
E ~ ±0.9MeV
For 4.5 turns
(E/E ~ ±0.5%)
z=2mm
A.Ghigo, M.Zobov CTFF3-004 (2001)
CSR in transient states
(z=2mm)
Analytic solution (steady state)
Lbend = 0.56m
Simulation
Simulation for Lbend = 1.5m
Magnet entrance
Static solution of parallel plates model works for CTF3 combiner ring because of the small bending radius, the wide chamber width.
black -> red -> magenta -> green -> blue-> cyan
ExitDIFFERENT COLOURS CORRESPONDING TO DIFFERENT POSITIONS ALONG THE BENDING
Loss factor
CSR+RWCS
R
RF deflector
Initial state
0.5 turn
1.5 turns
2.5 turns
3.5 turns
4.5 turns
Longitudinal phase space distribution (z, )z=3mm 0=5×10-3
Ne=2.33nC
Initial state
0.5 turn
1.5 turns
2.5 turns
3.5 turns
4.5 turns
Longitudinal phase space distribution (z, )z=2mm 0=5×10-3
Ne=2.33nC
E/E = 1%
z=2mm
z=1.5mm
z=1mm
z=3mm
Change of energy due to CSR
Energy spread vs. turn
Center of energy vs. turn
(Initial energy spread 0 = 5×10-3)
Uo = 40 eV @150MeV (in 4.5 turns E/E = 0.001 ‰ )
z=2mm
z=1.5mm
z=1mm
z=2.5mm
Longitudinal distribution at 4.5 turns
Energy distribution
— 1mm — 1.5mm — 2mm — 3mm
Bunch length
Energy spread vs. Bunch length
E/E = 1%
z=1.2mm=9.4×10-
3
If the bunch length is shorter than 1.2mm, the energy spread exceeds 1% at 4.5 turns.
4.5 turns
Summary (Agoh)
Parallel plates model works for the CTF3 combiner ring,
• because of small bending radius, wide chamber width.
• A preliminary study by A.Ghigo and M.Zobov (CTFF3-004) goes on well.
A short bunch is affected by CSR in the combiner ring.
• If the bunch length is 1.2mm, the energy spread is 0.94% (< 1%) after 4.5 turns, however, the energy distribution is significantly deformed.
• Considering a bunch compression after the combiner ring, minimum acceptable length may be around 2mm.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
R56 (m)
sigm
al (
mm
)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
R56 (m)
beta
l (m
m)
-0.2 -0.1 0 0.1 0.2 0.3-6
-4
-2
0
2
4
6
R56 (m)
alfa
-2 -1 0 1 2-0.04
-0.02
0
0.02
0.04
z (mm)
dp/p
-10 -5 0 5 10-0.04
-0.02
0
0.02
0.04
z (mm)
dp/p
-2 -1 0 1 20
50
100
150
200
250
z (mm)-10 -5 0 5 100
100
200
300
z (mm)
Linac outputCR input
R56 = -0.2 m
All previous simulations have been done with uncorrelated longitudinal distribution.Tom is now simulating the effects with different longitudinal phase space correlations obtained by tuning the chicane R56
Work in progress
Missing: start to end simulations for overall beam dynamics behaviour
• Longitudinal plane : linac acceleration, csr, wake fields, RF deflector effects, non-linear isochronicity, CSR
• Transverse plane: transverse non linearities, betatron beating from mismatch, chromaticity, high order terms, …
From linac20 cm – 667 psec
2 cm – 67 psec
Passage through Delay Loop
# turns 5 4 3 2 1 5 4 3 2 1
10 bunches microstructure at CR output
1
2
3
4
5
6
7
8
9
10
From Linac exit
L H V VHL
To Decelerating section input
2003
10-7
10-6
10-5
0.0001
0 2 4 6 8 10 12
exeyexey
emittances
bunch position
Dp/p = 1%
Dp/p = 0.5 %
0.0001
0.001
0.01
0 2 4 6 8 10 12
bunch length (mm)
dldl
bunch position
Dp/p = 1%
Dp/p = 0.5 %
Variation of transverse and longitudinal emittances along the bunch train (single particle effects)
PRELIMINARY ANALYSIS OF THE RFD INDUCED ENERGY SPREAD IN THE CR
a) The Panofsky-Wenzel theorem relates the RFD transverse deflecting voltage and the longitudinal electric field gradient;
b) The transverse deflecting voltage and the longitudinal one are 90 deg out-of-phase
t
Vx
The deflected bunch does not “see”, to the first order, any longitudinal electric field.
