The Physics and Applications of High Brightness Electron Beams - Erice, October 9-14, 2005...

The Physics and Applications of High Brightness Electron Beams - Erice, October 9-14, 2005The Physics and Applications of High Brightness Electron Beams - Erice, October 9-14, 2005

Simulations of coherent Simulations of coherent synchrotron radiation effects synchrotron radiation effects

on beam dynamicson beam dynamics

G. Dattoli, M. Migliorati, A. SchiaviG. Dattoli, M. Migliorati, A. Schiavi


0)s,z(WE

Ne

zs 0

2

Vlasov equation for longitudinal distributionVlasov equation for longitudinal distribution

z

3/13/23/10

||

dxdx

s,xd

xz

1

R3

2

4

1

'd'dz)s,','z()'zz(Ws,zW

CSR


)0,,z()s,z(E

Ne

zsexp)s,,z(

0

2

s

0'ds)'s,z(W)s,z(

TEO codeTEO code(Transport by Exponential Operators)(Transport by Exponential Operators)

the formal solution of the Vlasov equation can be the formal solution of the Vlasov equation can be written aswritten as

For small ‘s’ (steps), by using For small ‘s’ (steps), by using the the symmetricsymmetric split we obtain split we obtain

0),s,s

2

1z(A),s,s

2

1z(sA

2

1sz)s,,z(


smooth evolution smooth evolution of distribution of distribution

function (example function (example with CSR without with CSR without

noise)noise)


CSR effectsCSR effectswith initial noisewith initial noise


Example of microbunching due to Example of microbunching due to CSRCSR


Comparison with tracking codeComparison with tracking code

-20

0

20

40

60

80

100

120

-0.06 -0.04 -0.02 0 0.02 0.04

[1/m]

z [m]


Comparison with tracking codeComparison with tracking code

0

50

100

150

200

250

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

[1/m]

z [m]


Phase space distribution


0)s,z(WE

Ne

zs 0

2

Vlasov equationVlasov equation

z

3/13/23/10

CSR dxdx

s,xd

xz

1

R3

2

4

1s,zW

It is convenient to use dimensionless variables:It is convenient to use dimensionless variables:

0z

z'z

andand

0

'


0'

''s,'zF

'z

''

N

NC

's

'

1'd'dz'

2

3/10z

3/22000

e

RE4'CN

sRE4

Ne

3

2's

3/40z

3/2000

2

3/1

'z

3/1 dydy

's,y'd

y'z

1's,'zF


CSR instability thresholdCSR instability threshold

bunch)(Gaussian 4

6C

e

RE4CNN

3/1

2/5

2

3/10z

3/22000

th


CSR effect on dynamicsCSR effect on dynamics(Gaussian bunch)(Gaussian bunch)

If N / NIf N / Nthth and and s’ are the same, the s’ are the same, the

beam dynamics is the same beam dynamics is the same (fixed the type of distribution)(fixed the type of distribution)

sRE4

Ne

3

2's

3/40z

3/2000

2

3/1

2

3/10z

3/22000

th e

RE4CN

3/1

2/5

4

6C


N = 5e10σε0 = 7e-4σz0 = 7e-3 mη = 1.4e-3E0 = 1.5 GeVL = 1000 m

N = 6.35e10σε0 = 1e-3σz0 = 1e-3 mη = 5e-3E0 = 0.5 GeVL = 28 m

λ’ λ’

z’ z’


Preliminary simulations for SparxinoPreliminary simulations for SparxinoIn 80 cm the wake field does not perturb the uncorrelated gaussian

distribution (N/Nth ~ 3.2e-3).


Phase space grid 300x300


Fit of the distributionFit of the distribution(work in progress)(work in progress)


Initial distribution (2000x500)Initial distribution (2000x500)


Evolution of Evolution of longitudinal longitudinal distribution distribution

without noise without noise


Evolution of Evolution of longitudinal longitudinal distribution distribution with noise with noise


phase space distributionphase space distribution


Work to doWork to do

Improve the fit of initial distributionImprove the fit of initial distribution

Do a more realistic modelization of the Do a more realistic modelization of the compressor (e.g. add drifts, longitudinal compressor (e.g. add drifts, longitudinal space charge effect)space charge effect)

……

The Physics and Applications of High Brightness Electron Beams - Erice, October 9-14, 2005...

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