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On theboundarybetweencoherent

andincoherenteffects: a smallstudy

G. Franchetti, GSI

Space Charge 2019, CERN

4/11/2019G. Franchetti1

4/11/2019G. Franchetti2

1.Introduction

2.Debye length and collective effects 1D

3.An indicator for a beam

4.Summary and considerations

Introduction

4/11/2019G. Franchetti3

A dynamics driven by a global force deriving from the full beam: e.g. space charge

Collective Effects

Incoherent Effects

A dynamics in which each particle is driven by a force independent by other particles

Coherent effects

A dynamics where particle move coherently AEhaving a same property in the dynamics, same phase Coherent effects AEare originated by collective force Coherent effects AEare originated by incoherent forces: envelope oscillations without SC Goal of this small study AEexplore the relation of collective effects to the Debye length to be used as an indicator of the relevance of the incoherent effects

IfࣅD>> a thescreeningwill beineffective

andsingle-particlebehaviorwill dominate.

IfࣅD << acollectiveeffectsdue totheself

fieldsofthebeam will playan importantrole. Beam a1)theDebyelengthincreaseswithenergy

2) at sufficientlyhigh energythespace-

charge forcesbecomeinsignificantin comparisontotheexternalforcesacting on a beam.

Debye length in a beam

20/6/2018Giuliano Franchetti4

M. Reiser book

y x Typical textbook example:the one dimensional example

4/11/20195G. Franchetti

Vibration and Waves in physics, Iain G, Main, Cambridge University Press, 1978

Model including the bounding

potential z

Quadratic potential

z=0

Micro-plane = infinite plane with uniformly

distributed charged particles each with mass m, and charge q

4/11/20196G. Franchetti

Surface particle density

G. Franchetti IPAC19 orIOP Conf. Series: Journal of Physics: Conf. Series to appear

Electric field created by a

micro-plane z

Quadratic potential

Z=0

Infinite plane

uniformly charged

Constant field

4/11/20197G. Franchetti

zz'

Equation of motion

theinfiniteextensionofthemicro-planes.

4/11/2019G. Franchetti8

Space charge force for a full

distribution z

Quadratic potential

Z=0 microplanesplane density

4/11/20199G. Franchetti

z

Space charge force for a full

distribution z

Quadratic potential

Z=0 microplanesplane density

4/11/201910G. Franchetti

z

Equation of motion

With this equation it is possible to compute the evolution of micro-planes

Distribution of micro-planesDynamical coordinates

͞Beam distribution"

4/11/2019G. Franchetti11

Stationary distribution

In general a distribution depends on time

Stationary distribution AEkeep the same ͞character" (edžample͗ KV remains a KV)

In this example the distribution is

If all forces are linear, stationary distributions are possible if f() is function of the single particle invariant

Distribution is of type:

4/11/2019G. Franchetti12

Linearity of forces requires

z

Number of ͞particles" constant

4/11/2019G. Franchetti13

Generating the macro-planes

distribution An approximated distribution is obtained transforming two Gaussian

4/11/2019G. Franchetti14

Matched distribution

4/11/2019G. Franchetti15

Single particle equation for a

stationary distribution

For this distribution

With Debye frequency and

Using scaled coordinates

4/11/2019G. Franchetti16

Depressed focusing

Depressed Twiss parameters

For each particle of the

distribution

Optics and invariants

4/11/2019G. Franchetti17

Incoherent tune-shift

͞Debye" tune-shift

Oscillation frequencies

4/11/2019G. Franchetti18

Dynamics at the space charge limit

Space charge limit AE

Distribution: 1) uniform in

Lets add a perturbation to the equilibrium distribution:

Add some velocity to particles in a small region

4/11/2019G. Franchetti19

2) stationary

Equation of motion

Stationary solutionperturbation

Feature of the perturbation

( N-+ N+ = N0 )

For small perturbations

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Evolution of a density perturbation

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Matched distribution

1-1

Perturbing the velocities

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1-1

Motion of the perturbation

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Equation of motion of

micro-planes harmonic oscillator

Initial conditions

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Solution for central ͞particles"

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In phase space

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1-1 distribution at the space charge limit

G. Franchetti

Perturbing the velocities

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v

G. Franchetti

-11

Perturbing the velocities

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1-1 v

G. Franchetti

Perturbing the velocities

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1-1 z1 v

What happen

according of weather ?