zyx VcjV
~~
The stored bunches that passes out-of-phase with respect to the deflecting field “see” longitudinal electric field off axis that induces an energy spread.
x
t
Vz
FIRST ORDER ESTIMATION
eVVc xxRF
E ˆ
Induced energy spread
Average transverse beam dimension at the deflector
4102.0 EE
CR single passage in one RFD
Other effects to be evaluated (tracking):
a) multi-passage case;
b) RFD beam loading effects; D. Alesini
• Multipassage:180° betatron phase between RFDs: the effect is compensated at first order in each passage
x
x’
1st RFD
2nd RFD
t
Vz
X
Is it possible to measure these effects, discriminate among them, and consequently optimize the drive beam structure?
Tools• Runs with different currents (single particle against collective
effects)• Tuning R56 of chicane and/or TL1, keeping the changes
transparent (need of optimum optical model for every element)• Tuning of bunch logitudinal correlation with Linac • Measure of bunch length and energy spread in as many possible
points along the whole system
Do we have enough resolution in the diagnostics?
Type CTS + TL1 + CRM + TL2
CR Delivery
WCM 1 0 At CERN
BPM 2 5 October 2006
BPI 7 20 October 2006
BPR 1 1 At CERN
MTV 4 2 October 2006
PHM 0 1 At CERN
NCR 0 1 January 2007
Beam diagnostic
WCM = Wall Current Monitor
BPM = Beam Position M. ( 40 mm)
BPI = BPM from INFN (90 x 40 mm)
BPR = BPM (for bunch length behavior)
MTV = Ensemble camera & mirrors (for synchrotron light or Transition radiation)
PHM = Phase Monitor (Frequency meas.)
NCR = Nearly Confocal Resonator
TL1CR
MTVBPR
PHM
BPR.
NCR
WCM
MTV 0751MTV 0796
MTV 0796 0751
x (m) 0.4 1.7
Dx (m) 0.13 -0.08
xb (mm) ( = 0.34 rad) 0.37 0.76
xs (mm) (1% dp/p) 1.30 0.80
Isochronous configuration
Measure of beam size with synchrotron radiation monitors in two dipoles
SOME EXAMPLE
%6
5.4
75max
acc
x
p
p
cmA
cmD
1 1.5 2 2.5 350
100
150
200
250
bunch length (mm)
N tu
rns
# turns before hitting the vacuum chamber
5 % including betatron dimensions
Measure of energy loss by max # of turns in the crwithout extraction
(if the bunch length were kept constant but… )
Switch off the rf deflectors (time needed of the order of one turn) and let the beam spiralize inside
Sext off : 2° order terms :Non-isochronicity
Sext on: residual non isochronicity
10 turns -0.5 mm
20 turns -1.1 mm
50 turns - 3 mm
After few tens of turns at high current all bunches will have similar energy spread – will behave in a similar way. (6D simulations with CSR included must still be done)
Measure of energy loss with BPMs:
E/E = 10-3 (z = 1mm, 1 turn)
Dmax = 0.75 m =>
x = 0.75 mm from turn to turn
Measure difference in orbit between BPMs with dispersion and those with zero dispersion for the different parameter sets (low charge, high charge, short bunch, long bunch, etc)
Conclusions• Beam dynamics calculations in progress
(Cern, KEK, Frascati, …)
• Main aim: find the best configuration for power production
• Opportunity to use the 2007 commissioning as a test bench for emittance preservation, csr studies, .. applicable to other accelerators
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