analysis of the electron pinch during a bunch passage

Analysis of the Electron Pinch during a Bunch Passage

Elena Benedetto, Frank Zimmermann

CERN

ECLOUD04, 19/4/04 E.Benedetto, F.Zimmermann 2

Contents

• Analytical calculation of the electron density evolution during the passage of a proton bunch through the electron cloud (linear force approximation)

• Expression for the tune shift experienced by the protons into the bunch.

• Simulations: extension to non-uniform, e.g. Gaussian, transverse beam profiles, which give rise to non-linear forces on the electrons.

• Estimation of the tune spread from the simulations results.


Electron Pinch along the bunch

The electrons are accumulated around the beam center during the bunch passage (pinch)

The aim is trying to understand the mechanism of the slow emittance growth, that is probably caused by the tune shift and tune spread due to the electron pinch.

For this reason we compute the electron cloud density evolution during a bunch passage.


Highlights of the analytical calculations

Equation of motion of an electron in the bunch

potential

Time evolution of the electron density

Tune shift experienced by the protons

Linear force approximation

via Liouville theorem

Uniform Gaussian

For any longitudinal bunch profile

Initially Gaussian electron

distribution


Electron density evolution• Electron distribution in the phase space (the density is obtained

by integrating in the velocities). • In the linear force approximation, the horizontal and vertical

planes are uncoupled → factorization

tyytxxtyyxx yx ,,,,,,,,

20

20

20

20

'22

0000 '2

0,,,,

xx

exx eexxtxx

• Liouville Theorem + assumption of Initial Gaussian Distribution

• (x0, x’0) are obtained as a function of (x, x’) by inverting the solution of the Eq. of motion xxfxx ,, 00


Equation of motion of an electron in the bunch potential (1)

• Bunch distribution:

2

2

222

)(~

,~ r

r

r

bb e

zezr

z)(nσc

t z 1

3

22

0

22

2

22

2

12

,rm

le

r

te

rm

ltreE

dt

rdm

e

r

b

ebe

r

• We consider t=0 when the bunch enters into the cloud:

• Equation of motion ( the Electric field is obtained via Gauss Theorem):


• Solution in the form:

Equation of motion of an electron in the bunch potential (2)

02 xtωx e 2

22

r

ebe

crtt

• Horizontal component of the Eq.of motion + approximation of linear force (r«r):

00

00

)()(

)()(

xtxtx

xtxtx

1)()()()( tttt

• It can be inverted and inserted into the expression of the phase space density.


Tune shift

• The tune shift experienced by the protons (as a function of r and z) is:

r

zrezrEe

02

),(~

),(~

x

C

x ksdsQ )(41 x

E

cm

ek xe

px

,

2

txxxdtxn xx ,,,

• The tune shift is obtained from the electron density, which is computed by integrating the phase space distribution:

tyntxntrn yx , ,,

• Where Ee is the field produced by the electrons is (→from Gauss theorem):

r

e drrzrnzr0

'),'(~2),(~


Longitudinal Uniform Profile (1)

2

22

2 0

r

eb

e

e

cr

xx

xCsxtxtxx

xS

Cxtxtxx

eeee

ee

ee

cossin

sin1

cos

0

0

22

0222

0

220

2220'2

22

2

'2,,,,

CS

etyntxntyxn

e

CeS

re

eeyx

• Eq. of motion → harmonic oscillator :

• The electron density is:

• The solution can be easily inverted:


• The maximum tune shift is inversely dependent on the electron initial temperature:

• For ’0 « e it goes periodically to very high values when

20

2

'

1

4ˆ

epe

x

rLQ

20

220

22

22

20

22

20'

220

2 '21

1

1

4),(

SC

r

C

rLzrQ

e

e

eC

S

pex

• The tune shift is:

Longitudinal Uniform Profile (2)

0cos tC eKeep in mind

that this is only valid for the transverse linear force

approximation !!!


