Study of ep instability for a coasting proton b

eam

K. Ohmi, T. Toyama, G. Rumolo

ECLOUD04,

19-23 April, 2004,

Napa

Introduction Coasting beam traps electrons. Does instability always occur for coasting

beam? Does electron production rate play some

roles in the instability?

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Linear theory : Wake fieldProton beam interacts with electron cloud which is gathered at beam position.

We consider motion of beam and cloud centroids.

Coasting beam and cloud model.

Q factor becomes finite due to non-linearity and beta modulation.

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Effective impedance of electron cloud

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Instability threshold

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ez/c>>1 Coasting beam model 　

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Keil-Zotter theory

Threshold of neutralization factor for coasting beam

JPARC-MR PSR ISIS AGS FNAL-MI KEK-PSL(m) 1567.5 90 163 202 3319 339 50.9 1.85 1.07 1.2 128 12.8

Np(x1013) 33 3 3 1.8 3 0.3

p (x1011/m) 2.1 3.3 1.8 0.89 0.09 0.089

r(cm) 0.35 1 3.8 1 0.17 0.5 -0.0013 -0.187 -0.83 -0.65 0.002 0.022p/p(%) 0.25 0.4 0.5 0.5 0.03 0.3

eL/c 7739 195 69 226 6970 246

fth 0.0015 0.025 0.45 0.15 0.00055 0.05

Build-up time due to ionization for the instability Ionization (2x10-7 Pa)

Y1,i=8x10-9e-/(m.p)

= fth/cY1,i =0.4 fth

JPARC-MR 0.6ms PSR 10 ms instability is observed ISIS 180 ms no instability AGS booster 60 ms instability AGS 10 ms no instability FNAL-MI 0.2 ms KEK-PS 20 ms no instability

Nonlinear and many body problem

Not only proton beam but also electrons become unstable in the instability regime.

Electrons are more unstable than beam.

Electron diffusion

Saturation of beam amplitude (emittance)

t

t ebee dtttctssysy0

')'(sin)',()(

if beeti

b yTyey e ,

Image the physical systemLinear theoryBeam is represented by a string. Electrons are represented by a string or a macro-particle. Linear force acts between them. Exceeding a threshold, beam and macro-electron become unstable. The amplitude of electron is much larger than beam yb=eT ye>>ye. Trick of linear theory.Nonlinear and many body systemBeam is represented by a string. Electrons are represented by many macro-particles. Nonlinear force acts between them. Exceeding a threshold, beam and macro-electron become unstable. Landau damping of electrons is much faster than beam, e>>, ep/p. Electrons diffuse on the phase space immediately, and the power causes instability is lost.

Instability parameters for JPARC-MR

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2

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epe

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1)(3

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00

Z

ZrU e

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p

e=1.39x109 s-1

n = e/0 = 1150. Landau damping rate Threshold e~ 4.5x108 m-1

ne = 2.5x106 m-1T0-1 (P= 2x10-7 Pa,Ye=8x10-9 /m.p)

number of ionization electron created in a revolution time.

e/ne = 180 turn build-up time until the threshold

2101.13

/2

ppn

D

Electron diffusion

Electron oscillates 105 period until starting the instability.

Do electrons oscillate stably during so much long term?

The proton beam fluctuates or becomes unstable, electrons may disappear, with the result that the instability may saturate.

Electron motion in the beam potential

Fixed coasting beam

beam position modulation of 1mm

Red: fixed beam. Green: 10 turn. Blue: 100 turn

Simulation

Interaction between the coasting beam and electrons created by ionization or proton loss.

Electrons are added in every revolution by ne. Secondary emission for absorbed electrons is taken

into account. Beam is represented by a series of macro-particles

(1000) distributed along the longitudinal direction uniformly.

