6-D dynamics in an isochronous FFAG latticee-model

Main topic :

• Tracking code development :

3-D simulation of the field in an isochronous FFAG optics.

• Application to study of beam transmission by multiturn tracking.

Franck lemuet, Doctoral Student CERN/CEA

NuFact05, Frascati 24/06/2005

NuFact05, Frascati 23/06/2005

The ray-tracing method – Ingredients for magnet simulation

( ) ( ) ( ) ( ) ( ) [ ]

( ) ( ) ( ) ( ) ( ) [ ]

etc. equation), (Lorentz using

:Velocity

(1) : Position

...,

...!5

...!2

...!6

...!2

5

0

2

0001

6

0

2

0001

=′′′′×+×′=′′×=′

+Δ

′′′′′++Δ

′′+Δ′+≈•

+Δ

′′′′′++Δ

′+Δ+≈•

uBuBuuBuu

sMu

sMusMuMuMu

sMu

sMusMuMRMR

rrrrrrrrr

rrrrr

rrrrr

Z

XY

Z

Y

X

M

0

Reference

( )1). . (Fig frame reference

Zgoubi the in sderivative its and , field, the yields A kji kji ZYXBZYXB ∂∂∂∂ ++ /,,rr

model field

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

).,,3,1,...,,

,,,,

,,,

/

,...,

2

ZYXjiX

suX

ZYXBsusu

XX

ZYXBsB

suX

ZYXBsB

dsBd

ji

ii i

jiij ji

ii i

nn

or for stand (

etc. ,

: using

above the from derived are 1 Eqs. in needed sderivative The

=

′∂

∂+

∂∂

∂=′′

∂

∂=′

∑∑

∑rr

r

rr

r

Figure 1 :Position and velocity of a particle in the reference frame.

Integration of the Lorentz equation, based on Taylor series expansions

Design of the isochronous cell [Ref.G.Rees]

NuFact05, Frascati 23/06/2005

Original design

Lcell = 0.65 m

BF is a multipole

• gradient is dB/dx.

Zgoubi model:

BF magnet is a sector magnet

• gradient is dB/dr • sector angle value of half the cell deviation θ = 1/2 (360/45) = 4 deg

Figure 2: The electron model isochronous cell.

Figure 3: The sector magnet in the zgoubi model

Field models

NuFact05, Frascati 23/06/2005

Gradient profiles K(m-2) vs. x(m)

-0.015-0.01 -0.005 0.005 0.01 0.015

-1.5

-1

-0.5

-0.01 -0.005 0.005 0.01

-8

-6

-4

-2

-0.015-0.01 -0.005 0.005 0.01 0.015

4

5

6

7

The magnets’ gradient are constitutive of the design data, they are approximated using 4th degree polynomials.

( )( )( ) 48362

4732

4632

7228.1108.26.10056352.16892981.1

15832.355093291.1525472.8646395.3

7728.52.2470785.33497949.5216096.1

xExExxxK

rErrrrK

xExxxxK

BD

BF

bd

−−−−−=

++−+−=

−+++−=

The gradients are integrated to get the multipole coefficients of the field, as needed in the zgoubi data file.

( )( )( ) 56432

0

56432

0

56432

0

4456.35271012.33521762.8492981.1

89581.7137733637.5082365.4346395.3

1547.179.617662.11163975.2616096.1

xExxxxbxB

rErrrrbrB

xExxxxbxB

BDBD

BFBF

bdbd

−−−−−=

++−+−=

−+++−=

oo BD(+) o BF(,+) o bd() OBF(,+)bd()O

0.075 0.04 0.0750.04 0.04 0.04bd

BD

BF

Tracking in a cell

• T.O.F in a cell is 2.177 ns

NuFact05, Frascati June 2005

• Tunes in a cell

Isochronicity is better than a ps

11 12 13 14 15 16 17 18 19 20

0.002177

0.002177

0.002177

0.002177

0.002177

0.002177

0.002177

0.002177 TOF in a cell (microseconde) vs. Energy (MeV)

Tracking in a cell : vertical field

NuFact05, Frascati 23/06/2005

0.0 0.1 0.2 0.3 0.4 0.5 0.6

-.1

-.05

0.0

0.05

0.1

Bzo (T) vs. s (m)