• Eq. of motion:

• We look for a solution in the form:

• WKB approximation:

• The general solution can also be written as:

General longitudinal distribution (1)

; 0)(2 xtx e2

22 )(

r

ebe

crt

)()()( tiSetAtx

ee

e

e

e

2,2

3

)()(

)(

1)(

ttS

tStA

e

)()(

)(cos)(

)( 21 tsinSt

ctS

t

ctx

ee

t

e dtttS0

)()(


General longitudinal distribution (2)

• c1 and c2 are determined by the initial condition, so we obtain again a solution of the form:

• that can be inverted in order to get:

• The electron density is:

• And the tune shift:

xtdtcx

xtbxtax

)()(

)()(

0

0

)(2

2

)(2),( tD

r

etD

trn ee

2

022

02 ')()()( tbtdtD

4

0

2

4

31

4),(

r

OD

r

D

rLzrQ pe

x

00 ,, xxLxx

1)()()()( tctbtdta

In particular, we find the

expression of D(t) for a

longitudinal Gaussian

distribution


Vertical position vs. time for 6 electrons at different start amplitude, from 0.5b to 3b : linear force approximation (left) and Gaussian

transverse profile (right). Gaussian bunch shape in z.

Simulations: Linear and Gaussian force (2)

z/z z/z

vert

ical

pos

ition

[m

]

vert

ical

pos

ition

[m

]

(Parametres of LHC @ inj)


Electron density vs. time at the centre of the pipe, during the passage of a bunch, assuming a linear transverse force (Left) and a Gaussian transverse beam profile (Right). In green, the analytical results. A Gaussian bunch profile is assumed in z.


z/zz/z

00

(The head of the bunch is on the right)


Snap shot of radial distribution (∙r) at 4 different times during

the bunch passage: linear force approximation (left) and Gaussian transverse profile (right).


r/b r/b

ec-d

ensi

ty[

a.u

.]

ec-d

ensi

ty[

a.u

.]

z=- 3z

z=- 1.5z

z= 0z

z= 3z

z=- 3z

z=- 1.5z

z= 0z

z= 3z


Density enhancement during the bunch passage (non linear force)

Ec-density vs. Time, during the passage of a Gaussian bunch

Inside the bunch the density enhancement is about a factor 50.

z/z

/0

t0

t1 t3

t2


Horizontal phasespace at different time step:

t0 = when the bunch enters into the cloud (z=-3z)

t1 = first peak

t2 = first valley

t3 = last peak

t0 t1

t3t2

Density enhancement during the bunch passage (non linear force)


Estimated incoherent tune shift from the simulations (non linear force)

• The density enhancement at the centre of the bunch is about a factor ~ 50.

• A simple evaluation of the tune shift gives the value:

0

2

50

2

e

pe

px n

rLn

rLQ

~ 0.13

3110 106 mne

Where the unperturbed electron density is:


• The tune shift expected from an unperturbed cloud is about ~ 0.0025.

• The spread of the tune footprints computed via frequency map analysis from HEADTAIL simulations is ~20 times larger

• In our estimate we got ~50 times larger

Courtesy Papaphilippou

Estimated incoherent tune shift from the simulations (non linear force)


Summary• Analytical approach to investigate the cause of a

slow emittance growth due to the tune shift and tune spread from the electron pinch.– Analytical expression for the ec-density evolution

along the bunch and for the tune shift induced on the protons (linear force approximation)

– Numerical extension to non-linear force effects.

• The simulations (with the parameters of LHC @ inj.) show that the density enhancement inside the bunch is about a factor 50.– First estimation gives a tune spread of Q ≈ 0.13 (for

an initial ec-density of 6e11 m-3.


Ongoing and Future plans

• Continue analytical approach to model electron cloud phenomena

• Comparison with HEADTAIL simulations.

• Produce instability diagrams for the electron cloud

• Investigations about

– what happens after the bunch had passed

– Longitudinal discontinuities in the electron plasma.

analysis of the electron pinch during a bunch passage

Documents

bunch distribution

bunch passage elena

electron densitytune

proton bunch

electron initial temperature

maximum tune shift

bunch potentialtime

phase space density