Landau damping is considered as x=(1-D)x, where D=1.1x10-2/rev

Ye and growth

The simulation was performed for the following 5 cases,

Ye = 10-7 m-1 2 x 10-6 Pa thr. 20 turn

= 10-6 m-1 2 x 10-5 Pa thr. 2 turn

= 10-5 m-1 2 x 10-4 Pa thr. 0.2 turn (PSR loss, H- injection)

= 10-4 m-1 2 x 10-3 Pa thr. 0.02 turn

= 10-3 m-1 2 x 10-2 Pa thr. 0.002 turn

P=2 x 10-7 Pa was too low pressure to show interesting results.

Ye = 10-7 /m.p 2 x 10-6 Pa ne = 2.5x107 /mT0

** Horizontal axis is 10x Turn

This value is for 1/10 turn.

Ye = 10-6 /m.p 2 x 10-5 Pa ne = 2.5x108 /mT0

** Horizontal axis is 10x Turn

This value is for 1/10 turn.

Ye = 10-5 /m.p 2 x 10-4 Pa ne = 2.5x109 /mT0

** Horizontal axis is 10x Turn

The amplitude grows 1/10 x.

(PSR loss, H- injection level)

This value is for 1/10 turn.

Ye = 10-4 /m.p 2 x 10-3 Pa ne = 2.5x1010 /mT0

** Horizontal axis is 10x Turn

The amplitude grows ~x.

This value is for 1/10 turn.

Ye = 10-4 /m.p 2 x 10-2 Pa ne = 2.5x1011 /mT0

** Horizontal axis is 10x Turn

PSR loss x100 multipactoring level

This value is for 1/10 turn.

Ionization and instability

Strong instability is caused by high pressure P>103 -104Pa.

Ionization was too week as source of coasting beam instability for a normal vacuum P>106-107Ps .

If the electrons are supplied by H- injection foil or proton loss, instability can grow.

Electrons produced by injection Foil. Ye,F = 2 / rep/L. For rep=500, and L=300m (JPARC-RCS), Ye,F = 1.x 10-5 /m.p.

Electrons produced at the chamber wall, which are not trapped for static potential, may be important.

High efficiency electron source at the chamber wall Electrons are absorbed after one interaction

with the beam, if no beam perturbation. Perturbation due to beam motion may make tr

ap the electrons. Threshold e~ 4.5x108 m-1

Electron yield Ye,L = 4x10-6 /m.p (PSR value) ne = Ye,L Np=1.33x109 /mT0 The number of electron created in one revolution time alr

eady exceed 10 times of the threshold.

Ye = 10-6 /m.p ne = 2.5x108 /mT0

Electron line density

Beam amplitudes

Cloud size

***Turn number is 1/10 for the actual machine.

Red : created at the center

Blue : created at the wall

Electrons are not trapped, therefore the line density is very small.

Ye = 10-5 /m.p

ne = 2.5x109 /mT0 (PSR level)

Beam amplitude is 2% of the size, that is about half for electrons created at the center,

Red : created at the center

Blue : created at the wallElectrons are somewhat trapped, therefore the line density increase.

Ye = 10-4 /m.p

ne = 2.5x1010 /mT0

Beam amplitude is comparable with the size.

Red : created at the center

Blue : created at the wallElectrons are trapped. The line density increase clearly.

Conclusion We study a coasting beam ep instability caused by Ionizat

ion electron. Electrons are trapped basically, but they are diffused whe

n beam becomes unstable. The diffusion weakens the beam instability, therefore stab

ility is determined by the comparison with Landau damping.

Secondary electron is created by the diffused electron. The energy of the absorbed electron was small for a weak

instability of beam (x<x). Even consider the secondary effect, ionization electron ca

n not cause ep instability. High intensity source like Proton loss, which contributes a

fast build-up, can cause the instability.

Conclusion II We also study a coasting beam ep instability caused by el

ectron created at the wall. Electrons are not trapped basically, but they are trapped d

uring a short period or are accelerated when beam becomes unstable.

Secondary electron is created by the accelerated electrons.

Electrons with the high yield of PSR level due to proton loss may cause the instability.

Landau damping is treated as amplitude decrement in this model.

It is better to be treated as phase mixing=>G.Rumolo.