Further Studies on longitudinal dynamics in the CR

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