G. Franchetti

4/11/201930

G. Franchetti

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LD

G. Franchetti

4/11/201932

LD

G. Franchetti

Above the space charge limit

depressedlatticekz/kz0dynamics, whichacts incoherentlyoverall particles

Incoherent regime

a Debye dynamicscharacterizedby thetermwithʘ2/kz0whichinvolve theperturbationɷNо(z)

Collective regime

Equationofmotion

4/11/2019G. Franchetti33

Which regime dominates ?

Incoherent

Collective

In terms of optical functions

Incoherent

= rmssize = rmsvelocity is the Debye length

4/11/2019G. Franchetti34

Regimes and Debye length

Incoherent dominated regime

Collective dominated regime

4/11/2019G. Franchetti35

Distribution above

the space charge limit

4/11/201936

AEUse the stationary distribution

This is a density function f(ߝ

The x projection yields a uniform

distribution f(ߝ xG. Franchetti

4/11/201937

This distribution

Does not change

With time

Xrms= 1.73

G. Franchetti

4/11/201938

X

The projection of

This distribution

Does not change

With time and is

uniform

Xrms= 1.73

G. Franchetti

4/11/201939

Here we perturb the

Distribution and look

For ͞collectiǀe" effects

Xrms= 1.73

G. Franchetti

4/11/201940

Here we perturb the

Distribution and look

For ͞collectiǀe" effects

Xrms= 1.73

G. Franchetti

4/11/201941

Here we perturb the

Distribution and look

For ͞collectiǀe" effects

Xrms= 1.73

G. Franchetti

4/11/201942

Here we perturb the

Distribution and look

For ͞collectiǀe" effects

Xrms= 1.73

G. Franchetti

4/11/201943

Collective effects are related

to the Debye length the more intense the beam is, the higher is the tune-shift

Is the thermal

temperature

G. Franchetti

What is it

?In an particle accelerator beam

Only with the thermal velocity

4/11/201944

x -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 -4-2024

For a ͞linear correlation"

G. Franchetti

Debye length in beams

Matched beam distribution

Debye length (Reiser Book)

4/11/2019G. Franchetti45

limit

4/11/2019G. Franchetti46

Incoherent vs Collective

4/11/2019G. Franchetti470

1 2 3 4 5

00.050.10.150.20.250.3|Qx| / Qx0

f

Collective

Space charge

Incoherent

Space charge

Mixed

Regime

Qx0ઢQx0

PS-Exp.6.10.055.5

GSI-Exp4.30.054.6

SIS10018.70.53.05

SIS184.30.51.46

Possible dynamical regimes

4/11/2019G. Franchetti48

Collective EffectsIncoherent Effects

Space charge forces

create a collective beam response:

Envelope oscillations

Envelope instabilities

Coherent tunes

Space charge forces

acts only on particles like ͞edžternal forces"

Amplitude dependent detuning

Tune-spread

Modification of optics

Structure resonances

Feature

Example

Time scale

FastFast

Summary and comments

4/11/2019G. Franchetti49

1.Although the beam is a ͞non-neutral" plasma, the Debye length seems still

to have a meaning for collective effects

2.The Debye length relative to the beam are an indicator of the strength of the

collective beam response

3.Larger Debye length with respect to the rmsbeam size implies weak

collective beam Response AEmore relevance of incoherent dynamics

Comments

The relation of coherent resonances for non-uniform beam with Debye length is not yet derived from first principles [i.e. 1stfind non-uniform stationary beam distribution, On the other hand this small study suggest that taking as indicator of the incoherent effects makes some sense.quotesdbs_dbs41.pdfusesText_41

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