12.09.2005Yu. Senichev, Coloumb 2005, Italy1 HAMILTONIAN FORMALISM FOR HALO INVESTIGATION IN HIGH...

12.09.2005 Yu. Senichev, Coloumb 2005, Italy

1

HAMILTONIAN FORMALISM FOR HALO

INVESTIGATION IN HIGH INTENSITY BEAM

Yu. Senichev, IKP, FZJ


2

The problem definition The aim of this work is the investigation of the

behaviour of the smallest part of beam, which we call the halo!!!

For investigation of this phenomenon we applied Hamiltonian formalism together with standard theory of perturbation.

We investigate the non-linear resonances and their self-stabilising effect at the different initial distributions.


3

The halo definition In case, when Hamiltonian

has not the explicit dependence on the time the particle moves along the trajectory

When Hamiltonian has the explicit dependence on the time the particles oscillate around the time averaging curve:

P

Q

In itia l phase portra it

F ina l e ffective phase portra it

H a lo L ines o f the m axim um <H (p ,q ,t>

Figure III.3 : Phase space portra its.

In term ed ia te phase portra it

),(, 00 qpHqpf

),,(,, 000 tqpHtqpft


4

Model approximations The coasting beam is assumed

to have axial symmetry: solenoid and triplet channel

The central core (90-95% of the intensity) is unaffected by the halo.

4.320

Tue Aug 23 11:45:29 2005 OptiM - MAIN: - C:\optim\last\solenoide.opt

50

50

BETA

_X&Y

[m]

DISP

_X&Y

[m]

BETA_X BETA_Y DISP_X DISP_Y

4.440

Tue Aug 23 11:46:11 2005 OptiM - MAIN: - C:\optim\last\triplet.opt

50

50

BE

TA

_X

&Y

[m]

DIS

P_

X&

Y[m

]

BETA_X BETA_Y DISP_X DISP_Y

The core the halo


5

The core distribution the distribution is discribed by the binomial polynomial:

The space charge electrical field:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

4 10 4

8 10 4

0.0012

0.0016

0.002

0.0024

0.0028

0.0032

0.0036

0.0043.034 10 3

0

r1 r( )

r2 r( )

r3 r( )

r4 r( )

r15 r( )

11 10 3 r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2 104

4 104

6 104

8 104

1 105

1.2105

1.4105

1.6105

1.8105

2 105

1.61 105

381.576

Er5 r( )

Er2 r( )

11 10 3 r

122max1

1

0)1(

2

1),(

nn

nm

m

n

n

brf

avr r

r

A

Lf

ImrE

!)!(

!

nnm

mAnm

1

2max

2

2max

1

m

brf

av

r

r

r

m

Lf

Ir


6

Equation motion After the longitudinal coordinate normalizing

is the periodical coefficient

the space charge force:

,2

2

rFrKd

rd

fLz /

p

LeGK f

2

m

n

nnm

nsc rr

rA

r

CrF

1

22

max

12max )(

)1()(

,


7

Equation solution Solution is seeked in the form

where

Envelope equation

In case of the space charge

)()( 0RR iea

01

02max

1

30

0202

02

Ra

AC

RR

d

Rd msc

rRR 00 03

002max

1

030

20

R

r

Ra

AC

R

r

Rrr msc


8

Equation solutionEnvelope oscillation with

Particle in core oscillates with

2/1

2

2

01

12~

hh

h

02max

1

2a

ACh msc

hh 20 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

22

0.099

mu h( )

mu2 h( )

5.00 h 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

55

0

mu3 h( )