20 Mev

11 Mev

11 Mev

20 Mev

11 Mev

20 Mev

11 Mev

20 Mev

20 Mev

11 Mev

z

FRz

)cos(6

tan2

Sharp edge model

Vertical kick correction

Fringe field model

No overlapping

Tracking in a cell : closed orbits

NuFact05, Frascati 23/06/2005

0.1 0.2 0.3 0.4 0.5-.015

-.01

-.005

0.0

0.005

0.01

0.015

0.02

0.025

X (m) vs. S (m)

11 MeV

12.2 MeV

14 MeV

15.8 MeV

17 MeV

20 MeV

New fitting procedures

• We have enhanced fitting capabilities in Zgoubi in relation to FFAG design, for instance allow automatic adjustement of bi’s coefficients

so as to match tunes, or isochronism, etc …

• Automatic search of closed orbits and Twiss parameters, for given set of energies

• Etc …

( ) 5

5

4

4

3

3

2

210 xbxbxbxbxbbxB +++++=

NuFact05, Frascati 23/06/2005

Stability limits

NuFact05, Frascati June 2005

Two goals :1. Check symplecticity of the motion over the all energy span.2. Find the maximum stable amplitudes in both planes, as well as coupled

-.02 -.01 0.0 0.01 0.02-.04

-.03

-.02

-.01

0.0

0.01

0.02

0.03

0.04 11 MeV

-.02 -.01 0.0 0.01 0.02-.04

-.03

-.02

-.01

0.0

0.01

0.02

0.03

0.04 14 MeV

-.02 -.01 0.0 0.01 0.02-.04

-.03

-.02

-.01

0.0

0.01

0.02

0.03

0.04 x ’ (rad) vs. x (m)

20 MeV

pure

coupled

-.006 -.004 -.002 0.0 0.002 0.004 0.006-.008

-.006

-.004

-.002

0.0

0.002

0.004

0.006

0.008 11 MeV

-.006 -.004 -.002 0.0 0.002 0.004 0.006-.008

-.006

-.004

-.002

0.0

0.002

0.004

0.006

0.008 14 MeV

-.006 -.004 -.002 0.0 0.002 0.004 0.006-.008

-.006

-.004

-.002

0.0

0.002

0.004

0.006

0.008 z ’ (rad) vs. z (m)

20 MeV

Pure and coupled horizontal motion limits

Vertical motion limits (x = xco)

Amplitude detuning

NuFact05, Frascati 23/06/2005

-1. 0.0 1. 2.0.1

0.15

0.2

0.25

0.3

0.35

0.4 Nu_r vs. r (cm)

11 MeV

12.2 MeV

14 MeV 15.8 MeV

17 MeV

20 MeV

0.0 0.2 0.4 0.6 0.8

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Nu_z vs. z (cm)

11 MeV 12.2 MeV

14 MeV

15.8 MeV

17 MeV

20 MeV

Horizontal

Vertical

0.0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5 TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

0.0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5 TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

20 MeV17 MeV15.8 MeV14 MeV12.2 MeV11 Mev

6 nm

zx QQ 2

Beam transmission (1)

NuFact05, Frascati June 2005

A 10000 particles beam is launched for 15 turn acceleration (45 cells/turn), from 11 to 20 MeV.40 kV per cavity, no synchrotron motion. Cavities are put every three cells at the center of the long drift.

10.9 10.95 11. 11.05 11.10

20

40

60

80

100

120

Energy (Mev) spectrum

-.025 -.02 -.015 -.01 -.005 0.0

-.02

-.01

0.0

0.01

0.02

0.03

Eps/pi,Bta,Alp: 1.54E-05 0.472 7E-04 (MKSA)

x ’ (rad) vs. x (m)

-.004 -.002 0.0 0.002 0.004-.004

-.003

-.002

-.001

0.0

0.001

0.002

0.003

0.004 z ’ (rad) vs. z (m)

Eps/pi,Bta,Alp: 6.97E-06 2.02 -4E-03 (MKSA)

50 100 150 2000

0.2

0.4

0.6

0.8

1 N/10000 & E/20Mev vs. cavity #

-.004 -.002 0.0 0.002 0.004-.004

-.003

-.002

-.001

0.0

0.001

0.002

0.003

0.004 z ’ (rad) vs. z (m)

Eps/pi,Bta,Alp: 5.88E-07 1.80 -5E-02 (MKSA)

-.02 -.015 -.01 -.005-.03

-.02

-.01

0.0

0.01

0.02

0.03

Eps/pi,Bta,Alp: 1.63E-06 0.42 -4E-02 (MKSA)

x ’ (rad) vs. x (m)

-.212 -.208 -.204 -.20

2

4

6

8

10

12

14

16

18

dp/po , po = p14Mev Vs spectrum

Initial phase space

Envelopes

Initial phase spaces of transmited particles after the acceleration cycle.

50 100 150 200-.02

-.015

-.01

-.005

0.0

0.005

0.01

0.015

X mean orbit v.s cavity #

Turn 136 Nux=0.251

Losses histogram

50 100 150 200-.04

-.03

-.02

-.01

0.0

0.01

0.02

0.03

0.04 Z mean orbit v.s cavity #

Losses histogram

Beam transmission (2)

NuFact05, Frascati June 2005

-.018 -.016 -.014 -.012 -.01

-.008

-.006

-.004

-.002

0.0

0.002

0.004

0.006

0.008

0.01 x ’ (rad) vs. x (m)

Eps/pi,Bta,Alp: 2.82E-06 0.45 8E-04 (MKSA)

10.9 10.95 11. 11.05 11.10

20

40

60

80

100

120

Energy (Mev) spectrum

50 100 150 2000

0.2

0.4

0.6

0.8

1 N/10000 & E/20Mev vs. cavity #

-.004 -.002 0.0 0.002 0.004-.004

-.003

-.002

-.001

0.0

0.001

0.002

0.003

0.004

0.004-.004

-.003

-.002

-.001

0.0

0.001

0.002

0.003

0.004 z ’ (rad) vs. z (m)

Eps/pi,Bta,Alp: 2.60E-06 1.37 4E-03 (MKSA)

A 10000 particles beam is launched for 15 turn acceleration inside the acceptances obtained withthe previous run.

x11.3 mm mradxnormalised = 243 mm

mrad

z = 10.4 mm mrad z,normalised = 224 mm mrad

50 100 150 200-.02

-.015

-.01

-.005

0.0

0.005

0.01

0.015

X mean orbit v.s cavity #

Turn 136 Nux=0.251

Losses histogram

50 100 150 200

-.08

-.06

-.04

-.02

0.0

0.02

0.04

0.06

0.08

Z mean orbit v.s cavity #

Turn 136 Nuz=0.111

Losses histogram

Tunes : acceleration cycle

NuFact05, Frascati June 2005

0.0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5 TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

TUNE DIAGRAM NUZ

NUX

• A particle is launched at the injection energy (11 MeV) on its closed orbits.• Tunes are computed during the acceleration cycle, approximate to paraxial tunes due to the low energy detuning .

Extraction20 MeV

Injection11 MeV

Summary

NuFact05, Frascati 23/06/2005

•Efficient tools for tracking studies has been developed

To do :

• Focus on beam losses and correlations with resonnances crossing

• Study the new design of the e-model with insertions