2.00 h


9

Equation solution Using Courant and Snyder formalism together with

Floquet method, we have the selfconsistent equations system:

n

nm

n

nm

nsc

m

aACq

R

rR

d

d2max

12

1

1

0

20

20

202

212~cos21


10

Non-linear equation The right side of equation: ,,...,,

~,...,,, 123

1123

0 nn FFF

iea

dFa

dFad

d

dF

dFd

da

n

n

n

n

cos cos,cos,...,cos,cos~

2

1

coscos,...,cos,cos 2

1

sin cos,cos,...,cos,cos~

2

1

sincos,...,cos,cos 2

1

1232

01

0

1232

00

00

1232

01

0

1232

00

0

B-M=>


11

Non-linear equation solution Solution:

22

max22max

2001

12max

200

2max

22

1

200

02max

222

000

02max

122

00

1

)1(

~2cos2

1

~2sin2

1

n

nnm

m

n

nscm

sc

n

n

nnm

m

n

nsc

rrnrn

rnrnmsc

rn

rrnrn

rnrnmsc

rn

a

abA

a

CRbA

a

CR

a

abACRa

nda

aACR

R

ra

d

d

nda

aACR

R

r

d

da

Linear detuning Non-linear detuning


12

Isolated resonance New variables: and

Resonance condition at some

rn2

~

2a 2maxmax a max/ˆ

rrnr

rnrnm

scrn

nn

m

n

nm

nsc

rL

ndn

AC

RR

r

bAn

CR

nH

2cosˆ

2

1

ˆ1

ˆ2

~,ˆ

0max

200

0

2max

2000

1

02max

20000 ˆ1ˆ

2

~n

nnm

m

n

nscLres

rbA

CR

n

0ˆˆ


13

New Hamiltonian After canonical transformation with :

rrnr

rnrnm

scrn

m

n

nn

nm

nsc

ndn

AC

RR

r

bAnC

RH

2cosˆ

2

1

2

ˆˆ11,

00

max

200

0

2

2

20

max

200

0ˆˆˆ


14

Resonance width

m

n

nn

nm

n

rnr

rnrnm

widthres

bAn

dn

AR

r

2

20

00

0.

ˆ11

ˆ

ˆ

Th

e co

re

reso

nan

ce

w

idthH

alo

/N r

0

The core

The halo

No

n-p

ertu

rbed

bu

nch

Non-perturbed

resonance

width


15

Emittance growth Nr is the resonant

harmonic Emittance growth is

m

n

nn

nm

n

rnr

rnrnm

bAn

dn

AR

r

20

00

0

0 ˆ11

ˆ

1ˆ

ˆ


16

Phase oscillation The halo phase delay relative the core

rnrnm

scres

rnr

rnrnm

scrn

r

m

n

nn

nm

nscr

AR

rCR

dn

AC

RR

rUn

UnbAC

Rnd

d

00

max

max

200

00

max

200

00

02

20

max

2002

2

ˆ*

ˆ

2

1)ˆ( и 2 where

0sinˆˆ)1()1(2


17

Numerical simulation

Numerical simulations have been done for the periodical FDO channel

Maximum size of beam, normalised on in FDO channel at 100 mA (the lower curve) and 150 mA(the upper curve) micro-

pulse current.


18

Numerical simulation

-10.00 0.00 10.00

-8.00

-4.00

0.00

4.00

8.00

In itia l em ittance

F ina l em ittance

X , m m

dX /dZ , m rad

F igureIII.6: In itia l and final horizontal em ittances in FDO channel at tune space charge depression 50%

-10.00 0.00 10.00

-8.00

-4.00

0.00

4.00

8.00

In itia l em ittance

F ina l em ittance

X , m m

dX /dZ , m rad

F igure III.8: In itia l and final horizontal em ittances in FDO channel at tune space charge depression 75%


19

Conclusion the model of the halo creation was developed

the case without an external resonance was considered, and the beating of envelope is the source for the emittance growth

the analitical formula for the emittance growth has been derived

The analitical and the numerical results have been compared and the good agreement was observed

12.09.2005Yu. Senichev, Coloumb 2005, Italy1 HAMILTONIAN FORMALISM FOR HALO INVESTIGATION IN HIGH...

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